Power System Frequency Control: Modeling and Advances 0443184267, 9780443184260

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Power System Frequency Control

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Power System Frequency Control Modeling and Advances

Dillip Kumar Mishra Li Li Jiangfeng Zhang Md. Jahangir Hossain

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN 978-0-443-18426-0 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

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Contents

Contributors

xi

1

1

2

3

Fundamentals of load frequency control in power system Dillip Kumar Mishra, Li Li, Jiangfeng Zhang, and Md. Jahangir Hossain 1.1 Basic concepts 1.2 AGC in a modern area power network 1.3 Power network frequency loop 1.4 Individual model of the AGC system 1.5 Structure of the AGC system Appendix 1: AGC parameters and values References Controller design for load frequency control: Shortcomings and benefits Dillip Kumar Mishra, Li Li, Md. Jahangir Hossain, and Jiangfeng Zhang 2.1 Introduction 2.2 Traditional control design 2.3 Shortcomings of the traditional controller 2.4 The need for an advanced control method 2.5 Controller 2.6 Objective function Appendix References Transient/sensitivity/stability analysis of load frequency control Dillip Kumar Mishra, Tapas Kumar Panigrahi, Prakash Kumar Ray, and Asit Mohanty 3.1 Introduction 3.2 Transient analysis 3.3 Sensitivity analysis 3.4 Stability analysis References

1 2 2 6 8 14 15

17 17 17 27 27 27 32 36 38

41

41 42 64 66 95

vi

4

5

6

7

8

Contents

Significance of ancillary devices for load frequency control Dillip Kumar Mishra, Prakash Kumar Ray, Asit Mohanty, and Tapas Kumar Panigrahi 4.1 Introduction 4.2 Thyristor-controlled series capacitor (TCSC) 4.3 Static synchronous series compensator (SSSC) 4.4 Unified power flow controller (UPFC) 4.5 Interline power flow controller (IPFC) 4.6 Summary References Challenges and viewpoints of load frequency control in deregulated power system Dillip Kumar Mishra, Li Li, Jiangfeng Zhang, and Md. Jahangir Hossain 5.1 Introduction 5.2 Transient response analysis of AGC with a deregulated environment 5.3 Summary References Battery energy storage contribution to system frequency for grids with high renewable energy sources penetration Giuliano Rancilio, Andrea Vicario, Marco Merlo, and Alberto Berizzi 6.1 Introduction 6.2 The fast frequency regulation 6.3 The proposed methodology 6.4 Results 6.5 Conclusions References The power grid load frequency control method combined with multiple types of energy storage system Kezhen Liu, Guo Liu, Min Dong, and Jing He 7.1 Introduction 7.2 Model of load frequency control 7.3 Model of PPS-HESS combined control 7.4 Design of controller 7.5 Analysis of simulation 7.6 Conclusion References Sophisticated dynamic frequency modeling: Higher order SFR model of hybrid power system with renewable generation Jianbo Yi, Xiujie Zheng, Changxuan Liu, and Qi Huang 8.1 Introduction 8.2 Frequency dynamic response characteristics

97

97 98 100 106 109 114 114

117 117 120 130 131

133 133 135 138 145 148 148

151 151 152 154 157 160 165 166

169 169 170

Contents

8.3 8.4 8.5 8.6

Traditional second-order SFR model Modeling and analysis of the higher order SFR model Higher order SFR model of hybrid power systems Correction of mixture proportion parameter in higher order SFR model 8.7 Summary Acknowledgment References

9

10

11

Application of neural-network based variable fractional order PID controllers for load frequency control in isolated microgrids Komeil Nosrati, Vjatseslav Skiparev, Aleksei Tepljakov, Eduard Petlenkov, Yoash Levron, and Juri Belikov 9.1 Introduction 9.2 Isolated HMG configuration and mathematical modeling 9.3 (FO)PID controllers, actions, and tuning rules 9.4 The proposed (FO)PID-based LFC: Multiagent NN-based online tuning approach 9.5 Simulation results 9.6 Conclusion Acknowledgments References Coordinated tuning of MMC-HVDC interconnection links and PEM electrolyzers for fast frequency support in a multiarea electrical power system Georgios Giannakopoulos, Arcadio Perilla, Jose Rueda-Torres, Peter Palensky, and Francisco Gonzalez-Longatt 10.1 Introduction 10.2 Theoretical background 10.3 Optimization problem formulation 10.4 The mean variance optimization algorithm 10.5 The test system 10.6 Simulation study and results 10.7 Discussion 10.8 Conclusions References Under-frequency load shedding control: From stage-wise to continuous Changgang Li 11.1 Introduction 11.2 Under-frequency load shedding: Concepts and cases 11.3 Performance of continuous under-frequency load shedding 11.4 Implementation of continuous under-frequency load shedding

vii

173 176 187 191 201 201 202

203

203 204 207 209 212 214 215 215

217

217 219 224 226 228 232 241 244 245

249 249 250 252 255

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Contents

11.5 Applications 11.6 Conclusions References 12

13

14

15

Emergency active-power balancing scheme for load frequency control  Urban Rude z, Tadej Skrjanc, and Rafael Mihali c 12.1 Introduction 12.2 Electric-power system response to active-power imbalance 12.3 Emergency active-power balancing 12.4 Challenges and a way forward References Keeping an eye on the load frequency control implementation using LabVIEW platform Dillip Kumar Mishra, Asit Mohanty, Prakash Kumar Ray, and Tapas Kumar Panigrahi 13.1 Introduction 13.2 Overview of LabVIEW 13.3 Elements and functions 13.4 Control system toolbox 13.5 Case study References An overview of the real-time digital simulation platform and realization of multiarea multisource load frequency control model using OPAL-RT Dillip Kumar Mishra, Prakash Kumar Ray, and Asit Mohanty 14.1 Introduction 14.2 Real-time emulator 14.3 Why use real-time simulation 14.4 RT-LAB system architecture 14.5 Real-time validation steps 14.6 Real-time study using OPAL-RT 14.7 Conclusions References Design and testing capabilities of low-inertia energy system-based frequency control using Typhoon HIL real-time digital simulator Sourav Kumar Sahu, Dillip Kumar Mishra, Soham Dutta, and Debomita Ghosh 15.1 Introduction 15.2 Type of real-time configurations in Typhoon HIL environment

260 263 264

265 265 266 272 284 285

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289 290 291 295 301 308

311 311 311 312 312 314 316 319 319

321

321 322

Contents

15.3 15.4

Cost and fidelity analysis of different configurations Flow chart of workflow for Typhoon HIL real-time simulation 15.5 Communication protocols 15.6 Results and analysis: Active distribution network under study 15.7 Conclusion References

