Phasors for Measurement and Control (Power Systems) 3030670392, 9783030670399

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Power Systems

Gerard Ledwich Arash Vahidnia

Phasors for Measurement and Control

Power Systems

Electrical power has been the technological foundation of industrial societies for many years. Although the systems designed to provide and apply electrical energy have reached a high degree of maturity, unforeseen problems are constantly encountered, necessitating the design of more efficient and reliable systems based on novel technologies. The book series Power Systems is aimed at providing detailed, accurate and sound technical information about these new developments in electrical power engineering. It includes topics on power generation, storage and transmission as well as electrical machines. The monographs and advanced textbooks in this series address researchers, lecturers, industrial engineers and senior students in electrical engineering. **Power Systems is indexed in Scopus**

More information about this series at http://www.springer.com/series/4622

Gerard Ledwich Arash Vahidnia •

Phasors for Measurement and Control

123

Gerard Ledwich School of Electrical Engineering and Robotics Queensland University of Technology Brisbane, QLD, Australia

Arash Vahidnia School of Engineering RMIT University Melbourne, VIC, Australia

ISSN 1612-1287 ISSN 1860-4676 (electronic) Power Systems ISBN 978-3-030-67039-9 ISBN 978-3-030-67040-5 (eBook) https://doi.org/10.1007/978-3-030-67040-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The book examines what inferences about the operation of a power system can be made from Phasor Measurements and what control actions can be taken. The work on Phasor Measurement Units (PMUs) by the authors started in 1995 with the development of PMUs in Queensland was based on the use of GPS by Arun Phadke in the area of power system protection. This work showed that high-accuracy measurements of power system angles were possible and that system dynamic modal frequencies and damping could be extracted. This initial work was extended to a National Electricity Market measurement system for Australia in collaboration with transmission utilities. Further refinement of the algorithms showed how useful angle measurements could be. Other researchers in the USA also developed algorithms with similar performance for long signal analysis but there was some bias for inference of angle over short time segments. One of the aspects that emerged from the high accuracy of angle measurements was the ability to extract dynamic models of the major loads. For Sydney, composite load as well as some industry loads, the result was the portion of load which consisted of induction motors through the day. This work required the development of tools for the identification under closed loop. The desire arose for the use of this new PMU tool to develop better controls of power system dynamics. Most of the control designs for power system are based on linear processing of local voltage and frequency to infer corrections to excitation system or Static Var Compensator (SVC) controls. Because SVCs are able to make a near-instantaneous change in power flow, we were able to compute the control which would make a strong reduction of the total kinetic energy of machines following a disturbance. This proved a robust design for SVC controls with some refinement for the first swing which redirected the focus to machines at most risk of separation. For an application with low numbers of measurements, groups of generators are needed to be aggregated to give an accurate model of the lower frequency interarea modes of a power system. For the Australian system, this largely resulted in one aggregate machine in each state with some justification of two aggregates in Queensland. Because of the history of development of the power systems separately in each state, there were generally stronger connections within a v

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state and weaker between states. Nonlinear controls based on this were found to give good damping and attenuation for the first swing and for interarea modes that were largely independent on the operating point. Nonlinear Kalman filters were able to combine PMU measurements from several points and robust control was demonstrated. The extrapolation to control through excitation needed a new concept of inverse filtering. The system's Kalman filter with the kinetic energy derivative was able to infer power flow changes that would be needed for interarea modes over the next second. Inverse filtering can then be applied to infer the input to the excitation of a machine to give the required power flow changes. This work was extended to the contribution from wind farms and HVDC. This concept was also adapted in SVC controls where the dynamics of the load were identified through PMUs and was used for adaptive wide-area control of SVCs to compensate for the load dynamics. Further developments were in the analysis of travelling waves at 400–1000 km/s. The implications were that some disturbances could be better anticipated. Much of the work was on the transmission system, but there were impacts of the distribution system becoming clearer. The estimation of voltages and angles down the feeders with very few measurements was desirable but needed engineering knowledge to create pseudo-measurements to help refine the task. This has been done in several places but the contribution discussed here is the representation of voltages and currents directly in the Kalman filter equations for load flow. The transmission’s approach had been to develop 2000 measurements to infer states at 1000 buses at one time instant. The Kalman approach uses pseudo-measurement and a time series to infer a rolling estimate of states. The PMU measurements can be used directly in the Complex Kalman distribution state estimator. Scalar measurements are much more common and the recursive nature of the filtering permitted angle measurements to be recursively refined. The PMU measurements are becoming more standard and the applications in both transmission and distribution are growing. The high accuracy of PMUs opens the door to many techniques including inferring Thevenin Voltage to refine the operation of solar inverters. The issue of weak grids for connection of inverters is current and the tools of the PMU yield insights into this area as well. The authors are grateful for the hard work and inspiration of many former research students and colleagues who have helped progress the work. Brisbane, Australia Melbourne, Australia

Gerard Ledwich Arash Vahidnia

Acknowledgements

The material in this book has emerged from 25 years of work with some very good power engineering researchers. I would like to acknowledge the contributions from my colleagues: Ed Palmer for early work on the kinetic energy control design and on corrections for time delays Tekwai Chan for understandings of first swing control on critical links Arash Vahidnia for work on aggregation and on control with unclear separations of areas Riza Goldoost-Soloot for developing the tools to work when generator controls had delays by using inverse filters Keqian Hua for extending inverse filtering to dynamic load control with SVCs Tianya Li for taking the ideas of travelling waves in power systems to better control ideas Tanya Parveen for identification of power system loads Carlos Moyano for load modelling and extension to industry loads Mehdi Shafiei and Alan Lewis for work on distribution estimation. Thanks is also expressed to the industry co-workers who assisted in the testing of these systems and researchers who developed the hardware. In addition, I, Gerard, acknowledge the support from my wife Cathy Ledwich and the inspiration from the Holy Spirit.

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Contents

1

Phasor Measurements for Identification and Control 1.1 How a Phasor Measurement Unit Works . . . . . . 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Identification of Power System Dynamics . . . . . . . . . . . . 3.1 Identification of Dynamics . . . . . . . . . . . . . . . . . . . . 3.1.1 Identification with White Input . . . . . . . . . . 3.1.2 Power System Dynamics-Forming Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 State Estimation . . . . . . . . . . . . . . . . . . . . . 3.1.4 Forming Areas for Aggregation . . . . . . . . . . 3.2 Aggregation of Power System Areas . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Aggregation Process Based on Generator Coherency . . . . . . . . . . . . . . . . . . . . . . . . .

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Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Load Model . . . . . . . . . . . . . . . . . . . . . Measurement Approaches . . . . . . . . . . . . . . . . . 2.3.1 Tap Changer . . . . . . . . . . . . . . . . . . . . 2.4 Load Modelling . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Traditional Load Modelling . . . . . . . . . 2.4.2 Inadequacy of the Traditional Model . . . 2.4.3 Closed-Loop Feedback Load Modelling . 2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Motor Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Real Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2.3 Simulation on Test Systems . . . . . . . . . . . . . . . . . . 3.2.4 Summary of Aggregation Process . . . . . . . . . . . . . . 3.3 Aggregation for Machines Without Clear Delineation of Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Mathematics of Aggregation Where Modes Are Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Example of Aggregation with Poorly Separated Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Uniform 2D Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Process of Aggregation . . . . . . . . . . . . . . . . . . . . . . 3.4 HVDC Control Based on Kinetic Energies . . . . . . . . . . . . . . 3.4.1 Control of 2D Aggregated Systems . . . . . . . . . . . . . 3.4.2 Non Uniform Impedance Set for Linear Connection of Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Control Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 DC Link to Island . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Load Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Estimation of Criticality of a Disturbance . . . . . . . . . . . . . . . 3.6.1 Summary of Model Reduction with Imprecise Definition of Areas . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Introduction to Identification from Ambient Noise . . . . . . . . 3.7.1 Identification from White Noise . . . . . . . . . . . . . . . 3.7.2 Frequency Domain Fit to the Spectrum . . . . . . . . . . 3.7.3 Prony or Extended Prony Fit to the Time Domain Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Autocorrelation Analysis . . . . . . . . . . . . . . . . . . . . . 3.7.5 Pole Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . 3.7.6 Effect of Damping Term . . . . . . . . . . . . . . . . . . . . . 3.8 Identifying Forced Oscillations . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Forced Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Direct Control of Power System Dynamics . . . . . . . . . . . . 4.1 Wide Area Control Approach . . . . . . . . . . . . . . . . . . 4.1.1 Application Considerations . . . . . . . . . . . . . . 4.1.2 Controls Acting Behind a Delay . . . . . . . . . . 4.2 First Swing Stability of Critical Links . . . . . . . . . . . . 4.2.1 Control Law Based on Remote Measurements 4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Nonlinear Dynamic State Estimation of the Reduced System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Robustness to System Dynamic Loads . . . . . . 4.3.2 Transfer Capacity Improvement . . . . . . . . . . .

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4.3.3 4.3.4 4.3.5

Robustness to Time Delays in Controls . . . . Time Delay Compensation in Kalman Filter Proposed Non-linear Delay Compensation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Implementation and Testing of Algorithm . . 4.3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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Indirect Control of Power System Dynamics . . . . . . . . . 5.1 The Need of Wide Area Damping Control: Inverse Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Wide Area Damping Controllers . . . . . . . . . . . . . . 5.3 Comparison Between SVCs and Excitation Systems 5.3.1 Excitation System . . . . . . . . . . . . . . . . . . . 5.4 Inverse Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Controller Design Steps . . . . . . . . . . . . . . 5.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Inverters Operating in Power System in Weak Grids . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Simple Model of Ideal Current Source and Quadrature Voltage Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Estimator Based Controllers of Inverters in Weak Grids (Switch Averaged) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Case 1: Model with Unknown 50 Hz Phase and Angle of Terminal Volts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Case 2 Broadcast of Reference Angle . . . . . . . . . . . . . 6.6 Case 3 Projecting to Vs Using Inverter Current . . . . . . 6.7 Offset the Current Reference with Respect to Inferred Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Kalman Filter Based PLL . . . . . . . . . . . . . . . . . . . . . . 6.9 LQR/Kalman Control in Switching Inverter . . . . . . . . . 6.9.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Extreme Case of Step Impedance . . . . . . . . . . . . . . . . . 6.11 Summary of Performance of Inverters . . . . . . . . . . . . . 6.12 Wide Area Control of Power Systems with Inverters and Synchronous Generators . . . . . . . . . . . . . . . . . . . .

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6.13 Hierarchy of Control Efforts . . . . . . . . . . . . . . . . . . . . . 6.14 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Inverters to Add Damping by Reducing Output When Velocity in the Area Is High . . . . . . . . . . . . . . . . . . . . . 6.16 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Summary of Inverter Synchronization . . . . . . . . . . . . . . 6.18 Distributed Versus Lumped Battery Compensators . . . . . 6.18.1 Introduction to Distributed Controllers . . . . . . . . 6.18.2 Battery Controller . . . . . . . . . . . . . . . . . . . . . . . 6.18.3 Case 1 Line of Generators . . . . . . . . . . . . . . . . 6.18.4 Case 2 Interconnected Generators . . . . . . . . . . . 6.18.5 Summary of Lumped Vs Distributed Batteries . . 6.19 Distributed Control by Batteries with a Spread of Delays 6.20 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Travelling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Electromechanical Wave Propagation . . . . . . . . . . . . . . . 7.2 Attenuation of Electromechanical Waves Using SVC . . . 7.3 Frequency Range and Modal Analysis . . . . . . . . . . . . . . 7.3.1 Forward Wave Analysis . . . . . . . . . . . . . . . . . . 7.4 Test Systems and Simulations . . . . . . . . . . . . . . . . . . . . 7.5 Artificial Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 IEEE Benchmark System (Simplified Australian System) 7.7 Summary of Travelling Waves . . . . . . . . . . . . . . . . . . . 7.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Identification of Dynamics of Inverters and Loads in Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Summary of Inverter Identification . . . . . . . . . . . . . . . . . . 8.2 Identification of Load Dynamics Using PMUs and SVCs . 8.3 Load Adaptive Wide-Area Controlled SVCs . . . . . . . . . . . 8.4 Implementation of Adaptive Wide-Area Controller . . . . . . 8.5 Identification of a Set of Thevenin Impedances from Residential PV Inverters . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Example of Identification of Thevenin Impedance in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Applications of Thevenin Identification . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Phasors for Distribution . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . 9.2 Distribution State Estimation . . . . . . . . . 9.3 Forecast Aided Complex State Estimator

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9.3.1 Augmented Complex Kalman Filter . . . . . . . . . . 9.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Weighted Least Squares (WLS) Formulation . . . 9.3.4 Network Layers . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Scalar Measurements . . . . . . . . . . . . . . . . . . . . 9.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Summary of Distribution State Estimation . . . . . . . . . . . 9.5.1 Contribution of Distribution Loads and Batteries to System Stability While Assisting Local Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Voltage Controllers . . . . . . . . . . . . . . . . . . . . . 9.5.3 Centralized Control of Real Power . . . . . . . . . . 9.5.4 Load Modulation Only . . . . . . . . . . . . . . . . . . . 9.5.5 Battery and Load Modulation . . . . . . . . . . . . . . 9.5.6 Using Local Angle Measurement . . . . . . . . . . . 9.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Chapter 1

Phasor Measurements for Identification and Control

Power Systems relying on synchronous generators have been traditionally monitored and controls using measurements of voltage, current and power. One key factor affecting the power flow between generators is the angle between their rotors. This is because the power flow between portions of the network depends on the voltage angle between them. With the development of phasor measurement units (PMU) by Arun Phadke in 1988 [1] this angle term became available as a real time measurement and control term. Initially this measurement of angle was used to improve the protection of transmission lines. The big benefit step for the development of PMU was the use of GPS satellites to provide well synchronized accurate time references. As shown in Fig. 1.1 the references signals give a common reference, such that the phase of voltage signals can be referenced to a common signal. For the early deployment in Australia a simple laptop based unit was included to provide a relocatable unit (Fig. 1.2). Because of the importance of these PMU measurements standards were developed for their implementation [2, 3]. In this book it is shown how the PMU enables. 1. Dynamic load identification 2. Identification of oscillation mode frequencies and damping of oscillations between areas 3. Interpretation of power system transients as travelling waves 4. Modelling large power systems as a smaller group of generators 5. Developing nonlinear control designs for large power systems The initial applications were measurement and post disturbance analysis but the benefits will expand greatly with wide area control as in Fig. 1.3. The reason for the very wide set of applications is firstly that angle dynamics play a critical role in keeping the power system together. The second factor is that the method of forming the phasor estimate from hundreds of measurements in each mains cycle is that this angle can be measured from the bus voltage with an accuracy far exceeding the original voltage measurements. In addition, the process substantially reduces the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_1

1

2

1 Phasor Measurements for Identification and Control

Fig. 1.1 GPS signals giving references for phase measurements Fig. 1.2 Laptop based mobile PMU implementation

Fig. 1.3 Wide area PMU measurements for control

Controller PMU Measurements

Controlled Device

1 Phasor Measurements for Identification and Control

3

noise on the measurements such that angle measurements have a negligible level of high frequency noise interference. The consequence of this very low noise property is that the small variations of angles caused by customer load changes of the order on 1 degree can be measured with high accuracy. With customer load changes continually exciting the oscillations between generators it becomes possible to characterize these oscillations without any explicit disturbance required to be injected. The same is true concerning the dynamics of the loads. The concept is that most dynamics seen in large loads in response to frequency and voltage changes in the power system arise from induction motors. The simple process is to characterize the dynamics of the load response to voltage and frequency as a transfer function [4]. A little more care is needed with large city loads because while the dynamics from the rest of the power system can affect the load power, similarly the voltages and frequency of the power system is affected by loads. The process to perform this identification under closed loop is discussed in Chap. 2. Most identification starts from the idea of an input–output transfer function. If you can measure the system input and the output you can form a transfer function to characterise the dynamics. For the power system problem it is not feasible to measure the changes of all customer loads to form this type of identification. One technique that has arisen is frequency domain decomposition. Here the frequency and damping peaks of the energy in frequency domain can be found and thus form a system model. This process implicitly assumes that there is a white noise process driving the measured response and this process is flat in frequency domain. This implies that any peaks in the output are due to the system. In the case of electricity there is a general rise in the use in the morning and a falling off later at night. But there is little relation between when one customer chooses to switch on a motor to when another’s toaster switches off. In measuring most loads this change in load over one time step is uncorrelated with the change of loads at another time step. There will always be exceptions such as a steel rolling mill, when the load increases at a particular time as the billet enters the rolls there will be a corresponding decrease at a later time directly related to the length of the billet. Similarly for an air conditioner operating between 23 and 25 °C; there will be a definite pattern between turning ON and OFF. However from measurements at bulk supply for Sydney or from smart meters from domestic premises in Brisbane there is essentially zero correlation of the changes of load switching between time steps. This is behind the integral of white noise model noted in [5]. So in the frequency band of interest for generator dynamics (0.1 to 2 Hz) loads can be represented as an integrator driven by white noise. Thus the observed dynamics can be used to identify system oscillation modes once the integrator action is taken into account. Some tools for this identification are considered in Chap. 3. In considering the oscillations between generators, the ones causing most limitation on power system operation are the lower frequency (0.1 to 1 Hz) terms characterizing oscillation between groups of generators. Because synchronous generators are traditionally designed with damper bars which can provide damping of the higher frequency oscillations, the lower frequencies are of the most concern. Designing

