Dynamic Security of Interconnected Electric Power Systems - Volume 2: Dynamics and stability of conventional and renewable energy systems [2] 978-3-659-80714-5

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Power System Dynamic Security - Volume 2

This book is written for engineers involved in the operation, and control of electric power systems. The book also provides information and tools for researchers working in the fields of power system security, stability, operation and control. It consists of two volumes. In the first volume, the traditional techniques for stability and dynamic equivalence are presented. In addition, an overview of the main drivers and requirements for modernization of the traditional methods for online applications is discussed. The second volume includes advances in the security, stability, control, and stabilization of electrical power systems. The given material is written in graded complexity for facilitating their inclusion in undergraduate, postgraduate, and technical training courses. For online dynamic security systems, a major part of this volume includes the derivation, analysis, and stabilization of the SMIB equivalence of power systems. This volume also includes the operation and dynamical characteristics of variable renewable power generation (wind and solar-PV) as well as their dynamical interactions with power systems and the interconnection requirements are presented in details.

Mohamed EL-Shimy

Dynamic Security of Interconnected Electric Power Systems – Volume 2

Dr. M. EL-Shimy is currently an Assoc. Prof. in the department of Electrical Power and Machines – Ain Shams University. He is also an electromechanical specialist, and a freelance trainer. He is a technical reviewer for some major journals and conferences. Please visit http://shimymb.tripod.com and http://goo.gl/tCNBLU for more details

EL-Shimy

978-3-659-80714-5

Dynamics and stability of conventional and renewable energy systems

DOI: 10.13140/RG.2.2.36832.07683

DYNAMIC SECURITY OF INTERCONNECTED ELECTRIC POWER SYSTEMS – VOLUME 2 Dynamics and stability of conventional and renewable energy systems

Mohamed EL-Shimy Ain Shams University – Cairo, Egypt Contact info: [email protected] [email protected] 002 01005639589 http://shimymb.tripod.com

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

To my wife Hala, and my children (Sara, Malak, Omar, and Aly)

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M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

PREFACE This book is written for engineers involved in the operation, control, and planning of electric power systems. In addition, the book provides information and tools for researchers working in the fields of power system security and stability. The book consists of two volumes. The first volume provides traditional techniques for the stability analysis of large scale power systems. In addition, an overview of the main drivers and requirements for modernization of the traditional methods for online applications is discussed. The second volume (i.e. thus volume) provides techniques for offline and online stability and security studies. In addition, the impact of variable generation on the stability, and security of power systems is considered in the second volume. The book covers some essential aspects related to the modeling, simulation, and analysis of power systems. Its contents are useful for educational and research objectives as well as training of engineers and high level technicians. Fast and online assessment of the dynamic security and stability of power systems are given significant attention in this volume. The stability of grid-connected variable renewable energy sources is presented considering wind and solar-PV technologies. In addition, the impact of these renewable sources on the transient stability of power systems is presented with sufficient details for providing an understanding of various dynamic phenomena. Several numerical examples and elaborated case studies are given to enhancing the capability of the engineers in the modeling, simulation, analysis, and control of power systems. Corrective actions for transient stability preservation and restoration are also presented with a focus on the load shedding for restoring and enforcement of power system stability. For these targets, the minimization of load shedding is considered as a techno-economical solution for solving stability problems associated with a sudden drop in the power generation. These sudden drops may be caused by several reasons such as forced outage of generating units, or the intense reductions in the renewable resources. The load shedding minimization for dynamic security preservation is also considered in this book. Reasonable approaches for online security and corrective actions studies are presented. These approaches are based on the electromechanical equivalence of power systems that is based on online 3

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

measurements. Single-machine infinite-bus (SMIB) equivalences of power systems are presented. These equivalences can be estimated and studied at high speeds that are reasonable for online decision making and implementation of controlled changes or corrective actions. Consequently, the modeling, the simulation, the control of the SMIB are given in details in this volume. These issues are presented considering the steady and various dynamic states as well as various levels of the modeling details. Many toolboxes of the MATLAB and the power system analysis toolbox (PSAT) are described and used in for implementing various models and simulations. The first volume 1 consists of three chapters and starts with a detailed overview of the operational requirement of recent and future power systems considering the integration of variable generation resources into the electricity grid (chapter 1). The fundamentals and advances of power system security requirements are also presented in chapter 1. An overview of the electromechanical equivalence techniques is presented in chapter 2. An improved coherency-based equivalence technique is presented in this volume (chapter 3). The presented technique uses the traditional data sets for the construction of the equivalence. In addition, the concept of remote areas is introduced for the maximization of the dynamic model reduction of very large-scale systems. Several case studies are presented for the evaluation, validation, and analysis of the presented theories and models. This volume consists of 13 chapters and 9 appendices. The numbering of chapters in this volume starts from 4 as the first three chapters form the first volume. The titles of various chapters and appendices in this volume are: - Chapter 4: dynamic modeling, simulation, control, and analysis of simplified power systems - Chapter 5: measurement-based SMIB equivalence of multi-machine systems and transient stability - Chapter 6: load shedding and online applications - Chapter 7: damping improvement and stabilization through facts-pod controllers - Chapter 8: overview, modeling, and performance analysis of gridconnected wind energy sources M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 1. ISBN 978-3-659-71372-9, LAP LAMBER Academic Publisher, Germany, 2015. 1

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M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

- Chapter 9: dynamical impacts of wind power and damping enhancement - Chapter 10: overview, characterization, and modeling of solar-pv systems - Chapter 11: modeling, dynamic analysis, and control of grid-connected solar-PV generators - Chapter 12: consistence of wind power technologies with the fault ride-through capability requirements - Chapter 13: overview of the operational requirements of grid connected solar-PV systems - Appendix 1: steady state modeling of the SMIB considering the effect of the excitation control - Appendix 2: linearized transfer function of a simulink model, and root locus based controller design - Appendix 3: modeling of individual and aggregate loads in power systems - Appendix 4: fundamentals of speed governors - Appendix 5: transient droop compensators for speed-droop governors of hydroelectric units - Appendix 6: interrelation between the dynamic and the static security - Appendix 7: numerical integration methods - Appendix 8: data of the detailed two-area system - Appendix 9: nomenclature of the symbols used in chapter 10 and 11 The following table summarizes the main subject(s) associated with each chapter and appendix. This table is also valuable in organizing the reading and the use of the manuscript. I would like to express my gratitude for some of my students that contribute in executing some of the numerical examples and case studies presented in this volume of the textbook. Significant parts of the numerical examples of the analysis of wind energy conversion systems are partially performed by Eng. Nisma Ghaly and Eng. Omnia Rassem. The simulation results FACTS-POD design and analysis are partially performed by Eng. Mohamed Mandour. Some of the results of the modeling and analysis of the solar-PV generators are performed by Eng. Hossam Khairy. Finally, some of

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the literature review of solar-PV technologies is performed by Eng. Taha Abdo.

Volume 2

Volume 1

Nomenclature and data

Grid codes

Impacts and Stability control

Modeling and simulation

Renewable energy

Fundamentals

Renewable sources

Stability analysis & control Conventional sources

Online

Offline

Introductory

Equivalence

Ch. 1 Ch. 2 Ch. 3 App. Ch. 4 Ch. 5 Ch. 6 Ch. 7 Ch. 8 Ch. 9 Ch. 10 Ch. 11 Ch. 12 Ch. 13 App. 1 App. 2 App. 3 App. 4 App. 5 App. 6 App. 7 App. 8 App. 9

Generally, nothing is either absolutely perfect or final. Therefore, please do not hesitate to send me your feedback, and suggestions. M. EL-Shimy, Nov. 2015

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TABLE OF CONTENTS (VOLUME 2) Preface Table of contents Chapter 4 Dynamic Modeling, Simulation, Control, and Analysis of Simplified Power Systems 4.1 Introduction 4.2 Identification and characteristics of steady and dynamic states of power systems 4.3 Frequency dynamics of power systems 4.4 Linearized classical modeling and simulation of the traditional SMIB system 4.4.1 The swing equation 4.4.2 The transient power-angle characteristics 4.4.3 Linearization and linearized analysis 4.4.4 Time-domain solutions of the linearized model 4.4.5 Small-signal stability and eigenvalues of the state matrix 4.5 Inclusion of field circuit dynamics in the linearized model of the SMIB system 4.6 Including AER and PSS in the linearized third-order model of the SMIB system 4.7 Power System Stabilizers (PSS) 4.8 Frequency response and Speed Governors 4.8.1 Frequency response of a governor-less generator 4.8.2 Types of speed governors 4.8.3 Supplementary control action for unit output power control Chapter 5 Measurement-based SMIB Equivalence of Multi-Machine Systems AND Transient Stability 5.1 Introduction 5.2 Measurement-based electromechanical equivalence 5.2.1 Stage I: Inertia estimation at the interface buses 5.2.2 Stage II: Electrical parameters of the equivalent generator 5.3 SMIB equivalence of multi-machine systems 7

3 7 13 115 13

17 23 24 31 34 51 64 65 80 86 90 93 97 108 117 197 117 122 124 125 127

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5.4 Transient stability and the Equal Area Criterion (EAC) 5.4.1 Classification of power system stability 5.4.2 The equal area criterion (EAC) and advancements 5.4.3 Transient Stability Enhancement 5.5 Measurement-based SMIB Equivalence of Multi-machine power systems – A case study Chapter 6 Load Shedding and Online Applications 6.1 Introduction 6.2 Voltage Stability and UVLS – an overview 6.2.1 Basic Definitions 6.2.2 Classifications of voltage stability studies 6.2.3 Connection between Voltage Stability and Rotor Angle Stability 6.2.4 Fundamentals of Voltage stability analysis 6.2.5 UVLS schemes 6.3 UFLS – AN OVERVIEW 6.4 Dynamic load shedding based on online measurements Chapter 7 Damping Improvement and Stabilization through FACTS-POD controllers 7.1 Introduction 7.2 FACTS devices – Definitions 7.3 Modeling and Modal Analysis 7.3.1 Linearized Modeling for POD design 7.3.2 TCSC Dynamic Modeling and Control 7.4 POD Design and Tuning 7.4.1 Frequency Response Method 7.4.2 Residue Method 7.5 Study System 1 – SMIB system 7.6.1 Impact of the TCSC on the small- signal stability 7.6.2 Observability and controllability of various input signals 7.6.3 POD designs 7.7 Case Study 2 – Weakly interconnected multi-machine system 7.8 Placement of FACTS Devices 7.8.1 Preliminary mathematical modeling 8

131 133 135 170 193 199 246 199 202 202 205 207 207 230 234 237 247 306 247 249 254 254 256 258 259 262 265 267 267 268 274 286 287

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7.8.2. General form of static-sensitivity analysis 291 7.8.3 TCSC Placement Problem 293 7.8.4 Study System 298 7.8.5 Implementation and Results 299 Chapter 8 307 Overview, Modeling, and Performance Analysis of Grid382 connected Wind energy Sources 8.1 Introduction 307 8.2 Wind Energy Conversion Technologies (WECTs) 310 8.3 Modeling of Wind Turbine Generators (WTGs) 315 8.3.1 Wind turbine model and its controls 317 8.3.2 Shaft model 319 8.3.3 Modeling of Generators, and their controls 321 8.4 Case study 1 - Steady-state characteristics and steady-state 332 stability of DFIG 8.5 Case study 2 - Transient performance of FSWT based on SCIG 341 considering the single-mass model (rigid rotor) 8.6 Case study 3 – Design of Pitch Angle Controllers 353 8.7 Case study 4 – Modal Analysis of FS-IGIB System 366 383 Chapter 9 418 Dynamical Impacts of Wind Power and Damping Enhancement 9.1 Introduction 383 9.2 Impact of wind power on the inertia of power systems and the 383 equivalence SMIB 9.3 Impact of wind power on the stability of power systems 387 9.4 Response of WTGs in comparison with Conventional 399 Synchronous Generators 9.5 Design of PODs in the presence of wind power generation 411 419 Chapter 10 477 Overview, Characterization, and Modeling of Solar-PV Systems 10.1 Introduction 419 10.2 Qualitative Overview of solar-PV technologies 422 10.3 Quantitative Characterization of solar-PV technologies 425 10.4 Fundamentals and Applications of solar-PV systems 442 10.4.1 Solar-PV arrays 442 10.4.2 PCU and MPPT 449 9

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10.4.3 Earth-Sun Geometry and Observer Angles 457 10.4.4 Sun tracking mechanisms 463 10.4.5 Modes of operation of solar-PV generators 466 10.5 Modeling of grid-connected solar-PV generators 469 10.5.1 Resource Assessment and Energy Production (RA&EP) 470 Models of solar-PV modules 10.5.2 Equivalent Circuit based Models (ECBMs) of solar-PV 473 generators 10.6 Case Studies 477 Chapter 11 479 Modeling, Dynamic Analysis, and Control of Grid-Connected 527 Solar-PV generators 11.1 Introduction 479 11.2 Modeling of solar-PV generators for dynamic analysis 481 11.2.1 Solar-PV array modeling 482 11.2.2 DC-DC Converters and MPPT control 493 11.2.3 DC-AC Converters (Inverters) and Power control 508 11.3 Matlab-Based Modeling and Simulation of the grid-connected 514 Solar-PV generator 11.3.1. Modified MPPT techniques 516 Chapter 12 529 Consistence of Wind Power Technologies with the Fault Ride540 through Capability Requirements 529 12.1 Introduction 12.2 FRT Capability Requirements 530 12.3 Numerical Examples 534 Chapter 13 541 Overview of the Operational Requirements of Grid Connected 556 Solar-PV Systems 13.1 Introduction 541 13.2 Grid code and standard requirements 543 13.2.1. Normal operation requirements 543 13.2.2. Under grid disturbance requirements 552 13.2.3 SCADA Integration Requirements 555 Appendix 1: Steady state modeling of the SMIB considering the effect 557 10

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of the excitation control Appendix 2: Linearized transfer function of a simulink model, and root locus based controller design Appendix 3: Modeling of individual and aggregate loads in power systems Appendix 4: Fundamentals of speed governor Appendix 5: Transient droop compensators for speed-droop governors of hydroelectric units Appendix 6: Interrelation between the dynamic and the static security Appendix 7: Numerical integration methods Appendix 8: Data of the detailed two-area system Appendix 9: Nomenclature of the symbols used in Chapter 10 and 11

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591 595 617 621 625 629 633 637

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Chapter 4 Dynamic Modeling, Simulation, Control, and Analysis of Simplified Power Systems 4.1 Introduction In the first volume of this book, the construction and evaluation of the coherency-based electromechanical equivalence is presented. The presented method uses the traditional approach which relies on the availability of the required data sets of system parameters as well as its variables. The method presumes that the required data sets are complete and accurate; however, not only the operating conditions of power systems vary with time, but also the parameters of various components; for example, aging and operating stresses affect the values of the parameters of various components and may cause significant change in their recorded historical values. In addition, the information provided by the data sets is highly uncertain and may suffer incompetence or at least they may not be in agreement with the actual system data. The modeling and analysis presented in the first volume consider only conventional power systems. The integration of recent generation technologies causes changes in the dynamic behavior and interaction between system components. Therefore, the impact of these technologies as well as advanced modeling approaches and methods for online stability and security studies are presented in this volume. The problems associated with the availability and accuracy of the data sets of power systems as well as the estimation of the operating conditions are highly reduced by the use of synchronized online monitoring of power systems. Therefore, electromechanical equivalency techniques that solely use online measurements instead of the traditional data sets are presented in this volume. The method can be effectively executed online and this becomes possible due to the availability of WAMS and PMUs. The WAMS and PMUs are recently introduced in power systems for time-synchronized measurements, telemetry, and recording of critical quantities (such as state variables, and status of components). Consequently, accurate online analysis and management of large-scale power systems becomes possible. An overview of the functionality and characteristics of these systems is summarized in the first volume of the book (chapter 1 and 2). 13

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The objectives of this chapter also include the analysis of some critical issues associated with the modeling, stability, security, operation, and control of power systems.

4.2 Identification and characteristics of steady and dynamic states of power systems A great variety of different dynamics occurs in power systems. The physical origin and the time scale of these dynamics are very different. If the time derivative of any system’s quantity is non zero, then the system is conceptually is in a dynamic state. In that sense, a power system will never reach a steady state condition because there are normal continuous changes in the system loads. This is too strict from engineering, mathematical, or even physical points of view. A power system may be subjected to many changes that may not be considered as disturbances. For example, slow and small changes in the system loading is a situation where a power system is considered in a steady state rather than a dynamic state; however, sudden and large changes in the system loading activate the system’s dynamic state. Generally, sudden changes are disturbances that cause fast changes in the system operating conditions. From mathematical point of view, the magnitude of a disturbance defines the appropriate mathematical modeling approach of a power system. If the magnitude of a disturbance is large, then modeling a power system using differential equations is common for handling the large nonlinearities in the performance of power systems. On the other hand, linearized models are generally acceptable for modeling power systems subjected to small disturbances. In the two situations (i.e. large and small disturbances), the desired accuracy level, the simulation time, and the available computer memory defines the model details. More modeling details are always associated with higher accuracy; however, these higher accuracy levels are at the expenses of larger simulation time and memory requirements as well as mathematical complexity. In addition, highly detailed models may result in too large number of time dependent quantities. Consequently the results may be difficult to be understood and clear conclusions may be also difficult to be extracted. Therefore, reduced order models with reasonable simplifications may be appropriate for fast assessment of the stability of power systems. This is of special importance when the high accuracy is not a 14

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

main target and reasonable accuracy as well as high simulation speed is demanded (e.g. online applications). According to this discussion, mathematically and physically, a power may be in one of three states; steady state, dynamics associated with small disturbances, and dynamics associated with large disturbances. The big questions are then: what is the threshold magnitude of a disturbance that makes it large disturbance ? What is the maximum magnitude of a slow change at which the system is considered in a steady state ? Actually, there are no universal values of these values as they are highly dependent on the parameters, topology, operating conditions, and the scale of the power system. The reaction (or the response) of a power system to a specific change defines the type of the change and the corresponding state of the system; however, the power balance mismatch associated with a given change is suggested in this book as an indicator to the type a change and the corresponding state of a power system. Generally, the steady state usually fulfills the secure operation requirements i.e. the normal state requirements (stated in volume 1/chapter 1). Switching actions such as distribution capacitor bank or small load switching may cause limited local disturbances that may not propagate through wide parts of a power grid and vanish fast. Therefore, their impact on the power system is insignificant. Changes in a power system may be classified into three categories, I.

Slow changes with small magnitudes : the reaction of a power system to

such changes is insignificant and the system may be considered in a steady state. In this case, time-independent mathematical models are sufficient for representing power system. These types of models are called ‘load flow models ’. The system is then considered in a steady state if it could continually keep the power balance or in other words the active and reactive power mismatch (P , and Q ) is practically zero. The power mismatches are defined by,

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II.

III.

Fast (or sudden) changes with small magnitudes: In this case, the ratio of the active and reactive power mismatches to the respective total generation ( are small. In this case linearized modeling of the system nonlinearities is usually acceptable. For the linearized modeling to be acceptable, the steady-state deviation of the system variables should be small (e.g.  10% of the initial steady state values). Fast (or sudden) changes with large magnitudes: In this case, the ratio of the active and reactive power mismatches to the respective total generation ( are large. System nonlinearities should be carefully modeled for such kind of disturbances. Large disturbances usually cause violations in the normal state requirements and the system in this case is said to be in an emergency state or extreme state as defined in chapter 1 of the first volume of this textbook.

It should be noted that the impact of active power mismatches is significantly different in comparison with reactive power mismatches 2. Active power mismatches significantly affect the frequency of the system while their impact on the voltage is minor. On the other hand, reactive power mismatches have a significant impact on the voltage while their impact on the system frequency is minor. From disturbance spread point of view, active power mismatches cause widespread impact on the system performance while reactive power mismatches have a local impact. For example, an active power mismatch causes frequency upset all over the power grid while a

2

Further readings:  Kundur, P. (1994). Power system stability and control (Vol. 7). N. J. Balu, & M. G. Lauby (Eds.). New York: McGraw-hill.  Venikov, V. A. (1977). Transient processes in electrical power systems. Mir Publishers.  Machowski, J., Bialek, J., & Bumby, J. R. (1997). Power system dynamics and stability. John Wiley & Sons.  Savulescu, S. C. (Ed.). (2014). Real-time stability in power systems: techniques for early detection of the risk of blackout. Springer.  Kundur, P., Paserba, J., Ajjarapu, V., Andersson, G., Bose, A., Canizares, C., & Vittal, V. (2004). Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. Power Systems, IEEE Transactions on , 19(3), 1387-1401. 16

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reactive power mismatch causes local voltage problems at the mismatch bus and its vicinity. 4.3 Frequency dynamics of power systems Understanding of the frequency dynamics of power systems 3,4,5,6 is essential for several issues including frequency control design and estimation of the system inertia through measurements. Consider a power system with several conventional synchronous generators. If this power system is subjected to a disturbance such as a sudden outage of a generator, then dynamical changes in the system start instantaneously. These dynamics are mainly caused by the instantaneous power imbalance between the instantaneous generation and consumption of electric power. Consequently, the remaining synchronous generators are subjected to acceleration and deceleration effects. Since a generator’s turbine governor cannot increase the input mechanical power instantaneously, hence the additional power has to be extracted from the kinetic energy stored in the rotor. Since each generator has an inertia, then its power angle (or the angle of its induced emf;

) cannot change as rapidly as needed for exactly restoring the power balance. Consequently, a disturbed generator will be subjected to the power and speed swings (or oscillations) which are of decaying amplitudes in stable dynamics. Due to the strong connection between the mechanical and electrical frequency, the changes in the rotor speed results in changes in the electrical frequency. As an example, the frequency behavior of a generic power system with frequency controls following a loss of generation is shown in Fig. 4.1. Initially, the system electrical loading is higher than the mechanical input power inputs. Therefore, the speed of generators and grid frequency decrease. Consequently, the remaining generators increase their output power by 3

Ravalli, P. (1986). Frequency Dynamics and Frequency Control of a Power System, Particularly Under Emergency Conditions (Doctoral dissertation, University of Melbourne). 4 Andersson, G. (2012). Dynamics and control of electric power systems. Lecture notes 227-0528-00, ITET ETH, EEH - Power Systems Laboratory, ETH Z¨urich 5 Eremia, M., & Shahidehpour, M. (Eds.). (2013). Handbook of electrical power system dynamics: modeling, stability, and control (Vol. 92). John Wiley & Sons. 6 Wall, P., Gonzalez-Longatt, F., & Terzija, V. (2012, July). Estimation of generator inertia available during a disturbance. In Power and Energy Society General Meeting, 2012 IEEE (pp. 1-8). IEEE. 17

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

releasing kinetic energy stored in their rotors (Fig. 4.1: from t = 0 sec to 10 sec). As a result, the power imbalance is reduced while the turbine governors’ increase the input mechanical power to the generators. The lowest frequency (or the frequency nadir) is reached when the mechanical power equals to the electrical power). Since at the frequency nadir the frequency is still below the normal value, then the governors increase the mechanical power input to the generators. Therefore, the initially lost kinetic energy values from the rotors are now starting to be restored and the system frequency increases. Eventually, the power balance and the frequency are restored in the system has sufficient capacity to compensate the lost generation. This control action is called primary Automatic Load Frequency Control (ALFC)7. This frequency control loop causes a stationary frequency error determined by the droop characteristics (or function) of the turbine governor. Fine tuning of the system frequency is achieved by another control loop called the secondary ALFC. These control loops are shown in Fig. 4.2. The overall objective of the ALFC is to maintain an active power balance (or load tracking) at the generator buses by maintaining a constant frequency. Consequently, the ALFC systems collectively maintain active power balance and frequency on the system wide basis. The response of the primary ALFC loop is faster in comparison with the secondary ALFC loop. The primary ALFC loop responds to frequency (or speed) error signal, which is an indirect measure of the active power imbalance. Via a speed governor and control valves, the steam (or hydro) flow is regulated with the intent of achieving an active power balance to relatively fast load fluctuations. On the other hand, the secondary ALFC loop maintains fine adjustment of frequency (i.e. it eliminates the small frequency error which still remaining after the action of the primary ALFC loop); however, the secondary ALFC is insensitive to rapid load and frequency variations, but it responds to drift-like (or slow) changes that take place over a period of minutes. The mechanism by which a generator supplies power to the grid is summarized in the following points. Consider a generator operating in its normal state, delivering a steady state active power P go MW to the network. In this case,

7

Elgard, O. I. (1982). Electric energy systems theory. New YorkMc Graw-Hill. 18

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 The generator currents and rotor magnetic field create a constant

electro-mechanical decelerating torque Tgo = P go/mo, where mo is the rational speed of the turbine-generator in rad/sec. The generator torque Tgo is also referred to the counteracting electromagnetic torque ( Teo)

 The turbine delivers a constant accelerating torque TTo which, if expressed in turbine power, amount to P to = mo Tto. The turbine torque is also referred to the mechanical input torque ( Tm).  Under the stated conditions, both torques (and powers) are in complete balance and the frequency fo is maintained constant i.e. Tt = Te.

Assume now this equilibrium is suddenly upset by an electrical load

change, P g. This a typical situation when either the local load is changed by

an active power P D or the transmission power flow is changed by P line (which may be caused by faults) or both.  As a result the generator power changes instantaneously with the amount of P G electrical power balance require that P g = P D + P line

 In the moment following this electrical change no changes take place in the turbine torque due to the time delay associated with the response of the prime mover. Consequently, the turbine-generator experience a slight torque or power imbalance.

 If the generator’s load change is positive (load increase), the turbinegenerator will decelerate, and if P g is negative, the turbine-generator will accelerate. In either case the generator frequency will undergo a change f, which thus becomes the indicator of the existing power imbalance.

 The ALFC loops are designed to maintain the power balance by an appropriate adjustment of turbine torque .

The frequency shown in Fig. 4.1(a) is the system frequency which is the average frequency of the system. As shown in Fig. 4.3, the frequencies of the individual generators can be regarded as comparatively small variations over the system frequency. The system frequency is also called the frequency of the Center of Inertia (COI). 19

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(a)

(b) Fig. 4.1: Frequency and power dynamics of an interconnected power system as subjected to a generator outage. (a) Frequency dynamics. (b) Active power and mechanical power dynamics

Fig. 4.2: Model for the ALFC loops 8 8

Elgerd, O. (1981). Control of electric power systems. IEEE Control Systems Magazine , 2(1), 4-16. 20

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Fig. 4.3: Average frequency of the system and frequencies of individual generators The frequency dynamics can be classified according to the strength of the disturbances in two categories. The first category includes dynamics associated with small disturbances (such as normal small load variations) while the second category concerns with large disturbances (such as faults). Despite the strength of the disturbance, the frequency deviations are associated with the power imbalance in the system. Practical examples 9 of each category are shown in Fig. 4.4 and 4.5. Most utility systems set a maximum threshold of ±1% frequency variations for continuously supplying system loads. Therefore, frequency variations of amplitudes less than ±1% will not be considered as a problem and no control is required for such cases; however, larger variations need to be controlled for restoring the system frequency to the normal bandwidth.

9

Eremia, M., & Shahidehpour, M. (Eds.). (2013). Handbook of electrical power system dynamics: modeling, stability, and control (Vol. 92). John Wiley & Sons. 21

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Fig. 4.4: Typical small frequency variations 7

Fig. 4.5: Large frequency variations resulting from a generation trip of 1250 MW in the WECC on May 18, 20017 As previously mentioned, the inertia of the generators, and the ALFC control loops as well as other controls such as the Automatic Generation Control (AGC) define the frequency control actions as illustrated in Fig. 4.6. In the following sections, the modeling, analysis, and computer simulation of the dynamics of power systems will be presented considering simplified models and simple systems. This chapter considers the SMIB while multimachine power systems will be considered in the next chapter. The main objective of the chapter is providing the fundamental understanding of the dynamics of power systems, control design, system simulation, and analysis.

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(a)

(b) Fig. 4.6: Frequency variations; (a) Variations under small and large disturbances; (b) various ranges of frequency deviations and associated control actions 4.4 Linearized classical modeling and simulation of the traditional SMIB system The Single Machine Infinite Bus (SMIB) system is popularly used in the simplified analysis of the performance of a generator connected to an infinite bus (a bus with constant voltage magnitude and frequency; represents a large power grid connected to the generator); see Fig. 4.7. 23

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Fig. 4.7: The traditional SMIB system In the following it is assumed that, 1. The normal-state automatic controls (shown in Fig. 4.8), such as Automatic Excitation Regulators (AER) and governors, are not active. This assumption is valid for the first instants of a transient process due to the inherent time delay associated with controlled quantities and their control systems. 2. The network active power losses are neglected. This assumption is commonly acceptable as the X/R ratio of power transmission systems is high values (typical value ranges from 3 to 10 for HVAC cables and from 5 to 10 or more for HVAC overhead line). Therefore, the transmission network, which is commonly constructed as an overhead system, is usually considered as a pure reactive network. 3. The synchronous generator is represented by classical model (shown in Fig. 4.9). With this model, the transient emf of the generator is assumed to be constant and the field circuit dynamics are neglected during the transient process. An assumption which is valid for the initial stage of the transient process; however, in reality the transient emf and the machine equivalent reactance changes with the time during the transient process. The assumption is shown to provide high simplicity with reasonable accuracy. 4.4.1 The swing equation

For stability analysis, the shaft connecting the rotating elements in a turbine-generator is usually assumed rigid and the free-body rotation is assumed. Therefore, the total inertia of the rotating elements is simply the sum of the individual inertia of various connected rotating elements such as the turbine, the rotor, and the exciter. The equation of motion of such rotating 24

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

system obeys the Newton’s second law of rotational motion which takes the form,

where J is the total moment of inertia in kg.m2, m is the rotor shaft velocity in rad/s, Tt is the turbine or input torque in N.m, Te is the electromagnetic or the counteracting torque in N.m, and TD is the damping torque in N.m.

(a)

(b) Fig. 4.8: Normal-state controls for conventional synchronous generators; (a) block diagram; (b) connection layout

25

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Fig. 4.9: Classical dynamic model of conventional synchronous generators This equation is called the equation of motion or the swing equation. The term represents the rotor acceleration. According to the torque balance presented on the RHS of eq. (4.3), the rotor will accelerate if the RHS term of eq. (4.3) is positive while rotor deceleration is associated with a negative value of that term. In case of perfect torque balance (i.e. the RHS of eq. (4.3) = 0), then the rotor will rotate at a fixed speed which is the synchronous speed. It is clear from eq. (4.3) that the impact of torque imbalance on the rate of change of the rotor speed is inversely proportional to the monument of inertial. Therefore, for the same torque imbalance subjected to two identical generators except their moment of inertia, the generator having a higher moment of inertia is less sensitive to torque imbalance in comparison with the generator with lower inertia. It is also important to note that the changes in the turbine torque are significantly slow in comparison with the electromagnetic torque. This is attributed to the long time constants associated with the prime mover system while the electromagnetic changes are almost instantaneous. Since the power angle ((shown in Fig. 4.9) represents the rotor position with respect to a synchronously rotating reference frame axis (see Fig. 4.10), then the rotor speed as a function of the rate of change of (takes the form,

26

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Fig. 4.10: the power angle  In eq. (4.4), m presents the mechanical angle of the rotor in mech. rad.,

m presents the rotor speed in mech. rad/s. Based on (4.4) can be written as . It is clear that the deviation in the rotor speed can be represented by . The damping torque in (4.3) can be presented by,

where Dm is the damping torque coefficient in N.m.sec. Substituting (4.4) in (4.5) and the resulting equation as well as eq. (4.4) in (4.3), then

Since, the synchronous speed is a constant value, then

The term the term becomes,

presents the net mechanical shaft torque (Tt) and presents the rotor speed deviation , then (4.7)

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M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Since the power = torque x angular speed, then . If small disturbances are consider ed, then the speed deviations may small enough for neglecting its impact on the power-torque interrelation. Therefore, and eq. (4.8) becomes the swing equation,

where is the damping torque coefficient and Mm = J sm which is the angular momentum of the rotating elements at the synchronous speed. P m is the shaft net power input to the generator, and P e is the electromagnetic power which is equal to the air gap power. It is worthy to be noted that the angular momentum Mm is related to kinetic energy (K.E) stored in the rotating elements in the synchronous speed. This K.E is usually called the inertia constant H in MJ. The inertia constant H ‘quantifies the K.E stored in the rotating elements at the synchronous speed ( ) in terms of the seconds it would be taken by the generator to provide an equivalent amount of energy when operating at rated power’. With Sn defined as the rated MVA of the generator, the inertia constant H and the momentum Mm take the form,

Therefore, the momentum equals to double the value of the K.E of the rotating elements. The momentum can also be presented in terms of the mechanical time constant . The mechanical time constant can be explained by considering a generator at rest and suddenly a mechanical torque equals to is applied to the shaft. In this case, the rotor speed will be linearly increased and the rotor will take a time to reach the synchronous speed. 28

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

The previous modeling of the swing includes mechanical quantities while it is preferable to express the model using electrical quantities. To do so, the following relations are to be used, the mechanical angle = number of pole pairs (p) x electrical angle , and mechanical angular velocity ( ) = number of pole pairs x electrical angular velocity ( ) i.e.

Since all the upcoming equations will include electrical quantities, the suffix e will be removed. Equation (4.9) can be transformed to include the electrical quantities as follows,

or

where The swing equation can be represented in p.u by dividing (4.16) by Sn,

In this representation, the inertia constant

in seconds as well

as the inertia constant H . The synchronous speed s = 2fo in rad/s and fo is the normal state frequency in Hz. Therefore, or M = 2 H with the synchronous speed is taken as 1.0 p.u. The power values are in p.u while the angles are in rad. Therefore, (4.17) can be written as, 29

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

H d 2  Pm  Pe  PD  f o dt 2

(4.18)

Although we assumed that the damping coefficient is a constant value such that the damping power takes the form , the damping coefficient is a nonlinear function of the power angle highly dependent on its value. For small deviations in the rotor speed, it can be shown that 10,





PD  D d sin 2   D q cos2    D ( )

(4.19)

The function D () takes the form shown in Fig. 4.11. It is depicted from the figure that the damping increases with the increase of the power angle till a power angle of 90o. Afterward, the damping declines. It is known that the maximum allowable steady state power angle of an uncontrolled cylindrical pole machine is 90o; higher angle values cause instability of the machine. In most types of power systems studies, the average value D av of the damping coefficient is used instead of representing the actual variations of the damping coefficient.

Fig. 4.11: Variations of the damping coefficient with the power angle The value of the damping coefficient D depends on many factors such as:

10

Machowski, J., Bialek, J., & Bumby, J. R. (1997). Power system dynamics and stability. John Wiley & Sons. 30

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

1. The induction motor action (i.e. the electromagnetic damping) due to the damper winding and rotor body eddy currents. 2. Speed – torque characteristics of the prime mover. 3. Parameters of the excitation control system. 4. The load characteristics. 5. The loading conditions. The instabilities that may be seen with small disturbances are always attributed to low damping of oscillations. 4.4.2 The transient power-angle characteristics

Referring to eq. (4.18), the electrical power P e, can be represented by the transient power-angle characteristics of the system. Considering cylindrical and salient pole generators, the transient power angle characteristics will be derived in the following part. Given that all resistances neglected, the vector diagram of Fig. 4.12 represented the SMIB (Fig. 4.7) with cylindrical (non-salient) pole machine while Fig. 4.13 shows the vector diagram of the system considering a salient pole generator. It will be used for obtaining the transient power-angle characteristics. The details of the derivations are presented in Appendix 1. In addition, the steady state powerangle characteristics of various types of conventional synchronous generators can be found in Appendix 1. The equations are derived based on the vector diagrams of voltage and currents. As shown in Appendix 1, the steady state and transient power-angle characteristics of various machine types and their corresponding plots are shown in Table 4.1. The power-angle characteristics representing the SMIB system is described by equation (4.21) for non salient pole generators and by (4.23) for salient pole generators. In comparison with the steady state characteristics of the non-salient pole generator, it is seen from Fig. 4.15 and 4.17 that the maximum angle increased above 90 o. This is attributed to the impact of saliency during the transient state even with non-salient pole machines. Detailed explanations can be found in reference 11. In addition, the saliency 11

Krause, P. C., Wasynczuk, O., Sudhoff, S. D., & Pekarek, S. (2013). Analysis of electric machinery and drive systems (Vol. 75). John Wiley & Sons. 31

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

causes an increase of the maximum power. For simplification, it is assumed in the following modeling and analysis that,

Fig. 4.12: vector diagram of the SMIB system with cylindrical pole generator

Fig. 4.13: vector diagram of the SMIB system with salient pole generator 32

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Table 4.1: Power-angle characteristics of the SMIB system considering non-salient and salient pole generators Steady state characteristics E qV Xd

sin 

P

(20)

EV X d' 

sin  

V2 1 1   '   sin 2 (21) 2  Xd Xd 

Non-salient pole generator

P 

Transient characteristics ' q

Fig. 14

Xd

V2 1 1  sin      sin 2 (22) 2  X q  X d  

P

E q'V X d' 

1  V2 1 sin    '   sin 2 (23) 2  X d  X q  

Salient pole generator

P

E qV

Fig. 15

Fig. 16

Fig. 17

 The difference (’) shown in Fig. 4.12 and 4.13 is small enough to consider that, E q'  E ' cos(   ' )  E '

33

(4.24)

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

 It is also assumed that the saliency effects on the characteristics are negligible. Therefore, pure sinusoidal characteristics are obtained. Therefore, the transient power-angle characteristics of salient and nonsalient pole machines can be represented approximately by, E 'V Pe  ' sin   Pmax sin  xd

(4.25)

Fig. 4.18 illustrates a comparison between exact and approximate power – angle characteristics. It can be seen that the approximations result in an increase in the maximum power and reduction of the maximum angle to 90 o. As a result, the stability of the system is overestimated; however, the mathematical treatment will be simplified. For performance demonstration purpose, the approximations are acceptable; however, practical applications should consider the impact of these approximations on the results.

Fig. 4.18: Exact and approximate power-angle characteristics 4.4.3 Linearization and linearized analysis

The swing equation (18) can be now written as, H d 2 d    sin P P D  m max  f o dt 2 dt 34

(4.26)

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

The swing equation is then a nonlinear function of the power angle ; however, for small disturbances, the swing equation may be linearized with little loss of accuracy. The swing equation can also be written in the form of two first-order differential equations as,

 

fo

  

H

Pm  Pe  D 

(4.27)

where  is a differential operator i.e. d/dt operator. Consider a small deviation oin power angle from the initial operating point

o, i.e.



(4.28)

Linearizing (4.27) about an initial operating condition represented by o as follows:

 

fo

H   

Pm  Pe  D 

(4.29)

where Pe 

Pe 

o

  Pmax cos  o (  )

(4.30)

The term P max cos o is known as the synchronizing power coefficient. The higher the synchronizing power coefficient, the more stable the system. The P s is given by: Ps  Pmax cos  o

Therefore, (4.29) becomes, 35

(4.31)

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

 

fo

H   

 Pm  Ps   D  

(4.32)

Equation (4.32) can be written matrix form:

fo   fo   D Ps    fo  d    H H       H  Pm (4.33) dt          0  0   1 Which takes the standard state-space form form: 

 x  A x  B u

(4.34)

It is clear that the state matrix A is dependent on the system parameters D, H, X’d, and the initial operating condition represented by the value of E’ and o. The block diagram representation of the linearized swing equation is shown in Fig. 4.19. This diagram can be implemented, for example, on the Simulink for the simulation of the small-signal performance of the SMIB.

Fig. 4.19: Blok-diagram of the SMIB system with classical model linearization

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M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Example Simulation and analysis of the linearized classical model of the SMIB system As an example , the SMIB system shown in Fig. 4.7 consists of a 50-Hz

equivalent generator having inertia constant H = 10 sec and a transient reactance of 0.3 p.u is connected to an infinite bus through a purely reactive circuit with Xe = 0.4 p.u. The generator is delivering power of 0.6 p.u at 0.8 lagging power factor to the infinite bus at 1.0 p.u voltage. The damping power coefficient is assumed to be D = 0.1 p.u. The linearized model of the system is implemented on the simulink as shown in Fig. 4.20. Based on the models presented in Appendix 1, various values of the variables are

calculated. It is found that o = 18.5940o, E’ = 1.3172 p.u, and P s = 1.7835 p.u. The following simulations are performed to show the response of the system and for the analysis of the system’s frequency response. In addition, the impact of various quantities on the system performance is demonstrated. The system linearized model is implemented on the simulink as shown in Fig. 4.20. The following cases are considered.

Fig. 4.20: The simulink model of the linearized dynamic model of the SMIB system of the numerical example – the machine is represented by the classical model 37

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

 Case #1 : The response of the system as subjected to a sudden decrease in the mechanical power input of 10%. The disturbance is started at t = 0.5 sec from the simulation time. This is to show the initial conditions in the pre-disturbed conditions. The results are shown in Fig. 4.21.  Due to the disturbance, an oscillatory stable transient process occurs and the system settles to a new steady state point after about 5 seconds from the instant of the disturbance.  Since, the disturbance is a sudden reduction in the mechanical power input, it seems that the situation is impossible in reality due to the inherent relatively long time constant of the prime mover subsystem; however, reduction in the mechanical power input as a disturbance is used here for only showing the transient response of the SMIB system.  The disturbance causes a negative mismatch in the power balance (defined by the RHS of the swing equation; eq. (4.17)). Therefore, the generator initially decelerates and its speed as well as its frequency is reduced. It is clear from the transient power-angle characteristics (eq. (4.25)) that the electrical active power is directly proportional to the sine of the power angle. Therefore, as expected, the disturbance initially causes reduction in the power angle due to the input power reduction. The electrical power is also reduced. The damping power - which is mathematically a scaling of the speed deviation by the damping coefficient (see eq. (4.19)) - is shown, as expected, to be in-phase with the speed deviation. In addition, the electrical active power and the power angle are in-phase.  Case #2 : in this example, the impact of the damping on the system response is evaluated. The original value of the damping coefficient of the system is assumed to be 0.1; new values are given to the damping coefficient for the same disturbance of case #1 and comparative analysis is performed. The new values are 0.2, 0.0, and -0.1 and the results are shown in Fig. 4.22 and Fig. 4.23. The value of the damping coefficient associated with each curve is shown in the figures.

38

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Time (s) Fig. 4.21: Response to a step change of -10% in the mechanical power input

Time (s) Fig. 4.22: Impact of the positive values of the damping coefficient on the linearized transient performance of the SMIB system – Classical model (D are the shown numbers) 39

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Fig. 4.23: Impact of the negative value of the damping coefficient on the linearized transient performance of the SMIB system – Classical model  It is clear from Fig. 4.22 that the amplitude and the settling time of the transient oscillations are highly dependent on the damping coefficient value. The higher the damping coefficient, the better the response (lower amplitudes and settling time). It is also shown that the frequency of the oscillations is insensitive to the damping coefficient.  Based on Fig. 4.22, and 4.23, it can be easily depicted that the positive damping results in damped oscillatory response (a stable situation) while the negative damping is associated with oscillatory response with increasing amplitudes (instability situation). The case of zero damping is associated with sustained oscillations of constant amplitude (critical or marginal stability).  Negative damping can be realized in reality in the situations where the parameters of the generator controllers are incorrectly tuned. In addition, positive damping is associated with some types of loads.  Zero damping is a sort of a theoretical assumption or a situation where negative damping sources compensate the positive damping of the system. Generators have the inherent damping capability due to their electric power 40

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

losses, and eddy current flow in the magnetic core of the machine as well as other energy losses such as windage, and friction.  Case #3 : in this study the impact of the generator’s inertia on the transient response is considered. In comparison with the data of case #1, in this case only the inertia constant of the generator is changed. The original value of the inertia constant H = 10 sec. In this case, two additional values are considered which are 5 sec, 20 sec, and 100 sec. The simulation results are shown in Fig. 4.24.  It is clear from the results that increasing the inertia reduces the magnitudes of the speed oscillations; however, the frequency and the settling time of the speed oscillations increase. This is attributed to the increase in the K.E stored in the rotating system with the increase of the inertia constant. This means that the energy needed to alter the speed of the generator. Consequently, the amplitude decreased with the increase in the inertia while restoration of the normal K.E takes more time, which results in increasing the settling time and reducing the frequency of the speed oscillations.  From the power angle deviations point of view and considering the increase in the rotor K.E energy with increase in the inertia, it is clear that the increase of the inertia increases the amplitudes of the angle oscillations. In addition, the settling time is increased and the frequency of the angle oscillations is reduced. Same comments can be given to the electrical power due to its tight relation to the power angle.  Although the increase of the inertia constant causes some drawbacks such as a reduction in the frequency of oscillations and increase of the settling time, the speed (or the frequency) deviations are reduced. This is of great importance in keeping the system synchronized. Therefore, with higher inertia, a disturbance is unlikely to move the system to the extremes state in comparison with a system with lower inertia.  From the damping power point of view and considering the impact of increase the inertia on the speed deviations, higher initial amplitudes of the damping power are available with lower inertia values. This is because the damping power is directly proportional to the speed deviations. Therefore, increasing the damping ratio of low inertia generators can be a good measure for enhancing its stability. This is can be achieved via Power System Stabilizers (PSS) for the conventional synchronous machines or generally 41

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

with Power Oscillation Damping (POD) controllers. PODs can be designed to improve the system damping and can be placed at any location on a power grid and they are usually integrated with FACTS devices such as SVCs or TCSCs12,13. The design of PSSs and PODs will be considered in the upcoming chapters of this textbook.

Fig. 4.24: Impact of the inertia constant on the linearized transient performance of the SMIB system – Classical model (H are the shown numbers)

M. Mandour, M. EL-Shimy, F. Bendary, and W. Ibrahim, “Damping of Power Systems Oscillations using FACTS Power Oscillation Damper – Design and Performance Analysis”, 16th International Middle East Power Systems Conference (MEPCON’14), Dec. 23-25, 2014, Cairo, Egypt, pp. 1- 8. 13 M. Mandour, M. EL-Shimy, F. Bendary, W.M. Mansour. Impact of Wind Power on Power System Stability and Oscillation Damping Controller Design. Industry Academia Collaboration (IAC) Conference, 2015, Energy and sustainable development Track, Apr. 6 – 8, 2015, Cairo, Egypt 12

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M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

 Case #4 : In this example the impact of the grid coupling strength on the performance of the SMIB sys tem is considered. The grid coupling strength is inversely proportional to the value of the network connecting the generator to the infinite bus. In the considered simplified model, the equivalent reactance network is the external reactance ( xe). If xe is small, then strong coupling is achieved while high value of xe is referred to weak coupling. With the SMIB system data as given in case #1, only the external reactance is given some new values. These considered values are 0.1 p.u, 0.4 p.u (the original value), 0.7 p.u, and 1.0 p.u. Consequently, the values of the initial power angle, the transient emf, and the synchronizing power are changed. Table 4.2 lists these values for various values of the external reactance. These values are shown in Fig. 4.25 and the simulation results are shown in Fig. 4.26.

xe (p.u)

0.1 0.4 0.7 1.0

Table 4.2: Impact of variations of xe on E’, o,and P s E’(p.u) P s (p.u) o (deg) 1.1916 1.3172 1.4318 1.5379

11.6190 18.5940 24.7751 30.4775

2.9180 1.7835 1.3000 1.0195

 The changes shown in Table 4.2 and Fig. 4.25 reveal some important stability and operational issues. Fig. 4.25(b) shows a general shape of the synchronizing power and transient power functions. This figure shows that increasing the electrical power is associated with an increase in the power angle and a decrease in the synchronizing power. In addition, Table 4.2 shows that increasing the external reactance results in a decrease in the synchronizing power. Therefore, the synchronizing power decrease with either an increase in the generator loading or decrease in the network coupling strength or both of them. Actually, Fig. 4.25(b) shows that the increase in the external reactance simultaneously increases the power angle and decreases the synchronizing power. As previously stated and also shown in Fig. 4.26, better transients are associated with higher synchronizing power. The phrase better refers to lower the amplitude of oscillations, and settling time. For example, in Fig. 4.26, the increase in the external reactance which 43

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

(as shown in Table 2) decreases the synchronizing power, results in an increase in the amplitude of the speed oscillations and reduction in their frequency of oscillations.

(b) Impact on small signal stability

(a) Values of main variables

Fig. 4.25: Impact of the external reactance on the variables and stability SMIB system  Fig. 4.25(b) also shows that increasing the external reactance requires a significant increase in the transient emf (the steady state emf also should be increased as shown in Table 4.1). This required increase in the emfs mandates an increase in the design rating of the generator excitation system. In addition, for a given design, increasing the emf reduces the active power production capability. This is because the active and reactive power of a generator are dependent quantities and related to the generator MVA rating by . From a practical point of view, the reactive power needed for compensating long lines is not only provided by the centralized generators. Instead, distributed reactive power compensators are installed along the long lines for reactive power support, voltage control, and voltage stability enhancement.

44

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Fig. 4.26: Impact of the grid-coupling strength on the linearized transient performance of the SMIB system – Classical model (xe are the shown numbers)  Case #5 : The given linearized model can be used for representing some other disturbances such as a sudden permanent disconnection of a line. Consider the SMIB system shown in Fig. 4.27 in which two lines connect the generator to the infinite bus. The generator normal frequency is 50-Hz generator and having an inertia constant M = 2H = 20 sec. The transient reactance equals to 0.3 p.u. The generator is connected to the infinite bus through a purely reactive circuit with XT = 0.2 p.u., and XL = 0.4 p.u. The generator is delivering power of 0.6 p.u at 0.8 lagging power factor to the infinite bus at 1.0 p.u voltage. The p.u damping coefficient is assumed to be 0.10. In this example, it is assumed that the line outage disturbance is small enough for the linearized model to be valid. 45

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Fig. 4.27: The SMIB system of case #5  Recalling that with the classical model, a constant transient emf is assumed during the transient process. Therefore, before and after the outage of the line, the transient emf will not be changed; however, the transient power-angle characteristics will be changed as shown in Fig. 4.28. In that figure, the curve number I refers to the pre-outage characteristics while the curve number II refer to the post-outage characteristics. Since, the outage of the line results in an increase in the transfer reactance (xe) between the generator and the infinite bus, the maximum power of curve II is low in comparison with curve I. This shows that the line outage can be simulated by an equivalent sudden change in the mechanical power. Numerical values and calculation methodology are shown below.

Fig. 4.28: Pre-outage and post-outage transient power-angle characteristics  Based on the models of Appendix 1, 46

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

P  I

E 'V I X12

I sin  Pmax sin

The pre-outage (I) system reactance = X d  X12 'I

I

= 0.3 + 0.4/2 + 0.2 = 0.7 p.u.

The transient emf E’ can be obtained using (34), it is calculated at prefault conditions. Q P I   I  E   o X12   V  o X12  V    V 2

2

'

Hence, E’ = 1.380 p.u. = constant. Therefore, P I = 1.97 sin 





I The initial operating angle =  oI  sin1 Po Pmax = 17.732

o

=

0.309485 rad. II = 0.9 p.u. The post-outage reactance = X12

The post-outage approximated characteristics is then: P

II



E 'V II X12

transient

power

angle

II sin  Pmax sin = 1.53 sin 

Based on Fig. 4.28, it can be concluded that at the instant of line tripping, a sudden change in the system topology occurs resulting in a sudden change in power-angle characteristics of the system. The operating point moves suddenly from point (a) to point (b) causing a sudden power imbalance between the mechanical input and electrical output of the generator; P mo. The operating point corresponding to point (b) cannot be stable due to the mismatch between mechanical and electrical power. As the difference is positive the rotor accelerate following the post-outage characteristics (II) and then after some power 47

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

oscillations the system reaches a new steady state operating point corresponding to point (c) in Fig. 4.28. Where





II is the final steady state angle (after power oII  sin1 Po Pmax

oscillations settles) = 23.089 o = 0.402975 rad. The initial change in generator input power



Pmo  P I  P II



  oI

I II  Pmax sin oI  Pmax sin oI

P mo = 1.97 sin( 17.732) – 1.53 x sin(17.732) = 0.1340 p.u

Fig. 29: Simulation of a line outage by an equivalent power imbalance Therefore, the considered disturbance can be simulated in the same way as

II cos  oI = 1.457 p.u. The results are the previous cases. In this case, Ps  Pmax

shown in Fig. 4.29. 48

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 From a physical point of view, the line outage causes an increase in xe. Therefore, the final steady state angle is expected to be high in comparison with the initial steady state angle. The response verifies this issue. In addition, the sudden increase in xe causes sudden decrease in the electrical loading on the generator. Consequently, it is expected that the frequency will increase just after the line outage.  Case #6 : simulation of a temporary outage of a line . This situation may be faced due to, for example, temporary fault or operation of Automatic Circuit Reclosers (ACR). In this case, the mechanical power input remains unchanged during the transient process. The disturbance in this case can be simulated by an equivalent sudden change in the power angle. As shown in Fig. 4.30, the initial operating angle of the system is ; corresponding to point Fig. 4.30: temporary line outage (a) on curve I. When the line is disconnected, the transient power curve changed to curve II and the operating point moves to point (b) on curve II. It is assumed that the line will return to service after a very short duration. In the figure, it is shown that the line

reconnected at an angle see which is corresponding to point (c) on curve II. Now, the system characteristic return to curve I and at the instant of line reconnection, the operating point moves to point (d) on the curve I. Due to the power mismatch, a transient process acts and afterward the system operating point settles at the initial operating point . The overall changes in the system due to the disturbance are null; however, the difference ( acts as a disturbance which is equivalent to the momentary outage of the line. In this case, the data of the system of case #5 is considered such that one of the lines is temporary disconnected and the equivalent sudden angle change equals to 10 degrees (0.17453 rad). The original simulink model is modified as shown in Fig. 4.31 by setting an initial non-zero angle which equals to o. Since the transient process will mainly act on the curve 49

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

I. Hence, = 1.8764 p.u. The results are shown in Fig. 4.32 which are in confirmation with the stated analysis of this case (Fig. 4.30). The response of the system for momentary disturbances is called zero-input response while the response for sustained disturbances is called forced response.

Fig. 4.31: The Simulink model for simulation of a momentary outage of a line in the SMIB system – classical model.

Fig. 32: Simulation of a momentary outage of a line in the SMIB system (zero-input response) – classical model. 50

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By the end of this numerical example, it is worthy to be noted that the conclusions derived from the analysis of the presented simple system and simplified models are still valid for larger system and more complex models. This is will be demonstrated later in this textbook.

4.4.4 Time-domain solutions of the linearized model

Fortunately, it is possible to derive the analytical time domain expressions that describe the solution of the linearized swing equation (4.26). The equation presents a linearized standard second order differential equation and its solutions will be presented in the following part. The solutions corresponding to various zero-input and forced responses will be considered. From the block diagram of Fig. 19, we have in the s-plane

1  f     o  Pm  Ps   D   (4.35) s  Hs  Eliminating  from (4.35) using s results in:

1 f     o Pm  Ps   Ds  s  Hs 

(4.36)

Rearranging (4.36) in the form of a quadratic equation results in s 2 (  ) 

fo D H

s(  ) 

fo Ps H

(  ) 

fo H

Pm

(4.37)

Therefore, the characteristic equation is given by: s2 

fo D H

s

fo Ps H

 0 (4.38)

Equation (4.38) is similar to the standard 2nd order differential equation of the general form: 51

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s 2  2 n s   n2  0

(4.39)

Therefore, the undamped natural frequency n in rad/sec is

fo

n 

H

Ps

(4.40)

and the damping ratio  is given by:



D 2

fo HPs

(4.41)

The equations describing the free-motion zero-input response (u = 0) of the system can then be derived as follows. Referring to the state equation (4.34), we see that the free motion (with zero input) is given by: 

 x  Ax (4.42)

Considering the SMIB system shown in Fig. 4.33, suppose that the breakers of one of the transmission circuits open and close quickly. As previously

described, the disturbance is equivalent to a small deviation  in the power

angle from the initial operating point o i.e.  = o + 

(4.43)

Fig. 4.33: SMIB system for the analysis of the zero-input response In this case u = Pm = 0, and the differential equation of motion becomes: d 2  dt 2

 2 n

d   n2   0 dt 52

(4.44)

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with the characteristics equation: s 2  2 n s   n2  0

(4.45)

The roots of the characteristic equation 1,2 (which are the eigenvalues of the state matrix A) are given by:



1, 2    n   n  2  1



(4.46)

Generally, eigenvalues can be represented as  =   j  (4.47) where frequency of oscillation is f = /(2) Hz.

It is known that the response depends on the roots of the characteristic equation. Three different cases can be realized as shown in Fig. 4.34.

Fig. 4.34: Possible conditions of the eigenvalues  Case 1: 1  2 and both are real. This is case occurs when  2 > 1. The general solution in this case is  (t )  A1e 1t  A2 e 2t

(4.48)

The constants A1 and A2 can be found from the initial conditions:

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 (0)  0 and  (0) 

d (t ) 0 dt t 0

Therefore, (4.47) becomes:

 (t ) 



 o 2 e 1t  1e 2t 2  1



(4.49)

(4.50)

which is an over-damped response. This case describes a situation where 1 = 1 and 2 = 2. In addition, 1 = 2 = 0. It is clear that three situations can be realized according to the values of the 1 and 2.

(i) 1 > 0 and 2 > 0. In this case, as shown in eq. (4.50), the time domain (TD) response is a rising exponent with time

constant  = 1/ as shown in Fig. 4.35. This situation, of course, is corresponding to an unstable response.

Fig. 4.35: Response with 1 > 0, 2 > 0 and  = 0 (ii) 1 < 0 and 2 < 0. In this case, as shown in eq. (50), the time domain (TD) response is a falling exponent with time

constant  = -1/ as shown in Fig. 4.36. This situation, of course, is corresponding to a stable response.

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Fig. 4.36: Response with 1 < 0, 2 < 0 and  = 0 (iii)

Either 1 > 0 or 2 > 0 or both of them > 0 . In this case,

as shown in eq. (50), the time domain (TD) response is a rising exponent similar to that shown in Fig. 4.35. This situation is corresponding to an unstable response.  Case 2: 1 = 2 =  and both are real. This is case occurs when  2 = 1. The general solution in this case is  (t )  e t  A1  A2t 

(4.51)

Using (4.51) to find the constants A1 and A2. Then the solution becomes:  (t )  e t  o 1  t 

(4.52)

which is a critically damped response. The situations described in case 1 are also valid in this case.  Case 3: 1 and 2 are complex conjugate. This is case occurs when  2 < 1 resulting in an oscillatory TD solution. The roots of the characteristic equation take the form:

1, 2    n  j d  55

(4.53)

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where d is the damped frequency of oscillations,

d  n 1   2

(4.54)

and the solution takes the form:  (t ) 

 o

1 2

e  nt sin( d t   )

(4.55)

where cos-1 . In this case, the TD response is oscillatory. And the response time constant is given by:



 n 1

(4.56)

If the response is stable then the response settles in approximately 4

times  Therefore, the settling time is s  4 . The frequency of the oscillations takes the form.  (t )  

 n  o 1

2

e  nt sin( d t )

(4.57)

The stability of the system in this case is governed by the signs of the real parts of the eigenvalues according to the following situations. (i) i > 0 and   0. The time domain response of this situation

is harmonic oscillations with exponentially rising amplitude (i.e. unstable response) as shown in Fig. 4.37. 

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Fig. 4.37: Response with i > 0 and   0 (ii) i < 0 and   0. The TD response of this situation is stable damped harmonic oscillations as shown in Fig. 4.38.

Fig. 4.38: Response with i < 0 and   0 (iii)

i = 0 and   0. The TD response of this situation is

harmonic oscillations with constant amplitude of o and

frequency  as shown in Fig. 4.39. The system in this case is called critically stable or in bifurcation.

Fig. 4.39: Response with i = 0 and   0 57

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Demonstrating problem The reader may use the following data for applying the time domain solutions. a) A 50-Hz generator having inertia constant M = 2H = 20 sec and a transient reactance of 0.3 p.u is connected to an infinite bus through a purely reactive circuit with equivalent reactance of 0.4 p.u. The generator is delivering power of 0.6 p.u at 0.8 lagging power factor to the infinite bus at 1.0 p.u voltage. Assume the per unit damping power coefficient is 0.14.

Consider a small disturbance  = 10o resulting of breakers open and quick close of one of the two lines. It is required to obtain and plot the functions

describing the motion of the rotor i.e. (t) and f(t).

Hint : in this case  = 0.2025 < 1. Hence, oscillatory solution governed by

(4.55) and (4.57). The plots of the solutions take the form shown in Fig. 4.40 and 4.41. This solution is in confirmation with the TD solution obtained with the Simulink.

Fig. 4.40: Angle response of problem 1

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Fig. 4.41: Frequency response of problem 1

b) The data for the system of Fig. 33 are: fo = 50 Hz. H = 9.95 sec. x’d = 0.30 p.u. xe = 0.40 p.u. V = 1.0 p.u. D = 0.15. Consider a small disturbance o = 10o. It is required to plot the response (t) and (t) for various values of the active power delivered to the infinite bus form, say zero to 0.95

Pmax. Also, plot the functions P(), Ps(), , n(). Assume that E’ = 1.4 p.u. is kept constant for all load values. Analyze the results.

Expressions describing the TD response of the SMIB as subjected to sudden changes in the mechanical power inputs can also be derived. The system response in this case is a forced response in which the disturbance is

sustained. Assume that the input power is increased by a small amount P m.

In this case P m  0 i.e. u  0. The linearized equation of motion of the system can be rewritten as: d 2 d  fo 2 2     Pm  u    n n dt H dt 2

59

(4.58)

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The characteristic equation of the system is given by (4.39). Equation (4.58) can be written in the state space form as,

The state variables are defined as x1 =  and x2 = . This equation can be written as and the output takes the form . The s-space of this model can be obtained using the Laplace transform such that . This is shown in eq. (4.60) while eq. (4.61) shows the solution of the state variables. The Laplace transform for a sudden (or step) change in the input u is equals to u/s which is called U(s) in eq. (4.60).

Based on (4.61), the state variables take the form,

The TD solutions of the s-space solutions (i.e. eq. (62) and (63)) take the forms,

 (t ) 

 f o Pm H n2

  1 n t   1 sin   e t  d   1  2  

and

60

(4.64)

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 (t ) 

fo Pm

H n 1  

2

e  nt sin d t 

(4.65)

Demonstrating problem The reader may use the following data for applying the time domain solutions. A 50-Hz generator having inertia constant M = 20 sec and a transient reactance of 0.3 p.u is connected to an infinite bus through a purely reactive circuit shown in Fig. 4.42. With XT = 0.2 p.u., and XL = 0.4 p.u. The generator is delivering power of 0.6 p.u at 0.8 lagging power factor to the infinite bus at 1.0 p.u voltage. Assume a per unit damping power coefficient is 0.14. While the system is operating under steady state condition, the breakers of one of the lines are suddenly open and remain open. Assuming the initial power imbalance caused by the disturbance is small enough to utilize linearized system equations, and assume that the transient emf E’ remains constant at prefault value. Obtain and plot the functions describing the motion of the rotor i.e .(t) and f(t).

Fig. 4.42: SMIB for demonstration of the forced response Solution hints. The disturbance can be simulated by as a step change in the

mechanical power input. The pre-disturbance transient power-angle characteristics take the form P  I

E 'V I X12

I sin  Pmax sin = 1.97 sin 





I = 17.732 o = 0.309485 rad. The initial operating angle =  oI  sin1 Po Pmax

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The post-disturbance approximated transient power-angle characteristics take the form,

P

II





E 'V II X12

II sin  Pmax sin = 1.53 sin 

The pre- and post-disturbance characteristics are shown in Fig. 4.43.

Fig. 4.43: Pre- and post-disturbance characteristics and equivalent disturbance The equivalent disturbance is the initial change in generator input power



Pmo  P I  P II



  oI

I II  Pmax sin oI  Pmax sin oI 0.1340 p.u

Since the system dynamics occurs along the post-disturbance power angle characteristics. Then, the synchronizing power is calculated as: II Ps  Pmax cos  oI = 1.457 p.u.

The responses are then shown in Fig. 4.44 and 4.45. 62

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Fig. 4.44: Angle response

Fig. 4.44: Frequency response

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4.4.5 Small-signal stability and eigenvalues of the state matrix

The previous analysis shows the interrelation between the system smallsignal stability and the eigenvalues of the system. In addition, it is shown that the shape of the response can be easily predicted from the eigenvalues of the system. Although the considered system is just SMIB and the adopted model is very simple, the derived conclusions are general regardless of the system size and the model complexity. Generally, the roots of the characteristic equation of a power system are the eigenvalues of the state matrix A. The time response of a dynamic system is dependent on the values of the eigenvalues. The eigenvalues of a state matrix A can also be found by solving: det(A - I) = 0 (4.66) In fact, expansion of the determinant of (4.66) gives the characteristic

equation. The n solutions of  are the eigenvalues of A. Each eigenvalue is corresponding to a dynamic mode and the time dependent characteristic of a

mode corresponding to an eigenvalue i is given by exp(i t). The stability of the system is determined by the eigenvalues as follows: 1. A real eigenvalue corresponds to a non-oscillatory mode, such that: a. A negative real eigenvalue represents a decaying mode. The larger its magnitude ,the faster the decay. b. A positive eigenvalue represents aperiodic instability. 2. Complex eigenvalues occur in conjugate pairs, and each pair corresponds to an oscillatory mode. The real component of the eigenvalues gives the damping, and the imaginary component gives the frequency of oscillation. Such that: a. A negative real part represents a damped oscillation. b. A positive real part represents oscillation of increased amplitude. Thus, the equilibrium points may be classified as follows:

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1. Stable if the real part of all eigenvalues are negative i.e. all eigenvalues

are on the left-half complex plane  , j.

2. Unstable if the real part of at least one of the eigenvalues is positive i.e.

at least one of the eigenvalues is on the right-half complex plane  , j.

3. Bifurcation o if the real part of at least one of the eigenvalues is zero

i.e. at least one of the eigenvalues lies on the imaginary axis complex plane  , j.

4.5 Inclusion of field circuit dynamics in the linearized model of the SMIB system The classical model of the synchronous machines assumes that the transient emf is kept constant (E’ = 0) during the transient processes. Therefore, the d-axis flux density is assumed to be constant during transient. Note that the transient emf is proportional to the direct axis flux linkages. Consequently, the field circuit electromagnetic dynamics are neglected. The linearized classical model can be considered as an elementary model for the analysis of the damping and synchronizing power of synchronous generators. Higher order models of the synchronous machines can simulate the field circuit dynamics14,15,16,17. The following simplifying assumptions are considered in the following model, 1. The effects of the damper windings (amortisseur effects) are neglected. These effects include additional damping; however, selecting an appropriate damping coefficient is a possible way of considering the amortisseur effects in a simple way. This assumption is quite acceptable when the rotor speed deviations are small. 2. The electromagnetic transient of the stator windings are neglected. Therefore, the time-dependent changes in the stator flux linkages (p)

14

Demello, F. P., & Concordia, C. (1969). Concepts of synchronous machine stability as affected by excitation control. IEEE Transactions on power apparatus and systems, 88(4), 316-329. 15 Kundur, P. (1994). Power system stability and control (Vol. 7). N. J. Balu, & M. G. Lauby (Eds.). New York: McGraw-hill. 16 Rogers, G. (2012). Power system oscillations. Springer Science & Business Media. 17 Padiyar KR. Power System Dynamics: Stability and Control: Anshan; 2004. 65

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are neglected18. It can be shown that neglecting these transients is compensated by neglecting the electromagnetic transients associated with the power network. Therefore, modeling the power network components by time-independent parameters is also acceptable. In addition, the phasor diagram can be used to define the stator equations (Fig. 4.13). 3. The machine saturation is neglected. 4. The resistances associated with the machine’s armature and the network are neglected. 5. For generality, a salient-pole machine model is considered. Given the stated assumptions, the machine model becomes a third order model instead of a second order model. The additional differential equation describes the field circuit dynamics. This model is as follows.

or

or

Kundur, P. ‘Chapter 5: Simplifications essential for large-scale studies – example 5.1’ in: ). Power system stability and control (Vol. 7). N. J. Balu, & M. G. Lauby (Eds.). New York: McGraw-hill.

18

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Note that the generator field voltage ( Efd) equals to 1.0 p.u for 1.0 p.u terminal voltage at open circuit conditions (i.e. on the air gap line). The linearized form of this model is defined by the Heffron-Phillips Constants as follows,

The block diagram representing the linearized model of the SMIB system with field circuit dynamics included is shown in Fig. 4.45. As shown in the figure, additional control systems can be included. These control systems include the prime mover and the excitation control systems as well as additional stabilization controls. In addition, the model can now simulate the machine terminal voltage. The equations representing these constant can be found through the linearization of the machine model represented by equations (4.67) to (4.75). In order to do so, linearization of equations (4.67), (4.68-a), (4.69-a), (4.71) to (4.75) are prepared as they will be needed during the derivation of the Kconstants.