Index

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322 324 325 325 330 330 331

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Contributors

Juri Belikov Department of Software Science, Tallinn University of Technology, Tallinn, Estonia Alberto Berizzi Politecnico di Milano, Department of Energy, Milano, Italy Min Dong Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming, Yunnan, People’s Republic of China Soham Dutta Department of Electrical and Electronics Engineering, Manipal Institute of Technology, Manipal, Karnataka, India Debomita Ghosh Department of Electrical & Electronics Engineering, BIT Mesra, Ranchi, Jharkhand, India Georgios Giannakopoulos Department of Electrical Sustainable Energy, Delft University of Technology, Delft, Netherlands Francisco Gonzalez-Longatt Department of Electrical Engineering, Information Technology and Cybernetics, University of South-Eastern Norway, Porsgrunn, Norway Jing He Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming, Yunnan, People’s Republic of China Md. Jahangir Hossain School of Electrical & Data Engineering, University of Technology Sydney, Ultimo, NSW, Australia Qi Huang School of Mechanical and Electrical Engineering, Chengdu University of Technology, Yibin, China Yoash Levron The Andrew and Erna Viterbi Faculty of Electrical & Computer Engineering, Technion—Israel Institute of Technology, Haifa, Israel Changgang Li School of Electrical Engineering, Shandong University, Jinan, China Li Li School of Electrical & Data Engineering, University of Technology Sydney, Ultimo, NSW, Australia

xii

Contributors

Changxuan Liu School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, China Guo Liu Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming, Yunnan, People’s Republic of China Kezhen Liu Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming, Yunnan, People’s Republic of China Marco Merlo Politecnico di Milano, Department of Energy, Milano, Italy Rafael Mihalic University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia Dillip Kumar Mishra School of Electrical & Data Engineering, University of Technology Sydney, Ultimo, NSW, Australia Asit Mohanty Department of Electrical & Electronics Engineering, Odisha University of Technology and Research, Bhubaneswar, Odisha, India Komeil Nosrati Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia Peter Palensky Department of Electrical Sustainable Energy, Delft University of Technology, Delft, Netherlands Tapas Kumar Panigrahi Department of Electrical Engineering, Parala Maharaja Engineering College, Brahmapur, Odisha, India Arcadio Perilla Department of Electrical Sustainable Energy, Delft University of Technology, Delft, Netherlands Eduard Petlenkov Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia Giuliano Rancilio Politecnico di Milano, Department of Energy, Milano, Italy Prakash Kumar Ray Department of Electrical & Electronics Engineering, Odisha University of Technology and Research, Bhubaneswar, Odisha, India Urban Rudezˇ University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia Jose Rueda-Torres Department of Electrical Sustainable Energy, Delft University of Technology, Delft, Netherlands

Contributors

xiii

Sourav Kumar Sahu Department of Electrical & Electronics Engineering, BIT Mesra, Ranchi, Jharkhand, India Vjatseslav Skiparev Department of Software Science, Tallinn University of Technology, Tallinn, Estonia Tadej Sˇkrjanc University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia Aleksei Tepljakov Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia Andrea Vicario Politecnico di Milano, Department of Energy, Milano, Italy Jianbo Yi School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, China Jiangfeng Zhang Department of Automotive Engineering, CLEMSON University, Clemson, SC, United States Xiujie Zheng Chengdu College of University of Electronic Science and Technology of China, Chengdu, China

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Fundamentals of load frequency control in power system

1

Dillip Kumar Mishraa, Li Lia, Jiangfeng Zhangb, and Md. Jahangir Hossaina a School of Electrical & Data Engineering, University of Technology Sydney, Ultimo, NSW, Australia, bDepartment of Automotive Engineering, CLEMSON University, Clemson, SC, United States

1.1

Basic concepts

In power systems, automatic generation control (AGC) plays a significant role in the control process in a power network to achieve the equilibrium between generation and load in the most cost-effective manner. AGC approach deals with frequency regulation and control of the power exchange and economic dispatch. It also adjusts the MW power of multiple generators at different power plants corresponding to the load change [1]. Principally, the AGC is a closed-loop-based scheme used to control both the frequency and exchange power. More importantly, two parameters, such as frequency and voltage, are crucial to the power network to balance the generation and load. Therefore, to control the frequency deviation, one of the essential control loops, i.e., load frequency control (LFC), has been used [2]. However, another loop, i.e., automatic voltage regulator (AVR), has been employed to manage the voltage level. The abovementioned control loop, namely LFC and AVR, is the extension of AGC, where both the frequency and voltage have been considered the control parameter. In most cases, researchers have assumed that the LFC is termed AGC [3]. Because frequency control is more important than the voltage, as far as frequency deviation is concerned, it can only cope with
0.5 hertz with the power frequency. And less than or more than that, the power network may be collapsed, or a blackout may happen. Hence, over the past few decades, a large body of research has been carried out to focus on load frequency control. And several advancement methods have been established to improve the dynamic response of the AGC during disruptive events [4]. An interconnected network means two or more areas are connected in a single platform. Similarly, in the power system context, the interconnection means the two or more control areas are incorporated through the tie-lines. The control areas are regulated by their control actions in the form of coherence principles, which must be assumed while doing the simulation. On top of that, a control signal can be derived from frequency change and tie-line power deviation, and this is termed an area control error (ACE). It indicates the need for generation either lowered or raised [5]. This chapter introduces the basics of AGC in modern area power networks, various control loops, individual models, and different structures. Besides, the facts about the Power System Frequency Control. https://doi.org/10.1016/B978-0-443-18426-0.00004-2 Copyright © 2023 Elsevier Inc. All rights reserved.

2

Power System Frequency Control

AGC and its role, objectives, primary functions, and specific features are outlined. The operation of the power network and its control range of frequency change with their droop characteristics are described.

1.2

AGC in a modern area power network

AGC offers an efficient scheme for monitoring the change in frequency and tie-line power to balance the generation. It can recognize generation levels by transmitting commands to the generating units which are under control mechanisms. The AGC response is completely trusting in commands and how they respond [6]. Eventually, the command responses rely on according to the unit types, fuels, control methods, and equilibrium points. The changes experienced by AGC over the past decade remain unprecedented on account of security measures, supervisory control, and data acquisition (SCADA) and coordinated control of load switching are the key factors of the energy management system (EMS), as shown in Fig. 1. The entire process of AGC is achieved remotely through the control center from each generating station, whereas the power generation is regulated by the governing system (turbine-governor) in the generation location. The detailed description of each unit is presented in Ref. [7].

1.3

Power network frequency loop

In an electric power network, frequency is one of the major factors for maintaining the load demand-generation balance. As discussed earlier, the permission frequency change in the system is
0.5 hertz, and more devotion prompts to blackout or power outage in the entire interconnected network. Hence, it is essential to keep within rigid Power System

SCADA and Load Coordination

AGC

Security measure

Workstation

Energy Management System (EMS)

Fig. 1 Modern EMS.

Fundamentals of load frequency control in power system

3

limits. More importantly, it has been taken as a favorable performance index to show generation-load balance characteristics. On the other hand, it affects the reliable, secure, and efficient operation of an electric power network. From the fundamentals, the rotation speed of the turbine is proportional to the frequency of the system irrespective of the constant pole. Hence, the change in the value of rotation speed can change the range of frequency level with the help of a speed governing the system, where the turbines (low pressure and/or high pressure) are coupled with each other. The speed changer plays a vital role in meeting the requirement of input steam to the turbine to balance the frequency and generation as well. The schematic diagram of the speed governing mechanism is shown in Fig. 2 [3]. In response to the frequency change, the control action can be made, and it has been categorized into three operations: primary, secondary, and emergency. In the primary operation phase, the deviation is assumed as very small, which can be alleviated through the primary loop. In the response of substantial-frequency change, the secondary control action comes into play, which has the capability to change the frequency level in a broader range, which is called secondary operation. However, with the highdisruptive events, or extreme events, both the control as mentioned above, action cannot restore the system frequency. In that scenario, emergency control action should be applied to get back to its system frequency, which is called emergency operation. The three control actions with the frequency range change are portrayed in Fig. 3.

Fig. 2 Speed governing system.

Power System Frequency Control

Frequency in Hz

4

1

2

1

Primary operation

2

Secondary operation

3

Emergency operation

3

Fig. 3 Three control operation of the power system.

1.3.1 Primary loop The active power generated by the generator depends on the mechanical energy produced by the turbine, which is different from different input sources. In thermal and hydro plants, the turbine is called a steam turbine and a hydroturbine. Besides, the input source is steam for thermal and water for hydro, which are the responsible input to the turbine, and it produces mechanical power. In spite of that, the input (steam or water) can be controlled through the regulator by opening and closing the valve to meet the active power demand [8]. The simple block diagram of the primary loop is illustrated in Fig. 4. In this loop, speed governor acts as a speed sensor, which senses the speed or frequency from the

Primary loop

Steam Generator (G)

Speed governor

Hydraulic amplifier

G

Turbine

Main valve

ω

Load

Fig. 4 Primary loop.