4

1 Phasor Measurements for Identification and Control

and implementing a non-linear control system explicitly considering hundreds of generators is a difficult task and if the generators naturally swing together in groups for these low frequency modes it is desirable to represent the system as a much smaller set of grouped generators. There are clear techniques for reducing a linear system to only its low frequency modes but here the process aims seeking to retain the nonlinear dynamics between generators so a model based on generators is being sought. This process of forming a lower order model is called aggregation and is discussed in Chap. 4. The techniques of nonlinear state estimation can be applied to measurements from the real system to infer the angle and velocity of the equivalent generator and this is also raised in Chap. 4. There are limitations to when a single generator reliably represents a group of generators and this is discussed in Chap. 3. When there is a power system representation with a limited number of machines and a state estimation process to enable full state measurements the question arises if a better control system can be developed both to damp small disturbances as well as suppressing transients for large disturbances. The concept that is drawn up on here is a Lyapunov energy model of the power system [6]. Initially, control actions such as a switching a load are considered which have an instant effect on power flows. In this case the control action is sought which gives a good reduction of overall system energy (Lyapunov Energy). The energy is expressed as the sum of Kinetic energy and potential energy. As discussed in Chap. 5, there is no immediate effect of the switched load on the potential energy so the control design consists of finding the control action to give a good reduction of the kinetic energy. It is found that control of Static Var Compensators, Series Compensators and shunt loads under this control gives rise to good damping of the low frequency system modes even when designed on the aggregated generators. When a large disturbance happens, some transmission links can be stressed to the extent they break, modifying the control to only focus on the kinetic energy of the aggregate generators adjacent to the link gives maximum contribution to retaining synchronism. This large and small signal control is seen in greater detail in Chap. 5. The control of power systems can make use of actions which do not have an immediate effect on power flows. The control of the magnetic field of a synchronous machine is called excitation control and can have a substantial influence on power system transients. There is a field winding with a time constant around 1 s between the voltage applied to the field and the resulting flux change. From the state estimate and the energy design it is known what would be the good pattern of flux changes if the flux could vary instantly. The concept of inverse filtering assists here, so knowing what the output is required to be, the input can be designed. The effect of output nonlinearities can also be addressed as discussed in Chap. 4. One of the properties of power systems are that they can behave as a set of masses with nonlinear springs between. The same way the starting of the locomotive can send a wave of motion down the set of carriages there are travelling waves in power systems that can be seen by PMU’s. Chapter 7 describes this and some of the control implications for power systems.

1.1 How a Phasor Measurement Unit Works

5

1.1 How a Phasor Measurement Unit Works The three phase instantaneous measurements Va, Vb and Vc can be transformed to a two phase representation using the Clark transformation [7]. ⎤⎡ ⎤  ⎡ ⎤ 1 −√21 1 − Vα Va 2 √ 2 ⎢ ⎥ ⎣ Vβ ⎦ = ⎣ 0 3 − 3 ⎦⎣ V b ⎦ 3 √1 √21 √1 2 Vo Vc 2 2 2 ⎡

Ignoring the zero sequence component, the three phase measurement can be expressed as complex number. The fundamental component can be found from V f = V α + j Vβ in terms of coefficients 1 C= 2π

2π V f (θ T /2π) · e jθ dθ 0

Over the period of the reference wave T. In the discrete time version of this, the phase of the three phase signal can be expressed as an angle with respect to the reference signal as well as determining the magnitude. If the reference signal is taken as a unity magnitude at 50 Hz then the angle of any voltage to the reference and the angle differences can be found by comparing the angle difference to the common reference. The big breakthrough was in using the GPS system to produce a very precise reference time signal [1] available at all locations in the power system. The big advantage of the Phasor measurement is that a phase locked loop can be used to generate a reference 50 Hz signal sampled at 10 kHz. Over a mains cycle there are then 200 measurements averaged to generate the phasor. Quantization noise and other random errors are thus able to be substantially reduced so that a measurement with 16 bits of accuracy can be effectively accurate to 20 bits. This comes from the averaging process reducing the variable by the square root of the number of averages which is approximately 24 or 4 extra bits of accuracy. In addition the measurement becomes robust against harmonics affecting the measurement. Thus very fine variations of angle can be accurately measured in the PMU. This led to use of the phasors in protection, and in post fault analysis. For wide spread adoption the set of standards for measurement have been developed [8] and to gather the data from many PMUs processes for data concentrators have developed [9, 10]. For the purpose of this book the phasors are treated as having zero measurement noise and quantization noise. When identifying power system dynamic models from ambient load perturbations there are angle variations much less then 1 degree but multiple modes can be extracted with much smaller amplitudes from the measurements.

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1 Phasor Measurements for Identification and Control

1.2 Outline Chapter 1 Introduction The precision of the PMU and the ability to show the critical state of a power system give rise to a new set of capabilities in the areas of model identification, system dynamics, model reduction, nonlinear control. This is extended in this book to the impact of high power inverters on system dynamics and the consequences of seeing the short term power system machine dynamics as having the characteristics of travelling waves. The availability of PMU’s has already made significant changes in power systems but the roll out of control is in its infancy. Chapter 2 Load Modelling The high precision of PMU measurements means that the small variations in power and angle can be used to identify dynamics of the loads in a power system. In much thinking of dynamic loads the research focuses on how the power system voltage and frequency affect the power drawn by the load. The key to successful identification is to recognize that the loads also affect the power system frequencies and voltages. Chapter 3 Identification of Power System Dynamics The basic concept is that customers are continually making switched changes in the load and in the short term these changes are largely uncorrelated with other customers or past loads. Of course there are general trends such that in the morning the load is generally rising but that is over a period of hours and the dynamics are examined on the basis of a few seconds. Chapter 3.1 Electrical loads affecting Power Systems Because most loads are switched on and off independently any change of load is nearly unpredictable in the short term. Thus the short term model of the loads is as the integral of white noise. This model permits identification of wide area system dynamics through this noise exciting the resonances of the machines. One aspect of loads that is not purely random switching is the dynamic response of induction machines to changes of system frequency. The system frequency affect the accelerating torque of all induction motors, at the same time the changes of the motor loads in the system affect the system frequency. This two way interaction needs tools of closed loop identification. Chapter 3.2 Aggregation of Power System Areas Introduction to aggregation through coherence of groups of generators. These groups are modelled as reduced order equivalent machines. The parameters of the dynamics or the reduced order system can be tracked using parameter identification tools. Chapter 3.3 Aggregation of Machines without clear separation When there is a strong of set of generators on a line or a 2D mesh of generators without a clear separation point, then the voltage prop concept needs to be included so that the model reduction reflects the power flow in the system. Voltage Props refers to supporting the voltage at intermediate points between equivalent generators to better match the full system. While a generator supports voltage and injects power,

1.2 Outline

7

a prop has no real power injection and no inertia. The power flow using a single aggregate generator to represent the area can fail to capture the power flow between areas unless the voltage profile forced by all generators is incorporated in the power flow. Chapter 3.4 HVDC Control based on Kinetic Energy The Maximum influence of total Kinetic energy enables a control design between islands linked by HVDC. Chapter 3.5 Load Estimation Shows how to obtain model of dynamic loads. Chapter 3.6 Identification of criticality of a disturbance The control action should change if there is a risk of separation. This section examines how to infer criticality of an event. Chapter 3.7 Introduction to identification from ambient noise The process of forming a model of power system dynamics using the load variations is a simple concept but there are signal processing pitfalls for short time identification Chapter 3.8 Identification of the presence of forced oscillations Normally power system dynamics are considered to be driven by the continual changes in the load. In one instance in Victoria there was a generator stabilizer that was reverse connected after maintenance, and this caused a continuing instability. This section is to show how to distinguish between such a forced oscillation compared to a load driven response. Chapter 4 Direct Control of Power System Dynamics It is difficult to design a nonlinear controller for the relatively fast dynamics of power systems. Model predictive control has proven a powerful tool for nonlinear dynamics but the computation load and the required data make it difficult to apply for the power system case. Chapter 4.1 Energy Based control A process which has shown itself powerful is the use of the Lyapunov Energy idea. Here a positive function is sought whose derivative is always negative. The application here is to use the portion of the energy which is directly affected by the control and to choose the control which makes the most negative change at this instant. The control design is to find this control for devices such as SVC, Series compensators, HVDC, switched shunt loads based on the maximal rate of reduction of Kinetic energy. Chapter 4.2 Corrections for first swing Control Damping of power system oscillations can be based on machine velocities but the first task of the controller is to ensure weak links are not broken during a first swing. The control needs to recognize the priority of responding to critical links and avoiding separation over the task of settling the system disturbances. Now the control action which maximizes synchronizing torque across critical link needs to override the maximum reduction in Kinetic Energy. Chapter 4.3 Estimation the system state and delay correction

8

1 Phasor Measurements for Identification and Control

To enable control acting on different parts of the system the state of the system needs to be known. This concept of a power system represent by equivalent generators means that the control needs to be based on the state of the equivalent system. Because the power flow between generators is based on a sine nonlinearity the state estimation is necessarily nonlinear. One concern in forming control over a wide geographic system is the concern for the effect of delays. In this section the process of compensation for delay is developed through modifications of the nonlinear Kalman filter. When a measurement is delayed benefit can be obtained by projecting the state estimate back to the time of the measurement making the correction based on the measurement then bringing the corrected estimate back to the present time. Chapter 5 Indirect Control of Power System Dynamics The control in Chap. 4 is based on controller which make an immediate and strong change in the power flow. When changing the excitation of a synchronous generator the change in power flow is slow to act. However there are two things that help the solution. First the dynamics within the generator are largely decoupled from the interarea dynamics and second there is a Kalman filter to predict the interarea oscillations. The transfer function between the control action and a parameter which directly affect the power flow is found. The inverse of this transfer function can be found for linear systems and for some nonlinear systems. Unfortunately this process of inverse filtering makes the control action dependant on future measurements and hence noncausal. The Kalman filter however can do a good job of predicting the future steps of the interarea oscillation and hence overcome this non-causal problem. For example the rotor flux of a synchronous machine in a simple classical model can directly affect the power flow. If the machine flux could directly set, the control design of Chap. 4 would be directly applicable as a function of aggregate angles and velocities. The inverse filtering would design the control sequence at the input of the excitation system to achieve the required flux, its dependence on future angles is addressed through the Kalman filter. When there are particular loads with an identifiable response, then these dynamics also need to be corrected for. The initial concept for SVC based control is that by raising the reactive power of the SVC the voltage is raised and thus the power to the load is raised. Strictly speaking this assumes a passive impedance model of the load while a motor component can have a delayed response which needs to the incorporated in the control. Chapter 6 Inverters Operating in a Power System As renewable energy proceeds in prominence and the coal fired power station retire then the dynamics associated with inverters which connect solar farms and wind farms rise in importance. There are no inherent inertia features inherent in inverter interfaced generation. The normal process is for a current controlled inverter to be synchronized to the local system voltage. When the line impedances are small the change of bus angle

1.2 Outline

9

with load is small and the synchronization process is perfectly acceptable. For sets of solar inverters in distribution the angle changes with load typically do not give rise to problems. When several large solar farms are connected along a high impedance line then each aims to injected power in phase with the voltage at its terminals. But the change of angle caused by the synchronization of other inverters can cause interactions between the Phase Locked Loops (PLL). Current practise is to model these inverters in great detail including the Pulse Width Modulation (PWM) switching. For control design a simplified model is required on which to understand these PLL instabilities and thus drive improvements. For transmission lines were have developed levels of modelling based on the required task. For lighting transients a travelling weave model of the line is used, for voltage rise analysis PI model of the line is used, while for machine transients lines are represented as a 50 Hz phasor impedance. For control design of inverters a similar gradation of models is desired. 1. Phasor modelling of the interaction of inverters. 2. Switch averaged Models of inverters. 3. PWM modelling of inverters. Starting from this understanding there can be a gradation of the control process. Initially design PLL with compensation of the line drop for each inverter caused by its own injection. This proceeds to a Kalman Filter based identification of the 50 Hz alpha -beta components of a Thevenin source. The performance of the control based on the state estimate or on a broadcast reference can be validated in a full PWM model. Using an LQR approach gives good confidence that the PWM and filter interactions are well controlled. The use of reference signals from the center of area or from the inferred Thevenin for an area the PLL modes are of significantly higher frequency and are well damped. Thus a similar model reduction process for removal of local machine modes, to focus on interarea dynamics, integrates inverter control to the energy concepts in Chaps. 4 and 5. Interarea control by modulation of inverter references is demonstrated in a PWM model. Chapter 7 Travelling Waves in Power Systems The model of the power system dynamics as a series of train carriages carries the idea that a bump on the end carriage would pass the jolt down the train. The speed of travel of the jolt given by the inertia of the carriages and the spring between them. Readers are familiar with the speed of light transients of transmission lines but these transients passing on the jolt between the angles of subsequent generators are comparatively slow. In the east coast of the US the speed of the wave is 400 miles per second. On the west coast with greater distance between generators and hence lower effective spring constant the speed is 600 miles per second. In Australia with fewer generators with lower inertia and longer distances the travel speed is more like 4000 miles per second.

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1 Phasor Measurements for Identification and Control

The travelling wave concept can help with control design particularly for the first swing of the system. Chapter 8 Identification of Dynamics of Inverters in a Power System Similar to the approach to find the dynamics of the induction component of large loads found by residue analysis in Chap. 2, similar techniques enable examination operationally of groups of inverters. The benefit of this identification is that the environment for any new inverters can be determined even where there are large numbers of existing inverters. The identification process is helped by the high accuracy measurement available from PMU’s. Identification techniques can also determine a model for dynamic loads so that the techniques of wide area control from Chap. 4 can be extended to loads with a dynamic phase shift. Another area where identification can assist in power system control lies in the identification of effective impedance in distribution systems. Droop control for voltage is improved by correction for the Thevenin impedance at the point of connection. Chapter 9 Phasors for Distribution Phasor measurements have shown analysis and control benefits for transmission that are now extending to distribution. This chapter shows a distribution state estimator based on phasor signals and then relaxes the requirements to incorporate scalar signals such as current or voltage magnitude. The process is strongly dependant on Kalman filters shown useful for interarea oscillations in Chap. 3 or in inverter control in Chap. 6. Chapter 10 Conclusions The high accuracy of PMU measurements makes fast estimation of the state and the dynamic response of a power system readily available. The motor content of loads can be identified as well as interarea dynamics. Nonlinear state estimators can tell us whether section of the network have critical issues. Nonlinear control design based on Kinetic Energy can provide both first swing improvements as well as overall nonlinear damping control. Inverters are rising in importance in the power system dynamics and Chap. 6 shows how these can be incorporated into the overall control philosophy. The expected reduction of conventional generation raises concerns about system control but with system state estimates available, fast command to customer loads and batteries can make a substantial impact on the control of critical faults in the power system.