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Fig. 4.45: Block diagram representing the linearized model of the SMIB system with field circuit dynamics included with approximate modeling of the machine damping (i.e. damper windings, eddy currents, electrical losses, friction, and windage ) Equation (4.67):

Equation (4.68-a):

Equation (4.69-a):

Equation (4.71):

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Equation (4.72):

Equation (4.73):

Equation (4.74):

Equation (4.75):

The constant K1 represents the change in electrical torque for a change in rotor angle at constant flux linkage in d-axis i.e. at Eq’ = constant.

Based on (4.82),

Based on (4.83) and (4.84),

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Substituting (4.89) and (4.90) in (4.88) and considering (4.87), results in:

Equations (4.70) shows that the term

equals to

.

Therefore, equation (4.91) becomes,

The constant K2 represents the change in electrical torque for a change in d-axis flux linkage at constant rotor angle i.e.

Based on (4.82):

Using equations (4.83), (4.84), and (4.73) we get,

Considering the s-space representation of eq. (4.85)

From which the values of K3 and K4 can be determined after eliminating

id using (4.83) and arranging the equation in the form of (4.77). 70

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It is clear that K3 represents the impedance factor that correlates the steady state and transient reactances while K4 represents the demagnetizing effect of change in rotor angle i.e. armature reaction as it is negative emf that oppose the transient emf that corresponding to Efd. Based on (4.78), the constants K5 and K6 are defined as follows.

The constant K5 Represents the change in terminal voltage with change in rotor angle

at constant

while the constant K6 represents the change in

terminal voltage with change in eliminating

and

at constant . Using (4.83) and (4.84) for

from (4.80) and (4.81) then substitute the resulting

equations in (4.79) for eliminating

and

. The definitions (4.98) and

(4.99) are then used to determine the constants. The results are:

The values of K1, K2, K3, K4, and K6 are usually positive while the value of K5 is usually negative. In fact, the increase in the machine active power loading (in other words, the increase in the machine power angle, 71

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reduces the terminal voltage) while the increase in the transient emf increases the terminal voltage. Given that

. Therefore, the

negative value of K5 and the positive value of K6 are expectable. The required initial conditions for estimating the K-constants are: EQo, Vo,

o, Vtdo, Vtqo, and Vto. These variables are determined based on the

mathematical models presented in Appendix 1. The value of Vo is supposed to be a known quantity as it is the infinite bus voltage magnitude while the other variable can be estimated as follows. P    Q   o X q    Vo  o X q   Vo Vo    2

E Qo

tan  

(Po /Vo )X q 

Vo  (Qo /Vo )X q 

(4.102)

(4.103)

P    Q Vto   o X e   Vo  o X e  Vo Vo    2

2

2

(4.105)

Vtqo Vto cos(o  eo ) (4.106)

Vtdo Vto sin(o  eo )

tan eo 

(Po /Vo )X e Vo  (Qo /Vo )X e

(4.107) (4.108)

Referring to Fig. 4.25, the synchronizing power coefficient ( P s ) is significantly affected by the inclusion of the field circuit dynamics. Generally, the synchronizing power coefficient is defined by:

With the field circuit dynamics neglected (i.e. the classical model), P s and the natural frequency of oscillations are given by,

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With the field circuit dynamics considered, by replacing

in (4.70) by

(4.77), the results is,

or

Therefore,

Equation (4.113) shows that at steady state (i.e. s  0), the synchronizing power coefficient is reduced by ( ) with the field circuit dynamics included in comparison with the classical model. This reduction is due to the demagnetizing effect of armature reaction. The reduction in P s is expected to have a negative impact on the settling time and the frequency of oscillations. Since the values of K1, K2, and K3 are usually positive, then it is possible to say that the due to the demagnetizing effect of armature reaction adds a positive damping (D AR). This can be easily seen by replacing s by j in (4.112) and decomposing the resulting equation into real and imaginary parts.

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Since becomes,

then

or

, then (4.115)

Equation (4.115) can be then written as

where Kd represent the armature reaction p.u damping coefficient while Ks represents the synchronizing power coefficient. Both coefficients are rotorspeed dependent and they are quadrature to each other as shown in Fig. 4.46; where

Fig. 4.46: Synchronizing and damping components of the electrical power

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The equivalent damping constant D eq can be obtained from,

Note: that if o is represented in p.u (i.e. equals to 1.0 p.u), then D = Kd. The natural frequency of oscillations in the absence of voltage regulator (Efd = 0) is then,

Note: that, if

which is popular situation, then

Example Simulation and analysis of the linearized model of the SMIB system considering field circuit dynamics The presented linearized third-order model is implemented on the Simulink as shown in Fig. 4.47. The parameters of the considered system and main variables of the considered system are given as follows such that the reactances, voltages, and power values are represented in p.u while the time

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and inertia constants are represented in seconds. The system data are typical values of generators in nuclear power plants19.

The steady state initial conditions and the K-constants as well as some other parameters are consequently determined using the presented model. Their values of the K-constants are,

Fig. 4.47: simulink model of the linearized model SMIB system considering field circuit dynamics The main objective of this example is to show the effect armature reaction damping. Since D is set to zero, the performance of the classical model was – 19

http://mkpalconsulting.com/ 76

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as shown in the previous simulations - sustained harmonic oscillations; however, with field circuit dynamics considered, the performance is oscillatory with decayed amplitudes. This is demonstrated by each of the following disturbances: 1. A step decrease in the mechanical power of 10% (Fig. 4.48 and 4.49) as an example of the forced response of the system due to sudden change in the load. 2. A step decrease in the field voltage of 10% (Fig. 4.50 and 4.51) as an example of the forced response of the system due to sudden change in the field voltage caused by a very fast acting excitation regulator , and 3. A sudden change in the power angle by 10 o (Fig. 4.52 and 4.53) as an example of the zero-input response of the system due to sudden change in the network topology. All simulations show clearly the damping effect of the field circuit dynamics. The model can provide many other simulations and analysis that may be implemented by the interested readers.

Fig. 4.48: Linearized response of the SMIB system for P m = -10% considering field circuit dynamics 77

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Fig. 4.49: Linearized response of the SMIB system for P m = -10% considering field circuit dynamics – the electrical power components

Fig. 4.50: Linearized response of the SMIB system for Efd = -10% considering field circuit dynamics 78

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Fig. 4.51: Linearized response of the SMIB system for Efd = -10% considering field circuit dynamics – the electrical power components

Fig. 4.52: Linearized response of the SMIB system for o = 10o considering field circuit dynamics 79

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Fig. 4.53: Linearized response of the SMIB system for o = 10o considering field circuit dynamics – the electrical power components 4.6 Including AER and PSS in the linearized third-order model of the SMIB system As stated in Appendix 1, an excitation system and an excitation control system are a combination of devices designed to generate a field current and control it by means of Automatic Excitation Regulators (AER) which is also called Automatic Voltage Regulator (AVR). An excitation control system is composed of three main components. These components are the field winding, exciter (i.e. a DC power source), and a regulator (AER or AVR). The AER controls the value of Efd as shown in Fig. 4.54. The regulator is nonlinear due to the presence of the Efd limiter.

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Fig. 4.54: Generic schematic of excitation control systems The excitation control system model can be integrated with the 3 rd order linearized model of the machine as shown in Fig. 4.55.

Fig. 4.55: SMIB 3rd order linearized machine model with excitation control The main objective of this model is to investigate the impact of the excitation control system on the damping of the system. There are many other applications in which the model of Fig. 4.55 can be used. Measurement-based estimation of the system inertia is an example that will be considered later in this chapter. Since measurement-based equivalence is a target of this book, no more detailed mathematical modeling will be given to the system of Fig. 81

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4.55; however, the interested reader may refer, for example, to Kundur’s textbook20.

Example Simulation and analysis of the linearized model of the SMIB system with excitation control system The system of Fig. 4.47 is considered in this example; however, an excitation control system is included in the system model as shown in Fig. 4.55. The combined transfer function of the exciter and the regulator is given the simple form,

With the gain of G (s) is set to zero, the effect of the excitation control system is deactivated. In this case, the effect of the effect of the field circuit dynamics is considered as shown in Fig. 4.47. In this example, it is assumed that the terminal voltage regulator is fast enough for neglecting the time delay effect of its time constant ( TR). In addition, it is assumed that the capacity of the AER is large enough for neglecting the AER output limiter; consequently, the AER becomes a linear network. A fast acting excitation control system with a time constant TA = 0.1 sec and gain KA = 50 (i.e. high gain) is considered. Proper selection of the gain KA is usually based on a stability criterion for the AVR loop21. A sudden decrease of 10% in the mechanical power input is considered as a disturbance. The effect of the excitation control system on the system damping in comparison with the unregulated system is investigated considering the mentioned gain (KA = 50) and a gain KA of a zero value. The simulations of the system with each of the mentioned gains are performed. The Simulink model is shown in Fig. 4.56 while the results are shown in Fig. 4.57. The yellow lines are corresponding to the KA = 50 while the other lines are corresponding to the KA = 0. It is clear from the 20

Kundur, P. (1994). Power system stability and control (Vol. 7). N. J. Balu, & M. G. Lauby (Eds.). New York: McGraw-hill. 21 Elgard, O. I. (1982). Electric energy systems theory. New YorkMc Graw-Hill. 82

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results that the considered AVR significantly reduces the system damping and causes instability due to negatively damped oscillations.

Fig. 4.56: Lumped simulink model of the system

Time (s) Fig. 4.57: Effect of the excitation system control in comparison with the unregulated system 83

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Reducing the AER gain improves the stability. This is shown in Fig. 58 with KA = 30 (yellow lines) and KA = 10. In both situations, the system is stable; however, the stability suffers from two issues: (1) The steady state error in the terminal voltage is high and its value is increased with the increase in the gain KA.; (2) The settling time of the oscillations is also high. Therefore, the stabilization gain (or damping enhancement) caused by the reduction of the AER gain is counteracted by the high steady state errors and the long settling time.

Fig. 4.58: Stabilization of the system by AER gain reduction 84

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The impact of the gain KA can be demonstrated and explained using the root locus of the system which is shown in Fig. 4.59. Fig. 4.59(a) shows the full root locus of the system while Fig. 4.59(b) shows the root locus focused near the imaginary axis for gain selection purposes.

(a)

(b) Fig. 4.59: Root locus of the regulated SMIB with AER – the effect of AER gain on stability and performance; (a) Full view; (b) Zoomed view

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It is clear from the figure that the critical value of the gain is 42.9 at which the system is critically stable. Higher values of the gain result in system instability while with lower values the system is stable but its damping is poor. Appendix 2 provides an overview of the extraction of linear transfer from a Simulink model and the linearized analysis of transfer function as well as gain selection based on the root locus analysis. 4.7 Power System Stabilizers (PSS) Power system stabilizers are supplementary stabilizing controllers of the AER. As shown in Fig. 4.57, AER can provide negative damping to power systems; a situation which leads to system instability. This AER caused instabilities depend on the system parameter, the operating conditions, and the AER structure as well as its parameters. For example, the responses of Fig. 58 are stable; however, the characteristics of the oscillations (settling time, overshoot, rise time… etc.) are poor. PSSs can also provide a tool for enhancing the system performance. Tuning of PSSs and the control design issues are out of the scope of this chapter; however, stabilization of the system of Fig. 4.56 will be considered. Generally, the main objective of PSSs is to artificially produce a damping torque, which in-phase with the rotor speed. This torque is called a supplementary stabilizing signal and the controllers used to generate these signals are known as power system stabilizers. Fig. 4.60 shows PSS integrated with the AER such that the speed deviation is the PSS input. The frequency or acerbating power may be also used as input signals to the PSS. As shown in Fig. 4.60(b), the speed deviation is used as an input to the PSS. The transducer measures and converts the input signal to voltage. The high frequency filter is designed to attenuate the generator’s torsional frequencies as well as high frequency measurement noise. The lead-lag stages are usually composed of identical a number of phase compensators. The function of the lead-lag stages is to provide an overall phase lead over the frequency range of interest to compensate for the lag produced in the generator-excitation system. The minimum number of phase compensators is one while the maximum known number is three while the most popular designs contain two compensators. The output signal of the lead-lag stages is then amplified by an amount Ks and sent through a washout stage. The washout prevents voltage offsets during steady-state or prolonged speed or 86

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frequency change. The output limiter prevents the stabilizer output signal from interfering with the action of the voltage regulator during severe system disturbances.

(a)

(b) Fig. 4.60: AER with PSS supplied from the speed deviation; (a) general block diagram; (b) the PSS details are shown

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Example Simulation and analysis of the linearized model of the SMIB system with excitation control system and simplified PSS With an AER gain of 50, the system of Fig. 4.56 shows instabilities due to the negative damping cause by the high-gain AER. In this example, PSS is used for system stabilization. The parameters of the PSS in this example are selected based on the experience; however, the parameters should be optimized for the best additional damping, performance enhancement, and stability. A simplified PSS is considered where the transducers and high pass filters are assumed to be fast enough for their transfer functions to be neglected. In addition, the effect of the PSS output limiter is neglected. One stage lead-lag compensator is considered. The parameters of the PSS are shown in Fig. 4.61 and the system response in comparison with the system performance without PSS (yellow lines) is shown in Fig. 4.62(a) and the PSS output is shown in Fig. 4.62(b).

Fig. 4.61: The linearized third-order model of the SMIB system with AER and PSS 88

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Although the PSS parameters in this example are roughly estimated (i.e. they are not well tuned) and the PSS model is simplified, the results show that the PSS stabilized the system even if the AER gain is too high (caused instability without the PSS). Better performance can be obtained by PSS optimized tuning; however, this issue is out of the scope of this chapter.

Time(s) (a)

Time(s) (b) Fig. 4.62: Performance of the linearized third-order model of the SMIB system with AER and PSS for a sudden drop in the mechanical power by 10% (a) and PSS output (b) 89

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4.8 Frequency response and Speed Governors The generator response to load changes is simply described by the equation of motion which is called the swing equation as described by equations (4.17) or (4.18). With the speed expressed in p.u, the linearized form of the swing equation can be written as,

or

where M = 2H , is the damping power caused by the damping associated with the machine’s electrical and mechanical losses as well as the control system and operating conditions. The damping power coefficient ( D ) relates this power to the p.u change in the rotor speed ( ) as shown in eq. (4.126) or (4.127). The transfer function of the linearized swing equation can be represented by the block diagram of Fig. 4.63. The damping power term is shown by dotted lines as it can be omitted from the block diagram if the machine electromagnetic transients (for example, the dynamics of the field circuit and damper windings as well as other dynamics associated with the controls).

Fig. 4.63: Transfer function of the linearized swing equation The electrical changes in the generator load are instantaneously reflected as changes in the electrical power. Consequently, the machine speed (or 90

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frequency) deviates due to the mismatch between the electrical and mechanical power. The load also responds to the changes in the frequency and voltage changes. These changes depend on the load mix supplied by the generator (see chapter 7 in Kundur’s textbook 22 and Appendix 3). Two popular static load models are usually used for load response modeling; the exponential load model and the ZIP load model. The exponential load model can be represented by (4.128) while ZIP takes the form of (4.129).

128)

where P o and Q o are the load active and reactive power at the nominal voltage Vo (usually 1.0 p.u). Kpf and Kqf represents respectively the active and

reactive load sensitivity to frequency variations. The exponents  and  represent respectively the load active and reactive power sensitivity to voltage variations. The coefficients p, p and p represents respectively the p.u share of constant power, constant current, and constant impedance in the

active power load while the coefficients q, q and q represents the same quantities for the reactive power load. The relation between these coefficients is,

From frequency sensitivity point of view, equations (4.128) and (4.129) show that the load can be represented by two components; insensitive component and sensitive component. Given that in p.u, f =  and 22

Kundur, P. (1994). Power system stability and control (Vol. 7). N. J. Balu, & M. G. Lauby (Eds.). New York: McGraw-hill. 91

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considering the load sensitivity to frequency variations, the changes in the electrical power can be represented by

where

is the frequency independent changes, (

frequency dependent load changes,

) is the

represents the load sensitivity to

frequency variations and also called the load damping constant. The load damping constant is in the range of 1% to 2%; and its value represents the percentage change in the load for 1% change in the frequency. For focusing the analysis on the load frequency response, the damping constant D will be neglected in the following analysis. Based on the presented load reaction to frequency variations, the swing equation can then take the form of equation (132) and can be represented by Fig. 4.64.

Based on block reduction theory, the block diagram of Fig. 4.64 can be reduced to Fig. 4.65 which can be represented as shown in Fig. 4.66. where

Therefore, at steady state (i.e. t   or s  0),

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Fig. 4.64: The linearized swing equation considering load components

Fig. 4.65: Simplified block diagram of the linearized swing equation considering load components

Fig. 4.66: Equivalent block diagram of the linearized swing equation considering load components 4.8.1 Frequency response of a governor-less generator

With that the load change ( load change (

(i.e. no speed governor; see Fig. 4.67), eq. (4.134) shows . Therefore, the steady state speed is the speed at which ) is exactly compensated by the frequency-dependent ). In addition, Fig. 4.66 reduced to Fig. 4.68.

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Fig. 4.67: SMIB with the speed governor deactivated

Fig. 4.68: Equivalent block diagram of the linearized swing equation considering load components and without a speed governor The time domain (TD) solution of the system is can be then obtained as follows,

If

= constant, then

and (4.135) becomes,

The inverse Laplace can be easily determined by partial fraction and the use of the Laplace transform table23,24 i.e.

23 24

http://tutorial.math.lamar.edu/Classes/DE/InverseTransforms.aspx http://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx 94

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This function is shown in Fig. 4.69 and the steady state speed deviation is then,

Fig. 4.69: Speed deviation for a sudden change in the load in governorless system As shown, a sudden increase in the load causes steady state reduction in the speed while a sudden reduction in the load causes a steady state increase in the speed or frequency). The torque-speed characteristics for a governor-less generator can be determined based on the torque-power relation,

The linearized form of (139) takes the form, 95

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For

,

or in p.u Therefore,

The integration constant can be obtained from the steady state initial conditions i.e. . Hence, in p.u,

This relation shows that the torque-speed relation in a governor-less system is linear. In addition, this relation is correct over a limited at rated speed (see Fig. 4.70). As shown, an increase in the frequency results in a reduction in the mechanical torque as the input power is constant.

Fig. 4.70: Torque-speed relation of a governor-less generator near rated speed

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4.8.2 Type s of speed governors

From the operational point of view, a power system must be capable of exactly supplying the load demand. Since power systems do not contain bulk power storage and the load also is a stochastic variable, then the total power generated should be sufficient to provide a power balance at every instant. The dilemma is that the time constants associated with the mechanical power changes are significantly larger in comparison with most load variations; however, this problem is solved by the proper operation of dispatchable generators as explained in the following. Consider a system with n-dispatchable generators. (n-1) generators are set to operate in the PV-mode while only one generator is set to operate in the slack mode. The daily load forecast is used as a main input data to the system operators. The continuous load forecast curve is transformed into a discrete function with a sampling time in the order of 30 min or 60 min. Then the load level at each time step is used to allocate the power over the generators operating in the PV-mode. Usually security-based economic dispatch is used to determine the appropriate load sharing between the PV-generator. Up to that point, the power balance requirement cannot be achieved due to the mismatch between the forecasted load curve and the generation as well as other mismatch sources such as forecast errors, power losses, and random events (e.g. forced outage of generators or sudden change in the load). Therefore, the remaining generator is set to operate as a slack generator. The slack generator is not actually has a specific load sharing demand. Instead, this generator operates as a power balancer for securing the power balance requirement and consequently keeping the system operating at an acceptable frequency. The described operation of the system can only be possible with the use of speed governors. Two main types of speed governors are available according to the desired operational function of a generator. PV-generators are required to share the load. Therefore, they are equipped with speed governors with droop characteristics. A slack generator is required to balance the system. Therefore, it is equipped with an isochronous governor. The ideal steady state speed-power characteristics of these types of governors are shown in Fig. 4.71. It can be seen from the figure that a droop governor reacts to changes in the frequency by opposite changes in the mechanical power. These two types 97

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of governors will be considered in the following. In addition, Appendix 4 provides an overview of the fundamentals of speed governors.

Fig. 4.71: Steady state characteristics of isochronous and droop governors A. Isochronous (constant speed) governors

The isochronous governor adjusts the valve/gate position for adjusting the generator speed (or frequency) to the desired value (usually the nominal frequency). This is shown in Fig. 4.72 where the turbine shaft speed is assumed to be identical with the rotor generator’s speed i.e. a single mass model is adopted.

Fig. 4.72: Block diagram of a generator with an isochronous governor In the figure, Y represents the valve/gate position. In load frequency response analysis, the generator model is represented by Fig. 4.65 or 4.66. Therefore, Fig. 4.72 becomes, 98

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Fig. 4.73: Generator with isochronous governor model for load-frequency response There many designs of steam and hydro turbines. Three popular designs are considered in the following analysis. These designs are non-reheat steam turbines, reheat steam turbines, and hydro-turbines. Details about these turbines can be found in Kundur’s textbook. Table 4.3 summarizes the transfer functions of these turbines as well as the typical values of their parameters.

Turbine Non-reheat steam

Table 4.3: Transfer functions of turbines Transfer function Typical parameters TCH = 0.3 sec

Reheat steam25

F HP = 0.3 TCH = 0.3 sec TRH = 7.0 sec

Hydro*

TW = 1.0 sec

* Important notes about hydroelectric turbines are presented in Appendix 5.

25

The model of the reheat steam turbine is applicable to non-reheat steam turbines but the value of TRH should be set to zero. 99

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EXAMPLE LOAD-FREQUENCY RESPONSE OF GENERATOR WITH AN ISOCHRONOUS GOVERNOR

In this example, the load-frequency response of a generating unit equipped with an isochronous governor is considered. The non-reheat steam turbine model shown in Table 4.3 will be simulated. The model shown in Fig. 4.73 is implemented on the Simulink (Fig. 4.74) and typical parameters are used. For all models a disturbance of 10% step increase in the load is applied. The typical parameters of the turbine transfer functions are shown in Table 4.3 while the rest of parameters are: H = 5 sec, D pf = 1.0 while the impact of the gain KG is tested by setting its value to 0.035, 0.1, 3.0, and 5.0; the results are shown respectively in Fig. 4.75 to 4.78.

Fig. 4.74: A generator with isochronous governor and non-reheat steam turbine The results show that with a gain of 0.035, a critically damped response is achieved. Higher values of the gain result in oscillatory performance while at a gain of 5.0 the oscillations become unstable. Therefore, proper design of the gain for a desired load-frequency performance is essential for stable and satisfactory dynamic performance of the governor. The most important characteristics of the isochronous governors are that the steady state errors in the speed (or frequency) are zero. This is due to the reset action of the integrator, which results in continuous control actions of the governor till the zero error in the speed is achieved.

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Time (sec) Fig. 4.75: Performance of a generator with isochronous governor and nonreheat turbine; KG = 0.035

Time (sec) Fig. 4.76: Performance of a generator with isochronous governor and nonreheat turbine; KG = 0.1

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Time (sec) Fig. 4.77: Performance of a generator with isochronous governor and nonreheat turbine; KG = 1.0

Time (sec) Fig. 4.78: Performance of a generator with isochronous governor and nonreheat turbine; KG = 3.0

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Time (sec) Fig. 4.79: Performance of a generator with isochronous governor and nonreheat turbine; KG = 5.0

B. Speed droop governors

The function and performance of an isochronous governor are satisfactory when the generator is operating off the grid or for grid balancing function; however, in a grid-connected operation, more than one isochronous governor generator results in problems in the load sharing between the generators and the overall stability of the system. Unless they have exactly the same settings, if two generators having isochronous governors operate in parallel they will fight each other to set the system to the individual settings of each of them. Since the optimal economic secure operation of generators requires specific optimal settings to be allocated to each generating unit. Therefore, using more than one isochronous generator is practically impossible. Stable load sharing between parallel units requires that a change in a generator frequency is to be compensated by its power output. In other words, an increase in the frequency should be compensated by a reduction in the power output and vice versa. This action can be achieved by the use of governors having speed-droop or speed-

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regulation characteristics26 (Fig. 4.80). The ideal steady state characteristics of a speed-droop governor are shown in Fig. 4.81. As shown in Fig. 4.80, the isochronous governor is transformed into a droop governor by adding a feedback loop with a constant gain R around the terminals of the isochronous governor. The value of R defines the droop of the governor as shown in Fig. 4.81. The droop defines the steady state change in the frequency due to a steady state change in the output power. For example, a droop of 2% means that 2% deviation in the frequency causes 100% change in the output power or the valve/gate position. As shown in Fig. 4.80(c), the governor equivalent model is a proportional controller with a gain of a value (1/R).

Example Load-frequency response of generator with a speed-droop governor In this example, the considered disturbance and the parameters of system of Fig. 4.74 are used for simulating a generator with a droop governor. The block diagram of Fig. 4.80(b) is implemented on the Simulink as shown in Fig. 4.82. The droop of the generator is set to 10% (0.1 p.u). The chosen value of the droop is high for clearly showing the steady state errors. Two values for the governor gain are selected to show the critical damped and oscillatory damped responses. These values are 0.035 and 1.0; the results are shown in Fig. 4.83 and 4.84. The results show that with either value of the governor gain, there is a steady state error in the frequency, output power, and valve position. In addition, this steady state error equals to the droop. It can also be easily seen that the settling time and overshoots associated with the generator equipped with a droop governor are much lower in comparison with the generator equipped with an isochronous governor. The addition of the droop constant to the governor control loop reduces the effect of the integrator and improves the dynamic performance; however, the steady state performance is degraded due to the steady state error.

26

There are some distinguishing characteristics of hydro-turbines that mandate an additional block called ‘transient droop compensation ’ to be added to the governor. Appendix 3 describes these special characteristics of hydro-turbines and the transient droop compensator. 104

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(a)

(b)

(c) Fig. 4.80: Droop Governors; (a) Block diagram of a generator with a droop governor; (b) Generator with a droop governor model for loadfrequency response; (c) Reduced block diagram model of the governor ( )

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Fig. 4.81: Steady state characteristics of a speed-droop governor

Fig. 4.82: Simulink model of a generator with a droop governor and nonreheat steam turbine

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Time (sec) Fig. 4.83: Performance of a generator with a droop governor and nonreheat steam turbine; KG = 0.035

Time (sec) Fig. 4.84: Performance of a generator with a droop governor and nonreheat steam turbine; KG = 1.0 107

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4.8.3 Supplementary control action for unit output power control

For generator output control (or prime mover control), droop governors require a sustained change in system frequency27. Therefore, speed-droop governors alone cannot restore the power system frequency to the predisturbance level. This is can be explained based on Appendix 4 and Fig. 4.85. It is clear from Fig. 4.85 that an increase in the load, the frequency (or speed) drops and consequently the governor increase the steam (or hydro) flow (see Fig. A3.1) for compensating the speed drop. Due to the droop characteristics of the governor, the frequency will not be totally restored;

however, a remaining error f will sustain. The governor will not restore the frequency to the original value; however, this is can be achieved by increasing the reference power ( P ref) settings of the governor by the use of the speed-changer (shown in Fig. A3.1). Consequently, the droop characteristics move upward as shown in Fig. 4.85. Proper increase in the reference power (P r ef) will result in restoring the frequency to the original value while the generator is supplying the higher load. Therefore, the speed-changer provides a supplementary control to the governor for adjusting its the settings.

Fig. 4.85: Operation of a droop-governor and the supplementary control action

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WECC Control Work Group. WECC Tutorial on Speed Governors. Une 20002. WECC. 108

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The load sharing between generators can also be provided by the droop governors and the speed-changers. If two generating units operating in parallel for supplying a load and both of them equipped with isochronous governors with a very slight difference in their speed settings (design parameter), then one of the units will try to carry the entire load and the other will shed all of its output. Therefore, neither load sharing nor control of sharing is possible when units operating in parallel have isochronous governors. This is can also be explained by considering two isochronous units coupled together on the same load and the speed settings are not the same.. Since there cannot be two different speeds (or frequencies) on one system, one unit will have to decrease its actual speed and the other unit will have to increase its actual speed to an average speed between the two units. The governor on the unit that decreased speed will move to increase steam to try to correct for the decrease in speed, and the governor on the other unit that increased speed will move to decrease the steam to try to correct for the increase in speed. The result will be that the unit with the higher speed setting will continue to take all of the load until it reaches its power limit, and the other unit will shed all of its power and become motored (driven by the other unit). Therefore, the system will become unbalanced when isochronous units coupled together. Consequently, in the power generation system, only one generator can be equipped with an isochronous governor while the rest of generators are to be equipped with droop governors.

Fig. 4.86: Load sharing between units with droop governors

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With droop governors, the load sharing is possible as shown in Fig. 4.86. As shown, the governors will share the load increase (P ) at a lower common frequency value. This new frequency is defined by

while P= P 1+ P 2. The system frequency can be restored by adjusting the speed changers (or the reference power settings). Consequently, the speedpower relation is changed such that the frequency is restored.

(a)

(b)

(c) Fig. 4.87: Droop Governors; (a) Block diagram of a generator with a droop governor and load reference control; (b) Generator with a droop governor model and load reference control for load-frequency response; (c) Reduced block diagram model of the governor with the load reference control ( ) 110

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When a generator is paralleled with a large utility grid, it is important to consider that: 1. The utility will act as an isochronous generator. Therefore, the utility will set the speed/frequency for any units being paralleled to it. Consequently, a simple isochronous unit cannot be paralleled to the utility. 2. When an isochronous generator is connected to a utility bus, the utility will determine the speed/frequency of the generator. If the governor speed reference is less than the utility frequency, the utility power will flow to the generator and motor the unit. On the other hand, if the governor speed is even fractionally higher than the frequency of the utility, the governor will go to full load in an attempt to increase the interface bus frequency. Since the definition of a utility is a frequency which is too strong to influence, the generator will remain at full load. 3. Droop governors provide the solution to this problem. The droop causes the governor speed reference to decrease as load increases. This allows the governor to vary the load with the speed setting since the speed cannot change. With the speed-changer considered, the block diagram of Fig. 4.80 is modified to that shown in Fig. 4.87.

Example Load-frequency response of generator with a speed-droop governor and load reference control In this example, the example of Fig. 4.82 is repeated considering the impact of step changes in the load reference control. The disturbance in this case is 10% increase in the reference power setting. The Simulink model of the system is shown in Fig. 88 while the results are shown in Fig. 4.89 and 4.90 for KG of 0.035 and 0.1 respectively. Based on the previous discussion, it is expected that the droop governor is not capable of restoring the system frequency; as shown, there are steady state frequency errors associated with the change in the power input to the generator. Since the input power 111

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increased by 10%, the positive steady state frequency error is appeared with both values of KG. The solution to this issue is demonstrated in the next example.

Fig. 4.88: Simulink model of a generator with a droop governor and load reference control with non-reheat steam turbine

Fig. 4.89: Performance of a generator with a droop governor and load reference control with non-reheat steam turbine; KG = 0.035 112

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Fig. 4.90: Performance of a generator with a droop governor and load reference control with non-reheat steam turbine; KG = 1.0 Example Load-frequency response of generator with a speed-droop governor and load reference control – AGC for restoring system frequency In this example, the example of Fig. 4.82 is repeated considering the impact of the load reference control in restoring the system frequency by adjusting the frequency-load relation. For adjusting the reference power (Fig. 4.87) to restore the system frequency, an integrator and a stabilization gain are connected between the speed deviation and the reference power.

Therefore, P r ef = -Kperf/s i.e. a feedback PI loop is formed. The integrator is used for keeping P r ef changed till the frequency error  becomes zero. The value of Kperf is set to 0.035 for control loop stabilization (see Appendix 2). The added reset or integral control of the reference power makes the 113

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system operates in an AGC mode for restoring the system frequency. The simulink model of the system is shown in Fig. 4.91 while the results are shown in Fig. 4.92 and 4.93.

Fig. 4.91: Simulink model of a generator with a droop governor and load reference control with non-reheat steam turbine – AGC mode

Time (sec) Fig. 92: Performance of a generator with a droop governor and load reference control with non-reheat steam turbine – AGC mode; KG = 0.035 114

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Fig. 93: Performance of a generator with a droop governor and load reference control with non-reheat steam turbine – AGC mode; KG = 1.0 It is clear from the results that the proper load reference control reduces the speed error signals to zero and also enhances the overall dynamic performance of the system.

This chapter presents the fundamentals of power system dynamics. Simplified models and system topology (i.e. the SMIB) are considered for their easy handling and simulation; however, the presented conclusions are quite applicable to larger systems from scale point of view and also to more complex models. The following chapters include further analysis of larger power systems. In addition, the dynamics of recent energy production technologies will also be presented.

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Chapter 5 Measurement-based SMIB Equivalence of MultiMachine Systems AND Transient Stability 5.1 INTRODUCTION It can be seen from Fig. 4.1 that a system subjected to a disturbance reacts to that disturbance by frequency dynamics. The system frequency is then the average of the frequency of individual generator in the system. Just after a disturbance, the system frequency dynamics (Fig. 4.1) can be characterized by two quantities; the frequency gradient ((df/dt), and the frequency deviation. In the previous chapter, the linearized modeling and simulation of various components and controllers in the SMIB system (Fig. 4.7 or 4.27) is presented. The model of Fig. 4.55 represents a synchronous machine and an AER. In that model, the synchronous machine is represented by the thirdorder linearized model i.e. the field circuit electromagnetic dynamics are considered. The PSS model is represented by Fig. 4.60 while the droop governor model with load reference control is represented by Fig. 4.87. The AGC supplementary control for restoring the system frequency is represented in Fig. 4.91. The stated models represented the overall structure of the SMIB system with machine controls included. This overall model is shown in Fig. 5.1. In this section, this model is simulated considering various levels of modeling details. The main objective is to investigate the impact of the dynamic response of various components in the SMIB system on the frequency dynamics of the system. The changes in the frequency gradient and frequency deviations are the main focus. This is because the values of these quantities will be shown to have a major impact on the measurement-based equivalence that will be presented in this chapter. A secondary objective of the considered simulations includes a more clear understanding of power system dynamics and their governing issues. The parameters of the considered system and main variables of the considered system are given as follows such that the reactances, voltages, and power values are represented in p.u while the time and inertia constants are represented in seconds. 117

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Fig. 5.1: The linearized 3rd order model of the SMIB system with various controllers included. Generator data :

The steady state initial conditions and the K-constants as well as some other parameters are consequently determined using the presented model. Their values of the K-constants are,

AER data : The transfer function of the AER is represented by equation

(4.125) and its parameters are: TSR = 0, TA = 0.1 sec, KA = 30 PSS data : The general block diagram of the PSS is shown in Fig. 4.60. In this example, one lead-lag compensator is considered while the transfer functions of the frequency transducer and the high frequency filter are neglected. Therefore, the transfer function of the PSS is shown in Fig. 5.2 while its parameters are, 118

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Fig. 5.2: Simplified transfer function of the PSS T1 = 0.16, T2 = 0.03, TW = 1.0, KS = 10 Turbine data: Non-reheat steam turbine is considered. The transfer

function of the turbine is stated in Table 4.3 (TCH = 0.3 sec) Governor data : KG = 0.035, R = 10% AGC (for restoring system frequency) loop data : KAGC = 0.035 The considered case studies are summarized in Table 5.1. The transmission network connecting the generator terminals to the infinite bus is constructed of a step-up transformer and two parallel lines. The considered disturbance is a momentary outage of one of the line s; a situation that can be theoretically represented – as explained in the previous chapter - by a sudden change in the power angle. This change in the angle is assumed to be 5 degrees. The system model is simulated considering the cases listed in Table 5.1 for the stated disturbance. A single disturbance is used for comparative analysis purposes. The results are shown in Fig. 5.3. The period of the TD simulations are set to 5.0 sec for illustrating some important characteristics of the transient processes. Table 5.1: Case studies for the system of Fig. 5.1 Case Conditions Case 1: Classical model K2 to K6 = 0.0, field dynamics neglected, no governor, no AER, and no PSS. D = 0.0 Case 2: Field dynamics No governor, no AER, and no PSS i.e. unregulated only machine Case 3: case 2 + AER No governor, and no PSS Case 4: case 3 + PSS No governor Case 5: case 4 + No supplementary control governor Case 6: the overall All controllers structure 119

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The most important conclusion that can be easily depicted from the results is that the characteristics of the initial frequency dynamics are the same regardless of the modeling details as well as the available controllers. In all cases, the initial frequency gradient, deviation, and nadir are the same. This can be easily explained based on the fundamentals given in the previous chapter. At the beginning of the transient process, the system dynamics are dependent mainly on the system inertia. This is because the changes in the electrical power response are instantaneous (Fig. 5.3(b)); the electromagnetic transient causes by, for example, the machine’s field circuit dynamics and the other dynamics caused by various controllers are inherently suffering from time delays.

(a)

(b) Fig. 5.3: Frequency and electrical power transients for various cases shown in Table 5.1

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These time delays depend on the parameters of the machine and thecontrollers as well as the initial conditions, and the severity of the disturbance. The mechanical power transients associated with the prime mover system and their controllers are shown in the previous chapter to be characterized by very large response time in comparison with the electromagnetic transient. Therefore, their effects will not act on the considered TD duration. The basic characteristics of the SMIB system and impact of various controllers are demonstrated in the previous chapter and the cases shown in Fig. 5.3. Although the derived conclusions were based on a simple system and linearized models, they are also applicable to larger systems with more complex details as will be shown later. In this section, an efficient simple method for estimating the equivalent system inertia constant at a specific is presented, performed, and evaluated. The method is solely dependent on synchronized bus frequency and power measurements under disturbed conditions 28,29,30,31,32 (refer to chapter 1/volume 1 of this textbook for more details about synchrophasor measurements). Based on the swing equation (4.17) or (4.18), the inertia constant M can be determined as,

PMUs can measure the AC power frequency (i.e. 50 Hz or 60 Hz) voltage and current waveforms at typical rate of 48 samples per cycle i.e. 2400 28

El-Shimy, M. (2015). Stability-based minimization of load shedding in weakly interconnected systems for real-time applications. International Journal of Electrical Power & Energy Systems, 70, 99-107. 29 Tsai, S. J., Zhang, L., Phadke, A. G., Liu, Y., Ingram, M. R., Bell, S. C., ... & Tang, L. (2007). Frequency sensitivity and electromechanical propagation simulation study in large power systems. Circuits and Systems I: Regular Papers, IEEE Transactions on , 54(8), 1819-1828. 30 Wall, P., González-Longatt, F., & Terzija, V. (2010, August). Demonstration of an inertia constant estimation method through simulation. In Universities Power Engineering Conference (UPEC), 2010 45th International (pp. 1-6). IEEE. 31 Chassin, D. P., Huang, Z., Donnelly, M. K., Hassler, C., Ramirez, E., & Ray, C. (2005). Estimation of WECC system inertia using observed frequency transients. Power Systems, IEEE Transactions on , 20(2), 1190-1192. 32 Zhang, Y., Bank, J., Wan, Y. H., Muljadi, E., & Corbus, D. (2013). Synchrophasor Measurement-Based Wind Plant Inertia Estimation: Preprint (No. NREL/CP-550057471). National Renewable Energy Laboratory (NREL), Golden, CO.. 121

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samples per second for 50 Hz systems and 2880 samples per second for 60Hz systems. Therefore, the changes in the mechanical power in a sampling

duration t can be neglected. This is due to the relatively slow mechanical system dynamics in comparison with electromechanical and electromagnetic dynamics. In addition, the changes in the rotor speed during are small enough for the P D = D  to be neglected. Just after a disturbance, typical inertial and frequency responses of power systems (Fig. 5.3) are initially governed by the system inertia as demonstrated in the previous examples. Therefore, the measurements used for estimating the inertia of the system should be just after a disturbance (i.e. at to+ where to is the disturbance instant).

Consequently, for a small sampling time of measurements ( t), the inertia constant M can be estimated at the location of measurements using,

In (5.2), the power and frequency are in p.u while the inertia constant and the time are in seconds.

5.2 Measurement-based electromechanical equivalence An improved approach for fast and flexible electromechanical equivalency for online applications will be presented in this chapter. The approach is based on the availability of WAMS and PMU ( see volume 1/chapter 1) at limited selected locations in the power system. Coherency grouping of generators is also required for the construction of the equivalence; however, the grouping is not necessary to be performed online. As demonstrated in the previous volume (chapters 2 and 3), a coherent group of generators remains coherent regardless of the disturbance type, location, severity; however, large changes in the system topology may cause upset of the coherency grouping. This issue can be solved by identification of the coherency grouping from the performance of the generators under various disturbed conditions raised from natural changes in the system. With this coherency identification approach, the coherency grouping remains up-todate without the need of performing a mathematical simulation for that purpose. Consequently, the accuracy and speed of coherency grouping will 122

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be significantly improved. Again, the identification step is only needed in situations of large topological changes in the power system. For simplifying the presentation of the approach, the system shown in Fig. 5.4 is considered. The considered interconnected power system is assumed to be composed of two radially connected areas. This system is said to be weakly interconnected if the transmission system interconnecting the twoareas are of a capacity is less than the capacity of the smaller area by more than 15% or 20%. The configuration of Fig. 5.4 is popular in interconnected systems and systems integrated with large amounts of renewable sources where the location of these sources are remotely placed in comparison with the grid access.

Fig. 5.4: Radial connection of power areas or systems The buses at which the tie-link is connected are called the interface buses . These buses are required to be equipped with WAM devices for online measurement of the voltage phasors at the interface buses via PMUs, frequency, and tie-line active and reactive power values. Either area 1 or area 2 may be considered as the study area (SA) while the other one is considered as the external area (EA). In the following modeling and analysis, the equivalency of both areas will be considered. The objective is the minimization of the dynamic order of the overall interconnected systems for dynamic security studies that will be considered in later in this book. It is assumed that the generators of each area are coherent. Therefore, their electromechanical aggregation is feasible. In this chapter, only conventional sources are considered while various renewable resources will be considered later in this book. The equivalency approach is based on two stages both of them is based on online synchronized measurements of the frequency, voltage phasors and tieline power flow. The first stage as shown in Fig. 5.5, involves the estimation 123

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of the equivalent inertia at the interface buses while the second stage (Fig. 5.6) involves the estimation of the electrical parameters of the equivalent generator of each area.

Fig. 5.5: Inertia estimation (stage I)

Fig 5.6: Electrical parameters of the equivalent generator (stage II) Classical generator models are considered for the equivalent generator. This simplification can be justified based on the application of the equivalent simplified models. These models are to be used for fast prediction of the transient stability of power systems, for example, using the energy functions or the equal area criterion (EAC). Therefore, the 0+ transient performances of the generators are of the major importance for the identification of the stability conditions. As shown in the previous chapter as well as this chapter, during the initial stages of a transient process (Fig. 5.3), the impact of generator controls may be neglected and the classical model may be considered as an adequate model for that purpose. 5.2.1 Stage I: Inertia estimation at the interface buses

The equivalent inertia of the coherent group of generators comprising an area can be estimated at the area interface bus. This is based on the use of the measurement of active power and frequency measurements on that bus. These measurements are performed under disturbed conditions. The measurements in a very short window after a disturbance are used for 124

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estimating the equivalent inertia. Since the inertia estimation is based solely on measurements, the internal structure of the considered area is not required and the area can be considered as a black-box. Therefore, the method is general and can be applied to any mix of generating technologies as well as large areas. This kind of flexibility cannot be provided with traditional methods of equivalency which are mainly based on system data sets. For the purpose of the estimation of the inertia at a specific interface bus, equation (5.2) is to be used. 5.2.2 Stage II: Electrical parameters of the equivalent generator

The equivalent generator is assumed to be represented by the classical model (i.e. fixed transient emf ( ) behind a transient reactance ( ) as shown in Fig. 4.9. Based on the standard model of the steady-state and approximate transient characteristics of synchronous generators (chapter 4 and Appendix 1), the following equations can be easily proved for the equivalent generator shown in Fig. 5.7 with ,

Fig. 5.7: An equivalent generator representing an area connected to the rest of the system through an interface bus.

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The interface quantities are assumed to be known from measurements through WAMs. These quantities are the interface bus voltage phasor ( Vt and t) and power flow (P t and Q t) from the considered area to the rest of the system. The unknown quantities associated with the equivalent generator are

its emf (Eg), the phase angle of the emf (g), and its reactance (Xg). The nature quantities are based on the measurements’ conditions i.e. steady state or transient. Equations 5.3 to 5.5 are to be solved simultaneously for estimating the unknown quantities of the equivalent generator. Since these equations are nonlinear transcendental equations, then the iterative approach is a good choice for their simultaneous solution. For this purpose, with the index ( i) represents an iteration number, the following equations are derived,

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With

is selected, equations (5.6) to (5.8) respectively can be used to

perform cyclic iterations. The initial value of the equivalent generator emf can be set to reasonable values such as 1.0 p.u or Vt. the termination criterion of the iterations process is where is each of the estimated quantities and is the desired accuracy tolerance (e.g. 10-5). This method is simple and straight forward in comparison to previous method such as that presented by Chow et.al 33,34. With the presented method in this section, the weakly interconnected power system in Fig. 5.4 can be reduced to the two-machine form presented in Fig. 5.5 or 5.6.

5.3 SMIB equivalence of multi-machine systems The online dynamic security assessment and control is required for enabling power systems to withstand unexpected contingencies in a secure way i.e. without causing system instabilities. Due to the complexity and inherent nonlinearity of power systems, the online dynamic security assessment and control is considered one of the major challenges in recent power systems. Chapter 1 and 2 (in Volume 1) discuss in details the speed and accuracy requirements for handling the stability or the dynamic security of power systems online. One of the most efficient and fast enough approaches for online security studies is the online based equivalence such as that presented in the previous section. With the possibility of reducing a multi-machine system to a single machine equivalence (SME), not only the speed of dynamic security assessment is significantly enhanced, but the fast stability analysis methods such as the Equal Area Criterion (EAC) - can be implemented on the SME or 33

Chow, J. H. (2013). Power system coherency and model reduction . London: Springer. Chow, J. H., Chakrabortty, A., Vanfretti, L., & Arcak, M. (2008). Estimation of radial power system transfer path dynamic parameters using synchronized phasor data. Power Systems, IEEE Transactions on , 23(2), 564-571.

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the equivalent SMIB system. The SME approach is a hybrid direct-temporal method for fast handling of the dynamic security or transient stability. The transient stability of power systems includes two main aspects; analysis and control. The speed of solution, its accuracy, and the nature as well as the strength of the disturbances define the appropriate modeling and solution techniques. For example, the EEA provides a very fast way for assessing the transient stability of the power system; however, in its original form, the size of the system is limited to one or two machines. Methods such as the Extended Equal Area Criterion (EEAC) can be used for extending the application limits of the tradition EEA. This section provides an improved EEAC by presenting the online-based SME. Therefore, an interconnected system such as that shown in Fig. 5.4 can be dynamically transformed to an equivalent SMIB (see Fig. 5.8) through the presented online equivalency approach.

Fig. 5.8: SMIB equivalence High-order nonlinear modeling of individual components forming a power system results in highly accurate results. The solution, in this case, is in the time domain; however, neither the execution time nor the volume of the results to be analyzed is acceptable especially for online applications. Other methods such as linearized analysis methods are significantly faster in comparison with the solution of nonlinear models; however, linearized models are only applicable for simulating small disturbances that result in small deviations in the operating conditions. Due to its high economical pressure and constraints, the deregulation of power systems causes near security limit operation of power systems. In addition, the deregulation leads to larger interconnections for energy security, and large power transactions for economical profit maximization. Therefore, the time needed for system monitoring and security analysis becomes too small for the current computational technologies and method to handle power 128

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systems in real-time. The SMIB equivalence of power system is shown to be a direct approach to enhancing the real-time handling of the stability and security of power systems 35. The derivation of the SMIB equivalence of a power system will be presented and evaluated in the following. The SMIB equivalence such that shown in Fig. 5.8 is to be determined based on the two-machine equivalence (Fig. 5.5 or 5.6) which is estimated as an equivalence to the multi-machine radially connected areas shown in Fig. 5.4. For simplicity, all resistances are neglected in this analysis too. This is assumption is also usual in the EAC based analysis. Based on the equivalent circuit shown in Fig. 5.6, the power mismatches at the interface buses can be represented by

where P max is the maximum power across the interconnection; P g1 and P g2 are respectively the power production of the equivalent generators of area 1 and area 2 respectively; P d1 and P d2 are the equivalent active power demands on area 1 and area 2 respectively; 12 is the phase angle difference across the

interconnection which is equal to 1 - 2. Therefore, the swing equation at each interface bus and the equation of the relative motion can be represented by,

Therefore,

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In addition, the equation of motion of the SMIB equivalence shown in Fig. 5.8 can be represented by

where

The equivalent transient reactance of the SME is simply the equivalent reactance of the two machines. Therefore, the SMIB equivalence of the system is presented by eqs. (5.14) to (5.16). Although, this model abstracts the original system (Fig. 5.4) in the very simple form shown in Fig. 5.8, the SMIB equivalence preserves some important characteristics in the twomachine equivalence as well as the original system. For example, eq. (5.16) can be rewritten in the following perturbation form,

It is clear from eq. (5.17) that a drop in P g2 increases the equivalent power (P eq) by M1/(M1+M2) while a reduction in P d2 reduces P eq by M1/(M1+M2). Therefore, changes of P g2 and P d2 have opposite impact on the value of P eq but its sensitivity to both of them is the same. This is also applicable to the impact of the changes in P g1 and P d1 on P eq; however, a drop in P g1 reduces P eq and a drop in P d1 increases P eq. In this case the sensitivity factor is M2/(M1+M2). As a direct application of the SMIB equivalence in security analysis is the estimation of the maximum sudden drop in the power generation P g2. In addition, the presented model can be used to determine corrective actions such as the minimum amount of load shedding for ensuring the stability of the system if the drop in the generation is higher than the maximum limit. The validity of this analysis as well as the entire equivalency approach and the efficiency of the EAC in assessing the stability of the 130

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system will be demonstrated later in this book considering the nonlinear models of power systems for stability analysis.

5.4 Transient stability and the Equal Area Criterion (EAC) A system is in a dynamic state if the time derivative of any system quantity (y) is non zero . A dynamic system can be described mathematically differential or difference equations. According to the IEEE/CIGRE joint task force on stability terms and definitions 36, power system stability can be defined as, “The ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact ”. This definition indicates that there are three conditions of system stability; stable, unstable, and critically stable. Based on section 4.4.4, stable and unstable conditions may show either non-oscillatory or oscillatory responses. The shape of the responses depends mainly on the system damping. Stable conditions as associated with positive damping while unstable conditions are associated with negative damping. Critical stability conditions are a special situation where the damping is absent. Fig. 5.9 illustrates various stability conditions. The IEEE/CIGRE definition of stability is accompanied with two important comments that are also illustrated in chapter 4. These comments are as follows:

 Comment 1: “ It is not necessary that the system regain the same steady state operating equilibrium as prior to the disturbance. This would be the case when e.g. the disturbance has caused any power system component (line, generator, etc.) to trip. Voltages and power flows will not be the same after the disturbance in such a case. Most disturbances that are considered in stability analyses incur a change in system topology or structure”. See Fig. 5.10.

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Kundur, P., Paserba, J., Ajjarapu, V., Andersson, G., Bose, A., Canizares, C., ... & Vittal, V. (2004). Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. Power Systems, IEEE Transactions on , 19(3), 1387-1401. 131

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(b)

(a)

(c) Fig. 5.9: Stability conditions; (a) Stable transients; (b) Unstable transients; (c) Critically stable transients (sustained transients)

Fig. 5.10: Illustration of comment 1  Comment 2: “It is important that the final steady state operating equilibrium after the fault is steady state acceptable. Other wise protections or control actions could introduce new disturbances that might influence the stability of the system. Acceptable operating conditions must be clearly 132

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defined for the power system under study”. For example, the steady-state

acceptable operating points of an unregulated non-salient pole synchronous generator connected to an infinite bus (see Appendix 1) are shown in Fig. 5.11; as illustrated, an increase in the mechanical power must be associated with an increase of the power angle for stable operation. Therefore, the lefthand side of the power-angle curve with respect to the 90o power angle is stable while the other side is unstable.

Fig. 5.11: Illustration of acceptable steady-state operating points 5.4.1 Classification of power system stability

Power system stability can be classified according to many criteria such as: type of disturbances, and the physical instability mechanism and time. According to the type the disturbances , the power system stability can be classified as:

 Local stability that defines the behavior of the system around the particular equilibrium point being analyzed. This type of stability is associated with small disturbances . In this case, the nonlinear models of power systems which are usually in the form of Algebraic Differential Equations (ADEs) are linearized around a specific operating point. Linearized system analysis and model analysis can then be performed for this type of disturbances.  Finite stability, which defines the behavior of the system in a finite region of space. If the region of interest is extended to all possible state space, it is called global stability for the system. Finite and global stabilities are usually associated with large disturbances . Due to the nonlinear 133

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behaviors of power systems to large disturbances, stability assessment is required in the simulation of the dynamic evolution of the system or energy based methods such as the Equal Area Criterion (EAC). According the physical instability mechanism and time , the stability problem can be classified according to Fig. 5.12.

Fig. 5.12: Classification of the stability problem according to the physical instability mechanism and time Rotor angle stability refers to “ the ability of synchronous machines of a power system to remain in synchronism after disturbance”. The ideal system

for the study of the rotor angle stability is the SMIB system. This type of stability studies can be classified according to the strength of the disturbance to either large disturbance stability (i.e. transient stability) or small-signal stability. The main difference between the two studies is the way power systems modeled. Small-signal stability also refers to local stability while transient stability is referred to either finite or global stability. Frequency stability refers to the ability of a power system to maintain steady frequency following a severe system disturbance resulting in a significant imbalance between generation and load . Simply, this type of stability studies concerns the ability of the system to sustain power balance.

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Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition . It concerns the ability to maintain/restore

equilibrium between load demand and load supply from the power system. Instability that may result occurs in the form of a progressive fall or rise of voltages of some buses. Small disturbance or static voltage stability analysis usually represents power systems of algebraic equations while large disturbance voltage stability (also called transient voltage stability) models similar to those used in the transient stability analysis. This classification shows specific studies of major interests, according to the system engineers, while the system dynamics due to one disturbance may show all kinds of classified studies. 5.4.2 The Equal Area Criterion (EAC) and advancements

The transient stability of power system can be generally studied by solving the ADEs representing power systems by numerical methods. Consequently, the time domain (TD) response of the system can be determined. According to the TD results, the system stability can be assessed. Fig. 5.9 illustrates some TD responses for various stability conditions. Despite the certainty of the results of this approach is high, it suffers from many drawbacks such as,

 The quality of the input data to the models are usually suffering from uncertainties.  The pre-disturbance operating conditions and exact topology of the system are also suffering from uncertainties.

 For a given disturbance in a typical power system, the simulation results are usually huge to be easily analyzed.

 Real-time dynamic security requires too fast decisions and corrective actions to be made. A situation that makes the conventional TD approach impotent to fulfill these time limits. The EAC37 ,38 is a graphical method for fast assessment of the transient stability and possibly fast analysis of the feasibility of some corrective 37 38

Kimbark, E. W. (1995). Power system stability (Vol. 1). John Wiley & Sons. Venikov, V. A. (1977). Transient processes in electrical power systems. Mir Publishers. 135

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actions. The method is traditionally used in the analysis of the SMIB system or two generators. In addition, the method suffers from oversimplifications such as classical modeling of synchronous generators with damping neglected; however, this issue is not critical if the method is used for assessing the first stage of the transient process (also called first swing stability). In addition, methods for enhancing the EAC modeling are constructed as will be shown in this section. This scope of stability analysis is shown to be indicative about the transient stability, but not sufficient under some conditions. The possibility to produce high quality equivalent models of power systems and the possibility, for a given large disturbance, to model a large scale power system by only two machines, extend the applications of the EAC to cover real power system sizes. For example, if a group of considered generators connected to a large system through weak link(s) (i.e. long transmission line(s)), then the study system can be transformed to SMIB if the group of generators shows sufficient coherency level between them. In another situation, if two groups of generators tied through weak link(s), the overall system can be represented by two generators given that each group show sufficient coherency between its generators. These situations will be clearly presented in this book. It is worthy to me mentioned that the equivalences presented in this book are independent on a specific disturbance. The Extended Equal Area Criterion (EEAC) is another well-known method for the Single Machine Equivalence (SME) of power systems; however, the produced equivalence is dependent on a specific disturbance or a set of disturbances. The MSE using the EEAC method can be determined based on classical dataset approaches or using online measurements. The core idea of the EEAC is based on the fact that regardless of the system size, the mechanism of loss of synchronism originates from the irrevocable separation of the system machines into two groups: critical machines, which comprise the units responsible for the loss of synchronism; and noncritical generators, that is, the remaining machines. Accordingly, when analyzing an unstable situation, the SME procedure starts with the identification of the critical group of machines as soon as the system enters the postfault phase. Therefore, the EEAC equivalence approach gathers temporal information about the system dynamics. An equivalent two machine system is then constructed such each group of machines are s reduced to one machine. Then 136

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the two machine equivalent system is reduced to the SMIB system by the Kimbark’s method. Originally the EEAC is based on datasets and mathematical modeling of various components in the detailed system while its recent applications are based on on-line measurements. A. Fundamental EAC concepts

Consider the SMIB equivalence shown in Fig. 5.8. The following assumptions are used in the traditional EAC transient stability method, 1. The synchronous machine is represented by the classical model. 2. The approximate transient power-angle characteristics (chapter 4) are used to represent the system. 3. The input torque to the synchronous machine is constant. 4. The angular momentum of the synchronous machine is constant. Therefore, the inertia constant is assumed to be constant. 5. The resistances of lines and machines are neglected. 6. All damping torques are neglected. Suppose the system of Fig. 5.8 is operating in the steady state delivering a

power of P o at an angle of 0. This is operating point a in Fig. 5.13. The initial power P o equals to the mechanical power input P m. Due to malfunction of the line, its circuit breakers open reducing the real power transferred to zero. This is also equivalent to a temporary 3-phase to ground short-circuit at the generator terminals. Consequently, the generator power-angle relation

becomes zero instead of the initial relation P e = P max sin . This is illustrated in Fig. 5.13. Due to its inertia, the speed of the rotor cannot be changed

suddenly i.e.  = d/dt = 0 at the instant of the fault. Therefore, the transition from the initial power-angle curve to the new power-angle relation is done with constant power angle i.e. vertical transition a  b. At point b, due to the power imbalance between the electrical and mechanical power, the speed of the generator starts to change. Since, the mechanical power is higher than the electrical power, the generator accelerates i.e. gaining kinetic energy (KE); the accelerating power P a becomes equal to P m; which gives rise to the rate of change of stored kinetic energy in the rotor masses. Therefore, the rotor will accelerate under the constant influence of non-zero accelerating 137

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power (equals to P m) and hence the load angle will increase in a continuous manner. Consequently, the operating point moves from b toward c.

Fig. 5.13: EAC basic concepts Now suppose the circuit breaker re-closes at an angle cl (point c). The power will then revert back to the normal operating curve. Therefore, the operating point moves suddenly from c to e. At point e, the electrical power will be more than the mechanical power and the accelerating power will be negative i.e. it becomes decelerating power. This will cause the machine to decelerate; however, due to the inertia of the rotor masses, the load angle will still keep on increasing along the path e  f. The increase in this angle may eventually stop if the rotor losses the excess KE during the acceleration stage. This is shown at point f where the rotor is at higher KE with respect to normal KE at the synchronous speed but at that point the KE gained due to the inertia of the generator is lost and the rotor starts to decelerate to move back along the f  e  d  a path. When the rotor reaches point a, its speed equals to the synchronous speed but also due to its inertia, the rotor continue to decelerate and moves along the path a  0 while it is subjected to accelerating power as P m > P e. This action continue till the rotor starts to accelerate again. The rotor in this case oscillate around point a (which is the post-disturbance steady state operating point) and settles at that point again after a period of oscillations (Fig. 5.14).

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Fig. 5.14: Stable oscillations It can be easily depicted from Fig. 5.13 that the stable situation is limited

by a rotor angle cr after which the subsequent operating points are associated with acceleration (P m > P e) rather than deceleration. The possible deceleration region (dehd) defines the maximum acceleration energy gain which can be absorbed by the system without causing loss of synchronism i.e. without causing instability. Otherwise, the system will be unstable. The KE gain due to the considered disturbance will be shown to be proportional to the area abcda. Note that, d d  d    2 dt  dt   dt 2

2  d    2    dt 

(5.18)

Hence, multiplying both sides of the swing equation by d dt and rearranging we get H d d  d     Pmo  Pe   f o dt  dt  dt 2

(5.19)

Multiplying both sides of (5.19) by dt and then integrating between two

arbitrary angles 0 and  we get

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H d   f o  dt 

2



0

  Pmo  Pe d 



0

(5.20)

The left-hand side (LHS) of (5.20) is proportional to rotor KE energy while its right-hand side (RHS) is the area under the curve given the range of the angle. Therefore, (5.20) represents that rotor KE. Now suppose the generator is at rest at 0. Therefore, at  = o, d/dt = 0. Once a fault occurs,

the machine starts accelerating. Once the fault is cleared at  = cl, the machine starts to decelerate. Therefore, the acceleration area is given by, A1  Aacc 

  Pmo  0d   0

cl

0

(5.21)

In a similar way, we can define the area of possible deceleration. A2  Apossdec 

  Pe  Pmo d   0

cr cl

(5.22)

Let Adec (or A3) be a deceleration area that counteracts the acceleration

area Aacc i.e. Adec = Aacc. If the fault is cleared such that Adec  Apossdec, then the system is stable and the maximum angle of swing is m. In this case, the

difference A = Apossdec – Aacc is considered as the transient stability margin (Kt); as described by (5.23), it can be used as an indicator of the stability. In

addition, as A increases, the stability is stronger. The value of A can be used also to study the effect of various factors on the transient stability.

As shown in Fig. 5.14, the system is unstable if Adec  Apossdec. If Adec = Apossdec, then the system is critically stable and the fault clearing angle is the

critical clearing angle limit (CCL) which is given the symbol ccl. In 140

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addition, the maximum angle of swing in this case equals to cr . This is represented by (5.24) while the general stability criterion is stated by (5.25).

  Pmo  0d     Pe  Pmo d 

ccl 0

cr

ccl

(5.24)

where

For the considered case shown in Fig. 5.13, the critical clearing angle

(CCA) ccl can be determined using (5.24) and the result is,

For the considered pure sinusoidal power-angle relation, Therefore, and then (5.27) can be written as,

Since

.

, then equation (5.28) can be reduced to, )

Based on (5.29), the relation between critical clearing duration ( ) and the generator loading can be clearly viewed. This relation is illustrated in Fig. 5.15. It is clear that the increase in the generator loading results in a significant decrease in the available clearing duration in radians. The usual units of time is seconds. Therefore, for the considered disturbance, the critical 141

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clearing time can be determined analytically by solving the swing equation during the fault period. During that period, P e = 0. Therefore, d 2  f o  Pmo H dt 2

(5.30)

d f f   o Pmo dt  o Pmo t dt H H 0 t

  t

fo

0

H

Pmo t dt 

s 4H

(5.31)

Pmo t 2   0

(5.32)

with s  2 f o , the time corresponding to an operating angle  such that o 

  ccl takes the form

t 

4H    0  s Pmo

(5.33)

and, the critical clearing time (CCT) is then, t ccl 

4H    0  s Pmo ccl

(5.34)

Fig. 5.15: Impact of initial loading on the available critical clearing 142

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Example 1 Critical clearing and transient simulation of a temporary three phase fault at the generator terminals In this example, a nuclear power station connected to an infinite bus through pure reactive transmission link is considered. The rating of the station is 2220 MVA and the link voltage is 400 kV; these values are the basis of the per unit quantities. The generator is with non-salient pole and its steady state reactance equals to 1.82 p.u while its transient reactance equals to 0.2 p.u. The transmission link reactance is 0.3 p.u. The generator inertia constant M is 8.0 seconds. The generator delivers 0.9 p.u active power to the system. At this operating point, the measured terminal voltage magnitude equals to 1.0432 and its phase angle is 0.261905 rad (15 o) while the infinite bus voltage magnitude is 1.0 p.u and its phase angle is taken as the reference angle. It is required to determine the critical clearing time of a temporary threephase at the terminals of the generator. In addition, it is required to evaluate the accuracy the result through TD simulation of the non-linear swing equation. The generator damping as well as all the losses is neglected. The critical clearing angle can be determined using (5.28) i.e.

Therefore, the initial transient power angle of the generator is to be determined. Based on chapter 4 and appendix 1,

The value of Po is given as 0.9 p.u while the value of V is given as 1.0 p.u. The transient emf is unknown but can be determined using,

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All values in this equation are known except the value of the reactive power Qo; however, this value can be determined using the measured terminal voltage phasor. Considering that,

The value of the reactive power injected to the infinite bus is 0.024097 p.u. Consequently, the value of E’ is 1.107584 p.u and its phase angle can be obtained from,

and it is found to be 0.41839 rad (23.96233o) as explained in Appendix 1, this value is approximately the power angle of the generator. Now the CCA can be calculated; its value is 1.547603 rad (88.6344 o). The corresponding CCT can be found using (5.33) and its value is then 0.252 seconds. For evaluating the results, the system is simulated using the Power System Analysis Toolbox39 (PSAT) as shown in Fig. 5.15. The three-phase fault is applied at the generator terminals at t = 0.5 second and cleared after a specific clearing time. The clearing time is set to three different values in seconds; Case 1: 0.2 < CCT, Case 2: 0.252 = CCT, and Case 3: 0.253 > CCT. This is to illustrate the validity of the results and also to study the performance of the system as well as to illustrate the causes of stability and instability from the TD prospective. The TD simulation results are shown in Fig. 5.16 to 5.20.

39

Milano, F., Vanfretti, L., & Morataya, J. C. (2008). An open source power system virtual laboratory: The PSAT case and experience. Education, IEEE Transactions on , 51(1), 17-23. 144

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Fig. 5.15: PSAT model of the study system The ID of each shown quantity refers to the clearing time case. It is clear from the results that the estimated TTC is accurate. The results also reveal some dynamic characteristics of power systems without considering various damping providers and impact of controllers. Approaching the CCT results in an increase in the amplitude of the angle oscillations and reduces their frequency. This is also shown in the voltage, speed, and active power responses. Clearing the fault at a time larger than the CCT results in an unlimited increase in the power angle. The transient voltage performance shows a voltage collapse when the clearing time is larger than the CCT. Therefore, the TVS as well as the angle are lost. The active power transients show a tight relation with the EAC. This is illustrated by the acceleration and deceleration areas from the power-time prospective. It is clear that the instability (case 3) is caused by the increase of the acceleration areas in comparison with the deceleration areas. In addition, in this case, the generator could not decelerate after the second swing and it continuously accelerates. In the other cases (i.e. 1 and 2), the acceleration areas are less than the deceleration areas. It is also interesting to see the (-) and (Pe-) trajectories for the considered cases. The results are shown in Fig. 5.21 and 5.22. As shown in these figures, that the stable cases (i.e. cases 1 and 2) are characterized by sustained oscillations around the initial steady state operating point. As shown in Fig. 5.21 and 5.22, the instability is shown as an unlimited increase in the speed and the outage of the generator power.

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Fig. 5.16: Power angles for cases 1 and 2

Fig. 5.17: Power angle for case 3

Fig. 5.18: Rotor speeds for cases 1 and 2 146

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Fig. 5.19: Terminal voltage

Fig. 5.20: Active power

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(a)

(b)

(c)

Fig. 5.21: The - trajectories for various cases

It is worthy to be mentioned here that in real systems, there are many sources of damping of oscillations (see chapter 4). Therefore, the amplitudes of the oscillations are reduced with time till the final steady state operating point is reached. For the considered zero-input disturbance, the final and initial operating points are the same. This is illustrated in Fig. 5.23

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(b)

(a)

(c) Fig. 5.22: The P e- trajectories for various cases

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(a) -t

(b) -t

(c) Pe-t

(d) - trajectories

(e) P e- trajectories Fig. 5.23: Stable transient response with damping considered

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B. EAC-based analysis of 3-phase faults away from the generator terminals

This section concerns with the EAC-based stability analysis of arbitrary balanced 3-phase faults in the transmission link connecting the generator to the infinite bus. As shown in Fig. 5.24, a balanced 3-phase fault occurs at the one of the transmission lines connecting the generator to the infinite bus. The fault is assumed to be occurred at a specific distance from the generator terminals over one of the lines. The fault is then provides a reactance aXl between its location and the terminal bus; where a is a factor proportional to the length of the faulty section w.r.t the terminal bus. The reactance of the remaining section of the faulty line is bXl where a + b = 1.0.