Fundamentals of load frequency control in power system

5

feedback through steam/hydro turbine and subsequently gives a command to the hydraulic amplifier to change the valve level either raise or low according to the desired frequency. In practice, the primary loop achieves a local programmed control that offers backup power in the response of frequency change. Though, it has been verified that this loop cannot restore the frequency at a rigid level when the power system has more than one area. Thus, a secondary loop must be employed to regulate the set point via the speed changer and controller.

1.3.2 Secondary loop The secondary loop consists of the primary loop with a speed regulator and controller, which is shown in Fig. 5. The importance of this loop is feedback control, which is fed from the frequency deviation output, and it adds to the primary loop to precisely control the frequency change. Meanwhile, the speed changer motor acts as a regulator after providing the set point command from the controller. In an actual power system scenario, the integral or proportional plus integral controller has been used as a dynamic controller. With this mechanism, the AGC restores the frequency level in response to the disruptive events within a broad range. However, in case of extreme events, even the secondary controller cannot restore the frequency, which may lead to a power outage; hence, the emergency loop must be considered [9].

1.3.3 Emergency loop The need for an emergency loop is in the extreme event cases such as major 3-phase faults, disasters, inevitable accidents, or other major catastrophes. In these cases, the frequency deviation is very large, which is very difficult to restore. Thus, load curtailment techniques shall be adapted to quickly manage the frequency deviation and evade the blackouts. Moreover, the emergency loop establishes better dynamic characteristics during the events with the load curtailment procedure. The main aims of the load curtailment are to cut down the least coupled load and make the reliable operation of the system from a contingency scenario to a normal operating state [9].

P Tie

Primary loop

Steam Generator (G)

Controller (I/PI)

Speed changer

Speed governor

Hydraulic amplifier

Turbine

G

Main valve Secondary loop

f

Frequency sensor Load

Fig. 5 Secondary loop.

6

1.4

Power System Frequency Control

Individual model of the AGC system

The AGC system comprises mainly controller, governor, turbine, generator, and load units. Each unit has been modeled is as follows [1].

1.4.1 Generator model The swing equation of synchronous generator with minute load change can be written as 2H d2 Δδ ¼ ΔPm  ΔPe ω dt2

(1)

With a small speed deviation, we have dΔ ωωs 1 ½ΔPm  ΔPe  ¼ 2H dt

(2)

Applying Laplace transformation in Eq. (2), we get ΔΩ ðsÞ ¼

1 ½ΔPm ðsÞ  ΔPe ðsÞ 2Hs

(3)

The parameters of the Eq. (3), H, Δ Pm , and Δ Pe are the inertia constant, mechanical power, and electrical power, respectively (Fig. 6).

1.4.2 Load model The load on the power system is given by ΔPe 5ΔPL + DΔω

(4)

In this case Δ PL, D, and Δ ω are the non-frequency sensitive load change, damping coefficient with frequency sensitive load change, respectively (Fig. 7).

∆Pe(s) _

∑ + ∆Pm(s)

Fig. 6 Generator model.

1 2Hs

∆Ω(s)

Fundamentals of load frequency control in power system

7

Fig. 7 Load model.

1.4.3 Turbine model The turbine model can be modeled from the input source to the output of mechanical energy. The input source is different from a different plant such as thermal, hydro, and gas is the steam, hydro, and gas turbine correspondingly. Further, the input can be presented as the change in valve power, i.e., Δ Pv. The transfer function model of the turbine is presented as GT ¼

ΔPm ðsÞ 1 ¼ ΔPv ðsÞ 1 + sT t

(5)

where Tt is represented as turbine time constant and the range between 0.2 and 2 s (Fig. 8).

1.4.4 Governor model The governor model can be defined as the feedback power Δ Pg with regulating parameter (R) to the valve power (Δ Pv). The transfer function model of the governor is presented as (Fig. 9) ΔPg ðsÞ ¼ ΔPref ðsÞ  Gg ¼

1 Δf R

ΔPv ðsÞ 1 ¼ ΔPg ðsÞ 1 + sT g

Fig. 8 Turbine model.

Fig. 9 Governor model.

(6) (7)

8

Power System Frequency Control

1.4.5 Tie-line model In an interconnected power network, the tie-line plays a vital role in increasing the number of areas. The frequency and tie-line both can present the mismatch characteristics of the connected areas. Thus, a proper power exchange can be done through AGC. The change in tie-line power Δ Ptie,i, can be written as 2 ΔPtie,i ¼

N X

ΔPtie,ij

j¼1 j6¼i

2π ¼ s

3

6X 7 N X 6 N 7 6 T ij Δf i  T ij Δf j 7 6 7 4 j¼1 5 j¼1 j6¼i

(8)

j6¼i

where Tij represents the synchronizing coefficient. For instance, if there is a two-area power system, which is interconnected with each other through tie-line, i.e., Tij ¼ T12. Similarly, for three areas, Tij ¼ T12, T13, and T23. For two area power system, Tij ¼ T12, and the Δ Ptie12 is formulated as (Fig. 10) ΔPtie12 ðsÞ ¼

1.5

2πT 12 ½Δf 1 ðsÞ  Δf 2 ðsÞ s

(9)

Structure of the AGC system

The AGC can be formed in a single-area or multiarea power system. The formation depends on the number of control area which is connected to the single area through tie-line. Furthermore, a diverse source of power generation can also be added to a single area to construct a multiarea network. There are many advantages of a multiarea network and mainly focusing on the reliability and power interchange. The detail description of LFC and its interconnection is explained as follows. LFC is termed as one of the breakthrough technology which has been employed over the past few decades to regulate the frequency and interchange power deviation in an interconnected power network by means of tie-line [1]. Here, the deviation in frequency is adjusted based on increasing or decreasing the individual generators in a group by designing suitable controllers. Many researchers have developed a

Fig. 10 Tie-line model.

Fundamentals of load frequency control in power system

9

variety of controllers for AGC and hence control the system frequency to maintain and to improve stability. The exchange power flow through the tie-line between the control areas can be achieved through frequency regulation during the normal and disruptive scenarios [2]. The main aim of the LFC is  To observe and regulate the frequency and throughput power deviation in an interconnected power network.  To monitor the tie-line exchange power between the control regions.

Even with a minute load change, the power system becomes an imbalance between generation and demand in their frequency range. This imbalance problem is due to kinetic energy extracted from the plant. Consequently, system frequency reduces. Seeing that power consumed by the load reduces slowly with the reduction of frequency. In bulk power networks, the balance can be achieved by all of them meeting at a point when the freshly added load is diverted by minimizing the power expended by the old load, which is directly related to the extraction of kinetic energy in the system. Undoubtedly, with the marginal frequency decrease we’re attaining this balance. In this way, some regulatory action takes place to meet the equilibrium point, and there is no requirement of governing operation. Notwithstanding, the level of the frequency deviation of this state is very high. To manage the large deviation, the governor plays a significant role. When the governor is put into operation, the generator’s output becomes imbalanced. Therefore, the equilibrium point can be achieved by means of load demand and power generation balance technique through governing action. It controls both raising and lowering the generation, which results in meeting the equilibrium point to evade the large deviation. Moreover, kinetic energy plays a vital factor, as it reduced the considerable amount, but not ultimately. Consequently, frequency deviation reduces but still require to meet the equilibrium point. However, in this action, the deviation reduction is minimal compared to those mentioned above. There is an increasing concern that this case mostly acquired the equilibrium from 10 to 12 s after adding loads. This type of governing action is named as the primary control. However, after the experience with governing operations, still, the system frequency does not match with the actual one. Thus, a new control technique must be needed to match with the actual frequency. In an initial attempt, engineers have applied an integral controller to further reduce frequency deviation, and this control action has been termed a secondary control loop. More importantly, after the operation of the governor, a secondary control loop plays into action to bring into the actual system frequency or closer to that.