References 1. A.G. Phadke, J.S. Thorp, History and applications of phasor measurements, in 2006 IEEE PES Power Systems Conference and Exposition (2006), pp. 331–335 2. I.P.E. Society (ed.), IEEE Standard for Synchrophasors Measurements for Power Systems, C37.118.1–2011 (IEEE, NY, USA, 2011)

References

11

3. IEEE (ed.), IEEE Standard for Synchrophasor Data Transfer for Power Systems C37.118.2– 2011 (IEEE Standards Association, NY, 2011) 4. H. Khalilinia, V. Venkatasubramanian, Recursive frequency domain decomposition for multidimensional ambient modal estimation. IEEE Trans. Power Syst. 32, 822–823 (2017) 5. G. Ledwich, E. Palmer, Modal estimates from normal operation of power systems, in 2000 IEEE Power Engineering Society Winter Meeting Conference (2000), pp. 1527–1531 6. M. Pai, Energy Function Analysis for Power System Stability (Springer, 1989) 7. C.J. O’Rourke, M.M. Qasim, M.R. Overlin, J.L. Kirtley, A geometric interpretation of reference frames and transformations: dq0, Clarke, and Park. IEEE Trans. Energy Convers. 34, 2070– 2083 (2019) 8. IEEE Standard for Synchrophasor Measurements for Power Systems, IEEE Std C37.118.1– 2011 (Revision of IEEE Std C37.118–2005), (2011) pp. 1–61 9. IEEE Standard for Phasor Data Concentrators for Power Systems, IEEE Std C37.247–2019, (2019) pp. 1–44 10. IEEE Standard for Synchrophasor Data Transfer for Power Systems, IEEE Std C37.118.2–2011 (Revision of IEEE Std C37.118–2005), (2011) pp. 1–53

Chapter 2

Load Modelling

2.1 Load For engineering analysis of power systems many different models are required covering many different time scales from the microsecond models of electromagnet transients to the years used in load forecasting models. For this book the focus is on representation of the dynamics of the power system for transient stability studies. Traditionally the modelling used is to represent the transmission lines, transformers and capacitor banks as in electrical steady state. The generators and excitation control systems are represented as differential equations with different degrees of precision in modelling of the turbine and boiler system. For the customer loads, planning studies often represent these as constant power loads. This is often justified since there are tap changing transformers between the transmission system and the majority of the loads which tend to correct the tapping for any changes in the transmission voltage. For the time scales of electromechanical transients, the tap changers do not have time to operate and variations in transmission voltage will directly affect distribution loads. For a load such as a water heater the load appears as a constant impedance load controlled by a hysteresis temperature controller. The effect of the temperature controller across hundreds of water heaters is in steady state to make them a constant power load. If the voltage on the system is reduced the temperature controller needs to operate with an increased duty cycle so that the instantaneous power to any one heater is reduced but the average ON time is correspondingly increased. However there is a time constant is these loads which means that a step reduction change of voltage will cause an initial drop of power to a group of loads but the regulator will restore it to constant power dependant on the time constant of the load. The same time constant effect will influence the response of refrigerators, air conditioners and ovens. For the short times of electromagnetic transient analysis this temperature regulation effect can often be ignored. The response of customer loads to voltage changes also includes loads such as televisions and computers. Many of these have switch mode power supplies capable © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_2

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2 Load Modelling

of operation from 90 to 240 V with a regulation time in milliseconds. For the purposes of electromechanical transients these loads can be treated as genuine constant power loads. Inverter driven air conditioners have a rectifier front end and a fast operating speed regulator for the output. As such they will tend not to be significantly influenced by system transients and should be represented as constant power loads. There are some genuine admittance loads but even the humble incandescent lamp is dependant on its temperature. As the voltage increases, the power increases but as the temperature increases the resistance also increases. The load presented by a transformer is highly dependant on its level of saturation. As the voltage increases above nominal the reactive admittance also increases with some corresponding increase in the magnetizing losses. The other significant class of loads that is the main subject of this work in the induction motor. This load is sensitive to both voltage and frequency changes. For large drops in the voltage there can also be a stall characteristic. This stalling is discussed in this chapter. Load type

Regulation

Model for electromechanical transients

Water heater, Oven, refrigerator

Hysteresis temperature control

Constant impedance load

Computer, amplifier, television

High frequency voltage regulator

Constant power load

Inverter driven Air conditioner

Fast regulator

Constant power load

Pool pump motor,

Affected by both voltage and frequency

Induction motor load

Washing machine

Time controller

Induction motor load

Incandescent lamps

uncontrolled

Nonlinear voltage response

Many of the small motor loads are becoming controlled by motor drives and are no longer influenced by small changes in system voltage and frequency.

2.2 Overall Load Model The composite load model can thus be composed of a constant power load, a constant impedance load and a set of induction motors of differing electrical parameters and inertial time constants. The main difficulty is to characterise the induction motors which are present in a given portion of the network for any particular time. There are two general approaches: survey (or component based) and measurement-based. The survey approach is characterized by surveys of customers to

2.2 Overall Load Model

15

know what appliances they possess and what is their pattern of use. The measurementbased approach is to determine what motors are running at any given time by measurement of the response of loads to changes of voltage and/or frequency.

2.3 Measurement Approaches 2.3.1 Tap Changer One approach to determine the motor load at a given time instant is to apply a voltage step through the use of tap changing on a distribution transformer. The transfer function from voltage to power change can give some idea of the time constants of the dynamic loads. There are two difficulties with this approach. The first is that with a range of motors in the composite load, it is difficult to extract the motor parameters from the voltage step. The other issue is that this is often a one-off test. So if the motor composition can be found it will be for one particular time instant. The test may be able to find the motor load at 4 pm for a Tuesday in summer but with no information about variation.

2.4 Load Modelling Power system load modelling is often expressed as a set of mathematical equations, which describe the relationship between the voltage and frequency at a given bus bar in the power system and the real and reactive power at the same bus bar. Algebraic equations are used in static load models and differential equations are used in dynamic load models [1]. Composite loads are the main type of loads present in the power system. However, the only frequency dependent loads are expected to be motors, especially induction motors, which are widely used in industries and consist of 60– 70% of total loads in the power system [2].

2.4.1 Traditional Load Modelling In the traditional approach to modelling of loads such as induction motors, it is assumed that a power system will affect the load only and the model contains only a feedforward component as shown in [9]. Figure 2.1 shows a common load model which is the frequency to power model (Another common load model is the voltage to power model, but it is not studied in this book). A transfer function from f to P can be inferred by observing the changes

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2 Load Modelling

Fig. 2.1 Traditional model

B(s)/A(s)

in power when there are changes in frequency. Measurements of observed parameter perturbations are available using PMU’s. Almost all previous research has performed load modelling based on traditional models which describe power system variations affecting the real and reactive power of the load [1, 3–5].

2.4.2 Inadequacy of the Traditional Model However, the general circumstance for the load model is that the load is affected by power system and the power system measurement is also affected by the load [6]. Customers switching loads ON and OFF cause continual variation in customer loads, which in turn causes angle and voltage magnitude changes at the supply bus. These angle changes will affect the measurements of frequency at the supply bus. It is a known fact that large increase of customer load can decrease bus voltage, decrease system frequency and excite inter-area modes. These facts are examples of the load’s effect on power system. This effect can be represented by a feedback term. A simple feedforward system without this feedback term is inaccurate in predicting a transfer function from f to P apart from very small load segments. Another evidence of the necessity of feedback structure is shown through the non-causal correlation between f to P [7]. When the load is significant for the power system strength then the feedback structures becomes important [6]. As seen if Fig. 2.2 the time instant t = 15 corresponds to zero delay and Fig. 2.2 shows power terms preceding and following frequency events. This non causal relation is because frequency changes are not fully responsible for power changes nor visa-versa. The two way interaction is summarised in Fig. 2.3 with the zero delay located at T = 15.

2.4.3 Closed-Loop Feedback Load Modelling A new load modelling technique, the closed loop feedback or residue approach, is developed in [8]. This technique considers the load’s effect on the power system as well as power system’s effect on the load [9]. The block diagram of the interaction between the power system and the load is shown in 3. The disturbance of the power system is indicated by two variables: W1 and W2 . The variations in the frequency measurement may not be fully predictable from past frequency measurements or load effect because of other loads causing system

2.4 Load Modelling

17

Fig. 2.2 Cross correlation between frequency and power

Fig. 2.3 Identification when there are multiple disturbances and feedback [7]

variations. The unpredictable frequency component is represented as a white noise input W1 , which is known as the white noise of the power system. W1 is also called the residual of frequency (unpredicted changes in frequency). Similarly there is a component of load variation that represents the unpredictable changes in load power by customers turning switches ON and OFF. This unpredictable load variation component is represented as a white noise input W2 , which is the white noise of the load. W2 is also called the residual of load (unpredicted changes in load). When all terms from the signal correlated to the past measurements are removed the residual of the signal is what remains. The goal of this processing is to find the best predictor for f and P. The white noise component as the residuals for f and P will thus be the white noise inputs W1 and W2 . Thus the process will be to find the transfer function from W1 to both f and P and the ratio will give the feed forward system B(s)/A(s), provided that W1 and W2 are uncorrelated. Similarly the transfer function from W2 to P and to f can be found and the ratio will give the feedback system D(s)/C(s).

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2 Load Modelling

If W2 = 0, then the transfer function between f and P would identify the feed forward system. If W1 = 0, then the transfer function between P and f would identify the feedback system. When both terms are present there is no clear separation between the effects [10]. Consider variables y(t) and u(t) (output y and input u) with representation in the s and frequency domain Y(s), U(s), Y(jω), U(jω), where (jω) is a fourier transformation of the sample data. Using the representation in s domain of the terms A(s), B(s), C(s) and D(s), the input and output can be expressed in the frequency domain as Y (s) =

B(s) U (s) + W2 (s) A(s)

(2.1)

U (s) =

D(s) Y (s) + W1 (s) C(s)

(2.2)

where W1 (s) and W2 (s) are the representations of the two noise terms in the s domain. C( jω) , and since A, B, C and D are supposed to be Defining ξ = A( jω)C( jω)−B( jω)D( jω) constant for each frequency value, Y W1 ( jω) = B( jω)ξ( jω)

(2.3)

U W1 ( jω) = A( jω)ξ( jω)

(2.4)

The frequency response transfer function of the feedforward term from input u(t) to output y(t) can be obtained through the input noise w1 (t) U Y ( jω) =

B( jω) Y W1 ( jω) = U W1 ( jω) A( jω)

(2.5)

The frequency response transfer function of the feedback term from output y(t) to input u(t) can be obtained through the output noise w2 (t).  ( jω) D( jω) Y U ( jω) = UY WW2 ( jω) = C( For detailed derivation please see [7]. jω) 2 The residue approach for the load modelling problem is able to give good estimates of the noise when output noise continues to drive the system. The residue method can be used for a low signal to noise ratio. In the residue method, the open loop transfer function is calculated based on input residue. Similarly, the feedback transfer function is calculated based on the output residue estimates. From the transfer function of power system to load, two main mechanical transient parameters, the inertia and torque damping factors can be estimated. The estimated error is minimal [9]. To identify the accuracy of the closed-loop load model with multiple noises, power system parameter data were collected from a substation PMU and analysed. The parameter value were also estimated from the transfer function of a real motor

2.4 Load Modelling

19

using residue method. The estimated and original values matched quite accurately with each other [6]. From Figs. 2.4 and 2.5, it can be seen that at low frequency, the real and reactive power of the load changes power system voltage and frequency. But the load cannot affect the power system parameters at medium to high frequency. Therefore, if the load is changed at low frequency, the power system parameters will be affected significantly. But at high frequency the load doesn’t affect the power system parameters [6].

Fig. 2.4 Substation data showing load P changes affecting frequency [6]

Fig. 2.5 Substation data showing load P changes affecting voltage [6]

20

2 Load Modelling Load changes affecting frequency

f

w1 +

Σ V

w3 +

Σ

Frequency f change affecting load P

Voltage v change affecting load P

+ w2 +

p

Σ +

-

Load changes affecting voltage

Fig. 2.6 Both frequency and voltage interacting with load

Both voltage and frequency can affect load so the feedback structure needs to recognize both loops as in Fig. 2.6 and there is now a new noise term w3 to be separated.

2.5 Example An industrial example of frequency to power relations shows the necessity of feedback term. In this case the variations of supply frequency affect the induction motors in the load but in the small network to which the load is connected variations of the load affect the system frequency. If a simple direct transfer function from f to P was identified, both the feedforward pole and feedback pole will appear in the transfer function. However, the transfer function of the feedforward term in Fig. 2.7 (Frequency changes affecting load) is significantly different from the transfer function of the feedback term in Fig. 2.8 (Load changes affecting frequency), each showing a different pole location. This clear separation of poles demonstrates that the separation of the feedforward and feedback term can be achieved perfectly and that it is necessary to separate then [7]. For a system with a feedforward term with a pole at 0.5 Hz and a feedback term with resonant poles at 1 Hz and damping 0.05, the direct identification of the system between u and y shows a model with poles at both feedforward and feedback locations, as shown in Fig. 2.9. When the noise term w1 and w2 are extracted and identified, the modelled system and the original system match very well as shown in Fig. 2.10. In Fig. 2.11 ratio 1 and ratio 2 are the calculated magnitude and phase of

2.5 Example

21

Fig. 2.7 Frequency changes affecting load [7]

Fig. 2.8 Load changes affecting frequency [7]

Fig. 2.9 Direct transfer function estimate U-Y [7]

Magnitude

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.5

1

1.5

2

2.5

3

Frequency Hz

3.5

4

4.5

5

22

2 Load Modelling Magnitude

2.5

ratio1 ratio2 ref1 ref2

2

1.5

1

0.5

0

0.5

1

2.5

2

1.5

3

Freq

Fig. 2.10 Quality of fit [7] 40 30

Magnitude

20 10 0 -10

1st order 3rd order 5th order

-20 -30 -40 -1 10

10

0

10

1

10

2

10

3

10

4

10

5

frequency in rad/s

Fig. 2.11 Transfer function of induction motor

the feedforward and feedback transfer functions [7].

2.6 Motor Loads When the 5th order model of an induction motor is examined there is a peak that is indicating the inertia of the motor. The lower the inertia the higher frequency of the peak. Also in general the area under the power vs frequency curve is an indication of the nominal motor rating. In [9] there is a correction process described where the

2.6 Motor Loads

23

motor ratings are more correctly identified. Figure 2.12 shows the f-P model for a range of motors and the change in the areas. Based on this calibrated set of curves the load components can be found for Sydney West measured loads. As seen in Fig. 2.13 the composite from a least squares fit is found adequately from a set of 3 motor sizes. Based on this identification the motor load composition changes through a day can be tracked. As seen in Fig. 2.14 the component of large motors increases during Frequency chnage affecting the real power change aggregated 11kw 15kw 4kw 7.5kw 18.5kw 22kw 30kw 45kw 180kw 630kw

1500

Magnitude

1000

500

0

-500

2

1

10

10

frequency rad/s

Fig. 2.12 Transfer function for a range of induction motors transfer function f-p real composite 180kw 15kw 630

0.1

magnitude

0.08 0.06 0.04 0.02 0 -0.02 10

2

frequency rad/s

Fig. 2.13 Transfer function of frequency affecting load power decomposed into 3 motor sizes

24

2 Load Modelling 1.2

different motor contribution in the 24hrs data in 90 min lemgth 180kw 15kw 630kw

1

coefficient

0.8 0.6 0.4 0.2 0 -0.2 00:00

04:00

08:00

12:00

16:00

20:00

time

Fig. 2.14 630, 15 and 180 kw motors percentage contribution in 24 h a day Xl

Infinite Bus

Passive loads

X2

Induction motor

Synchronous machine

Fig. 2.15 Composite load model

daytime hours [9]. The dynamic components of the load structure as in Fig. 2.15 can be extracted. The advantage of a genuine motor load model is that there is an expectation that even though the model is fitted based on small signals the response will be more faithful even through large voltage excursions.

2.7 Real Loads More motor loads are being fed through inverter interfaces. This small signal identification will see these motors as constant power loads, These buffered motors will not tend to act as dynamic loads in power system oscillations. During large disturbances there motor drives can trip but there is no indication of this from small signal models. As used in Chap. 3 the total load of major cities is driven by customer load switching and in the short term such a 10 min, the exact changes in load are unpredictable and can be modelled as white noise. With the changes of loads being white

2.7 Real Loads

25

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4 -5

-4

-3

-2

-1

0

1

2

3

4

5

Fig. 2.16 QQ plot showing deviation from a gaussian distribution

noise the loads themselves are modelled as the integral of white noise. As used in Chap. 9 this is also seen down to the level of distribution feeders and helps with distribution state estimation. Note that in general a combination of millions of residential customers from the central limit theorem can be represented as gaussian. When there are a small number of large industrial loads the central limit theorem does not hold and a gaussian mixture model is more appropriate (Fig. 2.16).