Fig. 5.24: The SMIB system with 3-phase fault on a transmission line As shown in Fig. 5.25, three stages characterize the fault process each of which is characterized by a power-angle curve. These curves are:

 The pre-fault curve (I); describes the initial power-angle curve of the system. The operating angle in this case is .

 The during-fault curve (II); describes the power-angle characteristics during the fault i.e. before clearing the faulty line.

 The post-fault curve (III); describes the power-angle characteristics after clearing the fault. If the system is stable, then the final steady state operating angle is . Otherwise, the system will

not reach a post-fault steady-state equilibrium point due to its instability. The figures also show the stable and unstable conditions from the P- and P-t prospective.

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(b) Stable conditions; P-t domain

(a) Stable conditions; P- domain

(c) Unstable conditions; P- domain

(f) Unstable conditions; P-t domain Fig. 5.25: EEA-based analysis of general balanced three-phase faults

The three curves are modeled by the same equation which is the approximate transient power-angle characteristics; however, in each of the three conditions the transfer reactance is different. This is illustrated in Table 5.2.

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Table 5.2: Various power-angle curves Curve (i)

System topology

i = I, II, and III (I) Pre-fault

(II) During-fault

(III) Post-fault

The values of X12 are determined using standard circuit reduction methods

(i.e. series, parallel, -Y, and Y- transformations) as will be illustrated in the following example. It can be easily shown that the CCA (i.e. the angle at which Aacc = Apossdec) takes the following form. Due to the nonlinearities of the system with the considered fault, the CT and CCT cannot be expressed by analytical expressions; however, numerical methods such as the trapezoidal rule or the Euler method may be utilized for obtaining TD solutions.

It is important to note that for the same CCA, the CCT may be changed from a system to another system. This is because the CCA is based on the P.E/K.E balance while the CCT is a function of the system inertia. The increase in the system inertia leads to increase in CCT while the CCA remains the same for the same electrical quantities, topology, and disturbance. It is of critical importance to understand that the dynamic security is mainly guaranteed by the transient stability; however, the transient 153

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stability is not a guarantee of the static security (or adequacy); see Appendix 6 for more details.

Example 2 Critical clearing and transient simulation of a three phase fault in the transmission system Consider a simple system in which a generator is connected to an infinite bus through a double circuit transmission line as shown in Fig. 5.26. The per unit system reactances that are converted into a common base, are also shown in this figure. Let us assume that the infinite bus voltage is 10. The generator is delivering 1.0 per unit real power at a lagging power factor of 0.9839 to the infinite bus. While the generator is operating in steady state, a three-phase bolted short circuit occurs in the transmission line connecting buses 2 and 4  very near to bus 4. The fault is cleared by opening the circuit breakers at the two ends of this line. The objectives of this example are to illustrate the analysis of the considered faults via the EAC and to determine the impact of the generator inertia on the permissible (i.e. critical) clearing time via TD analysis.

Fig. 5.26 SMIB with fault in the transmission system Let the current flowing from the generator to the infinite bus be denoted by I. Then the power delivered to the infinite bus is

From the above equation, 154

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I  1.0164  cos 1  0.9839 

 1.0164  10.3

p.u

The total impedance during the time when both the lines are operational (Fig. 5.27 (a)), the impedance between the generator and the infinite bus is j(0.3+0.1+0.1) = j0.5 p.. Then the generator internal emf is E   1.0  j 0.5 1.0164  10.3  1.224.625

Therefore the machine internal voltage is E = 1.2 per unit its angle is 24.625 or 0.4298 rad. The equivalent circuit of the system for various stages of the disturbance (i.e. pre-fault, during-fault, and post-fault) are shown in Fig. 5.27. In the during-fault phase the equivalent reactance X12 is determined by the two method described in section A1.4 of Appendix 1.

Fig. 5.27: Various transfer reactances; (a) Pre-fault; (b) During-fault; (c) Post-fault Based on Fig. 5.27, 155

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Pre  fault:

PeI 

During  fault:

1.2 sin   2.4sin  0.5 PeII  0.857sin 

1.2 sin   2.0sin  0.6 The three power-angle curves are shown in Fig. 5.28. From this figure we find that Post  fault:

PeIII 

1  

cr    sin 1    2.618 rad 2

Fig. 5.28: Power-angles curves for the three phases of operation of the system The CCA can be directly calculated using (5.36) or by the calculations of the areas as follows. The accelerating area is given by, Aacc 

 1  0.857sin  d 

ccl

 ccl  0.4298  0.857cos ccl  0.857cos  0.4298 0.4298

 ccl  0.857cos ccl  1.2089 and the decelerating area is Aacc 

 1  0.857sin  d 

ccl

 ccl  0.4298  0.857cos ccl  0.857cos  0.4298 0.4298

 ccl  0.857cos ccl  1.2089 156

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and the decelerating area is Apossdec 

  2sin   1d 

2.618

ccl

 2cos  2.618   2cos ccl  2.618  ccl  ccl  2cos ccl  0.8859

Equating the two areas we get

 0.323    1.8573 rad  106.41 1.1429   As mentioned earlier, the critical clearing angle depends on the system configuration. The critical clearing time, however, is dependent on the value of the inertia constant H (or M = 2H ). Usually numerical methods are employed in finding out the clearing time and for solving the swing equation in the TD. Appendix 7 presents some numerical integration methods for this purpose. In this example, the CCT is estimated using the MALAB ODE solver and the results are shown in Fig. 5.29. It is clear from the results that the CCT is very sensitive to the inertia constant. It can be seen that as the value of H increases, the CCT is also increasing; however, the critical clearing angle remains the same. This is obvious; as H increases, the response of the system becomes more sluggish due to larger inertia (see chapter 4). Therefore, the rotor takes more time to accelerate. The results best fit with the following function,

ccl  cos 1 

In this example, k = 0.1968.

Fig. 5.29: Impact of the inertia constant on the CCT

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C. EAC-based analysis of unbalanced faults

With unbalanced faults, the stability analysis is to be performed in the same way described in the previous sections; however, the impact of the unbalanced fault on the during-fault transfer reactance must be considered. The determination of the during-fault transfer is based on the standard fault analysis methods40,41. This requires the construction of the positive, negative, and zero sequence networks of the system, then a connection between these networks according to the considered fault. Table 5.3 summarizes the connection between various sequence networks according to various unbalanced short-circuit faults. Table 5.4 summarizes the zerosequence equivalent circuits of major system components.

SLG

LLG

LL

Sequence network connection

Representation

Fault

Table 5.3: Sequence networks for unbalanced S.C faults

40

Grainger, J. J., & Stevenson, W. D. (1994). Power system analysis (Vol. 31). New York: McGraw-Hill. 41 Nagrath, I. J. (2003). Modern power system analysis. Tata McGraw-Hill. 158

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Table 5.4: Sequence impedances of system components Impedances Z012

Generators

Lines

Balanced loads

Representation

Transformers*

Three single-phase units and five-legged core three-phase units

; Zo 

Three-legged core three-phase units

* The magnetization current and core losses represented by the shunt branch are neglected (they represent only 1% of the full load current). The transformer is modeled with the equivalent series leakage impedance Zl.

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Example 3 EAC for unbalanced faults Consider a simple system with two identical transmission lines, as shown in Fig. 5.30. A single-line-to-ground (SLG) fault is assumed in this example and the fault is cleared by disconnected the three phases of the faulted line.

Fig. 5.30: The SMIB with an LG fault

Fig. 5.31: During-fault transfer reactance for an SLG fault The pre-fault and post-fault transfer reactances can be found as demonstrated in the previous example. During the fault, the connection between the positive, negative and zero sequence networks for the SLG fault is shown in Table 5.3 and the sequence impedances of various components are shown in Table 5.4. For the considered system, the during-fault transfer 160

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reactance can be obtained as shown in Fig. 5.31. The rest of the EAC-based analysis or TD simulations can be performed using the resulting power-angle diagrams. It can be easily depicted that the during fault transfer reactance with the SLG fault will be high in comparison with a three-phase at the same location. Consequently, the peak of the during-fault will be higher and the acceleration area will be smaller. As a result, the impact of the SLG fault on the transient stability is lower than the impact of three-phase fault. D. EAC-based treatment of the damping

In the fundamental modeling for the EAC, it is shown than the machines are represented only by the swing equation with the damping neglected as well as all losses. Therefore, the original EAC only considers the electromechanical transients while the electromagnetic transients as well as the damping power are neglected. It is shown in chapter 4 and in this chapter that the damping has a key role the stabilization of power systems. The damping is provided by many causes including the machine design, fieldcircuit dynamics, tuning controllers, and various losses. In chapter 4, a simple representation of the damping power is given as

where  = -

o. During rotor acceleration, the speed deviation  > 0. Consequently, the damping power is positive. During the deceleration an opposite situation is

faced i.e.  > 0 and P D < 0. It is shown in Fig. 4.11, that the damping coefficient is a function of the power angle; however, the average value of the damping coefficient is usually used in the simplified stability and EAC analysis. Therefore, the damping power is linearly related to the speed deviation. Consequently, in the P- domain, it is shown by Willems42 that the effect of damping can be represented with straight inclined lines with a constant slop,

This is also valid in multi-machine power systems 43. In all cases, the generators are assumed to initially running at the synchronous speed. The 42

Willems, J. (1969). Generalisation of the equal-area criterion for synchronous machines. Electrical Engineers, Proceedings of the Institution of, 116(8), 1431-1432. 43 Willems, J. L. (1971). Direct method for transient stability studies in power system analysis. Automatic Control, IEEE Transactions on , 16(4), 332-341. 161

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modification in the EAC for inclusion of the damping effect is shown in Fig. 5.32 considering a sudden increase in the mechanical power input.

Fig. 5.32: Modified EAC for inclusion of the effect of the damping – sudden increase in the mechanical power In the original EAC, the stability condition is that A1  A2 while with the damping power considered the acceleration and deceleration areas are modified as shown in the figure and the stability condition becomes:

where

In this case,

(5.41)

It can be easily depicted from Fig. 5.32 and eq. (5.40) and (5.41) that A3 < A4. Therefore, the resultant increase in the acceleration area is less than the resultant increase in the possible deceleration area. Consequently, as expected, the system is more stable with the damping considered. In other 162

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words, neglecting the damping underestimates the system stability; a situation which does not cause harmful decisions and also enhances the avoidance of the system uncertainties. As shown in Fig. 5.33, the same approach can be used for either balanced or unbalanced faults.

Fig. 5.33: Modified EAC for inclusion of the effect of the damping – short-circuit faults In this case,

(5.41)

It is important to note that the single-machine equivalence presented in section 5.3, can be analyzed as presented in this section; however, the equivalence with damping considered will be valid if . Otherwise, it is better to neglect the effect of the damping. G. Demonstration of the impact of AER and PSS on the EAC stability

In the previous chapter, it is shown that the increase in the gain of AERs can result in system instability. This is due to the added negative damping to the system; however, with low AER gain the stability may be preserved, but the steady-state voltage error may be high. It is also shown that PSSs can be designed and tuned for improving the system stability. PSSs can

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be used for restoring the system stability when the instability is caused by negative damping caused, for example, by high-speed high-gain AERs. In this section, analysis of the impact of various controllers on the stability and the areas of the EAC will be presented considering stable and unstable conditions. The system of Example 1 is considered, but the following modeling and operational conditions are implemented:

 The generator is represented by the third-order dynamic (non-linear) model; however, the same initial power produced.

 The transmission system is constructed of two-parallel lines as shown in Fig. 5.34.

Fig. 5.34: SMIB system for demonstration of the impact of controllers From control points of view, the following cases are considered,  Case 1: The generator is unregulated.

 Case 2: The generator is equipped with high-speed low-gain AER (KA = 20).

 Case 3: The generator is equipped with high-speed low-gain AER (KA = 20) and tuned PSS.

 Case 4: The generator is equipped with high-speed high-gain AER (KA = 200) that is selected to cause instability due to its added negative damping. The considered disturbance is a rapid switching on one of the transmission lines due to a temporary fault on the line. The rapid switching is provided by Automatic Circuit Reclosers (ACR). The response of the system for each of the stated control conditions is simulated using the PSAT and the results are compared. A small simulation time is considered to show some major differences in the responses for various cases. The results are shown as follows. Fig. 5.35 shows that cases 1, 2, and 3 are stable while case 4 is unstable. It can also be shown that the low gain AER reduces the damping of the system 164

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in comparison to the unregulated case; however, the PSS is capable of increasing the damping to a level higher than that of the unregulated system. The presence of high gain AER shows sever reduction in the system damping and the net damping became negative causing the system instability. Fig. 5.36 shows that the high gain AER causes out of step of the generator while the other generators of the cases remain synchronized. It is also clear that, as previously explained, the initial gradient of the speed (or frequency) is not affected by the presence of the controller even in the unstable case. This is because the initial frequency gradient is mainly dependent on the generator inertia. Fig. 5.37 confirms the previous results and also shows that the speed and frequency convergence to – at the end of the transient process – to the initial point if the system is stable while divergence is detected when the system is unstable.

Fig. 5.35: (t) relations

Fig. 5.36: (t) relations 165

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(a) Case 1

(c) Case 3

(b) Case 2

Fig. 5.37: - trajectories

(d) Case 4

Fig. 5.38: Field input voltage

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Fig. 5.39: The transient emf (i.e. the internal voltage) Fig. 5.38 shows that case 1 is characterized by constant field voltage as the machine is unregulated. The field voltage in the other cases is timedependent due to the presence of the AER. In these cases, oscillatory transient variations of the field voltage are detected. These variations are with decaying amplitudes in stable cases, while raising amplitudes are detected in the unstable case. Fig. 5.39 shows the variations of the transient emf. In the unregulated machine case, it is clear that the variations of the transient emf are small in comparison with the regulated machine. Therefore, the constant transient emf assumption is valid in the unregulated systems. With the presence of the AER, the stable and unstable oscillatory responses are clearly shown.

Fig. 5.40: Terminals voltages of the generators Fig. 5.40 shows the variations of the terminal voltages. It is shown that the high gain AER (case 4) could not surpass the other cases, although it 167

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provided the highest field forcing at the beginning of the transient process; however, due to its inherent negative damping, the magnitudes of the voltage oscillations could not be bounded.

Fig. 5.41: Electrical active power Fig. 5.41 shows the electrical power as a function of time. These functions can be used to study the EAC in the time-domain as illustrated in Fig. 5.25. It is shown that the high-gain AER causes small reduction in the acceleration area and a relatively large reduction in the deceleration area. This is due to the negative damping provided by this AER. This phenomena can also be explained through the P- curves shown in Fig. 5.41. These curves are obtained from the TD simulations using appropriate time and other settings. The impact of positive and negative damping on the transient stability can be explained using the Willems’s method presented in the previous section. This is shown in Fig. 5.42. In comparison with a system with damping neglected, the positive damping increases both the acceleration and possible deceleration areas; however, the gain in the possible deceleration area is much higher. With negative damping, both areas are reduced, but the reduction in the possible deceleration area is very high. Therefore, the instability in this situation is very likely.

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Case 1

Case 2

Case 3

Case 4 Fig. 5.41: P- relations

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Unregulated system

System with negative damping System with positive damping Fig. 5.42: Impact of the sign of the damping coefficient on the stability

5.4.3 Transient Stability Enhancement

The main approaches for enhancing the stability of power systems include, but not limited to: 1. Reduction of the disturbing influence . For example, use of high-speed fault-clearing and rapid switching may be used for for minimizing the fault severity and duration. Disconnection of faulted phases only is also a way for reducing the impact of faults. This can be realized by the use of single-phase breakers. 2. Increasing the restoring synchronizing forces . This approach includes the proper design of additional controllers. These controllers may be designed for controlling the dynamic performance of individual generator or for a 170

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group of generator or on system-wide scope. The main objective of generator control for stability enhancement should reduce the accelerating area or increase the possible decelerating area or both of them. Generator controllers include PSS, fast-valving, and braking resistance. Network controllers include FACTS devices, and POD. 3. Enhancing the system configuration and operating conditions for stability enhancement. For example, reducing the power flow over long weak links and reducing the dependency between power areas are possible ways for enhancing the stability. Increasing the transmission as well as the generation reserve can serve as a strong defense to instabilities. 4. Proper choice of intrinsic limits and setting of operating limits for secure operation of system components . Protective devices and systems play a major role in providing that security. In addition, proper testing, modeling, simulation, and analysis of system components and system configurations are the main way of determining the proper settings. 5. Optimally set corrective actions for various contingencies for minimizing the impact of disturbances on the system intactness considering the techno-economic feasibility of the actions. Load shedding is an example of corrective actions that provide a tool for keeping system stability when large amounts of power production drops; however, as will be shown later, the load shedding should be optimized. The optimization includes the time of implementation, and the amount of load curtailment. In this section, some methods for the transient stability enhancement will be summarized; however, some other methods such as AER and PSS are already presented, while some other techniques such as FACTS-POD will be presented in the next chapters. Consider the SMIB system shown in Fig. 5.43.

Fig. 5.43: SMIB system with a fault at one of the transmission lines

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A. Fast Valving . The impact of fast changes in the mechanical power for

stability enhancement on the EAC-based stability is demonstrated in Fig. 5.44(a). As shown, these changes cause significant reduction in the acceleration area and simultaneously significant increase in the possible deceleration area. Consequently, the system stability is highly enhanced. The control action required within the turbine to produce such rapid changes in the mechanical power is called fast valving 44 which is also called early valve actuation. The sequence of operations encountered and characterizing durations of a fast valving process are shown in Fig. 5.44(b).

(a)

(b) Fig. 5.44: Turbine fast valving; (a) EAC analysis; (b) Sequence of valve operations 44

Edwards, L., Gregory, J. D., Osborn, D. L., Doudna, J. H., Pasternack, B. M., & Thompson, W. G. (1986). Turbine fast valving to aid system stability: benefits and other considerations. Power Systems, IEEE Transactions on , 1(1), 143-153. 172

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Fast valving requires a very fast response from the turbine. To be useful, the reduction in the mechanical power should be as soon as the fault occurred. For example, after 30% of the first swing period, which is in the order of a few tenths of a second following a disturbance. The mechanical power restoration to the initial pre-fault value should also be fast provided that the reduction causes system stability. For example, the mechanical power should be restored in less than a second. Due to their slow capability of dynamic changes in the mechanical power (see Appendix 5), hydro-turbines cannot usually provide fast valving. Steam turbines, on the other side, have faster response in comparison with hydroturbines and fast valving is possible. As shown in Fig. 5.44, the change in the mechanical power is a function of the rotor speed; however, additional supplementary control action should be added to the standard closed-loop STG governors. These controllers are needed for compensating the slow changes in the speed at the beginning of a transient process due to the STG inertia. Therefore, this supplementary control should be capable of identifying fault conditions and perform fast-valving within a reasonable time. B. Generator tripping 45 can be considered as an alternative to the fast

valving. Selective tripping of generators can be used in improving the stability of power systems during severe disturbances. By doing so, the power transfer to a fault is prevented through rapid trip of selected generators. Unless special facilities are provided, the generating-unit follows the normal sequences of shutdown and start-up. The look-up table method presents a fast way to deciding the generators to be tripped for dynamic security of power systems. In addition, the look-up table only needs to be updated in the cases of significant topological changes in the system. C. Rapid switching and high-speed fault clearing. Since the kinetic

energy gained by the generators during a fault is directly proportional to the duration of the fault, then reducing the fault clearing time is a way for 45

Karady, G. G. (2002). Improving transient stability using generator tripping based on tracking rotor-angle. In Power Engineering Society Winter Meeting, 2002. IEEE (Vol. 2, pp. 1113-1118). IEEE. 173

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enhancing the stability. The positive impact of rapid switching on the system stability is demonstrated in the previous section. Fig. 5.45 shows the impact of reducing the fault clearing time on EAC areas. As shown, the reduction the clearing angle results in a reduction in the acceleration area and an increase in the possible deceleration area. Consequently, the stability is enhanced.

Fig. 5.45: Impact of fault clearing reduction D. Reclosers . The use of reclosers also secures the system stability for the

momentary faults conditions. Two-cycle circuit-breakers, together with high speed relays and communication, are now widely used in locations where rapid fault clearing is important. Recently, a one-cycle circuit breaker by the Bonneville Power Administration (BPA) combined with a rapid response over-current type sensor is constructed for special applications that need very fast fault clearing. Recently, ultra-high-speed relaying systems for EHV lines are available. The operation of these relays is based on traveling wave detection. E. Dynamic braking by braking resistors is also shown to be an effective

method for stability enhancement using variable structure control 46. In this 46

Wang, Y., Mohler, R. R., Spee, R., & Mittelstadt, W. (1992). Variable-structure FACTS controllers for power system transient stability. Power Systems, IEEE Transactions on , 7(1), 307-313. 174

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method, an artificial electrical unity power-factor load is added at the generator terminals during a transient disturbance. The main objective of the dynamic braking is to increase the generator’s electrical power output. Consequently, the rotor acceleration is reduced. One form of dynamic braking involves temporary switching in shunt resistors for about 0.5 seconds following a fault for enhancing the stability nearby generator(s). Consider the SMIB system of Fig. 5.46. Without the braking resistor, the power-angle characteristics of the system take the form,

with the resistance included, the power-angle characteristics can be determined using the YBUS-based approach presented in volume 1/chapter 3 or the network reduction methods presented in Appendix 1; however, as the system size is small, the power-angle relations will be derived based on the basic circuit analysis theories. The generator current phasor takes the form,

The terminal voltage as a function of the infinite bus voltage takes the form,

The line current as a function of the generator and resistor currents takes the form,

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(a)

(b) Fig. 5.46: Braking resistors for stability enhancement; (a) System topology; (b) Equivalent circuit Eliminating Il from (5.44) using (5.45) and separating Vt from the resulting equation, the result is,

Eliminating Vt from (5.43) using (5.46) and separating Ig, the result is,

The apparent power can be then determined as,

Therefore, the active power production with the presence of the braking resistor takes the form, 176

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It can be easily shown that (5.49) is reduced to (5.42) when Rbr =  i.e. disconnected. As an example , the parameters and operating conditions of the system of Fig. 5.46 are P o = 0.6 p.u, Q o = 0.45 p.u, E’ = 1.3172 p.u, V = 1.0 p.u, x’d = 0.3 p.u, xe = 0.4 p.u, and M = 20 sec. Consequently, Vt = 1.3804 p.u, and P max = 1.8817 p.u. Using (5.49), the changes in the power-angle relation as affected by Rbr are shown in Fig 5.47. It is clear from the figure that reducing the braking resistance shifts-up the P- relation. The braking power consumption for a given resistance can be determined at the initial voltage applied to the resistance i.e.

The initial braking power is reduced with the increase in the barking resistor; see Fig. 5.47. As the braking resistors increased, the braking power decreased and consequently the impact of the barking resistor on the P- relation is reduced. Equation (5.49) can be simplified by considering that . The resulting equation is then,

where braking resistor; and

is the maximum power before inserting the

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(a)

(b)

(c) Fig. 5.47: Braking resistance; (a) Effect on the P- relations; (b) Braking power; (c) Increase in the maximum power with increase in the braking power

Considering the trigonometric relation,

with

and

; if the angle is within the first

quadrant than determined according to Fig 5.48.

. Otherwise, the angle is

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Fig. 5.48: Determination of the phase-shift in eq. (5.53) Based on (5.53), (5.51) can be written as, where

.In comparison with the detailed power-angle model (5.49), the approximate model (5.51) and its alternative form (5.54) nearly give the same power-angle characteristics. This is demonstrated using the sample system data of this section with Rbr = 2.0 p.u as shown in Fig. 5.49.

Fig. 5.49: P- characteristics with the presence of a braking resistor – various models and approximations 179

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The value of c3 is very close to unity as

.Furthermore, if

the phase-shift (5.55) is also neglected, the resulting equation takes the form,

Therefore, the P- relation with the presence of the braking resistor is reduced to the P- relation without the braking resistor added but shifted up by the factor KP . The stated assumptions show acceptable accuracy as shown in Fig. 5.49. It important to note that the braking power is of a constant value; however, it is not constant as the resistor is connected to a variable bus voltage. With the classical model considered (i.e. E’ assumed to be constant), the variations of the braking power with the power angle can be found using

where

The relation P br - is shown in Fig. 5.49 which reveals that the braking power is decreased with the increase of the power angle. The maximum value

of the braking power occurs at  = 0. This characteristics is beneficial in the transient stability enhancement as the acceleration area (which is usually at low power angles < 90 o) will be highly reduced due to the braking power. The impact of the braking power on the possible deceleration area is positive; however, its impact is minor as the values of the braking power at high angles are low. The impact of sudden addition of a braking load on the power-angle characteristics as well as the associated dynamics is illustrated in Fig. 5.50.

The figures show the insertion of a braking resistor at  = br . The model of (5.57) is used for this demonstration.

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Fig. 5.50: Dynamics of the insertion of a braking resistor At the insertion of the braking resistor (point a), the generator will decelerate and goes through oscillatory response till the operating point settles at point (c). The angle at point c is less than the initial angle due to the braking power. Braking resistors are connected to the system just after a fault that causes system acceleration. Usually, braking resistors are disconnected from the system after a short duration of a transient process. As an illustrative example, consider the SMIB system of Fig, 5.46. The system is subjected to a forced outage of one of the lines due a fault. The resistor in this case operated only during the acceleration and deceleration periods. As shown in Fig. 5.51(a), the braking resistor causes a decrease in the acceleration area and increase in the possible deceleration area. Therefore, it can enhance the system stability or even stabilize the system. During the second and possibly during higher order of swings, the braking resistor may also be connected to securing the transient stability. From the techno-economic feasibility point of view, braking resistors are only inserted during the acceleration periods i.e. during the increase of the system frequency (as shown in Fig. 5.51(b)). The braking control in this case, inserts the resistor when the frequency of the system increases above the normal. Therefore, the resistors are provided by bang-bang control (i.e. A control mode in which an element, such as a braking resistor is switched on and off) at appropriate moments. By doing so, the cost of the resistor and its heat dissipation capability are reduced. In addition, as shown in Fig. 5.49, it will be well utilized as its impact during the deceleration periods if effectively minor. 181

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The resistors can be switched by the use of either mechanical circuit breaker or electronic switches (Fig. 5.52). An example, of an electronic switch is the back-to-back thyristors configuration where the thyristors in this case are controlled by the integral-cycle control (ICC) algorithm. Unlike the phase angle control (PAC), the ICC allows continuous full connection and disconnection of the resistor. The PAC is suitable for devices such as thyristor controlled reactors (TCR) where partial amounts of the controlled inductance are desired, as a typical example, for voltage control function.

(a)

(b) Fig. 5.51: Impact of braking resistors on the transient stability; (a) The braking resistor operates during the acceleration and deceleration; (b) The braking resistor only operates during the acceleration.

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Fig. 5.52: Thyristor-switched braking resistor (TSBR) A special situation where the braking resistor becomes ineffective in enhancing the stability is when the fault occurs at the point of connection of the resistor with the system. In this case, switching on the resistor will not cause an increase in the electrical power during the fault. F. Fault Current Limiters (FCLs) are devices that are capable of limiting

fault currents47,48,49. This is achieved by providing impedance against the excessive increase of abnormal currents; for example, during faults. Resistive FCL can be considered as a series switched barking resistor. Therefore, resistive FCLs help to reduce the impact of faults on the transient stability of generators by increasing the electrical power consumption during fault conditions. Therefore, reduce the acceleration energy and enhance the transient stability. For this purpose, the FCL must be resistive or containing large resistance for consuming the active power during the during fault period which causes the stabilization effect. FCLs are also sometimes called Fault Current Controllers (FCC); however, they not a real controller that acts all the time. FCLs only acts when a fault occurs i.e. in the ‘during-fault’ period. In comparison with series current limiting devices such as reactors or high-impedance transformers, FCLs do not add significant impedance to the system during normal 47

Japan, I. E. E. (1999). State-of-the art and trends in fault current limiting technologies. IEEJ Report, (709). 48 Youjie, Z. X. X. L. M., & Xiaoning, X. (2004). The Development and Application of Fault Current Limiting Device [J]. Transactions of China Electrotechnical Society, 11, 000. 49 Noe, M., & Steurer, M. (2007). High-temperature superconductor fault current limiters: concepts, applications, and development status. Superconductor Science and Technology, 20(3), R15. 183

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conditions; the functional concept of FCLs is very similar to the surge arresters (SA). Both of them engage only during abnormal conditions while during normal operation, they have not a noticeable impact on the system performance. Fast valving and braking resistors are also of this nature. In comparison to FCLs, the reactors as fault limiting components cannot provide direct stability enhancing by reducing the acceleration area as they are not active power consumers. In addition, series reactors may lead to voltage stability problems especially during transients. From economical point of view, the cost of FCLs is highly compensated by the use of protective devices with lower ratings in comparison with the absence of FCLs. In addition, FCLs enhances the system stability and reliability. They also can be placed at any desired location in a power network at various voltage levels and for various system components. The main functional requirements of FCLs include:  Fully automated function.

 Invisible during the normal operation of power systems,

 Insert a large controllable and manageable impedance when a fault occurs,

 Operate as soon as a fault occurs or at least within the first cycle of the fault current,

 The recovery time should be as short as possible i.e. the FCL should return to its normal operating conditions within a short interval in the post-fault time,

 FCLs should be capable to re-engage in a short-duration for protecting the system from successive faults, and

 The coordination of the system protection should be shifted by the presence of FCLs.

From the technological point of view, there are two main categories of FCLs; superconducting and solid state FLCs. Technological details of FCLs are out of the scope of this book; however, the Yadav’s paper 50 entitled ‘Review on Fault Current Limiters’ is a good source about that issue. The 50

S. Yadav, G. K. Choudhary, and R. K. Mandal (2014).Review on Fault Current Limiters.IJERT 3(4), 1595 - 1603. 184

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common components of FCLs are (1) Fault detector ; (2) FLC controller ; (3) Limiting impedance . A typical characteristic of FCLs is shown Fig. 5.53. Typical values for the detection and starting times are 2 ms and 1 ms respectively. As shown in the figure, the impact of the fault detection and starting durations is a partial stress on the system by the fault current; however, the magnitude and duration of this stress are small.

(a)

(b)

(c) Fig. 5.53: Typical characteristics of FCLs; (a) Connection; (b) Characteristics; (c) Impact on fault currents The analysis of the inclusion of the FCLs and their impact on the transient stability can be performed in the same manner as the braking resistance; however, in this case the resistance is added in series with the faulty line during the during-fault period. Consider the system of Fig. 5.54 185

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which will be used to demonstrate the derivation of the during-fault characteristics. The system is subjected to a fault at a random point within one of the lines. The fault impedance is ZF i.e. a general unbalanced fault; the value of ZF is zero for balanced three-phase faults. Both lines are equipped with FCLs; however, the one on the faulty line is only considered in the analysis as the other FCL is not subjected to fault currents and consequently its impedance is nearly zero as shown in Fig. 5.53. The equivalent circuit of the system can be reduced to a simple form by successive Y- transformations are shown in Fig. 5.54. The resulting shunt impedances at the infinite bus are omitted as they have not an effect on the generator’s power flow.

(b)

(a)

(c)

(d)

Fig. 5.54: SMIB with fault current limiter and its equivalent circuit reductions The generator current and the during-fault power can be then represented by,

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With the absence of the FCL and for a three-phase fault (i.e. ZF = 0), the . Therefore, the difference between

during-fault power is then

the two characteristics is the power consumed by the FCL. The main impact of the FCLs on the stability is a reduction in the acceleration area. Therefore, the stability is enhanced by the presence of FCLs. As an example, the system of Fig. 5.55 is subjected to a momentary three-

phase fault at the generator’s terminals. In this special case, ZF = 0, Za = , and Zb = ZFCL + jX’d. Therefore, equation (5.62) is reduced to,

(a)

(b) Fig. 5.55: Impact of FCL on the transient stability Since in this case the during-fault power in a system without an FCL is zero, then (5.63) represents the FCL power consumption. This power 187

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consumption is also a constant in this special case; however, it is generally a function of the power angle. Due to the detection and starting periods, the action of the FCL is delayed by a small amount of time as shown Fig. 5.53. Therefore, initially, the during-fault power is zero; however, after this period, the during-fault power is moved upward by the amount as a result of the FCL. It should be noted that the equivalent power angle of the detection and starting

times (FCL) is dependent on the system inertia. Therefore, their effect on the power-angle curves can be neglected if the system inertia is high. As shown in Fig. 5.55, the acceleration area is significantly reduced. At the fault clearing instant, the current goes back to low values. Therefore, the FCL impedance is reduced to its normal near-zero value while the system characteristic returns to the initial one. Consequently, the possible deceleration area is not affected by the presence of the FCL while the overall stability is enhanced. Unlike the braking resistor, and fast valving, the FCL will not interact with the system speed variations. This is because FCLs respond to a current increase rather than frequency changes. Braking resistors and fast viewing, on the other hand, operates in a bang-bang mode i.e. they will interact during the entire transient process. Consequently, FCLs enhances only the first swing stability while braking resistors and fast valving can possibly enhance all the swings. The distinguishing feature of FCLs is that they can be placed at any location in a power network for fault current limiting. Therefore, FCLs may be found in the high-voltage bulk power grid as well as the low-voltage light-current distribution systems. In the next chapters, detailed analysis of the impact of FACTS devices on the stability of power systems will be presented. In addition, stabilization of power system through oscillation damping controllers integrated with FACTS devices will be designed. In this section, highlights about the impact of series and shunt regulated compensators on the system stability will be presented. G. Reduction of the transmission system reactance . The transmission

reactance has a significant impact on the maximum of a power curve. Therefore, reduction of the transmission reactance can be used for increasing 188

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the possible deceleration area and consequently enhancing the transient stability. The reduction can be achieved by either improving the design of transmission components or adding compensating components. Reducing leakage reactances of transformers and reducing the reactances of transmission lines are among the component design-based enhancement methods. Installing series capacitor compensators can be an effective way for the reduction of the transmission reactances. Generally, the main drawback of reducing the transmission reactances is the resulting increase of the fault level of the system. In addition, the costs associated for over-sizing power transformers for reducing their leakage reactances is not usually economical while the series capacitive compensations have a better relative economic feasibility. The main critical issues associated with series capacitive compensations are the possibility of resonance and sub-synchronous resonance51 (SSR). These phenomena results in limiting the amount of a series compensation that can be added to a transmission system. The switching of compensating devices should be carefully adjusted or optimized for enhancing the system stability52. H. Regulated shunt compensators . Regulated shunt compensators (such

as SVCs and STATCOMs) are devices that are mainly designed for controlling bus voltage magnitude; however, they are also results in an increase in the maximum transmission power transfer. Therefore, they conceptually have a positive impact on the transient stability; however, they may reduce the stability of a system if their controllers are not well tuned. I. Single-pole switching or independent-pole braking operations .

Independent-pole operation refers to the use of separate operating

51

Kumar, R., Harada, A., Merkle, M., & Miri, A. M. (2003, June). Investigation of the influence of series compensation in AC transmission systems on bus connected parallel generating units with respect to subsynchronous resonance (SSR). In Power Tech Conference Proceedings, 2003 IEEE Bologna (Vol. 3, pp. 6-pp). IEEE. 52

Rahim, A. H. M. A., & Al-Sammak, A. I. J. (1991, January). Optimal switching of dynamic braking resistor, reactor or capacitor for transient stability of power systems. In Generation, Transmission and Distribution, IEE Proceedings C (Vol. 138, No. 1, pp. 8993). IET. 189

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mechanisms for each phase of the circuit breaker 53,54. In such breakers, the three poles are mechanically separate and have separate tripping mechanisms. The outcomes of such mechanisms are the isolation of only faulty lines as well as reduction of the impact of failures in three-phase circuit breakers. Knowing that the majority of three-phase faults started as single line to ground faults, the value of singe-pole switching in reducing, from stability and economically, the impact of fault is high. J. Load shedding . Shedding of pre-selected loads is a possible way for

balancing the load and generation. Although, this action has a large economical impact due to the intentional disconnection of customers, sometimes it is the possible feasible way for keeping the system stability and synchronization55. Load shedding can also be used for balancing load and generation is a forced isolated area which may be developed during the extremes state of a power system (see chapter 1). There are three main categories of load shedding schemes 56. 1. Reactive load shedding is a fully automated scheme that depends on the value of the absolute frequency and look-up tables by which the amount of load to be shed are determined. 2. Proactive load shedding schemes are also fully automated shedding systems, but more sophisticated in comparison with the reactive schemes. In this scheme, not only the absolute frequency is used as an indicator for load shedding requirements, but also the rate of change of frequency or even the acceleration of frequency. These inputs are used by these schemes for initiating the load shedding process while the amount of load to be shed are determined via look-up tables.

53

Balu, N. (1980). Fast turbine valving and independent pole tripping breaker applications for plant stability. IEEE Transactions on Power Apparatus and Systems , 4(PAS-99), 1330-1342. 54 Taylor, C. W., Mittelstadt, W. A., Lee, T. N., Hardy, J. E., Glavitsch, H., Stranne, G., & Hurley, J. D. (1986). Single-pole switching for stability and reliability. IEEE transactions on power systems, 2(1), 25-36. 55

El-Shimy, M. (2015). Stability-based minimization of load shedding in weakly interconnected systems for real-time applications. International Journal of Electrical Power & Energy Systems, 70, 99-107. 56 Grewal, G. S., Konowalec, J. W., & Hakim, M. (1998). Optimization of a load shedding scheme. Industry Applications Magazine, IEEE , 4(4), 25-30. 190

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3. Manual load shedding is the slowest scheme in terms of reaction time in comparison with other schemes. The reaction time in this case is mainly dependent on the operator response time. This scheme has application where the operator has been informed of an impending or the likely loss of generation. Consequently, the operators can counteract the potential imbalance by shedding load. K. Islanding . Forced islanding is typically follows an unsuccessful

attempt to restore system frequency by shedding loads. Selective islanding can be performed manually for special severe situations (see chapter 1). Intentional Islanding 57, or controlled system separation , may be used to prevent a severe disturbance in one part of an interconnected system from propagating and causing a widespread system cascaded outages and breakup. Once an impending instability condition is detected system separation is initiated in a controlled manner by opening specific tie-lines. Considerations when assessing the possibility of implementing an islanding scheme include planning studies, stability studies, and fault-level studies. Planning studies ensure that there is no power imbalance within the proposed island and that steady-state stability is possible. Stability studies ensure that the island will maintain transient and voltage stability. Fault-level studies ensure that the new fault-levels in the islanded system are within acceptable limits. L. Parallel HVDC and HVAC transmission . Unlike, the standard HVAC

transmission systems, HVDC systems provide control of the power flow over the HVDC lines through their power electronic converters. In addition, operating HVDC line in parallel with HVDC line can be used for direct control of the power flow over the HVAC line 58. Therefore, they can be used for enhancing the system stability. Since the traditional control of HVDC lines does not provide either damping effect or synchronizing, supplementary 57

Pilo, F., Celli, G., & Mocci, S. (2004, April). Improvement of reliability in active networks with intentional islanding. In Electric Utility Deregulation, Restructuring and Power Technologies, 2004.(DRPT 2004). Proceedings of the 2004 IEEE International Conference on (Vol. 2, pp. 474-479). IEEE. 58

Rahman, H., & Khan, B. H. (2008). Stability improvement of power system by simultaneous AC–DC power transmission. Electric Power Systems Research , 78(4), 756764. 191

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control is needed for controlling the converters in order to improve the transient stability. For example, the power flow of the HVDC line can be controlled according to the frequency deviation.

(a)

(b)

(c) Fig. 5.56: Operation of parallel DC-AC transmission for transient stability enhancement As an illustrative example of the use of HVDC for enhancing the transient stability, consider the system of Fig. 5.56 in which HVDC line is operating in parallel redundant with HVAC line i.e. each of them can separately transfer the pre-fault power (P o ). Suppose that a three-phase fault occurs on the HVAC line very close to the generating station bus. Therefore, the duringfault power transfer is nearly zero. By its controls, the HVDC line increases its power flow such that the total initial power transfer is restored in the postfault conditions (see Fig. 5.56). If the DC line is replaced by an AC line, then this is action may be impossible due to the post-fault power transfer limit 192

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which is limited by the reactance of the available line. Therefore, as shown in Fig. 5.57, in comparison with an alternative HVAC line, the HVDC – with the proper control - results in a significant increase in the possible deceleration area while the original pre-fault operating point is restored.

Fig. 5.57: Impact of regulated HVDC line for transient stability control on the transient stability in comparison with uncontrolled HVAC line

5.5 Measurement-based SMIB Equivalence of Multi-machine power systems – A case study The presented equivalency approach is applied to the IEEE two-area system. As shown in Fig. 5.58, the system consists of two weakly interconnected areas. All generators are identical. In addition, all transformers are identical. The machine data are shown in Tables 5.5 while the rest of the system data and operating conditions are available at my paper59 entitled ‘Stability-based minimization of load shedding in weakly interconnected systems for real-time applications ’. Under the given loading conditions, area 1 is exporting 400 MVA to area 2. The initial power flow results are shown in Tables 5.6 while the line flow results are shown in Table 5.7.

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Table 5.5: Machine data of the two-area system on 900 MVA base Ra

Xd

Xq

Xl

Xd’

Xq’

Xd’’

Xq’’

Tdo’ Tqo’ Tdo’’

0.0025 1.8 1.7 0.2 0.3 0.55 0.25 0.25

8

0.4

Tqo’’

H

D

0.03 0.05 6.5 0

Table 5.6: Initial power flow results Bus Bus 01 Bus 02 Bus 03 Bus 04 Bus 05 Bus 06 Bus 07 Bus 08 Bus 09 Bus 10 Bus 11 Bus 12 Bus 13

V [p.u.] 1.03 1.01 1.03 1.01 0.995251 0.968819 0.955458 0.935301 0.983239 0.986062 1.004066 0.956271 0.98551

phase [rad] 0.575014 0.403633 0 -0.14569 0.46203 0.28216 0.130335 -0.13942 -0.40137 -0.26334 -0.10256 0.119751 -0.4196

P gen [p.u.] 6.9 7.1 6.34 7 -8.5E-14 -6.8E-13 -1.6E-12 2.64E-13 4.4E-13 -4.1E-13 -4.8E-14 1.03E-12 1.18E-12

P load [p.u.] 0 0 0 0 0 0 0 0 0 0 0 9.67 17.67

Q gen [p.u.] 2.52699 2.92232 1.92492 1.85999 -3E-15 8.86E-14 4.45E-13 9.96E-13 1.78E-13 1.53E-14 4.93E-15 1.44E-14 1.11E-13

Q load [p.u.] 0 0 0 0 0 0 0 0 0 0 0 -0.82891 -2.3993

Table 5.7: Initial line flow results From Bus Bus 05 Bus 08 Bus 06 Bus 07 Bus 07 Bus 13 Bus 11 Bus 10 Bus 08 Bus 07 Bus 01 Bus 02 Bus 04 Bus 03

To Bus Bus 06 Bus 09 Bus 07 Bus 12 Bus 08 Bus 09 Bus 10 Bus 09 Bus 09 Bus 08 Bus 05 Bus 06 Bus 10 Bus 11

Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14

P Flow [p.u.] 6.9 2.165 14 9.67 2.165 -17.67 6.34 13.34 2.165 2.165 6.9 7.1 7 6.34

194

Q Flow [p.u.] 1.674485 -0.12241 2.359337 -0.7259 0.46887 2.399305 1.233858 1.200513 -0.12241 0.46887 2.526995 2.922322 1.859988 1.924917

P Loss [p.u.] -8.9E-16 4.44E-16 0 -1.8E-15 0 0 -8.9E-16 0 4.44E-16 0 0 -2.7E-15 -8.9E-16 0

Q Loss [p.u.] 1.27240 0.59127 2.14749 0.10300 0.59127 0.32740 1.0345 1.84504 0.59127 0.59127 0.85251 0.96506 0.85881 0.69106

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The objectives of this case study are to estimate the electromechanical single-machine equivalence of each area and to determine the equivalent SMIB of the system.

(a)

(b) Fig. 5.57: The IEEE two-area weakly interconnected system; (a) Singleline diagram (SLG); (b) PSAT model

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The equivalent inertia and transient reactances of the study system are determined using the presented method. The disturbance is a temporary 3cycle three-phase fault started at t = 1 sec. The fault is applied on bus 8. The PSAT is used as a simulation environment and the PSAT model is shown in Fig. 5.57(b). The absolute angle (in radians) responses of the system is shown in Fig. 5.58 while the speed of the generators are shown in Fig. 5.59. These results reveal that the system is stable.

Fig. 5.58: The absolute angle responses (in radians) of the detailed system

Fig. 5.59: Rotor speed responses (in p.u) of the detailed system The estimations of the equivalent machine of each area at the associated interface bus are determined based on section 5.2. The outputs of the PMUs, bus frequency, and area active power measurements are shown in Fig. 5.60. On 100 MVA base, Table 5.8 shows the determined inertia values.

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(a)

(b)

(c)

(d) Fig. 5.60: Measurements at the interface buses

Table 5.8: Equivalent inertia (M) in sec, p.u transient reactances, and p.u transient emf (E’) Area 1 equivalence Area 2 equivalence SMIB equivalence Inertia Reactance 29.11

0.0236

E’

Inertia Reactance

1.047 53.15

0.0253

E’

Inertia Reactance

1.062 18.81

0.0122

In the next chapter, the determined equivalents will be evaluated using the EAC and TD. In addition, these equivalents will be used for demonstrating an EAC-based fast method for load shedding minimization.

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Chapter 6 Load Shedding and Online Applications 6.1 Introduction Load shedding is a controllable reduction of a pre-determined amount of the load power consumption according to specific shedding criteria . The pre-determined amount of load to be shed is traditionally determined according to an analysis of the dynamic security for a set of contingencies. In this case, look up tables are prepared and the implementation of the load shedding is performed according to them. A shedding look-up table is dependent on the operating conditions of the systems, its available topology, the available reserve, and the contingency. In addition, a look-up table must be updated for changes in the system. A look-up table determined for a given operating conditions and a specific set of single contingencies might be not valid in the situation of multiple contingencies such as cascaded outage of lines.60. The problems associated with look-up tables will be eliminated in this chapter through the online dynamic security analysis and load shedding estimation61. The main objective of the load shedding is correcting an abnormal system state (see chapter 1) to either the normal state or the alert state. For example, the balance of the generation and the load during abnormal operating conditions is a way for keeping the system stability. As explained in chapter 5, the load shedding can be applied manually or automatically. In addition, the automatic load shedding has many philosophies. For example, the automatic reactive load shedding depends on the value of the absolute frequency as a shedding criterion while with the proactive load shedding not only the absolute frequency is used as a shedding criterion but also the rate of change of frequency or even the acceleration of frequency.

60

Hafiz, H. M., & Wong, W. K. (2004, November). Static and dynamic under-frequency load shedding: a comparison. In Power System Technology, 2004. PowerCon 2004. 2004 International Conference on (Vol. 1, pp. 941-945). IEEE. 61 El-Shimy, M. (2015). Stability-based minimization of load shedding in weakly interconnected systems for real-time applications. International Journal of Electrical Power & Energy Systems, 70, 99-107. 199

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Based on its definition, the load shedding is a corrective action by which system overloads can be relieved. The system overloads can be classified according to its root causes as, 1. Active power shortage : in this case the available power production sources or generators are insufficient to meet the system demand. In this case, the main symptom of the system overload is a drop in the average system frequency. The shedding criterion in this case is the value of the system frequency. In the frequency drops below a predetermined value (see chapter 4), the load shedding is activated. This is called Under Frequency Load Shedding (UFLS). 2. Reactive power shortage : in this case the available capacity is capable to meet the load active power demand while the voltage at specific locations in the network is too low for appropriate power flow and load requirements. The shedding criterion in this case is the violation of the bus voltage magnitudes. For this purpose, it is called Under Voltage Load Shedding (UVLS). The UFLS and UVLS schemes can be considered as a special protection or wide area protection systems that attempt to minimize the impact of disturbances and prevent either brownouts or blackouts in power systems. Both types of load shedding should have shedding algorithms. A load shedding algorithm defines the method(s) by which several loads will be switched-off (and switched-on again) automatically to keep the power consumption below a defined secure level . Based on their time-frames and magnitudes of the associated changes in the operating conditions, system overloads may be also classified as, 1. Sudden and large changes : in this case a disturbance causes to the system to rapidly move from the initial stable operation conditions and the time needed to take appropriate corrective actions before a possible system collapse is very small. In this case, fully automated, fast, and appropriate corrective actions must be taken. 2. Slow and small changes : in this case a disturbance causes slow changes with small amplitudes. Consequently, manual corrective actions are possible. 200

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Based on this classification, the appropriate load shedding scheme can be determined. The UVLS schemes are integrated into utility electrical systems to operate as a last resort for the controllable shedding of specific amounts of loads at specific locations in the grid. This action can prevent the loss of a large amount of the load or the entire load due to uncontrolled cascading events. In contrast, the UFLS is designed for use in either the emergency or extreme states (see chapter 1). The main objective is to stabilize the balance between the available generation and load before or after an electrical island has been formed i.e. the UFLS drops enough load to allow the frequency to stabilize. Therefore, the UFLS helps to prevent the complete blackout, and allows faster system restoration in case of islanding. Typically, an UVLS responds directly to voltage conditions in a local area. The goal of a UVLS scheme is to shed load to restore reactive power relative to demand, to prevent voltage collapse and to contain a voltage problem within a local area rather than allowing it to spread in geography and magnitude. Load shedding is generally applied in steps. In each step, a specific amount of the load is dropped. If the first load-shedding step does not allow the system to rebalance, and voltage continues to deteriorate, then the next block of the load is dropped. The UFLS operates in the same manner. The online determination of the load to be shed seems impossible if the estimation of the shedding is based on the TD simulation of the full system with detailed models. This is due to the available time limits to do the dynamic security analysis, set decisions, take actions, and monitor the system response to the taken actions. In this chapter an extremely fast method will be presented for estimation of the minimum amount of load shedding for securing the transient stability of power systems. Consequently, the method is applicable for active power shortage problems. The method is based on the on-line based electromechanical equivalency and the single machine equivalence of power systems which are explained in chapter 5. The equivalent SIMB is then analyzed and the dynamic security analysis is performed using the standard EAC. The minimum amount of load shedding as a corrective action is also determined based on the EAC.

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6.2 Voltage Stability and UVLS – an overview In this section the fundamentals of voltage stability and methods for stability assessment will be presented. In addition, the UVLS schemes will be explained. 6.2.1 Basic Definitions

Historically, the analysis of power system stability pertained to the synchronous machine dynamics. On the other hand, in the analysis of voltage stability, the major concern is the voltage collapse, although the generators may remain in synchronism. As defined in chapter 4, Voltage stability refers to the ability of a power system to maintain steady acceptable voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition . In other words, the power system is voltage stable if voltages after disturbances are close to voltages at normal operating conditions. A power system becomes unstable when voltages uncontrollably decrease due to outage of equipment, increment in load, decrement in production or in voltage control. Voltage instability has been given much attention by power system researchers and planners in recent years, and is being regarded as one of the major sources of power system insecurity. Based on the CIGRE definition, the voltage instability is the absence of voltage stability, and results in progressive voltage collapse (or increase). The IEEE defines the voltage collapse as the process by which the sequence of events accompanying voltage instability leads to a blackout or abnormal low voltages in a significant part of the power system. Traditionally, voltage stability problems are associated with weak systems and long lines; however, voltage problems are now also a source of concern in highly developed networks as a result of heavier loading. Defining the power transfer corridor (PTC) as a transmission system separating the two portions of an interconnected power system (this type is called close PTC interface), or a subset of transmission circuits exposed to a substantial portion of the power exchange between two parts of an interconnected system (this type is called open PTC interface). The circuits connecting buses 7 and 9 in the two-area system of Fig. 5.57 is an example of the closed PTC interface. The power transfer limits of a PTC are limited by many constraints, including the system rotor-angle stability, voltage stability, and thermal (i.e. 202

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the N-1 static security transfer limit) limits. The impact of various constraining limits on the transfer limits is highly dependent on the operating conditions of the system where the system loading is the major governing factor62. This is illustrated in Fig. 6.1. It is clear that at light and medium loading conditions, the transient stability presents the main constraint. On the other side, at high loading conditions, the voltage stability becomes the dominant constraint. The secure operating limits are determined by the minimum value of various constraints.

Fig. 6.1: Secure transfer limits and various constraints Unlike the frequency stability, the voltage stability is generally a local problem; however, the consequences of voltage instability may have a widespread impact. The result of this impact is the voltage collapse , which usually results from a sequence of contingencies rather than from one particular disturbance. It leads to very low profiles of voltage in a major part of a power system. The voltage instability generally results in monotonically (or aperiodically) decreasing voltages. Sometimes the voltage instability may manifest as undamped (or negatively damped) voltage oscillations prior to voltage collapse (see Appendix 3). A possible outcome of voltage instability is loss of load in an area, or tripping of transmission lines and other elements by their protective systems leading to cascading outages. Loss of 62

Uhlen, K., Pálsson, M., Time, T. R., Kirkeluten, Ø., & Gjerde, J. O. (2002, June). Raising stability limits in the Nordic power transmission system. In 14th Power Systems Computation Conference. 203

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synchronism of some generators may result from these outages, or from operating conditions that violate field current limit. Several factors contribute to the voltage collapse such as increased loading on transmission lines, reactive power constraints, on-load tap changer (OLTC) dynamics and load characteristics. The problem of low voltages in steady state conditions should not be confused with voltage instability. It is possible that the voltage collapse may be precipitated even if the initial operating voltages may be at acceptable levels. Voltage collapse may be fast (due to induction motor loads or HVDC converter station) or slow (due to on-load tap changers and generator excitation limiters). The main reason for voltage instability is that the reactive power cannot be transmitted over long distances and has to be delivered directly to the point, which needs reactive-power support. The reasons for reducing reactive power transfer include, but not limited to,  The reactive power cannot be transmitted across large power angles, even with substantial voltage magnitudes . High angles are due to long lines and high power transfers. This is a physical constraint and can be simply explained by considering the 2-bus system shown in Fig. 6.2. With the line resistance neglected, the active and the reactive power transfer take the forms,

Fig. 6.2: An infinite bus supplying a load center via a transmission line

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where and transfer between node 1 and node 2.

is the maximum power

Fig. 6.3: Active and reactive power transfer characteristics It can be seen from Fig. 6.2, that either increasing the active power transfer or the angle separation between the buses, the reactive power transfer is reduced.  Active and reactive power losses constraints. Real losses should be minimized for economic reasons; reactive losses should be minimized to reduce investments in reactive power devices. Both active and reactive losses depend on reactive power transfer. Therefore, minimizing the reactive power transfer and keeping the voltage high is essential for loss minimization. This is can be explained by the following equations that can be derived from the system of Fig. 6.2.

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 Transfer of large amounts of reactive power will result in over-sizing the ampacity of the network components. 6.2.2 Classifications of voltage stability studies

The voltage stability is divided into short-term and long-term voltage stability and both types are load-driven as shown in Table 6.1. The distinction between long and short-term voltage stability is according to the time scale of load component dynamics. Short term voltage stability is characterized by components like induction motors, excitation of synchronous generators and devices like high voltage direct current (HVDC) or static VAR compensators. The time scale of short-term voltage stability is the same as rotor-angle stability. The distinction between these two phenomena is sometimes difficult, because voltage stability does not always occur in its pure form and it goes hand-to-hand with rotor-angle stability; however, the distinction between these two stabilities is necessary for understanding of the underlying causes of the problem in order to develop appropriate designs and operating procedures. The duration of long-term dynamics is up to several minutes. Table 6.1: Classification of power system stability according to the main driving forces, disturbance strength, and the time scale Time scale Generator-driven Load-driven

Pure rotor angle stability problem

Short Long

Pure voltage stability problem

Rotor-angle stability Voltage stability Small-signal Large-signal Small-signal Large-signal Voltage stability Frequency Stability Small-signal Large-signal

The voltage stability can be also classified according to the strength of the disturbances (see Table 6.1). Consequently, two main classifications are considered; small-disturbances and large-disturbances . Large-disturbance voltage stability examines the response of power system to a large disturbance, like faults, loss of load, or loss of generation. The ability to control voltages following large disturbance is determined by the system load characteristic and the interactions of both continuous and discrete controls and protections. The small-disturbance voltage stability considers the power 206

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system’s ability to control voltages after small disturbances like for instance small changes in load. It is determined by load characteristics, and controls at a given instant of time. The analysis of small-disturbance voltage stability can be done in steady state conditions by static methods like for examples load-flow programs; however, the voltage stability is a single problem on which a combination of both linear and non-linear tools can be used . 6.2.3 Connection between Voltage Stability and Rotor Angle Stability

Voltage stability and rotor angle stability are interlinked. Transient voltage stability is often interlinked with transient rotor-angle stability, and slower forms of voltage stability are interlinked with small disturbance rotorangle stability. There are examples of pure voltage stability; for instance a synchronous generator or large system connected by transmission line to asynchronous load. Pure angle stability, for example, identified in case of a remote synchronous generator connected by transmission lines to a large system. However, rotor-angle stability, as well as voltage stability, is affected by reactive power control. In particular, small disturbance instability involving aperiodical increasing angles was a major problem before continuously acting generator automatic voltage control regulators become available. We now can see the connection between small-disturbance angle stability and long-term voltage stability: generator current limiting prevents normal automatic voltage regulation. Voltage stability is concerned with load area and load characteristics. Rotor angle stability is concerned with integrating remote power plants to a large power system over a long transmission line. Voltage stability is basically load stability, and rotor angle stability is basically generator stability. For instance, if the voltage collapses at a point in a transmission system remote from loads, it is an angle stability problem. If the voltage collapses in a load area, it is probably mainly a voltage instability problem. 6.2.4 Fundamentals of Voltage stability analysis

Dynamic or transient voltage stability (TVS) includes non-linear modeling of power system components while static voltage stability involves steadystate representation of the system components. The load models and the TVS analysis are illustrated in the previous chapters and Appendix 3. Here in the static voltage stability modeling and analysis are explained. 207

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For the simple 2-bus system shown in Fig. 6.2, the system’s voltage and active power relation (V2 -P 2 characteristic) at the load bus can be determined

by writing equations (6.1) and (6.2) in a closed form by eliminating 12 and using,

, and

(2 is the power factor

angle at bus 2) as well as considering that

and

.

Without considering the thermal limit and other physical limits, it can be seen from (6.5) that the receiving end capability is a circle centered at (0, -B) and with radius A. This is shown in Fig. 6.4. It can be seen that the maximum range of possible reactive power flow occurs at zero active power flow. The reactive power range is reduced with the increase in the active power flow. The reactive power at the maximum active power flow is negative of a value –B.

Fig. 6.4: Ideal capability limits at the load bus The P2-V2 relation can be found by solving the quadratic equation (6.5);

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It can be seen from (6.6) that a real solution exists if . In addition, for a single value of the load bus voltage, two power solutions are available. The maximum loadability of the system is determined by the maximization of (6.6). The maximization conditions are and

. The corresponding voltage and power are called the

critical voltage (Vcr ) and the power transfer limit ( stability point of view respectively. These values are,

) from voltage

The function (6.6) is usually written for describing the relation between V2 and P2. Therefore, its preferable alternative form is,

This equation shows that positive real solutions are only possible if . The functions represented by (6.8) and (6.8) represent the equation of a parabola and their general form is shown in Fig. 6.5. Due to its shape, curve (b) is commonly called the nose curves.

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(b) (a) Fig. 6.5: Static voltage stability and practical limits; (a) P-V curve; (b) V-P curve. As shown in Fig. 6.5(b), for a single of the power consumption ( P o ), there are two values of the bus voltage; Va and Vb. This situation is physically impossible and one solution is stable ( Vb) while the other is unstable (Va ). The stability is explained using Fig. 6.6 where the active power P o is

increased by P . With an increase in the active power, the voltage drop over

the transmission line (V = I212Z12) increases. Therefore, the new operating point should be characterized by a lower voltage magnitude. As shown in the figure, the increase in the active power, results in a decrease in the voltage magnitude from b to b’ over the upper part of the V-P characteristics while the lower part shows an increase in the bus voltage magnitude from a to a’. In other words, the higher values of the voltage magnitude solutions are the stable conditions. Consequently, the positive sign of the inner root of (6.9) represents the stable solution. Therefore, the upper part is stable. The stability criteria can be then represented by,

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Fig. 6.6: Operating point static voltage stability From voltage stability point of view, the critical voltage and the maximum loadability limit are represented by (6.7) and (6.8); however, from load requirements point of view, the load bus voltage should be kept within an acceptable limits or bandwidth (see Fig. 6.5). Consequently, if bus 2 is supplying a load, then the load voltage requirement should be respected. This is required for proper operation of the load equipment and for preventing their malfunction. Typically, depending on the load-voltage sensitivity (see

Appendix 3) the allowable voltage bandwidth is 3%, or 5% or 10%.

Generally, the normal state demanded voltage ( VD) limits are represented by,

Usually, the critical voltage is much less than the allowable minimum load voltage; however, in the cases where the critical voltage is less than the minimum load voltage, the system operator restricts the bus voltage to the critical voltage rather than the load minimum voltage. The main operational objective is to simultaneously securing the voltage stability and the load voltage requirements . Due to these limits, not only the allowable bus voltage is modified but also the maximum loadability limit. For example, in Fig. 6.5 a significant reduction of the voltage stability based maximum loadability limit due to the load voltage limits. The maximum loadability limit is reduced from

to

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Equation (6.9) can be arranged to show a very important feature of the system. Considering that the short-circuit power at the grid access point (bus 2; also called point of common coupling (PCC)) is given by,

Consequently, (6.9) can be written as,

This relation shows that the V-P curve is universal and only dependent on the short-circuit capacity of the system and the load point power factor angle. The relation describes what is called the normalized V-P curve in which the load voltage is represented in p.u of the grid voltage while the load active power is represented in p.u of the grid short-circuit capacity. The V-P relation is illustrated in Fig. 6.7 considering the effect of the power factor of the receiving end.

Fig. 6.7: The normalized V-P relation as affected by the power factor at the receiving end 212

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The figure shows that the critical voltage locus is highly dependent on the power factor; from lagging to leading power factors, the critical voltage is increased. As previously explained, the operation of the system should secure satisfactory voltage level at the load bus while keeping the system stable. Therefore, the minimum of the critical voltage and the load minimum voltage limit is selected as the operation voltage i.e.

As an example, at the operating point h, the critical voltage is higher that the load minimum allowable voltage. Therefore, . Doing so ensures both the satisfaction of the load and the voltage stability. . In the situation where the critical

At point d,

voltage is very close or within the normal oper ation voltage, the monitoring of power system security becomes more complicated (see chapter 1) . These

situations can be realized when the system is over compensated by capacitive compensators. With high capacitive compensations (leading power factor), it is shown that the transfer limit is significantly increased but the system becomes more vulnerable to voltage collapse as the critical voltage also increases. An ideal situation is when the transfer limit is as high as possible while the critical voltage is as low as possible . This point will be considered later while discussing the voltage stability enactment and control options. So far, the V-P relation model is in agreement of many physical facts; however, the model as plotted in Fig. 6.7, did not show the effect of the power factor on the voltage at zero active power. An open-circuited transmission lines show receiving voltage value that is high in comparison with the sending end voltage. This is called the Ferranti effect and it is attributed to the effect of the line capacitance. Recalling, that the considered line model is simply a series impedance with shunt elements neglected, the resulting model will neither the Ferranti effect nor the accurate impact of the load power factor on the V-P relation. It is worthy to be mentioned here that the impact of the line resistance can be neglected in HV and EHV lines. This 213

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is because the X/R ratios at these high voltage levels are high. On the other hand, the X/R ratios in medium and low voltage circuits are not usually high enough for the resistance to be neglected. More sophisticated models are available for modeling these issues considering the distributed parameter models of transmission lines 63. For this purpose the line of Fig. 6.2 is represented by its long-line model parameters i.e. the characteristic (or surge impedance (Zo), the electrical line length (), and the surge impedance loading (P SIL) as shown in Fig. 6.8.

Fig. 6.8: Long line representation The standard performance equations for a long AC transmission line is written as,

The receiving end power (P 2 + j Q 2) is related to the receiving end current (I2) by:

Eliminating I2 from (6.13) using (6.14),

Through mathematical treatment of equation (6.15) and considering the phase angle of the voltage of bus 2 as a reference, the voltage solutions for equation (6.15) can be derived to take the form, 63

El‐Shimy, M. (2014). Modeling and analysis of reactive power in grid‐connected onshore and offshore DFIG‐based wind farms. Wind Energy, 17(2), 279-295. 214

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where Equation (6.16) gives the actual V-P characteristics of the system. It is plotted for an OH-HVAC and UG-HVAC (or submarine cable) lines with average parameters and the results are shown in Fig. 6.9.

(a)

(b) Fig. 6.9: Exact V-P relations; (a) OH-HVAC; (b) UG-HVAC cable It can be easily depicted from the exact relations that the impact of the Ferranti effect on OH-HVAC lines is minor and can be neglected in a simplified analysis. On the other hand UG-HVAC cables are characterized by 215

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a strong Ferranti effect. This is can be explained using the -model shown in Fig. 6.10.

Fig. 6.10: The -model for OH and UG transmission lines This model is valid for both overhead (OH) and underground (UG) transmission line; however; the major difference between the two transmission types is the value of the shunt admittance or capacitance. In OH lines, the shunt capacitance is very small in comparison with UG lines. This is mainly due to the insulation type which is just air in OH installations while it is a man-made insulating material in UG installations. The ratio of the UG capacitance to the OH capacitance is nearly equal to the relative permittivity

of the cable’s insulating material. Based on the -model, it is clear that the steady state normal reactive power production (i.e. the capacitive) from a line is nearly constant as the bus voltages are nearly fixed close to 1.0 p.u. On the other hand, the inductive power absorption is dependent on the line loading. At no load conditions, the series impedance current is corresponding to the current flow with the internal capacitances of the line and its value is very small. With the increase of the load, the inductive reactive power consumption increases. This is illustrated in Fig. 6.11 considering two identical lines; one of them is OH while the other is UG cable. Both lines are represented by the -model such that their terminal voltages are kept constant. It is shown that the UG line is reactive power producers for the entire load range. The OH line produces capacitive reactive power at light loading conditions while they consumes reactive power during the high loading conditions. The main cause of this effect is the shunt capacitances.

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Fig. 6.11: Reactive power vs. loading level; (a) OH lines; (b) UG cables The critical voltage shown in the previous analysis is corresponding to the system characteristics as the system is supplying constant power loads. As shown in Appendix 3, the loads have a verity of P-V and Q-V characteristics. The previous analysis only considers the system characteristics while the load-voltage sensitivity is not considered. The static representation of various load models is presented in Appendix 3. The load-voltage sensitivity has a significant impact on the operating point as well as the critical conditions. This is can be illustrated using the system of Fig. 6.2 such that V1 = 1.0 p.u and X12 = 0.06 p.u. The exponential load model of equation (4.128) is considered. The initial voltage Vo is set to 1.0 p.u while the initial power P o is set to have multiple values with a constant lagging power factor of 0.9. The

exponent  is set to be either 0.0 or 0.6 (note:  = 0.6 represents a composite load model; see Appendix 3). The model of equation (6.12) is used to simulate the system performance. For the considered load, the acceptable

voltage bandwidth is 10% around the unity voltage magnitude. The results are shown in Fig. 6.12.

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Fig. 6.12: Impact of load models on the voltage stability It can be seen from the figure that with  = 0.6, the increase in the load power (P o ) causes a reduction in the operating voltage. Up to a load power of 2.7, the load bus voltage magnitude do not violate the acceptable limits while higher load consumptions causes a violation of the lower limit of the load bus

voltage magnitude. It can be also depicted from the figure that with  = 0, the maximum power transfer and the critical voltage are found to be 5.223 p.u and 0.59 p.u respectively. Recalling that the critical voltage is defined as the point at which the system and load characteristics are tangent to each other. Therefore, it is shown that with  = 0.6, the critical voltage is reduced to 0.41 p.u and the maximum power transfer is also reduced to 4.7 p.u. Therefore, the nose point of the V-P curve is the critical voltage stability point only when a constant power load is considered while other load models will result in

different critical conditions. With  = 0.6, the reduction of the critical voltage is beneficial as the system becomes more stable at low voltage values in comparison with the constant power case; however, this benefit is counteracted by the reduction in the maximum power transfer. Considering the nose point as a critical operating point underestimates the system voltage stability; however, there will be a significant stability margin from the critical voltage point of view. On the other hand, the nose point overestimates the 218

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maximum power transfer. Therefore, it is recommended to improve the loading modeling as well as improve the voltage stability assessment for a secure operation64. The impact of contingencies or the N-1 security requirements on the static voltage stability will be illustrated considering the system of Fig. 6.2 with the reactance X12 represented by its details; a transformer and two parallel lines. This is shown in Fig. 6.13. As a numerical demonstration, the transformer reactance is taken as 0.01 p.u while the reactance of each line is 0.1 p.u. The load at unity voltage is taken as 1.0 p.u. In addition, the load power factor is fixed at 0.9 lagging. The load is modeled as a constant impedance ( = 2).