1.5.1 Power system interconnection and its significance The power system connects two or multiarea systems through tie lines; each area provides the control, and this tie-line between the areas permits the flow of power. While the change in load demand affects the output frequency in any interconnected regions of the system, tie-line power also gets disturbed. Hence, the disturbance affected by all

10

Power System Frequency Control

neighbor areas is monitored through the control action of the respective region to establish the predefined value of system frequency and tie-line power [10]. Each system has its area frequency deviation, tie-line power change, and area control error (ACE). The ACE can be described with frequency deviation (△ Fi), tie-line power change (△ Ptie), and frequency bias parameter (βi), expressed in Eq. (10). It is noted that the ACE is used as the input of the controller. ACE ¼ βi ΔFi + ΔPtie

(10)

Given all that has been mentioned so far, one may suppose that an appropriate control system is needed in the LFC wherein the two main parameters, frequency and exchange power, restore the actual value during the disturbance. But in practice, the interconnected power system of load frequency control is more important than an isolated single area system. Although, it is equally important for both isolated and interconnected power systems as per the theoretical knowledge. Today’s power system has all interconnections with their contiguous regions, and LFC is developed by combined effort. Some fundamental working practices of an interconnected power network are followed:  

The load demand of its own area is carried out by its own unit during the normal working conditions and addresses the scheduled percentage of the load demands of the neighbor region, as per agreement. They should have approved over implementation, monitoring, control methods, and unit in that area helpful during the normal and disturbing circumstances.

The major benefits of the power system interconnection are size and reserve capacity. In the context of the size effect of an interconnected network, frequency drop is usually low as compared to a single area network with the same load change. For instance, when there is a load addition, the energy needed is provisionally extracted kinetic energy from the plant. In general, the accessibility of power is more for bulk networks. This result, therefore, manages the lower frequency drop in interconnected systems. On the other hand, the reserve capacity must be reduced to evade the large deviations. To realize this to a large extent, the most vital factor, i.e., peak load demand, can be taken. For example, we know that the peak load demand can persist in any neighboring region at any time of the day. It is challenging to preserve within a few seconds; otherwise, it could bring the system’s instability. Generally, the relationship between the peak load and the average load in bulk power networks is lesser than in small-scale networks. It is, therefore, likely that such a low capacity reserve significantly makes the benefits through the interchange schedule [11].

1.5.2 Single-area model The combination of all the above models into a single platform can be made as a single area thermal power system is illustrated in Fig. 11. Moreover, the single area can add several diverse sources to meet the load demand due to the limitation of single-source output [3].

Fundamentals of load frequency control in power system

11

Fig. 11 Single-area model.

Fig. 12 Multiarea model.

1.5.3 Multiarea model The multiarea power system can be defined as the interconnection of two or more areas through a tie-line. A block diagram presentation of a multiarea power network is illustrated in Fig. 12. Each area has its own power generation capacities and is tied together for the necessities to meet the objectives of the interconnection. In general, power system interconnection deals with various generating sources like thermal, gas, nuclear, and hydro in association with renewable sources as well. Due to practical constraints, by considering the efficiency, nuclear stations are mostly preserved at baseload, which can provide maximum output power and notably no contribution toward the LFC scheme. However, gas power stations are best for supplying the varying load, which is particularly focused on merely peak demands. Following the recent power system market, integrating various energy sources into a single control area through their respective participation factors and other constraints is more convincing for the AGC study. Because the interconnection of power systems with AGC is more vital nowadays than the single area network.

12

Power System Frequency Control

Fig. 13 Multiarea diverse source model.

1.5.4 Multiarea extension model The multiarea diverse source-based extension model is demonstrated in Fig. 13 In this model, steam, hydro, and gas turbine-based system are considered, and both the areas are the same power rating as their participation. The parameter of the system is presented in Appendix 1. The mathematical expression of the multiarea power system is as follows [12]. The output power of area-1 and area-2 is given by Eqs. (11) and (12) PG1 ¼ PGt1 + PGh1 + PGg1

(11)

PG2 ¼ PGt2 + PGh2 + PGg2

(12)

The above two Eqs. (11) and (12) can be rewritten as PG1 ¼ PGt1 K t1 + PGh1 K h1 + PGg1 K g1

(13)

Fundamentals of load frequency control in power system

PG2 ¼ PGt2 K t2 + PGh2 K h2 + PGg2 K g2

13

(14)

It is noted that the sum of participation factors is equal to one, i.e., K t1 + K h1 + K g1 ¼ 1

or K t2 + K h2 + K g2 ¼ 1

(15)

By applying small load disturbance, Eqs. (11) and (12) becomes, ΔPG1 ¼ ΔPGt1 + ΔPGh1 + ΔPGg1

(16)

PG2 ¼ ΔPGt2 + ΔPGh2 + ΔPGg2

(17)

The AC tie-line power can be described as PtieAC ¼ P12

max

sin ðδ1  δ2 Þ

(18)

By applying small load disturbance, Eq. (18) becomes, PtieAC ¼ P12

max

sin ðΔδ1  Δδ2 Þ

(19)

Then the synchronizing coefficient (T12) is defined as T 12 ¼

dPtieAC dδ

T 12 ¼ T 12

max

(20) cos ðδ1  δ2 Þ

(21)

In recent days, the HVDC-link has been used to extend the power system in a broad context and dynamic stability enhancement as well. From Fig. 9, the AC tie-line power can be described as Ptie AC ¼

2πT 12 ðΔF1  ΔF2 Þ s

(22)

Moreover, similarly, the AC tie-line power can be described as Ptie DC ¼ The Ptie

12

K DC ðΔF1  ΔF2 Þ 1 + sDC

(23)

is written as

Ptie AC ¼ Ptie AC + Ptie DC

(24)

By applying small load disturbance, Eq. (18) becomes, ΔPtie AC ¼ ΔPtie AC + ΔPtie DC

(25)

14

Power System Frequency Control

With the consideration of both AC and DC-links, the area control error (ACE) can be written as ACE1 ¼ β1 ΔF1 + ΔPtieAC + ΔPtieDC

(26)

ACE2 ¼ β2 ΔF2 + a12 ðΔPtieAC + ΔPtieDC Þ

(27)

where a12 is presented as area size ratio and is given by a12 ¼

Pr1 ¼ 1 Pr2

(28)

Appendix 1: AGC parameters and values Symbols

Nomenclature

Values

i F PRi Bi R1 ¼ R2

Subscript referred to area i (1,2) Nominal system frequency (Hz) Rated power of area i (MW) Frequency bias parameter of area i (p.u. MW/Hz) Governor speed regulation parameter of thermal areas RTi (Hz/p.u. MW) Governor speed regulation parameter of hydro areas RHi (Hz/p.u. MW) Governor speed regulation parameter of gas areas RGi (Hz/p.u. MW) Speed governor time constant for thermal areas (s) Steam turbine time constant (s) Power system time constant (s) Power system gain (Hz/p.u. MW) Synchronizing coefficient between areas 1 and 2 (p.u.) Steam turbine reheat constant Steam turbine reheat time constant (s) Nominal starting time of water in penstock (s) Hydroturbine speed governor reset time (s) Hydroturbine speed governor transient droop time constant (s) Hydroturbine speed governor main servo time constant (s) Gas governor lead time constant (s) Gas governor lag time constant (s) Gas turbine valve positioner Gas turbine constant of valve positioner Gas turbine fuel time constant (s) Gas turbine combustion reaction time delay (s) Gas turbine compressor discharge volume-time constant (s)