2.8 Application PMU based identification of motor loads is seen to be possible, what is important to discuss here are issues in implementation and what benefit there is in using this approach. Motor loads have been important in getting good prediction of system damping for control of interarea damping. In the early days of interconnection, models were able to fit the system response adequately. With connection between states generating new frequencies of oscillation, previous models did not give a good prediction of the achieved damping. Introducing more complex load modes was able to match the observed damping. A lot of the difference was due to induction motor loads. Motors are not the only aspect of loads that impacts power system performance. In major frequency transients the question of whether wind farms and distributed PV inverters remain connected is becoming of importance. Unfortunately the small

26

2 Load Modelling

signal response examined here is unable to tell of the propensity of a given inverter to trip in a major disturbance. The question that rises is whether there is a good tool to measure how much load and how much PV generation has tripped. The change of the motor load fraction can be determined using the approach of this chapter. Is it possible to determine PV generation at any given time? If there were a single time constant which characterised the phase locking of the inverters there could be a path to examine the transfer function from angle changes to power changes but the spread of response makes that difficult. The hardware for implementation of motor load identification is based on PMU signals sampled significantly faster than any load dynamics so even small motors should be identifiable provided the point of connection contains many customer loads making continual switchings.

References 1. F.T. Dai, J.V. Milanovic, N. Jenkins, V. Robert, Development of a dynamic power system load model, in Seventh International Conference on AC and DC Transmission (London, UK, 2001) 2. P. Kundur, Power System Stability and Control (McGraw-Hill, New York, 1994). 3. IEEE_Task_force_for_dynamic_performance, Standard load models for power flow and dynamic performance simulation. IEEE Trans. Power Syst. 10(1995) 4. B. Khodabakhchian, G. Vuong, Modeling a mixed residential-commercial load for simulations involving large disturbances. IEEE Trans. Power Syst. 12 (1997) 5. Standard load models for power flow and dynamic performance simulation. IEEE Trans. Power Syst. 10(8), 1302–1313 (1995) 6. T. Parveen, G. Ledwich, E. Palmer, Model of Iinduction motor changes to power system disturbances, in Australasian Universities Power Engineering Conference AUPEC 2006 (Melbourne, VIC, Australia, 2006). 7. C.F. Moyano, G. Ledwich, Load modelling: induction motor, in Electric Power Systems in Transition (Nova Science Publisher Inc, 2010) 8. T. Parveen, G. Ledwich, E. Palmer, Induction motor parameter identification from operational data, in 2007 Australasian Universities Power Engineering Conference, 9–12 Dec. 2007 (Piscataway, NJ, USA, 2008), pp. 302–307. 9. T. Parveen, Composite Load Model Decomposition: Induction Motor Contribution (Queensland University of Technology, 2009) 10. C.F. Moyano, G. Ledwich, Load model: induction motors in large disturbance, in 2008 Australasian Universities Power Engineering Conference (AUPEC’08), 14–17 Dec. 2008, (Piscataway, NJ, USA, 2008), 6 pp.

Chapter 3

Identification of Power System Dynamics

To identify oscillations and damping system identification is required but to develop control processes low order models and a control approach applicable to nonlinear systems are desired. This chapter introduces the model reduction process and the foundations of energy based control.

3.1 Identification of Dynamics The technology of phasor measurements units provided a step jump in the accuracy of angle measurements in power systems. Now with 200 or more measurements per cycle, even transducer noise could be reduced to very low levels by averaging across mains cycles. In the first measurements of phasors across the Queensland system [1] the results showed a clear trend of the energy of angle signals falling as a function of frequency with resonant peaks at the dominant oscillation modes of the system. This was identified that the driving signal was the integral of white noise exciting the system resonances. The summation of millions of customer loads was showing that in the short time scale of tens of seconds that the changes of loading were uncorrelated with the pervious loading. The customers decision to switch on a toaster/light or for the fridge to turn off were uncorrelated with the total load that was being consumed at the time. The characteristic has been confirmed at the large city level and down to the individual feeder level that the changes in load were white noise and this the load can be represented as the integral of white noise. When the load consists of a very large number of identical loads the central limit theorem [2] says that the changes will be Gaussian white noise. For a residential feeder with large numbers of similar loads that proves true. For a city load the total load is not Gaussian since is combines the Gaussian noise of residential loads with another possibly Gaussian load of large industrial customers. An analysis of the composite load is Sydney shows the characteristic of

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_3

27

28

3 Identification of Power System Dynamics

the sum of Gaussian loads where the industrial load component shows a larger variance component associated with industrial loads switching ON and OFF combined with the lower variance component of residential loads. There are dynamics of city loads which means they are not pure integral of white noise. The induction motor component of the loads means that there is a response of the motor loads to the frequency changes in the rest of the power system. As an identification problem the loads are responding to system frequency changes as well as having load variations which cause broadband frequency changes. So the basic characteristic is of a load variation that is the integral of white noise giving rise to induction motor responses within the cities as well as resonant interactions between the generators.

3.1.1 Identification with White Input Much of the theory for the identification is dynamic systems starts from an assumption that the input is white noise. This means that the derivative of the angle signal needs to be examined to identify the dynamics of the generator response. Normally taking a derivative is avoided because it accentuates the noisy high frequency component of the signal. Because the PMU has a large level of averaging across the mains cycle the noise in the measurement of angle is very low and the derivative still contains a high level of signal to noise. As discussed later, there is occasionally an exception to this integral of white noise characteristic which is called a forced response when the load is generating a signal which is not random. For now the derivative of angle signal is driven by white noise is consider. Taking the Laplace transform of these signals the power system can be represented as a set of white noise inputs exciting the oscillations of the power system Y (s) = H (s)U (s) If the spectrum of the output signal Y is examined, the resonant peaks of the system will yield the transfer function of H because the spectrum of the white noise term U is flat in frequency domain. Now this is a true statement for a long signal sequence but when the signal is examined over a short time the spectrum of white noise is not flat and has a variance that is proportional to the square root of the length of the sequence. So while the spectrum of the input signal is ideally flat there is a variation that depends on the length of the signal. So the input spectrum is flat with random variations that are uniform with frequency. U ( jω)U ( jω)∗ = E + w(ω) Thus the spectrum of Y ( jω)Y ( jω)∗ = H ( jω)H ( jω)∗ (E + w(ω))

3.1 Identification of Dynamics

29

Thus the spectrum of finite length signals shows the properties of H but with an uncertainty given as H ( jω)H ( jω)∗ (w(ω)). Most of the spectral tools assume a system of the form Y ( jω)Y ( jω)∗ = H ( jω)H ( jω)∗ (E) + k(ω) where the spectrum is distorted by added white noise rather than multiplicative white noise. Let us consider the nature of short length signals driven by white noise in an example where the system H represents a system resonant at 1 Hz.

3.1.2 Power System Dynamics-Forming Parameter Estimates The autocorrelation sequence is a good way to summarize the consistent variations in system response. The time domain autocorrelation sequence maps to the impulse response of the system for infinite length sequences [3]. r (τ ) = E(y(t) · y(t + τ )) The fourier transform of this signal is the same as the spectrum of the signal. Thus by forming the autocorrelation of a signal in time and then taking the FFT gives a good representation of the signal spectrum [4]. The matlab function “invfreqs” is designed to find the transfer function which best fits an observed spectrum. As described in the MATLAB help, the spectrum is parameterized by a finite length transfer function b(s)/a(s). The coefficients of the polynomials a(s) and b(s) are found by a gradient search process that can be weighted in frequency. For a power system with several PMU’s the autocorrelations and cross correlations can yield estimates of the system transfer functions. For a power system with ‘n’ measurements there will be ‘n’ autocorrelations and potentially ‘n * (n − 1)’ cross correlations. For a given connected power system there will be only one set of poles to characterise the system and an appropriate form to model the set of spectra is B(s)/a(s) where a(s) is a scalar. A variation of the standard “invfreqs” can be generated to form this gradient search for the common denominator form of system. One point of interest is that adding additional measurements does not dramatically improve the estimates in the way expected for noisy measurements. Certain measurements will be richer in particular modes and will assist but because the PMU signals are of low noise, the additional measurements are of the same oscillations are the modal estimates are not dramatically improved.

30

3 Identification of Power System Dynamics

3.1.3 State Estimation Thus far in this chapter identification of system dynamics has been developed, knowing the frequencies at which it oscillates and the damping of the modes. This section is addressing the next step in the problem of knowing the angles of the equivalent generators representing the angles of the areas. When a power system operates there may be several PMU’s in an area to be represented by one equivalent generator. The way to move from multiple measurements at different points of the system, to a representation of the angles of the equivalent generators for all areas is through state estimation. For linear systems the state estimation can be posed as finding the best estimate of the state ‘x’ given the state space model in the form x˙ = Ax + Bv, y = C x + w

(3.1)

where the input noise ‘v’ and the measurement noise ‘w’ are unknown and E(vv’) = R and E(ww’) = Q. For the power system the issue is that the dynamics depend of the operating point of the system and a better representation is x˙ = (x) + Bv, y = C x + w

(3.2)

capturing the nonlinear relation of the system dynamics. For the problem in (3.2) the solution is through the use of a Kalman Filter [5] to solve for the value of the correction gain ‘L’ in   x˙ = Ax + L y − y , y = C x











(3.3)

For the real power system the operating point is not well characterize beforehand so the Eq. (3.3) is linearized around a common operating point to find a good value for ‘L’ and to use it in the equation   x˙ = (x ) + L y − y , y = C x











(3.4)

The reason this is possible is because the sine nonlinearity of the power transfer between busses is well known, so all terms in (3.4) are known. This approximate Kalman filter has worked in all systems evaluated and shows a good performance. Using (3.4) the state estimates for the angles of an area can be found given power or angle measurements in an area. The example in [6] showed an area with several PMU’s. Each of the measurements is helpful to form the area angle estimates. When one of the measurements is lost the state estimate can use one designed with the reduced measurements and the system can proceed with low error. The simplistic approach to average the angle measurements of an area, to form an angle estimate

3.1 Identification of Dynamics

31

can produce a step change in the estimate which can but detrimental to good closed loop control. Further aspects of State Estimation of nonlinear Power Systems is found in Sect. 4.3.

3.1.4 Forming Areas for Aggregation Aggregation is an essential step in developing controls for large power systems. There can be hundreds of generators in an interconnected power system and forming a control based on all individual generators poses a large and unnecessary overhead on the control implementation. Typically power system oscillations can be broken into two classes, interarea oscillations where groups of generators swings against one another and local oscillations where one or two generators swings against the rest of the system. The interarea oscillations are lower frequency often 0.1–0.5 Hz while local oscillations are typically 1−3 Hz. The local oscillations are typically of lower concern because the damper bars on the surface of the rotor provide good damping of higher frequency oscillations. The lower frequency modes are typically less well damped and active control can be necessary to provide a good dynamic response. So the aggregation problem is which generators should be grouped together to characterise system dynamics. Aggregation has often been used to provide a simple model of the remainder of the power system when undertaking studies of disturbances in a local area. The aim here is to group generators which largely swing together during disturbances showing the lower frequency interarea oscillations. Some faults will excite local oscillations but as discussed above these are of low concern, other disturbances excite largely an interarea response and this is the main concern here. The research shows that in most circumstances loads behave as the integral of white noise and this noise excites the oscillation of the system. To group generators, the approach taken here is to undertake time domain simulations of the power system with the major loads modelled as noise terms. Using a low pass filter on all generator velocity terms the correlation between all pairs of generators can be formed and those with high correlation across a representative range of disturbances will be designed as an area. For some disturbances there may only be 2 or 3 groups but across the range of disturbances there can be a larger number of groups required to capture the coherent behaviour. In the Australian example in [7] there were 5 groups identified which largely reflected state boundaries due to the history of development of the systems. Queensland showed evidence of two groups of generators because of a more dispersed pattern of generators. One aspect to be aware of is that the pattern of groups can change with operating point and thus the grouping process in some cases may need a range of operating points in forming the definition of groups.

32

3 Identification of Power System Dynamics

3.2 Aggregation of Power System Areas 3.2.1 Introduction Electrical generators have been successfully aggregated to define a reduced order system. The state of the reduced system can be used to control the lower frequency modes even for large disturbances. This control design can be very effective when the machine groups are well separated. The issue of control and estimator design for the case of more uniformly spread generators is addressed in this subsection. The power flow equations need to be modified from the standard approach based on the angle between the aggregate groups. The control designs are demonstrated on HVDC links in a uniform 2D system of generators and on a 1D non uniform set with shunt load modulation. Aggregation in power systems refers to a process to develop a reduced complexity system which mimics the dynamics of the original system at least for the interarea or low frequency terms. One of the motivations for aggregation is to simplify the external area such that simulations within your own area can be completed faster [8]. However the motivation in this chapter is to reduce the model order in order to design a simple control system for the interarea control of the power system. This chapter is particularly concerned with the case when there are no clear boundaries between the groups. The development of the PMU systems is such that the dynamic state of large power systems is measurable but the complexity of passing data about every generator in the system to every controller is daunting and unnecessary.

3.2.2 Aggregation Process Based on Generator Coherency Studies such as [9] successfully aggregate groups of generators. This occurs particularly cleanly where groups of generators are separated by transmission lines of higher impedance than within the group. Defining the areas in large inter-connected power systems consists of two steps, including finding coherent generators which form a group and assigning the nongeneration buses to these groups. The generators which have similar behaviour after the disturbances in the systems are called coherent generators and usually the term “coherency” refers to the coherency of generators in the slow inter-area modes. To study the generator coherency a classical machine model is usually considered to be sufficient [10, 11]: Ji

d 2 δi + Di ωi = Pmi − Pei dt 2 ωi =

dδi , i = 1, . . . , n dt

(3.5) (3.6)

3.2 Aggregation of Power System Areas

33

In the equations above Ji is the inertia ratio, Di is damping ratio, Pmi and Pei are the mechanical power and electrical power of the ith machine, ωi is the generator velocity, δi is the generator angle, and n is the number of generators. Electrical power Pei depends on the load modeling in the system and by modeling them as constant impedances the differential algebraic equations (DAEs) will be simplified. In most of the power system transient stability simulations, a disturbance is considered as a fault on a line or bus and the analysis is performed based on the single perturbation of the power system. In this approach instead of single disturbance in the system, the disturbance is distributed within the system to guarantee the excitation of all modes. This has been implemented by low level randomly changing of the load at all load buses which resembles the real load changes in power systems. This load change has the effect of a distributed disturbance in the power system, and will cause the generators to swing in a range of different frequencies related to different oscillation modes. By applying the mentioned load disturbance, the velocity changes of all the machines in the system can be studied and spectrum analysis is applied to the output velocity of the generators. The Discrete Fourier Transform (DFT) is a common method for spectrum analysis of discrete time signals and is seen in Eq. (3.7). Xf =

N −1 

x n H e−

i2π N

fn

, f = 0, . . . , N − 1

(3.7)

n=0

X f is DFT, x(n) is the sampled velocity of each generator, N is the number of samples, and H is the window function. The kinetic energy of machines plays a significant role in the importance of each mode in power systems. By having the kinetic energy of each machine and also the total kinetic energy of the system over the spectrum, the important modes with the higher kinetic energy and low frequency can be considered as the possible inter-area modes. The kinetic energy of machines and total kinetic energy of the system can be calculated as (3.8, 3.9), where E i is the kinetic energy of each machine and E T is the total kinetic energy of the system E i = 1/2Ji ωi2 ET =

n 

Ei

(3.8)

(3.9)

i=i

To obtain the kinetic energy of each machine and total kinetic energy of system respectively over the spectrum, the DFT of generator velocities ωi is calculated by (3.6) and obtained X f s are squared and multiplied by inertia ratio Ji according to (3.7) to define the spectral distribution of kinetic energy of each machine. Then referring to (3.8) the total kinetic energy across the spectrum can be easily calculated