(a)

(b) Fig. 6.13: A system for demonstration of a line outage contingency; (a) The initial topology (I); (b) the post-contingency topology (II) The static voltage stability is dealing with slow changes of small magnitudes. For this type of analysis, the transient associated with a contingency are neglected. For example, in the considered system, the 64

Rassem OM, EL-Shimy M, Badr MAL. Assessment of Static Voltage Stability Limits as Affected by Composite Load Models. ASJEE. 2008;2:201 - 9 219

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electromechanical and electromagnetic transients associated with a line outage contingency are neglected; the focus in the static analysis is on the initial and final static voltage stability. Therefore, the during-fault conditions will not be considered; however, these conditions are of major importance in dynamic analysis studies such as the transient stability and the transient voltage stability. In this illustration, a static outage of one of the lines is considered. In comparison with the pre-fault reactance (

), the post-fault

reactance ( ) is increased. As a result, the pre-fault V-P curve of the system is changed to the post-fault V-P curve as shown in Fig. 6.14.

(a)

(b) Fig. 6.14: Static voltage stability – contingency analysis using V-P curves 220

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It is clear from Fig. 6.14 that the contingency causes a reduction of the nose point from c to c’. From the critical voltage point of view, the nose point can be considered – as explained earlier – as a conservative indicator of the stability limit. As shown in the figure, the considered disturbance did not alter the nose point voltage; however, the nose point power is significantly reduced as a result of the increase of the transfer reactance X12. The contingency also moves the operating point from a to b while if a constant power load is considered, the load point will move from a to b’. It is clear that the voltage associated with point b is high in comparison with point b’. Therefore, the operating point is more secure with the constant impedance model in comparison with the constant power model. As explained earlier, the constant power model is a highly theoretical load while the constant impedance load and the exponential load model prove their capability to model actual load performance. In this demonstration, if the load power is increased to 1.8 p.u and bus

voltage deviations should be less than or equal to 10%, then the simulation of the system with a constant power load model result in a violation of the minimum voltage limit and the system is considered insecure. On the other hand, the constant impedance load model shows acceptable operation, but very close to the lower limit (i.e. an alert condition). If the load is increased to 8.0 p.u, the stability will be highly degraded as the operating point will be at the post-fault nose point. Higher power consumptions will cause static voltage instability. The popular method for the analysis and the assessment of the static voltage stability is usually through the V-P curves; however, the Q-V curves can also be used for this purpose. In addition, they also can provide some additional insight of the static voltage stability. A Q-V curve expresses the relationship between the reactive support Qc at a given bus and the voltage on that bus. As shown in Fig. 6.15, the Q-V curves can be determined by connecting a fictitious generator with zero active power at the load bus, then the reactive power of the compensator is recorded as the terminal voltage is being varied. Since it does not produce the active power, the fictitious generator can be treated as a synchronous condenser or a static VAR system (SVS). SVSs include SVCs, and STATCOMs.

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Fig. 6.15: Use of a fictitious generator to produce Q-V curves As previously explained, the V-P curves define the system characteristics while the load characteristics are merged with the system characteristics for finding the operating point. Unlike the V-P curves, the Q-V curves show the static characteristics of the combined system and load. The Q-V relation can be determined in a similar way as that presented in section 6.2.D but equations (6.1) and (6.2) should be written as,

Therefore,

Which can be written as,

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Equations (6.20) and (6.21) describe the Q-V curve considering a constant power load model. Real solution is only possible if

from

which the maximum demand can be found. The load can be represented, for example, by the exponential or the ZIP models as descried (4.128) and (4.129). As a numerical demonstration, the parameters of the system of Fig. 6.13(a) are considered. The Q-V curve for a load active power of 3.0 p.u while the load is operating at a constant lagging power factor of 0.9 is shown in Fig. 6.16.

Fig. 6.16: The Q-V curve of the demonstration system for 3.0 p.u active load @ 0.9 lagging power factor The figure shows the plots of the positive and negative roots of equation (6.20). The values of the positive roots do not provide meaningful operating conditions. The negative root allocates operating equilibrium for a single value of the injected reactive power from the hypothetical compensator. Of course, this situation is impossible and one operating point is stable while the other one is unstable. For a given value of Qc, the possible operating equilibrium is a and b. If the compensator capacitive power increased these operating points move such that a  a’ and b  b’. From the physical point of view, the increase in the capacitive power injection at the load bus should increase the voltage magnitude. As seen, the voltage at a’ is less than the 223

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voltage at a while the voltage at b’ is higher than the voltage at b. Therefore, the right-hand side of the negative root is stable while the other side is unstable. The operating point d presents the critical point. The stability considering the negative root of (6.20) is then defined as,

The value dQ c/dV2 is also called the voltage sensitivity factor (VSF). The VSF measures the change in voltage magnitude at a given node as a consequence of a reactive power injection in that node. Since we are dealing with a fictitious reactive power compensator, then the operating equilibrium of the actual system is either at point c or c’. Based on (6.10), the stable equilibrium point is c. It is then clear that the distance from point d (i.e. the critical point) and the horizontal axis presents the margin to the loss of a stable operating point. Therefore, this distance is called the reactive power margin (KQ ). With the load active power changed, a family of Q-V curves is obtained as shown in Fig. 6.17.

Fig. 6.17: Impact of increase in the load consumption on the Q-V curve and the static voltage stability; the compensator sizing is also shown

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It is clear from the figure that the increase of the load results in a reduction in the load bus voltage magnitude (the operating point moves from a to b to c as the load increase from 1.0 to 3.0 to 5.3 p.u). It is also shown that at a power demand of 5.3 p.u, one operating point is available which is point c. This point is called the critical point at which the reactive power margin is zero. Increasing the consumption above the critical value, no possible operating equilibrium can be achieved with zero compensation. This is shown in the fourth loading level (8.0 p.u). In this case, a capacitive compensation is essential for system operation. From voltage stability point of view, the minimum reactive power distance that can bring the system at least to a critical condition; see Fig. 6.17; however, this minimum value result in an insecure system operation and a higher value should be selected for achieving a reasonable reactive power margin and acceptable operating voltage magnitude. From the load satisfaction point of view, points b and c violate the lower voltage limit. Therefore, a capacitive compensator may be used to increase the bus voltage magnitude to an acceptable value. Considering the N-1 criteria or a contingency of a static outage of one of the lines in the system of Fig. 6.13, the Q-V curves for active power consumptions of 1.0 p.u and 3.0 p.u are illustrated in Fig. 6.18.

Fig. 6.18: Static voltage stability – contingency analysis using Q-V curves

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The results show that the lightly loaded operating conditions results in a secure system operation in the pre-fault and post-fault conditions. On the other hand, the heavily loaded system could not find an operating point in the post fault conditions. In addition, the initial operating conditions lightly violates the lower limit of the bus voltage magnitude. Practically, the impact of various contingencies on the voltage stability or voltage security can be performed considering the long-term (static) or short-term (transient) voltage stability. The main purpose of these studies is the determination of the voltage security level of power systems. The standard contingency analysis methods (chapter 1) are used for this purpose. In addition, the data required for the simulation of various contingencies can be based on offline datasets or online measurements. The online measurements are currently possible due to the availability of WAMs and PMUs (chapter 1). Fig. 6.19 shows a schematic diagram of voltage security assessment algorithms.

Fig. 6.19: Voltage security assement algorithms

If the system could not provide an acceptable operation (see Fig. 6.17 and 6.18), then the voltage control is required for that purpose. Based on Fig. 226

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6.20, the relation between the voltages V1 and V2 as a function of the voltage drop and voltage deviation takes the form,

(a)

(b) Fig. 6.20: Basics of voltage control; (a) Phasor diagram; (b) Control methods

where

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If the HV or the EHV transmission is considered, then the X/R ratio is high enough for the resistance to be neglected. Therefore, the voltage drop is mainly dependent on the reactive power flow over the line while the voltage deviation is mainly dependent on the active power flow. It can be seen from (6.24) that five variable affects the voltage drop; however, the impact of these variables on the voltage controls is not equal. For example, the control of the active power flow has a minor impact on the voltage drop as it is weighted by the line resistance which is typically low. In addition, improvement in the resistance is commonly attributed to its reduction. Therefore, the development in the wired transmission technologies is expected to provide a further reduction of the impact of the active power control on the voltage drop. The active power supplied from the system ( P 2 ) to the load can be altered, for example, by injecting active power directly to the load via a conventional diesel generator or a solar-PV generator. In this case, where P DG is the active power produced by a distributed generator connected at the load bus. On the other hand, controlling the reactive power has a significant impact on the voltage drop. This is because the high weight of the reactive power caused by the line reactance. The reactive power supplied from the system to the load (Q 2) can be controlled via direct injection or absorption of reactive power at the load bus. This is can be provided by passive or active compensators (Fig. 6.21). Passive compensators includes fixed capacitors (FC), and fixed reactors (FR) while active compensators include synchronous condensers, thyristor controlled reactors (TCR), thyristor switched capacitors (TSC), static var compensators (SVC), and static synchronous compensators (STATCOM). In this case, where Q c is the reactive power of a shunt connected compensator at the load bus.

Fig. 6.21: Shunt compensation

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The reactance itself can also be controlled by the use of either passive or active series compensators (Fig. 6.22). Fixed capacitors and thyrsitor controlled series compensators (TCSC) are commonly used for that purpose. In this case, where Xc is the reactance of the series compensator. Series compensators are usually capacitive for reducing the voltage drop and increasing the power transfer limit. Therefore, Xc is usually negative. Consequently, the use of series compensators is accompanied with dangers from resonance and sun-synchronous resonance (SSR). A situation that can be avoided by proper selection of the possible range of capacitive series compensations.

Fig. 6.22: Series compensation The use of under-load tap-changing (ULTC) transformers also provide a tool for a direct control of the load bus voltage magnitude. ULTC transformers can be controlled manually or automatically. Based on the expected load curve and load flow, timetables are usually constructed for the manual control of ULTC transformers. On the other hand, automatic ULTC transformers provide a semi-immediate correction of the bus voltage magnitude; however, their effectiveness is limited by the small range of tap changers. In addition, automatic ULTC transformers are usually subjected to frequent failures due to excessive operation of the tap-changers. These excessive operations may also be unnecessary in the situation of momentary voltage variations (such as voltage dip or swell) where the voltage returns to the acceptable range in a short-duration which is small in comparison with the time constant of the tap-changing mechanism. In addition, devices such as uninterruptable power supplies (UPS) and dynamic voltage restorers (DVR) provide a more effective way for securing sensitive loads from these voltage disturbances. Practically, as shown in Fig. 6.23, the control systems of ULTC transformers include a deadband and a time delay blocks. The deadband defines the acceptable limits of the voltage magnitude of the controlled bus. 229

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These limits are defined according to the load sensitivity. The time delay block serves mainly as a protection device for preventing the tap-changer from responding to short-duration voltage variations.

Fig. 6.23: Automatic control of ULTC transformers The Q-V and V-P curves can serve as a vital tool in the analysis of the impact of various voltage control alternatives as well as sizing them 65. 6.2.5 UVLS schemes

As a result of a disturbance, the voltage magnitude may drop to a preselected level for a pre-determined time. In such cases, the UVLS sheds a selected amount of loads at selected locations in the system. The objective is then to prevent the widespread of voltage problems or voltage collapse. System planning engineers perform numerous studies (see Fig. 6.19) using the V-P curves as well as other analytical methods for the proper determination of the minimum amount of load that must be shed for securing the system to retain voltage stability under credible contingencies. Various voltage control equipments provide a defense against voltage collapse; however, in the situations the system is subjected to severe disturbances or cascaded outages, various voltage control and voltage restoration equipment may fail to restore the normal state voltage range. In these situations, the load shedding provides an effective corrective action for 65

Van Cutsem, T., & Vournas, C. (1998). Voltage stability of electric power systems (Vol. 441). Springer Science & Business Media. 230

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preventing either the voltage collapse or the islanding of the system. This is illustrated in Fig. 6.24. The shown voltage problems are attributed to the insufficiency of the reactive power sources needed for restoring an acceptable voltage level. It should be noted that system operators usually shed load as the last resort. Under Voltage Load Shedding (UVLS) schemes, drops a load when the voltage gets too low. The dropping of load will alleviate the system by eliminating the current flowing to the dropped load. UVLS usually triggers distribution feeders to open when voltage of the bulk electric system is around 90%. Definite time relays usually act when all three phases show low voltage for around 10 seconds at 90% voltage magnitude, this would be after some of the ULTC transformers have been acted. Certain critical customers cannot be dropped from load despite the help it may present to the system. Critical customers include hospitals or customers that would loose lots of revenue from being dropped.

Fig. 6.24: Long-term (slow dynamic) voltage security and corrective actions The order of various corrective events following a contingency is shown in the figure. The events started as a response to a contingency such as a forced outage of a generator or a line. Erroneous operation or vandalism may also cause disturbances of the similar impact. 231

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There are two main UVLS schemes; the decentralized (also called distributed UVLS) and the centralized schemes. With a decentralized scheme, protection relays are installed at the loads that are candidate to be shed upon severe voltage problems. As voltage conditions at these locations begin to collapse, load assigned to that relay is automatically shed. On the other hand, a centralized scheme has undervoltage relays installed at key system buses within system areas. The trip information is transmitted to shed loads at various locations. Both schemes require high-speed and reliable communication for proper operation. Fig. 6.25 illustrates a typical distribution substation with integrated UVLS and UFLS special protections.

Fig. 6.25 A typical distribution substation with integrated UVLS and UFLS special protections In this system, the UV relay is installed on the HV side of the transformer for correct detection of the grid voltage. This is because the voltage on the secondary side is not a real indicative of the grid voltage due to the actions of the ULTC transformer (or any other load side voltage controllers). The UF relay is installed on the LV side of the transformer because the transformer does not affect the frequency while the LVPT is more economical in comparison with the HVPT. Fig. 6.26 shows examples of the setting of the UVLS considering a contingency. The results of Fig. 6.14 and 6.18 are used for this purpose. The operating margin is left for allowing the natural 232

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reduction of the load due to the load-voltage sensitivity. The main complication associated with the setting UVLS is the inaccuracies associated with the PT and the UV relay. Therefore, proper and secure setting should be chosen for considering the probable inaccuracies.

(a)

(b) Fig. 6.26 Contingency-based setting of the UVLS protection 233

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6.3 UFLS – AN OVERVIEW The frequency response and stability are briefly considered in the previous chapters. In this section an overview of the UFLS schemes and technologies will be presented. Generally, the system frequency is a good indicator of the power balance and overload conditions in a power system. The UFLS is the last resort for the treatment of serious frequency declines in power systems when subjected to large disturbances. Under the emergency state or the extreme state, the ability to maintain the power balance and stabilize the frequency is directly related to the effectiveness of the employed UFLS Strategy. An effective UFLS strategy should be capable of: (1) Restrain the frequency decline, (2) Restore the normal frequency, (3) Minimization of the load shedding, (4) Minimization of the frequency recovery time, (5) Minimization of the frequency fluctuations, and (6) Provide the desired protection functions as economical as possible. Typically, an UFLS scheme sheds the loads in several stages. In each stage a pre-defined amount of the load is disconnected and the shedding of the load is continued till the normal frequency (i.e. the power balance) is restored. This is illustrated in Fig. 6.27. In 50 Hz systems, the common practices by most utilities use 49.3 Hz (i.e. 1.4% drop in the frequency) as the first frequency step, and between 48.5 and 48.9 Hz for the last step 66. In proper dynamic control of power systems, a sufficient time delay must be left between the shedding of each load block. This is of major importance for monitoring the correct impact of disconnecting a load block on the system frequency and for avoiding excessive, unnecessary load shedding as well as for avoiding subjecting the system to over-frequency conditions due to the over - shedding of loads. The proper timing of shedding each load block should not only depend on the frequency but also the rate of change of the frequency. This results in an adequate time separation between the shedding 66

Hafiz, H. M., & Wong, W. K. (2004, November). Static and dynamic under-frequency load shedding: a comparison. In Power System Technology, 2004. PowerCon 2004. 2004 International Conference on (Vol. 1, pp. 941-945). IEEE. 234

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of blocks. Using small load shedding blocks in conjunction with shedding timing based on the rate of change of frequency can be an effective way for prevention of over-shedding. The blocks of load shedding shown in Fig. 6.27 can be selected based on two criteria; the static criterion and the dynamic criterion . Fig. 6.28 shows flowcharts describing the logic of each criterion. In the static criterion, fixed load blocks are disconnected in each load shedding stage. This criterion may reduce the impact and the effectiveness of the load shedding, especially in large disturbance conditions that are associated with a steep decline in the frequency. The dynamic load shedding is constructed for solving this problem. In the dynamic load shedding the amount of load to be disconnected at each shedding stage is dynamically selected based on the system frequency, the rate of change in the frequency, the voltage, and the severity of the disturbance(s). In other words, the amount of a load shedding block is a function of the magnitude of the power imbalance.

Fig. 6.27 UFLS conceptual operation – blocks of load shedding in k-stages scheme

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There are three main methods for the implementation of UFLS strategies. These methods are

(a) (b) Fig. 6.28: UFLS criteria; (a) Static (semi-adaptive) UFLS; (b) Dynamic (adaptive) UFLS 1. The traditional method; when the frequency is lower than the first setting value, the first level of load shedding will be implemented. If the frequency continues to decline, it is clear that the first load shed amount is insufficient. When the frequency is lower than the second setting value, the second stage of load shedding is then implemented. If the frequency continues to decline, the further load shed stages are activated until the normal frequency value is restored. The traditional method follows the static shedding criteria and the amount of load shedding per each shedding stage is determined based on the analysis of the worst possible expected events. Therefore, for 236

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less severe events, the first stage of shedding may result in an overshedding and may also cause over-frequency problems 2. The semi-adaptive method; to some extent, this method is similar to the traditional method; however, the specific amount of load to be shed is determined in terms of the measuring value of the rate of change of frequency. 3. The self-adaptive method follows the dynamic shedding criterion for more accurate estimation of the proper amounts of the load to be shed in each stage and the timing of each stage.

6.4 Dynamic load shedding based on online measurements As shown in Fig. 6.28, the major challenge in the practical implementation of the dynamic load shedding method is the proper estimation of the load to be shed in each stage. The load shedding timing can be easily determined based on the measured df/dt while the amount of the load to be shed requires deep analysis of electromechanical transient associated with the disconnection of a specific amount of load. This is mainly for avoiding over-shedding, which may lead to over-frequency problems. In addition, from economy, reliability, and power quality points of view, the load to be shed should be minimized. The minimum value of the load to be shed can be defined as the minimum amount of disconnected loads for restoring the system transient stability (or dynamic security). On the other hand, the estimation of the minimum load shedding using detailed electromechanical modeling of power systems requires huge time in comparison with the under frequency phenomena. In addition, the amounts of simulation results to be analyzed are very huge for reaching appropriate conclusions and decisions in a reasonable time. In this section, the value of the presented online-based SMIB equivalency (chapter 5) in the analysis and decision making considering the underfrequency fast phenomena will be demonstrated and evaluated. The two-area system of Fig. 5.57 will be used to demonstrate the use of the SMIB equivalence (Table 5.6) in the estimation of the online load shedding minimization67. The system consists of two weakly interconnected areas or 67

El-Shimy, M. (2015). Stability-based minimization of load shedding in weakly interconnected systems for real-time applications. International Journal of Electrical Power & Energy Systems, 70, 99-107. 237

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systems. It is assumed that the generators in area 2 are highly variable while area 1 comprises less variable power sources. In both areas sudden drop in the power generation may be attributed to forced outage of generators, or faults, or unavailability of the primary energy resources. In area 2, the variability of the power sources causes sudden changes in the output power of the generators. These sudden changes may cause emergency or extreme security and transient stability problems if the drop in the power sources is intense and rapid. In such situations, load shedding is implemented to ensure system stability by curtailing sufficient system loads for matching the available generation with the remaining loads and keeping the system stability.

Fig. 6.29: Equivalent representation of a sudden change in Pg2 for the system of Fig. 5.57 using its SMIB equivalence shown in Fig. 5.7 As early state, the load shedding is implemented as a last resort to protect the system against the dangerous decline of either the frequency or the voltage or both of them. In this analysis, the load shedding is implemented to protect the system interconnection against frequency declines that cause system transient instability. Due to the delay in the propagation of frequency changes in weakly interconnected systems, there is a tendency to localize the power adjustments following large contingencies. In addition, the localized power adjustments have a significant improvement on the system stability. Therefore, drop of generation in a specific area will be primarily

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compensated by load shedding within that area unless more necessary actions should be taken. With the SMIB equivalence of the considered interconnected system (Fig. 5.8 and Table 5.6), the maximum sudden drop in the power generation P g2 can be determined using the EAC. Consider that the system shown in Fig. 5.57 operates at initial conditions P g1o, P g2o, P d1o, and P d2o. As shown in Fig.

6.29, this is equivalent to operation at P eqo and o; refer to section 5.3 for associated the mathematical modeling . The value of P eqo equals to 4.330 p.u. This value is not dependent on the nature of the power sources. The value of 1 is 64.5o.

In the following the symbol 12 will be written as . Due to the variability of generation in area 2, it is assumed that a sudden drop in the generation (P g2 < 0) occurs. Consequently, P eq is suddenly increased by –M1 P g2/(M1 + M2). The corresponding acceleration area (Aacc) is the area (abca ), the deceleration area (Adec) is the area (cdec), and possible deceleration area

(Apossdec) is the area (cdfec) are shown in Fig. 6.29. On Fig. 6.29, o, 1, max,

cr indicate the initial steady state operating angle, the final steady state operating angle, the maximum angle of oscillations, and the critical transient

angle respectively. It is clear from the figure that cr and 1 are dependent

variables such that cr =  - 1. The stability of the system is governed by the ratio of the possible deceleration area and the acceleration area ( refer to section 5.4). The system is stable if this ratio is higher than one while it is unstable if the ratio is less than one. A unity ratio indicates critical stability conditions. The maximum or the permissible sudden drop in P g2 can be then determined by determining the value of P eq1 at which the Aacc equals to Apossdec. In this case max = cr . Therefore,

The integration in eq. (6.26) results in the following equation which is to

be solved for 1.

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In eq. (6.27), cr =  - 1, o = sin-1(P eqo/P max). The maximum permissible increase in P eq and the maximum generation drop in P g2 can be then found by

equals to 6.32 p.u. Consequently, the

is found to be

approximately -5.6 p.u. It is clear from (6.28) and (6.29) that the permissible changes are highly dependent on the initial operating conditions of the system. As shown in Fig. 6.30, a generation drops higher than results in system instability. This is can be mitigated by fast load shedding in the form of immediate reduction in P d2.

Fig. 6.30: Transient stability regain by load shedding following an emergency sudden drop in P g2. Recalling that the impact of sudden reduction in P d2 is a reduction in P eq

by M1P d2/(M1 + M2). Therefore, if the generation drop

then the minimum load shedding as shown in Fig. 6.30 is defined by

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Equation (6.30) for the considered parameters and variables is then,

It is known that the decrease in the mechanical driving power has the same impact on the rotor angle swing as that of increasing the electrical output power. In this analysis, eq. (6.30) depicts that the change electric power demand shows that the same impact as the changes in the mechanical power. The verification of the estimated maximum drop in power generation and minimum load shedding requirements is performed through time domain simulation of the two-machine equivalent system (shown in Fig. 5.6). The perturbation cases shown in Table 6.2 are considered. Table 6.2: The considered disturbances Case Perturbation 1 5.0 p.u drop in the generation of area 2 2 5.6 p.u drop in the generation of area 2 3 6.0 p.u drop in the generation of area 2 4 6.0 p.u drop in the generation of area 2 & 0.4 p.u simultaneous load shedding Case 1 represents a drop on the generation below the critical value (5.6 p.u). Case 2 represents the critical drop in the generation in area 2 while case 3 present situation where the drop of generation is higher than the critical values. Case 4 presents mitigation by the minimum amount of load shedding. The dynamic performances of the system are shown Fig. 6.31. These figures show the rotor angle swing of area 2 relative to area 1. The time-domain simulations are performed using the PSAT. The results ensure the accuracy of the presented method in predicting the system transient response and minimization of the load shedding requirements.

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(a)

(b) Fig. 6.31: Dynamic performance associated with sudden changes in the generation and load; (a) Cases 1, 2, and 4; (b) Case 3. From a practical point of view, the simultaneous drop in the power generation and the required load shedding is technically impossible. In addition, introducing a time delay in the load shedding process is useful for avoiding over-shedding of loads (Fig. 6.28). Therefore, the impact of delayed load shedding on the system stability and the validity of the presented results is investigated through the EAC and time domain dynamic simulation. In comparison with Fig. 6.30, the simultaneous load shedding with the values presented by eqs. (6.30 and 6.31) will not cause changes in the acceleration area (area abca ) shown in Fig. 6.32(a). Therefore, the system will be stable after the perturbation in the power generation and load. On the contrary, the delayed load shedding, as shown in Fig. 6.32(b), causes an increase in the acceleration area by the amount bfghb. Therefore, the minimum amounts of load shedding determined by eq. (6.30) will not effectively restore the system stability. Therefore, the 242

(eq. (6.28) and

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Fig. 6.32) must be reduced to

(shown in Fig. 6.32(b));

is

reduced by an amount that increase the possible deceleration area by an amount jcekj such that the increase in the acceleration area is precisely compensated by the increase in the possible deceleration area i.e. area jcekj = area bfghb.

(a)

(b)

(c)

(d)

Fig. 6.32: Impact of load shedding delay; (a) Simultaneous load shedding (shed = o); (b) Delayed load shedding (shed > o): energy balance; (c)

Delayed load shedding: shed < h; (d) Delayed load shedding: shed  h In Fig. 6.32, shed denotes the angle at which the load shedding takes place

while h is the angle defined by

where P eq1 is the

equivalent power due to the reduction in P g2. The load shedding delay time is nonlinearly related to the difference between the load shedding angle (shed) and the initial angle (o). Fig. 6.32(c) and 6.32(d) show two situations defined 243

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according to the relation between shed and h. These situations are shed < h

(Fig. 6.32(c)) and shed  h (Fig. 6.32(d)). The acceleration and possible deceleration areas are clearly illustrated in the figures. It can be easily shown that the angle 1 can be determined for both situations by solving eq. (6.32)

for 1. Equation (6.32) is determined by equalizing the acceleration and deceleration areas in each situation.

It is clear that eq. (6.32) will be reduced to eq. (6.27) if simultaneous load

shedding (i.e. shed = o) is applied. Based on the value of 1, Fig. 6.32) can be determined using

(shown in

. Consequently the

minimum reduction in peak (denoted by As a result, the minimum load shedding (

) is equal to

.

) can be determined using

. It is depicted from eq. (6.32) that the value of 1 is

dependent on the initial operating conditions of the system represented by o

and the shedding delay which is represented by (shed - o).

Fig. 6.33: Impact of load shedding delay on the minimum required load shedding – EAC approach

Considering P g2 of -6.0 p.u, Fig. 6.33 illustrates the impact of delayed load shedding on the minimum load shedding requirements. The results show that with the increase in the shedding delay, the minimum load shedding requirements are increased. The relation between the minimum load shedding and the shedding time

delay (rather than (shed - o)) can be determined by solving the nonlinear 244

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mathematical model representing the system or performing numerical integration considering the EAC-based classical model. This is can be easily adopted in contingency analysis/corrective action schemes in the system security center. The relation between the minimum load shedding and the shedding time delay is determined as shown in Fig. 6.34.

Fig. 6.34: Impact of shedding delay time on the minimum amount of load shedding – TD analysis The results shown in this figure are determined through TD dynamic simulation of the two-machine equivalence of the system considering a drop of 6.0 p.u in the power generation of area 2. Recalling that the maximum drop of generation in this area is found to be 5.6 p.u. Therefore, as determined, at least 0.4 p.u of area 2 load must be immediately disconnected. With the delayed load shedding, the results shown in Fig. 6.34 show that the minimum value of the load shedding must be increased for restoring the system stability. This is in agreement with the EAC based analysis shown in Fig. 6.32 and 6.33; however, with the time domain analysis, the shedding delay is represented in time rather than angles. In addition, the application and the accuracy of the EAC is limited to the initial stages of the dynamic processes; however, the EAC shows significant potential and accuracy in the fast estimation of the transient stability of power systems. The time domain responses of the two-machine equivalence considering zero cycles (case 5) and eight cycles (case 6) load shedding delay times are 245

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shown in Fig. 6.35. With the delayed shedding, the minimum load shedding is increased from 0.4 p.u (no delay) to 0.48 p.u. The transient associated with the load shedding action is clearly shown on the figure.

Fig. 6.35: Dynamic response of properly reduced loads with and without time delay.

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Chapter 7 Damping Improvement and Stabilization through FACTS-POD controllers 7.1 Introduction The power system stability has been recognized as an important problem for secure system operation since the early power systems. The importance of this phenomenon has emerged due to the fact that many major blackouts in recent years caused by power system instability. Therefore, there is a great need to improve electric power utilization while still maintaining reliability and security. As power systems have evolved through continuing growth in the interconnections and the increased operation in highly stressed conditions, different forms of power system instability have emerged (see chapters 4 and 5). Historically, the control of power systems is based on the generators’ controllers such as AVRs, PSSs, and ALFCs. Most power systems contain elements that help regulate the network’s power flow, such as phase changers, series compensation, and shunt compensation. Historically, these devices were mechanically switched and may not be capable of reacting fast enough to prevent cascading failures. With the demand of improving the power quality, the reliability, and the security of modern power systems, these controllers cannot fulfill the recent and future modernization requirements. Consequently, the need for better high-speed control of power flow led to an initiative by the Electric Power Research Institute (EPRI) to develop power-electronic based devices 68, employing high speed, high power semi-conductor technology, to help better regulate power flow. All these devices are collectively known as Flexible AC Transmission Systems (FACTS) devices. The family of FACTS devices includes high speed versions of traditional devices like phase changers, and series and shunt compensators, as well as devices other electronically controlled controllers. According to the Edris 69, the FACTS devices are defined as “ power electronic based controllers and other static equipment which can regulate 68

www.epri.com Edris, A. A. (1997). Proposed terms and definitions for flexible AC transmission system (FACTS). IEEE Transactions on Power Delivery, 12(4). 69

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the power flow and transmission voltage through rapid control action ”.

Currently, the FACTS devices are considered a powerful tool for enhancing both steady-state and dynamic performance of power systems 70. The advent of the FACTS devices has required additional efforts in modeling and analysis, requiring engineers to have a wider background for a deeper understanding of power system's dynamic behavior. From an economical point of view, the FACTS provides an excellent alternative for increasing the transmission capacity by improving the efficient use of the existing power grid. This can be accomplished through increased control. One of the most promising recent decentralized network controllers is the family of (FACTS) devices. FACTS devices have been shown to be effective in controlling power flow and damping power system oscillations . By controlling power flow of an individual line, power can be redirected to/from various parts of the power grid. Redirecting power flow allows for utilization of power lines that physics of power flow alone would not allow. The variable series compensation is highly effective in both controlling power flow in the line and in improving stability. With series compensation, the overall effective series transmission impedance from the sending end to the receiving end can be arbitrarily decreased thereby influencing the power flow. This capability to control power flow can effectively be used to increase the transient stability limit and to provide power oscillation damping. Therefore, this chapter will present the design of a supplementary controller called Power Oscillation Dampers (PODs) for FACTS-based series compensation with a main objective of enhancing the damping of power systems71. A variety of design methods can be used for tuning the parameters of various power system controllers. The most common conventional techniques are based on frequency response 72 [5], pole placement73 [6], eigenvalues sensitivity74 [6, 7] and residue method 75 [8]. 70

EL-Shimy M. Multi-objective Placement of TCSC for Enhancement of Steady-State Performance of Power System. Scientific Bulletin - Faculty of Engineering - Ain Shams Uni. 2007;42(3):935 - 50. 71 M. Mandour, M. EL-Shimy, F. Bendary, and W. Ibrahim, “Damping of Power Systems Oscillations using FACTS Power Oscillation Damper – Design and Performance Analysis”, MEPCON 2014. 72 N. Martins and L. Lima, "Eigenvalue and Frequency Domain Analysis of Small-Signal Electromechanical Stability Problems," IEEE Symposium on Application of Eigenanalysis and Frequency Domain Method for System Dynamic Performance, 1989. 248

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Due to their popularity, POD designs are presented in this chapter using the frequency domain and residue methods for control design. The Thyristor Controlled Series Compensator (TCSC) that belongs to the family of FACTS devices is considered for its significant impact on both power flow control (see Fig 7.1) and damping of oscillations. The TCSC are mainly used for power flow control and as active series compensators for AC power transmission lines; however, with the POD as a supplementary controller to the TCSC, the capability of TCSCs in damping power oscillations will be presented. The small signal stability of power systems as affected by TCSC devices and PODs are evaluated and compared with the base power system where no FACTS devices are included. Both modal analysis and time domain simulation (TDS) are presented to show the impact of the designed PODs on damping the electromechanical oscillations in power systems. Several examples are given to show the impact of POD input signals on the design and system response.

7.2 FACTS devices – Definitions FACTS devices are being carefully studied and installed for their fast and accurate control of the transmission system voltages, currents, impedance and power flow. The fundamental objective of FACTS devices is to improve power system performance without the need for generator rescheduling or topology changes . Therefore, they found numerous applications in deregulated as well as regulated power systems. The fast development of reliable power electronic devices facilitates the construction of the FACTS devices. The main objectives of FACTS devices include, but not limited to:  Increase the power transfer capability of a transmission network in a power system,

73

B. C. Pal, "Robust pole placement versus root-locus approach in the context of damping interarea oscillations in power systems," IEE Proceedings on Generation, Transmission and Distribution, vol. 149, Nov 2002. 74 R. Rouco and F. L. Pagola, "An eigenvalue sensitivity approach to location and controller design of controllable series capacitors for damping power system oscillations," IEEE Transactions on Power Systems, vol. 12, Nov 1997. 75 R. Sadikovic, et al., "Application of FACTS devices for damping of power system oscillations," presented at the IEEE Power Tech, St. Petersburg, Russia, June 2005. 249

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 Provide the direct control of power flow over designated transmission routes,  Provide secure loading of transmission lines near their thermal limits, and  Improve the damping of oscillations as this can threaten the security or limit usage of the line capacity.

FACTS devices include multiple technologies that can be applied individually or in coordination with other devices to control one or more interrelated power system parameters such as series impedance, shunt impedance, current, voltage and damping of oscillations. These controllers were designed based on the concept of FACTS technology known as FACTS Controllers. According to the IEEE76, the term FACTS is defined as “The Flexible AC Transmission System(FACTS) is a new technology based on power electronic devices which offers an opportunity to enhance controllability, stability and power transfer capability of AC Transmission Systems ”. For power system security, various control strategies can be implemented to FACTS devices to guarantee the avoidance and survival of emergency conditions and to operate the system at lowest cost. FACTS controllers are historically introduced for solving specific power system problems; however, with research efforts, their applications extend their original objectives. From a chronological point of view, FACTS devices can be classified into three generations as shown in Fig. 7.1. The figure shows some common examples of the FACTS devices in each generation. The classification also shows the method of connecting various devices to the power system. The hybrid-connected FACTS are having series and shunt connected components. The abbreviations are defined in Table 7.1. As shown in Fig. 7.1, there are a large variety of FACTS devices currently available. These devices comprise shunt devices, series devices, and hybrid or (combined) devices. Each of these devices can be characterized by different characteristics during the steady state and transient operation of each of them. In addition, each device has a number of operational modes . 76

NGHingorani, L. G. (2000). Understanding facts concepts & tech nology of flexible AC transmission system. NewYork, IEEEPowerEngineering Society. 250

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For example, the TCSC can be controlled to provide either a constant power control or a constant admittance control (see section 7.3.2).

(a)

(b)

(c) Fig. 7.1: FACTS devices; (a) Classification; (b) Concepts of power flow control using FACTS devices (active and also reactive); (c) Generalized model of FACTS devices 251

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Table 7.1: Definition of the abbreviations in Fig. 7.1 as well some other devices

1st Generation

2nd Generation

3rd Generation

SVC TCR TSC TSR

Static VAR compensator Thyristor controlled reactor Thyristor switched capacitor Thyristor switched reactor Thyristor controlled series capacitor or TCSC compensation TCSR Thyristor controlled series reactors TCPST Thyristor controlled phase shifting transformer TCPAR Thyristor controlled phase angle regulator STATC Static synchronous compensator OM SSSC Static synchronous series compensator BESS Battery Energy Storage System UPFC Unified power flow controller IPFC Interline power flow controller GUPFC Generalized unified power flow controller HPFC Hybrid power flow controller

Fig. 7.1(c) shows a generalized model for FACTS devices. According to its design and operational characteristics, some of the FACTS devices contain all the shown components while some other FACTS devices have some of them. The definitions of various variables and parameters shown in Fig. 7.1 are as follows, N

ID of the FACTS device.

i

Mode

Sending end bus number. Terminal end (or sending end) bus number (0 for pure shunt connected devices such as SVCs and STATCOMs). Control mode.

Pref Qref

Reference or desired active power flow arriving at the terminal end bus. Reference or desired reactive power flow arriving at the terminal end bus.

Vref

Voltage set point at the sending end bus. Maximum shunt element current at sending end bus at unity voltage. Some devices such as TCSC do not contain a shunt element. Maximum value of the shunt element susceptance. This parameter is not

j

Ishmax Bshmax

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Bshmin

P tr max VSEmax VSEmin Vsemax Isemax Xse Xsemax Xsemax

applicable to some devices such as STATCOMs while it is a main parameter in SVCs. This value is also not available in pure series devices such as TCSCs. Minimum value of the shunt element susceptance. This parameter is not applicable to some devices such as STATCOMs while it is a main parameter in SVCs. This value is also not available in pure series devices such as TCSCs. Maximum bridge active power transfer between the series element and the shunt element. This power transfer is possible in hybrid devices while it is not available in either pure series or pure shunt devices. Maximum acceptable voltage magnitude at the terminal end bus. Minimum acceptable voltage magnitude at the terminal end bus. Maximum series voltage. Maximum series current in MVA at unity voltage (default 0.0) Reactance of the series element. This reactance is a dummy reactance in some devices and used in certain solution states Maximum value of the reactance of the series element. Minimum value of the reactance of the series element.

The proposed generalized FACTS model has a series element that is connected between the two shown buses and a shunt element that is connected between the sending end bus and ground. The shunt element at the sending end bus is used to hold the sending end bus voltage magnitude to Vref subject to the sending end shunt current limit Ishmax or other limits such as the shunt susceptance operational limits. In the steady state analysis, the shunt is handled in power flow solutions in a manner similar to that of locally controlling synchronous condensers and continuous switched shunts for the operation within the linear control regions. The shunt element is considered as a fixed susceptance when the shunt element hits its operational or control limits. The series element is mainly controlling the power flow on a line through appropriate changes. These changes include, for example, change in the line reactance by TCSC or phase separation in TCPAR. One or both of the shown elements (i.e. the shunt and the series) may be used depending upon the type of device. An adaptation of the generalized model for a specific device is a simple matter. UPFCs have both the series and shunt elements active, and allows for the exchange of active power between the two elements. SSSCs are modeled by deactivating the shunt element i.e. by setting both the maximum shunt current limit and the 253

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maximum bridge active power transfer limit to zero. STATCONs and SVCs are modeled by setting the terminal end bus to zero (i.e. the series element is deactivated). IPFC are modeled by using two consecutively numbered series FACTS devices. By setting the control mode, one device will be assigned, as the IPFC master device while the other becomes the slave device. Both devices have a series element without a shunt element. The conditions of the master device define the active power exchange between the devices. The mathematical modeling of some of these devices will be presented in the next sections and chapters. According to the nature of the analysis and the simplifying assumptions, some of the models will be applicable for dynamic studies while the others will be applicable for static studies.

7.3 Modeling and Modal Analysis 7.3.1 Linearized Modeling for POD design

The power systems are dynamic systems that can be represented by differential algebraic equations in combination with non-linear algebraic equations. Hence, a power system can be dynamically described by a set of n first order nonlinear ordinary differential equations. These equations are to be solved simultaneously. In vector-matrix notation, these equations are expressed as,

where:

,

, , , is the order of the system, is the , number of inputs, and is the number of outputs. The column vector x is called the state vector and its entries are the state variables. The vector u is the vector of inputs to the system, which are external signals that have an impact on the performance of the system. The output variables are those that can be observed in the system. The column vector y is the vector of system output variables, referred as output vector and is the vector of nonlinear functions defining the output variables in terms of state and input variables. 254

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The design of POD controllers is based on linear system techniques. After solving the power flow problem, a modal analysis is carried out by computing the eigenvalues and the participation factors of the state matrix of the system. The dynamic system is put into state space form as a combination of coupled first order, linearized differential equations that take the form,

where represents a small deviation, is the state matrix of size ,B is the control matrix of size , is the output matrix of size , and is the feed forward matrix of size The values of the matrix D define the proportion of input which appears directly in the output. The eigenvalues of the state matrix can be determined by solving Let be the ith eigenvalue of the state matrix A; the real part gives the damping, and the imaginary part gives the frequency of oscillation. The relative damping ratio is then given by:

If the state space matrix A has n distinct eigenvalues, then the diagonal matrix of the eigenvalues (Λ), the right eigenvectors ( ), and the left eigenvectors (Ψ) are related by the following equations.

In order to modify a mode of oscillation by a feedback controller, the chosen input must excite the mode and it must also be visible in the chosen output [8]. The measures of those two properties are the controllability and 255

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observability, respectively. The modal controllability

and modal

observability ( ) matrices are respectively defined by,

The mode is uncontrollable if the corresponding row of the matrix

is

zero. The mode is unobservable if the corresponding column of the matrix is zero. If a mode is neither controllable nor observable, the feedback between the output and the input will have no effect on the mode. 7.3.2 TCSC Dynamic Modeling and Control

The TCSC as shown in Fig. 7.2(a) can be defined as capacitive reactance compensator which consists of a series fixed capacitor (FC) bank shunted by a thyristor-controlled reactor (TCR) in order to provide a smoothly variable series capacitive reactance. When placed in series with a transmission line as shown in Fig. 7.2(b), the TCSC can change the power flow on the line as a result of the changes made by the TCSC on the line reactance; the following algebraic equations approximately govern the power flow on a line connecting buses k and m when the line resistance is neglected.

The TCSC can be controlled to provide either a constant power control or a constant admittance control. The constant power control scheme is shown in Fig. 7.3(a). In this case, the state variables of the TCSC are and . Therefore, the state space model of the constant power regulator takes the form,

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where:

,

,

, and

(a)

(b) Fig. 7.2 TCSC structure and control modes: (a) Basic structure, (b) A line with TCSC The constant admittance regulator for TCSC takes the form shown in Fig. 7.3(b). In this case, one state variable ( ) represents the TCSC and the state space model takes the form,

The constant admittance operation of the TCSC will be considered in the following analysis for compensating differences between the reactances of two parallel transmission lines.

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(a)

(b) Fig. 7.3 Control modes of TCSC: (a) Constant power regulator; (b) Constant admittance regulator

7.4 POD Design and Tuning The POD controller is designed using two methods. These are the frequency response method and the residue method. The main design objective is to achieve a predefined damping level of the electromechanical oscillations. The general control diagram of the power system controlled by the POD is shown in Fig. 7.4. The structure of the POD controller (Fig. 7.5) is similar to the classical power system stabilizer (PSS) described in chapter 4. The controller consists of a stabilizer gain, a washout filter, and phase compensator blocks. The washout signal ensures that the POD output is zero in steady-state. The output signal vPOD is subjected to an anti-windup limiter and its dynamics are dependent on a small time constant Tr (in this analysis Tr = 0.01 s). The gain Kw determines the amount of damping introduced by the POD and the phase compensator blocks provide the appropriate phase leadlag compensation of the input signal.

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Fig. 7.4 General feedback control system

Fig.7.5: Scheme of the POD controller 7.4.1 Frequency Response Method

The POD controller is designed using the frequency response method through Nyquist plots of a given Open Loop Transfer Function (OLTF).The Nyquist criterion allows to assess the closed-loop stability of a feedback system by checking the OLTF poles and plotting its frequency response. Closed-loop stability of the open-loop unstable system is obtained by ensuring an anti-clockwise encirclement of the (-1) point of the complex plane in the Nyquist plot of the OLTF after applying feedback compensation. The main steps of the procedure for POD design using the frequency response method can be described by a flowchart as shown in Fig. 7.6. The main design steps in the POD design using the frequency response method can be summarized as follows 77, 1. Eigenvalue analysis. In this design, the critical modes of the uncompensated system (i.e. without the POD) are identified based on eigenvalues and the participation factors of the state matrix. The participation factors ( ) of the state variables to each eigenvalue are computed by using right and left eigenvectors. If 77

and

represent

H. M. Ayres, I. Kopcak, M. S. Castro, F. Milano and V. F. d. Costa, "A didactic procedure for designing power oscillation damper of facts devices", Simulation Modelling Practice and Theory, vol.18, no.6 June 2010 259

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respectively the right and the left eigenvector matrices (Eqs. (7.6) to (7.8)), then the participation factor

of the

state variable to the

eigenvalue can be defined as,

2. State-space form. In this step, all output and input matrices ( ,

, , and ) are determined. The observability and controllability as defined by Eqs. (7.9) and (7.10) can be determined based on these matrices.

3. Nyquist analysis. In this step, the value washout filter time constant is randomly selected between 1 and 20 Sec then the Nuquist plot of the uncompensated loop including the washout filter is constructed. The required phase compensation  is then determined from the constructed Nyquist plot. The objective is to obtain a good phase margin based on the critical frequency . 4. Compensator blocks tuning. Based on the value of

that is determined in the previous step, the parameters of the phase compensator blocks are determined in this step using,

where is the number of the lead-lag blocks and is the frequency of the critical mode to be damped. The value of is usually one or two; Fig. 7.7 shows a POD with two lead-lag blocks (i.e. = 2) which is considered in this paper. In this layout, T3 and T4 are equal to T1 and T2. 5. Damping ratio adjustment. In this step, the root locus plot of the

compensated system is used to determine the value of Kw that provide 260

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an acceptable damping ratio (i.e.  ≥ 10%). The POD design is completed by completing this step; however, further adjustment of the design can achieved by fine tunning the POD parameters as described in the next step.

\

Fig. 7.6: Flowchart describing the frequency response method 6. Fine tuning of the POD design. The POD parameters have to be specified and chosen to fulfill specific performance parameters. The 261

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damping is one of the most important performance parameters; however, the performance is also governed by many parameters such as the maximum rise time ( ), the maximum overshoot (

), the

desired damping ratio ( ), and the settling time ( ). The fulfillment of these performance parameters can be achieved by fine tuning of the POD parameters keeping in mind that the damping ratio is the main specification in power system control design and, for large power systems, 10% of damping is considered sufficient for POD controllers. 7.4.2 Residue Method

The residue method for POD design will be described based on the general feedback control system shown in Fig. 7.7. The transfer function of the system is G (s) and the feedback control is H (s). The open loop transfer function of a SISO system78 is

can be expanded in partial fractions of the Laplace transform in terms of the matrix , the matrix, the right eigenvectors, and the left eigenvectors as:

Each term in the nominator of the summation is a scalar called residue. The residue for a particular mode gives the sensitivity of the eigenvalue of that mode to the feedback between the output and the input of the SISO system. The residue is the product of the mode’s observability and controllability. When applying the feedback control, eigenvalues of the initial system are changed. It can be proved that when the feedback control is applied, movement of an eigenvalue is calculated by, 78

R. Sadikovic, et al., "Application of FACTS devices for damping of power system oscillations," presented at the IEEE Power Tech, St. Petersburg, Russia, June 2005. 262

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Fig. 7.7 Closed-loop system with POD control.

It can be observed from (7.23) that the shift of the eigenvalue caused by a feedback controller is proportional to the magnitude of the residue. For improving the damping of the system, the change of eigenvalue must be directed towards the left half side of the complex plane. This is can be achieved by the use of the FACTS-POD controller. The compensation phase angle

required to move an eigenvalue to the left in parallel to the real

axis is illustrated in Fig. 7.8. This phase shift can be implemented using the lead-lag function of the POD represented by Fig. 7.5 and equation (7.25). The parameters of the lead-lag compensator are determined using,

Fig. 7.8 Shift of eigenvalues with the POD controller.

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Fig. 7.9 Flowchart describing the residue method

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where: is the phase angle of the residue , is the frequency of the mode of oscillation in rad/sec, is the number if compensation stages (in this paper, ). The controller gain location

is computed as a function of the desired eigenvalue

according to Eq. 7.29.

The flowchart summarizing the previous design procedures is shown in Fig.7.9.

7.5 Study System 1 – SMIB system The study system is shown in Fig.7.10. This system will be studied and analyzed with the aid of the Power System Analysis Toolbox (PSAT), the Simulink and the control system toolbox of Matlab 2012a. The study system consists of four 555 MVA, 24 kV, 60 HZ units supplying power to an infinite bus through two transmission circuits. The four generators are represented by one equivalent generator that is represented by the second order dynamic model. On 2220 MVA and 24 kV base, the transient reactance of the equivalent generator ( ) is 0.3 p.u, its inertia constant (H ) is 3.5 sec, and its damping coefficient (D ) is 10 in p.u torque/p.u speed. The initial conditions of the system in p.u on the 2220 MVA, 24 KV base are , , P = 0.9, and Q = 0.3 (overexcited).

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Fig. 7.10 The study system with the p.u network reactances are shown on 2220 MVA base The TCSC will be placed on line 2 as shown in Fig.7.11 and will be used to compensate the difference between the reactances of line 1 and line 2 of the study system. In this case, series compensation ratio is 0.462. The input variables to the PSAT block for modeling the TCSC are: sec, p.u, p.u.

Fig. 7.11 The study system after connecting the TCSC 7.6 CASE STUDY 1 – THE SMIB SYSTEM The results will be presented through studying the system described in Fig. 7.10 in three scenarios as shown in Fig. 7.12. In the Time Domain Analysis (TDS), the considered small-signal disturbance is a +10 % step increase in the mechanical power input ( P m) to the equivalent generator of the study system. The changes in the mechanical power will be started at t = 2 sec.

Fig. 7.12 Study Scenarios 266

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7.6.1 Impact of the TCSC on the small- signal stability

Tables 7.2 and 7.3 show the system dominant eigenvalues and their participation factors of scenario 1 and scenario 2 respectively.

Table 7.2: Scenario 1 - dominant eigenvalues and participation factors

Participation factors  Most Associated States (%) δ1 ω1 0.5 0.5 δ1, ω1 -0.71429  j7.6085 1.2163 9.34 Eigenvalues

f (Hz)

Table 7.3: Scenario 2 - dominant eigenvalues and participation factors

Participation Factors f  Most associated states (Hz) δ ω X1_TCSC (%) δ ,ω 0 -0.71429j8.0854 1.2919 8.79 0.5 0.5 -100 0 100% 0 0 1 X1_TCSC Eigenvalues

It is clear from these tables that both scenarios are stable; however, the eigenvalues of the system are changed as an effect of adding the TCSC to the system. The TCSC adds a non-oscillatory eigenvalue. The frequencies of the oscillatory modes of the system with TCSC are increased by 6.216% in comparison with the system without the TCSC while their damping ratios are reduced by 5.888%; (the percentage changes are calculated according to: % change = 100*(new value – old value)/old value). Therefore, the inclusion of the TCSC degrades the system stability. The damping ratio is less than 10%. Therefore, inclusion of POD is recommended to elevate the damping ratio to a value higher than or equal to 10%. POD designs according to the frequency response and residue methods are presented in the next sections; the objective

is to increase the damping ratio to an acceptable value i.e.  ≥ 10%. Various input signals to the POD will be considered. In addition, the observability and controllability of them will be determined using equations (7.9) and (7.10). 7.6.2 Observability and controllability of various input signals

The observability and controllability of candidate feedback signals to the POD will be determined. Based on Fig. 7.10, the varies feedback signals are the current across the transformer, the sending end active power, and the sending end reactive power. The modal controllability 267

and modal

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observability ( ) matrices associated with the considered feedback signals are shown in Table 7.4. Table 7.4: Modal observability and controllability of various feedback signals Feedback signal The current across the transformer The sending end active power The sending end reactive power

Modal observability C’ matrix [ 1.3707 1.3707 -0.2843] [1.223 1.223 -0.2785 [0.1438 0.1438 0.0209

Modal controllability B’ matrix  0.08  j 0.93     0.08  j 0.93     100  

Considering the critical electromechanical modes shown in Table 7.3 (highlighted by gray shading), it is depicted from Table 7.4 that all the considered signals are observable and controllable. Highest observability is associated with the current across the transformer feedback signal followed by the sending end active power then the sending end reactive power. Due to space limits, POD designs will be presented considering only the current across the transformer as a feedback signal; however, the presented design algorithms are general and can be applied to design PODs considering any acceptable feedback signal. 7.6.3 POD designs

Based on the flowcharts presented in Fig. 7.6 and 7.9, POD designs using the frequency response and residue methods are respectively presented in this section. Designs with each of the considered feedback signals will be determined. The initial stages of the design i.e. building the input and output matrices, analysis of the eigenvalues, modal controllability, and modal observability are already performed in the previous subsections. The washout filter time constant (Tw) is chosen to be 7. This value is arbitrary selected between 1 and 20. A) The Frequency Response Method

With the transformer current as an input signal to the POD, the Nyquist plot (for positive frequencies) of the uncompensated OLTF (predesign) and the compensated OLTF (post-design) is shown in Fig. 7.13. 268

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Fig. 7.13 Nyquist plots of SMIB system with and without POD It is depicted from Fig. 7.13 and Table 7.3 that the OLTF for the system is stable, but presents poorly damped poles. For a good POD design, the resulting polar plot should be approximately symmetric with respect to the real axis of the complex plane. Based on the Nyquist plots, the value of

the angle  required to relocate the critical frequency is 100.23o. Therefore, using equations (7.19) and (7.20), the parameters of the lead-lag compensators are T1 = 0.3408 sec. and T2 = 0.0449 sec. The gain Kw is determined based on the root locus of the system including the POD. The Matlab control system toolbox is used to construct the root locus as shown in Fig. 7.14. The gain Kw is determined by dragging the critical mode to an acceptable damping ratio which is chosen to be higher than 10%. As shown in Fig. 7.14, the value of the damping of the critical mode in the compensated system is set to 15.63% and the corresponding gain is 0.0641. The transfer function of the POD is then takes the form,

With the POD connected to the system shown in Fig. 7.11 as shown in Fig. 7.15, the design will be evaluated by both the eigenvalue analysis and the TDS of the compensated system. The results of the eigenvalue analysis of the compensated system is shown in Table 7.5 which indicates that the minimum damping of the system is improved to 15.63% as set by the POD design. Tables 7.2 and 7.3 show respectively that the damping of the system 269

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without TCSC is 9.34% and 8.79% in the uncompensated system with TCSC. This ensures the success of the POD design for improving the damping of the system.

Fig. 7.13 Root locus of the compensated system and selection of the gain Kw

Fig. 7.15 Modelling of the SMIB in the 3rd scenario Table 7.5: Frequency domain method based eigenvalue analysis of the compensated system f  Eigenvalues Most associated states (Hz) (%) X1_TCSC, V3_POD -12.3288 j61.9306 10.05 19.53% -13.0749+j0 0 100% V2_POD δ ,ω -1.2394j7.8266 1.2612 15.63% -0.14262+j0 0 100% V1_POD The TDS is performed considering a 10% step increase in the mechanical power input to the equivalent synchronous generator. This 270

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disturbance started at t = 2 sec. The simulation is performed using the Matlab control system toolbox. The responses of the systems of the three scenarios shown in Fig. 7.12 are compared as shown in Fig. 7.16.

(a)

(b) Fig. 7.16 TDS for 10% increase in the mechanical power: (a) Rotor angles; (b) Rotor angular speeds. It is depicted from Fig.7.16 that the POD improves the dynamic performance of the system through increasing the system damping, decreasing the overshoots, and decreasing the settling time. B) The Residue Method

The design is based on the flowchart of Fig. 7.9. With the transformer current as an input signal to the POD, the residues for all eigenvalues of the system without POD should be obtained to determine the residue of the most critical mode. This is shown in Table 7.6. Afterward, the POD parameters can be determined as previously described. The transfer function of the POD is then takes the form: 271

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Table 7.6 Residues of the eigenvalues Eigenvalues Residues -0.1043-0.71429+j8.0854 j1.281 -0.71429-j8.0854 0.1043+j1.281 -100 -28.4305

Table 7.7 shows the eigenvalue analysis of the system after connecting the POD to the system which indicates the improvement in the system damping in comparison to the systems of scenarios 1 and 2. The TD responses as various scenarios subjected to the considered disturbance are shown in Fig. 7.16. The results validate the POD design using the residue method which results in approximately the same TD response of the system. Table 7.7 Residue method: Eigenvalue analysis of the compensated system Most f  Eigenvalues associated (Hz) (%) states X1_TCSC, 8.3614 58.57% V3_POD -30.7726j42.5808 δ ,ω -1.2397j7.9995 1.2884 15.33% -12.4271+j0 0 100% V3_POD -0.14255+j0 0 100% V1_POD C) Further Analysis

In this section a summary of some other related results will be presented to show the effect of some critical issues in damping of oscillations in power systems. These issues are the impact of POD input signal and the value of the time constant of the washout filter (Tw) on damping of power system oscillations. The impact of POD input signal is shown in Fig. 7.17 which indicates that better dynamic performance can be achieved with the transformer reactive power as a feedback signal while the other feedback signals (i.e. the transformer current and the transformer active power ) have the same impact on the dynamic performance of the system. Therefore, 272

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careful choice of the input signal is important for damping maximization through POD design. High damping can be achieved with the transformer reactive power as an input signal because of the less control loop stability restrictions on the POD parameters in comparison with other signals.

(a)

(b) Table 7.17 TDS for 10% increase in the mechanical power with various feedback signals: (a) Rotor angles; (b) Rotor angular speeds. The results shown in Table 7.7 are obtained with Tw = 7. Although the literature recommended to select a random value for Tw between 1 to 20 sec, detailed analysis shows that the acceptable range of Tw is dependent on the system parameters and operating conditions. This is demonstrated in Table 7.8 for the same design conditions shown in Table 7.7. Three values of Tw are shown. These values are 1 sec, 7 sec, and 14 sec.

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Table 7.8: Impact of Tw on the dynamic performance Tw = 1 Tw = 7 Tw = 14 f f f    (Hz) (Hz) (Hz) (%) (%) (%) 27.54% 9.5167 19.53% 10.05 9.15% 16422.0 100% 0 100% 0 100% 0 16% 1.2593 15.63% 1.2612 16.21% 1.2566 100% 0 100% 0 100% 0 Table 7.8 shows that for all the considered values of Tw, the damping ratio of the critical electromechanical modes can be successfully increased to values higher than 15% which is practically acceptable damping level. The interesting part here is that, as shown in Table 7.8, that increasing Tw results in decreasing the damping ratios and increasing the frequencies of some of the electromechanical modes that was originally not critical (i.e. their damping ratio was higher than 10%). It is also shown that high value of T w such as 14 could result in creating new critical modes in the compensated system. Therefore, careful selection of Tw should be considered in the initial stages of the design. It is also important to know that a suitable value of Tw for a specific system may be not suitable for another system. In addition, the impact of Tw on the dynamic performance is also sensitive to the operating conditions of a power system.

7.7 Case Study 2 – Weakly interconnected multi-machine system The two-area system that is considered in the previous chapter is considered here to but with some structural and data changes for the inclusion of FACTS devices and consideration of higher-order models of the generators.

Fig. 7.18: The two area study system 274

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The new study system is shown in Fig. 7.18. The detailed system data are listed Appendix 8. In the original system, each area consists of two synchronous generator units. The rating of each synchronous generator is 900 MVA and 20 kV. Each of the units is connected through transformers to the 230 kV transmission line. There is a power transfer of 400 MW from Area 1 to Area 2. The PSAT model of the two area system is shown in Fig. 7.19.

Fig. 7.19 The PSAT model of the study system This system will be studied considering three scenarios. In the first two scenarios, the synchronous generators will be controlled by AVRs only (i.e. without PSS) and the PODs in these scenarios will be designed for the purpose of improving the small signal stability by increasing the critical and unacceptable damping ratios to acceptable levels (>=10%). In the first scenario, one PODfor one SVC will be designed and the SVC will be installed in the central bus between the two areas (bus 8). In the second scenario, two PODs for two SVCs will be simultaneously designed. In this case, the old capacitors located at bus 7 & bus 9 will be replaced by SVCs. In the third scenario, the PSS will be included in the system and the POD will be designed to improve the system stability in the presence of the PSS. The description of study scenarios is shown in Fig. 7.20. The basic structure of SVC is shown in Fig. 7.20(a). The SVC is connected to a coupling transformer that is connected directly to the ac bus whose voltage is to be regulated. The SVC is composed of a controllable shunt reactor and a fixed shunt capacitor(s). When placed in shunt with the ac bus as shown in Fig. 7.20, total susceptance of SVC can be controlled by 275

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controlling the firing angle of thyristors; however, the SVC acts like fixed capacitor or fixed inductor at the maximum and minimum limits.

Two Area Study System

Scenario 1 With AVRs and one POD

Scenario 2 with AVRs and two PODs

Scenario 3 with AVRs, PSS and POD

Fig. 7.19 Study scenarios of the two area system The SVC regulators can be controlled to provide either two models. The first one assumes a time constant regulator, as depicted in Fig. 7.21(a). In this model, a the dynamics of the SVC takes the form,

(a)

(b) Fig. 7.20 SVC structure and control modes: (a) Basic structure, (b) A bus with SVC 276

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The model is completed by the algebraic equation expressing the reactive power injected at the SVC node,

As shown, the regulator has an anti-windup limiter. Therefore. the reactance bSVC is locked if one of its limits is reached and the first derivative is set to zero. The second model, as depicted in Fig. 7.21(b) takes into account the firing angle α, assuming a balanced, fundamental frequency operation. Thus, the model can be developed with respect to a sinusoidal voltage. The differential and algebraic equations are then,

The state variable α also undergoes an anti-windup limiter. A. Modal analysis of the original system (no SVCs)

Before starting the study scenarios, the system eigenvalues without SVCs will be obtained to show the impact of the added SVCs on the system stability. The system eigenvalues that have the lowest damping ratios will be included in Table 7.9. Table 7.9: Dominant eigenvalues and participation factors of the test system without FACTs Eigenvalues

f (Hz)

 (%)

-0.54657j6.5963

1.0534

-0.56151j6.7847 -0.06528+j3.3288

1.0835 0.5299

Most associated states

Status of Eigenvalues

8.25%

δ2 ,ω2

Unacceptable

8.25% 1.96%

δ4 ,ω4 δ3 ,ω3

Unacceptable Critical

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(a)

(b) Fig. 7.21 Control modes of SVC: (a) SVC Type 1 Regulator; (b) SVC Type 2 Regulator It is clear that the system without FACTs is stable but presents three poorly damped eigenvalues with damping ratios 8.25%, 8.25% and 1.96%. Hence, the system has very poor damping of oscillations. B. Scenario 1

In this scenario, an SVC will be added to the system at bus 8; Fig. 7.22. The synchronous generators will be controlled by AVRs but without PSSs.

Fig. 7.22: Modelling of first scenario

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The eigenvalues of the system with SVC which have the lowest damping ratios will be shown in Table 7.10. In comparison with Table 7.9, it is clear that the SVC degrades the system damping. Table 7.10: Dominant eigenvalues and participation factors of the test system in Scenario 1 with SVC Eigenvalues -0.53283j6.6197 -0.5361j6.831 -0.05755+j3.5689

f (Hz)

 (%)

1.057 8% 1.0905 7.77% 0.56808 1.6%

Most associated states

Status of Eigenvalues

δ2 ,ω2 δ4 ,ω4 δ3 ,ω3

Unacceptable Unacceptable Critical

Table 7.10 Root locus of the compensated system and selection of the gain Kw With the SVC, the critical eigenvalue has a damping ratio 1.6% while the two unacceptable eigenvalues have damping ratios 7.77% and 8%. Since the damping ratio is less than 10%. Therefore, inclusion of POD is required to elevate the damping ratio to a value higher than or equal to 10%. The frequency response method will be used for that purpose. The POD washout filter time constant will be chosen as Tw=1 sec. By testing all possible POD stabilizing signals according to its own root-locus leads to that there is no any signal that can achieve the design objective. As demonstration, the POD is designed with the sending current between Bus 6 and Bus 7 are used as 279

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stabilizing signal to the POD. Based on the Nyquist plot, the value of the

angle  required to relocate the critical frequency is found to be be 103.71 o. Therefore, using equations (7.20) and (7.21), the parameters of the lead-lag compensators are T1 = 0.0969 sec. and T2 = 0.8104 sec. The POD gain (Kw) is selected based on the root-locus of the system as shown in Fig. 7.23. It is shown that with a gain of 1.21 the 1.6% damping ratio becomes 11% while the 7.77% and 8% damping ratios will not be increased to the acceptable levels. The transfer function of the POD then takes the form,

With the POD connected as shown in Fig. 7.11, the eigenvalues are shown in Table 7.11. The results indicates that it is not possible to increase all the damping ratios to accepted level by designing one POD in the two area system which does not contain PSS.

Fig. 7.11Modelling of the two area system in the 1st scenario with POD Table 7.11 Dominant eigenvalues and participation factors of the test system in Scenario 1 with SVC & POD Eigenvalues

f (Hz)

 (%)

-0.54297j6.8326 -0.53477j6.6161 -0.39502j3.4775

1.0909 1.0564 0.55701

7.9% 8% 11%

280

Most associated states

Status of Eigenvalues

δ4 ,ω4 δ2 ,ω2 δ1 ,ω1

Unacceptable Unacceptable Accepted

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C. Scenario 2

In this scenario, the two old capacitors installed at Bus 7 and Bus 9 are replaced by two SVCs for the purpose of designing two PODs for the system. The two area system with the installed SVCs is shown in Fig. 7.12. The synchronous generators will also be controlled by only AVRs as the previous scenario.

Fig. 7.12 Modelling of the two area system in the 2nd scenario The eigenvalues of the system with SVCs which have the lowest damping ratios will be shown in Table 7.12. Table 7.12 Dominant eigenvalues and participation factors of the test system in Scenario 2 with SVCs Eigenvalues -0.52663j6.6178 -0.5382j6.8191 -0. 0.03599+j3.7071

f (Hz)

 (%)

1.0566 8% 1.0887 7.92% 0.59003 0.45%

Most associated states

Status of Eigenvalues

δ2 ,ω2 δ4 ,ω4 δ1 ,ω1

Unacceptable Unacceptable Critical

According to Table 7.12, the presence of the SVCs degrads the system in comparsion with the original topology and scenario 1. There is a critical eigenvalue with damping ratio 0.45% and two unacceptable eigenvalues with damping ratios 7.92% and 8%. The design of the PODs will be achieved in two stages. In the first stage, one POD is designed and the system minimum damping ratio is determined. If the minimum damping is acceptable, then there is no need for the inclusion of the second POD. Otherwise, the second POD is designed for achieving acceptable damping ratios. 281

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For the first POD, the sending current between Bus 10 and Bus 9 is chosen as a stabilizing signal to the POD for the SVC located at bus 9. The POD transfer function is then found to be,

With this POD connected to the system, the system eigenvalues are shown in Table 7.13. Table 7.13 Dominant eigenvalues and participation factors of the test system in Scenario 2 with SVCs & POD1 Eigenvalues -0.53872j6.818 -0.52629j6.6183 -0.38416j3.7492

f (Hz)

 (%)

1.0885 7.92% 1.0567 8% 0.59982 10.1%

Most associated states

Status of Eigenvalues

δ4 ,ω4 δ2 ,ω2 δ3 ,ω3

Unacceptable Unacceptable Acceptable

It is depicted from Table 7.13 that the first POD has a positive impact on the critical eigenvalues since it has the ability to increase the damping of the critical eigenvalues to 10.2% and it has no significant impact on the other eigenvalues. The design objective of the second POD is to increase the unacceptable damping ratios to accepted level. The root-locus of all possible stabilizing signals to the second POD are determined to select the best signal which can improve the damping ratios of the rest of eigenvalues to accepted levels. Unfortunately, through testing all possible POD stabilizing signals, no signal can be found suitable for achieving the design target; however, the best found POD has the following transfer function and the system than has the eigenvalues shown in Table 7.14 which are unacceptable but slightly improved.