– 60 2000 0.425 2.4

TGi TTi TPi KPi T12 Kri Tri TWi TRi TRHi TGHi Xi Yi ci bi TFi TCRi TCDi

2.4 2.4 0.08 0.3 20 120 0.0433 0.3 5 1.0 5 28.75 0.2 0.6 1.0 1.0 0.05 0.23 0.3 0.2

Fundamentals of load frequency control in power system

15

Symbols

Nomenclature

Values

KDC TDC tsim a12 NP G F CR Ui ACEi ΔFi ΔPLi ΔPTieij ΔPGti ΔPRti ΔXti ΔPGhi ΔXhi

HVDC power system gain (Hz/p.u.) HVDC power system time constant (s) Simulation time (s) PR1/PR2 Number of population size Number of generation Step size Crossover probability Control signal to the ith area Area control error of area i Incremental change in frequency of area i (Hz) Incremental step load change of area i Incremental change in tie-line power between areas 1 and 2 (p.u.) Deviation in thermal turbine output Deviation in intermediate state of reheat turbine Deviation in steam turbine governor output Deviation in hydro turbine output Deviation in output of mechanical hydraulic governor of hydroturbine Deviation intermediate state of hydro turbine governor Deviation in gas turbine output Deviation in intermediate state of fuel system combustor of gas turbine Deviation in intermediate state of fuel system and combustor of gas turbine Deviation in valve positioner of gas turbine Deviation in Intermediate state of speed governor of gas turbine Deviation in total power output of ith area

1 0.2 70 1 100 50 0.8 0.8 – – – – – – – – – –

ΔPRHi ΔPGgi ΔPRgi ΔPFci ΔPVPi ΔXgi ΔPGi

– – – – – – –

References [1] O.I. Elgerd, C.E. Fosha, Optimum megawatt-frequency control of multiarea electric energy systems, IEEE Trans. Power Appar. Syst. 4 (1970) 556–563. [2] H. Bevrani, Robust Power System Frequency Control, Springer, 2014. [3] O.I. Elgerd, Electric Energy Systems Theory: An Introduction, US DOE, OSTI.GOV, https://www.osti.gov/biblio/5599996%7D,%20journal, 1982, 5599996. [4] A. Ashok, S. Sridhar, A.D. McKinnon, P. Wang, M. Govindarasu, Testbed-based performance evaluation of attack resilient control for agc, in: 2016 Resilience Week (RWS), IEEE, 2016, pp. 125–129. [5] N. Jaleeli, L.S. VanSlyck, D.N. Ewart, L.H. Fink, A.G. Hoffmann, Understanding automatic generation control, IEEE Trans. Power Syst. 7 (3) (1992) 1106–1122. [6] P. Kumar, D.P. Kothari, Recent philosophies of automatic generation control strategies in power systems, IEEE Trans. Power Syst. 20 (1) (2005) 346–357.

16

Power System Frequency Control

[7] H. Bevrani, T. Hiyama, Intelligent Automatic Generation Control, CRC Press, 2016. [8] H. Bevrani, Real power compensation and frequency control, in: Robust Power System Frequency Control, Springer, 2009, pp. 1–23. [9] H. Bevrani, G. Ledwich, Z.Y. Dong, J.J. Ford, Regional frequency response analysis under normal and emergency conditions, Electr. Power Syst. Res. 79 (5) (2009) 837–845. [10] D.K. Mishra, T.K. Panigrahi, P.K. Ray, A. Mohanty, Performance enhancement of AGC under open market scenario using TDOFPID and IPFC controller, J. Intell. Fuzzy Syst. 35 (5) (2018) 4933–4943. [11] H. Bevrani, Y. Mitani, K. Tsuji, Robust decentralized AGC in a restructured power system, Energy Convers. Manag. 45 (15–16) (2004) 2297–2312. [12] K. Lim, Y. Wang, R. Zhou, Robust decentralised load–frequency control of multi-area power systems, IEE Proc. Gener. Transm. Distrib. 143 (5) (1996) 377–386.

Controller design for load frequency control: Shortcomings and benefits

2

Dillip Kumar Mishraa, Li Lia, Md. Jahangir Hossaina, and Jiangfeng Zhangb a School of Electrical & Data Engineering, University of Technology Sydney, Ultimo, NSW, Australia, bDepartment of Automotive Engineering, CLEMSON University, Clemson, SC, United States

2.1

Introduction

The controller is the key element of a control system to compare the controlled value to the desired value. From the comparison, if any deviation exists, the controller adjusts it. It is basically a device that generates the control signals to alleviate the deviation of the real value from the required value to either zero or an inconsiderable level. The controller ensures the plant’s efficiency and smooth run. Indeed, the frequency is a sensitive parameter of the power system, which needs to be maintained within rigid limits. Thus to control the frequency, AGC is considered an essential part of the power system, where the controller plays an important role. This chapter explains various control techniques with mathematical modeling and block diagram illustrations. Further, error functions are discussed, which have been used as an objective function in AGC.

2.2

Traditional control design

The transfer function-based model of the two-area AGC by means of the state-space equation is portrayed in Fig. 1 [1]. In general, the state-space equation is given by Eq. (1) x_ ¼ Ax + Bu + τd

(1)

where A is a state matrix of order n  n, B is a control matrix for n number of state variables, and m number of inputs of order n  m, and τ is expressed as a disturbance matrix. Diverse variables have been demarcated as State variable: x1 ¼ ΔF1 , x2 ¼ ΔPZt1 , x3 ¼ ΔPg1 , x4 Z¼ ΔF2 , x5 ¼ ΔPt2 , x6 ¼ ΔPg2 , x7 ¼ ΔPtie12 x8 ¼

ACE1 dt, x9 ¼

ACE2 dt:

Power System Frequency Control. https://doi.org/10.1016/B978-0-443-18426-0.00007-8 Copyright © 2023 Elsevier Inc. All rights reserved.

18

Power System Frequency Control

Fig. 1 Two-area thermal AGC.

Control inputs: U1 ¼ Δ Pc1, U2 ¼ Δ Pc2; U ¼ [U1 U2]T Load disturbances input: PL1 ¼ Δ PL1, PL2 ¼ Δ PL2; PL ¼ [PL1 PL2]T Using the state-space equation, input matrix A is of the order of 9  9, the control matrix B is of order 9  2, and the disturbances matrix τ is of an order of 9  2. The investigation plant can be described with the following state equation: x_ ¼ Ax + Bu

(2)

Y ¼ Cx + Du

(3)

Output equation is given by

Nonetheless, for the feedback control system, the matrix D presumed to zero. Therefore, the output equation becomes Y ¼ Cx

(4)

where C is called an output matrix of order 2  9. The values of the matrix can be intended through the assistance of a referral paper [2], and the matrix with the corresponding value is addressed in the appendix section. By adopting these control techniques, the control inputs are being feedbacked to the system and have a linear combination of all nine states.