34

3 Identification of Power System Dynamics

by adding the obtained kinetic energy of each generator. Then the obtained total kinetic energy is checked for low frequency to find the important inter-area modes. As it is seen in the test results, the inter-area modes usually have the low oscillation frequency between 0.1 and 1 Hz while the kinetic energy analysis will also assist definition of the inter-area oscillation frequencies. Finding the generators which are coherent for the inter-area frequencies, the coherent generators will be found and generator grouping is achieved. For the chosen disturbance, the velocity of generators can be considered as the random process and these signals are analysed in the inter-area frequency band. Therefore, the generators velocity signals are low pass filtered using a low pass digital signal filter such as Chebyshev filter to exclude the higher frequency oscillations. In this case for the filtered velocities, signal correlation coefficient is calculated as below: N ri j =  N k=1

k=1 (

−

−

f ωik − f ωi )( f ω jk − f ω j ) −

2

( f ωik − f ωi )

N k=1

( f ω jk − f

(3.10) − 2 ωj)

where f ωik and f ω jk are the k th element of filtered velocity signal of machine i −

−

and j, f ωi and f ω j are the sample means of these signals, and ri j is the correlation coefficient between mentioned velocity signals. The resulting ri j is a real number such that −1 < ri j < 1 and its sign shows if the pairs are positively or negatively correlated. Calculating the correlation coefficient for the low pass filtered velocity of generators as mentioned above, coherent groups of generators are identified for a large power system. Based on the results for each pair of generators, highly positive correlated generators will define the coherent groups of generators. This method is very efficient in large power systems as highly correlated generators in the inter-area frequencies are considered to form groups. In some cases, some generators may not only correlate with one group but also have higher correlations with other groups. In such cases and also for more accurate grouping and characterizing inter-area oscillations between areas, further analysis is performed on velocity of generators. To achieve this, the coherence and cross spectral density functions will be applied on the velocity changes of machines. The coherence (magnitude-squared coherence) [4] between two velocity signals related to machine i and j is calculated as: Ci j ( f ) =

   Pi j ( f )2 Pii ( f )P j j ( f )

(3.11)

where Pi j ( f ) is the cross spectral density of velocity of generators i and j, Pii ( f ) and P j j ( f ) are the power spectral density (PSD) of ωi and ω j calculated as (8, 9). It can be inferred that cross spectral density and PSD functions are Discrete Fourier Transforms of cross-correlation Ri j (n) and auto-correlation Rii (n) functions which

3.2 Aggregation of Power System Areas

35

can be obtained by (3.14, 3.15) where E is the expected value operator [3] Pi j ( f ) =

N −1 

Ri j (n)e−

i2π N

fn

, f = 0, . . . , N − 1

(3.12)

Rii (n)e−

i2π N

fn

, f = 0, . . . , N − 1

(3.13)

n=0

Pii ( f ) =

N −1  n=0

Ri j (n) = E ωi (m + n)ω j (m + n)∗

(3.14)

Rii (n) = E ωi (m + n)ωi (m + n)∗

(3.15)

The coherence function (3.11) defines the coherency of the velocity output of machines across the frequency range and defines how the machines are correlated. This coherence function is not sufficient to define the coherent group of generators as it does not indicate if the generators are positively or negatively coherent in each frequency. To overcome this issue, the angle of cross spectral density function Pi j ( f ) can show the angle that generators are correlated in each frequency at the range of inter-area modes. So if the angle is close to zero the machines are positively coherent and belong to a group and if it is close to 180°, means the machines are negatively coherent for that frequency so they cannot be grouped and belong to different machine groups. Comparing the coherence functions and cross spectral density angle between pairs of generators, the inter-area oscillation frequency band of every group of generators can be defined. In this approach, the generators in one group are highly and positively coherent with each other in that frequency band, while they are highly but negatively coherent to generators belonging to the other group. Therefore, generators in each area along with the inter-area oscillation frequency between different areas will be defined. This method is also applicable for the generators located in the area boundaries which can have high correlations with both of the neighbouring areas. These generators are checked for the specific inter-area frequency band of two areas to see their relation to each of the generator groups. In this case if the disputed generators are positively coherent at the inter-area oscillation frequency band with all of the generators in a group, these generators can be considered as a part of that group. To define the concept of general coherency, the load and generation have to be randomly changed for several cases to check the coherency of generators over different loading scenarios. The purpose is to check if the grouping of generators is robust against the load changes in the system or grouping will change by changing the load in network. It is proven that load changes can affect the coherency characteristics of the power systems. In this case, some generators specially the generators in the boundaries may be swapped between the neighbouring areas.

36

3 Identification of Power System Dynamics

Defining the coherent generator groups is the essential part of area detection in power systems. The second part will be to identify the non-generator buses located in each area. To determine which load and switching bus should be considered in an area, the angle swing of all load buses are obtained. For this reason, the loads can be considered as constant impedances and by having the generators angle swings the angle swings in load buses are obtained. I = YV 

IG IL





Y 11 Y 12 = Y 21 Y 22

(3.16) 

VG VL

(3.17)

V L = Y −1 22 Y 21 V G

(3.18)

δ L = angle(V L )

(3.19)

V G and I G refer to the voltage and current injection at generators while V L and I L are the voltage and current at non-generation buses and Y is admittance matrix of the system. As the inter-area oscillation frequency bands of neighbouring areas are known, the load buses located in each of the areas can be defined. Similar to the generators, the angle swing in non-generation buses are low pass filtered using a low pass digital signal filter to include only the low frequency inter-area modes and, signal correlation coefficient is calculated for the filtered angle swings.  Nl rl i j = 

k=1 (

 Nl k=1

−

−

f δ ik − f δ i )( f δ jk − f δ j ) 2

−

( f δ ik − f δ i )

 Nl k=1

( f δ jk − f

− 2 δj)

(3.20)

Similar to (3.10) f δ ik and f δ jk are the k th element of filtered angle swing signal −

−

of non-generation buses i and j, f δ i and f δ j are the sample means of these signals, Nl is the number of non-generation buses, and rl i j is the correlation coefficient between these signals. Calculating the correlation coefficient for the filtered angle swing of non-generation buses, highly positive correlated buses are considered to be in the same area. This method is applicable to large systems to assign highly correlated buses to each area. To exactly define the border lines (lines connecting the areas) and border buses (buses at each end of border lines) coherence function is calculated for each pair of the non-generation buses using (3.12–3.15) and replacing the angle swing δ L with generator velocity ω. Each of the neighbour border buses connected through a border line belong to different areas. Therefore, the target is to find the neighbour non-generator buses which are negatively coherent in the inter-area oscillation frequencies and the border lines are obtained consequently.

3.2 Aggregation of Power System Areas

37

In some cases same as the generators, it might be difficult to assign some of the non-generation buses to any area. These buses will usually swap between the areas depending on the load and generation changes in the system. Therefore, the border lines and buses may change due to the changes of the load and generation in the system.

3.2.3 Simulation on Test Systems To verify the proposed method, it is evaluated on two test systems, a simple 4 machine system and modified 68 bus, 16 machine NPCC system [12].

3.2.3.1

Test Case 1

The first case is a simple 4 machine test system as illustrated in Fig. 3.1 with classical model of generators, and the transmission lines are modelled as constant impedances with zero resistance with the line reactance of 0.1 pu of all lines except for line connecting buses 6 and 7 which has the reactance of 0.5 pu. The system is considered to be operating in an operating point and the disturbance is applied on the system by randomly changing the loads in load buses. In this case the load change is considered to be a random change with 1% variance at each bus and applied at each time step which is 0.01 s. As mentioned in previous section, the load change has the effect of continuous distributed disturbance on the system and causes the generators to swing as illustrated in Fig. 3.2 by velocity swings of the generators over a 10 s period. To obtain the oscillation frequencies of the generators, spectrum analysis is applied on the velocity output of the generators. The simulation is performed on the system with a long simulation time and repeated to increase the accuracy of the obtained spectrum. For a more accurate result, DFT is calculated in several time frames and the average results are obtained as the spectrum output of the velocity swings shown in Fig. 3.3 along with the spectrum distribution of kinetic energy of whole system. The kinetic energy spectrum can define the important system oscillation frequencies and for this system three major oscillations can be observed. Inter-area modes usually have the frequency below 1 Hz and this example system has only 1 inter-area

Generator 1

5

1

6

7 3

Generator 2

Generator 3

Generator 4

2

Fig. 3.1 Simple 4 machine test system

4

38

3 Identification of Power System Dynamics -4

8

x 10

Gen 1 Gen 2 Gen 3 Gen 4

Velocity Change (pu)

6

4

2

0

-2

-4

-6

0

1

2

3

4

5

6

7

8

9

10

Time

Fig. 3.2 Velocity changes in generators due to the disturbance

mode with the frequency of around 0.53 Hz. By checking the spectrum results of the generators velocity it can be observed that only in the inter-area oscillation frequency all the machines are oscillating while in all other major oscillation frequencies not all of the machines are participating in the oscillations which shows that the mentioned inter-area oscillation frequency is the sole inter-area mode of the system. The system has only one inter-area mode which means it can be considered as two areas so the next step is to find the generators in each of the areas. Referring to the spectrum analysis, higher frequencies can be filtered from the velocity signals to preserve only the inter-area mode and correlation coefficient ri j for the low pass filtered signals will define the generator grouping in the system which is shown in Table 3.1. It can be inferred from Table 3.1 that generators 1 and 2 are highly and positively correlated while being negatively correlated to generators 3 and 4 which means they belong to a same group. Similar result is observable for generators 3 and 4 which means these two generators are also forming a group. To clarify the grouping as mentioned in section II, the coherence function and the angle of cross spectral density between pairs of generators are calculated as the generator in same groups are highly and positively coherent in the inter-area frequency band. The results are illustrated in Fig. 3.4. The coherency results also validates the grouping obtained previously as it can be observed from Fig. 3.4 only the generator groups 1, 2 and 3, 4 are highly and positively coherent in the inter-area frequency band which is around 0.53 Hz. These results also show that at the inter-area frequency, machines belong to other groups are highly but negatively coherent which validated the defined inter-area frequency

3.2 Aggregation of Power System Areas

39

0.1 0.09 Gen 1 Gen 2 Gen 3

0.08

Gen 4

Magnitude (pu)

0.07 0.06 0.05 0.04 0.03 0.02 0.01

0

0.5

0

2

1.5

1

2.5

3

3.5

4

2.5

3

3.5

4

Frequency (Hz) -3

x 10

1

Kinetic Energy (pu)

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

Frequency (Hz)

Fig. 3.3 Spectrum analysis of the generator velocity changes; and distribution of kinetic energy of whole system across the spectrum Table 3.1 Correlation coefficient of low-pass filtered generator velocity signals

i, j

ri j

i, j

ri j

1, 2

0.987

2, 3

−0.769

1, 3

−0.788

2, 4

−0.705

1, 4

−0.725

3, 4

0.959

40

3 Identification of Power System Dynamics 1 0.9 0.8

Coherence

0.7 0.6 0.5

1, 2 1, 3 1, 4 2, 3 2, 4 3, 4

0.4 0.3 0.2 0.1 0

0

0.2

0.1

0.3

0.4

0.6

0.5

0.7

0.8

1

0.9

Frequency (Hz) 180

1, 2 1, 3 1, 4 2, 3 2, 4 3, 4

160

Coherency Angle

140

120

100

80

60

40

20

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Fig. 3.4 Magnitude-squared coherence and angle of coherency between pairs of generators in test system 1

to be the inter-area mode of system. Both of the proposed methods resulted in the same solution and defined the coherent group of generators. The next step is to define the load buses in each area and boundary buses between areas which will lead into finding the boundary lines connecting the areas. This system consists of two neighbouring areas with one inter-area mode with the frequency of approximately 0.53 Hz. The angle swings in all of the non-generation buses are calculated using (3.16–3.19) and filtered to preserve only the inter-area

3.2 Aggregation of Power System Areas Table 3.2 Correlation coefficient of low-pass filtered angle swings of load buses

i, j

41 rl i j

5, 6

0.999

5, 7

−0.969

6, 7

−0.960

frequency band. The correlation coefficient (3.20) is then calculated for these filtered signals and highly correlated non-generation buses are defined to be included in each area as shown in Table 3.2. It can be inferred from the results in Table 3.2 that load buses no. 5 and 6 are highly and positively correlated in the inter-area frequency band which means they belong to the same group while they are highly and negatively correlated with bus no. 7. The buses 6 and 7 are neighbour buses and as they are negatively correlated they can be called border buses and the line connecting these two buses as border line. In this case calculating the coherency function for the angle swings might not be necessary as the correlation results defined the load bus grouping with a high accuracy.

3.2.3.2

Test Case 2

In the second case, the proposed method is simulated and tested on a more complex system containing 68 bus and 16 machines In order to apply this method and run the simulations, the system loads and line impedances are modified to accelerate the simulation process but general system properties remained unchanged. The system is operating at a stable operating point and in this case the simulation is performed using random walk load deviation of 2% of the load amount at load buses at each time step which is considered to be the same as previous case. This random load change disturbs the system as the distributed disturbance and the generators will swing similar to the case 1 illustrated in Fig. 3.3. Running the simulation for a long simulation time, the oscillation frequencies of the generators can be obtained by applying the spectrum analysis on the obtained generator velocity changes. The average DFT is calculated for several simulation time frames similar to the previous case to obtain the spectrum analysis of the generator velocities which is illustrated in Fig. 3.5 along with the distribution of total kinetic energy of system across the spectrum. The kinetic Energy spectrum shows the important and high energy low frequency inter-area oscillations which are mostly below 1 Hz. Unlike the previous case, it is not easy to find the inter-area frequencies based on the kinetic energy spectrum. Referring to generator velocities spectrum, some higher frequency oscillations in the range of 1−1.5 Hz can be observed which is present for most of the generators and regardless of the frequency and lower energy might be considered as interarea modes. Therefore for more accurate assumption about the range of inter-area frequencies more analysis is required on the velocity signals.

42

3 Identification of Power System Dynamics 0.9

0.8

Gen 1 Gen 2 Gen 3 Gen 4 Gen 5 Gen 6 Gen 7 Gen 8 Gen 9 Gen 10 Gen 11 Gen 12 Gen 13 Gen 14 Gen 15 Gen 16

0.7

Magnitude (pu)

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

Frequency (Hz) 0.8

0.7

Kinetic Energy (pu)

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.5

1

1.5

Frequency (Hz)

Fig. 3.5 Spectrum analysis of the generator velocity changes; and distribution of kinetic energy of whole system across the spectrum in 16 machine system

Considering the inter-area modes to be in the range of below 1 Hz, the velocity signals are low pass filtered to exclude all the higher frequencies. The correlation coefficient is then calculated for every pair of these filtered signals to find the highly correlated signal which is given in Fig. 3.6. Based on the calculated value of ri j in the table above, it can be inferred that generators 1–9 are highly and positively correlated in the mentioned frequency range which means they can be considered as a group. Generators 11–13 also have a high correlation and can be considered as another group of generators. The correlation

3.2 Aggregation of Power System Areas

43

1 0.9 0.8

Coherence

0.7 0.6 0.5

1-2 1-4 1-8 1-10 1-12 4-13 4-10 10-13

0.4 0.3 0.2 0.1 0

0.5

0

1

1.5

1

1.5

Frequency (Hz) 180

1-2 1-4 1-8 1-10 1-12 4-13 4-10 10-13

160

Coherency Angle

140 120 100 80 60 40 20 0 0

0.5

Frequency (Hz)

Fig. 3.6 Coherence and angle of coherency between selected pairs of generators in test system 2

coefficients related to machines 14–16 are not significantly high and these generators can not be grouped with any of other machines. The only remaining machine is generator 10 which has a high correlation with both of the defined generator groups but as it has a higher correlation with its neighbouring machine 11 and can be considered as a part of second group. This classification is based on the low pass filtered signals for the frequencies below 1 Hz while by changing the frequency cut off band for several cases from 1 to 1.5 Hz the general grouping did not change. For a more accurate grouping especially for the generators that have high correlation with generators in two groups, the coherence function and the angle of cross

44

3 Identification of Power System Dynamics

spectral density between pairs of generators are calculated. The generators belong to the same area are highly and positively coherent in the range of inter-area frequency band. Therefore, the disputed generators can be checked for the inter-area oscillation frequency of the two groups so these generators will be assigned to the group of generators which are highly and positively coherent with them for that frequency. The results of coherence function and the angle of cross spectral density between selected pairs of generators are illustrated in Fig. 3.6. The result of the coherency analysis provides useful information about the system grouping and inter-area oscillation frequency range. It can be inferred that generators which are assumed to be in the same group based on the previous results, are not highly and positively coherent for the frequencies more than 1 Hz. Therefore, no big group of generators can be found to be coherent in the frequencies over 1 Hz which means those frequencies are not in the range of inter-area modes. Furthermore, the results provide more accurate results for the boundary generators which were difficult to be assigned to a group such as generator number 10 in this case. Based on the results shown in Fig. 3.6, machines 1−9 and 11−13 are oscillating against each other with the frequency around 0.62 Hz and in that frequency generator 10 has higher coherency with the first group while having higher coherency in other frequencies in the range of inter-area modes with the second group. Therefore, machine 10 is highly coherent with both of the groups and depending on the oscillation mode it can be assigned to one of the two groups. In Fig. 3.7 the result of system grouping for generators and load buses is illustrated. The proposed area detection process, similar to other methods, finds the generator grouping and area borders for a specified load and generation level. To provide a more general approach, the system steady state load and generation is randomly changed for several cases and in each case the area definition process is followed. The results showed that, generators and buses which had very high correlation still remain together regardless of random steady state load and generation change of up to 30% within the system, but the generators and buses which are not highly correlated to any area such as generator 10 may swap from one area to another area depending on the load change.