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Table 7.14 Dominant eigenvalues and participation factors of the test system in Scenario 2 with SVCs & PODs Eigenvalues -0.55513j6.8251 -0.61789j6.6169 - 0.31606 j 2.7477

f (Hz)

 (%)

1.0898 8.2% 1.0577 9.3% 0.4402 11.4%

Most associated states

Status of Eigenvalues

δ4 ,ω4 δ2 ,ω2 δ3 ,ω3

Unacceptable Unacceptable Acceptable

D. Scenario 3:

In this scenario, another control will be added on the synchronous generators of the two area system shown in the previous scenario. PSS will be added to generator 4 as shown in Fig. 7.13. The PSS is added to generator 4 for the purpose of improving one of the unacceptable damping ratios and the others will be increased by the POD.

Fig. 7.13 modelling of the two area system in the 3rd scenario The eigenvalues of the system with SVCs which have the lowest damping ratios are shown in Table 7.15. Table 7.15 Dominant eigenvalues and participation factors of the test system in Scenario 3 with SVCs Eigenvalues

f (Hz)

 (%)

-1.6434j7.5784

1.2342

-0.52845j6.6191 - 0.13075j 3.8027

Most associated states

Eigenvalue Status

21.17%

δ4 ,ω4

acceptable

1.0568

7.95%

δ2 ,ω2

Unacceptable

0.60557

3.4%

δ3 ,ω3

Critical

According to Table 7.15, there is a critical eigenvalue with damping ratio 3.4% and unacceptable eigenvalues with damping ratios 7.95%. It is clear 283

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that the PSS has a positive impact on the damping; however, the damping ratios are still unacceptable. Therefore, inclusion of POD is performed to elevate the damping ratio to a value higher than or equal to 10%. It is found that the sending current between Bus 5 and Bus 6 as a stabilizing signal to the POD can be used for achieving this target. Based on the Nyquist plot, the

value of the angle  required to relocate the critical frequency will be -21.77o. The parameters of the lead-lag compensators are T1 = 0.3184 sec. and T2 = 0.2172 sec. The POD gain (Kw) is selected based on the root-locus of the system. With a gain of 0.0767 the 3.4% damping ratio becomes 18.1% while the 7.95% damping ratio becomes 10%; see Fig. 7.14.

Fig. 7.14 Root locus of the compensated system and selection of the gain Kw The transfer function of the second POD then takes the form,

This chapter presented successful designs of PODs for enhancing the stability and stabilization of power systems. The design approach was based on conventional methods in which the design is carried out at a specific operating point. The controllers are then performs well at that specific operating point as 284

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well as operating points in the close proximity of the design point; however, power systems are subjected to large normal changes of the operating conditions during its normal operating conditions. Therefore, the capability of the designed controllers to serve their objectives and functions are doubtful when there are large deviations of the operating conditions in comparison with the design-based operating conditions. Therefore, the use of intelligent and adaptive systems that combine knowledge, techniques and methodologies from various sources are essential for the real-time control of power systems79. These methods are out of the scope of the current volume of the book; however, a brief survey is given here for interested readers and researchers. In practice different conventional control strategies are being used for POD design as described in this chapter; however, handling of system nonlinearities is the main advantage of the AI design techniques over the conventional design techniques. Artificial Intelligence techniques such as Fuzzy Logic, Artificial Neural networks and Genetic Algorithm can be applied for POD design, which can overcome the limitations of conventional controls. The POD based FACTs devices such as UPFC and STATCOM has been designed by neural network80,81 in a multi-machine power system to suppress the power system oscillations and to enhance the transient stability. The POD has also been designed by fuzzy logic 82,83 to improve the damping the power system oscillations. The genetic algorithm 84,85 79

Patel, R. N., S. K. Sinha, and R. Prasad. "Design of a Robust Controller for AGC with

Combined Intelligence Techniques." Proceedings of World Academy of Science: Engineering & Technology, Vol. 47, No.1, pp. 95-101 , 2008. 80

D. Ravi Kumar, N. Ravi Kiran " Hybrid Controller Based UPFC for Damping of Oscillations in Multi Machine Power Systems," International Journal of Engineering Research & Technology (IJERT) , ISSN: 2278-0181, Vol. 2, Issue 10, October - 2013. 81 Mozhgan Balavar,” Using Neural Network to Control STATCOM for Improving Transient Stability” Journal of Artificial Intelligence in Electrical Engineering, Islamic Azad University, Ahar, Iran, Vol. 1, No. 1, June 2012. 82 Dakka Obulesu, S. F. Kodad, B V. Sankar Ram, " Damping of oscillations in multimachine integrated power systems using hybrid fuzzy strategies," International Journal of Research and Reviews in Applied Sciences, Vol. 1, Issue 2, November 2009. 83 Prasanna Kumar Inumpudi and Shiva Mallikarjuna Rao N,” Development of a Fuzzy Control Scheme with UPFC’s For Damping of oscillations in multi machine integrated power systems” International Journal of Engineering Research and Applications (IJERA), Vol. 1, Issue 2, pp.230-234. 84 Aghazade, A.; Kazemi, A.”Simultaneous coordination of power system stabilizers and STATCOM in a multi-machine power system for enhancing dynamic performance”4th 285

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method can be also used to design the POD and the POD gain was obtained through the minimization of an objective function based on the damping ratio. Moreover, the GA-based methodology can be used to solve power system problems86 such as the inter-area congestions and oscillations issues in a HV Transmission system. Recently, many researches focus on the robust control design of the POD to enhance the system dynamic performance. H∞ technique has been applied87 to design the POD in large wind power plants (WPPs) to improve small signal stability by increasing the damping of electromechanical modes of oscillation. While in 9 , the optimization of POD parameters is formulated based on a mixed H2 H∞ control88, and carried out under all system outage events such as line tripping, and load/generation shedding.

7.8 Placement of FACTS Devices The placement and the selection of FACTS devices in power networks are mainly determined by the problem to be solved. Usually, the fundamental function(s) of a device is used for the allocation of a device or a set of devices for solving problems. For example, solution of bus voltage magnitude problems are usually allocated to devices such as SVCs, STATCOMs, and ULTC transformers. On the other hand, treatment of problems associated with power flow control and line flow overloads or congestions are usually allocated to, for example, TCSCs. The primary functions of various FACTS devices for implementing the proper corrective actions to common problems are illustrated in Table 7.16. The use of FACTS devices in their traditional International Power Engineering and Optimization Conference (PEOCO), IEEE, pp: 13 – 18, Shah Alam, 23-24 June 2010 85 M. Mandour, M. EL-Shimy, F. Bendary, and W.M. Mansour. Design of power oscillation damping (POD) controllers in weakly interconnected power systems including wind power technologies. Accepted for publications in the JEE, Oct. 2015. 86 A. Berizzi, C. Bovo, V. Ilea,” Optimal placement of FACTS to mitigate congestions and inter-area oscillations” PowerTech, 2011 IEEE Trondheim, pp: 1 – 8, Trondheim, 19-23 June 2011 87 J. Mehmedalic, T. Knuppel and J. Ostergaard,” Using H∞ to design robust POD controllers for wind power plants” Universities Power Engineering Conference (UPEC), 2012 47th International, IEEE, pp.1-6, London, 4-7 Sept. 2012 88 Issarachai Ngamroo and Tossaporn Surinkaew,” Optimization of Robust Power Oscillation Dampers for DFIG Wind Turbines Considering N-1 Outage Contingencies” Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), 2014 IEEE PES, Istanbul, October 12-15 286

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control form in the enhancement of power system damping, as shown in this chapter, requires additional supplementary control actions (i.e. the PODs). Supplementary control for FACTS devices (not considered in Table 7.16) can be provided for making them effective in enhancing the voltage stability, transient stability, dynamic security, and reliability. Some devices have inherent control and capabilities to contribute in enhancing the system dynamics. In this section, the multi-objective placement of TCSC in power networks will be considered89. The basic objectives of this section is to quantify the effect of controllable series reactances on power systems and to find a simple, but accurate method for multi-objective placement of TCSC based on sensitivity analysis for enhancing static performance of power systems. This section presents a new multi-objective placement technique of TCSC based on sensitivity analysis for: 1) Maximum relief of network loading (or congestion), 2) Power flow control, 3) Maximum reduction in active and reactive losses in a particular line, and 4) Maximum reduction in active and reactive power loss in the entire network. The OPF with FACTS constraints is used to show the validity of the placement technique to relief network congestions. The preliminary mathematical modeling of a system with TCSC, and a general form of staticsensitivity analysis are first considered then the placement criterion will be presented. A case study is then presented. 7.8.1 Preliminary mathematical modeling

The power system AC power flow model can be found in any power system analysis text book for example. The fundamental model is based on the system section shown in Fig. 7.15. the symbols in that figure are defined as,

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EL-Shimy M. Multi-objective Placement of TCSC for Enhancement of Steady-State Performance of Power System. Scientific Bulletin - Faculty of Engineering - Ain Shams Uni. 2007;42(3):935 - 50 287

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Table 7.16: Primary applications of some major FACTS devices

Increase the transfer limit

Limited transmissi on power

Xl

Adjust reactances

Adjust phase angles

Adjust phase angles

Reduce line Xl

Limit the short circuit current AND increase line Xl

288

TCPAR

UPFC

SSSC

TCSC

STATCOM

SVC

TCR

TSC

Problems Low voltage at heavy loading High voltage at low loading Low voltage following a contingenc y

Supply Q

Power distribution over parallel line

High voltage following a contingenc y

Absorb Q

Transmiss ion circuit overloadin g

Supply Q

Absorb Q

High short circuit currents

Stability Short circuit power

Corrective action(s)

Reduce line

Load flow reversal

Load flow

Thermal limits

Voltage limits

Main Subjects

FACTS devices

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Fig. 7.15: Power flow model; (a) Single-line diagram; (b) Detailed representation in terms of network elements. bsi: Susceptance of all shunt devices connected to bus i (e.g. capacitor or inductor banks) yij = 1/zij = yji = gij + j bij: series admittance of line m connecting bus i

and bus j. ysij = ysji = gsij + j bsij: shunt admittance of line m. Vi: bus voltage magnitude of bus i.

i: bus voltage phase angle of bus i. m: line index. SGi: apparent power generated at bus i = P Gi + jQ Gi. SDi: apparent power demanded from bus i = P Di + jQ Di.

The active and reactive power injected to bus i are given by,

Pi  PGi  PDi V i 2 g ii V i Q i  QGi  Q Di  V i 2bii V i



j K (i )







V j g ij cos  ij  bij sin  ij (7.43)





V j g ij sin  ij  bij cos  ij (7.44)

j K (i )

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where K(i) is the set of all buses connected to bus i; ij = i - j; g ii 

g

jK ( i )

sij

 g ij

;

bii 

( b

jK ( i )

sij

 bij )  bsi

 The active and reactive power flow on line m from bus i to bus j are given by, Plm  Pij V i 2 g ij  g ijV iV j cos  ij  bijV iV j sin ij (7.45) Q lm  Q ij  V i 2 (bsij  bij )  bijV iV j cos ij  g ijV iV j sin ij (7.46)

The active and reactive power losses on line m are given by, PLm  Pij  P ji  g ij (V i 2 V j2 )  2g ijV iV j cos ij (7.47) Q Lm  Q ij  Q ji  V i 2 (bsij  bij ) V j2 (bsij  bij )  2bijV iV j cos ij (7.48)

The network active and reactive power losses are given by: S L  PL  jQ L   PLm  j  Q Lm (7.49) nl

nl

m 1

m 1

where nl is the number of network lines. Now, consider a line with a TCSC as shown in Fig. 7.16. The TCSC can be defined as capacitive reactance compensator which consists of a series fixed capacitor (FC) bank shunted by a thyristor-controlled reactor (TCR). This structure is required for providing a smoothly variable series capacitive reactance. See Fig. 7.17. The line power flow, line losses are obtained by modifying equations (7.45) to (7.48) by replacing bij by (bij+ bclm) where bclm is the susceptance of the TCSC on line m, and equals to the inverse of the TCSC reactance xclm. This is the static-reactance model of a line with TCSC.

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Fig. 7.16: A line with TCSC

Fig. 7.17: Thyristor-Controlled Series compensation or capacitor (TCSC) 7.8.2. General form of static-sensitivity analysis

The static power flow equations (7.43) and (7.44) can be written in a generalized vector form as, F (X ,U , D )  0

(7.50)

where F is a vector function; X is the dependent or the state variable vector; U is the independent, input, or control variables; D is the demand variables. If the system is consisted of N buses such that buses 2 to m are PVbuses, buses m+1 to N are PQ-buses, and bus 1 is the slack bus then, X T   2 U T  PG 2

 N V m 1

PGm V 2

D T   PD 1

PDN

Vm

QD1

V N  (7.51) x cl 1

x c lnl 

(7.52)

Q DN  (7.53)

It should be noted that the control vector U may contain other variables such as switch status of switched- capacitor or inductor banks, phase-angle of ULTC or TCPAR. The solution of (7.50) is the state vector X from which the output variables can be calculated. These output variables include active and reactive line power flow, active and reactive line power loss, system active and reactive power loss, and output power of slack bus generator. All the 291

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desired output variables form the output vector W. For a given operating point (Xo, U o, D o), the output vector Wo can be determined as a function of that operating point. This function takes the general form, W o  (X o ,U o , D o ) (7.54)

Applying the first-order Taylor expansion to (7.54) for a change U in the control vector and linearizing (i.e. 1st order sensitivity). This results in the following equation,

W  X X  U U  D D

(7.55)

Assuming that D = 0 and multiplying the 1st term by U/U , equation (7.55) becomes, X W   X U  U U U X     X  U  U U      X K XU  U  U

 K WU U

(7.56)

 where KXU is state-to-input sensitivity matrix, and KWU is the output-to-input sensitivity matrix. The state-to-input sensitivity matrix KXU is obtained from the 1st order sensitivity of equation (7.50) with D = 0. This results in the following equations K XU  FX1FU

(7.57)

KUW  X FX1FU  U

(7.58)

Equation (7.56) represents a general form of output-to-input static-sensitivity analysis. This general formulation will be used as the core for the multi-

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objective placement of TCSC in power system for steady-state performance enhancement. 7.8.3 TCSC Placement Problem

The selection of input and output variable depends on the study being carried. Consider an N-bus system with win nl-lines such that a TCSCs are placed on all lines and let the vector Xcl be the vector of TCSC reactances i.e. X clT  U T  x cl 1

x clm

x c lnl 

(7.59)

The output vector W takes the form, W T  | Pl 1 | ... | Plnl | | Q l 1 | ... | Q lnl | PLl 1 ... PL lnl Q Ll 1 ... Q L lnl PL Q L  (7.60)

The modulus of line active and reactive power flow is taken for avoiding the error and the misunderstanding of the sensitivity results. For example, if the modulus is not taken and the sensitivity of power flow on line m’ due to placement of a TCSC on line m is negative, this will give one of the following meanings:  The effects of the TCSC is to reduce the flow on line m’, or

 The flow on line m’ is increased in opposite direction i.e. from bus j to bus i if the sensitivity is calculated for flow on line m’ from bus i to bus j.

As the modulus of line active and reactive power flow is taken, a negative sensitivity will only mean that the flow on line m’ is reduced due to placement of TCSC on line m, and this flow is increased with positive sensitivity. The output-to-input static-sensitivity is obtained using the general form (7.56) where the matrices required in the calculation of the output-to-input static-sensitivity matrix KUW are explained in the following. The Jacobian matrix F X is the Newton-Raphson power flow standard Jacobian matrix and takes the form, 293

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 P      F   X   Q   (7.61)    V 

The Jacobian matrix F U takes the form:

 P   Q    FU  X l     P X cl     X cl  (7.62) Q X   cl   A general element in the P/Xcl matrix (and similarly Q/Xcl) takes

the form P i/xclm. The derivatives P i/xclm and Q i/xclm are obtained by differentiating equations (7.43) and (7.44) after modifying them due to the existence of the TCSC as previously described. These derivatives takes the form:

Pi X clm





 lm V i 2 V iV j cos  ij  lmV iV j sin  ij   for i  slack, and m  K (i ) (7.63)  Otherwise  0





 lm V i 2 V iV j cos  ij   lmV iV j sin  ij  Q i  for i  K (PQ ), and m  K (i ) (7.64) X clm  0 Otherwise  Where K(PQ ) is the set of all PQ-buses.

 lm  lm

2r (x  x clm ) g lm  2 lm lm x clm (rlm  (x lm  x clm )2 ) 2

(7.65)

r lm2 -(x lm  x clm )2 b lm (7.66)   x clm (rlm2  (x lm  x clm )2 )2

z lm  rlm  jx lm 294

(7.67)

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The vectors X and U takes the form:  X  W X and U  W U

where the input vector U is defined in equation (7.59), the output vector W is defined in equation (7.60), and the state vector X is defined in equation (7.51). Theses derivatives can be easily found by direct differentiation similar to that performed in equations (7.63), and (7.64). With series compensation the overall effective series transmission impedance from the sending end to the receiving end can be arbitrarily decreased thereby influencing the power flow on the entire lines causing redistribution of the power flow all over the network. A basic constraint on placement of TCSC is that the placement will not cause transmission congestions (line overload); a situation that can initiate a cascaded failure event. Therefore, this reduces system security and reliability. Congestions also results in reduction in the competition opportunity in deregulated electricity markets, load shedding, and finally system collapse. One way to avoid such situation is to use the sensitivity of line real power flow performance index (PPI) to line power flow as restraining factor on placement of TCSC on electrical networks90. The PPI itself can be used to describe the severity of system loading during normal and contingency conditions91. This PPI is given by,

 PPI    lm m 1  2n nl

  Plmo    max    Plm 

2n

(7.68)

where P lmmax is the rated continuous capacity of line m; n is an exponent; lm is a real non-negative weighting factor which may be used to reflect importance of line m; P lmo is the base-case power flow on line m. The value of PPI is small when all network lines normally loaded (i.e. significantly below their thermal limits), and its value increases as the network lines loading increases. The PPI becomes high when there is a single major congestion or a number of small line loading violations. The inability to discriminate between a single major congestion and a number of small line S.N. Singh, A.K. David, “Optimal Location of FACTS Devices for Congestion Management”, Electric Power System Research, Vol. 58, 2001, pp. 71-79. 91 G.C. Ejebe, and B.F. Wollenberg, “Automatic Contingency selection”, IEEE, PAS, Vol. 98, No. 1, 1979, pp. 92-104.

90

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loading violations is called masking effect. Because a single major congestion may have severe effect on system security as compared to a large number of small violations, the masking effect should be omitted. This is can be achieved through proper selection of the exponent n. A value of n > 1 may be a solution leading to reduction of the probability of occurrence of the

masking effect. Herein, n is taken equals to 2, and lm is taken equals to unity for all lines (equal importance ). The sensitivity of line real power flow performance index PPI to line power flow as affected by placement of TCSC on line m’ is represented by the index  Plcm’ which is defined by,

PPI  lcm '  x cl '



x cl ' 0

 1  Plm |  max   Plm  x clm ' 4

nl

P

3 | Plmo m 1

(7.69) 

The value of  P is anticipated as follows

 A positive value of  P indicates that the network loading increases with the considered location of the TCSC,  A negative value of  P indicates that the network loading decreases with the considered location of the TCSC.

Hence, for maximization of network loading relief (or maximization of network congestion relief) TCSC should be placed on the location corresponding to the most negative value of  P . Although calculation of  P is

initiated to represent a basic constraint on the placement of TCSC on

electrical networks, the most negative value of  P itself satisfy the objective of maximization of network congestion relief. This can be proven using a network with severe congestions such that the traditional optimal power flow OPF algorithms cannot relief this congestion without load curtailment. Then, the optimal placement of TCSC is implemented on the system based on the

most negative  P-criteria and a modified OPF algorithm incorporating FACS devices with power flow control constraints92 is utilized to show the effect of the TCSC on the relief of severe network congestions. G Shaoyun, and T S Chung, “Optimal Power Flow Incorporating FACTS Devices with Power Flow Control Constraints”, Electrical Power & Energy Systems, Vol. 20, No. 5, 1998, pp. 321-326

92

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By inspecting equation (7.68) that was extensively used in reliability, security, and stability studies, it is clear that the line loading is represented by its maximum (rated) active power flow which is far from the reality. This is because the actual limits of a line is given as either rated current flow ( Ilmmax), or rated apparent power flow (Slmmax). Hence, for sack of accuracy an apparent power performance index (SPI) is constructed instead of the PPI. The SPI takes the form,

   S  SPI    lm   lmo max  m 1  2n   S lm  nl

2n

2 2  lm   Plmo  Q lmo    max S lm m 1  2n    nl

   

2n

(7.70)

The sensitivity of line apparent power flow performance index SPI to line power flow as affected by placement of TCSC on line m’ is represented by the index  Slcm’ which is given by (for n = 2, and lm = 1):

SPI  lcm '  x cl '

 nl

S

x cl ' 0

m 1

2 S lmo

 1   max   S lm 

4

 Plm Q lm   | Q lmo |  | Plmo |  (7.71)    x x clm ' clm '   

The value of  is treated in the same manner of treating the value of S

 P. In this analysis both  S and  P are used and compared. In doing so, in the calculation  P the line active power limit is taken equals to its apparent power limits as widely used based on the assumption that the line power flow is

mostly active power. But this justification is not general.The objectives of the TCSC placement in this analysis are,    Maximum relief of network loading (or congestion),

 Power flow control,  Maximum reduction in active and reactive losses in a particular line, and

 Maximum reduction in active and reactive power loss of a power network. The proposed procedure to solve the stated multi-objective placement problem is to construct a set of placement solutions that satisfy each of the 297

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objectives and then an overlapping approach is used for identifying the solution(s) that satisfy all the constraints. This procedure will be illustrated in the results section. 7.8.4 STUDY SYSTEM

The multi-objective placement and sensitivity analysis are performed on the six-bus system shown in Fig. 7.18. The system consists of 6-buses, 4generators, and 7-lines. Line indexes (m) are shown within a circle. The system bus-data and base-case AC power flow, line-data, base-case line power flow and loses, and generator-cost functions and generator limits are shown in Tables 7.17, 7.18, 7.19, and 7.20 respectively, all in p.u on 100MVA, 138-kV base. A TCSC is assumed to be placed on all lines such that 0.7xlm  xclm  0 to avoid overcompensation.

Fig. 7.18: Six-bus Study System Table 7.17: Bus Data and Base-Case AC Power Flow Bus No. Type 1 Slack 2 PV 3 PV 4 PV 5 PQ 6 PQ

V  (deg.) 1.020 0 1.040 2.734 1.010 -0.364 1.030 2.695 1.009 -0.313 0.953 -2.919

298

PG 0.22 0.99 0.86 1.30 0 0

QG 0.14 0.21 0.25 0.09 0 0

PD 0.70 0.45 0.50 0.30 0.35 1.00

QD 0.1 0.1 0.1 0.1 0.1 0.3

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Table 7.18: Line Data Line Line index, m From bus To bus 1 1 2 2 1 5 3 2 4 4 3 5 5 3 6 6 4 5 7 4 6

r lm

xlm

blm

Slmmax

0.04 0.08 0.04 0.08 0.08 0.08 0.08

0.08 0.16 0.08 0.16 0.16 0.18 0.16

0.02 0.04 0.02 0.04 0.04 0.08 0.04

1.0 1.0 1.0 1.0 1.0 0.5 1.3

Table 7.19: Base-Case Line Power Flow and Loses Line index, From To P ij Q ij P ji m bus i Bus j 1 1 2 -0.537 0.014 0.548 2 1 5 0.057 0.023 -0.057 3 2 4 -0.008 0.123 0.009 4 3 5 -0.001 -0.011 0.001 5 3 6 0.361 0.165 -0.348 6 4 5 0.301 -0.03 -0.294 7 4 6 0.690 0.159 -0.652

Q ji

P Lm

Q Lm

-0.013 -0.064 -0.143 -0.030 -0.178 -0.007 -0.122 Losses

0.01110 0.00040 0.00067 0.00001 0.01293 0.00684 0.03839 0.07034

0.02220 0.00080 0.00130 0.00001 0.00259 0.01540 0.07680 0.14240

Table 7.20: Generator-Cost Functions and Generator Limits* P Gmin P Gmax    2 $/h $/MWh $/MW h MW MW 373.5 7.62 0.0020 10 100 388.9 7.57 0.0013 20 280 194.3 7.77 0.0019 20 200 253.2 7.84 0.0013 20 300 *Generator cost functions takes the form C( PG )    PG  PG2 $/h

Gen. on bus 1 2 3 4

7.8.5 Implementation and Results

It should be noted that some of the lines will be rejected from

placement list based on the values of  in equation (7.69) or (7.71). The calculation of  P requires the calculation of line active power flow sensitivity

to TCSC location and the calculation of  S requires the calculation of both line active power flow and line reactive power flow sensitivities to TCSC location. The line active power flow sensitivity matrix is shown in Table 7.21. The values can be used as a guide for TCSC placement for line active power flow control as shown in Table 7.22.

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Table 7.21: Line active power flow sensitivity matrix (Plm/xclm’) Loc. 1 Loc. 2 Line 1 0.9375 0.2875 Line 2 0.46875 0.15 Line 3 0.975 0.3 Line 4 0.11875 0.0375 Line 5 0.11875 0.0375 Line 6 -0.32222 -0.1 Line 7 -0.13125 -0.04375

Loc. 3 Loc. 4 Loc. 5 Loc. 6 Loc. 7 -0.125 0 0.45 -0.8875 -0.75 -0.05625 0 0.225 -0.44375 -0.375 -0.1375 0 0.475 -0.925 -0.775 -0.01875 -0.00625 0.61875 0.28125 0.91875 -0.01875 -0.00625 0.61875 0.28125 -1.05625 0.038889 -0.005556 0.377778 0.677778 -0.63333 0.0125 0 -0.66875 -0.3 1.1

Table 7.22: Priority sets of placemat of TCSC for line active power flow control based on line active power flow sensitivity matrix of Table 7.21 Location of TCSC Placement for Line, m Increase active power flow Decrease active power flow 1 1, 5, 2 6, 7, 3 2 1, 5, 2 6, 7, 3 3 1, 5, 2 6, 7, 3 4 7, 5, 6, 1, 2 3, 4 5 5, 6, 1, 2 3, 4 6 6, 5, 3 7, 1, 4 7 7, 3 5, 6, 1, 2 The priority locations in Table 7.22 for placement of TCSC are arranged in priority sets for TCSC placement i.e. from the most effective locations to the least effective locations for placement of TCSC for line active power flow. The diagonal elements of the sensitivity matrix of Table 7.21 represents the self-sensitivity to active power flow i.e. sensitivity of line active power flow as a result of placement of a TCSC on that line, while the off-diagonal values represent the mutual-sensitivity i.e. i.e. sensitivity of line active power flow as a result of placement of a TCSC on another line. It is clear from Table 7.21 that the active power flow on a line can be increased, or decreased, or not affected significantly by placement of TCSC on another specified line. For example, the most effective TCSC location for reduction of active power flow on line 1 is placement of TCSC on line 6 as the value of sensitivity is the most negative index while placement of TCSC on line 1 results on the maximum increase of active power flow on line 1 as affected by the TCSC as the value of sensitivity is the most positive index. It is also shown that placement of TCSC on line 4 has no significant effect on the active power flow on line 1. An interesting situation appear with placement of TCSC, for example, on line 3, the active power flow sensitivity 300

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on line 3 as affected by TCSC on that line results in active power flow reduction (this verified through full AC power flow). Hence, it is not a general rule that placement of a series capacitor on a line within a network increases the active power flow on that line, actually the full AC power flow shows that the reactive power flow on line 3 increases by placing a TCSC on that line, this is clarify the importance of using the most negative  S-criteria

instead of most negative  P-criteria as a constraint explained earlier for enhancing the system static security. In addition, the full AC power flow shows that the placement of TCSC has insignificant effect on bus voltage magnitudes. The value of  P as affected by TCSC location is shown in Table 7.23 which shows that placement of TCSC either on line 5 or line 6 increases network loading ( P > 0) with maximum increase associated with placement of TCSC on line 5. Therefore, both lines are rejected from the TCSC placement set.

3

4

P

-0.00249

-0.00179

-0.00272

Placement priority

2

4

5

3

5

6

7 -0.31602

2

0.13766

1

0.18693

Location, m

-0.0053

Table 7.23: The value of  P as affected by TCSC location

Rejected Rejected 1 (Opt.)

The optimal placement of TCSC for network loading relief based on the most-negative P-criteria is a TCSC placed on line 7. The effectiveness of placement of TCSC on other lines for network loading relief is arranged as a priority list as shown in the table. Hence, one of these location (i.e. placement on lines 1, 2, 3, 4, 7) will be the selected location if it satisfy all the desired objectives. The correctness of the rejections will be verified by calculation the performance index PPI for separate placement of TCSC on line 7 and line 5 and simulating the changes in PPI through multiple loadflow as the TCSC reactance changes all over its range. Fig. 7.19 shows these changes in the PPI, it is clear from that figure that placement of a TCSC on line 7 decreases the PPI all over the range of variation of the TCSC and the opposite occurs with placement of TCSC on line 5. 301

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Fig. 7.19: Changes in the PI through Multiple Loadflow The line reactive power flow sensitivity matrix is shown in Table 7.23 and can be used as a guide for TCSC placement for line reactive power flow control this shown in Table 9 arranged in priority sets for TCSC placement i.e. from the most effective locations to the least effective locations for placement of TCSC for line active power flow. Table 7.23: Line reactive power flow sensitivity matrix (Qlm/xclm’) Line 1 2 3 4 5 6 7

Loc. 1 Loc. 2 Loc. 3 Loc. 4 Loc. 5 Loc. 6 Loc. 7 0.6125 0.15 -0.075 -0.0125 0.25 -0.4875 -0.4125 -0.23125 0.1125 0.025 -0.01875 -0.10625 0.24375 0.18125 0.5 0.1625 1.8 0 0.25 -0.4625 -0.3875 -0.05625 0.08125 0.0125 -0.0375 -0.3125 -0.1625 0.53125 -0.05625 -0.01875 0.00625 0 0.13125 -0.1375 -0.08125 -0.12222 0.0388889 0.011111 0.016667 0.144444 0.383333 -0.24444 0.04375 0.0125 -0.00625 0 -0.3875 0.10625 -0.33125

Table 7.24: Priority Sets of Placemat of TCSC for line reactive power flow control Based on Line reactive Power Flow Sensitivity Matrix of Table 7.23 Location of TCSC Placement for Line, m Increase reactive power flow Decrease reactive power flow 1 1, 5, 2 6, 7, 3, 4 2 6, 7, 2, 3 1, 5, 4 3 3, 1, 5, 2 6, 7 4 7, 2, 3 5, 6, 1, 4 5 5, 3 6, 7, 1, 2 6 6, 5, 2, 4, 3 7, 1 7 6, 1, 2 5, 7, 3

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4

Rejected

3

6

7 -0.34721

2

5

0.15468

Placement priority

4

0.19112

S

3

-0.00213

2

0.00180

1

-0.00024

Location, m

-0.00825

Table 7.25: The value of  S as affected by TCSC location

Rejected Rejected 1 (Opt.)

Based on the values of the line active- and line reactive- power flow sensitivity matrices, the sensitivity of line apparent power flow performance index SPI to line power flow as affected by placement of TCSC on line m’ is calculated and these values and TCSC placement priority are shown in Table

7.24 and 7.25. Based on the most-negative  S criteria an additional line is rejected from TCSC placement list which is line 3. It is logical that the results

obtained from  S criteria are more accurate than those obtained from  P criteria. Therefore, the final TCSC placement list consists of 4 locations on lines {7, 1, 4, 2} arranged according to their effectiveness in relieving network overloads (congestions). Based on that, the rejected locations are identified in Table 7.22 and Table 7.24 by non-bold numbers. In addition, from this point the rejected locations will not be mentioned in the following placement lists. The line- active power loss and reactive power loss sensitivity matrices are shown in Table 7.26 and Table 7.27 respectively. Those matrices can be used as a guide for TCSC placement for line active and reactive power loss reduction as shown in Table 7.28. Hence, the overlapping placement decisions of active and reactive power loss reduction represent the candidate locations of TCSC for line apparent power reduction. 

Table 7.26: Line active Power loss Sensitivity Matrix (PLlm/xclm’) Line Loc. 1 Loc. 2 Loc. 3 Loc. 4 Loc. 5 Loc. 6 1 0.04 0.0125 -0.005 0 0.02 -0.0375 2 0.003125 0.001875 0 0 0.00125 -0.00188 3 0.005 0.00125 0.01875 0 0.0025 -0.005 4 0 0 0 0 0.00125 0 5 0.005 0.001875 -0.01937 0 0.04 0.0125 6 -0.01444 -0.004444 0.001667 0 0.017778 0.031667 7 -0.0125 -0.004375 0.00125 -0.28125 -0.07937 -0.02875 303

Loc. 7 -0.03125 -0.00188 -0.005 0.000625 -0.06063 -0.02833 0.106875

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Table 7.27: Line reactive Power loss Sensitivity Matrix (QLlm/xclm’) Line 1 Line 2 Line 3 Line 4 Line 5 Line 6 Line 7

Loc. 1 Loc. 2 Loc. 3 Loc. 4 Loc. 5 Loc. 6 Loc. 7 0.6125 0.15 -0.075 -0.0125 0.25 -0.4875 -0.4125 -0.23125 0.1125 0.025 -0.01875 -0.10625 0.24375 0.18125 0.5 0.1625 1.8 0 0.25 -0.4625 -0.3875 -0.05625 0.08125 0.0125 -0.0375 -0.3125 -0.1625 0.53125 -0.05625 -0.01875 0.00625 0 0.13125 -0.1375 -0.08125 -0.12222 0.0388889 0.011111 0.016667 0.144444 0.383333 -0.24444 0.04375 0.0125 -0.00625 0 -0.3875 0.10625 -0.33125

Table 7.28: Priority Sets of Placemat of TCSC for line active, reactive, and apparent power loss reduction Location of TCSC Placement for Line, m Decrease line Decrease line Decrease line active power loss reactive power loss apparent power loss 1 7 7, 4 7 2 7 1, 4 NO 3 7 7 7 4 NA* 1, 4 NO 5 7 7, 1, 2 7 6 7, 1, 2 7, 1 7, 1 7 4, 1, 2, 7 NO * NA: Not Applicable ** NO: No Overlapping (in this case one line power loss can be reduced via a sigle TCSC)

Table 7.29: Network power loss sensitivity matrices and placement priority sets Loc. 1 Loc. 2 Loc. 3 Loc. 4 Loc. 5 Loc. 6 Loc. 7 PL 0.0005 0.00125 0.0175 -0.00063 -0.00625 -0.00278 -0.00437 Sensitivity PL Priority Rejec Rejec Rejec 2 Rejec Rejec 1 (Opt.) sets QL -0.2875 -0.00625 0.0125 0 -0.175 -0.08889 -0.51875 Sensitivity QL Rejec Rejec Rejec Rejec 2 3 1 (Opt.) Priority sets SL Rejec Rejec Rejec Rejec NO NO 1 (Opt.) Priority sets

The network active, and reactive power loss sensitivity matrices are shown in Table 7.29. Those matrices can be used as a guide for TCSC placement for network power loss reduction. Hence, the overlapping placement decisions of active and reactive power loss reduction represent the candidate locations of TCSC for line apparent power reduction. 304

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Based on the presented results, the optimal placement of TCSC network loading relief maximization and network apparent loss minimization is a TCSC is placed on line 7. Tables 7.22 and 7.24 gives a priority list for TCSC placement for line active and reactive power flow control which is very valuable, especially, in deregulated electricity markets where controllable power flow paths are an important requirements for maximization of electricity markets deregulation. Although, placement of TCSC on line 7 minimize the overall network power loss, Table 7.28 lists a priority sets for placement of TCSC for power loss reduction on a particular line. To evaluate the proposed technique in the congestion relief reduction, the load parameters of Table 7.17 are doubled and a traditional OPF algorithm (with no load curtailment option) is applied to solve the system with no TCSC installed. The results in this case is that the OPF could not get an acceptable operating point for the system because line 5 is congested and the system operating cost is 6371.1 $/hr. To solve this problem a TCSC is placed on two locations separately, the first location on line 7 (the optimal location) and on line 3 (the rejected location with lowest positive  S). The FACTS-OPF algorithm of is applied and the results show that with a TCSC placed on line 7, an acceptable operating point is obtained with no congestions and the system operating cost is 6368.58 $/hr. With a TCSC placed on line 3, the congested line remains congested. By forcing the reactance of the TCSC on line 3 to a value of -0.1xl3, additional congestion is occurred on line 6 and the system violations increases. These results prove that the optimal placement of TCSC is on line 7 and also prove the accuracy of the most-negative  S-criteria over the most-negative  P-criteria.

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Chapter 8 Overview, Modeling, and Performance Analysis of Gridconnected Wind energy Sources 8.1 Introduction Generally, the electric power is produced through an energy conversion process in which a primary energy source is converted into electric power. As illustrated in Fig. 8.1, the primary energy resources can be classified according their capability of replenishment (or renewal) into two broad categories; non-renewable and renewable energy resources 93. The conventional or non-renewable energy resources are available on the earth in limited quantity and will be vanished in the future. On the opposite side, renewable resources are natural sources of energy that are continually renewed, or replenished by nature, and hence will never run out. Energy plays an important role in the national security of any given country as a fuel to power the economic engine 94. Therefore, the energy security or in other words the access to cheap energy and availability of energy resources for the future is one of the major challenges in the energy sector. Non-renewable energy resources are distributed in an uneven way throughout the world while many types of renewable energy resources such as wind and solar energies are available for all locations on the earth. Therefore, renewable energy resources can contribute in enhancing the worldwide energy security, reducing the energy threat, and crisis. In addition, renewable energy sources are sustainable due to their low pollution levels in comparison with conventional sources of energy as well as their natural availability. It should be noted that the phrase renewable and the phrase sustainable are different. “ Renewable energy includes all those natural sources which can be replaced by natural ecological cycles. On the other hand sustainable energies are those energies that help this world in meeting its present needs without depleting ability of serving next generation’s needs.

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Maczulak, A. (2010). Renewable Energy: Sources and Methods. New York: Infobase Publishing, 2010 94 Sovacool, B. K.; Brown, M. A. (2010). Competing dimensions of energy security: An international perspective. Annual Review of Environment and Resources, 35, 77-108. 307

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Both kinds of energy leaves approximately zero effect on environment and remain in same form throughout existence on this universe ” 95.

Fig. 8.1: Conventional and renewable energy resources As an electrical power source, renewable energy sources suffer from some significant problems in comparison with conventional energy sources 96 (see chapter 1). These problems set a major challenge on the reliable use of renewable energy resources as electrical power sources especially when they significantly contribute by a large amount in the energy mix of power systems. A major change in the energy mix (see chapter 1) of power systems caused by the integration of large amounts of variable renewable energy sources raises a number of challenges regarding grid stability, reliability, security, power quality and behavior during fault conditions. Consequently, elaboration of specific technical requirements or grid codes for the connection of large amounts of variable renewable energy has been constructed. The grid codes stipulate that these energy sources should

95

Difference between renewable and sustainable energy, http://www.solarpowernotes.com/difference-between-renewable-and-sustainableenergy.html#.UnAsrlPnURQ 96 Hand, M.M.; Baldwin, S.; DeMeo, E.; Reilly, J.M.; Mai, T.; Arent, D.; Porro, G.; Meshek, M.; Sandor, D. (Eds. 4 Vols). (2012). Renewable Electricity Futures Study NREL/TP-6A20-52409. Golden, CO: National Renewable Energy Laboratory 308

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contribute to power system operation and control in a similar way the conventional generating systems do 97,98. Given that it is not techno-economically feasible or may be impossible to store massive amounts of electric power, the operational problems are mostly related to the characteristics of the renewable energy resource. Some major renewable energy resources such as wind and solar (shown in Fig. 8.1) are inherently variable and intermittent. Therefore, despite of the capability of predicting them, such grid-connected variable and intermittent energy sources cannot provide the main operational requirements of power systems such as dispatch, unit commitment, security, and reliability. Consequently, significant R&D is needed for solving these operational problems. It is worthy to be mentioned that some renewable resources such as biomass, biofuel, and hydropower with reservoir are characterized by low variability and high dispatchability. Therefore, such energy sources have better operational characteristics in comparison with other sources. Essential grid code requirements are related to frequency, voltage and behavior in case of grid faults. The most common requirements include 99 active power control, frequency control, frequency and voltage acceptable ranges, voltage control, voltage quality, fault ride-through (FRT) capability, power plant modeling, and communication and external control. The main aim of the requirements is to ensure that renewable power plant do not adversely affect the power system operation and control with respect to security of supply, reliability, and power quality. The fulfillment of the grid code requirements is mostly related to the technological advances of the renewable power generators; however, the fulfillment of the operational requirements need more sophisticated efforts because these requirements are mostly related to the characteristics of the primary renewable resources .

97

Tsili, M.; Papathanassiou, S. (2009). A review of grid code technical requirements for wind farms. IET Renewable Power Generation, 3(3), 308-332. 98 El‐Shimy, M. (2012). Modeling and analysis of reactive power in grid‐connected onshore and offshore DFIG‐based wind farms. Wind Energy. http://dx.doi.org/10.1002/we.1575, Dec. 6, 2012 99 Holttinen, H.; Bettina Lemström, V. T. T.; Meibom, F. P.; Bindner, H. (2007). Design and operation of power systems with large amounts of wind power. State-of-the-art report. VTT (Espoo))(= VTT Working Papers, 82). URL: http://www. ieawind. org/AnnexXXV/Publications W, 82 309

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Probably the hydrogen storage and hydrogen technologies will play a key role in the future of variable renewable energy sources100. In chapter 1, an overview of various energy sources and their operational characteristic is given. Both renewable and non-renewable energy sources are considered. In this chapter, detailed overview of the technological aspects of wind and solar-PV energy production technologies will be presented. In addition, the grid-interconnection requirements (i.e. the grid codes) for connecting these energy sources to the grid will be explained considering various international codes.

8.2 Wind Energy Conversion Technologies (WECTs) For thousands of years, man has utilized wind energy to sail ships, grind grain and pump water. The kinetic energy in the wind can be converted to a mechanical power using windmills. Historically, windmills are created from thousands of years for direct use of mechanical power in applications such as grinding grains, and pumping water. Windmills were in use in ancient Egypt fifty centuries ago for grinding flour and water pumping windmills have been recorded in Kautalya’s Arthashastra, indicating their existence in India from 400 BC. Recently as the late 20 th century, the wind power is used in Europe for electric power production through double energy conversion systems called wind turbines. Generally, as shown in Fig. 8.2, two energy conversion processes comprise a wind turbine. The first process is the conversion the wind kinetic energy to a mechanical energy Fig. 8.2: Wind energy conversion processes which is converted to electrical energy in the second energy conversion process through an electricity generator. As shown in Fig. 8.3 to 8.10, wind turbines can be classified according to many indicators 101,102 such as the axis of rotation (Fig. 8.3), airflow path 100

Parfomak, P. W. (2012). Energy storage for power grids and electric transportation: A technology assessment. Congressional Research Service, Tech. Rep. R, 42455 101 EL-Shimy, M. (2013). Probable power production in optimally matched wind turbine generators. Sustainable Energy Technologies and Assessments, 2, 55-66 310

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relative to the turbine rotor (Fig. 8.4), turbine capacity (Fig. 8.5), drivetrain condition (Fig. 8.6), power supply mode (Fig. 8.7), location of the wind

turbine (Fig. 8.8), type of generator (Fig. 8.9), and operating concept (Fig. 8.9). Fig. 8.3 shows the classification of wind turbines based on the direction of the axis of rotation i.e. the horizontal axis wind turbine (HAWT) and the vertical axis wind turbine (VAWT). Due to the low aerodynamic efficiency of the VAWT, the HAWT dominate the wind turbine industry and the market.

Fig. 8.3: Classification of wind turbines according to axis of rotation

Fig. 8.4: Classification of HAWTs according to airflow path relative to the turbine rotor 102

Spera, D. A. (2009). Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering . American Society of Mechanical Engineers 311

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According to the airflow path relative to the turbine rotor , the HAWT can be classified into upwind and downwind wind turbines as shown in Fig. 8.4; the majority of HAWT is upwind. The classification according to the turbine capacity is shown in Fig. 5. Turbine capacities are designed according to the intended applications; however, most recent designs are multimegawatt wind turbines for large-scale wind farm applications. Table 8.1 provides a brief comparison between various WECTS. Although Fig. 8.9 and Table 8.1 shows a large variety of generator/concept mix, a recent technological status 103 shows the DFIGbased VSWT present about 78% of the wind power installed capacity, 12% for Fig. 8.5: Classification of wind full-scale converter based VSWT, 5% for turbines according to turbine the limited variable speed concept, and 5% capacity for the SCIG-based FSWT.

Fig. 8.6: Classification of wind turbines according to drivetrain conditions

103

Fig. 8.7: Classification of wind turbines according to power supply mode

EL-Shimy, M. (2013). Probable power production in optimally matched wind turbine generators. Sustainable Energy Technologies and Assessments, 2, 55-66 312

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Fig. 8.8: Classification of wind turbines according to location of the wind turbine

Fig. 8.9104: Classification of wind turbines according to generator types and operating concepts 104

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Table 8.1: WTG configurations and their comparison Configuration

Gen. type

Conv. size

Advantages

Disadvantages

Fixed speed operating concept (geared)

* No slip-rings/brushes * Simple and robust design * Low cost * High reliability and durability

NA

SCIG

* Gearbox related problems * Low aerodynamic efficiency * High mechanical loading * High electric power pulsations * No reactive power capability

Semi-variable speed operating concept (geared)

Partialscale

WRIG

* Improved speed range * Reduced mechanical stresses * Reduced electric power pulsations * Improved aerodynamic efficiency

* Slip-rings/brushes related problems * Limited speed (above the synchronous speed) * Higher power losses in the rotor circuit * Gearbox related problems

Variable speed geared concept

BDFIG

Partial-scale

DFIG

* Control of active and reactive power * Improved aerodynamic and overall efficiency

* Limited speed range depending on the converter size * Slip-rings/brushes related problems in partial-scale converter configurations * Gearbox related problems

* Control of active and reactive power * Improved aerodynamic and overall efficiency * Full speed range * No brushes

* Gearbox related problems

Full-scale

BDFRIG

PMSG

SCIG

EESG

Full-scale

Variable speed gearless conecpt

* Control of active and reactive power * Improved aerodynamic efficiency * Full speed range

* Relatively complex control

PMSG

FSWT LVSWT VSWT PSC FSC SCIG DFIG

Fixed Speed Wind Turbine. Limited Variable Speed Wind Turbine. Variable Speed Wind Turbine. Partial-Scale Converter. Full-Scale Converter. Squirrel-Cage Induction Generator. Doubly-Fed Induction Generator.

WRIG

Wound-Rotor Induction Generator.

DC BDFIG BDFRIG PMSG

Direct Current Generator. Brushless DFIG. Brushless Double-Fed Reluctance Induction Generator. Permanent Magnet Synchronous Generator.

EESG

Externally Excited Synchronous Generator.

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Fig. 8.10 illustrates a typical configuration and main components of geared HAWT105,106.

Fig. 8.10: A typical configuration and main components of geared HAWT

8.3 Modeling of Wind Turbine Generators (WTGs) The list of symbols used in the modeling of the considered WTGs is shown in Table 8.2. The models of various system components are summarized here and they are based on107,108,109,110,111,112,113.

105

EL-Shimy, M. (2010). Alternative configurations for induction-generator based geared wind turbine systems for reliability and availability improvement. Proc. of IEEE 14th international middle-east power conference MEPCON2010. Cairo, Egypt.(pp. 617-623). IEEE 106 Ciang, C. C.; Lee, J. R.; Bang, H. J. (2008). Structural health monitoring for a wind turbine system: a review of damage detection methods. Measurement Science and Technology, 19(12), 122001 107 Milano, F. (2010). Power system modelling and scripting . Springer. 108 Milano, F. (2008). Documentation for PSAT version 2.0.0, 2008. 109 EL-Shimy, M. (2010) Steady State Modeling and Analysis of DFIG for Variable-Speed Variable- Pitch Wind Turbines. Ain Shams Journal of Electrical Engineering, 1, 179 - 89 110 Slootweg, J. G.; Kling, W. L. (2002). Modelling and analysing impacts of wind power on transient stability of power systems. Wind Engineering, 26(1), 3-20 315

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Table 8.2: List of Symbols

GB

Pitch control gain Number of blades Number of poles Power rating (MVA) Pitch control time constant (s) Number of machines that compose the wind park Gear box ratio Air density (kg/m3)

Stator resistance (p.u) Rotor resistance (p.u) Rotor leakage reactance (p.u) Magnetizing reactance (p.u) Stator leakage reactance (p.u)

p

Rated frequency (rad/s)

cp

Stator reactance =

Ar

Rotor reactance = Blade pitch angle (deg) Turbine performance coefficient or power coefficient Tip speed ratio Area swept by the blades (m2)

vbt

Blade tip speed (m/s)

Electrical torque (p.u)

v

Mechanical torque (p.u)

P

Wind upstream the rotor i.e. The wind speed (m/s) The mechanical power extracted from the wind

Wind turbine angular speed (p.u) Generator rotor speed (p.u) Relative angle displacement of the two shaft (rad)

Machine rotor inertia constant (MWs/MVA) Wind turbine inertia constant (MWs/MVA) Shaft stiffness (p.u)



Rated output mechanical power uc

Cut-in wind speed (m/s)

ur

Rated wind speed (m/s) Furling or cut-out wind speed (m/s) Shunt capacitor conductance Converter voltage Generator reactive power Rotor reactive power on the rotor side Convertor reactive power on the grid side Stator ( ) and rotor ( ) currents

Shadow effect factor

uf

Stator flux Converter current Generator active power Rotor active power on the rotor side Convertor active power on the grid side Stator ( ) and rotor ( ) voltages Voltage behind transient reactance d_q components

bc

111

Voltage at generator terminals

El-Shimy, M.; Badr, M. A. L.; Rassem, O. M. (2008, March). Impact of large scale wind power on power system stability. In 12th International Middle-East Power System Conference MEPCON 2008. Aswan, Egypt. (pp. 630-636). IEEE 112 EL-Shimy, M. (2013). Probable power production in optimally matched wind turbine generators. Sustainable Energy Technologies and Assessments, 2, 55-66 113 Ghaly, N.; EL-Shimy, M.; Abdelhamed, M. Parametric study for stability analysis of grid-connected wind energy conversion technologies. In 15th International Middle-East Power System Conference MEPCON 2012. Alexandria, Egypt. (pp. 1-7). IEEE 316

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8.3.1 Wind turbine model and its controls

The modeling of the pure mechanical items of the wind turbine is independent of the generator configuration. Fig. 8.11 illustrates a general schematic diagram of a geared grid-connected WECS. The output mechanical of the WTG can be mainly controlled by either stall-control or pitch-control of the turbine blades.

Fig. 8.11: Generic Grid-connected geared WECS In the stall controlled turbines, the turbine blades are firmly bolted to the turbine hub and the rotor speed is adjusted using a gearbox attached between the turbine and generator shaft while the pitch angle is fixed at a value that results in limiting the power output at high wind speed by blades stalling. In pitch control, the stall angle of the blades is controlled via pitch angle variations to limit the power output to a set value. Stall controlled turbines are mainly implemented in the FSWT concept; however, the current technological standard utilizes pitch-control for both FSWT and VSWT concepts. The stall controlled turbines are not currently common due its problems such as gearbox wear and blade design complications. The considered model here is for pitch-controlled wind turbines which are currently dominating the wind turbine industry. This is because this model is adequate for the speed control of recent wind turbines. In this model, turbine blades can rotate in order to reduce the rotor speed and output power in case of high wind speeds. The angular position of the blades is called pitch angle. The mechanical power

extracted from the wind is a function of the 317

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wind speed power

, the rotor speed

and the pitch angle

. The mechanical

model can be approximated by

The tip speed ratio is the ratio between the blade tip speed wind upstream the rotor . The tip speed ratio is defined by

and the

(a)

(b) Fig. 8.12: Wind turbine; (a) Power curve and operational regions of a pitch controlled wind turbine; (b) Pitch angle control scheme

The

curve is popularly approximated as follows

where 318

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The power curve and the operational regions for a typical pitch-controlled wind turbine are shown in Fig. 8.12(a). As shown, there are four operational regions each of which has its own characteristics and limitations. In region 1, the wind speeds are insufficient to run the turbine. Therefore, the output power is zero. Region 2 starts when the wind speed exceeds the cut-in value and extends up to the rated wind speed at which the rated output power is produced. In this region, maximization of the power extraction is the main concern. Therefore, the pitch angle is kept zero. This is achieved by an antiwindup limiter in the pitch angle controller. In the third region, the rotor speed and consequently the power extraction is limited to the rated power to avoid overloading the turbine. This is achieved on the turbine level by using the pitch angle control which adjusts the pitch angle to an appropriate controlled value. In region 4 where the wind speeds reach dangerous limits, the turbine is taken out of service for protection against mechanical damage. A schematic diagram of a pitch-angle controller is shown in Fig. 8.12(b) and it can be described mathematically by

where  is a function which allows varying the pitch angle set point only when the difference

exceeds a predefined value

depending on the operating concept of the WTG. In the Fixed-Speed Wind Turbine (FSWT) concept the turbine drive a Squirrel-Cage Induction Generator (SCIG) that is directly connected to the grid expect during start-up where a soft-starter is used to minimize the startup

stresses on the system and the grid. In this operating concept,  should not exceed 1% or 2% above the synchronous speed and the speed is kept constant within this range. In the Variable-Speed Wind-Turbine (VSWT) concept, the allowable speed range is high. For example it is 30 around the synchronous speed in DFIG-based systems. 8.3.2 Shaft model

The consideration mechanical dynamics of the shaft depends on the operational concept of the WTG. In FSWT concept, the shaft dynamics should be considered due to its significant impact on the WTG performance 319

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including power and voltage fluctuations. Therefore, popularly a two mass model is adopted to represent the shaft in the FSWT concept. In the VSWT concept, the available controls minimize or prevent the impact of the shaft dynamics. Therefore, usually the shaft dynamics are not considered in the modeling of the WTG and one mass model is adopted. In the two-mass model (Fig. 8.13), the shaft dynamics are represented by,

where is the displacement of the two shafts, is the electrical torque, and is the mechanical torque which can be represented by

Fig. 8.13: Two-mass representation of WTG shaft

A periodic torque pulsation can be added to Tt to simulate the tower

shadow effect. The shadow-effect frequency depends on the rotor speed t, the gear box ratio GB, and the number of blades nblade. The torque pulsation model is

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In the VSWT concept, the Voltage Source Converter (VSC) controls can effectively damp shadow effect modes and shaft oscillations. If the control is efficient enough, the shaft can be considered rigid, i.e., = and can be modeled using,

8.3.3 Modeling of Generators, and their controls A. SCIG model and control

The equivalent circuit of a SCIG (with a single-cage) is shown in Fig. 8.14. In comparison with a conventional induction motor, the induction generator currents are positive if injected into the network. The equations are formulated in terms of the real (d) and imaginary (q) axes, with respect to the network reference angle. Considering the third-order model of the machine, the equations of the machine are as follows,

(a)

(b) Fig. 8.14: SCIG; (a) Equivalent circuit of a SCIG; (b) dq-coordinates

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The link between the voltages, currents, and state variables is modeled as follows,

where and

is the stator reactance, . Note that

, is the rotor reactance

In the synchronously rotating reference frame, the link between the network and the stator voltages of the machine is as follows,

The active and reactive power flow can be determined by

where is the fixed capacitor conductance which is connected to the generator terminals. The value of this capacitor can be determined at the initialization step to impose the desired initial bus voltage level. In the FSWT concept, the only available control is the pitch angle control and no additional control is provided to the system. The impact of the pitch-controller SCIG-based FSWT is illustrated in Fig. 8.15. With zero pitch-angle, the turbines produce its maximum power at a wind speed of 10 m/s while the output power is reduced to 0.725 (i.e. 27.5% reduction) when the pitch-angle becomes five degrees. The characteristics shown in the figure 322

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are obtained by simulating equation (1) using MATLAB SimPowerSystems Toolbox.

(a) (b) Fig. 8.15: Impact of pitch-angle control on the output power of a FSWT: (a) Power-speed characteristics at zero pitch-angle; (b) Power-angle characteristics at a pitch-angle of five degrees B. DFIG model and control

Fig. 8.16 shows the basic building blocks of a DFIG-based VSWT. The system consists of a wound-rotor induction machine controlled by two backto-back connected PWM converters with a controllable DC intermediate link scheme. The first converter is an AC-DC PWM-converter called the rotorside-converter (RSC) connected between the rotor AC-windings (via slip rings and brushes) and the DC-link. The second converter is a DC-AC PWMconverter called the grid-side-converter (GSC) connected between the ACgrid (i.e the DFIG stator windings) and the DC-link. A wind turbine (WT), with energy-control via the pitch-angle controller, is coupled to the generator shaft via a gearbox. The RSC controller provides control of both the WT output power and the DFIG reactive power output or terminal voltage or power factor. The main function of the GSC control is to regulate the DClink voltage as well as possible other control functions such as the generation or absorption of reactive power. In power control, the turbine output is controlled in order to follow a pre-defined power-speed characteristic corresponding to the maximum wind-energy-capture tracking called the maximum power tracking characteristics (MPTC). This achieved through integration between the RSC and the pitch-angle controllers. 323

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Fig. 8.16: Basic building blocks of a DFIG-based VSWT

Fig. 8.17: Speed-power characteristic of VSWT wind turbine at zero pitch-angle

Fig. 8.18 Equivalent circuit of a DFIG 324

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The speed control is aimed to maximize the power production of the wind turbine. Fig. 8.17 shows the dependence of the mechanical power produced by the wind turbine on the wind speed and the turbine rotor speed . In addition, the maximum mechanical power locus for each wind and rotor speeds is illustrated on the figure. This curve is used for defining, for each value of the rotor speed, the optimal mechanical power that the turbine has to produce. For super-synchronous speeds, the reference power is fixed to 1.0 p.u to avoid overloading the generator. For < 0.5 pu, the reference mechanical power is set to zero. For , the detailed ( , ) characteristic is,

This is can be simplified to Due to the fast dynamics of the stator flux comparison with the grid dynamics, the electromagnetic equations of the DFIG are usually represented by a steady-state model. The equivalent circuit of a DFIG is shown in Fig. 8.18. The three – phase stator and rotor windings of an induction machine can be represented by two sets of orthogonal fictitious coils. The DFIG is controlled in a rotating d-q reference frame, with the d-axis aligned along the stator-flux vector. The machine stator and rotor voltages are functions of the stator and rotor currents as well as the rotor speed . The d-q representation of the machine is as follows,

The links between stator fluxes and generator currents are modeled using 325

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The network interface is modeled as follows,

The generator active and reactive power productions depend on the stator currents ( voltages (

), the converter currents ), and the converter voltages

, the stator as follows,

The above expressions can be rewritten as a function of stator and rotor currents and stator and rotor voltages. In fact, the converter powers on the grid side are represented by (8.31) and (8.32) while the converter powers on the rotor side are represented by (8.33) and (8.34).

Assuming a loss-less converter model, the active power of the converter coincides with the rotor active power, thus = . The reactive power injected into the grid can be approximated by neglecting stator 326

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resistance and assuming that the d-axis coincides with the maximum of the stator flux. Therefore, the powers injected in the grid can be simply represented by,

C. Steady-state performance

It is clear from Fig. 8.14 and 8.14, that the model of the DFIG includes the model of the SCIG. If the rotor circuit of a WRIG is shorted, the machine becomes conceptually similar to a SCIG. Therefore, the following analysis is given for the DFIG as it is the general case. The 3rd order fundamental frequency model for the DFIG with flux oriented control scheme is considered here for extraction of the steady state performance characteristics 114.

Fig. 8.19 dq-representation of DFIG The 3rd-order model is based on the following assumptions. (1) Neglecting magnetic saturation; (2) Single-mass representation of all rotating masses; (3) Stator electromagnetic transients as well as stator resistance are neglected; (4) 114

EL-Shimy M. Steady State Modeling and Analysis of DFIG for Variable-Speed Variable- Pitch Wind Turbines. ASJEE. 2010;1:179 - 89 327

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Generator current and power convention is used; (5) Synchronous reference

frame. In addition, for stator flux-oriented control scheme, the stator flux s

is aligned to the q-axis (i.e. = 0) as shown in Fig. 8.19. This is providing decoupling between active and reactive power output control. Based on the above assumptions and model requirements, the DFIG 3 rd-order model is as follows.

where

is the equivalent transient reactance of the machine

and s is the slip of the rotor.

Based on equations (8.41) and (8.42), it is noticed that stator active and reactive power equations are decoupled due to the stator flux oriented-control scheme.

It is assumed that the GSC operates at unity power factor for all operating conditions i.e. Q r = 0. In addition, it is assumed that both 328

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mechanical losses and converter losses are neglected. The rotor active power is also neglected under steady-state conditions. Therefore,

The stator active and reactive power represented by equations (8.41) and (8.42) can be obtained in terms of the rotor voltage d- and q-components by solving, under steady-state conditions, equations (8.39) and (8.40) for the rotor current d- and q-components. These are results in the following equations,

Therefore, by substituting (8.47) in (8.41) and (8.48) in (8.42), the stator active and reactive power characteristics are obtained and take the form,

For assessing the steady-state stability, based on the DFIG P s -s characteristics, for a given operating point on the P s-s characteristics of an 329

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induction machine the dP s /ds criteria can be used. Since generator power convention is used in the machine modeling, a steady-state stable operating point requires that dP s /ds < 0. Based on (8.49), dP s /ds take the form,

where

Since we are dealing with DFIG characteristics control via rotor injection, it is valuable to determine the values of the slip at zero output power (sR) as well as the generator-mode critical slip scr _gen. The former is obtained by setting P s = 0 in (8.49), and the later is obtained by setting dP s /ds = 0 in (8.51). Two solutions for scr are obtained, where the value of scr corresponding to positive P s is the generating-mode critical slip scr_gen and the other (corresponding to negative stator power) is the motoring-mode critical slip scr_mot. The value of sR and scr _gen can be computed using,

where

,

,

The control strategy of DFIG system includes controlling the active and reactive power injections from the generator. For active power control, optimum operating point tracking of the VSWT is considered to provide maximum energy capture from wind. The objective of tracking control is to keep the turbine on this optimum tracking curve as the wind speed varies. 330

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A general optimal tracking strategy for DFIG system is shown in Fig. 8.20 and can be represented by the following equation,

for for for for for for

Based on Fig. 8.20, the function of the pitch-angle control is to limit the turbine output energy at higher than rated wind speeds through variable-pitch operation. For lower than rated wind speeds the wind energy capture is maximized through fixed-pitch operation with pitch-angle set to zero degrees.

Fig. 8.20 Optimal tracking strategy for DFIG system Two methods are used, for DFIG converters, to control output realpower production of the DFIG according to the optimal tracking characteristics 115. These are (i) current-mode control, and (ii) speed-mode control. In current-mode control, that can be considered as a standard tracking mode, the electrical real-power output from the DFIG is controlled 115

R. Pena, J.C.Clare, and G. M. As her, Doubly fed induction generator using back-toback PWM converters and its application to variable-speed wind-energy generation, IEE Proc.-Electr. Power Appl, Vo1 143, No. 3, May 1996, pp. 231-24 331

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according to the generator rotor speed. In speed-mode control, the generator rotor speed is controlled according to a desired electrical real-power output allowing extra merits over current-mode control in allowing flexible dispatchable-like operation of the DFIG system; however, the available wind resource defines the maximum available generation. In the current-control mode as illustrated in Fig. 8.16, the RSC is used to set the appropriate electrical active power output reference of the DFIG to follow the optimal tracking characteristics shown in Fig. 8.20 according to equation (8.41). In addition, the RSC can also be used to regulate the reactive power output from the DFIG according to equation (8.42). The steady-state rotor circuit d-q current components required for active- and reactive- power control are obtained directly from equations (8.49) and (8.50). However, the steady-state d-q voltage components impressed on the DFIG rotor circuits for active- and reactive- power control can be obtained for a given rotor speed and demanded reactive power output through simultaneous solution of equations (8.49) and (8.50) where the DFIG active power output is calculated from the tracking characteristics of Fig. 8.20. These d-q voltage components take the form,

where

8.4 Case study 1 - Steady-state characteristics and steady-state stability of DFIG Consider DFIG of 2MW, 690V, 50Hz. The DFIG p.u parameters are listed in Table 8.3, which also include definitions of DFIG parameters and symbols.

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Table 8.3 DFIG parameters Parameter Symbol Value, p.u Stator resistance Rs 0.00488 Stator leakage reactance Xs 0.09241 Rotor resistance Rr 0.00549 Rotor leakage reactance Xr 0.09955 Magnetizing reactance Xm 3.95279 Inertia constant (lumped) H 3.5, sec.

A 2MW variable-pitch variable-speed wind turbine with characteristics shown in Fig. 8.21 (obtained using the VSWT aerodynamic model available within SimPowerSystems Toolbox). The turbine tracking characteristics data are given in Table 8.4.

Fig. 8.21 VSWT power characteristics at zero-pitch angle and optimal tracking characteristics Fig. 8.21 Tracking characteristics data Parameter Rotor speed at point A Rotor speed at point B Rotor speed at point C Rotor speed at point D Output Power at point C Output power at point D

333

Symbol Value, p.u 0.7 r A

r B r C r D PC PD

0.71 1.2 1.21 0.73 1

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In the following steady-state simulation, the DFIG terminal voltage is assumed to be kept constant at 1 p.u which can be realized practically when the DFIG is connected to an infinite-bus through a network with infinite strength. A. Operation over full tracking characteristics

With generator-rotor speed varies from 0.6 p.u to 1.3 p.u, the demanded active power output from the DFIG is obtained according to the optimal tracking characteristics shown in Fig. 8.21 and the DFIG reactive power output is set to zero. The resulting rotor voltage and current components are shown in Fig. 8.22 while Fig. 8.23focuses on the rotor voltage components only.