Controller design for load frequency control

19

The nine-state that being feedback is expressed as x1, x2, x3, ……x9, and the control inputs U1 and U2 can be represented as U 1 ¼ k11 x1 ,k12 x2 ,k13 x3 ,k14 x4 ,k15 x5 ,k16 x6 ,k17 x7 ,k18 x8 ,k19 x9

(5)

U 1 ¼ k21 x1 ,k22 x2 ,k23 x3 ,k24 x4 ,k25 x5 ,k26 x6 ,k27 x7 ,k28 x8 ,k29 x9

(6)

Lastly, the input of the control equation can be transcribed as U ¼ kx

(7)

where k is called as FeedBack Gain matrix of 2  9 order, which is given by  k¼

k1,1 k1,2 k1,3 k1,4 k1,5 k1,6 k1,7 k1,8 k1,9 k2,1 k2,2 k2,3 k2,4 k2,5 k2,6 k2,7 k2,8 k2,9

 (8)

The value of feedback gain matrix k should be found out from the designing the control law in the definition of optimal control problem and at which the performance index will be decreased, whereas the switches from initial to an infinite state which is equal to x(0) 6¼ 0 and x(∞) ¼ 0, respectively. Mostly, the performance index PI is in quadratic nature and taken as [2]: PI ¼

1 2

Z

∞

 xT Qx + uT Ru dt

(9)

0

In Eq. (9), Q and R are represented as a state-weighing matrix, control-weighing matrix, and which are positive definite, symmetric, and real in character and designated as identify matrix opposite dimension in power system model under examination.  2 β  1  0  0   0  Q ¼  0  0  β  1  0  0

0

0

0

0

0

β1

0

0

0

0

0

0

0

0

0 0

0 0

0 β22

0 0

0 0

0 β2

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

β2

0

0

2

0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

R¼

1 0

0 1

 0   0  0   0  0   0  0   0  1

(10)

(11)

The eccentricities of ACE around the steady-state errors are minimized. Thus, ACE can be written as Eqs. (12) and (13) for area-1 and area-2, respectively.

20

Power System Frequency Control

ACE1 ¼ B1 ΔF1 + ΔPTie ¼ B1 x1 + x7

(12)

ACE2 ¼ B2 ΔF2 + α12 ΔPTie ¼ B2 x4  x7

(13)

Ð The deviations of ACE dt about the steady-state errors are lessened. Here in our system, these deviations are x8 and x9 also, the control inputs (U1 and U2) deviations in which the steady-state errors are lessened. So, optimal control law can be found out as A, B, R, and Q matrices are calculated priory and assumed as U ¼  kx where K is the feedback gain matrix, and it is given as k ¼ R1BTS In this case, S is a matrix with real, symmetric, and positive definite in nature that can be attained by decomposing of matrix Riccati equation and is presented by AT S + SA  SBR1 BT s + Q ¼ 0

(14)

Hence with feedback, the overall closed-loop equation becomes: x_ ¼ Ax + BðkxÞ ¼ ðA  BkÞx ¼ Ac x

(15)

where Ac is called a closed-loop system matrix which is nothing but (A  Bk). The Eigenvalue of the matrix Ac demonstrates the system stability with a feedback controller. It is reported that, if the magnitude of real increases negatively, then the system becomes more stable. Eigen Value of A ¼ 0.0000 + 0.0000i 0.0000 + 0.0000i 13.052 + 0.0000i 12.9566 + 0.0000i 0.5480 + 3.1428i

0.5480  3.1428i 0.6414 + 2.3337i 0.6414  2.3337i 1.6052 + 0.0000i

From the above observation, it is clear that the implementation of an optimal control strategy that two values out of all are zero suggests the system is marginally stable. The Eigenvalues of the closed-loop system matrix Ac are: 13.0594 + 0.0000i 13.0758 + 0.0000i 1.034 + 3.4078i 1.034-3.4078i

1.4791 + 2.5810i 1.4791-2.581i 1.3521; 0.7439 0.6887 + 0.000i

Here all the Eigenvalues of Ac are having negative real parts, which signifies the system is a stronger notion of convergence after applying the optimal control strategy.

Controller design for load frequency control

21

2.2.1 Control design with reheat In the second attempt, reheat is added into the two-area thermal power system as depicted in Fig. 2, which is explained as follows. State variable: x1 ¼ ΔF1 , x2 ¼ ΔPt1 , x3 ¼ ΔPr1 x4 ¼ ΔPg1Z, x5 ¼ ΔF2 , x6 ¼ ΔP Z t2 , x7 ¼ ΔPr2 , x8 ¼ ΔPg2 , x9 ¼ ΔPtie12 , x10 ¼ ACE1 dt, x11 ¼ ACE2 dt Control inputs: U1 ¼ Δ Pc1, U2 ¼ Δ Pc2; U ¼ [U1 U2]T Load disturbances input: PL1 ¼ Δ PL1, PL2 ¼ Δ PL2; PL ¼ [PL1 PL2]T With the help of state variables, the input matrix A is the order of 11  11, the control matrix B is of an order of 11  2, and the disturbances matrix τ is of an order of 11  2. The open-loop system matrix A has the Eigenvalues and is given by 0.0000 + 0.0000i 0.0000 + 0.0000i 12.6985 + 0.0000i 12.6923 + 0.0000i 0.1716 + 2.5868i

0.1716 2.5868i 2.0178 + 0.000I 1.0223 + 0.7052i 1.0223 0.7052i 0.0968 0.4068

It is observed just before implementing the optimal control strategy that two values out of all are zero suggesting the system is marginally stable.

Fig. 2 Two-area thermal reheat AGC.

22

Power System Frequency Control

The Eigenvalues of the closed-loop system matrix Ac: 12.6995 + 0.0000i 12.6933 + 0.0000i 0.4538 + 2.6334i 0.4538  2.6334i 2.0199 + 0.0000i 1.1766 + 1.0781i

1.1766 1.0781i 0.839 + 0.0000i 0.2703 + 0.1338i 0.2703 0.1338i 0.2937 + 0.0000i

Here all the Eigenvalues of Ac are having negative real parts, which signifies the system is a stronger notion of convergence after applying the optimal control strategy.

2.2.2 Extension of two area with reheat and HVDC-link The transfer function model of the proposed plant is represented by state-space presentation, as shown in Fig. 3 and described in Eq. (16) as x_ ¼ Ax + Bu + τd

(16)

Concerning the Eq. (16), A is represented as a state matrix (n  n), B denotes aa a control matrix for n numeral of state variables and m numeral of inputs (n  m), and τ is expressed as disturbance matrix. Diverse parameters have been determined as State variables: x1 ¼ ΔF1 , x2 ¼ ΔPtie12 , x3 ¼ ΔF2 , x4 ¼ ΔPFGT1 , x5 ¼ ΔPRT1 , x6 ¼ ΔXT1 , x7 ¼ ΔPGH1 , x8 ¼ ΔXH1 , x9 ¼ XRH1 , x10 ¼ ΔPGH1 , x11 ¼ ΔPFC1 , x12 ¼ ΔPVP1 x13 ¼ ΔXG1 , x14 ¼ ΔPGT2 , x15 ¼ ΔPRT2 , x16 ¼ ΔXT2 , x17 ¼ ΔPGH2 , x18 ¼ XH2 , x19 ¼ XRH2 , x20 ¼ ΔPGH2 , x21 ¼ ΔPFC2 , x22 ¼ ΔPVP2 Z x23 ¼ XG2 , x24 ¼

Z ACE1 dt, x25 ¼

ACE2 dt, x26 ¼ ΔPtieDC

Control inputs: U1 ¼Δ Pc1, U2 ¼ Δ Pc2; U ¼ [U1 U2]T Load disturbances input: PL1 ¼ Δ PL1, PL2 ¼ Δ PL2,; PL1 ¼ PL [PL1 PL2]T The state-space equation is expressed in Eqs. (17)–(41), and its transfer function block diagram is presented in Fig. 3 is given below. x_ 1 ¼ 

kp1 kp1 kp1 kp1 kp1 1 x  x + x + x + x  ΔPL1 T p1 1 T p1 2 T p1 4 T p1 7 T p1 10 T p1 x_ 2 ¼ 2πT 12 x2  2πT 12 x3

(17) (18)

Controller design for load frequency control

23

Fig. 3 State-space model of multiarea-multisource power system [3].