3.2.4 Summary of Aggregation Process Based on simulation of a system with random walk load perturbations, the low frequency coherence makes it possible to divide generators into groups which remain aggregated during disturbances as shown in Fig. 3.8. The intermachine modes at higher frequencies are ignored. From this process an identification of the equivalent machines can be performed as discussed in [13]. In some cases where the impedances between areas are similar to those between areas, the process of forming equivalent machines needs to be refined as discussed in Sect. 3.3.

3.3 Aggregation for Machines Without Clear Delineation of Areas

45

Fig. 3.7 Generator grouping and defined areas for 16 machine test system

3.3 Aggregation for Machines Without Clear Delineation of Areas In the case for Fig. 1 when the lines within the areas have impedance 0.1 pu and the line impedance between areas is 0.7 pu the aggregation results in a power flow between the aggregated machine of each area the power flow differs from conventional equations because of an α term [14, 15]. Pi j = E i E j sin(αi j δi j )/ X i j

(3.21)

for this case the αi j can become as low as 0.75 corresponding to a peak stability angle of 1.33 * π/2 (Fig. 3.9). As the impedance between the areas becomes commensurate with the impedance within the areas, then the power flow is no longer well described by the simple sine equation of the angle between the areas. One approximation is to lump the inertia to a single equivalent generator in each area but to retain the generator busses as voltage support points. Consider the two machine phasor diagram with unlimited voltage support in the middle of the line (Fig. 3.10). In this case the power flow reaches its maximum when δ1 = δ2 = 90°. The middle bus voltage Vm is propping the voltage up by reactive injection and the

46

3 Identification of Power System Dynamics

Fig. 3.8 Grouping of generators into equivalent machines

3.3 Aggregation for Machines Without Clear Delineation of Areas

47

Fig. 3.9 The longitudinal three area system

Vm

Fig. 3.10 Phasor diagram of 2 machine system with mid line voltage support

I2.

I1.

E

δ1

δ2

V

maximum angle between E and V is 180°. When the middle voltage prop is behind a higher impedance the reactive injection reduces. Thus the magnitude of the backbone voltage Vm decreases and the maximum power flow reduces. For the system in Fig. 1 with uniformly spaced perfect voltage propping there are 11 line segments that can each reach 90° giving a total of 990° between the ends. In a time domain simulation where the generator and connection impedance is 10 times the backbone line segment impedance, the time domain plot shows angles between the ends of the system reaching nearly 500° (Fig. 3.11). The issue is how to reduce the system to set of 3 machines to represent the lowest 2 modes of oscillation but to have a line model such that the peak angle phenomena is similar to the original system. This cannot be achieve using only 3 standard generators and further voltage support is required. In the simplest uniform case it is assumed that the voltages of the prop busses are spread linearly in angle between the aggregate generators.

3.3.1 Mathematics of Aggregation Where Modes Are Spread In [9, 16] groups of synchronous machines are aggregated to one equivalent machine in each area. When there is not a clean separation with a high line impedance between areas it has proven useful to still aggregate system dynamics but to solve for the power flow with voltage sources at some intermediate busses. The simplest form is to have the angle spread of the interarea mode giving the pattern of bus angles and

48

3 Identification of Power System Dynamics Angle response of 13 machine longitudinal system 300

Voltage Phase Angle at Bus Terminal (deg)

Fig. 3.11 Simulation of 13 machine longitudinal system (initial conditions of low frequency mode)

200 100 0 -100 -200 -300 -400 -500

0.5

1

1.5

2

2.5

Time (sec)

voltage sources but to have only the one differential equation for the area angle. If the bus angles vary linearly between areas then the voltage sources can be determined and then the network can be reduced based on that set of sources. For example when the angles associated with the eigenvector for the low frequency oscillation is normalized such that the centre angles are +1, −1, W = [1 d1 d2 … dn − 1] then for the case where the angles of the areas are δi , δ j the angles of the areas are

δi − δ j · W/2 + δi + δ j /2. These additional buses at the real generator locations are operating only as voltage props and at each step of simulation can be reduced using standard bus reduction to the buses of the equivalent area machines. The power flow to and from these equivalent machines can be found from this reduced equivalent system. So using a set of dynamic equations for the aggregated machines Ji δ¨i = Pi −



Pi j

In a simple case where W = [1 0.33 −0.33 −1] where the line impedance between the areas is 0.3 pu and where 2 props are equally spaced and connected with an impedance of 0.2 (Fig. 3.12). Here the equivalent impedance from i to j is 0.52 pu, from ‘i’ to ‘a’ is 0.42 and from ‘i’ to ‘b’ is 1.05. These impedances are found from elements of the reduced admittance matrix eliminating the busses on the main line between the areas. Then the angle difference of δi to prop ‘a’ is 0.66 of δi j and for ‘b’ it is 0.33 of δi j so the power flow from ‘i’ consists of three terms       Pi j = sin δi j /xi j + sin 0.66δi j /xia + sin 0.33δij /xib The second term has a peak at 140° while the last peaks at 270°. Together Fig. 3.13

3.3 Aggregation for Machines Without Clear Delineation of Areas

49

Fig. 3.12 Sample 2 area problem with 2 voltage props

Fig. 3.13 Power transfer with voltage props

5 P1 P2 P3 Pt Pd

4 3 2 1 0 0

20

40

60

80

100

120

140

Angle deg

shows a power flow with a peak beyond 90°. The three terms of Pij are labelled P1, P2, P3 in fig X, while the total is Pt and without props Pd. Note that for small angles there is no difference caused by the props, this becomes visible for large angles.

3.3.2 Example of Aggregation with Poorly Separated Areas In Fig. 3.11 the results of as transient simulation of a 13 machine system are shown. This system has a set of identical generators with a connection impedance 10 times the backbone line impedance between connections. To focus on the low frequency modes an initial velocity pattern corresponding to the lowest frequency mode is used as the transient. As expected the angle between the ends of the system open well beyond the 90° of single machine systems of naive aggregated systems. If Fig. 3.14 the 13 machine system is aggregated into 3 equivalent machines. The difference here is that the remaining generators are replaced by zero inertia ideal voltage sources behind an impedance. When there are a set of reactive sources along a reactive impedance line the angle of these sources vary close to linearly between the angles of the aggregate machine at either end by inclusion of these voltage props.

50 Fig. 3.14 Three machine equivalent including voltage props

3 Identification of Power System Dynamics Angles deg 300 2 7 12

200 100 0 -100 -200 -300 -400

0.5

1

1.5

2

2.5

Because these voltage props are connected by relatively low impedance they are quite successful in supporting the backbone line voltages for large angle differences. Figure 3.15 shows these voltages nearing their limits during the transient. The control design for the low frequency mode control is thus developed on the 3 machine equivalent using the maximal rate of reduction of kinetic energy approach developed in [17–19]. In the case for this chapter a HVDC link between two dimensional mesh of generators is also considered. Fig. 3.15 Voltage magnitude on backbone, three machine equivalent

Volts 1

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

0.95

0.9

0.85 0

0.5

1

1.5

2

2.5

3.3 Aggregation for Machines Without Clear Delineation of Areas

51

3.3.3 Uniform 2D Mesh In power systems such as east coast of USA or in Europe the system consists of a mesh of very many generators. As seen in the FNET transient recorded in [20] the system can be approximated by a roughly uniform travel velocity of the travelling waves giving a characteristic such as a drumskin. The difference is that power systems have a nonlinear connection between the elements. This system is related to the issue of damping oscillations of space structures using piezo elements because there is no air to provide the simple damping in space. As discussed in [21, 22] there are many different control designs but they all end up based on developing a lumped model and then a control is designed for the finite order reduced model. Following that approach this chapter aims to develop a strategy of developing a coarsely lumped model of power systems and designing the control system based on this nonlinear model.

3.3.4 Process of Aggregation From coherency in low frequency or from eigenanalysis, form the areas and place the total inertia at the inertia weighted center of the area with impedance determined by the inertia weighted impedances. Now form the reduced admittance matrix retaining all former generator busses. From a low stress power flow or from modes determine the pattern of angles on these retained generator busses is set by the eigenvectors of the original generator busses for a range of power flows. The equivalent impedance and the α factor can thus be obtained for the link. Thus the geographic connection between areas will be mostly preserved and the location and angles of voltage prop points determined in terms of the eigenvectors of the prop points. This can be simplified for the control design using the α factor. The aggregate generator angles will then have large disturbance dynamics similar to the original system.

3.4 HVDC Control Based on Kinetic Energies Consider a HVDC link linking area ‘i’ and ‘j’ in a 2D system such as in Fig. 3.16. The acceleration of area ‘I’ based on the value of the HVDC link power flow ‘Uij’ is given by Hi δ¨i = P gi − Pli − Ui j  −

E i E k sin(αik δik )imag(Y r edik )

(3.22)

k f or all other ar eas and local pr ops

The derivative of the kinetic energy of area ‘i’ with respect to Uij is found from

52

3 Identification of Power System Dynamics

Fig. 3.16 2D mesh of generation

d 21 H i δ˙i2 dt

= Hi δ˙i δ¨i

(3.23)

¨

Hi δi And from ∂∂U = −1 in (3.23) the total kinetic energy derivative with respect to ij the transfer power is

∂

  2 d 21 Hi δ˙i2 +H j δ˙ j dt

∂Ui j

= −H i δ˙i − H j δ˙ j

(3.24)

This implies that the control of the DC link flow should have a term proportional to the weighted sum of area velocities. In the normal case the local nominal frequency is taken as reference.

3.4.1 Control of 2D Aggregated Systems If the focus of the control design is low frequency modes then the measurements should reflect the angles and velocities of the set of aggregated machines. Following [23] the state estimate of the reduced system can be found from multiple measurements. Here a simple inertia weighted average of the velocities across the area is used which then reduced the measurement to emphasize the low frequency modes. The aggregation of the full system is given in the uniform case by having a load flow based on the uniform spread of voltage props between the mesh of areas. The example considers the regular mesh network of 20 * 20 generators reduced to a 4 * 4 set of aggregate generators. To give an example of control, a DC link is connected from the center of aggregate area in row 1 column 2 to row 4 column 2.

3.4 HVDC Control Based on Kinetic Energies

53

This 2D example considers the system in an initial condition with velocities to create a disturbance at the lowest frequency with components mainly in the X direction but with some disturbance in the Y direction. The link is controlled based on the estimate of the angle of a 4 * 4 group of machines but is applied to the full order system with the HVDC link connected to a specific generator. The velocity of the uniform aggregated group is estimated by the average of the 16 machines in the group. The HVDC link is able to provide good damping in the X direction but is unable to influence the disturbance in the Y direction. There is no intermachine oscillations within an area created by this control because it is based on the velocity average (Fig. 3.17). The angles taken on a slice of system along the direction of the DC link (X direction) in Fig. 3.18 show a good damping compared to the uncontrolled case. Angle plot deg: time = 6 sec

100

50

0

-50

-100 5 10

20 15

15

10 20

5

Fig. 3.17 Plot of angles in uniform 2D mesh of generators with HVDC link active angles coa

3 2 1 0 -1 -2 -3

0

1

2

3

4

5

6

Time

Fig. 3.18 Plot of all angles with slow convergence across Y axis and good control of X

54 Fig. 3.19 Control action of HVDC link

3 Identification of Power System Dynamics HVDC action 20 15 10 5 0 -5 -10 -15 -20 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Time (sec)

Note that the Y axis modes are uncontrolled by this link action because the DC link cannot change the power flow in the Y direction Fig. 3.19.

3.4.2 Non Uniform Impedance Set for Linear Connection of Generators Here a case of non uniform impedances between generators is considered with backbone impedances zb = j * [0.1 0.2 0.1 0.1 0.3 0.15 0.12 0.2 0.1 0.12 0.15 0.1], and connection impedances zc = j * [0.02 0.02 0.025 0.015 0.03 0.01 0.015 0.02 0.022 0.018 0.018 0.02 0.02]*3 for the strong connection case. The connection impedances are comparatively low so there is strong voltage support and an α factor of 0.6 and a peak transfer of P = 3.6 corresponding to X = 0.28 pu as in Fig. 11. For area 1–2 the peak is at 102° with power transfer of 3.01 corresponding to an α factor of 0.88 and X = 0.33. In the weakly connected case the connection impedances are increased by 3.3 giving a much weaker α factor of 0.75 and equivalent impedance of 0.42. This shows that the strength of connection of the generators can substantially re-inforce the backbone voltage and reduce the effective impedance between areas (Figs. 3.20, 3.21, 3.22 and 3.23). Note that the fault at bus 10 occurring at t = 15 in the full model case shows a fuller range of frequencies while the aggregate can only represent fewer frequencies.

3.4 HVDC Control Based on Kinetic Energies Fig. 3.20 Power between area 2 and 3 as function of effective power transfer (strong connect)

55 Power angle 2-3

4 3.5 3 2.5 2 1.5 1 0.5 0 0

Fig. 3.21 Dynamics of the 13 machine system-step loading

20

40

60

80

100

120

140

160

all angles

3

2

1

0

-1

-2

-3

-4 0

5

10

15

20

25

30

35

3.4.3 Control Case The control design for the aggregated system is based on the estimate velocities of the areas. In general the interarea angles are expected to oscillate at a low frequency while the local oscillations are higher frequency and more intrinsically well damped. For this case the fault duration is such that the area 3 separates from areas 1 and 2 and that there is a later separation for these areas (Fig. 3.24). Now consider the case of a modulated load in bus 12 of area 3. The aggregation design indicates that the peak steady angle between areas 2 and 3 is 150°. Based on the control design in [24] when a single link is at risk then only the kinetic energy of generators either side of that link is relevant. In this case then the influence of the controller in area 3 is only evaluated by the inertia weighted velocity in area 3, for

56 Fig. 3.22 Dynamics of reduced equivalent system step loading

3 Identification of Power System Dynamics angle

3

1 2 3

2

1

0

-1

-2

-3 0

5

10

15

20

25

30

35

time

Fig. 3.23 Dynamics of reduced equivalent system for fault at t = 15

angle

2

1 2 3

1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0

5

10

15

20

25

30

35

time

first swing control the maximum stability is achieved by keeping the control at full until the angle has swung past the risk area. In Fig. 3.25 the control action is responding to overall area velocity but addressed both interarea oscillations as well as local (Figs. 3.26 and 3.27). Modulating loads in areas 2 and 3 (Fig. 3.28).

3.4 HVDC Control Based on Kinetic Energies

57

Fig. 3.24 Angles of machines for 0.6 s fault on bus 10

Fig. 3.25 Control action taken for load modulation at bus 12

modulated load

1

0.5

0

-0.5

-1

-1.5

-2 0

5

10

15

For controls in areas 2 and 3 and fault on 5 between areas 1 and 2, the same control laws apply but because the fault does not stress the link then only pure delta dot control is needed (Figs. 3.29 and 3.30). If the controller is only in area 2, 2–3 separation cannot be stopped but area 12 can be kept together (Fig. 3.31).