Fig. 8.22 Operation over full tracking characteristics with Qs = 0; rotor current and voltage components As depicted from equations (8.41) and (8.42) and Fig. 8.22 and Fig. 8.23(b), the resulting rotor current components that are decoupled from each other are linearly related with the demanded active and reactive power for all rotor speeds. Based on Fig. 8.23(a), for rotor speed below 0.7 p.u where both active and reactive power demands are zero, the Vqr is of a fixed negative value independent on the rotor speed while Vdr is linearly related and decreasing with the rotor speed. The described changes of Vdr are also applicable for the rotor speed range with positive values of slip at sub 334

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synchronous speeds and negative value at supersynchronous speeds. The sign of Vqr is negative for speeds up to 1.13 p.u and positive for supersynchronous speeds above 1.13 p.u while its value is zero at the synchronous speed. Although Vqr is of a fixed value independent on the rotor speed for demanded zero output active power, its value is increasing with the rotor speed above 1.2 p.u with fixed demanded output active power of 1.0 p.u. This is shown as steady-state instability as depicted from Fig. 8.24 that shows dP s/ds values for the entire tracking characteristics under different demanded reactive power output settings. Therefore, it can be concluded that positive sign of Vqr indicates steady-state instability; however, these results are highly dependent on the considered parameters and may significantly vary for different parameters. Since dP s/ds is positive for rotor speeds above 1.13 p.u (for Q s = 0), the DFIG is unstable in this region. This is occurring with different values of reactive power output demanded at rotor speeds higher than 1.13 p.u. The causes of the steady-state instability for rotor speeds greater than 1.13 p.u can be explained by calculating the DFIG maximum steady-state power, critical generator slip, and the slip at zero power output as shown in Fig. 8.25 with the instability zone focused in Fig. 8.26. Based on Fig. 8.24, Fig. 8.25 and Fig. 8.26 for stable operating points where dP s/ds < 0, the demanded rotor speeds and active power demands are less than the critical generator speed and the maximum generator power respectively. In addition, for rotor speeds between 1.13 p.u and 1.2 p.u, the demanded active power output is higher than the maximum generator output. In addition, for all the stable operating points, the demanded rotor speeds are higher than the rotor speed at zero active power output. The steady-state unstable operating points are realized by positive dP s/ds and positive Vqr . These points are characterized by violating the critical generator speed or the maximum generator power or both as depicted from Fig. 8.23 and Fig. 8.26. The considered reactive power output demands do not significantly affect the steady-state stability of the DFIG as shown in Fig. 8.24. Therefore, the

considered DFIG should not run at supersynchronous speeds ( r max shown in Fig. 8.20) higher than 1.13 p.u in order to achieve stable steady-state operation. In addition, positive sign of rotor voltage d-component can be considered as an indicator of DFIG steady-state instability and injection of positive Vqr should be prohibited to avoid hindering steady-state stability. 335

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(a) (b) Fig. 8.23 Operation over full tracking characteristics with Qs = 0; (a) Rotor voltage components; (b) Rotor currents and powers

Fig. 8.24 Operation over full tracking characteristics with different Qs; Steady-state stability

Fig. 8.25 Operation over full tracking characteristics with Qs = 0 - Causes of steady-state instability

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Fig. 8.26 Operation over full tracking characteristics with Qs = 0 - Focus on the steady-state instability region B. Solo effects of rotor voltage components on DF IM performance

The solo effects of each of the d- and q- components of the rotor voltage on the steady-state performance and steady-state stability of the doubly-fed induction machine (DFIM) are analyzed based on the assumption of unity terminal voltage. The analysis considers the motor as well as generator operation modes. The resulting steady-state responses are plotted for a limited range of rotor speeds (slips) for better visualization. However, due to the careful selection of the scale range, this will not affect the derived conclusions. With Vqr = 0, the effects of variations of Vdr on the DFIM Ps-s characteristics, steady-state stability, and critical limits are shown in Fig. 8.27-a, Fig. 8.27-b, and Fig. 8.27-c respectively. The following are depicted from Fig. 8.27. With positive Vdr , the DFIM Ps-s characteristics shifted to right (Fig. 8.27-a) and a significant extension of the steady-state stability region, with respect to zero rotor voltage case, occurs in subsynchronous speeds as positive Vdr increased. In addition, a significant reduction of the stability region in supersynchronous speeds occurs as positive Vdr increased. In addition, a significant linear increase of the maximum power in the generating mode as positive Vdr increased; this is in contrary to the motoring mode maximum power, which is very close to zero with all positive values of Vdr . Therefore, with positive Vdr , wide range of subsynchronous speeds are available for steady-state stable operation of the DFIM. However, due to the significant increase in generating-mode maximum power and the significant 337

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reduction in motoring-mode maximum power, positive Vdr improves the generating-mode of the DFIM for subsynchronous speeds. The opposite is obtained for negative Vdr . Therefore, with negative Vdr , wide range of supersynchronous speeds is available for steady-state stable operation of the DFIM. However, due to the significant increase in motoring-mode maximum power and the significant reduction in generating-mode maximum power, negative Vdr improves the motoring-mode of the DFIM for supersynchronous speeds. Generally, linear increase of the slip at zero power output is obtained with increasing Vdr . Fig. 8.28 shows the solo effect of Vqr with Vdr = 0. Based on the figure, neither positive nor negative Vqr affect either the critical slips of the DFIM or the slip at zero power output. With positive Vqr , the steady-state stability region of rotor slips is normally within the critical slip values corresponding to generating and motoring modes of operation; however, with negative Vqr the steady-state stability region of rotor slips is inverted. With positive Vqr , the steady-state stability of both generating and motoring modes of operation are enhanced via increase in the maximum power; however, this is valid for subsynchronous rotor speeds in the generating mode and supersynchronous speeds in the motoring mode. With negative Vqr , enhancement in the steady-state stability through increase in the output power occurs for supersynchronous speeds higher than 1.03 p.u in the generating mode and for subsynchronous speeds less than 0.97 p.u in the motoring mode. Within the mentioned speed range the machine is unstable in either modes of operation. It is clear from Fig. 8.28-c that an approximately linear transitions between the stable and unstable regions of rotor speeds occur as Vqr is reduced from 0.05 p.u to -0.05 p.u.

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(a)

(b)

(c) Fig. 8.27 Solo effects of Vdr on DFIM performance and stability; Vqr = 0 339

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(a)

(b)

(c) Fig. 8.28 Solo effects of Vqr on DFIM performance and stability; with Vdr =0 340

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The above analysis of the solo effects of rotor voltage components on the steady-state performance and steady-state stability of DFIM show that infinite number of machine characteristics can be obtained through mix selection of Vqr and Vdr even with limited amplitude of rotor voltage Vr . Each of these characteristics has its own merits and steady-state stability limits.

8.5 Case study 2 - Transient performance of FSWT based on SCIG considering the single-mass model (rigid rotor) This case study presents a detailed analysis of the transient response considering two control situations of a pitch controlled wind turbine (see Fig. 8.30). In the first situation, the pitch angle control loop is working properly while in the second situation the pitch controller is stalled due to an internal failure. The first situation will be called Active Stall Control (ASC) while the second situation will be called Stalled (or disconnected) Controller (SC). The failure may be a disconnection in the closed loop pitch-control system or problems in the hydraulic pitch changer. The objective is to demonstrate the dangers associated with losing the only available controller which is responsible of limiting the output power of the turbine during abnormal conditions. These abnormal conditions include high wind speed in comparison with the rated wind speed or a network fault. The single-mass model will be utilized in this example for simplification; however, this is only applicable to rigid shafts while the usual shafts of FSWTs are soft and the multi-mass model is recommended for accurate analysis. A detailed comparison between the stated situations is conducted. Two types of disturbances are considered. The first is an intense change in the wind speed and the second is a three-phase fault at the induction generator terminals. A. Study system and modeling requirements

The study system of Fig. 8.29 is implemented on SimPowerSystems based Matlab Toolbox. The low-speed shaft dynamics and the pitch-angle rate limiter are neglected. This is to allow investigating the FSWT system dynamics under design improvement in both the low-speed shaft material stiffness and the rate of change of pitch angle capability. High rates of pitch341

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angle change may be obtained via fast acting hydraulic/mechanical actuator designs that capable of handling high speed change in pitch angle variations either using PI controllers or neural network based controllers 116. From technical and operational points of view, appropriate delay-logic should be included in the pitch-angle control loop. This delay may be optimized based on the existing magnitude and rate of change of the wind speed considering also a statistical survey analysis of wind speed patterns. This is to avoid excessive unnecessary operation, and wear out of the pitch-angle control system. The basic building blocks needed to simulate the study system of Fig. 8.29 are shown in Fig. 8.30.

Fig. 8.29: Study System

Fig. 8.30: Basic building blocks of study system model

M. EL-Shimy, “ Modeling and Control of Wind Turbines Including Aerodynamics”, Scientific Bulletin - Faculty of Engineering - Ain Shams Uni. Vol. 41, No. 2, June 30, 2006 116

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The equations describing each of the shown components are already stated

in the previous sections. For stalled controller, the pitch angle  is fixed to zero regardless of the available wind speed. This concept is used for implementing the model in the SimPowerSystems toolbox. In the SimPowerSystems library, the induction generator is represented by 5 th order dynamic model. Therefore, stator dynamics as well as network dynamics are included in the results. In addition, a pitch angle controller of a PI configuration is also included in this library. The system model initialization is conducted by simulating the undisturbed system and then appropriate initial conditions are recorded in the initial conditions slot of each block. This method shows better model initialization to steady-state conditions in comparison with the model initialization method described in the toolbox manual. The data of the wind turbine and the pitch-angle controller are shown in Table 8.22. The turbine power characteristics at zero and at five degrees pitch-angles are plotted and they are shown in Fig. 8.31 and Fig. 8.32 respectively. Table 8.22: Wind turbine data For both stall and active stall controlled turbine Nominal Mechanical Power (MW) Base wind speed* (m/s) Maximum power at base wind speed (pu of nominal mechanical power) Base rotational speed (pu of base generator speed) For Active stall controlled turbine Pitch angle controller gain, kp Pitch angle controller gain, ki Maximum pitch angle (deg)

1.5 10 1 1 5 25 40

*The base wind speed is the mean value of the expected wind speed.

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Fig. 8.31: Turbine power characteristics at zero pitch angle

Fig. 8.32: Turbine power characteristics at five degrees pitch angle The controlled stalling effect using pitch-angle control is depicted from Fig. 8.31 and Fig. 8.32. Based on Fig. 8.31, with zero pitch-angle the nominal mechanical of the turbine occurs at the base wind speed, which obeys the turbine data. However, based on Fig. 8.32, with five degrees pitch angle the output power is limited due to blades stalling results in 0.725 pu mechanical power output at the base wind speed. Induction generator data are shown in Table2 with pu values are based on generator ratings. The p.u data of the system based on the turbine rating are shown in Table 8.23 and Table 8.24. B. Simulation results

For showing the capability of the considered FSWT systems under large disturbances originates from the nature or from the system faults, two types of disturbances are considered. The first is an intense change in the wind 344

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speed and the second is a three-phase fault at the induction generator terminals. Table 8.23: Induction generator data Nominal Power (MVA) 1.6667 Nominal L-L rms voltage (V) 575 Nominal frequency (Hz) 60 Stator resistance (pu) 0.01 Rotor resistance (pu) 0.014 Stator leakage reactance (pu) 0.1 Rotor leakage reactance (pu) 0.098 Magnetizing reactance (pu) 3.5 Inertia constant including the wind turbine (pu) 3 Number of pole 6

Table 8.24: Network data Transformer 1 2 Nominal power (MVA) 2 47 Nominal frequency (Hz) 60 60 Turns ratio 25e3/575 120/25 Resistance (pu) 0.0017 0.0053 Reactance (pu) 0.05 0.160 Transmission Line 0.0825 Resistance () Reactance ()

0.1037

Grid Nominal voltage (kV) 120 nominal frequency (Hz) 60 S.C level (MVA) 2500 S.C ratio 10 Compensating capacitor Nominal voltage (V) 575 Rated capacity (kVAr) 700

B.1. Wind Speed Disturbance

Starting at the base wind speed (i.e. the mean value of the expected wind speed = 1 pu), 60% step increase in the wind speed occurs at t = 1 second and sustained at this level for two seconds till this change in wind speed is vanished in step manner at t = 3 second. The selected base wind speed is 10 m/s, which is also corresponding the maximum mechanical power 345

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output. The considered wind speed disturbance is shown in Fig. 8.33. This disturbance is applied to the considered situations i.e. the stalled controller (SC) and the active stall control (ASC). The corresponding response of the mechanical torque (Tm) and the electrical torque (Te) for both situations are shown in Fig. 8.34.

Fig. 8.33: Wind speed disturbance

Fig. 8.34: Torque response for the wind speed disturbance Based on Fig. 8.34, the mechanical torque and electrical torque of the SC turbine nearly follow the wind speed pattern. Therefore, upon turbine/generator protection failure, this may be representing a hazard to the system security during high wind speeds. However, with ASC, a significant and continuous drop in the torque is obtained resulting in lower system stress during disturbance. This results from the pitch angle control that limits the 346

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generator output power to its nominal value for high wind speeds. The pitchangle variation is shown in Fig. 8.35. The maximum rate of the required change in the pitch angle is about 9.0 deg/s.

Fig. 8.35: Pitch-angle variation for the wind speed disturbance The generator active/reactive power response and the generator terminal voltage response for the wind speed disturbance of Fig. 8.33 are shown in Fig. 8.36 and Fig. 8.37 respectively.

Fig. 8.36: Generator active/reactive power response for the wind speed disturbance Based on Fig. 8.36, a better response of active power generation is obtained as a result of the pitch-angle control. With the controller stalled, the violation of the rated power of the turbine is sustained during the considered wind speed disturbance while the ASC results in smaller violations that are decreasing with time. In addition, with pitch-angle control, the generator bus 347

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reactive power demand is greatly reduced resulting in lower generator terminal voltage fluctuations (Fig. 8.37) during high wind speed disturbance. Therefore, better power quality is obtained and less effect on local sensitive load equipment is resulted from the ASC. If the acceptable steady state

terminal voltage magnitude is 5%, then with the SC, the generator is subjected to an unacceptable drop in its terminal voltage while the situation is absent with the ASC.

Fig. 8.37: Generator terminal voltage response for the wind speed disturbance

Fig. 8.38: Generator slip response for the wind speed disturbance The generator slip during wind speed disturbance is shown in Fig. 8.38. It is shown that in either wind turbine control conditions, the generator is stable and did not temporarily operate as motor during the considered wind speed 348

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disturbance; however, a better response is obtained with the ASC. Better generator stability is obtained with ASC as depicted from the torque-slip trajectory of Fig. 8.39.

Fig. 8.39: Torque-slip trajectory for wind speed disturbance B.2. Three-Phase Fault at Generator Terminals

The analysis the considered FSWT systems during severe fault disturbances is considered in this section. A 5-cycle, 3-phase (LLLG) fault started at t = 1.0 sec is applied at the generator terminals and the criticalclearing time (CCT) is determined. The results show that CCT is 6-cycle (0.1 second) for both SC and ASC conditions. Unlike the considered wind speed disturbance, the fault will not cause an increase in the generated power during the fault conditions. Therefore, the impact of the pitch angle control did not reflect on the CCT. The generator terminal voltage response for both stall and active stall controlled turbine systems is nearly identical and is shown in Fig. 8.40. The mechanical torque response comparison and pitch-angle response of the ASC are shown in Fig. 8.41 and Fig. 8.42 respectively.

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Fig. 8.40: Generator terminal voltage response for the 3-phase fault

Fig. 8.41: Mechanical torque response for the 3-phase fault

Fig. 8.42: Pitch-angle response for the 3-phase fault Based on Fig. 8.41, a significant reduction in the turbine mechanical torque output is resulted with the ASC due to pitch-angle control as depicted 350

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from Fig. 8.4; however, this significant reduction in the mechanical torque due to the fault will not be reflected significantly on other system variables especially the active/reactive power generated as will be illustrated below. Therefore, it is clear that the generator and the network mainly govern the electromechanical transient process and, to high extent, independent on the turbine control. This is depicted from the generator torque-slip trajectory shown in Fig. 8.43; however, the benefit of the input mechanical torque reduction due to pitch-angle control is a consequent significant reduction of the torsional stress applied to the shaft during the transient process.

Fig. 8.43: Torque-slip trajectory for the 3-phase fault

Fig. 8.44: Electrical torque response for the 3-phase fault

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The responses of the electrical torque, active power generated, and reactive power generated, and generator slip are shown in Fig. 8.44 to Fig. 8.47 respectively.

Fig. 8.45: Active power response for the 3-phase fault Based on Fig. 8.43 to Fig. 8.47, insignificant gain in the overall system response is obtained with the ASC for the considered 3-phase fault at the generator terminals; however, both types of control results in stable FSWT system without the complications associated with RSC and the voltage recovery following a close fault associated with DFIG based systems 117.

Fig. 8.46: Reactive power response for the 3-phase fault Mustafa Kayikçi and J. V. Milanovic´, “Assessing Transient Response of DFIG-Based Wind Plants—The Influence of Model Simplifications and Parameters”, IEEE Transactions On Power Systems, Vol. 23, No. 2, May 2008, pp. 545-554 117

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Fig. 8.47: Generator’s slip response for the 3-phase fault

8.6 Case study 3 – Design of Pitch Angle Controllers118 This section presents a model for a variable-pitch, constant-speed horizontal-axis wind turbines including aerodynamics and mechanical parts. In addition, the pitch angle control design is considered. Two techniques of control are applied; the PID control and ANN-NARMA-L2-based control. The main objective of both techniques is to keep the speed of the turbine constant at a desired level. A. Turbine Description and Preliminary Modeling

The wind turbine rotor is connected to a generator. The generator output can be controlled to follow the commanded voltage. The wind turbine has pitchable blades to control the aerodynamic power extracted from the wind. In addition, there is a mechanical speed-changer (i.e. gearbox) between the low-speed rotor shaft and the high-speed generator shaft. The low-speed shaft is driven by the turbine blades, which generates aerodynamic power. The high-speed shaft is loaded by the electric generator in the form of an electrical load. As the wind speed fluctuates, the wind turbine is controlled by changing the pitch angle to fix the rotor speed following the variation of the wind speed. Therefore, the wind-turbine-generator (WTG) system converts rotational energy to electrical energy, which is may be directly supplied to the

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utility grid at the distribution level or to the HV grid. The choice depends on the sacle of the wind farm and the intended application. A.1. Available Wind Power Modeling

The kinetic energy, U of a packet of wind of mass m flowing at upstream speed u in the axial direction of the wind turbine is given by,

1 1 U  mu 2  ( Ax)u 2 2 2

(8.57)

where A is the cross-sectional (swept) area of the wind turbine, is the air density and x is the thickness of the wind packet. The wind power, P w in the wind, which represents the total power available for extraction, is given by, Pw 

Therefore,

Pw 

dU dt

(8.58)

dx 1 1 Au 2  Au 3 dt 2 2

(8.59)

The mechanical power, P m extracted from the available power in the wind P w is expressed by the turbine power coefficient of performance C P which is a

nonlinear function of tip speed ratio  and pitch angle . Therefore, Pm  C p  ,  Pw

(8.60)

In ideal conditions, the turbine cannot extract more than 59% of the total power of undisturbed tube of air with cross sectional area equals to wind

turbine swept area. This called Betz limit. The tip speed ratio  is a variable that combines effect of rotational speed of the turbine and wind speed. It is defined as the ratio between the rectilinear speed of the turbine tip (R) and the wind speed (u).

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

R

(8.61)

u

where R is the maximum radius of the wind turbine swept area. The following equation can be used to approximate the C P curve, C p ( ,  )  0.22116K  0.4  5e 12.5 K

where

K

1 0.035  3   0.08   1

(8.62)

(8.63)

Variation of CP with tip speed ratio at constant pitch angle and variation of C P with pitch angle at constant tip speed ratio are shown in Fig. 8.48 and Fig. 8.49 respectively. As shown in Fig. 8.48, the maximum value of Cp reduced with the increase in the pitch angle. In addition, the tip-speed ratio at maximum Cp increases with the decrease in the pitch angle. This clarifies the impact of the variations of the pitch angle on the power coefficient which consequently affects the power production. The wind-turbine rotor performance can also be evaluated as function of the coefficient of torque C q. As the wind power P w equals to the product of

the aerodynamic torque TA and the rotor rotational speed , the torque coefficient can be related to the power coefficient by, C p ( ,  )  C q ( ,  )

(8.64)

Based on (8.59), (8.60), (8.61), and (8.64), the aerodynamic torque that turns the rotor shaft takes the form,

1  TA   AR C q ( ,  )u 2 2 

355

(8.65)

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Fig. 8.48: Variations of the power coefficient with the tip-speed ratio at constant pitch-angle

Fig. 8.49: Variations of the power coefficient with the pitch angle at constant tip-speed ratio A.2. Wind Turbine Linearized Modeing for Controller Design

The most special feature about wind turbines is the fact that, unlike other generation systems, the power inflow rate is not controllable. In most conventional power generation systems, the fuel flow rate, or the amount of energy, applied to the generator controls the output voltage and frequency; however, the wind speed varies with time and so does the power demand. Therefore, conventional generation systems can be referred to as controlled energy sources, while the wind is an uncontrolled energy resource . The power demand is an uncontrolled energy sink. Sometimes the wind speed can be very high resulting in power generation that exceeds either the rating of the turbine or the demanded power. This 356

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might lead to the turbine exceeding its rotational speed limit and subsequently the turbine may be damaged. On the other hand, the wind speed can be too low for any power production and therefore alternative energy sources should be used. This forms a hybrid-electric power source. The wind-turbine-generator (WTG) linearized model will now be derived for the use in the pitch-angle control design. The model is divided into two main parts. The first part is the wind turbine, which included a turbine rotor on a low-speed shaft, a gearbox and high-speed shaft. The inputs for this part of the plant are the wind speed and the blade pitch angle while the outputs are the high-speed shaft angular rotation and the mechanical power, Pm. The second part is the electric generator whose input is constant angular rotation from the turbine plant and whose output is electrical power. Fig. 8.50 shows a block diagram of the wind turbine system.

Fig. 8.50: Grid-connected WTG The equation of motion of wind turbine system is given by,

JT

d  T A T L dt

(8.66)

where JT is the equivalent combined moment of inertia of the rotor, gearbox and both the low-speed and high-speed shafts; TL is the wind turbine load torque representing the input torque to the electrical generator and opposed by its electrical torque. For the purpose of dynamic analysis and for designing a linear controller, such as PID controller, equation (8.66) is linearized around an initial

operating point (uo, o, o). Substituting for TA in (8.66) using (8.65), then the linearized form of (8.66) takes the form,

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T d  u      L dt JT

(8.67)

The parameters , and are calculated at the initial operating conditions

(uo, o) and they are given by:



 dC 1 ARuo  2C qo  o q  2J T d 





dC 1 AR2u o q d 2J T

dC 1 ARuo2 q d 2J T

   o

(8.68)

(8.69) o

(8.70) o

The magnitude of  and  respectively show the relative weight of the effect of wind speed and pitch angle on the wind turbine dynamics. In s-space, (8.67) takes the form:

 ( s ) 

1 s 

 T  u ( s )   ( s )  L  JT  

(8.71)

Equation (8.71) represents the linearized form of the wind turbine transfer function. The turbine power output is given by, Pm  TA

(8.72)

The linearized form of output power equation (8.72) takes the form:

Pm   o TA  TAo 

358

(8.73)

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Based on (8.67) and (8.74), the block diagram of wind turbine plant is given in Fig. 8.51.

Fig. 8.51: Linearized model of the WT A.3. Wind Turbine Control A general block diagram for wind turbine pitch-control control system is shown in Fig. 8.52.

Fig. 8.52: General block diagram of the pitch-control system for WTS The transfer function of a hydraulic actuator that changes the blade pitch angle can be represented by first-order transfer function, G A( s ) 

kA  ( s )   c ( s ) s  k A 359

(8.74)

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In this case study two controllers are considered; a PID and a neural network NARMA-L2 controllers (that are available in the MATLAB). PID controllers regulate the error, or difference between the measured input and the desired input (). This error value along with its derivative and integral with respect to time provides a signal to the actuator(s), which affects the controlled plant. The PID controller is a linear, single-input single- output controller limited to three gains. The transfer function of the PID controller is given by,  c ( s ) k GC ( s )   k p  I  kD s (8.75)  ( s ) s The central idea of the NARMA-L2 neurocontroller is to transform nonlinear system dynamics into linear dynamics by canceling the nonlinearities. There are typically two stages involved when using neural networks for control. The first stage is system identification in which a neural network model of the plant to be controlled is developed by training a neural network to represent the forward dynamics of the system. The second stage is the control design in which the neural network plant model is used to design (or train) the controller. B. ANALYSIS AND CONTROL D ESIGN

The mathematical model in previous sections is applied to develop the response of controlled and uncontrolled wind turbine plant. The wind turbine parameters are given in Table 8.24. Table 8.24: Wind turbine parameters Rated Power (kW) 20 Radius (m) 5 2 Drive train inertia (kg.m ) 1270 Gear ratio 11.43 Operation angular speed (rpm) 105 Rated wind speed (m/s) 11.7 Cut-in speed (m/s) 6.5 Furling speed (m/s) 23

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B.1. Uncontrolled Response of Wind Turbine Plant

With initial operating point is (uo = 7 m/s, o = 10.5 rad/s, o = 9 deg.) the

parameters , and  calculated; their values are 0.2071, -0.0668, and – 0.0298 respectively. With uncontrolled wind-turbine plant, the response to a unit step in wind speed, a unit step in pitch angle, and a 10% step increase in load torque are determined and shown in Fig. 8.53, Fig. 8.54, and Fig. 8.55 respectively. These responses are mainly dependent on the rotational inertia of the wind turbine plant, the scaling factors  and and the parameter . The

parameter  represents the wind turbine aerodynamic characteristics and it does not affect the wind turbine plant inputs.

(a) (b) Fig. 8.53: Uncontrolled response to a unit step in the wind speed

(a) (b) Fig. 8.54: Uncontrolled response to a unit step in the pitch angle

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(a) (b) Fig. 8.55: Uncontrolled response to a unit step in the load troque The shown simulation results verify the conceptual correctiness of the model. For example, an increase in the wind speed results in an increase in the mechanical power and the speed while an increase in the pitch angle causes reduction in both the speed and the power. The increase on the load torque has a similar impact to the increase in the pitch angle but with a different time constant. B.2. Controlled Response of Wind Turbine Plant

PID and neural network NARMA-L2 controllers are used to compensate

the wind turbine speed deviations by changing the pitch angle . Based on Routh-analysis of the wind turbine transfer function and trial and error approach, the gains of PID controller are selected to be (k P = 60, kI = 50, and kD = 20). The plant response to a unit step in wind speed, and a 10% step increase in load torque are shown in Fig. 8.56, and Fig. 8.57 respectively. It is shown that PID controller succeeded in keeping wind turbine speed and output power. In order to show the validity of the PID controller, a variable wind speed is assumed as shown in Fig. 8.58, the plant response to this variable wind speed is shown in Fig. 8.59. It is clear that plant response suffers from a small amount of control errors due to the fast changes in the wind speed.

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(a)

(b)

(c) Fig. 8.56: PID controlled response to a unit step in the wind speed

(a)

(b)

(c) Fig. 8.57: PID controlled response to a unit step in the load troque

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Fig. 8.58: Variable wind speed

(a)

(b)

(c) Fig. 8.59: PID controlled response to variations in the wind speed The neural network NARMA-L2 controller with on-line training is used instead of PID controller. The plant response to a unit step in wind speed, a 10% step increase in load torque, and variable wind speed are shown in Fig. 8.60, Fig. 8.61, and Fig. 8.62 respectively. 364

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(a)

(b)

(c) Fig. 8.60: ANN controlled response to unit step in the wind speed

(a)

(b)

(c) Fig. 8.61: ANN controlled response to 10% increase in the load torque 365

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(a)

(b)

(c) Fig. 8.62: ANN controlled response to variable wind speed Although a small amount of control errors are obtained when using PID controllers, high accuracy of plant response to follow the objective of a nearly zero speed deviation is obtained with neural network NARMA-L2 controller. The main critical issue associated with the presented use of the NARMA-L2 controller is the required high rate of the pitch angle changes. These high rates require fast acting actuators and mechanical withstand capability of the affected components.

8.7 Case study 4 – Modal Analysis of FS-IGIB System119 This section presents a linearized model for a simple system consisting of an induction generator (IG) connected to an infinite bus through a linear passive transmission network. The linearized model is obtained by linearizing the third-order dynamic model presented in section 8.3.3.B. This model will be used for studying the small signal stability of the induction generator and modal analysis of the system as affected by the network parameters and 119

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initial operating conditions. The considered network parameters are the series impedance of the interconnecting network and the shunt capacitive susceptance at the generator terminals. A. System Modeling

The system under study consists of an induction generator connected to an infinite bus (IGIB) through a linear passive transmission network as shown in Fig. 8.63. This system is practically acceptable for modeling the integration of wind-energy conversion systems utilizing induction generators with largescale power systems.

Fig. 8.63: Induction generator infinite-bus (IGIB) system

Fig. 8.64: 3rd order model for induction machine

Fig. 8.65: Equivalent circuit of induction machines B. Nonlinear model for the IGIB system

The third-order model of the induction generator presented in section 8.3.3.B will be presented here using compact forms using the phasor or matrix representations. The machine equations are based on the generator 367

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current direction convention. The stator electromagnetic transients are neglected using the 3rd order induction machine model in the synchronously rotating reference frame axis, Fig. 8.64. Therefore, the stator windings equations are represented by algebraic equations. The 3 rd order machine model considers two rotor windings on the d- and q- axis. Hence, two state variables define the electromagnetic transients of the rotor. The steady state equivalent circuit of induction machines is shown in Fig. 8.65. Based on Fig. 8.64, and Fig. 8.65, the stator algebraic equations take the form (8.12) and (8.13). These equations can also be represented in the phasor form (8.76) or the matrix form (8.77). Equations (8.14) and (8.15) can be either combined in the phasor form of (8.78) or the matrix form (8.79). The rotor electromechanical transients can be represented by (8.80). E ' Vt  I s ( rs  jx' )

 E d'   rs  '  '  E q   x

where  

(8.76)

x '   I d  V d       rs   I q  V q 

 pE d'  1/ To' s o   E d'   0   I d   '   '     I  ' pE   s T 1/   q   o o   E q   0   q 

(8.79)

(x ss  x ' ) , and p is a deferential operator (i.e. d/dt). To'

2Hps  Te  Tm Where

(8.77)

Te  E d' I d  E q' I q

(8.80)

(8.81)

A general linear passive network connecting the induction generator to an infinite bus (IB) with voltage magnitude Eb and with zero phase-angle can be 368

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represented as hybrid two-port network120. If, as shown in Fig. 8.66, the induction generator is connected to bus 1 and the IB bus is bus 2, the hybrid two-port network model of the network takes the form, Vt  h11 I a  h12 E b

(8.82)

Fig. 8.66: General linear passive network connecting IG to IB The parameters h11 and h12 are related to the reduced admittance matrix of the network by the following equations; keeping in mind that induction generator and IB buses are only the retained buses in that matrix: h11 

1 Y11

and h12 

 Y21 Y11

(8.83)

Generally, h11 and h12 can be expressed as: h11  z R  jz I 



h12  h1  jh2 

(8.84)

For induction machines, Kron’s transformation is not needed and the machine can be directly analyzed in the common DQ-frame121. Therefore, the following expression is generally valid, Vd   z R V     q   zI

 z I   I d   h1  E b    z R   I q  h2 

(8.85)

The output active and reactive power supplied to the grid are presented by

120

K R Padyiar. Power System Dynamics: Stability and Control. Interline Publishing Pvt Ltd; 1996 121 B. M. Nomikos, C. D. Vournas. Evaluation of Motor Effects on the Electromechanical Oscillations of Multimachine Systems. IEEE Bologna PowerTech Conference, Bologna, Italy 2003; (June 23-26) 369

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Ptogr id  Vd I d  Vq I q

(8.86)

Qtogr id  Vq I d  Vd I q

(8.87)

Linearizing equations (8.77), (8.79), (8.80), and (8.85) and eliminating the non-state variable Vd, Vq, Id, and Iq, the following state input equation is obtained as,

 p E d'   c 5    '  p E q    c 7  p s   k 1    2H

o E qo'   E d'   0      ' ' o E do   E q    0   T m  (8.88) c8     k2 0   s   1    2H  2H c6

This takes the general state-space input equation px  Ax  Bu , where, c5 

1  c3 To'

c 6  s o o  c 4

c 7  s o o  c1

c8 

' ' k1  I do  E do c1  E qo c3

1  c 2 To'

' ' k2  I qo  E do c 2  E qo c4

c1 c 2   rs  z R c c    '  3 4   x  z I

x '  z R   rs  z r 

1

The initial conditions are calculated from the steady-state system model with the initial slip is calculated separately using,

so  max ( so1 , so 2 ) so1,2 

Vto2 rr

 2Tmo xt2



rr xt

(8.89) Vto4 2 2 xt 4Tmo

370

1

(8.90)

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An accurate value of so can be obtained through an iterative procedure by setting the initial guess of Vto to 1.0 p.u and updating it after solving the system equations under steady-state conditions. The linearized system equations are represented by the block diagram of Fig. 8.67 which can be implemented on the SIMULINK for the linearized time domain (TD) simulation of the system.

Fig. 8.67: Linearized IGIB system block diagram for linearized TD simulation Equation (8.88) describes the connection between of the rotor mechanical equations and stator as well as the rotor electrical equations including the network block. The terminal voltage and stator current block is obtained eliminating Vd and Vq from (8.77) using (8.85) and linearizing the resultant

equation to get Id and Iq then Vd and Vq are obtained from the linearization of either (8.85) or (8.77). The output power block is obtained directly by linearizing (8.86) and (8.87). The terminal voltage and stator current vectors are obtained from the relations, Vt  Vd2  Vd2

(8.91)

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 V   t   Vt  tan 1  q  V  q 

I a  I d2  I q2

(8.92)

(8.93)



a   I a  tan 1 I q I d



(8.94)

C. Problem Formulation and Solution Algorithm

For ensuring dynamic stability as well as acceptable operating conditions, the shunt-capacitor and the equivalent external reactance should be carefully selected. This is can be stated as, For a specified initial conditions Search for bc and ke s.t.

(8.95)

Vtmin  Vt  Vtmax

real(eig(A ))  0

The value of the external line impedance (r e+jxe) is defined as a percentage of the induction generator transient impedance (r s+jx’s) by the variable ke defined as: ke 

re 

jxe 

r

s

 jx s'



(8.96)

The problem stated by (8.95) can be used in either the design stage or for the analysis of an existing system. A general solution algorithm for (8.95) is illustrated in the flowchart of Fig. 8.68, which is implemented on MATLAB, with the terminal voltage limits of the induction generator are assumed to be Vtmin = 0.9, Vtmax = 1.1

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Fig. 8.68: IGIB SSSA algorithm Flowchart D. System Parameters

The induction generator parameters data on 6.45 MVA, 25 kV, 50 Hz base are listed in Table 8.25.

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Table 8.25: Induction generator data Parameter Value Unit Stator resistance, r s 0.03 p.u Stator leakage reactance, xs 0.27 p.u Rotor resistance, r r 0.03 p.u Rotor leakage reactance, xr 0.22 p.u Magnetizing reactance, xm 8.06 p.u Total inertia Constant*, H 3.6 sec * Includes the wind turbine inertia constant

E. Results and Discussion

In this part, the results for the following cases are illustrated and discussed:  With infinite network strength (ke = 0, bc = 0) the effects of variations of Tmo (i.e. initial conditions) on system stability are investigated.

 With finite network strength (ke > 0) the effects of bc on system stability are investigated for various values of ke. E.1. Infinite network Strength

The eigenvalues (’s) of the system as affected by the input mechanical torque with Eb = 1 p.u are listed in Table 8.26 and their real-part values are illustrated in Fig. 8.69. The maximum of the real-part of the eigenvalues and the torque-slip relation are shown in Fig. 8.70 and Fig. 8.71 respectively. Based on Table 8.26, for low input mechanical torque values up to 0.3 p.u, only real eigenvalues exist; however,with further increase in the input mechanical torque, two of the three eigenvalues become complex numbers. The critical stability is obtained at 1.0 p.u input mechanical torque as also depicted from Fig. 8.69, and Fig. 8.70. These results are in agreement with

the Te/s criterion which is based on the torque-slip characteristics of the induction generator for a generator model that is based on the generator current direction convention (shown in Fig. 9.71). The case where Tmo = 0.9 p.u is selected as a base-case for subsequent analysis.

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Table 8.26: Eigenvalues ( s) of the system as affected by the input mechanical torque with Eb = 1, ke = 0 Tmo (p.u) 0.001 0.1 0.3 0.5 0.7 0.9 1

1

2

3 -20.8466 +j0 -16.242 +j0 -5.3816 -21.246 +j0 -5.4474 +j0 -15.7768 -19.1165 +j0 -18.264 +j0 -5.0897 -19.1336 +j4.8918 -19.1336 -j4.8918 -4.203 -19.7408 +j8.4033 -19.7408 -j8.4033 -2.9885 -20.5146 +j13.1638 -20.5146 -j13.1638 -1.4409 -21.2351 +j20.6006 -21.2351 -j20.6006 0

+j0 +j0 +j0 +j0 +j0 +j0 +j0

Fig. 8.69: Real-part of the eigenvalues for infinite network strength

Fig. 8.70: Maximum of the real-part of the eigenvalues for infinite network strength

Fig. 8.71: Torque-slip relation for infinite network strength 375

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E.2. Finite network Strength

The effects of external network strength on the system stability and generator terminal-voltage magnitude are investigated in two phases. First, without shunt capacitor installed at generator terminals. Second, with shunt capacitor installed at generator terminals. With varying ke for zero shunt capacitive susceptance at generator terminal, the eigenvalues of the system and the generator terminal voltage are shown in Table 8.27. Fig. 8.72 shows the variations of both the maximum of the real-part of the eigenvalues and generator terminal voltage for different values of ke at bc = 0.

Fig. 8.72: Effect of ke on maximum of the real-part of the eigenvalues and the generator terminal voltage for Eb = 1, bc = 0, and Tmo = 0.9 Table 8.27: Eigenvalues (’s) of the system as affected by ke with Eb = 1, bc = 0, and Tmo = 0.9 ke 0 0.1 0.2 0.3 0.4 0.5 4

re 0 0.003 0.006 0.009 0.012 0.015 0.12

xe 0 0.048 0.096 0.144 0.192 0.24 1.92

-20.5146 -19.5752 -18.7859 -18.1128 -17.5315 -17.0236 -10.9804

1 +j13.1638 +j12.9693 +j12.7755 +j12.5913 +j12.4205 +j12.2638 +j10.7593

-20.5146 -19.5752 -18.7859 -18.1128 -17.5315 -17.0236 -10.9804

2 -j13.1638 -j12.9693 -j12.7755 -j12.5913 -j12.4205 -j12.2638 -j10.7593

3 -1.4409 -1.3506 -1.2666 -1.1863 -1.1087 -1.0332 0.0105

+j0 +j0 +j0 +j0 +j0 +j0 +j0

Vto 1 0.9694 0.9389 0.9087 0.8792 0.8504 0.3449

Based on Table 8.27 and Fig. 8.72, it is clear that the reduction in the network strength (i.e. increase of ke) provokes both system stability and generator terminal voltage in an approximate linear manner. With ke greater than 0.3 p.u the generator terminal voltage drops below the minimum

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allowable value which is 0.9 p.u. In addition, very high values of ke (i.e. very weak external network) results on system instability. Now, the combined effects of bc and ke on the system stability and the generator terminal voltage are investigated for ke = 0.4 p.u (corresponding to voltage level violation problem) and 4.0 p.u (corresponding to both voltage level violation and stability problems). By varying bc with ke = 0.4 p.u, the eigenvalues of the system and the generator terminal voltage are shown in Table 8.28. Fig. 8.73 shows the variations of both the maximum of the realpart of the eigenvalues and generator terminal voltage for different values of bc at ke = 0.4 p.u. Table 8.28: Eigenvalues (’s) of the system as affected by bc with Eb = 1, ke = 0.4, and Tmo = 0.9 1

bc

0 0.1 0.2 0.4 0.8 1 1.2

-17.5315 -17.4622 -17.3903 -17.238 -16.8946 -16.5966 -16.4883

+j12.4205 +j12.442 +j12.4647 +j12.5145 +j12.6347 +j12.7489 +j12.7928

2

-17.5315 -17.4622 -17.3903 -17.238 -16.8946 -16.5966 -16.4883

-j12.4205 -j12.442 -j12.4647 -j12.5145 -j12.6347 -j12.7489 -j12.7928

3

-1.1087 -1.1451 -1.1832 -1.2649 -1.4534 -1.6204 -1.6816

Vto

+j0 +j0 +j0 +j0 +j0 +j0 +j0

0.8792 0.8942 0.9097 0.9424 1.0154 1.0779 1.1005

Fig. 8.73: Effect of bc on maximum of the real-part of the eigenvalues and the generator terminal voltage for Eb = 1, ke = 0.4, and Tmo = 0.9 Based on Table 8.28 and Fig. 8.73 it is clear that the increase of bc enhances both the generator terminal voltage level and the system stability. Moreover, the maximum limit of bc is determined by the maximum allowable voltage level at generator terminals. Therefore, it is clear that the role of 377

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shunt capacitor bank at the induction generator terminals is not only enhances the generator terminal voltage but also improves system dynamic stability; however, this role is hindered by lowering external network strength as shown in the following. By varying bc at ke = 4.0 p.u, the eigenvalues of the system and the generator terminal voltage are shown in Table 8.29. Fig. 8.74 shows the variations of both the maximum of the real-part of the eigenvalues and generator terminal voltage for different values of bc at ke = 4.0 p.u. Table 8.29: Eigenvalues of the system as affected by bc with Eb = 1, ke = 4.0, and Tmo = 0.9 bc

0 0.1 0.3 0.5 1 1.2 1.3 1.4 1.5

Vto 1 2 3 -10.9804 +j10.7593 -10.9804 -j10.7593 0.0105 +j0 0.3449 -9.8377 +j10.9833 -9.8377 -j10.9833 0.0812 +j0 0.3641 -6.0232 +j11.8514 -6.0232 -j11.8514 0.3761 +j0 0.4068 -1.7876 +j12.8235 -1.7876 -j12.8235 0.6748 +j0 0.4548 -7.2939 +j11.5301 -7.2939 -j11.5301 2.5833 +j0 0.5676 -4.6959 +j6.3858 -4.6959 -j6.3858 -0.8555 +j0 0.5846 -3.9976 +j1.9998 -3.9976 -j1.9998 -0.5162 +j0 0.5825 -9.2755 +j0 3.7197 -j0 -0.6827 +j0 0.5734 -11.3805 +j0 8.486 -j0 -0.4805 +j0 0.5581

Fig. 8.74: Effect of bc on maximum of the real-part of the eigenvalues and the generator terminal voltage for Eb = 1, ke = 4.0, and Tmo = 0.9 Based on Table 8.29 and Fig. 8.74, it is clear that the increase of bc to 1.2 p.u increases the generator terminal voltage and brings the system to stable operation. However, this stable operating point is not valid due severe violation in the generator terminal voltage. Further increase of bc brings the system back to unstable operating points with reduction the generator 378

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terminal voltage which results from the voltage instability. Therefore, based on Fig. 8.73 and Fig. 8.74, the role of shunt capacitor bank at the induction generator terminals in enhancing both generator terminal voltage and system stability is hindered by lowering the external network strength. As the external network strength increases, properly selected shunt capacitor bank at the induction generator terminals can provide good voltage regulation and better stabilization. However, in case of very weak external network strength, neither adequate voltage regulation nor stabilization effects of the capacitor bank can be provided. In this case, acceptable operation of the system can be achieved by distributed capacitive compensation over the transmission line. For reasonable external network strength, it is recommended to use at the terminals of the generator thyristor switched capacitor banks (TSC) with optimized control algorithm instead of fixed shunt capacitor banks (FC). This is can provide optimal stabilization in addition to providing adequate voltage control for different operating points of the system. E.3 Time Domain Simulation

The block diagram of Fig. 5 is implemented on SIMULINK for providing time domain simulations of the system. The considered disturbance is a 10% step increase in the mechanical power input at t = 1.0 sec. In order to compare the system response to the prescribed disturbance, the following cases are considered:  Base-case: Eb = 1, Tmo = 0.9, bc = 0, ke = 0.

 Case#1: Eb = 1, Tmo = 0.9, bc = 1.2, ke = 0.4

 Case#2: Eb = 1, Tmo = 0.9, bc = 1.0, ke = 4.0

The variations of Te and s are shown in Fig. 8.75, the Te-s

trajectories are shown in Fig. 8.76, the variations of P togrid are shown in Fig.

8.77, and the variations in Q togrid are shown in Fig. 8.78.

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Fig. 8.75: Variations of Te and s

Fig. 8.76: Te-s trajectories

Fig. 8.77: Variations of P togrid 380

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Fig. 8.78: Variations of Q togrid The time domain simulation as shown in Fig. 8.75 to Fig. 8.78 validate the modal analysis results as the base-case and case#1 are stable and case#2 is unstable. The following point presents a summary of the results and conclusions, 1. The reduction in the network strength (i.e. increase of ke) provokes both system stability and generator terminal voltage in approximately linear manner. 2. The role of shunt capacitor bank at the induction generator terminals is not only enhances the generator terminal voltage but also improves system stability. However, this role is hindered by lowering external network strength. 3. As the external network strength increases, properly selected shunt capacitor bank at the induction generator terminals can provide good voltage regulation and better stabilization. However, neither adequate voltage regulation nor stabilization effects of the capacitor bank can be provided in case of very weak external network. For very weak links, distributed compensators instead of terminal compensators are required. 4. The TSC at the generator terminals instead of the usual fixed capacitors is recommended for optimized injection of the reactive power at the generator terminals. As shown, the proper amount of the capacitive compensation at the terminals is highly dependent on the system parameters and the operating point. 381

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The next chapter will present more analysis of the impact of wind power on power systems. In addition, the combined impact of FACTS devices and wind power on the dynamic stability of power systems will also be demonstrated. The design of POD for enhancing the stability of power systems that contain wind power sources will also be given.

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Chapter 9 Dynamical Impacts of Wind Power and Damping Enhancement 9.1 Introduction In this chapter a series of case studies will be presented for demonstrating some critical aspects about the dynamical characteristics of power systems as affected by wind power. In the previous chapter, it is shown that the FS-SCIG WT needs a controllable capacitive compensation at its terminals for voltage control and stability enhancements. In addition, FACTS devices may be used in power systems for many additional reasons such as power flow control. Therefore, the impact of FACTS devices such as SVCs and TCSCs on the dynamics of power systems in the presence of wind power will also be presented. The chapter also includes the stabilization and stability enhancement in power systems that contain wind power generators.

9.2 Impact of wind power on the inertia of power systems and the equivalence SMIB In chapter 5, the two-machine and single-machine equivalences of the weakly interconnected areas of Fig. 5.57 is presented in section 5.5. The power generation technologies in that system are totally conventional power generators. In this section, the same system is considered; however, wind power generation is integrated with the system122 as shown in Fig. 9.1; 150 MW of the conventional power generated in area 2 is replaced by wind power. Therefore, the 700 MW power generations from G4 in the first scenario becomes 550 MW in the second scenario. The wind farm is formed of 100 identical fixed-speed squirrel-cage induction-generator (FS-SCIG) based

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El-Shimy, M. (2015). Stability-based minimization of load shedding in weakly interconnected systems for real-time applications. International Journal of Electrical Power & Energy Systems, 70, 99-107. 383

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wind turbine generators (WTGs) each of 2 MVA rating. The parameters of the WTGs123,124,125 are listed in Table 9.1. Table 9.1: Wind turbine and induction machine parameters Value Parameter SCIG DFIG Number of blades 3 Rated wind speed, u [m/s] 16 Blade length [m]

75

Inertia constant of turbine H wr [kWs/kVA]

2.5

Inertia constant of machine H m [kWs/kVA]

0.5

Shaft stiffness for FSWT126 [p.u.]

0.3

------

Pitch control gain [p.u]

10

Pitch control time constant [s]

2

Voltage control gain Kv [p.u]

------

10

Power control time constant Te [s]

------

0.01

Gear box ratio[int -]

[41/89]

Number of poles

4

Rated voltage [kV]

0.69

Frequency [Hz]

60

Resistance of the stator, RS [p.u.]

0.01

Resistance of the rotor, Rr [p.u.]

0.01

Leakage inductance of the stator, xs [p.u.]

0.1

Leakage inductance of the rotor, xr [p.u.]

0.08

Mutual inductance, xm [p.u.]

3

123

El-Shimy M, Badr M, Rassem O. Impact of large scale wind power on power system stability. In: Power system conference, 12th international middle-east power conference (MEPCON). Aswan, Egypt: IEEE; 2008. p. 630–6. 124 El-Shimy M, Ghaly N, Abdelhamed M. Parametric study for stability analysis of gridconnected wind energy conversion technologies. In: 15th International middle-east power conference (MEPCON). Alex., Egypt: IEEE; 2010. p. 1–8. 125 EL-shimy M, Ghaly B. Grid-connected wind energy conversion systems: transient response. In: Encyclopedia of energy engineering and Technology. 2nd ed., vol. IV. CRC Press, Taylor & Francis Group; 2014. p. 2162–83 126

Multi-mass models are not considered for DFIG because of its controls which significantly cancel the effect of shaft stiffness by decoupling the mechanical and electrical power 384

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Fig. 9.1: The two-area system with wind power integration For the same disturbance stated in section 5.5 (i.e. A temporary 3-cycle three-phase fault started at t = 1 sec applied on bus 8), and the PSAT is used as a simulation environment, the system equivalent inertia at the interface buses and the equivalent inertia of the SMIB equivalence of the system are determined based on the method of section 5.2. The results are shown in Table 9.2. The impact of the wind power on the equivalent inertia constants and the transient reactances is illustrated in Fig. 9.2. Table 9.2: Equivalent inertia (M) in sec, p.u transient reactances, and p.u transient emf (E’); all values are on 100 MVA base Area 1 equivalence Area 2 equivalence SMIB equivalence ’ ’ Inertia Reactance E Inertia Reactance E Inertia Reactance 29.15 0.0237 1.034 47.61 0.0243 1.050 18.08 0.0120 It can be seen from these results that the wind power generators alters the frequency, transient, and steady state responses of the system. This is illustrated by the changes in the equivalent inertia constant and the equivalent transient reactance. The impact of wind power is significantly appearing on area 2 where the wind power is available. Since the equivalent inertia constant and the equivalent transient reactance of this area are reduced, then the stability of the system is degraded by the presence of wind power. This is can be demonstrated by determining the maximum drop in the power generated by the same way presented section 385

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6.4. Without wind power, the maximum drop in the generation is found to be 5.6 p.u while it becomes 4.9 p.u with the presence of the wind power.

(a)

(b) Fig. 9.2: Impact of wind power on the equivalency; (a) equivalent inertia constant; (b) equivalent transient reactance The difference between the two values demonstrates the stability reduction caused by the wind power inclusion in the system. Due to the nature of the wind as an input energy source, the wind power is inherently variable and also suffers from some intermittency. Therefore, a power system with large amount of wind power is expected to be subjected to sustained changes in the power generation. The magnitude and sign of these changes depends mainly on the changes in the wind speed. Therefore, without a 386

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feasible method for bulk energy storage, integrated large amounts of variable generation (such as wind and solar-PV) in power systems may results in significant provoking of the stability, the security (static and dynamic), the reliability (or the power supply continuity), and the energy security. The hydrogen as an energy storage medium and an energy carrier (see chapter 1) may be considered as a perfect solution for the future system with very large amounts of variable generation.

9.3 Impact of wind power on the stability of power systems The behavior of a power system is significantly determined by the behavior and the interaction of the generators that are connected to it. The grid itself consists mainly of passive elements, which hardly affect the behavior of the system, and as for the loads, only those in which directly grid coupled motors are applied have a significant impact on the behavior of the system. In wind turbines, the generating systems differ from the conventional directly grid coupled synchronous generator which is traditionally used in power plants. Due to their different characteristics, these generating systems interact differently with the power system in comparison with the conventional synchronous generators. The change in this interaction determined mainly by the degree of penetration of wind power generation in the system. In the analysis of wind generation-system interaction a distinction is made between local and system wide impacts of wind power127,128 ; this shown in Fig. 9.3. Local impacts of wind power are the impacts that occur in the (electrical) vicinity of a wind turbine or a wind park. These impacts can be attributed to a specific turbine or park, i.e. of which the cause can be localized. These effects occur at each turbine or park, independently of the overall wind power penetration level in the system as a whole. Wind power locally has an impact on the following aspects of a power system: branch flows and node voltages, protection schemes, fault currents and switchgear ratings, harmonics, and flicker. System wide impacts , on the other hand, are J.G. Slootweg, W.L. Kling, “Modelling and analysing impacts of wind power on transient stability of power systems”, Wind Engineering, v.26, n.1, 2002, pp.3-20. 128 J.G. Slootweg, W.L. Kling, “The impact of large scale wind power generation on power system oscillations”, Electric Power Systems Research, v.67, n.1, 2003, pp.9-20

127

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impacts of which the cause cannot be localized. They are a consequence of the application of wind power that can, however, not be attributed to individual turbines or parks. Nevertheless, they are strongly related to the penetration level in the system as a whole. However, in opposition to the local effects, the level of geographical spreading of the wind turbines and the applied wind turbine type are less important. Apart from the local impacts, wind power also has a number of system wide impacts, because it affects: dynamics and stability, reactive power generation/voltage control possibilities, and system balancing: frequency control and dispatch of the remaining conventional units.

Fig. 9.3: Impacts of wind power on power systems This section presents a detailed analysis of the impact of large scale FSSCIG based wind farms on both the transient voltage stability (TVS) and the transient stability (TS) of electric power systems. With the PSAT as a simulation environment, the following problems have been analyzed:  Different penetration of wind power impact on TS and TVS following a major fault in the transmission system,

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 Determination of the acceptable wind power penetration level in power systems without deteriorating both transient stability and voltage stability of the system.  The effect of SVC on the system stability is studied. The mathematical modeling of various is as presented in chapter 8. The IEEE 9-bus system shown in Fig. 9.4 represents the interconnected power system used for this study. This system is considered the “original system” and consists of 9 buses, 3 synchronous generator, 6 lines, 3 transformers, and 3 load centers. Each generator is equipped with an AVR of the standard IEEE model 1 type with its block diagram shown in Fig. 9.5. The original system data on a 100 MVA base obtained from the Anderson & Fouad textbook129.

Fig. 9.4: The Original 9-bus system

Fig. 9.5: Standard IEEE Model 1 AVR P.M. Anderson & A.A. Fouad, “Power System Control and Stability”, Galgotia publications, 1981 129

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The FS_SCIG wind farm has been modeled as an equivalent induction generator that is connected to the original system at bus 7 via the two transformers and a line as shown in Fig. 9.6. The SCIG parameters are shown in Table 9.1.

Fig. 9.5: The 9-bus system with a wind farm connected at bus 7 and an SVC connected to bus 8

Fig. 9.7: SVC Regulator An SVC at bus 8 has been also added to the original as the reactive power a controllable source. The SVC is modeled in the simulation program as a limited linear controlled susceptance. The regulator has an anti-windup limiter, thus within the operating limits the SVC performs automatic voltage control at the connected node and outside the operating limits the SVC susceptance is locked. The block diagram of the SVC regulator is shown in 390

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Fig. 9.7. The SVC regulator parameters on a 100 MVA base are: the regulator time constant, Tr = 10 sec.; the regulator gain, Kr = 100; the reference voltage, Vref = 1.00 p.u.; the maximum susceptance, bmax = +1 p.u., the minimum susceptance, bmin = -1 p.u. This study system can be considered as a simple model for studying impact of large scale wind power on power system stability. After the load flow analysis, the focus is to perform dynamic analyses to verify that the system remains transiently stable under different contingencies and different levels of wind power production. The system loads are kept fixed at the values of the original system. For setting a base for comparison between the system transient response with and without wind generation, a 3 cycle (i.e. 0.05 sec, for 60 Hz) duration fault is simulated on line 5-7 very close to bus 7 (the fault is started at t = 0.5 sec) in the original IEEE 9-bus system (i.e. 0.0 % penetration level, and no SVC). The power angles and bus voltage magnitude (shown for buses 7 and 8 only to avoid crowding of plots) variations are shown in Fig. 9.8 and Fig. 9.8. Based on these figures, following the applied fault the system is dynamically stable and the system is able to restore the voltage.

Fig. 9.8: Original System (0.0 % penet. level, and no SVC) Power Angle Variations

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Fig. 9.9: Original System (0.0 % penet. level, and no SVC) Bus Voltage Magnitude Variations As a matter of investigation, the simulation is performed with the same fault location and fault duration but with the SVC connected at bus 8 (0.0 % penetration level, and with SVC). The power angles, and bus voltage magnitude variations are shown in Fig. 9.10 to Fig. 9.11. In addition, the variations of the SVC susceptance, voltage magnitude at bus 8, and the net reactive power injected to bus 8 are shown in Fig. 9.12.

Fig. 9.10: Power Angle Variations (0.0 % penet. level, and with SVC)

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Fig. 9.11: Bus Voltage Magnitude Variations (0.0 % penet. level, and with SVC)

Fig. 9.12: Variations of b, V8, Q 8 (0.0 % penet. level, and with SVC) Based on these simulation results, and considering the results associated with absence of the SVC, following the applied fault the system is transient stable and the system is able to restore the voltage; however, the results show a reduction in the damping of the system as a result of the inclusion of the SVC. This appears as less settling time and larger magnitudes of oscillations in the power angles. This impact is in agreement with the model analysis presented in chapter 7 and these problems can be avoided by proper FACTSPOD controllers. Although, the damping of the system is reduced, the transient voltage stability is enhanced at least in terms of the initial post fault voltage magnitudes. 393

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Now, the impact of wind power generation on the system transient and voltage stability is considered. Different penetration levels of wind power generation are considered such that a part of the power produced by the synchronous generator at bus 2 is replaced by an equal amount of wind power generation at bus 12 and the system is simulated for the previously described fault location and fault duration. The question to be answered is: what is the limit of wind power penetration level to keep the system transient and voltage stability for the considered contingency? The following cases are considered:  Case 1: No SVC, 0.63 p.u wind power generation (19.7 % total penetration level, 24.55% of the replaced conventional synchronous generation)

 Case 2: SVC, 0.63 p.u wind power generation (19.7 % total penetration level, 24.55% of the replaced conventional synchronous generation)

 Case 3: SVC, 1.40 p.u wind power generation (43.8% classical penetration level, 77.94% of the replaced conventional synchronous generation) The power angles, and bus voltage magnitude variations for each of these cases are shown in Fig. 9.13 to Fig. 9.18.

Fig. 9.13: Case 1 - power angle variations

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Fig. 9.14: Case 1 - bus voltage magnitude variations

Fig. 9.15: Case 2 - power angle variations

Fig. 9.16: Case 2 - bus voltage magnitude variation

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Fig. 9.17: Case 3 - power angle variations

Fig. 9.18: Case 3 - bus voltage magnitude variations Based on the responses of cases 1 and 2, it is clear that both the transient voltage stability of the system is enhanced with the SVC installed; however, the transient response of the power angles shows reduction in the system damping. In both cases the system is stable. With penetration level of 77.94% (case 3), the system becomes unstable both in transient stability and voltage stability. The instability phenomena can be viewed as an insufficiency of reactive power compensation; this can be detected form the variations of the SVC susceptance (shown in Fig. 9.19 and Fig. 9.20 for cases 2, and 3 respectively). Based on Fig. 9.20, it is clear that the SVC susceptance hits its upper limit and locked at that value which means that the reactive power limits of the SVC are reached. Therefore, the system instability in this situation is a voltage instability phenomenon in nature which is clear from Fig. 9.18. Based on the presented study, the maximum penetration level is less than 43.8%.

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Fig. 9.19: Case 2 - variations of b, V8, Q 8.

Fig. 9.20: Case 3 - variations of b, V8, Q 8. For the investigation of the maximum penetration level, case 4 is considered. In this case, the wind generation is set to be 1.2 p.u and the generation of bus 2 is 0.4 p.u with the wind total penetration level of 37.55%, which represents 60.13% of replaced conventional synchronous generation. The penetration level of case 4 is determined by a trial and error approach such that the system becomes critically stable at the stated penetration level. The power angle variations and the bus voltage magnitudes are shown in Fig 9.21 and Fig 9.22. The variations of the SVC susceptance and reactive power are shown in Fig. 9.23.

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Fig 9.21: Case 4 - power angle variations

Fig 9.22: Case 4 - bus voltage magnitude variations

Fig 9.23: Case 4 - variations of b, Q8 398

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Recalling that the wind power integrated to the original system is not an added power generation but it is a replacement of the conventional power by the same amount of the added wind power. In addition, with the absence of the wind power, the original system is stable for the same disturbance regardless of the impact of the SVC on the system damping. Therefore, the instability of the system at the maximum wind power penetration level is attributed to the characteristics of the FS-SCIG. The squirrel-cage induction machines consumes reactive power reactive power either during motoring or generating modes. Therefore, the steady-state reactive power loading on the system increases with the increase in the penetration level. In addition, further increase of reactive power loading occurs during the fault conditions due to the reduction of the SCIG terminal voltage. Consequently, the system fails to survive due to the transient voltage instability that is mainly caused by the insufficiency of the reactive power. Providing more reactive power during the fault by an SVC with higher rating is a possible way for enhancing the transient stability of the system while the associated reduction in the damping can be overcome by the use of PODs. This is will be illustrated later in this chapter. It should be noted that the maximum penetration level is highly dependent on the point of connection of the wind farm to the system, the considered contingency, and the WTG technology. For example, if the point of connection of the considered wind farm becomes bus 9 instead of bus 7, then 100% of the power generation at bus 3 can be replaced by wind power generation without causing system instability.

9.4 Response of WTGs in comparison with Conventional Synchronous Generators In this section a detailed analysis of the steady state as well as the transient response of FS-SCIG and DFIG wind power technologies in compassion with conventional synchronous generators will be presented. In addition, the effect of static and dynamic load models will also be considered. The study system is this section shown in Fig. 9.24 in which a power plant is connected to the stiff-grid through a transmission system and a local load. The transmission system consists of two transformers and two parallel lines. The local load is connected to the system through a transmission line. The 399

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generators of the power plant are made of conventional synchronous generators (SGs), or FS-SCIGs, or DFIGs. Therefore, the system shown in Fig. 9.24 may be considered as three separate systems where the only difference between them is the type of the generating technology. The topology of the system is popular in many industrial loads where the local power generation is installed for several reasons. These reasons include but not limited for, 1. Enhancing the reliability of the grid as a main power source. The local power source can either operates in a backup source (i.e. supplying power only when the grid power is unavailable) or continuously working in parallel with the grid. In this study, continuous parallel operation is considered. 2. Reducing the energy consumption from the grid and consequently reducing the electricity tariff. In this case, the power sources operate continuously in parallel with the grid such that it provides a large portion of the local load. 3. Supplying the local load with the possibility of exporting power to either nearby loads or to the grid. In this case, the local generation should be carefully sized for fulfilling the local load requirements as well as energy export contracts with the nearby loads or the grid. The local generation can be considered as a sort of an independent power producer (IPP) and the power export to the grid is subjected to the national feed-in-tariff structure. The grid in this case, purchases the surplus local energy while the possible local power shortage is purchased from the grid. 4. Apart of the active power production, the local generator may be mainly used for modifying the power factor (PF) of the local load for meeting juristic contracts with the grid as a part of the interconnection requirements. This is common in large industrial loads where their low power factor may affect the voltage regulation and stability of the grid.

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Fig. 9.24: The study system Conventional generation or renewable generation technologies may be used for the local power generator. The need of local energy storage is determined by the objectives of installing it and the degrees of the variability and intermittency of the primary source of energy. For example, a power source for enhancing the power supply reliability of the local load should itself be characterized by a high reliability. If the utilized power source is variable or intermittent, then sufficient local energy storage should be provided for fulfilling the target reliability level. The variability and intermittency (see chapter 1) are not only characterizing some renewable energy sources such as wind and solar resources but also characterize conventional energy sources. For example, if the local generator utilized petroleum oil and if the local load is located in remote areas relative to the location of the oil supply. Therefore, oil periodic transportation and local oil storage are required for feeding the generator. Consequently, the degree of the availability of the oil defines the variability and intermittency of the local power source. The study system is modeled and simulated through the PSAT simulation software tool. The models of the generating technologies and loads are available in the previous chapter as well as the manual of the PSAT. On 100 MVA bases, Table 9.3 shows the p.u parameters of the synchronous generator while Table 9.1 shows the parameters of the SCIG and the DGIG. The lines and transformer data are shown in Table 9.4. The impressed load on the system (connected to bus 5) is 4.0 +j 1.0 p.u. Various static and dynamic 401

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load models are considered for evaluating the system performance, studying the impact of load model variations, and comparing the generating technologies. The parameters of the considered loads are shown in Table 9.5. Table 9.3:Synchronous generator data. Parameter

Value

Machine dynamic order

4

Stator resistance (r a )

0.0000

Leakage reactance (xl)

0.0000

Direct-axis steady-state reactance [xd, x'd, x''d] Direct-axis open circuit time constants in sec. [T'do, T'' d0] Quadrature-axis reactance [xq, x' q, x'' q] Quadrature-axis open circuit time constants in sec. [ T'qo, T'' qo] Inertia in sec. (M=2H) and damping in p.u

[0.8958, 0.1198, 0.0000] [6.00, 0.00] [0.8645, 0.1969, 0.0000] [0.5350, 0.0000] [12.80, 0.00]

Table 9.4: Data of lines and transformers Value

Parameter

L2-3(1) and L2-3(2) L3-5 T1 T2 Length in km 10 3 Reactance in p.u (H/km) 0.002 0.001 0.0625 0.0576 Resistance in p.u (ohm/km) 0.0002 0.0001 0 0 Susceptance in p.u (F/km) 0.0035 0.00175 -

Bus 4 is modeled as a PV-bus with a unity voltage setting while the maximum active power setting is selected for this bus. The maximum active power injected to bus 4 is chosen such that the system will remain stable after being subjected to a specified disturbance. The considered disturbance in this case is a 3-cycle temporally three-phase fault at the midpoint of L2-3(2) started at t = 0.5 sec. The maximum power values for each of the load models shown in Table 9.5 are shown in Table 9.6 and Fig. 9.25. In the following sections, both static and dynamic analysis of the system will be presented. The static analysis includes load flow (LF) and static voltage stability (SVS) while the dynamic analysis includes transient stability (TS), dynamic voltage stability (DVS), and small-signal stability (SSS). 402

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Table 9.5: Load models Case ID Static models130 Dynamic models

1 2 3 4

Constant impedance model: α=2.0 , = 2.0 Composite load model: α=1.5 , =2.5 70% composite static load model + 30% induction motors with the 3rd order model 30% composite static load model + 70% induction motors with 3rd order model

It is clear from Table 9.6 and Fig. 9.25 that maximum power associated with SGs and DFIGs is sensitive to the load models while the SCIG is not sensitive. The presence of the induction motor load causes a decline in the maximum power level of the SG and increase for DFIG. For SG, the reduction in the maximum power level increases with the increase in the amount of induction motor loads while the DFIG is not significantly sensitive to that issue. The impact of the amount of induction motor loads on the maximum power of the FS-SCIGs is negligible. It is also depicted that the synchronous generator provides the highest maximum power followed by the DFIG then the SCIG. The results also show that the maximum power setting for all generators and all load models are 0.6 p.u which is limited by the SCIG. Fixing this power setting to allow proper comparison between various generation technologies in the following studies. This is because the only difference between the systems is the technological structure of the generators. Table 9.6: Maximum power settings Case

Max. Power level SG DFIG SCIG

Case 1

3

1

0.6

Case 2

3

1

0.6

Case 3 2.7

1.3

0.6

Case 4 2.6

1.3

0.6

130

The static load is represented by the exponential model given by and where P , and Q are respectively the load active and reactive power at a load bus voltage V while P o, and Qo are respectively the load active and reactive power at an initial voltage Vo (which is usually 1.0 p.u). For more details, see Appendix 3. 403

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Fig. 9.25: Maximum power settings A. Load flow analysis

The load flow analysis is an inspection of the static performance of power systems. In the standard load flow analysis, the loads are represented by a constant PQ-model while the generators are represented by PV-buses. The stiff node is defined as a standard independent voltage source (i.e. as a constant voltage magnitude and phase angle) for representing a very large system in comparison with the considered system. In the load flow analysis, the stiff node acts as a slack or a swing bus. The principal information of power flow analysis is to find the magnitude and phase angle of voltage at each bus and the active and reactive power flowing in each transmission line. This is valuable in checking the intrinsic and operational limits of the system (see chapter 1) under the given operating conditions. In this analysis the

maximum acceptable bus voltage deviation is 10%. Numerical iterative techniques are usually used for solving the load flow problem. In this section, the Newton-Raphson’s method is used for solving the load flow problem while the trapezoidal rule is used for the numerical integration in the TD analysis. Fig. 9.26 shows the power flow results for all load models for various generator types. The results show that acceptable voltage levels are obtained for various conditions. Recalling that the local load impressed on the system is 4.0 +j 1.0 p.u at unity voltage at bus 5; however, this value is changed according to the actual bus voltage magnitude and the model of the load. The results show that the magnitude of bus 5 voltage is highly dependent on the 404

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load model. This is because the power consumption of various load models is sensitive to the bus voltage by different relations. As can be seen, the dynamic load models have higher impact on the bus voltage magnitudes in comparison with the static load models. Consequently, the drop in the load bus voltage is higher with the dynamic loads in comparison with the considered static loads. Due to the presence induction motors, the reactive power consumption is very high in comparison with the considered static loads. Since the generator bus is represented as a PV-bus, the impact of the technologies of the generators and their actual capability limits on the power flow are absent.

(a)

(b) Fig. 9.26: Some power flow results B. Transient analysis

In this section, the impact of the fault Clearing Time (CT) on the TS and the TVS of the system are considered. In addition, the fault Critical Clearing 405

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Time (CCT) is determined. The considered disturbance is a temporarily 3-

phase fault at midpoint of the line L2-3(2); see Fig. 9.24. Various load models and generation technologies are considered. The fault starts at t = 0.5 sec and the CT is 3-cycles (0.06 sec). Fig. 9.27 shows the effect of load models with the same generation technology on the bus voltages while Fig 9.28 shows the effect of various generation technologies with the same load model on the bus voltages.