α12 T p2 kp2 kp2 kp2 kp1 1 x  x + x + x + x + x T p2 2 T p2 3 T p2 14 T p2 17 T p2 20 T p1 26 kp2  ΔPL1 T p2

x_ 3 ¼ 

1 x + x_ 4 ¼  T r1 4



 kt1 kr1 kt1 k k  x5 + r1 t1 x6 T r1 T t1 T t1

(20)

1 1 x + x T r1 5 T t1 6

(21)

1 1 1 x + x + U R1 T g1 1 T t1 6 T g1 1

(22)

x_5 ¼  x_ 6 ¼ 

(19)

24

Power System Frequency Control

  2kh1 T R1 2 2kh1 2kh1 x  x7 +  x_ 7 ¼  x T GH1 R1 T RH1 4 T w1 T w1 T GH1 8   2kh1 T R1 2k 2kh1 T R1 +  h1 x  U T GH1 T RH1 T GH1 9 T GH1 T RH1 1 T R1 1 x  x + T GH1 R1 T RH1 1 T GH1 8 T R1 + U T GH1 T RH1 1

x_ 8 ¼

x_9 ¼ 



1

T GH1



 T R1 x T GH1 T RH1 9 (24)

1 1 1 x  x + U R1 T RH1 1 T RH1 9 T RH1 1

kg1 kg1 T CR1 1 x + x  x T CD1 10 T CD1 11 T F1 T CD1 12   1 1 T ¼ x11 + + CR12 x12 T F1 T F1 T F2

(25)

x_ 10 ¼ 

(26)

x_11

(27)

X1 c 1 X1 x  1x + x + U b1 R1 Y 1 1 b1 12 b1 13 b1 Y 1 1     X1 1 1 1 X1 ¼   x +  x U1 1 Y 1 13 Y1 Y12 R1 Y 1 2 R 1 Y 1 x_ 12 ¼ 

x_13

(23)

x_ 14 ¼ 

1 x + T r2 14

   kt2 kt2 kR2 k k  x15 + t2 R2 x16 T R2 T t2 T t2

(29)



(30)

1 1 x +  x T t2 15 T t2 16

(31)

1 1 1 x  x + U R2 T g2 3 T g2 16 T g2 2

(32)

x_ 15 ¼  x_16 ¼ 

(28)

  2kh2 T R2 2 2kh2 2kh2 x3  x17 +  x T w2 T GH2 R2 T RH2 T w2 T GH2 18   2kh2 T R2 2k 2kh2 T R2 +  h2 x  U T GH2 T RH2 T GH2 19 T GH2 T RH2 2

x_ 17 ¼ 

x_ 18 ¼

T R2 1 x  x + T GH2 R2 T RH2 1 T GH2 18 T R2 + U T GH2 T RH2 2 x_ 19 ¼ 



1

T GH2



(33)

 T R2 x T GH2 T RH2 19

1 1 1 x  x + U R2 T RH2 3 T RH2 19 T RH2 2

(34) (35)

Controller design for load frequency control

kg2 kg2 T CR2 1 x + x  x T CD2 20 T CD2 21 T F2 T CD2 22   1 1 T ¼ x21 + + CR22 x22 T F2 T F2 T F2

x_20 ¼ 

(36)

x_21

(37)

X2 c 1 X2 x  2x + x + U b2 R2 Y 2 3 b2 22 b2 23 b2 Y 2 2     X2 1 1 1 X2 ¼   x3  x23 + U2 Y2 Y2 Y22 R2 Y 2 2 R2 Y 2 x_ 22 ¼ 

x_23

25

(38)

(39)

x_ 24 ¼ β1 x1 + 1⁎ x2 + 1 ⁎x26

(40)

x_25 ¼ α12 x2 + β2 ⁎ x2 + α12 ⁎ x26

(41)

kdc k 1 x  dc x  x T dc 1 T dc2 3 T dc 26

(42)

x_26 ¼

By implementing the above Eqs. (17)–(42), it is determined that the input matrix A is of an order of 26  26, the control matrix B (26  2), and the disturbances matrix τ (26  2). The general state, output equation is mentioned earlier, Eqs. (3)–(5). The values of the matrix can be intended through the assistance of the referral thesis [4], and the matrix with corresponding value is addressed in Appendix. By adopting this control technique, the control inputs are being feedbacked to the system and have a linear combination of all 26 states. The twenty six state that actuality feedback is expressed as x1, x2, …x26, and the control inputs (U1 andU2) can be represented as U 1 ¼ k11 x1 þ k12 x2 þ k13 x3 þ k14 x4 þ k15 x5 þ k16 x6 þ k17 x7 þ k18 x8 þ k19 x9 þ k1,10 x10 þ k1,11 x11 þ k1,12 x12 þ k1,13 x13 þ k1,14 x14 þ k1,15 x15 þ k1,16 x16 þ k1,17 x17 þ k1,18 x18 þ k1,19 x19 þ k1,20 x20 þ k1,21 x21 þ k1,22 x22 þ k1,23 x23 þ k1,24 x24 þ k1,25 x25 þ k1,26 x26 U 2 ¼ k21 x1 þ k22 x2 þ k23 x3 þ k24 x4 þ k25 x5 þ k26 x6 þ k27 x7 þ k28 x8 þ k29 x9 þ k2,10 x10 þ k2,11 x11 þ k2,12 x12 þ k2,13 x13 þ k2,14 x14 þ k2,15 x15 þ k2,16 x16 þ k2,17 x17 þ k2,18 x18 þ k2,19 x19 þ k2,20 x20 þ k2,21 x21 þ k2,22 x22 þ k2,23 x23 þ k2,24 x24 þ k2,25 x25 þ k2,26 x26

The control input equation can be presented as Eq. (4) The feedback gain matrix K is expressed as  K¼

k1,1

k1,2

k1,3

k1,4

k1,5

k1,6

:

: : :

: :

: : :

: :

k1,18

k1,19

k1,20

k1,21

k1,22

k1,23

k1,24

k1,25

k1,26

k2,1

k2,2

k2,3

k2,4

k2,5

k2,6

:

: : :

: :

: : :

: :

k2,18

k2,19

k2,20

k2,21

k2,22

k2,23

k2,24

k2,25

k2,26



The value of feedback Gain matrix K should be found out from the designing the control law in the definition of optimal control problem and calculated performance indices will be lessened, whereas the plant changes from an initial state to infinite-state as mentioned earlier. Mostly, the performance index PI is in quadratic nature and taken as PI ¼

1 2

Z

∞

 xT Qx + uT Ru dt

(43)

0

In this case, the ACE can be written as ACE1 ¼ B1 ΔF1 + ΔPTie ¼ B1 x1 + x2

(44)

ACE2 ¼ B2 ΔF2 + α12 ΔPTie ¼ B2 x3  x2

(45)

The deviations of ACE about the steady-state errors are lessened. Here in our system, these deviations are x24 and x25 also, the control inputs (U1 and U2) deviations about the steady-state values are lessened. So, the optimal control law can be found as A, B, R, and Q matrices are calculated priory and represented in Eq. (7), and K is in Eq. (9). The Riccati equation and closed-loop equation are presented in Eqs. (14) and (15). From the closed-loop matrix, the Eigenvalue signifies the stability of the system by means of showing the magnitude of the real part of the Eigenvalue. If the values are more negative, then the system is called greater stability. The matrix A,B, and τ and their values are shown in Appendix 2.