58 Fig. 3.26 Angle difference between areas 2 and 3

3 Identification of Power System Dynamics Diff area 12 and 23 (deg)

200

Diff 12 Diff 23

180 160 140 120 100 80 60 40 20 0

5

10

15

Fig. 3.27 Angles after applying control based on area velocity

3.4.4 DC Link to Island If there is a DC link to an island with 2 machines then maximizing the derivative of kinetic energy results in a control law u = −α(H i δ˙i − H j δ˙ j ) where velocities are measured with respect to the relevant center of area of the two areas. When this is applied with the DC link between area 3 and one of the two island machines the frequency in the island (Fig. 3.32).

3.4 HVDC Control Based on Kinetic Energies

59

modulated load

0.5

Area2 Area3 Separate

0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

5

10

15

Fig. 3.28 Modulating loads

Diff area 12 and 23 (deg)

130

modulated load

0.6 Diff 12 Diff 23

Area2 Area3

0.4

120

0.2

110 0

100 -0.2

90 -0.4

80

-0.6

-0.8

70 0

5

10

15

0

5

10

15

Fig. 3.29 Control for different fault location, diff angles and modulating load

The angle differences are small but the frequency excursions in the island should be no larger than in the main group.

60

3 Identification of Power System Dynamics

Fig. 3.30 Control for different fault location-all angles

Fig. 3.31 Control with limited controllers

3.5 Load Estimation When an estimator of the dynamics of the system is sought the value of the loads is typically not fully available. One path is to estimate the load components from the angles and velocity estimates of the areas. To the system matrix three new states need to be adjoined satisfying L˙ i = wi As discussed in [25] the customer load changes are in the short term random so the load is modelled as a random walk. The load estimates are added to the equations

3.5 Load Estimation

61

Fig. 3.32 Control using DC link

in (2). For small peturbations the system model can be formed and the Kalman gain determined. Using the equations for the power flow between areas based on the alpha factor with the correction factor based on the Kalmanidentification gain times the measurement error. If the measurements are taken as the angle and velocity area averages the full state can be estimated. As seen in the figure the angles and velocities can be well estimated (Fig. 3.33). During the fault the equivalent loads are poorly estimated but the estimates recover after the fault. In addition the loads can be also estimated but with higher error because there are no direct measurements. The estimated net load in area 3 is 3.

3.6 Estimation of Criticality of a Disturbance For a link given in area 1 and voltages assumed close to 1p.u. the peak steady angle becomes αi j δi j = π/2 Considering only the 2–3 link the total energy [26] can be summarized as V =

1  2 H δ˙ + E V [cos(αδ) − cos(αδs )]/x − Pm (δ − δs ) 2

where Pm is the load in the area and δs is the stable equilibrium point. For a steady state flow into area 3 of 3.6 at an angle of 100° even with area 1 energy it doesn’t balance (Fig. 3.34).

62

3 Identification of Power System Dynamics State estimates and area averages

3

2

1

0

-1

-2

-3 0

5

10

15

Velocity estimates and area averages

1

loads estimator

4 3

0.5 2 1

0 0 -1

-0.5

-2 -3

-1

-4

-1.5

-5

0

5

10

15

0

5

10

15

Fig. 3.33 Estimation of areas Fig. 3.34 Energy during transient

Energy

10

KE PE1 PE2 tot

8 6 4 2 0 -2 -4 -6 -8 0

5

10

15

3.6 Estimation of Criticality of a Disturbance Fig. 3.35 Equal area criterion

63

4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Let us use the equal area criterion to assess the link between areas 2–3 in the immediate post fault period. The equilibrium point is δs = 1.6 and a Pm = 3. For the power angle curve between areas 2 and 3 with the alpha factor the curve follows (3.1). In the example for the case of unit voltages the deceleration area can be found as the area starting from δs = 2.25 to δm where δm = 3.63 (Fig. 3.35). δm Ar ea = δs

sin(α · δ23 ) − Pm x23



Thus the area of deceleration is 0.56 so for any velocity which gives a Kinetic energy exceeding the deceleration area there is a high risk of separation unless the control is vigorous. In the example the kinetic energy at the start is 0.711 but there is control action and the system fails to separate.

3.6.1 Summary of Model Reduction with Imprecise Definition of Areas Using the eigenvectors to obtain the effective low frequency power angle between groups of machines, the correction factor and equivalent impedances can be determined. This can be found for both 1D and 2D system under uniform or arbitrary impedance cases. Forming non-linear state estimates from groups of machines, the low order model can yield successful low frequency control design for very high order power systems.

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3 Identification of Power System Dynamics

3.7 Introduction to Identification from Ambient Noise The inherent low noise of PMU measurements means that small variations in a power system can reliably indicate aspects of system performance. This factor combined with the continual customer load variation and its randomness properties permits the identification of system dynamics. The same variation combined with an understanding of the key parameters of load dynamics opens the possibilities of on-line load dynamic identification. Even though the normal variations in voltages and frequencies are small, this chapter summarizes how the key properties of PMU signals open a new field of power system identification. Results are given for both synthetic data and real data from the power system. Phasor measurement units were introduced for protection purposes. Given an accurate timing signal the phasor value of a 3 phase measurement could be evaluated. Intrinsic to the processing was a large number of samples being reduced to a single data point. The consequence of this is that the measurement noise of the PMU signal is very low. There can still be offset errors caused by the CT or VT ratios but the high frequency measurement noise is not an issue with PMU data. If you are using a 10 bit A/D converter the expected quantization error is approximately one part in 1000. If the internal PMU sampling and processing was 200 times the mains frequency and the noise on each sample was independent the variance would be reduced by approximately the square root of the number of samples. If the measurement was stationary over the cycle this gives variance reduction of approximately 14 which is almost 4 extra bits of precision in the measurement. This means that very small variations in power systems are able to be reliably processed from PMU data. This aspect became apparent in the early days of PMU analysis [27–29] and led to the ability of identification of the system oscillation modes. In the short time scale of a few seconds, it is almost impossible to predict which loads will turn ON or OFF. An aggregated load measurement with hundreds of customers shows almost perfectly random changes in the level of the load. Thus the change in the aggregate loads can be treated as white noise at least out to a few hertz. The longer term trends of morning load increases are predictable but at a longer time scale. Thus the changes in load power can be modelled as white noise, and the changes in system angles or voltages or power flow can be seen as the output of a system driven by white noise. The other component of identification treated here is that the continual small perturbations of system voltage and frequency can be used to identify the load dynamics. The normal variations are usually much lower than 1% but the low measurement noise of PMU data enables accurate modelling of the load response.

3.7 Introduction to Identification from Ambient Noise

65

3.7.1 Identification from White Noise The spectra of PMU signals obtained from a number of sites support the concept that the noise properties of the loads can be represented as the integral of white noise. From [30] one of the early measurements in 1998 in Australia, the angle differences in power systems showed a resonance at known system frequencies as seen in Fig. 3.36. The electromechanical resonances rise from a background with a background that increased at low frequencies. This rise in the energy of the background is repeated in the US data from [31] seen in Fig. 3.37. This change of the background angle response 50

40

30

20 Brisbane - Bcomb 10 Cairns - Bcomb 0 0

0.5

1

1.5

2

2.5

Hz

Fig. 3.36 Averaged spectra of angles between Queensland nodes

Fig. 3.37 Ambient autospectra for BPA power interchanges, tests of 09/04/97

3

66

3 Identification of Power System Dynamics

Fig. 3.38 Log plot of angle spectra of Qld data

with frequency would show a 1/f characteristic if the load could be represented as the integral of white noise. The data from Fig. 1 is shown in a loglog plot in Fig. 3.37 shows the straight line characteristic with the expected resonant peaks confirming that the changes in load can be modelled as white noise (Fig. 3.38). Using this model of the load changes, identification of power systems dynamics driven by white noise can follow a number of paths. 1. Frequency domain fit to the spectrum 2. Prony or extended Prony fit to the time domain data 3. Time domain weighted Fit to autocorrelation. The next sub-sections give the background to each of these approaches in turn.

3.7.2 Frequency Domain Fit to the Spectrum For white noise w(s) driving a system with transfer function G(s) the transfer function relations are Y (s) = G(s) · w(s)

(3.25)

For white noise w(s) driving a system with transfer function G(s) the transfer function relations are Y (s)∗T Y (s) = w(s)∗T G(s)∗T G(s)w(s) = G(s)∗T G(s)w(s)∗T w(s) Y (s)∗T Y (s) = G(s)∗T G(s){R + v(s)}

(3.26)

3.7 Introduction to Identification from Ambient Noise Fig. 3.39 Autocorrelation spectrum: data length = 10000 s

67

60 50 40 30 20 10 0

0.5

1

1.5

2

2.5

3

Freq Hz The average value of w(s)∗T w(s) is a constant R as a function of frequency, and the random component v(s) has a constant variance as a function of frequency. Thus the spectrum of the autocorrelation will have a mean value determined by the transfer function [32]. The variability of any finite length measurement will be scaled by the same transfer function as the average. The variability of the spectrum can be reduced by averaging across close frequencies but the opportunity is limited when the data length is low. If the data length approached infinity then the averaging can reduce the variation of the spectra towards zero as seen the 10 sets of data in Figs. 3.39 and 3.40. The issue then is what are the methods in time or frequency domain to process the autocorrelation signal knowing the noise properties of the signal. In this section the frequency domain approach is examined starting from the algorithm invfreqs in matlab. This has been corrected for the use of discrete signals and extended for multiple measurements with common poles. The process takes the spectrum of multiple auto and cross correlation signals and performs and approximate gradient search over the parameters of a finite order model to get a weighted fit in freq domain. Since the noise level is roughly scaled by the magnitude of the response the weighting of the fit by the reciprocal of the measured signal magnitude will match the assumptions of a least squares fit. For random noise taken over a finite time the average of the spectrum is flat over all frequencies and the variance of the spectrum is also flat. When such an input signal is applied to a linear system, the variance of a peak is proportional to the height of a peak thus the damping uncertainty is highest for poorly damped modes. The low variance of the 10000 s based spectra in Fig. 3.39 contrasts with the high variance at the peak of the 100 s based spectra in Fig. 3.40. If the autocorrelation signal is scaled by e−t/τ before computing the spectra the damping is changed predictably but the variance of the estimate is also reduced. This

68

3 Identification of Power System Dynamics

Fig. 3.40 Autocorrelation spectrum with data length = 100 s (10 sets)

0.8

0.6

0.4

0.2

0

0.5

1

1.5

2

2.5

3

Freq Hz

Fig. 3.41 Specta including an exponential weight on autocorrelation

4.5 4 3.5 3 2.5 2 1.5 1

0.5

1

1.5

2

2.5

Freq Hz

exponential weighting can then be used to correct the estimated poles by a shift in the REAL direction of 1/τ (Fig. 3.41).

3.7.3 Prony or Extended Prony Fit to the Time Domain Data One alternate approach is to use least squares approaches to fit an AR model to the time domain sequence. Since the input noise is not known, direct fitting of an ARMA model is not possible. To account for the actual plant zeros a higher order model may be used as in the theory of [33] as implemented in [34]. yk+1 = a1 yk + a2 yk−1 + a3 yk−2 + a4 yk−3 + wk

(3.27)

3.7 Introduction to Identification from Ambient Noise

69

Using the 4th order AR model the residual error in the fit is not white as seen in Fig. 3.42 there is still a spike of coherence associated with both resonant frequencies. Extending the model order to at least 19 as in [33, 34] produces an estimate with a residue close to white. The variation of the quality of fit as a function of the model order is seen in Fig. 3.43. Fig. 3.42 Spectra of the 4th order Prony model fit to time domain data

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1

2

3

4

frequency in Hz Fig. 3.43 Ratio of error or Prony fit to weighted autocorrelation fit as function of Prony order

Prony fit error /Autocorrelation fit error 7 1Hz pole 1.5Hz pole

6

5

4

3

2

1

0

5

10

15

Prony Length

20

25

70

3 Identification of Power System Dynamics

3.7.4 Autocorrelation Analysis For white noise input and for a data length approaching infinite the autocorrelation of the output approaches the impulse response of the system [32] as seen in one example of Fig. 3.44. Instead of seeing a simple exponential decay of two sinusoids in Fig. 3.45, this correlation set for finite data length shows a background noise level with oscillations near the system frequencies. Direct fitting of an impulse response to this result would indicate a much lower damping than the actual system. The initial portion of data is more able to be trusted because it is much higher than the noise level. In this context the signal to noise ratio of the autocorrelation signal will decrease exponentially at the decay rate of the impulse response which in this case is near 2 s. Given a reasonable knowledge of the damping level then the optimal weighting of the time domain fit can be as a simple least squares fit that is exponentially weighted by the damping envelope. This same idea can be extended into frequency domain fitting of the autocorrelation signal. A search in the space of the system parameters can be conducted to find the best fit to the spectra from the autocorrelation. If the autocorrelation signal is multiplied by the exponential weighting signal before transfer into the frequency domain then the noise properties improve. The estimate show a higher damping but can be corrected by 1/τ knowing the exponential envelope parameter τ . One of the remaining problems of frequency domain fitting is where a low damped signal is close in frequency domain to a poorly damped signal. Fig. 3.44 Autocorrelation of system output for long data sequence (10000 s)

auto1 15

10

5

0

-5 0

5

10

Time (sec)

15

20

3.7 Introduction to Identification from Ambient Noise Fig. 3.45 Autocorrelation of system output for short data sequence (100 s)

71 auto1

0.2 0.15 0.1 0.05 0 -0.05 -0.1 0

5

10

15

20

Time (sec)

3.7.5 Pole Fitting Results In time domain analysis, the simple Prony approach can be used to find the dynamics coefficients of the system in response to disturbances small enough to give a linear response but large enough to be significantly above the normal variations in measurements. When the background noise level is high or when the sample rate is high compared with the oscillation modes, simple Prony analysis is notorious for having poor estimation properties. One approach is for extended Prony where a higher order model is fitted such that the model length becomes a reasonable fraction of the impulse response length [33, 34]. The roots of the high order model include good estimation of the roots of the original system. To illustrate the estimation properties consider the system with the following parameters. The system frequencies are 1, 1.5 Hz with damping terms 0.15, 0.08 respectively. The base case autocorrelation is based on a length of 100 s data sampled at 10 Hz. The decay envelope is assumed to be τ = 3. In Fig. 11 the data length is set at 10,000 samples while Fig. 12 uses 1000 samples. The pole estimates shown are from time domain extended Prony fitting [33, 34], from weighted least squares fitting to the autocorrelation sequence, and from spectral fitting to the exponentially weighted autocorrelation [29, 35]. Figures 3.46 and 3.47 show that long data length reduces the variance in the estimates. When τ = 3000 in Fig. 3.46 the estimation in both the autocorrelation time domain and frequency domain fit is essentially disabled. Note now that the frequency domain fit has high variance but low bias, there is an increased bias of fit to time domain autocorrelation fitting (Fig. 3.48).