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 9.27: Transient voltage response as affected by load models for the same generation technology; a, b, and c: generator terminal voltage; d, e, and f: load terminal voltage

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(d)

(e)

(e)

(f)

(f)

(g)

(g)

(h)

Fig. 9.28: Transient voltage response as affected by generating technologies for the same load model; a, b, c, and d: generator terminal voltage; e, f, g, and h: load terminal voltage

It can be seen from these results that the transient voltage responses associated with the considered static loads models is nearly identical. The transient voltage responses associated with the dynamic load models are highly different than that of the static load models. In addition, the dynamic load models cause significant degradation in the transient voltage responses 407

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in terms of the transient deviations, and the settling time of the transient. It is also shown that the negative impacts of the dynamic load models on the transient voltage response increases with the increase in the percentage of the induction motor load in the composite load structure. For the considered disturbances, stable equilibrium points are reached in the post-fault conditions. Therefore, the stated CT is less than the CCT of any of the load and generation technological conditions. It is also clear that the transient voltage response of the DFIG is better in comparison with the conventional SG and the FS-SCIG. The worst transient voltage responses are associated with the SCIG. This is can be attributed to the voltage control capability of the DFIGs and the SGs. On the other side, the SCIGs cannot provide either a controllable reactive power neither they have a reactive power capability131,132,133. As these generators are equipped with fixed capacitors at their terminals, their voltage support during steady state conditions is limited while they did not provide contributions in enhancing the transient voltage support. The fixed capacitors of the SCIGs are usually sized based on the initial conditions for providing an acceptable terminal voltage magnitude by reducing the reactive power loading on the system. As stated in the previous chapter, controllable reactive power support (for example, SVCs or STATCOMs) is recommended to be provided to the FSSCIG for enhancing the voltage control as well as the system stability. The transient responses of the active and reactive power of various generators as affected by load models are shown in Fig. 9.29 while Fig. 9.30 shows comparative transient power responses between various generating technologies for the same load models. The comments on the transient voltage responses are also applicable to the transient responses of the active and reactive power; however, the impact of the FS-SCIG is clearly illustrated. The results show that the SCIG presents an initial high reactive power loading during the fault period and its remains M. Ahmed, M. EL-Shimy, and M. Badr, “Advanced modeling and analysis of the loading capability limits of doubly-fed induction generators,” Sustainable Energy Technologies and Assessments, vol. 7, pp. 79-90, 2014. 132 EL-Shimy M. Reactive Power Management and Control of Distant Large-Scale GridConnected Offshore Wind Power Farms. International Journal of Sustainable Energy (IJSE), 2012. Available online: Mar 20, 2012. Volume 32, Issue 5, pp. 449 - 465, 2013. 133 M. EL-Shimy, “Modeling and analysis of reactive power in grid-connected onshore and offshore DFIG-based wind farms”, Wind Energy, 17:279 – 295, 2014 131

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performing as a reactive power loading during the post fault transients. This is attributed, as previously stated, to the reactive power characteristics of the induction machines. This also explains the degraded transient voltage responses associated with the SCIGs.

(a)

(e)

(b)

(f)

(c)

(g)

(h) (d) Fig. 9.29: Transient active power responses of the generators as affected by load models for the same generation technology; a, b, c, and d: active power; e, f, g, and h: reactive power

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On the other hand, both SGs, and DFIGs presents a momentary reactive power loading on the system during the fault period. Afterward, the voltage (or reactive power) control actions of these generators provide a reactive power support to the grid. Consequently, the transient voltage responses associated with these generators are better in comparison with the SCIG. In addition, the voltage collapse with these generators is less likely in comparison with the FS-SCIGs.

(a) (d)

(b)

(e)

(f) (c) Fig. 9.30: Transient active power responses of the generators as affected by generation technologies for the same load model; a, b, and c: active power; d, e, and f: reactive power

The determination of the CCT for the stated faults is determined by increasing the CT till the instability is detected. Table 9.7 and 9.31 shows the value of CCT for various load models and various generation technologies. 410

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CCT (s)

Table 9.7: CCT values Case SG DFIG SCIG Case 1 0.99 -----134

1.23

Case 2 0.99 Case 3 0.47

-----1.42

1.23 0.65

Case 4 0.21

0.22

0.1

1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00

SG DFIG SCIG

case 1

case 2

case 3

case 4

Case ID

Fig. 9.31: Critical clearing time for 3 based system with the different load models The results show that the CCT is highly dependent on the generation technologies and the load models. In addition, as shown in the previous chapters, the CCT is also highly dependent on the initial conditions, the system inertia, and the strength of the grid-coupling link. With the static load models, the DFIG remains stable till the CT is set to 20 sec while the other technologies show a specified CCT. The CCT of the SCIG is higher than the CCT of the synchronous generator. This s can be attributed to the impact of the AVR which for some parameters should be equipped with PSS for stability enhancement of the SGs. The changes in the parameters of the static load model did not show a significant impact on the CCT. With the dynamic load models considered, the CCT is generally reduced with the increase of the amount of induction motors in the load composition.

9.5 Design of PODs in the presence of wind power generation The basic traditional methods for the design of PODs are presented and implemented in chapter 7. In this chapter, the frequency response method will 134

Till CT = 20 sec, the system stable 411

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be utilized for the POD design. The presence of wind power, as previously explained, alters the dynamics of the systems and causes changes in the eigenvalues. Therefore, no additional POD design techniques are required for the design of POD considering the impact of wind power; however, the impact of wind power on the system dynamics will demonstrated via the modal analysis for more clarification of their impacts. In this section, the impact of wind power on the dynamic stability of weakly interconnected power systems and the design of the required PODs are considered135,136,137. Two situations are covered. The first one is the replacement of conventional power by wind power while the second one includes the addition of wind power to an existing conventional power generation system. The considered system is composed of the two weakly interconnected areas of chapter 7. In this system, the area 1 is exporting power to the area 2. This situation defines the main objective of integrating the wind power. As shown in Fig. 9.32, the wind power is integrated to area 1 for reducing the fuel consumption and consequently reducing the GHG emissions. This objective is based on the fact that area 1 has surplus power. On the other hand, area 2 is importing power from area 1 over a weak link. This is due to the shortage in the power generation in area 2 in comparison with its loads. Therefore, the main objective of the wind power integration in this case is the reduction of the dependency of area 2 on area 1. This is achieved also via clean energy sources; however, the main problem associated with wind power is the availability of the wind resource. In the following analysis, the wind power integration to area 1 will be considered as first option of wind power integration while the wind power integration to area 2 is considered as the second option. One option will be considered at a time i.e. the analysis

135

M. Mandour, M. EL-Shimy, F. Bendary, and W.M. Mansour. Design of power oscillation damping (POD) controllers in weakly interconnected power systems including wind power technologies. Accepted for publications in the JEE, 2015 136 M. Mandour, M. EL-Shimy, F. Bendary, and W.M. Mansour. The Design of POD Considering Conventional and Renewable Power Generation. International Electrical Engineering Journal (IEEJ), Vol. 6 (2015) No.7, pp. 1962-1972 137 M. Mandour, M. EL-Shimy, F. Bendary, W.M. Mansour. Impact of Wind Power on Power System Stability and Oscillation Damping Controller Design. Industry Academia Collaboration (IAC) Conference, 2015, Energy and sustainable development Track, Apr. 6 – 8, 2015, Cairo, Egypt. 412

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and the POD design will consider a situation where only one option will be implemented.

Fig. 9.32: The two-area system with wind power integrations For the same reasons stated in chapters 7 and 8 as well as in this chapter, the traditional fixed capacitors of the original system (Fig. 7.18) are replaced by SVCs as shown in Fig. 7.13 and 9.32. This is also important for handling the reactive power support needed with the presence of wind power generators. Both FS-SCIG and DFIG technologies are considered. The modeling of various technologies is presented in chapter 8 while the data of various WTGs as well as the system are shown in Appendix 8. A. Sizing of the wind farms

In this context, the maximum wind penetration (MWP) is defined as the maximum value of the wind power after which the system will lose its stability due to the impact of the wind power. Two methods can be applied for the determination of the maximum wind penetration level. The first method depends on the response of the system for a specific fault defined by three parameters; location, duration, and type. In this context, this method will be called the time-domain simulation (TDS) method. The main problem associated with the TDS method is the high dependency of the determined MWP level on the chosen disturbance as well as the parameters of the 413

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disturbance. The second method overcomes this problem. In the second method, the MWP level is determined based on the modal analysis of the system. In this case, the MWP level is defined as the maximum amount of integrated wind power before provoking the small-signal stability of the power system. Although, the evaluation of the stability of this system in this method is limited to small disturbances, the results are independent on a specific disturbance. Of course, the MWP level is highly dependent on many factors such as the WTG technology, the topology of the hosting system and its parameters as well as the considered operating. Usually, the peak loading level of the system is considered as the design operating point and also the point at which the MWP level is determined. Fig. 9.33 shows the various values of the MWP level considering the location of the wind farm, the WTG technology, and the operation philosophy. In the conventional power replacement philosophy, the amount of wind power added to area 1 is reflected as a reduction in the conventional power generation in that area. In the wind power addition philosophy, the wind power added to area 2 do not affect the original convention power generation in that area. The results show that the MWP level of the FS-SCIG is the same for the power replacement and power addition options. On the other side, higher power in the power addition option with DFIGs is achieved in comparison with the power replacement option of the same WTG technology. For both options, the MWP level of the DFIG is much higher than the FS-SCIG MWP level. This attributed to the better control and stability characteristics of the DFIGs in comparison with the FS-SCIGs.

MWP level (MW)

600 500 400 300 200 100 0 FS-SCIG

DFIG

FS-SCIG

DFIG

Po er repla e e t

Po er additio

MWP level for area 1

MWP level for area 2

Fig. 9.33: MWP level 414

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B. Modal analysis

The system without any wind power integration is considered for inspecting the impact of the wind power on the eigenvalues of the system. The dominant eigenvalues as well as the frequency and the damping of the dominant modes of the system without wind power integration are shown in Table 7.12 and also shown here in Table 9.8 for the convenience of the reader. For the MWP level shown in Fig. 9.33 and for values slightly higher than them, the dominant eigenvalues as well as the frequency and the damping of the dominant modes are shown in Tables 9.9 and 9.10 for the power replacement and power addition options of the FS-SCIG respectively. Tables 9.11 and 9.12 shows the same quantities for the power replacement and power addition options of the DFIG. Table 9.8: Dominant eigenvalues and participation factors without wind power integration Eigenvalues -0.52663j6.6178 -0.5382j6.8191 -0. 0.03599+j3.7071

f (Hz)

 (%)

Status of Eigenvalues

1.0566 1.0887

8% 7.92%

Unacceptable Unacceptable

0.59003 0.45%

Critical

Table 9.9: Dominant eigenvalues and participation factors for power replacement in area 1 by FS-SCIGs Eigenvalues P1+P2=1260MW -0.59442j6.5443 P12=140MW -0.15025j3.821 -0.57982j6.5303 P1+P2=1250MW -0.14699j3.8239 P12=150MW -0.42722j7.0026 0.12483+j0

f (Hz) 1.0458 0.6086 1.0434 0.6090 1.1166 0

Eigenvalue  Status (%) 9.04% unacceptable 3.9% Critical 8.88% unacceptable 3.86% Critical 6.08% unacceptable ------Unstable

Table 9.10: Dominant eigenvalues and participation factors for power addition in area 2 by FS-SCIGs Eigenvalues P12=140 MW -0.59524j6.5765 415

f (Hz) 1.051

 (%) %9.02

Eigenvalue Status unacceptable

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-0.17052j3.9459 0.6286 %4.31 1.05 %9 -0.60136j6.5699 -0.16879j3.9522 0.62958 %4.3 P12=150 MW -0.57935j6.9841 1.1154 %8.28 0.17508+j0 0 -------

Critical unacceptable Critical unacceptable Unstable

Table 9.11: Dominant eigenvalues and participation factors for power replacement in area 1 by DFIGs Eigenvalues P1+P2=900MW -0.81087j6.2776 P12=500MW -0.20784j4.0177 -0.85172j6.2167 P1+P2=850MW -0.22005j4.0365 P12=550MW -0.12055j1.7216 0.008 j0

f (Hz) 1.0074 0.64028 0.99866 0.64339 0.27467 0

 (%) 12.8% 5.16% 13.5% 5.4% 7.04% ------

Eigenvalue Status acceptable Critical acceptable Critical unacceptable Unstable

Table 9.12: Dominant eigenvalues and participation factors for power addition in area 2 by DFIGs Eigenvalues P12=350 MW

 (%) %12.3 %6.65 %13.06 %7.1 %16.6 --------

Eigenvalue Status acceptable Critical acceptable Critical acceptable Unstable

7 6

5 4 3 2 1

No Wind

FS-SCIG

Addition

Replacement

Addition

Replacement

Two SVCs

0 FCs; No SVCs

Damping of the critical mode (%)

P12=400 MW

-0.78117j6.3636 -0.27893j4.182 -0.83732j6.2965 -0.30421j4.2103 -0.22003j1.3624 0.000j0

f (Hz) 1.0204 0.66706 1.0109 0.67183 0.21964 ---------

DFIG

Fig. 9.34: The damping of the critical mode 416

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The impact of various options of the wind power integration as well as the technological impact of the WTGs on the damping of the critical mode of the system are shown in Fig. 9.34. The figure also shows the damping of the critical modes without wind power integration; in this case, the impact of the SVCs (see chapter 7) is also demonstrated. It can be seen from these tables and Fig. 9.34 that: 1. The SVCs causes a decline in the system damping in comparison with the fixed capacitors (FCs). It should be noted that the flexibility in the reactive power support and voltage control provided by SVCs is of high importance in the secure operation of power systems. The problem associated with the reduction of damping can be solved by the POD as a supplementary SVC controller for enhancing the system stability. 2. The wind power technologies enhance the system damping if their penetration level is less than the MWP level. Increasing the wind power penetration level to values higher than the MWP level causes instability of the system. This is can be attributed to the reduction of the system inertia caused by the wind power technologies (see section 9.2). Therefore, the integration of wind power causes two opposing effects. The positive effect is the enhancement in the system damping while the negative effect is the reduction in the system inertia. Therefore, even if the total power generation is unchanged due to replacing conventional power by wind power, the replacement level is limited due to the opposing impacts of wind power on the system stability. This is also demonstrated in section 9.2. 3. The impact of DFIGs on the system damping is better in comparison with the FS-SCIGs. In addition, the power addition option has a better impact on the system damping in comparison with the power replacement option. For the given study system, none of the wind power options could enhance the minimum system damping to an acceptable level (i.e. 10% or more). Therefore, PODs are designed for various options and system

417

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structures. The design is based on the same algorithm provided in chapter 7 and acceptable minimum damping ratios are achieved 138,139.

138

M. Mandour, M. EL-Shimy, F. Bendary, and W.M. Mansour. The Design of POD Considering Conventional and Renewable Power Generation. International Electrical Engineering Journal (IEEJ), Vol. 6 (2015) No.7, pp. 1962-1972 139 . Mandour, M. EL-Shimy, F. Bendary, W.M. Mansour. Impact of Wind Power on Power System Stability and Oscillation Damping Controller Design. Industry Academia Collaboration (IAC) Conference, 2015, Energy and sustainable development Track, Apr. 6 – 8, 2015, Cairo, Egypt 418

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Chapter 10 Overview, Characterization, and Modeling of Solar-PV Systems 10.1 Introduction The solar energy is classified as one of the promising renewable-energy worldwide. In the power and energy discipline, the sunlight energy can be involved in energy conversion processes through three main evolving technological categories. As shown in Fig. 10.1, these categories are Photovoltaics (SOLAR-PV), Concentrated Solar Power (CSP), and Solar Heating and Cooling (SHC). This classification is based on the energy conversions processes involved in the solar energy conversion stream.

Fig. 10.1: Fundamental classification of solar energy technologies The classification shown in Fig. 10.1 illustrates only the fundamental categories of solar energy technologies; however, each of the shown classes can be classified to a large number of technologies. These three ways of harnessing the sun are complementary. Therefore, developers should carefully assess their needs, project constraints, meteorological conditions, and environmental impact when choosing a specific solar technology to 419

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use140. This chapter focuses on the solar-PV technologies, which generates electricity through the direct conversion of sunlight. Over the last decade, solar-PV technologies have shown the potential to become a major source of power generation in the world with a strong and continuous growth even during times of financial and economic crisis. This growth is expected to continue in the years ahead as worldwide awareness of the advantages of solar-PV increases. Now, solar-PV is the third most important renewable energy in terms of global installed capacity (after hydro and wind power). At the end of 2009, the world’s SOLAR-PV cumulative installed capacity was approaching 23 GW 141. In 2010, the global photovoltaic market was almost doubled; it was exceeding 40 GW 142. In 2011, more than 70 GW are installed globally and could produce 85 TWh of electricity every year. This energy volume is sufficient to cover the annual power supply needs of over 20 million households143. The year 2012 was another historic one for solar photovoltaic (solar-PV) technology, which has experienced outstanding growth over the past decade. The world’s cumulative solar-PV capacity is currently more than 102 GW 144. This capacity can annually produce at least 110 TWh of electricity and save more than 53 million tones of CO2. Solar energy is expected to play a crucial role in meeting future energy demand through clean energy resources. Existing studies expect the longterm growth (e.g., until 2050) of solar energy vary widely based on a large number of assumptions 145. For example, EPIA/Greenpeace produces the most ambitious forecasts for future solar-PV installation146. The study assumes that if sufficient market support mechanisms are provided, a dramatic growth of 140

El Chaar, L., & El Zein, N. (2011). Review of photovoltaic technologies. Renewable and Sustainable Energy Reviews, 15(5), 2165-2175. 141 Ecogeneration, (2010). Soaring to new heights:Solar PV in Australia. In ecogeneration pp. 1-53, Great Southern Press Pty Ltd, Melbourne. 142 Solarbuzz. Available from: www.solarbuzz.com 143 Masson, G., Latour, M., & Biancardi, D. (2012). Global market outlook for photovoltaics until 2016. European Photovoltaic Industry Association , 5. 144 Masson, G., Latour, M., Rekinger, M., Theologitis, I. T., & Papoutsi, M. (2013). Global market outlook for photovoltaics 2013-2017. European Photovoltaic Industry Association, 12-32. 145 Timilsina, G. R., Kurdgelashvili, L., & Narbel, P. A. (2012). Solar energy: Markets, economics and policies. Renewable and Sustainable Energy Reviews, 16(1), 449-465. 146 European Photovoltaic Industry Association (EPIA). (2011). Solar generation 6: Solar photovoltaic electricity empowering the world. Retrieved February, 23, 2012. Greenpeace International & European Photovoltaic Industry Association: Brussels, Belgium. 420

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solar-PV would be possible, which will lead to worldwide PV installed capacity rising from around 40 GW in 2010 to 737 GW in 2020 to 1845 GW in 2030 to 3256 GW in 2040 to 4669 GW by 2050. The capacity would reach over 1000 GW in 2030 even with a lower level of political commitment. Another study by the International Energy Agency (IEA) 147 estimates solar power development potential based on two scenarios that are differentiated on the basis of global CO2 emission reduction targets. In the first scenario, where global CO2 emission in 2050 are restricted at 2005 level. In this case, the global solar-PV capacity is estimated to increase from 11 GW in 2009 to 600 GW in 2050. In the second scenario, where global CO2 emission is reduced by 50% from 2005 levels in 2050. This is estimated to approach or exceed 1100 GW capacity in 2050. Like solar-PV, predictions are available for CSP technology. The International Energy Agency (IEA) predicts that CSP capacity could reach 380 GW to 630 GW, depending on global targets for GHG mitigation. The lower value represents to the scenario of limiting global CO2 emissions in 2050 at 2005 level, whereas the upper value refers to the scenario to reduce global CO2 emissions in 2050 by 50% from 2005 levels. A study by Greenpeace, The European Solar Thermal Power Industry Association (ESTIA), and the International Energy Agency (IEA) 148 expected that the global CSP capacity would expand by one hundred-fold to 37 GW by 2025 and then skyrocket to 600 GW by 2040. On the same way another study by the Greenpeace, and International and European Renewable Energy Council (EREC) expected that global CSP capacity could reach 29 GW in 2020, 137 GW in 2030 and 405 GW by 2050 149. Chapter 10, 11, and 12 are complementary to each other and can be used as course handling solar-PV generators. Their main objective is to present overview, modeling, simulation, control, and grid-interconnection requirements of solar-PV systems. They may be considered as an extended 147

OECD/IEA (2008). Energy Technology Perspectives: Scenarios and Strategies to 2050; International Energy Agency: Paris, France. 148 Aringhoff, R.; Brakmann, G.; Geyer, M.; Teske, S.; Baker, C (2005). Concentrated Solar Thermal Power- Now! Greenpeace International, European Solar Thermal Power Industry Association (ESTIA): Amsterdam, the Netherlands, IEA SolarPACES Implementing Agreement. 149 Teske, S.; Zervos, A.; Scha¨fer, O (2007). Energy revolution: a sustainable world energy outlook; Greenpeace International-European Renewable Energy Council (EREC): Amsterdam, the Netherlands. 421

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chapter (or part of this book). Therefore, sometimes the phrase ‘this chapter’ will be used for referring to a specific one of these chapters or all the three chapters. Chapter 10 provides an overview of various solar-PV technologies. In addition, detailed quantitative characterization of various technologies is presented. Various models related to the solar-PV projects from the electrical energy prospective are given. In addition, dynamic modeling and analysis of grid-connected solar-PV power plants are provided in details in chapter 11. An enhanced Maximum Power Point Tracking (MPPT) technique is also presented in chapter 11. Analysis of the grid codes for interconnecting solarPV and wind power generators will be presented in chapter 12.

10.2 Qualitative Overview of solar-PV technologies The history of solar-PV technologies starts in 1839 150, when Alexander-Edmund Becquerel observed that electrical currents arose from certain light induced chemical reactions. Later on in 1877, the photovoltaic effect in solid Selenium was observed by Adams and Day. Fritz in 1883 developed the first photovoltaic cell and its efficiency was less than 1%. Discontinuous evolution of PV technologies is observed since those ancient times. Collapse and growth rates in the PV industry have been observed in many periods. For example, the years immediately following the oil-shock in the 1970s saw much interest in the development and commercialization of solar energy technologies; however, this incipient solar energy industry of the 1970s and early 1980s has been collapsed due to the sharp decline in oil prices and a lack of sustained policy support 151. Solar energy market growths as well as technological evolution have regained momentum since the early 2000s, exhibiting phenomenal growth recently. The total installed capacity of solar based electricity generation has increased to more than 40 GW by the end of 2010 from almost negligible capacity in the early nineties 152.

150

Chapin, D. M., Fuller, C. S., & Pearson, G. L. (1954). A new silicon p‐n junction photocell for converting solar radiation into electrical power. Journal of Applied Physics, (25), 676-677. 151 Bradford, T. (2006). Solar revolution: the economic transformation of the global energy industry. MIT Press Books, 1. 152 REN21. Global Status Report 2005–2011; Renewable Energy Policy Network for the 21st century (REN 21) 2005–2011: Paris, France. 422

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Currently, there is a wide range of PV cell technologies on the market. Depending mainly on the basic material used and their level of commercial maturity, PV cell technologies are usually classified into three generations as illustrated in Fig. 10.2; namely, first generation, second generation, and third generation technologies.

Fig. 10.2: PV cell technologies The first generation of PV technologies is mainly made of crystalline structure which uses silicon (Si) to produce the solar cells. These cells are connected together to make PV modules. The first generation technologies are not vanished rather they are constantly being developed to improve their capability and efficiency153. In general, the efficiency of crystalline silicon (cSi) modules range from 14% to 19% and they have a relatively high production cost and subsequently high selling price154. The relatively high costs result from the complex and numerous production steps involved in the silicon wafer and cell manufacturing as well as the large amount of highly purified silicon feedstock required. Continued cost reductions are possible through improvements in materials and manufacturing processes, and from economies of scale if the market continues to grow, enabling a number of high volume manufacturers to emerge. As shown in Fig. 10.2, the first

153

IRENA (2012). Renewable energy technologies: Cost analysis series- solar Photovoltaics. pp. 45, International Renewable Energy Agency, Abu Dhabi. 154 Poullikkas, A. (2010). Technology and market future prospects of photovoltaic systems. International Journal of Energy and Environment , 1(4), 617-634. 423

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generation technologies include the mono-crystalline silicon (mono-Si), polycrystalline silicon (poly-Si), and Gallium Arsenide (GaAS). The Mono-Si technology has higher efficiency in compassion with other crystalline PV materials. The efficiency claims from manufacturer are normally lies between 15% and 17%; however, mono-crystalline solar cells can have an efficiency higher that 20%. The Mono-Si cells are the most commonly used; constitute about 80% of the market and will continue to be the leader until a more efficient and cost effective PV technology is developed. The development of the poly-Si is mainly for the reduction of the flaws, and contamination as well as crystal structure improvement 155. Therefore, multi-Si modules will continue to retain a large portion of the market in the future. The GaAs is a compound semiconductor form by Gallium (Ga) and Arsenic (As) that has similar crystal structure as silicon. Compared to silicon based solar cells, GaAs has higher efficiency and less thickness. The GaAs is normally used in concentrator PV module and for space application since it has high heat resistance and high cell efficiency reaching about 25 to 30%; however, GaAs material and manufacturing process can be costly in comparison with silicon based solar cells 156. In comparison with the first generation technologies, the second generation technologies hold the promise of reducing the cost of PV array by lowering material and manufacturing costs without affecting the lifetime of the cells as well as their environmental hazards. Thin film technology is the core of the second generation technologies. Unlike crystalline solar cells, where pieces of semiconductors are sandwiched between glass panels to create the modules, thin film panels are created by depositing thin layers of certain materials on glass or stainless steel substrates using sputtering tools. The advantage of this methodology lies on the fact that the thickness of the deposited layers which are a few microns (smaller than 10 m) thick compare to crystalline wafers which tend to be several hundred microns thick. In addition, it is possible to film deposited on stainless steel sheets which allows the creation of flexible PV modules. As a result the overall manufacturing

155

Manna, T. K., & Mahajan, S. M. (2007, May). Nanotechnology in the development of photovoltaic cells. In Clean Electrical Power, 2007. ICCEP'07. International Conference on (pp. 379-386). IEEE. 156 Mah, O. (1998). Fundamentals of photovoltaic materials. National Solar Power Research Institute, 1-10. 424

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costs are lowered. Technically, reducing the thickness of the PV layers reduces the solar radiation absorption and consequently reduces the efficiency of thin film cells in comparison with crystalline cells; however; this can be bypassed by using appropriate alloys for enhancing the efficiency. In addition, the material flexibility of thin film technologies facilitates the building-integrated PVs (BIPV)157. These distinguishing characteristics of thin film technologies along with their temperature withstand-ability increase the market share of these technologies to about 15% - 20%. The main commercially available types of thin-film solar cells are the Amorphous Silicon (a-Si and a-Si/µc-Si), Cadmium Telluride (CdTe), Copper-IndiumSelenide (CIS), and Copper-Indium-Gallium-Diselenide (CIGS). The third-generation PV technologies are at the pre-commercial stage and vary from the technologies under the demonstration (e.g. multi-junction concentrating PV) to novel concepts still require basic R&D activities (e.g. Quantum-dots (QD) PV cells, QD meta-materials are a special semiconductor system that consists of a combination of periodic groups of materials molded in a variety of different forms. They are on the nanometer scale and have an adjustable band-gap of energy levels performing as a special class of semiconductors). Some third-generation PV technologies are beginning to be commercialized, but it remains to be seen how successful they will be in taking market share from existing technologies. There are four types of thirdgeneration PV technologies: Concentrating PV (CPV), Dye-sensitized solar cells (DSSC), Organic solar cells, and Novel and emerging solar cell concepts. Due to lack of sufficient information, the third generation technologies will not be considered in this paper.

10.3 Quantitative Characterization of solar-PV technologies The previous section provided a qualitative overview of various solar-PV technologies. In this section, various technologies are characterized through a quantitative approach. The results are then very in selecting and matching a specific technology and module to a desired application considering various technical and economic constraints.

157

Sheikh, N. M. (2008, March). Efficient utilization of solar energy for domestic applications. In Electrical Engineering, 2008. ICEE 2008. Second International Conference on (pp. 1-3). IEEE. 425

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With the available PV technological alternatives and the large number of PV modules in each technological alternative, the design engineer faces a challenge in selecting the best PV module under a specified project design criteria and constraints. Therefore, this paper provides a decision aid through site-independent quantitative characterization of various PV technologies and modules. In this regard, two stages are performed 158,159.  In the first stage, various PV technologies are characterized as shown in Fig. 10.3 by their peak wattage (Wp) per unit area (Wp/area), efficiency, lifetime, degradation rate, temperature coefficient, and cost. These characterization parameters are used to arrange various technologies according to various goals of technology selection. The goals include Wp/area maximization, efficiency maximization, lifetime maximization, degradation minimization, temperature coefficient minimization, and cost minimization. The best PV technology is determined based on the achievement of these selection goals. Within a specific PV technology, the characteristics of commercially available PV modules differ significantly.

Fig. 10.3: Parameters of technological characterization of PV modules T. Abdo, M. EL-Shimy, “Quantitative Characterization and Selection of Photovoltaic Technologies”, MEPCON 2014. 159 M. EL-Shimy, T. Abdo, “PV Technologies: History, Technological Advances, and Characterization’, In Sohail Anwar (ed.) Encyclopedia of Energy Engineering and Technology, Taylor & Francis - CRC Press, 2014.

158

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 In the second stage, the best PV module for each of the considered technologies is determined. The PV module selections goals are the peak wattage per unit area (Watt-peak/area), and the efficiency. For each PV technology, the best module is determined according to the fulfillment of the objectives i.e. maximization of the Watt-peak/area and efficiency. The global best PV module is then determined. It is worthy to be mentioned that the presented selection and characterization approach is site-independent. This is a point of strength in comparison with the capacity factor (CF) based selection 160 which is a sitedependent method and it is mainly used for optimal site matching of solarsolar-PVs. Since, each solar-PV technology has its own features that limit its practical application (for example, thin film technologies are significantly suitable in building integration solar-PV in comparison to crystalline technologies due to its flexibility), then the final selection of PV technology and module should be based on the project model. The characterization of PV technologies and modules requires a huge amount of data. Therefore, the availably of the required data in an accurate form is a challenge. The RETScreen product database161 is used herein as a data source. This database provides an up-to-date information about the Wp, area, and efficiency data of various PV technologies and modules. In addition, the database includes almost all the commercially available PV modules; 1210 PV modules where 332 modules are mono-Si, 769 modules are poly-Si, 78 modules are a-Si, 14 modules are CdTe, and 17 modules are CIS. The included PV technologies in the RETScreen database are mono-Si, poly-Si, a-Si, CdTe, and CIS. Therefore, the first and second generation technologies are available in the RETScreen database. The rest of the required data for assessing the lifetime, degradation rate, and module cost of

160

Salameh, Z. M., Borowy, B. S., & Amin, A. R. (1995). Photovoltaic module-site matching based on the capacity factors. Energy Conversion, IEEE Transactions on, 10(2), 326-332. 161 "Renewable energy project analysis software", in RETScreen International Clean Energy Decision Support Centre, Minister of Natural Resources Canada. 427

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various PV technologies is obtained from162,163,164 respectively. In this section analysis of the available data, characterization , and selection criteria are presented.  First Stage: Technological Characterization At this stage, various PV technologies are characterized considering the characterization parameters shown in Fig. 10.3. The ranges (maximum, average, minimum, and standard deviation) of each characterization parameter are determined for each technology. The standard deviation () shows how much variation or spreading exists for the average, or expected value. A low standard deviation indicates that the data points tend to be very close to the mean value while high standard deviation indicates that the data points are spread out over a large range of values 165. A module efficiency depends primarily on the type of the PV cell; however, within various types there are wide variations in module efficiency from manufacturer to another, depending on the manufacturing processes used. Based on analysis of the RETScreen PV database, Fig. 10.4 shows the results of PV efficiency analysis. It is depicted from Fig. 10.4, that the efficiency of the mono-Si PV technology is the highest followed by Poly-Si then CIS then CdTe then a-Si. The standard deviation of the efficiency values is less than 2% for all technologies; however, the lowest standard deviation is associated with the a-Si technology and the highest is associated with the poly-Si technology. This means that the efficiency of the first PV generation is still the highest in comparison with the rest of technological generations. This is attributed to the maturity of the manufacturing process. The modules and their manufacturers that are associated with the maximum and minimum efficiency values for each PV technology are shown in Table 10.1. It is shown in Table 10.1 that manufacturers that produce PV modules with the highest efficiency are Sunpower, Canadian solar, Q-cells, and abound. 162

Sherwani, A. F., & Usmani, J. A. (2010). Life cycle assessment of solar PV based electricity generation systems: A review. Renewable and Sustainable Energy Reviews, 14(1), 540-544. 163 Jordan, D. C., & Kurtz, S. R. (2013). Photovoltaic degradation rates—an analytical review. Progress in photovoltaics: Research and Applications, 21(1), 12-29. 164 PVXChange, http://www.pvxchange.com/ 165 DasGupta, A. (2010). Fundamentals of probability: a first course . Springer Science & Business Media. 428

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Fig. 10.4: Efficiency analysis of various PV technologies Table 10.1: PV modules with the lowest and highest efficiencies and their manufacturers Lowest efficiency

PV tech.

 (%)

Mono

7.58

Poly

6.07

a-Si

1.81

CdTe

4.86

STP005S12 ND-070 ERU Uni-Pac 10W(24V) CX-35W

CIS

6.60

CIGS 50W

Model

Manuf.

 (%)

Suntech

Highest efficiency Model

Manuf.

19.62

SPR-320EWHT

Sunpower

Sharp

17.59

CS 6A 190W

Canadian Solar

Uni-Solar

8.16

SN2-145.0W

Q-Cells

Q-Cells Centennial Solar

10.07

AB1-72

Abound

11.33

SL 1-85W

Q-Cells

The efficiency of solar PV cells varies with their operating temperature. Most cell/module types display a decrease in efficiency as their temperature increases . The PV temperature coefficient (µ expressed in % /°C), is defined by:

where (T) is the efficiency of the solar cell at an operating temperature T,

and (Tref) is the efficiency of the cell at the reference temperature Tr ef (usually 25oC). 429

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The ideal value of the temperature coefficient is zero. With this value, the efficiency is insensitive to temperature variations. It can be seen from equation (10.1) that for a specific operating and reference temperature, the efficiency decrease as the temperature coefficient increase. Therefore, the sensitivity of the PV efficiency increases with the increase in the temperature coefficient. The value of the temperature coefficient depends primarily on the cell type and it is difficult to find this value on manufacturer data sheets. Therefore, in the absence of this information the default values shown in Fig. 10.3 may be used as recommended by the RETSCreen. It is depicted from Fig. 10.3 that the efficiency values of the thin film technologies expect the CIS are less sensitive to temperature variations in comparison with the crystalline silicon. The efficiency of the CIS technology has the highest sensitivity to temperature which is significantly higher than the crystalline silicon sensitivity.

Fig. 10.5: Default temperature coefficient values for various PV Technologies The Wp/A or the Watt-peak/area is the peak power capacity of a PV module (in Wp) over its frame area. The high value of the Wp/A indicates better land use and higher density of the energy production. The Wp/A can be used as a selection index for PV modules 166. Based on analysis of the RETScreen PV database, Fig. 10.6 shows the results of Watt-peak/area analysis. 166

El-Shimy, M. (2009). Viability analysis of PV power plants in Egypt. Renewable Energy, 34(10), 2187-2196. 430

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Fig. 10.6: The WP /A of various PV technologies It is depicted from Fig. 10.6, that the Watt-peak/area of the mono-Si PV technology is the highest value followed by Poly-Si, CIS, CdTe and a-Si technologies. This means that the first PV generation is still has the highest Watt-peak/area in comparison with the other technological generations. The modules and their manufacturers that are associated with the maximum and minimum Watt-peak/area values for each PV technology are shown in Table 10.3. It is shown in Table 10.3 that manufacturers that produce PV modules with the highest Watt-peak/area are Q-cells, and abound. Table 10.3: The highest and lowest WP /A for PV technologies and their manufacturers PV tech

Wp/A

Mono 71.43 Poly 15.00 a-Si

18.10

CdTe 48.61 CIS

66.67

Low Wp/A Model Manuf. STP005S-12 Suntech YL15(17)P Yingli Solar Uni-Pac Uni-Solar 10W(24V) -CX-35W Q-Cells Centennial CIGS 50W Solar

200.00 200.00

High Wp/A Model Q6LM-1680 Q6LTT-1640

Manuf. Q-Cells Q-Cells

81.46

SN2-145.0W

Q-Cells

100.00

AB1-72

Abound

113.33

SL 1-85W

Q-Cells

Wp/A

The ability to predict output power over the course of time is important in the prediction of the long-term performance of various PV technologies. 431

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Among the key drivers in the PV industry growth are the efficiency (with which sunlight is converted into power) and how this relationship changes over time (i.e. the degradation rate). The quantification of power production decline or diminishing over a time course is known as a degradation rate167,168,169. The degradation occurs due to chemical and material processes associated with weathering, oxidation, corrosion, and thermal stresses. Financially, the degradation of a PV module or system is equally important, because a higher degradation rate translates directly into less power produced and therefore, reduces future cash flows. Solar-PV systems are often financed based on an assumed (0.5 – 1.0) % per year degradation rate 170; however, 1% per year is used based on modules warranties171. This rate is faster than some historical data given for silicon PV. In a study172, more than 70% of 19–23 year-old c-Si modules had an annual degradation rate of 0.75% which is still less than the 1% year assumed. Technically, the degradation mechanisms are important to be understood because they may for the long run lead to failure. The classification of degradation mechanisms through experiments and modeling can lead directly to lifetime improvements. Typically, a drop in the output power production by 20% at standard conditions is considered as a failure that requires partial of full replacement of the PV modules; however, this is not an agreement with the definition of a failure from the reliability point of view. This is because the modules are still able to produce electricity. In

M. EL-Shimy, “Analysis of Levelized Cost of Energy (LCOE) and grid parity for utility-scale photovoltaic generation systems”, 15th International Middle East Power Systems Conference (MEPCON’12), Dec. 23-25, 2012, Alexandria, Egypt, pp. 1- 7. 168 M. Saed, M. EL-Shimy, and M. Abdelraheem, “Photovoltaics Energy: Improved modeling and analysis of the Levelized Cost of Energy (LCOE) and Grid Parity”, Sustainable Energy Technologies and Assessments, vol. 9, pp. 37-48, 2015. 169 Jordan, D. C., & Kurtz, S. R. (2013). Photovoltaic degradation rates—an analytical review. Progress in photovoltaics: Research and Applications, 21(1), 12-29. 167

170

Campbell, M., Aschenbrenner, P., Blunden, J., Smeloff, E., & Wright, S. (2008). The drivers of the levelized cost of electricity for utility-scale photovoltaics. White Paper: SunPower Corporation . 171 Zweibel, K., Mason, J., & Fthenakis, V. (2008). A solar grand plan. Scientific American, 298(1), 64-73. 172 Skoczek, A., Sample, T., & Dunlop, E. D. (2009). The results of performance measurements of field‐aged crystalline silicon photovoltaic modules. Progress in Photovoltaics: Research and Applications, 17(4), 227-240. 432

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addition, a high efficiency module degraded by 50% may still have a higher efficiency than a non-degraded module from a less efficient technology. Fig 10.7, shows the degradation rates of various solar-PV technologies. These summarized rates are the long-term degradation rates and they do not include short-term, light-induced degradation rates. The light-induced degradation rate effect is generally ascribed to boron-oxygen (B-O) defects in the cell itself which is a defect complex that is formed by prolonged exposure of solar cells to light and is accompanied by a reduction in the minority-carrier lifetime in the bulk of the cell. This degradation in the lifetime of the wafer is thought to be fully recoverable upon annealing, which causes the dissociation of the B-O complex173.

Fig 10.7: Degradation rates of various PV technology In Fig 10.7, the denotations “pre” and “post” refer to a date of installation prior to and after the year 2000, respectively. The selection of year 2000 was chosen such that the number of data points in each category is approximately equal. It is depicted from this figure that the crystalline Si technologies show low degradation rates in comparison with other technologies for pre-2000 and post-2000 categories; however, the figure shows that there are a significant reduction in degradation rates of the thin-film technologies in the analysis the post-2000 in comparison with the pre-2000 installations. 173

Sopori, B., Basnyat, P., Devayajanam, S., Shet, S., Mehta, V., Binns, J., & Appel, J. (2012, June). Understanding light-induced degradation of c-Si solar cells. In Photovoltaic Specialists Conference (PVSC), 2012 38th IEEE (pp. 001115-001120). IEEE. 433

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Generally, the working life of an asset is the life for which it continues to perform its tasks effectively. It is often true that the operation and maintenance (O&M) costs rise with the age of the asset. Since annual capital costs tend to decline and annual O&M costs rise, there is a minimum average cost per year at which point it is considered the economic life of the asset. This is called the economic lifetime. By the end of the economic life, the asset is then replaced or refurbished, since thereafter it becomes more expensive to run the asset174. The financial life of a solar PV system is usually considered to be the guarantee period offered by the manufacturer which is often 20–25 years; however, researches show that the life of solar PV panels is well beyond 25 years even for the older technologies. The current technologies are likely to improve lifetime further175. The O&M costs are due to replacing inverters (usually every 5 or 10 years), occasional cleaning, and electrical system repairs. It is expected that some of these costs, such the inverter, will decrease with time. It should also be noted that the life of many conventional power plants is much longer than the rated lifetime since they tend to be theoretically refurbished or re-commissioned indefinitely, then the same could be true of solar-PV plants. Therefore, the economic life of the system depends on the acceptable energy output which is mainly dependant on the degradation rate. On the other hand, the predicted or expected lifetime of a photovoltaic module is one of the main factors which define the levelized cost (LCOE) of PV solar electricity176. Extensive effort has been made for more than 20 years to define, predict and assure the lifetime of PV solar modules from the manufacturers, the research laboratories and international standards bodies. The effort to assure the functionality, quality and lifetime of PV modules is introduced by the International Electro-technical Commission (IEC) through accelerated test procedures called type approval tests . These test procedures 174

Leblanc, S., & Roman, P. (2005). Project Management: An Engineering Economics Perspective. Pearson Custom Publishing. 175 Skoczek, A., Sample, T., & Dunlop, E. D. (2009). The results of performance measurements of field‐aged crystalline silicon photovoltaic modules. Progress in Photovoltaics: Research and Applications, 17(4), 227-240. 176 M. Saed, M. EL-Shimy, and M. Abdelraheem, “Photovoltaics Energy: Improved modeling and analysis of the Levelized Cost of Energy (LCOE) and Grid Parity”, Sustainable Energy Technologies and Assessments, vol. 9, pp. 37-48, 2015. 434

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seek to assure, based on accelerated climatic testing, that the PV modules which successfully pass these tests would reasonably be expected to survive for 20 years in field operation. These tests however do not give a precise measure of the lifetime 177. The commonly used three technology types of solar-PV systems had been discussed based upon the lifetime assessment. The lifetimes have been estimated for amorphous, mono-crystalline, polycrystalline PV systems. It is found that the average lifetime for each technology is approximately to be the same (26 year). A PV module warranty usually refers to two different issues. The first is the warranty for materials and workmanship , which usually ranges from 1 to 5 years. The second is the warranty on the power produced by the PV module which typically guarantees that after the first 10–12 years the output power of the module will be at least 90% of its initial nominal power and that after 20– 25 years of operation the output power of the module will be at least 80% of its initial nominal power178. Recently, the manufacturers have redefined the concept of output power warranty which is now referred to as the minimum nominal power taking into account module power tolerances, and not with respect to the nominal power. The measurement tolerance is also quantified as an extra 3% of uncertainty179. The average PV module power warranty has increased from 5 years before 1987 to 25 years since 1999. Obviously, these warranties are not simply the result of thorough tests in the field because the modules have not been in the market long enough, but they are probably the combined outcome of empirical approaches, field tests of limited durations and a set of particular degradation tests that each manufacturer developed for this purpose. In this book, the effective lifetime (L.Teffective) of a PV module is defined as the span time at which the module will be able to effectively produce power such that by the end of the effective lifetime the energy production drop is 177

Dunlop, E. D., Halton, D., & Ossenbrink, H. (2005, January). 20 years of life and more: where is the end of life of a PV module?. In Photovoltaic Specialists Conference, 2005. Conference Record of the Thirty-first IEEE (pp. 1593-1596). IEEE. 178 Vázquez López, M., & Rey-Stolle Prado, I. (2008). Photovoltaic module reliability model based on field degradation studies. Progress in Photovoltaics: Research and Applications, 16(5), 419-433. 179 Cereghetti, N., Bura, E., Chianese, D., Friesen, G., Realini, A., & Rezzonico, S. (2003, May). Power and energy production of PV modules statistical considerations of 10 years activity. In Photovoltaic Energy Conversion, 2003. Proceedings of 3rd World Conference on (Vol. 2, pp. 1919-1922). IEEE. 435

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equal to 20%. In order to calculate the L.Teffective of a selected PV technology

module, the degradation rate of the PV technology should be known. Based on the given definition, the effective lifetime (L.Teff) in years can be easily determined based on Fig. 10.8 which shows the energy production decline with time; where d is the percentage degradation rate of the PV module.

Considering the post-year 2000 values of degradation rates shown in Fig 10.7 and equation (10.2), the determined effective lifetime values are shown in Table 10.4 and Fig. 10.9. This results also shows the percentage change of the effective lifetime in comparison with an assumed financial lifetime of 25 years (L.Tfinancial). The percentage change in the lifetime (L.T) is calculated using,

Fig. 10.8: The effective lifetime of a PV module

Table 10.4: Effective and percentage change of the lifetime of various PV technology PV Tech. Mono-Si Poly-Si a-Si CdTe CIS

ΔL.T (%) Minimum Average Maximum Minimum Average Maximum 30 41 51 20 64 104 23 30 41 -8 20 64 15 19 26 -40 -24 4 21 37 101 -16 48 304 12 18 34 -52 -28 36 L.Teff (years)

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It is depicted that the minimum effective lifetime of mono-Si is higher (30

years; L.T = 20%) than the financial 25 year lifetime while it is lower than that value in the rest of the technologies. The lowest minimum effective

lifetime is associated with the CIS modules (12 years; L.T = -52%). The

minimum effective lifetime of poly-Si (23 years; L.T = -8%) is close to the financial lifetime while it is significantly lower in the a-Si technology (15 years; L.T = -40%). From the average effective lifetime point of view, the results show that all the considered technologies, except the a-Si and CIS, have effective lifetimes higher than the financial lifetime. Highest average

effective lifetime is associated with the mono-Si (41 years; L.T = 64%) and the lowest one are associated with the CIS (18 years; L.T = -28%) which is still lower than the financial lifetime. It is also shown that the average

effective lifetime of the a-Si modules (19 years; L.T = -24%) is less than the financial lifetime. From the maximum effective life time point of view, the results show that all the considered technologies, except the a-Si, are having effective lifetimes that are higher than the financial lifetime. The highest maximum effective lifetime is associated with CdTe modules (101 years; L.T = 304%) while the lowest maximum effective lifetime is associated with a-Si modules (only 26 years; L.T = 4%).

Effective lifetime (years)

120 L.Teff (years) Min. L.Teff (years) Av.

100 80 60 40 20 0

Mono-Si

Poly-Si

a-Si

CdTe

CIS

L.Teff (years) Min.

30

23

15

21

12

L.Teff (years) Av.

41

30

19

37

18

L.Teff (years) Max.

51

41

26

101

34

Fig. 10.9: Effective lifetime of various solar-PV technologies

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If the minimum effective lifetimes are considered pessimistic and the maximum effective lifetimes are considered optimistic, then the average effective lifetime is considered the probable one. Therefore, Table 10.4 and Fig. 10.9 shows that both the a-Si and CIS modules will not survive to the end of the financial lifetime while the rest of technologies will survive for durations higher than the financial lifetime. Driven by advances in technology and increases in manufacturing scale, the cost of PV cells has declined steadily since the first solar cells were manufactured180. Although the cost of electricity produced from PV systems is still higher than the other competing technologies, this cost is expected to continue to decrease steadily. The cost of PV installation was USD 2 per unit of generating capacity in 2009 which came down to about USD 1.50 in 2011. According to industry analysis, this price is expected to be lowered in the future. These potential reductions in the cost, combined with the simplicity, versatility, reliability, and low environmental impact of PV systems, should help PV systems become highly utilized sources of economical, premiumquality power over the next 20 – 30 years181. Due to the high installation costs, the recent solar electric prices are at approximately USD 0.30/kWh, or around 2–5 times the average residential electricity tariffs (the actual ratio depends on the location of the installation and local electricity rates). On the other hand, solar energy can provide the basic energy needs for houses in remote and rural areas at a fraction of the cost spent on traditional electricity. Thus, the cost of electricity from PV systems is relatively cost effective in that and similar applications. As expected, the demand for solar powered systems is very high in countries with high electricity tariffs. On the other hand to get insight about the PV module prices, the pvXchange website provides a closed trading platform that provides spot market prices of various PV technologies. The prices of the year 2012 are shown in Fig. 10.10. It is clear from Fig. 10.10 that the prices of all technologies decline with time even on a monthly basis. In addition, the prices of the crystalline modules made in Germany and Japan are comparable but they are 180

Swanson, R. M. (2009). Photovoltaics power up. Science, 324(5929), 891-892. Devabhaktuni, V., Alam, M., Depuru, S. S. S. R., Green, R. C., Nims, D., & Near, C. (2013). Solar energy: Trends and enabling technologies. Renewable and Sustainable Energy Reviews, 19, 555-564. 181

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significantly higher than the prices of the crystalline modules made in China. The prices of the a-Si modules are the lowest followed by the Cds/CdTe modules then the a-Si/-Si modules and the China crystalline modules, then Germany and Japan crystalline modules. Although the prices of the a-Si modules are the lowest as shown in Fig. 10.10, they are also having a low average effective lifetime as shown in Table 10.4. Fig. 10.10 also shows that China crystalline modules offer a very low price in compassion to other crystalline modules and at the same time they offer a high average lifetime as shown in Table 10.4 (for the crystalline technologies). That is a probable reason for the high potential of the China PV market and production.

Fig. 10.10: Module prices of various PV technologies for year 2012 B. Second stage: module selection

In this stage, the best PV module in each of the considered technologies is determined. The PV module selections goals are the Wattpeak/area, and the efficiency. For each PV technology, the best module is determined according to the fulfillment of the objectives i.e. maximization of the Watt-peak/area and the efficiency. For each of the considered PV technologies, the top modules that fulfill each of the stated objectives are shown in Tables 10.5. As expected, it is not necessary that the PV module that has the maximum efficiency will also have the highest Watt-peak/area. Of course high efficiency results in high energy production and high Watt439

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peak/area results in lower land use. Therefore, the selection of an appropriate module will be based on the design requirements and limitations. In addition, since each PV technology has its own features that limit its practical application, and then the final selection of PV technology and module should be based on the project model. For example, thin-film technologies are the best for BIPV while crystalline technologies as well as thin-film technologies are suitable for other standalone and grid-connected applications. Tables 10.5: The best PV modules in various technologies Manufacturer

Efficiency() Wp/A (Watt/m2)

Module

Mono-crystalline Sunpower

SPR-320E-WHT

19.62

197.53

Q-Cells

Q6LM-1680

16.80

200.00

Poly-Crystalline Canadian Solar

CS 6A 190W

17.59

146.15

Q-Cells

Q6LTT-1640

16.40

200.00

8.16

81.46

10.07

100.00

11.33

113.33

a-Si Q-Cells

SN2-145.0W CdTe

Abound

AB1-72 CIS

Q-Cells

SL 1-85W

In the mono-Si technological category, it is clear from Tables 10.5 that the Sunpower mono-Si-SPR-320E-WHT offers the highest efficiency (19.62%) with a Watt-peak/area of 197.53 Watt/m2 while the Q-cells mono-Si-Q6LM1680 module offers the highest Watt-peak/area (200 watt/m2) with an efficiency of 16.8%. Since the Watt-peak/area gained from the Q-cells module is insignificant in comparison with the Sunpower module and the efficiency of the Sunpower module is significantly higher than the Q-cells module, then the best found mono-Si module is the Sunpower mono-Si-SPR320E-WHT. In the poly-Si technological category, it is clear from Tables 10.5 that the Canadian Solar poly-Si-CS 6A 190W offers the highest efficiency (17.59%) with a Watt-peak/area of 146.15 Watt/m2 while the Qcells poly-Si-Q6LTT-1640 module offers the highest Watt-peak/area (200 watt/m2) with an efficiency of 16.4%. Since the Watt-peak/area gained from 440

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the Q-cells module is significant in comparison with the Candian Solar module and the efficiency of the Canadian Solar module is not much higher than the Q-cells module, then the best found poly-Si module is the Q-cells poly-Si-Q6LTT-1640. As shown in Tables 10.5, the best found modules in the a-Si, CdTe, and CIS technological categories respectively are the Q-cells a-Si-SN2-145.0W, Abound CdTe-AB1-72, and Q-cells CIS-SL 1-85W. A comparison between the efficiency and the Watt-peak/area values of the best found modules in various technologies are illustrated in Fig. 10.11. In order to show them clearly on the same scale with the Watt-peak/area, the efficiency values are multiplied by 10. It is clear from Fig. 10.11 and Table 10.5 that the highest efficiency is associated with the mono-Si module followed by the poly-Si module, then CIS module, then the CdTe module, then the a-Si module. This is in conformance with the general results shown in Fig. 10.4. In addition, it is clear that the Watt-peak/area of the mono and poly crystalline modules are compared with a slight gain in the poly-Si module. Therefore, the highest Watt-peak/area is associated with the crystalline modules followed by the CIS module, then the CdTe module, then the a-Si module. This is also in conformation with the general results shown in Fig. 10.6.

Fig. 10.11: Comparison between the best found modules

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10.4 Fundamentals and Applications of solar-PV systems Fig. 10.12 shows a basic scheme of solar-PV electric power generators and load types as well as examples of the energy storage options. Depending on the system requirements, some of the shown components may be absent. As shown in the figure, the main components of a solar-PV generator are: (1) Solar-PV array; (2) Sun tracking mechanism; (3) Maximum Power Point Tracking (MPPT) controller; and (4) Power Conditioning Unit (PCU). 10.4.1 Solar-PV arrays

The solar-PV array is made up with a number of solar-PV modules (or panels). These modules are electrically connected for providing specific current and voltage ratings. Based on the rating of one module, the number of series connected modules defines the voltage rating of the solar-PV generator while the number of parallel modules defines the current rating of the generators. PV-modules are commercially available as a unit while the connection between them is based on the system design and operational requirements. In the manufacturing process, PV-modules are formed by series/parallel connection of a number of solar-PV cells (see Fig. 10.13). These internal connections and the current/voltage rating of the cell technology define the rating of the PV-module. The main function of the solar-PV generator is to transform the solar energy to DC electrical energy. Details about the physics of this transformation are outside the scope of this book; however, the fundamentals of this energy conversion process as well as details of the technical requirements for connecting cells and modules are available at many resources including the pveducation.org website 182. As shown in Fig. Fig. 10.14(a), two main dependent characteristics can be used for describing the performance of solar cells. These characteristics are the I-V and P-V curves. The open-circuit voltage (Voc) and short-circuit current (Isc) are illustrated at the figure. As shown in this figure, there is a specific operating point at which the power extracted from the cell is the maximum value. This point is called the maximum power point (MPP); P mp. The voltage and current values at the MPP are Vmp and Imp. In an ideal theoretical solar cell, the Vmp = Voc and Imp = Isc; however, realistic cell do not provide this relation. The fill factor (FF) defines the deviation of this relation 182

http://pveducation.org/ 442

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with respect to the ideal cell; FF = (Vmp Imp)/( Voc Isc). The efficiency () of the cell is determined as P mp/(H t A) where the solar irradiance incident on the plan of the cell (H t) in unit power/m2 and A is the cell surface area exposed to the solar radiation in m2. The value of P mp equals to Vmp Imp.

(a)

(b) Fig. 10.12: Solar-PV systems; (a) Generators and their possible load types; (b) Energy storage options The characterizing operating points ( Voc, Isc, Vmp, Imp, P mp) are usually determined and provided by the manufacturer at specific conditions called Standard Test Conditions (STC) which are mainly 1000 W/m2 solar 443

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radiations incident on the plan of the solar-PV cell at 25oC (77oF) of average air temperature and 1.5 air mass (or atmospheric density) 183,184.These characterizing operating points ( Voc, Isc, Vmp , Imp, P mp) are sensitive to both the incident solar radiation at the cell surface and the surface temperature of the cell. The effects of the solar irradiance and the temperature of the cell characteristics are illustrated in Fig. 10.14(b) to (d) respectively. As shown in Fig. 10.14(b), the changes in the irradiance have a significant impact on the short-circuit current and a minor impact on the open-circuit voltage. Increasing the irradiance increases the short-circuit current and the open-circuit voltage as well as the maximum power. The MPP voltage is increased with the increase in the irradiance. It is clear from Fig. 10.14(c) that the open-circuit voltage rapidly increases with the increase of the irradiance from zero. Shortly after that rapid increase, the open circuit voltage becomes less sensitive to the changes in the irradiance. The short-circuit current shows a linear direct relation with the changes in the solar irradiance. On the other hand, the changes in the cell temperature (Fig. 10.14(d)) have an insignificant impact on the short-circuit current and a significant impact on the opencircuit voltage. Increasing the temperature slightly increases the short-circuit current and significantly reduces the open-circuit voltage. Consequently, the maximum power is reduced with the increase in the temperature and the MPP voltage is also reduced. The characteristics of either modules or arrays can be easily derived from the characteristics of a cell. For example, a module having Ncells cells 183

IEC 61215, IEC 61646 and UL 1703 The solar-PV manufacturers usually use the STC to determine the characteristics of solar-PV modules. One the other hand, the PVUSA Test Conditions (PTC) which is sometimes also called performance test conditions is developed to test and compare solarPV systems as part of the PVUSA ( Photovoltaics for Utility Scale Applications) project, which was undertaken by a partnership between the US Department of Energy and several major utilities. PTC and STC are different test basis; however, the PTC is considered to be a better reflection of real operational conditions in comparison with the STC. This is because PTC ratings are based on conditions that are closer to typical outdoor operating environments. Therefore, PTC ratings are more realistic than nameplate (i.e. STC) ratings. Unlike the STC ratings which are measured values, the PTC ratings are calculated values. The Normal Operating Cell Temperature (NOCT) ratings are adopted by power utilities for more accurate estimation of the performance of solar-PV modules in the real world conditions; 800 W/m2 of sunlight irradiance at the plan of the module, 20°C (68°F) average air temperature, and 1.0 m/s (2.24 miles/hour) of wind speed with the back side of the solar panel open to that wind breeze. A method of estimating the PTC from NOTC is stated at http://goo.gl/ro1gX2 
 184

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connected in series and Mcells cells connected in parallel, will then approximately have,

(a)

(b) Fig. 10.13: Solar-PV. (a) Cell, module, and array; (b) Connection of cells in a module; 36 cells are connected in series for producing a rated voltage of 12 V.

Similarly, an array consists of Nmod modules connected in series and Mmod modules connected in parallel, will then approximately have,

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(b)

(a)

(c)

(d)

(e) Fig. 10.14: Characteristics of solar-PV arrays; (a) Characteristics of a cell; (b) Impacts of change in the solar irradiance; (c) Changes in Isc and Voc with the changes in the irradiance; (d) Impacts of changes in the temperature; (e) Characteristics of the Sanyo HIT 215W module at 25 oC. 446

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Solar-PV cells can be represented by equivalent circuits such as that shown in Fig. 10.15. This circuit includes a series resistance ( Rs) and a diode in parallel with a shunt resistance ( Rsh). The letter V represents the voltage at the load. This equivalent circuit can be used for an individual cell, for a module consisting of several cells, or for an array consisting of several modules.

(a)

(c) (b) Fig. 10.15: Equivalent circuit of PV cells; (a) Circuit diagram; (b) Impact of the series resistance; (c) Impact of the shunt resistance

The impacts of the series and the shunt resistances on the I-V characteristics are also shown in the figure. It is clear that increasing the series resistance decreases the MPP voltage and MMP current. Consequently, the maximum power and the efficiency are reduced with the increase in the series resistance. On the other hand, increasing the shunt resistance has an opposite impact in comparison with the impact of the series resistance. The figure also shows a way of determining these resistances from measured I-V curves. The series resistance is determined as the negative of the slop of the I447

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V curve at open-circuit conditions while the shunt resistance equals to the negative of the slop at short-circuit conditions. In an ideal cell, the shunt resistance is infinite while the series resistance is zero. As shown in Fig. 10.14(a), the overall resistances of a cell can be represented by the characteristic resistance (Rch) which is the negative slop of the straight line connecting the origin and the MPP current and voltage. This resistance is of major importance in the operational optimization of solar cells. According to the maximum power transfer theory, a cell supplying a resistive load of a resistance equals to the cell’s characteristics resistance will be supplied by the maximum power . This is important in either matching the load and the solar-PV array or the MPPT. It is also clear from the shown characteristics that the characteristic resistance is sensitive to the irradiance and the temperature as well as the series and shunt resistance of the cell. There are two tracking control systems for optimizing the performance of solar-PV generators. These tracking systems are the MPPT and the sun tracking. The ideal MPPT185 is a control system (see Fig. 10.16) that extract the maximum power for all operating conditions. This is can be achieved by controlling the load resistance at the terminals of the array to exactly match the array characteristic resistance. In another way, the MPPT modifies the terminal voltage of the array to be of the same value as the MPP voltage for maximum power transfer. The sun tracking system changes the position of the surface of the array for maximization of the irradiance incident at the plan of the modules. Both tracking systems will be briefly explained in the following.

Fig. 10.16: MPPT by DC-DC boost converters 185

The MPPT is an electronic DC to DC converter that optimizes the match between the solar-PV array (or module) and the connected load or the grid 448

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10.4.2 PCU and MPPT

The PCU is a functional block that handles one or more functions in one physical unit or more units. The main functions of the PCU includes the DC-AC inversion, harmonic filtering and possibly some other power quality (PQ) enhancements, and surge protection. According to the load solar-PV generator, DC and AC power are available; however, supplying AC loads from the solar-PV generator requires a conversion of the DC produced power to the required AC supply specifications. Therefore, inverters are needed for supplying AC loads. As shown in Fig. 10.12, solar-PV generators may be operated in the off-grid or grid-connected modes. In the off-grid mode, no inverters are needed if the entire system is DC; however, inverters are needed in the grid-connected operation. The MPPT function may also be included in the PCU block. The DC to AC inversion is provided by inverters. Currently, there are two options in the power inversion process; either centralized or micro inverters. With centralized inverters, large solar-PV power plants can be connected to one inverter station for handling the DC-AC conversion. With micro inverters, each module is integrated with a micro inverter. In comparison with micro inverters, the centralized inverters offer reduction in both the power conversion costs and the associated power losses; however, micro inverters has some unique features that cannot be provided in centralized-based inverter installations. These unique features include the capability of operating each module at its MPP and also performing maintenance and repair without interrupting the operation of the entire solar-PV plant. Therefore, micro inverters also enhance the overall availability of the solarPV output energy. There are many techniques for the MPPT 186,187 control. These techniques include (Fig. 10.17(a)) the constant voltage, open circuit voltage , short circuit current, perturb and observe (P&O), and incremental conductance (IC ) methods. The rank of an MPPT technique is usually presented as a percentage of the maximum power extracted at various irradiance (and temperature) conditions in comparison with the ideal MPPT algorithm. For 186

Dolara, A., Faranda, R., & Leva, S. (2009). Energy comparison of seven MPPT techniques for PV systems. Journal of Electromagnetic Analysis and Applications, 2009. 187 Faranda, R., & Leva, S. (2008). Energy comparison of MPPT techniques for PV Systems. WSEAS transactions on power systems, 3(6), 446-455. 449

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example, an MPPT technique with a rank of 85% indicates that the extracted maximum power is 85% of the actual available maximum power at various input conditions to the solar-PV array. Fig. 10.17(b) also illustrates a comparison between the approximate rankings of various MPPT techniques considering the variations of the irradiance values around the STC irradiance at a constant STC standard temperature. It should be notes that the dynamic performance and the hardware requirements of an MPPT technique are not of low importance in comparison with its rank. This issue will be considered later in this chapter.

(a)

(b) Fig. 10.17: MPPT; (a) Methods; (b) Approximate ranking at irradiance levels close to the STC value and a constant STC temperature

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In the constant voltage (C.V) method , the average MPP voltage is used for the MPPT. As shown in Fig. 10.14 and 10.18, the changes in Vmp due to the changes in the irradiance are not very large; however, the impact of the temperature various on the MPP is significant. Consequently, in this method, the average value of Vmp is determined and the solar-PV array terminal voltage is fixed at this average value which is also called the reference voltage Vref.

(a) (b) 10.18: Changes in the MPP; (a) Impact of irradiance; (b) impact of temperature The reference voltage may also be chosen to be equal to the Vmp at the STC conditions. Fig. 10.19 shows the constant voltage MPPT algorithm where k is the sampling rate of the measurements and D is the duty cycle of the DC-DC boost converter188 (Fig. 10.16); the output voltage is proportional to the duty cycle. The C.V method is the simplest MPPT technique; however, it has low rank in comparison with the other techniques when the irradiance is close to the STC value; however, the rank of the C.V method surpasses the P&O as well as the IC methods at low irradiance levels. This is due to variable sensitivity of the change of the MPP point with the irradiance.

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https://en.wikipedia.org/wiki/Boost_converter 451

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Fig. 10.19: Constant voltage MPPT algorithm In the open circuit (O.C) voltage MPPT technique, the reference voltage of the boost converter is set to an approximate MPP voltage. The variations in the MPP voltage are related to the variations in the O.C voltage

approximately by Vmp = OC VOC. Therefore, Vref = OC VOC. Typical value of

OC is between 0.7 to 0.8. The main disadvantage of this method is the necessary of interrupting the output power for determining the open circuit voltage according the variations in the irradiance and temperature. Fig. 10.20 shows the O.C voltage based MPPT algorithm. As shown from Fig. 10.17, the O.C voltage based MPPT is having a better ranking in comparison with the constant voltage and the S.C current based MPPT methods. In the S.C current, the MPPT is based on the current instead of the voltage. The MPP current is determined as a fraction of the S.C current i.e.

Imp = SC ISC. Therefore, Iref = SC ISC while the range of SC is between 0.9 to

0.98. Fig. 10.21 shows the S.C current based MPPT algorithm.

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Fig. 10.20: O.C voltage based MPPT algorithm

Fig. 10.21: S.C current based MPPT algorithm 453

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The Perturb and Observe (P&O) MPPT method is a search algorithm. In the P&O method searches for the maximum power point by perturbing the solar-PV voltage (or Current) and detecting the consequent change in solarPV power output. The direction of the change is reversed when the solar-PV power decreases. The main P&O MPPT concept is illustrated in Fig. 10.22; however, there are many forms of the P&O MPPT algorithm 189. Although the ranking of the P&O method is very close to the ideal MPPT (Fig. 10.17), the operation of the algorithm depends on the continuous disturbance of the solar-PV terminal voltage. In addition, these perturbations are not stopped even if the MPP is achieved; the operating point at the MPP conditions continuous to oscillate around the MPP as a result of the MPPT algorithm. Therefore, the dynamic performance of the P&O method reduces the value of the ranking of this method.

Fig. 10.22: Basic concepts of the P&O based MPPT algorithm 189

Dolara, A., Faranda, R., & Leva, S. (2009). Energy comparison of seven MPPT techniques for PV systems. Journal of Electromagnetic Analysis and Applications, 2009 454

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The P&O method also suffers from operational failure in the cases of sudden changes in the irradiance or the temperature. This problem is illustrated in Fig. 10.23. Suppose that the solar-PV array operates at the MPP (a) at specific meteorological conditions. The P&O algorithm then perturbs this operating point by a small change in the terminal voltage (V). With the meteorological conditions unchanged, the operating point goes to point (b). Therefore, the power is reduced and the MPP (a) is restored in the second MPPT cycle. Now consider a situation where the meteorological conditions are rapidly changed during the sampling period when the system operates at point (a). The power curve moves from curve (1) to curve (2). The perturbation of the operating point then causes point (a) to move to point (c). Since, the power at (c) is higher than the power at (a), the MPPT then considers (c) as the MPP while it is not the actual MPP at the new meteorological conditions. This divergence action will be sustained if the meteorological conditions remain changed rapidly.