Controller design for load frequency control

2.3

27

Shortcomings of the traditional controller

Over the past few decades, many researchers have described the major drawbacks of the traditional controller, which are summarized, is as follows. l

l

l

Sluggish operation. Due to the nonlinear properties of various system components, the traditional controller, such as an integral controller, fails to stabilize. For example, the nonlinearity function such as reheat turbine, governor dead band (GDB), and generation rate constraints (GRC) in thermal plants are really hard to withstand with an integral controller. Through the daily cycle, the load changes continuously, and accordingly, it adjusts the operating points, which is termed an intrinsic characteristic of a power network. To achieve better performances, the integral gain of the controller needs to be tuned repetitively to meet the equilibrium point. Further, the integral gain would also be necessary to provide a better transient response with less settling time during the dynamic period. However, it is practically hard; hence, the traditional controller is referred to as a fixed type. It can be only suitable for a low-scale system, where the transient response is infrequent.

Thus, the implementation of the control technique must be appropriate in response to the transient operation of the power network. Therefore, this control action contributes significantly to research on power system dynamics by applying the different controllers in multiarea power networks.

2.4

The need for an advanced control method

The application of the advanced control method offers significant support to AGC studies. Recently, the power system networks are increasing their complexity and interdependencies. Besides, power engineers are focusing on breakthrough technologies for socio-economic benefits and stable, reliable operation. Hence, advanced control methods are the only solution that can deliver high adaptability performance in response to the varying scenarios. Furthermore, a large deviation causes the system to be unreliable, and the transient response behaves more oscillatory and takes more time to settle. Therefore, the most appropriate controller should be employed according to the scenario. The key to advanced control methods is safety, reliability, stability, and quick recoverability. To take these factors come into action means, setpoints must be varied according to the requirement for safe operation. Subsequently, the system delivers continuity of power supply without any interruption, i.e., reliability, and the system offers stable performance, which is called stability. Further, fast restoration must occur to balance the system frequency and power, called quick recoverability.

2.5

Controller

The controller plays a significant role in a power system to overcome the oscillation problems and swiftly restore the frequency deviation. There are various types of controllers, and each controller has its unique characteristics, such as proportional gain

28

Power System Frequency Control

holds the capability to reduce the upswing time, but it cannot eliminate the steadystate error (SSE). Similarly, an integral gain possesses the ability to eliminate the SSE, but it may not change the oscillations. Likewise, the derivative gain is able to enhance the system’s stability, drop the overshoot, and refine the transient response. All these controllers have a single input and a single output signal. The input signals have the error, and the output signal has a controlled signal after the process of controlling parameters with the reference signal.

2.5.1 PI controller PI controller is the combination of two control action principles, i.e., proportional (P) and integral (I). P deals with the steady-state error (SSE) through a proportional measure of response, which can be minimized. On the other hand, I deal with errors and time adjusting. Therefore, the PI controller can take challenges to minimize error and settling time. It is a simple controller and has less cost. The block diagram of the PI controller is presented in Fig. 4 [5]. The mathematical expression of the PID controller is given by Eq. (46) GPI ¼ K P +

KI s

(46)

where KP and KI are the gains of the controller. However, it suffers some drawbacks, which cannot be concerned with transient oscillation. Therefore, to address this issue, power engineers have been developed other types of controller such as proportional, integral, and derivative (PID), fractional order PID (FOPID), tilt integral derivative (TID), integral double derivative (IDD), and with filter coefficient (NC).

2.5.2 PID controller In the modern area of control theory applications, it achieved a major breakthrough over the past few decades. Many control theories and practices have been established to a large extent as optimal, predictive, fuzzy, and neural control. Notwithstanding,

KP Ref. + _

Error

Feedback

Fig. 4 PI controller.

+

∑ KI s

+

Control output

Controller design for load frequency control

29

due to the valid application of the PID controller, it has have been achieved a milestone of industrial applications over the century. More importantly, the PID controller offers the superiority of robust operation to fulfill the requirements of the industrial process, simple design principles, and implementations [6]. Controllers that do not have a derivative term (i.e., PI) show a delayed response, large disturbances, large transportation delay, and noises in the system. However, in the PID controller, proportional (KP), integral (KI), derivative (KD) gains as design specifications that provide superior flexibility in controller design. In this controller, derivative action has a central role, which can significantly enhance the characteristics during the transient operation. The block diagram of the PID controller is presented in Fig. 5. The mathematical expression of the PID controller is given by Eq. (47) GPID ¼ K P +

KI + KDs s

(47)

where KP, KI, and KD are the gains of the controller.

2.5.3 IDDF controller As the name suggests, the integral double derivative filter (IDDF) controller is the combination of integral and double derivative gain with a filter coefficient. It has three gain blocks, namely KI, KDD, and NC. The importance of double derivative control can reduce the oscillation quickly as compared to the single one. Besides, the filter coefficient can further improve the system response. It is important to note that this type of control can be used when the system undergoes a large deviation [7,8]. The mathematical expression of the IDDF controller is given by Eq. (48), and the block diagram is shown in Fig. 6. GIDD ¼

KI + K DD s2 s



NC s + NC

 (48)

where KI, KD, and NC are the gains of the controller KP

Ref.

+ _

+ Error

Feedback

Fig. 5 PID controller.

KI s

Control output

∑ +

KDs

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Power System Frequency Control

Fig. 6 IDDF controller.

2.5.4 TIDF controller TIDF stands for a tilt integral derivative filter, which is the extension of the PID controller with the filter coefficient. It provides an enhanced loop compensator and possesses the advantages of the PID controller by the provision of proportional gain with KP tilt component, i.e., s1=n [9]. This controller is herein stated as a TID controller. Where n is a nonzero real number and the range. Moreover, it provides a simple tuning with better rejection ratio, lesser impacts on parameter variation as compared to PID [10]. The mathematical expression of the PID controller is given by Eq. (49), and the block diagram is shown in Fig. 7. GTIDF ¼

KP s1=n

+

  KI NC + KDs s s + NC

where KP, KI, KD, and NC are the gains of the controller

Fig. 7 TIDF controller.

(49)

Controller design for load frequency control

31

2.5.5 FOPID controller Recent advancements in the field of control theory have led to a renewed interest in transient performance enhancement through the PID controller with the aid of fractional calculus. It is important to note that the orders of derivative and integral are noninteger types [11]. This distinction is further exemplified in fractional calculus using a mathematical procedure, and it is a nonlocal calculus. It offers an exceptional mechanism for the characterization of memory and the inherited properties of several resources and processes. Besides, these dynamics are taken into consideration; it turns out the key benefit of a fractional derivative as compared to classical controllers. Furthermore, the PID controller has been reformed using the concept of fractional calculus, and with that, two control actions such as derivative and integral. The formation of derivative and integral gain with two more degrees of the tunable parameter gives the greatest extent of flexibility and, therefore, further enhances the performance of a classical PID controller. The FOPID controller is the same as the PID controller except for the two more tunable parameters. Moreover, due to the tunable parameters, it is otherwise called two-degree-of-freedom parameters, i.e., μ and λ, the FOPID controller is referred to as a two-degree-of-freedom PID (2-DOF-PID) controller, is shown in Fig. 8 [12]. It is interesting to note that the FOPID controller can act as a PI, PD, or PID controller with the variation of the value of μ and λ from 0 to 1, and it is defined via μ versus λ diagram shown in Fig. 9. The mathematical expression of the PID controller is given by Eq. (50), and the block diagram is shown in Fig. 8. GFOPID ¼ K P +

KI + K D sμ sλ

(50)

where KP, KI, and KD are the gains of the controller, and λ, and μ are the tunable parameters. In Fig. 9, the value of μ and λ establishes the PD, PI, PID, and FOPID controller. Table 1 signifies that different control actions can be made with a FOPID controller, which can be used in different industries according to the requirements. Therefore, this controller has been referred to as one of the best controllers of its performances and measures.

Fig. 8 FOPID controller.

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Power System Frequency Control

Fig. 9 μ versus λ diagram of FOPID. Table 1 Value of controller gains in terms of control actions.

2.6

Controller

μ

λ

Transfer function

PD PI PID FOPID

0 1 1 0