Fig. 10.23: Failure of the P&O MPPT during rapid changes in the meteorological conditions The incremental conductance (IC) based MPPT algorithm is based on the fact that dP/dV = 0 at the MPP (Fig. 10.18). Since P = VI, then dP/dV = d(VI)/dV = I (dV/dV) + V(dI/dV) = I + V(dI/dV) = 0. Therefore at the MPP, I/V = -(dI/dV) i.e. at the MPP, the conductance of the solar-PV equals to its 455

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incremental conductance. In addition, to the left of the MPP, dP/dV >0 which resulting in I/V + (dI/dV) > 0 while I/V + (dI/dV) < 0 holds if the operating point is to the right of the MPP. Therefore, information about the conductance and the incremental conductance leads to the identification of the location of the operating point on the P-V curve as well as the correct direction of change in the voltage for reaching the MPP. If I/V + (dI/dV) > 0, then the voltage should be increased while with I/V + (dI/dV) < 0, the voltage should be decreased. The ranking of this method is comparable with the P&O method while each of them has different dynamic performance as will be illustrated later. Fig. 10.23 illustrates the basic concepts of the IC based MPPT.

Fig. 10.24: Basic concepts of the IC based MPPT algorithm Later in this chapter the dynamics associated with the P&O and IC MPPT methods (as the most popular MPPT techniques) will be presented. In 456

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addition, the control algorithms of these methods will be enhanced for better dynamic performance. 10.4.3 Earth-Sun Geometry and Observer Angles

The location of the sun is very important for solar energy project. The location is a function of the time of the day, the day in the year, and the location on the earth. Every day the sun moves from the east to the west between sunrise and sunset. In addition, the sun also moves from north to south throughout the course of the year. If the position of the sun is measured every day at solar noon it would be at a different angle every day. Tracking the sun position is important for more extraction of the solar energy incident on a solar energy conversion machine (such as solar-PV or solar CSP). Sun tracking mechanisms are the tools used to do so. In this section the fundamental geometry of the sun-earth relation is presented while the next section provides an overview of the sun tracking options and their impact on the energy production.

Fig. 10.25: Daily rotation of the earth

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The earth continuously rotates about its axis of daily rotation as shown in Fig. 10.25. The line connecting the north and south poles is called the earth’s axis. This axis is perpendicular to the plane of the equator ; however, it is not perpendicular to the plane of the earth’s orbit plan. The tilt or obliquity of the earth’s axis to a line perpendicular to the plane of its orbit is currently about 23.5°. The plane of the sun is then defined as the plane at which the center of the sun is located and at the same time parallel to the equator plane. The elliptic cycle is then the annual cycle at which the earth passes alternately above and below the plane of the sun. Fig. 10.26 illustrates the plane of the sun and the elliptic cycle as well as the earth’s motion through the space.

(a)

(b) Fig. 10.26: Motion of the earth; (a) Plane of the sun; (b) Motion through space The sun declination angle (δ) is defined to be that angle between the equator plane and the line connecting the center of the earth and the center of 458

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the sun. This is illustrated in Fig. 10.27. as shown in the figure, the declination angle various between 23.5o (at the summer solstice) and -23.5o (at the winter solstice).

(a) (b)

(d) (c) Fig. 10.27: Sun declination; (a) angle definition; (b) angle variations; (c) at summer solstice; (d) at winter solstice Consider Fig. 10.28, where a random location (P) on the earth’s surface is shown. The geocentric latitude angle () of the random location P is the angle between the lines joining that location to the center of the earth and the equatorial plane. Therefore, all locations at the same latitude experience the same geometric relationship with the sun . The semicircles along the surface of the earth joining the north to the south poles are called lines of longitude. The longitude line crossing the Greenwich city at England is called the prime meridian. For the shown point (P) the latitude is +o north relative to the

equatorial plane while the longitude of the point is +o east relative to prime 459

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meridian. Fig. 10.29 illustrates various latitude and longitude lines on the earth’s surface.

Fig. 10.28: latitude (), longitude (), and prime meridian

Fig. 10.29: Latitude and longitude lines and their references The solar noon is defined to be that time of day at which the Sun’s rays are directed perpendicular to a given line of longitude. Therefore, the solar noon occurs at the same instant for all locations along any common line of 460

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longitude. Since one revolution of the earth is equivalent to 24 hours and

360o, then 15o of longitudes are passed during one hour. The solar noon occurs one hour earlier for every 15 degrees of longitude to the east of a given line and one hour later for every 15 degrees west of a given line. The sun hour angle (ω) is the angular distance between the meridian of the observer (P) and the meridian whose plane contains the sun’s rays (Fig. 10.30). Thus, the hour angle is zero at the local noon (i.e. when the sun reaches its highest point in the sky). At this time the sun is said to be due south at the northern hemisphere and due north in the southern hemisphere. Since the meridian plane of the observer contains the sun, the hour angle increases by 15 degrees every hour. At the local noon, the hour angle is then zero.

Fig. 10.30: Sun hour angle The previous angle shown in Fig. 10.30 (declination, latitude, and hour angle) are said to be the earth-sun angles. There are some additional angles required for the estimation of the solar radiation incident on the plan of a solar-PV module. These angles are shown in Fig. 10.31 and described in the following. The zenith angle ( z) is the angle between the line that points to the sun and the vertical — basically, this just defines the location of the sun is in the 461

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sky. At sunrise and sunset this angle is 0º while at the solar noon this angle is zero.

Fig. 10.31: Main observer angles The solar altitude angle (αs) is the angle between the line that points to the sun and the horizontal (i.e. the tangent of the earth’s surface). It is the complement of the zenith angle. At sunrise and sunset this angle is 0º while at the solar noon this angle is 90. The solar azimuth angle ( s) is the angle between the line that points to the sun and south. Angles to the east are negative. Angles to the west are positive. This angle is 0º at solar noon. It is probably close to -90º at sunrise and 90º at sunset, depending on the season. This angle is only measured in the horizontal plane; in other words, the height of the sun is neglected The angle of incidence ( ) is the angle between the line that points to the sun and the line that points straight out of a solar-PV module (it is also called the line that is normal to the surface of the module). This is the most important angle. If the module orientation is controlled for minimization of the incident angle, then the solar energy incident on the module will be maximized. Consequently, the energy production will be maximized. This 462

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control action is the basis of all sun tracking control systems. In some cases, the modules are kept fixed at a specific angle. For this purpose, the tilt angle () should optimized for the maximization of the energy production and the seasonal load variations 190. The surface azimuth angle ( ) is the angle between the line that points straight out of the module’s surface and south. It is only measured in the horizontal plane. Again, east is negative and west is positive. If a module pointed directly south, this angle would be 0º. The collector slope or tilt angle ( ) is the angle between the plane of the solar collector and the horizontal. If a module is placed horizontally, then it is 0º. 10.4.4 Sun tracking mechanisms

A sun tracking mechanism is defined as the way that used to orient a solar-PV module towards the sun for increasing the PV system performance. The normal tracking mode is the fixed mode in which the PV module is fixed at a typical angle called the tilt angle and does not track the daily sun movement. The optimal tilt angle of a fixed PV array is related to the local climatic condition, geographic latitude and the period of its use 191. Many studies have been performed to get the optimum tilt angle that relates to the maximum energy in different places. Previous studies show that the average yearly optimum tilt angle is close to the latitude of the location; however, the tilt angle can be optimized for a specific load pattern and specific set of objectives. For example, maximization of the energy production and minimization of the size of the required solar-PV generator for a specified load pattern are popular set of objectives. Generally, there are two main sun tracking modes; the single-axis (also called one-axis) and the dual-axis (also called two-axis) tracking shown in Fig. 10.32.

190

El-Shimy, M. (2013). Sizing optimisation of stand-alone photovoltaic generators for irrigation water pumping systems. International Journal of Sustainable Energy, 32(5), 333-350. 191 Ulgen, K. (2006). Optimum tilt angle for solar collectors. Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 28(13), 1171-1180. 463

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The single-axis tracker pivots on it an axis to track the sun, where the solar-PV modules rotate from east to west to track the daily sun movement192. The modules face the east in the morning and the west in the afternoon (Fig. 10.33). The tilt angle of this axis can be optimized in the same way of optimizing the tilt angle of the fixed modules or can be set to be equal the latitude angle of the location. In consequence for this type of sun tracking, a seasonal adjustment of the tilt angle is recommended for achieving the maximum energy production. The single-axis tracking can be implemented by several methods. These include horizontal single-axis trackers (HSAT), vertical single-axis trackers (VSAT), tilted single-axis trackers (TSAT) and polar aligned single-axis trackers (PSAT)193.

(a) (b) Fig. 10.32: Basic types of sun tracking systems; (a) Single-axis; (b) Dualaxis The two-axis trackers follow the sun in both azimuth and altitude (i.e. elevation) by having two-axes of rotation. Consequently, the PV module rotates from east to west to track daily sun movement and north or south to track seasonal sun movement throughout the year. Therefore, the sun's rays are kept normal to the module surface. Considering their ability to combine two rotational motions described around perpendicular axis, they are able to follow very precisely the sun path along an annual period. Therefore, the 192

System Advisor Model (SAM). SAM's Help system, National Renewable Energy Laboratory (NREL), September 2013 193 http://www.pvresources.com/PVSystems/Trackers.aspx 464

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dual-axis trackers are more efficient than the single one; however, the selection of the sun tracking option is based on many factors. These factors include the economic feasibility and the benefits of the tracking system in terms on the energy production. As will be shown later, the impact of various tracking options on the energy production and the LCOE is highly depended on the location of the solar-PV project194. The dual-axis tracking has two common implementations; tip-tilt dual-axis trackers (TTDAT) and azimuthaltitude dual-axis trackers (AADAT). Recently, a combination between solar trackers and mirrors is used to maximize the energy production. The most common combination uses V-trough mirrors with trackers.

Fig. 10.33: Daily motion of the single-axis trackers PV module tracking has the advantages of increasing the PV system’s output by about 40%; however, tracking equipment requires additional costs (cost of equipment and O&M costs) and it needs more land for tracking equipment as it must be spaced out to avoid the problem of shading 195. In comparison with the fixed PV modules, it is founds 196 that the one-axis 194

Said, M., EL-Shimy, M., & Abdelraheem, M. A. (2015). Photovoltaics energy: Improved modeling and analysis of the levelized cost of energy (LCOE) and grid parity– Egypt case study. Sustainable Energy Technologies and Assessments, 9, 37-48. 195 Kumar, A., Thakur, N., Makade, R., & Shivhare, M. K. (2011). Optimization of tilt angle for photovoltaic array. International Journal of Engineering Science and Technology, 3(4), 3153-3161. 196 Ong, S., Campbell, C., Denholm, P., Margolis, R., & Heath, G. (2013). Land-use requirements for solar power plants in the United States. Golden, CO: National Renewable

Energy Laboratory. 465

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trackers increases the land use by about 13%, while two-axis trackers increases the land use by about 40%. Based the average market price is the year 2014, the minimum, maximum, and the average costs in US$/W p of single-axis trackers are 0.36, 0.4, and 0.38 while their values for dual-axis trackers are 0.4, 3, and 0.96 US/Wp. The annual operating and maintenance costs of the tracking systems are in the range of 12 US$/kWp-year to 25 US$/kWp-year197. 10.4.5 Modes of operation of solar-PV generators

Based on Fig. 10.12(a), solar-PV generators can be operated in either the off-grid or grid-connected modes. In the off-grid operation, the loads may be non-deferrable or deferrable. For non-deferrable loads the continuous energy balance is a major challenge especially when the solar-PV arrays are the only source of power production. In this case energy storage is essentially required to supply the essential and critical portions of the load. The critical loads should not be interrupted while the essential loads can be interrupted for a short-duration while the non-essential loads can be interrupted for longer durations. The classification of various loads dependents on the importance on the load equipment in the overall function(s) of the nondeferrable loads as well as the hazardous situations and safety issues that can be resulted from the interruption of the load equipment. On the other hand, non-deferrable loads can be shifted in time without affecting the main target of the load. For example, water irrigation pumping is a typical deferrable load where the irrigation process can be shifted in time without significant impacts on the crops. Energy storage is not generally required for deferrable loads; however, non-electrical energy storage may be used for enhancing the utilization of the solar-PV generator. Energy storage in the form of pumped water in storage tanks or pumped storage energy production systems is examples of non-electrical storage options. In grid connected applications, the size of the solar-PV generator relative to the size of the rest of the grid is among the method of determining the need of energy storage. The energy storage is basically need for load following and energy balance requirements. Therefore, in isolated and small 197

http://solarprofessional.com/articles/products-equipment/racking/pv-

trackers 466

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grids, the energy storage may be needed while with large grids (e.g. utility) grid, the energy storage is not essentially required. Fig. 10.12(b) shows various options of energy storage. The figure shows two main classification of energy storage devices or systems. The electrochemical energy storage is provided by various types of batteries while the non-electrical energy storage can be provided by many options. These include the pumped storage, compressed air energy storage (CAES), and the hydrogen. Details about various classes are outside the scope of this book; however, due to its importance, the hydrogen as energy storage and an energy carrier will be explained in this section. Among the main critical challenges of renewable energy systems are the variability and intermittency of common renewable energy sources such as wind and solar (see chapter 1). Conceptually, appropriate sizing of variable resources accompanied sufficient energy storage can solve these operational problems; however, it is neither economical nor technically possible to construct bulk power energy storage by the use of traditional electrochemical options. Therefore, very large scale integration of variable renewable energy sources to power systems is not impossible from operation and security points of view.

Fig. 10.34: Use of hydrogen as an energy storage and energy carrier for enhancing the reliability of variable power sources

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The use of variable and intermittent electrical energy sources can be used for hydrogen production (as shown in Fig. 10.34). The produced hydrogen can be then stored in tanks at a specific compression level 198. The availability of this stored hydrogen facilitates the controllable use of the overall energy produced from an originally non-controllable resource (such as wind speed or solar irradiance). In addition, the hydrogen can be moved over long geographical distance in the same way of moving the oil and natural gas. Unlike the conventional electrical power transmission, the hydrogen energy transmission is not associated with energy losses or geographical barriers. The amazing issue here is the use of the hydrogen. It can be used directly for thermal energy production (like the natural gas in home and industrial applications as well as supplying vehicle). In addition, indirect use of hydrogen for electrical energy production is possible either through fuel cells or hydrogen gas turbines. Advances in the research and industry of hydrogen related technology for energy storage, energy transmission, and electrical energy production are needed for enhancing the security, safety, economy, and feasibility of such systems. Variable renewable energy sources can also be used for many other objectives. For example, reduction of the utility grid energy tariffs is possible by reducing the dependency on the utility grid through installing on-site, for example, solar-PV generators (Fig. 10.35). In this case, a utility customer will not only be capable of reducing the electricity tariff but also can export power to the grid. Usually feed-in-tariff structures are available in world-wide countries for this purpose. The structures allow independent power producers to sell power to the utility grid. Usually, the prices of the renewable energy exported by the customers are higher than the retail prices of the utility energy. This attracts the load-side integration of renewable resources even in the residential class. With the use of energy storage, these renewable sources can also be used for enhancing the load supply reliability as well as reducing the possible energy interruptions cause by the utility grid. The size of the required energy storage should be minimized. Therefore, the load should be classified

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Afgan, N. H., & Carvalho, M. G. (2004). Sustainability assessment of hydrogen energy systems. International Journal of Hydrogen Energy, 29(13), 1327-1342. 468

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according to its importance and the sizing of the storage should be based on the critical and essential load classes.

Fig. 10.35: Use of solar-PV supply for reducing the electricity tariff and possibly for enhancing the main supply reliability

10.5 Modeling of grid-connected solar-PV generators Based on the objectives of the mathematical modeling of solar-PV systems, two modeling categories can be realized. The first category of models is basically used for the solar resource assessment and a rough estimate of the energy production models. In this context, the first category of models is called Resource Assessment and Energy Production (RA&EP) Models 199. These models are useful in the initial stages of solar-PV projects where the accuracy of the results is not of the major importance. For these models, high details about the meteorological conditions of the project location are required while the modeling of the solar-PV cells is highly simplified; the PV modules are represented mainly by their efficiency and temperature coefficient values. Techno-economical feasibility analysis software tools such as the RETSCreen utilize the energy production models. The second category of models simulates the PV-system in high details from the prospective of the PV-cell and the time step of the meteorological date requirements. The second category models are called in this context 199

El-Shimy, M. (2013). Sizing optimisation of stand-alone photovoltaic generators for irrigation water pumping systems. International Journal of Sustainable Energy, 32(5), 333-350. 469

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equivalent circuit based models 200 (ECBM). The Solar Advisor Model201

(SAM) software tool uses such models for estimating the performance of solar-PV power plants. The time step of the meteorological data required for the RA&EP models are usually long term averaged values on daily, or monthly, or even annual basis. For the ECBMs, the time step of the meteorological data is usually on hourly or less time basis. This is required for achieving a reliable accuracy level. The performance analysis of solar-PV generator is performed using historical long-term average meteorological data when using the EPMs while the ECBMs can be used for simulating the generator using either the historical or the existing (online) data. The nomenclature is listed in Appendix 9. This nomenclature will be used in the rest of this chapter and the next chapter. 10.5.1 Resource Assessment and Energy Production (RA&EP) Models of solar-PV modules

As shown in Fig. 10.12(a), the energy available from a solar-PV generator is the energy output from the PCU. (10.4) where S is the overall solar collector area of the PV-generator given by . The number of modules m is defined by should be approximated to the nearest higher integer number.

,which

The average module temperature is given by (10.5) where

is a correction factor to account for the effect of off-optimal tilt

angle of the PV-generator on the temperature of the PV-modules. 200

Said, M., EL-Shimy, M., & Abdelraheem, M. A. (2015). Photovoltaics energy: Improved modeling and analysis of the levelized cost of energy (LCOE) and grid parity– Egypt case study. Sustainable Energy Technologies and Assessments, 9, 37-48. 201 https://sam.nrel.gov/ 470

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(10.6) The approximate optimal tilt angle of a PV-generator and the declination angle are defined by, (10.7) (10.8) The total daily radiation in the plane of the PV-generator equals to the summation of the hourly irradiance in the plane of the PV-generator i.e. . The estimation of involves the determination of the hourly beam (or direct) and diffuse irradiance on a horizontal surface for all hours of an average day having the same daily global radiation as the monthly average. The definitions of the global solar radiation of the horizontal surface (GSR)202 and its components i.e. the direct and diffuse radiations are illustrated in Fig. 10.36. The monthly mean hourly global radiation on a horizontal surface H can be correlated to monthly mean daily global radiation on a horizontal surface using a proper mathematical correlation model r t i.e. . Inappropriate solar correlation model selection for a specific design site is a major cause for inaccurate system sizing of solar-PV generators. Therefore, it is recommended to use the most appropriate correlation model for the design site in sizing PV-generators203. Two accepted correlation models are considered in this book. The best model is to be selected for the considered study site. These correlation models are the Liu and Jordan (LJ) given by equation (10.9) and Collares-Pereira and Rabl (CR) given by equation (10.10). The coefficients of the CR model are defined in equations (10.11)

202

Abdo, T., & EL-Shimy, M. (2013). Estimating the global solar radiation for solar energy projects–Egypt case study. International Journal of Sustainable Energy, 32(6), 682-712. 203 Yesilata, B., & Firatoglu, Z. A. (2008). Effect of solar radiation correlations on system sizing: PV pumping case. Renewable Energy, 33(1), 155-161. 471

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and (10.12). The solar hour angle and the sunset hour angle by equations (10.13) and (10.14) respectively.

Fig. 10.36: GSR, direct, and diffuse radiations

(10.11) (10.12) (10.13) (10.14) 472

are given

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The hourly beam (or direct) and diffuse irradiance, on a horizontal surface, are given by equations (10.15) and (10.16) with being the Liu and Jordan (LJ) correlation model. Equation (10.16) is valid for . The hourly irradiance in the plane of the PV-generator is then defined by equation (10.17). (10.15)

(10.16) (10.17) where the tilt factors for beam Rb and diffuse Rd radiations are defined by

10.5.2 Equivalent Circuit based Models (ECBMs) of solar-PV generators

To be familiar with the models of solar-PV cells, modules, and arrays, it is recommended to read section 11.2.1 in chapter 11; however, this is not essential if the reader knows the fundamental models of photovoltaics. The Sandia PV Array Performance model204 consists of a set of equations that provide values for five points on a module's I-V curve (which is the relation between the output voltage and current curve from the PV module as illustrated in Fig. 10.37 and a database of coefficients for these equations.

204

King, D. L., Boyson, W. E., & Kratochvil, J. A. (2004). Photovoltaic Array Performance Model, Sandia National Labs. 473

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The values of these coefficients are stored in the Sandia Modules library 205. These coefficients have been empirically determined based on a set of manufacturer specifications and measurements taken from modules installed outdoors in real, operating systems. The five points include the three classical points on the IV curve, i.e. short-circuit current (Isc), open-circuit voltage (Voc), and the maximum-power point voltage and current (Vmp and Imp respectively). The remaining two points are the current ( Ix) at the half of the open-circuit voltage and the current (Ixx) at a voltage midway between Vmp and Voc.. Sandia Model is described by the following equations.

Fig. 10.37: PV module characteristics (IV curve: red; power curve: blue) showing the five points required by the Sandia performance model

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System Advisor Model (SAM). SAM's Help system. Available: http://sam.nrel.gov 474

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The California Energy Commission (CEC) Performance Model uses the University of Wisconsin-Madison Solar Energy Laboratory's five-parameter model with a database of module parameters maintained by the CEC for the California Solar Initiative. The database of this model tends to be more up-todate than Sandia model database. Therefore, this model used in this chapter. The CEC model uses the equivalent circuit shown in Fig. 10.38 for modeling PV cells. This circuit includes a series resistance and a diode in parallel with a shunt resistance. The letter V represents the voltage at the load. As previously explained, this circuit can be used for an individual cell, for a module consisting of several cells, or for an array consisting of several modules. The basic equations of the model are,

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Fig. 10.38: Equivalent circuit representing the PV cell used in the ECE model. Note: the light current (IL) is also called the photocurrent (Iph)

The flat plate simple efficiency model is a simple representation of module performance that requires the user to provide the module area, a set of conversion efficiency values, and temperature correction parameters. It is 476

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recommended to use this model for the modules that do not exist in the CEC module database. The model calculates the module's hourly DC output, assuming that the module efficiency varies with radiation incident on the module as defined by the radiation level and efficiency table.

10.6 Case Studies For reducing the volume of the book, the case studies related to the presented characteristics, and modeling of solar-PV systems will be presented in some external references by the main author of this book. The characteristics and tilt angle optimization of standalone solar PVgenerators can be found in reference206. In addition, this reference provides an improved method for sizing minimization of solar-PV generators supplying deferrable loads. The considered load is irrigation water pumping. In reference207, many studies are presented. The most relevant study is the detailed analysis of the effectiveness of various sun tracking options is presented. The effectiveness is identified based on the impact of various sun tracking options on the energy production and the LCOE considering a wide range of latitudes. Detailed analysis of the dynamic performance of grid-connected solarPV generator will be considered in the next chapter. This chapter as well as the next two chapters can be considered as an extended chapter handling most of the technical and economical issues associated with solar-PV systems.

206

El-Shimy, M. (2013). Sizing optimisation of stand-alone photovoltaic generators for irrigation water pumping systems. International Journal of Sustainable Energy, 32(5), 333-350. 207 Said, M., EL-Shimy, M., & Abdelraheem, M. A. (2015). Photovoltaics energy: Improved modeling and analysis of the levelized cost of energy (LCOE) and grid parity– Egypt case study. Sustainable Energy Technologies and Assessments, 9, 37-48. 477

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Chapter 11 Modeling, Dynamic Analysis, and Control of GridConnected Solar-PV generators 11.1 Introduction The integration of large amounts of solar PV power in to the electricity grid presents new challenges 208. These challenges are mainly related to the modeling, control, performance, technical impacts, operational impacts, and techno-economic feasibility. One of the main operational and control targets of solar PV generators is the extraction of the maximum available active power. This available power is not only depends on the solar-PV generator but also highly related to various meteorological conditions. The solar irradiance and temperature are the main environmental input variables that affect the power production. The sun tracking and the MPPT are also among the main factors that affects the energy extraction. In the previous chapter, various MPPT techniques are presented while in this chapter the dynamic performance of grid-connected solar-PV generator equipped with some selected MPPT methods will be presented. In addition, enhanced control of the generator MPPT for better dynamic performance will also be presented. The modeling of solar PV cells and generators is considered in many previous researches 209,210,211,212,213. These researches provide methods for estimating the parameters of solar PV cells, modules, and arrays. In addition, mathematical models for the simulation of the behavior and performance are presented considering variations in the temperature and solar irradiance. 208

European Photovoltaic Industry Association. (2014). Global market outlook for photovoltaics 2014–2018.[Online] Available: http://www. epia. org/filead min/user_upload. Publications/44_epia_gmo_report_ver_ 17_mr. pdf. 209 De Soto, W., Klein, S. A., & Beckman, W. A. (2006). Improvement and validation of a model for photovoltaic array performance. Solar energy, 80(1), 78-88. 210 Chenni, R., Makhlouf, M., Kerbache, T., & Bouzid, A. (2007). A detailed modeling method for photovoltaic cells. Energy, 32(9), 1724-1730. 211 Tian, H., Mancilla-David, F., Ellis, K., Muljadi, E., & Jenkins, P. (2012). A cell-tomodule-to-array detailed model for photovoltaic panels. Solar Energy, 86(9), 2695-2706. 212 Chouder, A., Silvestre, S., Sadaoui, N., & Rahmani, L. (2012). Modeling and simulation of a grid connected PV system based on the evaluation of main PV module parameters. Simulation Modelling Practice and Theory, 20(1), 46-58. 213 Yang, Y., Chen, W., & Blaabjerg, F. (2014). Advanced Control of Photovoltaic and Wind Turbines Power Systems. In Advanced and Intelligent Control in Power Electronics and Drives (pp. 41-89). Springer International Publishing. 479

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Generally, as shown in the previous chapter, the output power of PV generators is nonlinear and varies according to the available level of irradiance and temperature. In addition, at certain operating point, the PV module produces its maximum output power; the MPP. The MPPT controllers are developed to derive the maximum power from solar PV generators214,215,216. In comparison with other MPPT techniques, the perturb and observe (P&O) technique is found to be simpler in structure and easier in implementation. In addition, the P&O-based MPPT is widely used in many installations. The operation of P&O-based MPPT controllers is solely dependent on the output DC voltage and current values at the solar PV generator terminals. Therefore, no meteorological sensors are needed for its operation and implementation. Despite these advantages, the conventional P&O exhibits a trade-off between the tracking speed and steady state oscillation. However in this chapter a new method is introduced to solve the problem of output power oscillation caused by perturb and observe technique. After extracting the extracted maximum DC active power from the solarPV plant, it is required to inject this active power in its specified AC form along with the demanded reactive power to the power grid. This task can be done by the inverter. Stability analysis is required to identify the effect of solar-PV plants on the grid, and the response of the solar-PV plant to the sudden changes in grid parameters (voltage, frequency… etc). In this chapter the dynamic performance of a grid connected PV generator will be investigated. Various types of disturbances are considered. These disturbances include changes in meteorological conditions i.e. the solar irradiance and the temperature. Grid disturbances such as voltage sag, voltage swill, frequency fluctuations and strength of the grid coupling are also important stability studies.

214

Salam, Z., Ahmed, J., & Merugu, B. S. (2013). The application of soft computing methods for MPPT of PV system: A technological and status review. Applied Energy, 107, 135-148. 215 Onat, N. (2010). Recent developments in maximum power point tracking technologies for photovoltaic systems. International Journal of Photoenergy, 2010. 216 Ahmed, J., & Salam, Z. (2014). A Maximum Power Point Tracking (MPPT) for PV system using Cuckoo Search with partial shading capability. Applied Energy, 119, 118130. 480

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11.2 Modeling of solar-PV generators for dynamic analysis Configurations for the grid-connected solar-PV-generator (SPVG) are shown Fig.11.1. The SPVG is connected to the grid via a transformer at a bus called the point of common coupling (PCC). For grid interconnection, the DC power produced by the SPVG is transformed to a specified AC power according to the grid specifications. This is performed by a power electronics based grid interface. The grid interface performs three main functions: (1) MPPT; (2) Control of the active and reactive power injections to the PCC; (3) Filtration of harmonics and protection against current and voltage surges.

(a)

(b) Fig. 11.1: Grid-connected solar-PV generator block diagram; (a) With DCDC converter for MPPT control; (b) Without DC-DC converter 481

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SPVGs are usually composed of many components as shown in the figure. The first component is a solar array that converts solar energy into DC electricity. The sun trackers are providing sun tracking control for maximization of the solar resource extraction. In this chapter, the dynamics associated with the sun tracking systems are neglected. This assumption is valid during the short period of study as no change in the sun position is assumed during that period. According to the characteristics and design of the system, the MPPT control may be achieved either by controlling a DC-DC converter (Fig. 11.1(a)) or directly controlling the inverter which is the DC-AC converter (Fig. 11.1(b)). In both situations, the output voltage (and/or the current) of the SPVG is adjusted according to the MPPT algorithm (see chapter 10). The DC-link provides a stabilization facility to the input DC voltage to the inverter. The inverter is not only providing conversion from DC power to AC power and possibly MPPT control but also provides control of the active and reactive power injections to the grid. There are some controllers for achieving the sun tracking, MPPT, and power control. The connection filter block provides a tool for enhancing the power quality of the output power and also provides some protection capabilities such as surge protection. In this chapter, the configuration shown in Fig. 11.1(a) is considered. The modeling of various components and the dynamic simulation of the system as well as control enhancements will be presented. It is also assumed that the array is controlled in a centralized manner. 11.2.1 Solar-PV array modeling

The I-V and P-V characteristics presented in the previous chapter can be found using measurements. Mathematical modeling can also be used for representing these characteristics and it will be considered in this section. The solar-PV array model will be presented starting from the ideal solarPV cell model. An ideal solar- PV cell can be represented by a current source connected in parallel with a diode; Fig. 11.2. This is because the cell generates current when it is illuminated and acts as a diode when it is not. The physical structure of a solar cell is similar to that of the popular P-N junction diode. When this junction is subjected to sunlight, the photons may pass straight through the junction if the photon energy is lower 482

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than the silicon energy band gap value 217 which is 1.1 eV or reflect off the surface or absorbed by the silicon if the photon energy is higher than the silicon band gap value 218. The absorbed photons generate an electron-hole pair and sometimes heat depending on the band structure. For solar-PV cells, the absorbed photons generate electron-hole pairs. As a result, an open-circuit voltage appears at the terminals of the junction. If the junction is connected to an external circuit, a current will flow on the external circuit causing recombination of the electrons and holes. If the load resistance is zero, then the current is called the short-circuit current. This process is called the lightgenerated or the photo-generated current which is directly proportional to the solar irradiance and it is demonstrated in Fig. 11.3 which is based on the http://pveducation.com.

(b) (a) Fig. 11.2: Solar-PV cell; (a) Ideal equivalent circuit; (b) Circuit symbol As shown in Fig. 11.2(a), the cell output current ( I) takes the form,

217

https://en.wikipedia.org/wiki/Band_gap

218

In solid-state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in electron volts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. In solar-PV cells, the optical band gap determines the portion of the solar spectrum a cell can absorb. A semiconductor will not absorb photons of energy less than its band gap. In addition, the energy of the electron-hole pair produced by a photon is equal to the band gap energy. The band gap of silicon is 1.1 eV and its 0.67 eV for Germanium. 483

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where Iph is the light-generated current (A) which equals to the shortcircuit current and ID is the diode current (A). The diode current takes the form,

Therefore, the cell’s output current takes the form,

The value is called the thermal voltage because of its units and because it is only dependent on the temperature. Defining VT as,

Therefore, equation (11.3) can be written as,

where Io is the saturated reverse current or the leakage current of the diode

(A); q is the electron charge (= 1.602  10-19 C); k is the Boltzmann constant

(= 1.38  10-23 J/oK); V is the cell output voltage (V); Tc is the cell operating temperature (oK); a is the diode ideality factor. The ideality factor (also called the emissivity factor) is a fitting parameter that describes how closely the diode's behavior matches that behavior predicted by theory. In theory, the p-n junction of the diode is assumed to be an infinite plane and consequently no recombination occurs within the space-charge region. A perfect match to theory is indicated when a = 1. When recombination in the 484

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space-charge region dominate other recombination, however, a = 2. Most solar cells, which are quite large in comparison with the conventional diodes, can be approximately considered as an infinite plane. Therefore, the cell

usually exhibits a near-ideal behavior under STC i.e. a  1 at STC. As illustrated in Fig. 11.4, the value ideality factor is also technological dependent219,220. The reverse saturation current ( Io) can be determined at open-circuit conditions where I = 0 and V = Voc. Consequently, the reverse stautaion current can be determined based on the datasheet parameters at STC conditions using,

Fig. 11.3: The light-generated current 219

Bellia, H., Youcef, R., & Fatima, M. (2014). A detailed modeling of photovoltaic module using MATLAB. NRIAG Journal of Astronomy and Geophysics, 3(1), 53-61. 220 Tsai, H. L., Tu, C. S., & Su, Y. J. (2008, October). Development of generalized photovoltaic model using MATLAB/SIMULINK. In Proceedings of the world congress on engineering and computer science (Vol. 2008, pp. 1-6). Citeseer. 485

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Fig. 11.4: Impact of solar-PV technologies on the diode ideality factor

or

Later, the impact of changes in the irradiance and the temperature on the cell characteristics as well as the impact of the cell losses will be presented. In the equivalent circuit of Fig. 11.2(a) which describes an ideal cell, losses are neglected. In reality, the losses reduce the output power and affect the overall I-V characteristics of the cell. The resistivity of the semiconductor material and the contacts can be represented by a series resistance (Rs ) connected at the terminals of an ideal cell as shown in Fig. 11.5. In this case equation (11.6) can be written as,

The voltage at the terminals of the diode ( VD) is related to the cell’s terminal voltage by . Therefore, (11.9) becomes, 486

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Fig. 11.5: Equivalent circuit of a solar-PV cell with semiconductor and contact losses (also called one-diode four-parameters model) The I-V relation and the open-circuit voltage can be then defined by,

Around the STC, the value of the photocurrent is nearly equals to the short-circuit currents i.e.

In the following, the impact of the variations in the solar irradiance and temperature will be presented. Therefore, the values of various variables and parameters at STC (1000 W/m2 and 25oC) will be illustrated by the suffix ref. The reference values are usually given in the datasheets of Photovoltaics. The short-circuit current at a temperature Tc is then,

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where H t is irradiance at the plane of the solar-cell which dependent on the tilt angle of the cell. KTi is temperature coefficient of the short-circuit current (/oK). The change in the cell temperature (K) is defined by,

The change in the short circuit current as affected by changes in the irradiance and temperature is then,

The reverse saturation current at STC can be found at open-circuit conditions (i.e. I = 0) using (11.13)

i.e.

The changes in the reverse situation current as affected by temeperature changes is given by,

where Eg is the band-gap energy of the semiconductor material; 1.1 eV for silicon. Based on (11.19), (11.18) is then

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The output power is then takes the form,

The value of the -1 in (11.6) or (11.7) is small enough in comparsion with the exponential term. The -1 can be then neglected if more simplifications are desired. Under this assumption, the series resistance can be determined as,

This is illustrated in Fig. 10.15 based on measured characteristics. The value of this resistance is usually determined at STC; however, its value is dependent on the temperature and the irradiance as depicted from the stated models. Better representation of the losses can be achieved by adding a shunt resistance as shown in Fig. 11.6. This equivalent circuit is called the onediode five-parameters circuit model. Therefore, in this model, two parasite resistances are considered; the series resistance (Rs) and the shunt resistance (Rsh) such as Rsh >> Rs is a usual relation. Therefore, Isc  Iph.

Fig. 11.6: Equivalent circuit of a cell considering the losses In this case, the current balance is stated as,

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Consequently (11.10) becomes,

As shown in Fig. 10.15, the STC values of the series and shunt resistances can be approximated by,

As shown in Fig. 10.13 and 11.7, a solar-PV module is constructed from a number of identical cells. Series and parallel connections of the cells respectively define the voltage and current ratings of the module. The presented cell models can be used to describe the module characteristics by modifying the cell characteristics according to the number of series and parallel cells within a module. The module voltage and current are then defined by,

Fig. 11.7: Solar-PV module with Ns series cells and Np parallel cells.

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The equivalent circuits of the module considering various models are shown in Table 11.1. This table illustrates the equivalent circuit of the ideal model (a), the single-diode four-parameter model (b), and the single-diode five-parameter model (c). The previous models of a cell are also valid for representing a module provided that all the quantities are modified according to the connections of the cells within the module. The characteristics of the module are illustrated in Fig. 11.8.

Fig. 11.8: Characteristics of a solar-PV module consists of Ns series cells and Np parallel cells The solar-PV array model can be derived based on the same concepts. Therefore, an array with Ms series modules and Mp parallel modules can be represented also as shown in Table 11.1. For the five-parameter model,

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Table 11.1: Solar-PV array models Representation

(a) Ideal model

Model

,

(b) Four-parameter model

Note:

(c) Five-parameter model

Note:

Note:

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If the impedance of the wiring connecting the modules is negligible, then

The accuracy of the one-diode five-parameter model is evaluated considering the I-V characteristics of the Sanyo-HIP-225HDE1 module. The module parameters are shown in Table 11.2. The simulation will be performed at the STC temperature while the irradiance is varied from 200 W/m2 to 1000 m2. The results are then compared with the datasheets of the considered module. This is shown in Fig. 11.9. 11.2.2 DC-DC Converters and MPPT control

Considering the configuration shown in Fig. 11.1(a) where a DC-DC converter is controlled for providing the MPPT requirement. Generally, the MPPT controls the DC-DC converter for regulating the input voltage at the 493

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array’s MPP and also capable of providing load matching for the maximum power transfer by controlling the input resistance of the array to match the load resistance. Various MPPT algorithms are described in Chapter 10. In this chapter two popular MPPT algorithms will be considered. These algorithms are the P&O and the IC. In addition, these algorithms will be enhanced for better dynamic performance of the SPVG. Table 11.2: Technical data of the Sanyo-HIP-225HDE1 module Quantity Symbol Value Number of series cells per module Ns 60 Number of paralell cells per module Np 0 Maximum power (W) P mp 225 Maximum power voltage (V) Vmp 33.9 Maximum power current (A) Imp 6.64 Open circuit voltage (V) Voc 41.8 Short circuit current (A) Isc 7.14 Temperature coefficient of Pmax (% / oC) KTPmax -0.3 Temperature coefficient of Vo.c (V / oC) KTv -0.105 Temperature coefficient of Is.c (mA / oC) KTi 2.14

(b)

(a)

Fig. 11.9: I-V characteristics of the Sanyo-HIP-225HDE1 module; (a) Simulated by the five-parameter model; (b) Manufacturer measured data The comparison indicates the accuracy of the presented model.

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Generally, DC to DC converters belong to a facility of converters called switching mode DC-DC converters 221. In comparison with their input voltage, DC-DC conveerters are capable of increasing or decreasing their output voltage in a controlled manner. In addition, the DC-DC converters are capable of providing impedance matching. DC-DC converters have a wide range of applications in DC power supplies. The salient applications can be found in battery equipped systems such as laptops, hybrid electric cars, and UPSs. In addition, they have a significant role in SPVG for various modes of SPVG operations (see chapter 10) as well as in SPVGs with battery storage. The main function(s) of DC-DC converters is achieved by fast switching of linear passive components such as inductors and capacitors; however, the switching itself is provided by nonlinear controlled electronic fast switches such as MOSFETs. In the MPPT, the duty cycle of the DC-DC converter is controlled for achieving the MPP according to the MPPT algorithm. The MPPT feedback control can be based on two variables 222; the terminal voltage of the array or the array output power. In the terminal voltage MPPT control, the terminal voltage of the array is used as the feedback control signal. The MPPT is achieved by regulating the array’s terminal voltage for matching it to a desired level which should be the MPP voltage. The main problem with this control approach is its insensitivity to wide changes in the irradiance and the temperature as the desired voltage level setting is fixed and valid for a narrow range for meteorological variations. Therefore, the terminal voltage based control is not suitable for the SPVG applications; however, it is very suitable in satellite systems where the solar irradiance is nearly constant. The main tracking problems associated with the terminal voltage based control is avoided with the use of the output power as a feedback control signal. In this case, the MPPT control forces the dP/dV to zero. Therefore, the MPPT control system will work satisfactory for any solar-PV system as its operation is not mainly dependent on the

221

George, K. (2006). DC Power Production, Delivery and Utilization: An EPRI White Paper. Available at: h ttp://www. epri. org [Accessed: 2.4. 12] . 222 Hua, C., & Shen, C. (1998, May). Study of maximum power tracking techniques and control of DC/DC converters for photovoltaic power system. In Power Electronics Specialists Conference, 1998. PESC 98 Record. 29th Annual IEEE (Vol. 1, pp. 86-93). IEEE. 495

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characteristics of the array; however, this method maximizes the power to the load rather than the power from the array. The basics and various types of DC-DC converters will be presented in the following. Three types of DC-DC converters will be considered; buck (step down converter), boost (step up converter), and buck-boost converters. The fundamental DC-DC converter for stepping down the DC input voltage is shown in Fig. 11.10 where R is the equivalent load resistance . As shown in Fig. 11.10(a), switching on and off of the switch in a periodic manner causes as shown in Fig. 11.10(b) reduction in the output voltage in comparison with the input voltage. The switch control is usually achieved by pulse width modulation (PWM) in which the switch is closed for an on duration (ton) and open for an off duration (toff) in a periodic way. The PWM control signal is shown in Fig. 11.10(c). With a switching periodic time Ts, the duty cycle (D ) of the switch is defined as,

The output voltage (vo(t)) and its average value (Vo) are then represented by,

It is clear from Fig. 11.10(b) and (11.41) that the output DC voltage contains harmonics due to the switching action (called switching harmonics) as shown in Fig. 11.11. The switching frequency is related to the switching periodic time by,

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(a)

(b)

(c) Fig. 11.10: Fundamentals of DC-DC step down control; (a) conceptual switching circuit; (b) Power voltage waveforms; (c) PWM and lowpass filter

Fig. 11.11: Switching harmonics in the output voltage These harmonics may cause degradation or even malfunction of the operation of the supplied DC load (in this case, the load is the inverter input port). Therefore, either elimination or reduction of these harmonics to a safe level defined by the load sensitivity is required (the reader may refer to the 497

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IEEE 519223 standard for more details). This problem can be solved by the use of lowpass filters as shown in Fig. 11.10(c). This filter passes the DC component while attenuates the switching harmonics. The filter action is governed by its L and C components. The series inductor (L) presents zero impedance to the DC component of the output voltage while it presents increasing impedance with the increase in the switching frequency. Therefore, the series inductor attenuates significant amounts of the harmonics. On the other hand, the shunt capacitor presents an infinite impedance to the DC component while it traps the harmonics to the ground due to the reduction of its impedance. The corner frequency ( fc) of the filter (see Fig. 11.12) is defined as,

To be effective in attenuating the switching harmonics, the corner frequency of the filter should be less than the switching frequency (i.e. fc 0) results in increase in the steam flow and consequently increase to the turbine power input. The opposite action occurs with upward movements in the control valve (xE < 0). In the following analysis of the operation of the system, downward movements are considered positive while upward movements are considered negative. The changes in the position of the control valve are due to one of the following causes: 1. Indirect automatoc feedback actions of the prime mover due to changes in the speed . Due to changes in the position of the connection point B (xB), resulting from changes in speed; where . Therefore, increase in the speed causes point B to move upward while speed reductions cause downward movements in point B. This is achieved by the action of the centrifugal forces of the 618

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rotating flyballs as the governor rotates by the same speed of the turbine. An increase in the speed , for example, of the turbine causes the centrifugal forces of the flyballs to move upward. As a result, the connection point B moves upward (xB < 0). Since the position of points A and D are yet unchanged, the change in the position of B causes a change in the position of point C in the opposite direction.

Therefore, point C moves downward (i.e. xC > 0). Yet, the position of the control valve is unchanged. Therefore, a change in the position of

C causes a change in the position of D in the same direction (i.e. xD > 0). Consequently, the pilot valve of the hydraulic amplifier moves downward and the resulting flow of pressurized oil cause the control

valve to move in the upward direction (xE < 0). Therefore, the steam flow will decrease causing the input power to the turbine to decrease. The speed of the turbine is the decreased and point B starts to move in the downward direction which reduces the flow of the pressurized oil. The operation continues till s balance (equilibrium point) is achieved. 2. Indirect automatic feedback actions due to variations of the steam pressure. 3. Direct control action by the changes in the speed changer (or power command) settings . This action may be manual or automatic. Automatic actions are the reaction of the Automatic Generation Control (AGC) where the governing system acts as the primary control loop of the AGC. The movements shown in Fig. A4.1 are corresponding to a raise power command in which the change position of point A is forced by the supplementary control action of the speed changer . The servomotor operates in a direction the results in downward (power increase command) or upward (power reduction command) of point A. The direction of rotation of the servomotor is determined by the polarity of its DC input voltage and it speed of motion is determined by the magnitude of its input voltage. This voltage control action can be achieved locally at the generating unit or from remote locations through communication channels.

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In the governing system shown in Fig. A4.1, the interaction between the governing system and the control valve is the cause of the droop. In the isochronous governors (Fig. A4.2), these interactions are absent.

Fig. A4.2: Isochronous Governor The droop in speed-governors is essential for stable operation. Consider an isochronous governor; an increase in the load is associated with a reduction in turbine speed . Therefore, the governor will respond by increasing the steam (or any primary input source) in order to restore the initial speed. Due to the inertial effect of the generating system, the changes power production lags the changes in the steam flow. Therefore, the governor actions will cause the control valve to open more than needed. Consequently, the speed will increase above the original setting. Consequently, the governor reacts by closing the control valve; however, the closing action will also be more than needed due to inertia/power lag characteristics. The overcorrection of the speed in both directions is usually cause instability. This instability problem can be eliminated with the speed-droop action. As the load increases, the magnitude of the speed reduction will be reduced due to the feedback between the control valve and the governor (see Fig. A4.1). Therefore, when the governor moves to correct for the speed decrease caused by the increased load, it will be correcting to a lower speed setting. This lower speed setting prevents the speed from overshooting and causing instabilities.

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Appendix 5 Transient droop compensators for speed-droop governors of hydroelectric units A schematic of a hydroelectric generating unit is shown in Fig. A5.1. The performance of a hydraulic turbine is influenced by the characteristics of the water column feeding the turbine which include,

Fig. A5.1: Schematic of a hydroelectric power unit 1. Water inertia . The effect of water inertia is to cause changes in turbine flow to lag behind changes in turbine gate opening. 2. Water compressibility. 3. Pipe wall elasticity in the penstock. The effect of elasticity is to cause traveling waves of pressure and flow in the pipe. This phenomenon is referred to as water hammer. Typically, the speed of propagation of such waves is about 1200 meters/sec. Traveling wave model is required only if penstock is very long. The representation of the hydraulic turbine and water column in stability studies usually assumes that: (a) the penstock is inelastic, (b) the water is incompressible, and (c) hydraulic resistance is negligible. The velocity of the water in the penstock is given by

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where U is the water velocity, G is the gate position, H is the hydraulic head at gate, and Ku is a constant of proportionality. The turbine mechanical power is proportional to the product of pressure and flow. With Kp is a proportionality constant, the mechanical power is given by,

The acceleration of water column due to a change in head at the turbine, characterized by Newton's second law of motion, may be expressed as

where L is the length of the conduit, A is the pipe area, ρ is the mass density, a g is the acceleration due to gravity. The term ρLA represents the mass of

water in the conduit while the term ρagH represents the incremental change

in the pressure at turbine gate. For small displacements about an initial operating point (subscript "o") it can shown that,

where

Tw is referred to as the water starting time. It represents the time required

for a head H o to accelerate the water in the penstock from standstill to the velocity U o. It should be noted that Tw varies with the load. Typically, Tw at full load lies between 0.5 s and 4.0s and its average value is 1.0 sec. Equation (A5.4) represents the classical transfer function of the turbine-penstock system. It shows how the turbine power output changes in response to a change in gate opening for an ideal lossless turbine. The TD response of the transfer function can be obtained by the inverse Laplace transform applied to eq. (A5.4); the TD function is,

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This TD function is plotted as shown in Fig. A5.2 for a unit step change in the gate position for a turbine with a time constant of 1.0 second. As shown in the figure, an initial surge of power reduction of 2.0 p.u (i.e. opposite to the direction of the gate direction and with double the magnitude of the gate change) is shown at the instant of the unit step increase in the gate position. This surge is due to the inertia of the water that prevents immediate change in the water flow. Consequently, the sudden change in the gate position is initially translated to a reduction in the pressure across the turbine. As a result, the initial surge of power reduction appears. With time, the water accelerates and the power starts to compensate the initial surge and finally the power reaches the new steady state value which equals to + 1.0 p.u.

Fig. A5.2: Response of a hydro-turbine to a unit step increase in the gate position Based on the transient characteristics of hydro-turbines, the simple speed-droop regulator shown in Fig. 80 will not be able to compensate the inertial characteristics of water flow (Fig. A5.2). For stable control performance, a large transient (temporary) droop with a long resetting time is therefore required to compensate the inertial response of water. This is 623

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accomplished by the provision of a rate feedback or transient gain reduction compensation as shown in Figure A5.3.

Fig. A5.3: Transient droop compensator for hydro-turbines

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Appendix 6 Interrelation between the dynamic and the static security It is of critical importance to note that the dynamic security is mainly guaranteed by the transient stability; however, the transient stability is not a guarantee of the static security (or adequacy). The static security (see volume 1/chapter 1) concerns mainly with the capability of the system – within the framework of selected N-k adequacy level - to provide acceptable steadystate operating conditions while all of its components operating within their operational and intrinsic limits. As an illustrative example, consider the system of Fig. 5.24. If the

disconnection of one of the lines leads to transient stable conditions if A acc  Apossdec; however, the new steady state operating point may violate the loading capability limit of the remaining line. Therefore, for the same disturbance (or contingency), the system may be dynamically stable but statically insecure due to the insufficient rating of the post-disturbance remaining circuits. In cases where the dynamic security (i.e. the transient stability) is not satisfied, there are many methods for enhancing the transient stability of power systems as will be presented in this book. The static security is out of the scope of this book. Therefore, it is recommended to read a textbook like the Allan J. Wood book 265; however, there are many corrective actions that can be taken for handling static insecurity problems. Fig. A6.1 and A6.2 summarizes some of these corrective actions. The corrective actions can be classified according to the implementation cost and according to the implementation time. From cost point of view, alternative corrective actions may be cost-free, with low-cost, and with highcost. From the implementation time point of view, corrective actions may be classified to immediate, actions with small time-delay in the order of minutes of hours, and actions with large time delay.

265

Wood, A. J., & Wollenberg, B. F. (2012). Power generation, operation, and control. John Wiley & Sons 625

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Fig. A6.1: Corrective actions for power flow overloads

Fig. A6.2: Corrective actions for voltage magnitude problems

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Cost-free options mainly dependent on adjusting the operation of the existing infrastructure while it has no impact of the optimal economic dispatch (OED) or the security constrained optimal power flow (SCOPF) or power transactions. Activation of the automatic AVR function of ULTC transformers is an example of cost-free corrective actions. These are also examples of immediate actions. Non-cost-free options usually include changes in the system economic operation or addition of new infrastructure. For example, generation shift and adjusting power transactions have a negative impact on the system economy. Planning-based corrective actions such as installing new line are usually associated with large investment and time of implementation.

(a)

(b) Fig. A6.3: OED-based static security analysis; (a) Pre-fault; (b) Steadystate post-fault operating conditions For a system that is dynamically secure, the static security may be adjusted through proper and defensive operation of power systems. For example, suppose that the system of Fig. 5.24 is subjected to a three-phase fault at one of the transmission lines where the transient stability analysis reveals that the system is stable. Now suppose that the system is supplying a load of 1200 MW, the maximum line loading is 400 MW, and the generator maximum active power is 800MW. Therefore, the parallel lines are not fully redundant. Based on the OED, the power production allocated to the local generator is 500 MW while the rest of the load is supplied from the rest of the system as shown in Fig. A6.3. Now of one of the line is subjected to a fault 627

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and consequently disconnected (i.e. forced outage) or disconnected for maintenance purpose (i.e. selective outage), the remaining line becomes overloaded as shown in Fig. A6.3. Therefore, the disconnection of the faulty line causes disconnection of the other line due to overloading. In larger systems, such situations can cause cascaded outages which may lead to system blackout. The SCOPF, on the other hand, allocates the generators for the system to be statically secure and defensive. For example, the system of Fig. A6.3 is to be operated based on the SCOPF under the N-1 security framework. This shown in Fig. A6.4. In this case, the generator allocations is not only based on cost minimization but also considering the N-1 requirements. Therefore, the generator power production becomes 400 MW for overcoming the outage of one of the lines. Consequently, the system becomes statically secure.

(a)

(b) Fig. A6.4: SCOPF-based static security analysis; (a) Pre-fault; (b) Steadystate post-fault operating conditions The shown corrective action affects the economy due to the shift in the power generation. Therefore, it is a non-cost free action; however, proper action may include increasing the transfer capacity by adding new line(s) such the local generator can operate at full load without violating the static security requirements. In this case, the time frame of implementing the corrective action is in the order of months or years and the associated costs may to very large. 628

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Appendix 7 Numerical integration methods In this appendix the characteristics of various numerical integration methods will be presented and their potentials and drawbacks will be summarized. As an initial step the differential equations to be solved are represented as a set of first-order differential equations or in the state-space form. For example, with damping power neglected, the swing equation of the SMIB system can be represented as,



o 2H

 Pm  Pe ( (t ))  

(a )

(A 6.1)

(b )

For various conditions of a fault (see chapter 5), P e(δ(t)) is given by the power flow equations:

where i represented either the pre-fault (I), during-fault (II), or post-fault (III) transfer reactance between the generator’s internal emf and the infinite bus. Note that in multi-machine power systems (see volume 1 of this textbook), each generator has a swing equation similar to that presented by (A6.1) while the electrical power takes the form,

Pei  E i2G ii   E i E jY ij cos(ij   i   j ) j 1 j i

(A 6.3)

where Ei, E j are the internal bus voltage magnitudes that are having angles of δi and δj, respectively. Yij= G ij+ jBij=Yij/_ ij is the element in the ith row, jth column of the appropriate Y-bus matrix. Depending on the particular time interval we are integrating, the “appropriate Y-bus” will be YI, YII, or YIII, corresponding to pre-fault, during-fault, and post-fault conditions, 629

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respectively. This equation is also applicable for the SMIB system if the losses are not neglected. The solution of (A6.1) can be found by integrating both sides of each equation. This action results in the form,

 (t )    ( )d   

o

t

t

0

2H 0

 Pm  Pe ( ( ))d 

 (t )    ( )d     ( )d  t

0

(a ) (A 6.4)

t

(b )

0

Note that there is a cross-coupling between the parts of A6.4 i.e. δ appears in the integrand of (a), and that ω appears in the integrand of (b). Since, the required solutions are (t) and (t), then a general closed form solutions cannot be obtained. Therefore, numerical integration is a suitable approach for solving such a problem. Let x1=ω and x2=δ, then (A6.4) can be written as,

x 1  f 1 (x 1, x 2 )    x  f (x ) x 2  f 2 (x 1 , x 2 ) 

(A 6.5)

It is important to note that, a power system with any scale and any configuration can be represented by the form of equation (A6.5). in solving

(A6.5), the boundary conditions are the initial values (0) = oI and (0) =

o. Therefore, the problem is an initial value problem. The fundamentals of the numerical integration for initial value problems can be demonstrated considering a single-state-variable; however, the treatment of many state-variable will be the same. For a single-state-variable, the problem is represented by,

x  f (x (t )),

x (0)  x 0

Hence, it is required to solve,

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(A6.6)

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x (t )   x ( )d    f (x ( ))d  t

t

0

(A 6.7)

0

Numerical integration methods utilize discrete time formulations rather than continuous time models. Therefore, (7) is written as,

x (kT ) 



 f (x (k  ))d 

kT

0



kT T

0

f (x (k  ))d   x ( kT T )

 x (kT  T ) 



kT kT T



kT kT T

f (x (k  ))d 

f (x (k  ))d 

(A 6.8)

In (A6.8), k is an integer number. Hence, the approach is comprehended as follows: “Given the value of x at t = kT-T i.e. the last known value of x, to determine value of x at the next time sample t = kT, the integral of (A6.8) must be determined”. This integral defines the change in x from the last step to the next in an iterative manner, i.e.,

x 



kT kT T

f (x (k  ))d 

(A 6.9)

Therefore, solving the considered problem requires only the ability to compute the integral in (A6.9). The value of the discrete time step (or sampling time) is T. Generally, the accuracy of the numerical integration is better with the reduction in the sampling time. There are many numerical methods for that purpose. There are two main classes of numerical integration methods. The first class is the explicit method such as the forward Euler, predictor-corrector, and Runge-Kutta methods. They are so-called because the evaluate x(kT) explicitly as a function of values at previous steps and their derivatives. All 631

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explicit methods are numerically unstable , i.e., they have a bounded stability

domain. Explicit methods are therefore must utilize small step-sizes to work correctly. The second class of integration methods that does not have this problem is called implicit methods. The implicit methods include the trapezoidal rule and the backward rule. Implicit methods require a value of the function x(kT) at a future time step. Implicit methods are good for “stiff” problems, where. Stiff problems are often characterized by large differences between the real parts of system eigenvalues . Details and implementation examples of these classes and methods are available at Hairer 266.

266

Hairer, E., & Wanner, G. (1996). Solving ordinary differential equations ii: Stiff and differential-algebraic problems second revised edition with 137 figures. Springer series in computational mathematics, 14. 632

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Appendix 8 Data of the detailed two-area system The system is shown in Fig. A8.1. The system consists of two similar areas connected by a weak tie. Each area consists of two synchronous generator units. The rating of each synchronous generator is 900 MVA and 20 kV. Each of the units is connected through transformers to the 230 kV transmission line. The generator parameters in per unit on the rated MVA and KV base are shown in Table A8.1. Each step up transformer has as impedance of per unit on 900 MVA and 20/230 KV base. The transmission system nominal voltage is 230 KV. The line lengths are identified in Fig. A8.1. The parameters of the lines in per unit on 100 MVA, 230 KV base are r = 0.0001 pu/km, xL = 0.001 pu/km, and bc = 0.00175 pu/km. The operating loadings on various generators are listed in Table A8.2. The loads and reactive power supplied by the shunt capacitors at buses 7 and 9 are shown in Table A8.3. The AVR data are shown in Table A8.4 while the PSS data are shown in Table A8.5. The parameters of the SVCs are listed in Tables A8.6 and A8.7.

Fig. A8.1: Two-area power system with wind power integration

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Table A8.1: Generator data

Table A8.2: Loading of generators Machine

(MW)

(MVAr)

G1

700

185

G2

700

235

G3

719

176

G4

700

202

(pu)

Table A8.3: Loads and reactive power compensations Bus

(MW)

(MVAr)

(MVAr)

7

967

100

9

1767

100

ZIP Model P

Q

200

Constant

Constant

350

current

impedance

Table A8.2: AVR data 20 0.055 0.36 0.125 1.8 0.0056 1.075 0.05

Table A8.2: PSS data 20

10

0.05 0.02

3

5.4

Table A8.6: Parameters of SVC at Bus 7 Regulator time constant Tr (sec) Regulator gain Kr (pu) Reference voltage (p.u.) Bmax (pu) Bmin (pu) 634

0.1 75 1 5 -5

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Table A8.7: Parameters of SVC at Bus 9: Regulator time constant Tr (sec) Regulator gain Kr (pu) Reference voltage (p.u.) αmax (rad) αmin (rad) Integral deviation Kd (pu) Transient time constant T1 (sec) Measurement gain Km (pu) Time delay Tm (sec) Inductive reactance Xl (pu) capacitive reactance Xc (pu)

0.1 75 1 3.14 -3.14 0.001 0 1 0.01 0.2 0.1

The parameters of the FS-SCIG and the DFIG integrated with the system (chapter 9) are listed in the following tables. Table A8.8: Parameters of SCIGs Power rating Sn (MVA) Voltage rating Vn (kV) Frequency rating fn (Hz) Stator resistance Rs (p.u.) Stator reactance Xs (p.u.) Rotor resistance Rr (p.u.) Rotor reactance Xr (p.u.) Magnetization reactance Xm (p.u.) Initial constant Hm (kWs/kVA) Number of poles p Number of Blades nb Gear box ratio GB

60 0.48 60 0.01 0.1 0.01 0.08 3 3 4 3 1/89

Table A8.9: Parameters of DFIGs Power rating Sn (MVA) Voltage rating Vn (kV) Frequency rating fn (Hz) Stator resistance Rs (p.u.) Stator reactance Xs (p.u.) Rotor resistance Rr (p.u.) Rotor reactance Xr (p.u.) Magnetization reactance Xm (p.u.) Initial constant Hm (kWs/kVA) Voltage Control Gain Kv (p.u) Pitch Control Gain (p.u) Time constants (s) Power Control Time Constant (s) Number of poles p Number of Blades nb Gear box ratio GB 635

900 20 60 0.01 0.1 0.01 0.08 3 3 10 10 3 0.01 4 3 1/89

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Table A8.10: Wind speed model Wind model type Average wind speed (m/s) Air density ρ(kg m3) Filter time constant τ(s) Sample time for wind measurements t Scale factor for Weibull distribution c Shape factor for Weibull distribution k

636

Mexican_hat 16 1.225 4 0.1 20 2

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Appendix 9 Nomenclature of the symbols used in Chapter 10 and 11 Cf EH Es EA

g H Hb Hd Ht

HTE i m

n NOCT

PModule

Qdc Rb Rd rd rt

S t to Tc Tr Ta  

Correction factor to account for the effect of off-optimal tilt angle of the PVgenerator on the temperature of the PV-modules. Monthly mean daily hydraulic energy demand (kWh/day) Monthly mean daily radiation on a horizontal plane (terrestrial radiation) in (kWh/m2/day) Actual monthly mean produced energy available for water pumping (kWh/day) Load matching factor to PV generator characteristics Acceleration due to gravity (m/s2) = 9.81 m/s2 Monthly mean hourly global radiation on a horizontal surface (kW/m2) Beam or direct component of H Diffuse component of H Hourly irradiance in the plane of the PV-generator (kW/m2) Total daily radiation in the plane of the PV-generator (kWh/m2/day) Monthly mean daily global radiation on a horizontal surface (kW/m2/day) Diffuse component of Total pumping head (m) The month of a year (i = 1 for January) Monthly average clearance index (dimensionless) Number of PV modules The day of a year (n = 32 for February 1) Nominal Operating Cell Temperature (oC) Nominal capacity or peak Watt of a PV- generator (Wp) Optimal capacity of a PV-generator (Wp) Rated power of a PV module (Wp) Total daily demanded water flow required for irrigating the crops (plant water requirements) in (m3/day) Tilt factor for beam radiation Tilt factor for diffuse radiation Ratio of total hourly to total daily diffuse radiation Ratio of hourly total to daily total values of monthly mean global radiation on a horizontal surface (dimensionless) Overall solar collector area of the PV-generator (m2) Local time on the 24-hour basis. Solar noon (hr) Temperature of the PV module (oC) Reference temperature of the PV module ( oC) = 25oC Mean monthly ambient temperature (oC) Diffuse reflectance (or ground albedo) of the ground Actual tilt angle (deg) 637

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* p 

Optimal tilt angle (deg) Temperature coefficient of the PV module ( oC-1) Declination angle (deg) Motor–pump unit efficiency Inverter efficiency Miscellaneous PV-array losses (wiring losses) Irrigation efficiency (shows to what extent the water enters an irrigation system is exploited) Irrigation system efficiency (account for energy losses due to friction of water in the irrigation system) Efficiency factor of the PV pumping system Overall efficiency of the PV generator PV module efficiency at reference temperature. Density of water (kg/m3) = 1000 kg/m3  Latitude (deg)  Solar hour angle (deg)  Sunset hour angle (deg) s Sandia Model Short Circuit Current (A) Current at the maximum-power point (A) Current at module voltage V = 0.5⋅ , defines 4th point on I-V curve for modeling curve shape (A) Current at module voltage V = 0.5⋅( + ), defines 5th point on I-V curve for modeling curve shape (A) Open-circuit voltage (V) Voltage at maximum-power point (V) Power at maximum-power point (W) Fill Factor Number of cell-strings in parallel in module Boltz a s o sta t, . 8 E-23 (J/K) Elementary charge, 1.60218*10-19 (coulomb) Cell temperature inside module (°C) Reference cell temperature, typically 25°C Reference solar irradiance, typically 1000 W/m2 Ther al oltage per ell at te perature . The effe ti e solar irradia e. This alue des ri es the fra tio of the total solar irradiance incident on the module to which the cells inside actually respond. Empirically determined coefficients relating to effective irradiance, , . + = 1, (dimensionless) Empirically determined coefficients relating to effective irradiance, , . is dimensionless and has units of 1/V. , Empirically determined coefficients relating to effective irradiance, . + = 1, (dimensionless) , Empirically determined coefficients relating to effective irradiance, . + = 1, (dimensionless) 638

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E piri ally deter i ed diode fa tor asso iated ith i di idual ells i the module, with a value typically near unity, (dimensionless). Fraction of diffuse irradiance used by module Empirically determined polynomial relating the solar spectral influence on to air mass variation over the day, where: Empirically determined polynomial relating optical influences on solar angle-of-incidence (AOI), where: at ( 1000 W/m2, , at ( , ) (A) at ( , ) (V) at ( , ) (V) at ( , ) (A) at ( , ) (A) Normalized temperature coefficient for Normalized temperature coefficient for

o

C,

to

= 0) (A)

, (1/°C) , (1/°C) 2

Temperature coefficient for module at a 1000 W/m irradiance level, (V/°C) Coefficient providing the irradiance dependence for the temperature coefficient, typically assumed to be zero, (V/°C) Temperature coefficient for module maximum-power-voltage as a function of effective irradiance, . Usually, the irradiance dependence can be neglected and is assumed to be a constant value 2

Temperature coefficient for module at a 1000 W/m irradiance level, (V/°C) Coefficient providing the irradiance dependence for the temperature coefficient, typically assumed to be zero, (V/°C) CEC Model Ideality factor (eV) Absolute air mass Ideality factor at SRC (eV) Total irradiance on horizontal surface (W/m2) Beam component of total irradiance on horizontal surface (W/m 2) Diffuse component of total irradiance on horizontal surface (W/m2) Effective irradiance incident on the tilted surface (W/m2) Effective irradiance at SRC (W/m2) Predicted current (A) Light Current (A) Light Current at SRC (A) Current at maximum power point (A) Current at maximum power point at SRC (A) Diode reverse saturation current (A) Diode reverse saturation current at SRC (A) Incidence angle odifier at i ide e a gle θ I ide e a gle odifier at θ = 8o Air mass modifier 639

M. EL-Shimy. Dynamic Security of Interconnected Electric Power Systems – Volume 2. LAP, 2015.

Air mass modifier at SRC (=1) Number of cells in series Predicted power (W) Ratio of beam radiation on tilted surface to that on a horizontal surface Series resistance (ohm) Series resistance at SRC (ohm) Shunt resistance (ohm) Shunt resistance at SRC (ohm) Cell temperature (oC) Cell temperature at SRC(oC) Assigned voltage (V) Voltage at maximum power point (V) Voltage at maximum power point at SRC (V) Temperature coefficient for short circuit current (A/ oC) Slope of the PV panel (Deg.) Open circuit voltage temperature coefficient (V/ oC) Material band gap energy (eV) Incidence angle, angle between the beam of light and the normal to the panel surface (Degree) Zenith angle, angle between the vertical and the line to the sun (beam) (Degree) Ground reflectance factor Other quantities Life of the project (years) Year number; t = , , …, T Net annual cost of the project for year t ($) The energy produced in year t (kWh) Initial investment and cost of the system for year t ($) Maintenance cost for year t ($) Operational cost for year t ($) Interest expenditure for year t ($) Discount rate (%) The initial rated energy output (kWh/year) Degradation rate (%) Depreciation rate (%) Tax rate (%) Residual value (%) The net present value of the cost factors till the loan payment ending period ($) The net present value of the cost factors after loan payment ending period to the end of the effective lifetime of the project ($). The total energy production till the end of the effective lifetime period (kWh). Effective lifetime of the PV technology (years). Financial lifetime of the PV project (years). No. of inverter replacements. Inverter capacity (W) Inverter cost ($/W) Per unit energy produced in year t (p.u) 640

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