72

3 Identification of Power System Dynamics

0.9 0.85

o Autocorr

0.8

* Prony

IMAG

0.75

x Freq

0.7

v Armax

0.65

^ N4SID

0.6

+Actual

0.55 0.5 0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

REAL

Fig. 3.46 Discrete frequency domain pole estimates (10,000 samples)

0.9

o Autocorr * Prony

0.85 0.8

x Freq

IMAG

0.75

v Armax

0.7

^ N4SID

0.65

+ Actual

0.6 0.55 0.5 0.45 0.5

0.55

0.6

0.65

0.7

0.75

0.8

REAL

Fig. 3.47 Discrete frequency domain pole estimates (1000 samples, τ = 3)

3.7.6 Effect of Damping Term The autocorrelation time and frequency domain fit is influenced by the parameter τ . Figure 3.49 shows the effect of changes in τ on the variance of the two poles estimates from the actual. Fitting in the frequency domain has a high variance, autocorrelation fitting shows a bias in the pole estimates. For short data lengths there is a consistent benefit in time domain fitting to the autocorrelation but as the data length increases in Fig. 3.51 there is not a consistent

3.7 Introduction to Identification from Ambient Noise

73

0.9

o Autocorr

0.85

* Prony 0.8

x Freq

IMAG

0.75

v Armax

0.7

^ N4SID + Actual

0.65 0.6 0.55 0.5 0.45 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

REAL

Fig. 3.48 Discrete frequency domain pole estimates (1000 samples, τ = 3000) Average rms error

Auto Freq Prony ARMAX N4SID

10

-2

10

3

No of samples

10

4

Fig. 3.49 Summary of performance for short term models

benefit for this example. The two curves are for the variance errors in the estimates of the two poles (Fig. 3.50). The quality of the fitting is influenced by the proximity of the set of poles to one another and by the damping ratio. It is shown that when the frequencies moved from 1, 1.5 to 1, 1.25 the quality of all estimations deteriorates, particularly frequency domain fitting (Fig. 3.52). The variance ratio between the extended Prony and the weighted time domain autocorrelation fit variance ratio is now 1.63 2.04 for the high and lower frequency

74

3 Identification of Power System Dynamics

Fig. 3.50 Error in pole location (1000 samples)

Autocorrelation Time and Freq fitting Time Freq

0.06 0.05 0.04 0.03 0.02 0.01

1

2

10

10

Exp damping Fig. 3.51 Error in pole location with data length

Autocorrelation Time and Freq fitting 0.03 Time Freq

0.025 0.02 0.015 0.01 0.005 0 2 10

Fig. 3.52 Discrete frequency domain pole estimates (1000 samples, f2 = 1.25)

3

10

o Autocorr

0.8

* Prony

0.75

x Freq

0.7 IMAG

4

10 Data length

+Actual

0.65 0.6 0.55 0.5 0.45 0.6

0.65

0.7

0.75 REAL

0.8

3.7 Introduction to Identification from Ambient Noise

75

poles respectively. In Fig. 3.53 when the damping of both poles is reduced by 20% the variance reduces. In Fig. 3.54 when only the higher frequency pole damping is reduced then the variance of its estimate is also reduced (Fig. 3.55). Overall the estimation of oscillation modes is aided by use of the exponential weighting for both time and frequency domain fitting to finite autocorrelation data as well as the low noise properties of PMU data. In [23] the frequency domain fitting was applied to real measurements from the Eastern Australia Grid (Fig. 3.54), showing the effectiveness of the variance reduction using the exponential weighting. Fig. 3.53 Discrete frequency domain pole estimates (damping reduction = 20%)

o Autocorr

0.9

* Prony

0.8

x Freq

IMAG

+Actual

0.7 0.6 0.5 0.5

0.6

0.7

0.8

0.9

REAL

Fig. 3.54 Discrete frequency domain pole estimates (damping reduction of 1.5 Hz pole = 20%)

0.9

o Autocorr * Prony

0.8

IMAG

x Freq 0.7

+Actual

0.6 0.5

0.5

0.6

0.7 REAL

0.8

0.9

76

3 Identification of Power System Dynamics Original magnitude data

0.7

y1 y2 y3

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

Frequency (Rad/s)

Fig. 3.55 Spectrum found from measured cross correlated angle data [23]

3.8 Identifying Forced Oscillations Identifying oscillation modes in power systems has made good use of the properties of ambient variations of customer loads. A basic assumption is that the changes in load are not significantly correlated with past values in the frequency band of interest (0.1–1 Hz). There can be small correlations of loads such as periodic switching of refrigerator loads but typically the individual correlations are too small to bias the mode estimates. Generally there is a load trend of load increasing in the morning but this trend is very small in the frequency band of interest. Figure 3.56 shows the correlation of changes in load for a set of residential loads 30

1

0.7 20 0.4 10

0.1

10 Fig. 3.56 Correlation of load changes

20

30

-0.2

3.8 Identifying Forced Oscillations

77

over 30 time steps. There is a small component of correlation but very small in comparison to the load self correlation. This property of randomness of load changes continues up to the city level. For large industrial loads there can be cases of significant correlation such as the repeated load in a rolling mill. There have been cases where another case emerges where an error in the control of generator can give rise to a significant power flow and this is often represented as a forced oscillation. To identify the source of the oscillation can be as simple as observing which generators are associated with the largest power variation. In general a forced oscillation can be caused by a periodic load which can trigger a large response oscillation between generators. Power system modes are a form of low damped resonance and the resulting oscillations may not be well mapped to the source of the disturbance. As shown in the early part of this chapter the acceleration in area ‘i’ is related to the load in area ‘i’.    sin δi − δ j ω0 Ji δ¨i = Pi − L i − Xi j j For small changes in loads the length ‘n’ vector of machine angles ‘a’ and the vector of velocities ‘v’ this can be linearised in the form ˙    a 0 I a 0 = + [u] v AD v B where ‘u’ is the length ‘n’ vector of load changes. In discrete time this equation becomes     Q M a 0 a = + [u]k N P B v k v k+1 As described in Sect. 4.3 a Kalman filter can be formed from the available measurements. The problem is that the kalman process assumes that the input is white noise. Given measurements of the angles and velocities can be solved with one delay for the load variations in the areas as u k = B −1 {vk+1 − N ak − Pvk } Normally the energy of the load variation ‘u’ is directly related to the size of load and time of day as well as uncorrelated with past values. Where there is a forced oscillation a strong autocorrelation of the load will be seen in the forcing area (Fig. 3.57).

78 Fig. 3.57 Vector of forced responses

3 Identification of Power System Dynamics vector of reff

0.04

7 5

0.03 9

0.02 0.01

30 5 7 9 2 1 4 6 8

0

1

3

2 10

-0.01 -0.02

4

-0.03

8 6

-0.04 -0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

3.8.1 Forced Oscillations When the loads or connected generators create a forced oscillation it is often in the form of a particular frequency of disturbance. In contrast the oscillations that are excited by the random load disturbances may have a large response but the phase varies in time. This is a major distinction which enables the separation of random response from the forced response. If the Kalman estimate is first taken of the angles of all areas as in Chap. 4 this provides a very low noise representation of the angles of an area. The velocity can also be extracted either from the Kalman estimate or from differencing of the angle estimates giving velocities as in Fig. 3.58. From the fft of these area velocities over ten or more seconds, the random phase components Fig. 3.58 Angle response with random signals in all area and a sine forcing signal in area 3

angles

0.6 1 2 3 4 5 6 7 8 9 10

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 0

5

10

15

20

25

30

35

40

3.8 Identifying Forced Oscillations

79

will tend to average towards zero while the forced components will be reinforced over time as in Fig. 3.59. The system in this example consists of 10 machines in straight line separated by equal impedances with identical inertias as illustrated in Fig. 3.58. This system with impedances of X = 0.2 J = 2 has resonances at Hz while the forcing signal is at 0.5 Hz so the system response from the small forcing signal will be large at 0.5 Hz as seen in Fig. 3.60. The linearized model of the areas as discussed in Chap. 4 can be represented as x˙ = Ax + Bu

(3.28)

vel

0.8

1 2 3 4 5 6 7 8 9 10

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

5

10

15

20

25

30

35

40

Fig. 3.59 Velocities of the areas from the state estimate

Freq-velocity 0.3

0.25

0.2

0.15

0.1

0.05

0

0.2

0.4

0.6

Hz

Fig. 3.60 Fast Fourier transform of area velocities

0.8

1

1.2

80

3 Identification of Power System Dynamics

When it is suspected that this forcing signal could come from a particular area, it is good to examine the system response from each of the areas for a forcing signal at 0.5 Hz from each area. Taking the Laplace transform of (5.1) gives the relation (s I − A)x = Bu Which yields x = (s I − A)−1 Bu In a system with ‘n’ areas there are ‘n’ load signals and at the test signal frequency of s = j2π · 0.5 the vector of response for the forcing function in each area can be found and shown as a plot where the angle of the largest response is aligned with the zero angle (Figs. 3.61 and 3.62). Plotting this for each area Fig. 3.60 shows a decaying response as the further removed from the exciting signal. For this case the pattern of response for excitation in area 3 with the next strongest responses in areas 2, 3, 4 and 5. When there is a substantial random component then there will be some response measured for finite length fft’s so the fft of the velocities will show some portion of the pattern not perfectly aligned with the expected pattern. To identify the area with the forcing signal the magnitude of the response can be a reasonable guide but in the general case it is good to match the response vector (Figs. 3.63 and 3.64). In cases where the forcing signal is small, a longer length signal will need to be examined using the fft to accentuate the forced response. Fig. 3.61 Vectors showing the response to disturbing signal in area 3

6

vector of response

10 -3

5 4

2

1

7 0

9

10

3

8

-2 6 -4 4 2 -6 -0.01

-0.005

0

0.005

0.01

0.015

0.02

3.9 Application Fig. 3.62 Mode of angle responses for forcing signal in each area in turn

81 response for different inputs

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

-0.1 1

Fig. 3.63 Vector of mode responses

6

2

3

4

5

6

7

8

9

10

vector of response

10 -3

5 4

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-2 6 -4 4 2 -6 -0.01

-0.005

0

0.005

0.01

0.015

0.02

3.9 Application The ability to form a dynamic model of a power system from on-line operation needs many seconds of operation to form the required identification. This chapter showed that good models can be formed while in steady state which predict the system response well for many disturbances. As long as the major portions of the system remain intact the identification will be valid. Issues can arise when there are major splits in the network during the fault. When the separation is between the aggregate machines then provided the split is known models of the new sections of the network can be formed. In the case where major sections are lost that cannot be represented by the original equivalent machines there is a new issue.

82 Fig. 3.64 Vector of identified response

3 Identification of Power System Dynamics vector of reff

0.04

7

0.03 0.02

5

9

0.01 1

0 2

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-0.01 -0.02

4 8

-0.03 -0.04 -0.06

3

6

-0.04

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0

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0.06

0.08

In [36] for a single machine change of operating point, a linear system model could be re-identified during a half cycle of an oscillation. The question is whether to rely on identification to form the system model and state estimate or to keep track of breaker status to reform dynamic models. The tools of this chapter show how identification can be formed for groups of synchronous machines. Chapter 6 shows how inverters can be incorporated into system dynamics once the local machine and inverter interactions are well damped and able to be ignored for interarea oscillations. The remaining questions are for inverters and load which do impinge on machine dynamics. For characterization of groups of inverters Chap. 8 gives some insights. For loads there is an expectation that the tools of on-line identification can incorporate the load damping provided by motors.

References 1. G. Ledwich, E. Palmer, Modal estimates from normal operation of power systems, in 2000 IEEE Power Engineering Society Winter Meeting Conference (2000), pp. 1527–1531 2. P. Billingsley, Probability and Measure (Wiley, New York, Chichester, Brisbane, Toronto, Singapore, 1995) 3. G.J. Miao, Signal Processing in Digital Communications (Artech House, Norwood, 2007). 4. M.S. Roden, Analog and Digital Communication Systems, 4th edn. (Prentice Hall, Upper Saddle River, NJ, 1996). 5. B.D.O. Anderson, Optimal Filtering (Dover Publications, Newburyport, 2012). 6. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Identification and estimation of equivalent area parameters using synchronised phasor measurements. IET Generat. Trans. Distrib. 8, 697–704 (2014) 7. A. Vahidnia, L. Meegahapola, G. Ledwich, R. Memisevic, Application of wide-area controls in Australian power system, in 2017 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), 26–29 Sept. 2017 (Piscataway, NJ, USA, 2017), p. 6

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8. R.J. Frowd, J.C. Giri, R. Podmore, Transient stability and long-term dynamics unified. IEEE Power Eng. Rev. PER 2, 40–41 (1982) 9. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Wide-area control through aggregation of power systems. IET Generat. Trans. Distribut. 9, 1292–1300 (2015) 10. M. Pai, Energy Function Analysis for Power System Stability (Springer, 1989) 11. P. Kundur, Power System Stability and Control (McGraw-Hill, New York, 1994). 12. J. Wenyun, J. Qi, K. Sun, Simulation and analysis of cascading failures on an NPCC power system test bed, in 2015 IEEE Power & Energy Society General Meeting (2015), pp. 1–5. 13. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Identification and estimation of equivalent area parameters using synchronised phasor measurements. IET Generat. Trans. Distribut. 8, 697–704 (2014) 14. Y. Xue, Y. Zhang, Z. Gao, P. Rousseaux, L. Wehenkel, M. Pavella, et al., Dynamic extended equal area criterion-part 2. Enlbeclding fast valving and automatic voltage regulation, in Proceedings of Joint International Power Conference Athens Power Technology (1993), pp. 896–900 15. A. Vahidnia, G. Ledwich, Y. Mishra, Correction factors for dynamic state estimation of aggregated generators, in 2015 IEEE Power & Energy Society General Meeting, 26–30 July 2015 (Piscataway, NJ, USA, 2015), pp. 1−5 16. A. Vahidnia, G. Ledwich, Y. Mishra, Correction factors for dynamic state estimation of aggregated generators, in 2015 IEEE Power & Energy Society General Meeting (2015), pp. 1–5 17. J. Fernandez-Vargas, G. Ledwich, Variable structure control for power systems stabilization. Int. J. Electri. Power Energy Syst. 32, 101–107 (2010) 18. E. Palmer, Multi-Mode Damping of Power System Oscillations (University of Newcastle, Australia, 1998). 19. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Wide-area control of aggregated power systems, in 2013 Australasian Universities Power Engineering Conference (AUPEC), 29 Sept.-3 Oct. 2013 (Piscataway, NJ, USA, 2013), p. 6 20. Current, Event Replay of 26.02.2008 Florida blackout caused by substation short circuit fault near Miami, ed: Current (2013) 21. C. Zhihuang, C. Li, Dynamic and variable structure control for dual-arm space robot to track desired trajectory in workspace, in 2006 Chinese Control Conference (2006), pp. 1547–1551 22. M. Nohmi, M. Uchiyama, Dynamics and 3-axes control of a spacecraft with flexible structures, in Proceedings of 35th IEEE Conference on Decision and Control, vol. 3 (1996), pp. 2695–2700 23. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Generator coherency and area detection in large power systems. IET Generat. Trans. Distribut. 6, 874–883 (2012) 24. K.S. Chandrashekhar, D.J. Hill, Cutset stability criterion for power systems using a structurepreserving model. Int. J. Electri. Power Energy Syst. 8, 146–157 (1986) 25. F. Moyano, G. Ledwich, Load modelling: induction motor, in Electric Power Systems in Transition, ed. Nova Science Publisher Inc (2010) 26. K.R. Padiyar, Power System Dynamics: Stability & Control (Anshan, 1996) 27. J.H. Chow, Time-Scale Modeling of Dynamic Networks with Applications to Power Systems (Springer, New York, 1982). 28. J.H. Chow, R. Galarza, P. Accari, W.W. Price, Inertial and slow coherency aggregation algorithms for power system dynamic model reduction. IEEE Trans. Power Syst. 10, 680–685 (1995) 29. H. Kim, G. Jang, K. Song, Dynamic reduction of the large-scale power systems using relation factor. IEEE Trans. Power Syst. 19, 1696–1699 (2004) 30. G. Ledwich, E. Palmer, Modal estimates from normal operation of power systems, in IEEE Power Engineering Society Winter Meeting, vol. 2 (2000), pp. 1527–1531 31. J.S. Pires de Souza, A.M. Leite da Silva, An efficient methodology for coherency-based dynamic equivalents [power system analysis]. Generat. Trans. Distribut. IEE Proc. C 139, 371–382 (1992) 32. D. Chaniotis, M.A. Pai, Model reduction in power systems using Krylov subspace methods, in IEEE Power Engineering Society General Meeting, 2005, vol. 2 (2005), p. 1412

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33. N. Ramaswamy, G.C. Verghese, L. Rouco, C. Vialas, C.L. DeMarco, Synchrony, aggregation, and multi-area eigenanalysis. IEEE Trans. Power Syst. 10, 1986–1993 (1995) 34. B. Marinescu, B. Mallem, L. Rouco, Large-scale power system dynamic equivalents based on standard and border synchrony. IEEE Trans. Power Syst. 25, 1873–1882 (2010) 35. H. You, V. Vittal, W. Xiaoming, Slow coherency-based islanding. IEEE Trans. Power Syst. 19, 483–491 (2004) 36. A. Ghosh, G. Ledwich, O.P. Malik, G.S. Hope, Power system stabilizer based on adaptive control techniques. IEEE Trans. Power Apparat. Syst. PAS 103, 1983–1989 (1984)

Chapter 4

Direct Control of Power System Dynamics

4.1 Wide Area Control Approach Large power systems can be visualized as a set of masses connected by nonlinear springs. The masses refer to the inertia of the generators all meant to be rotating at the same frequency. The springs correspond to the power transferred across inductive transmission lines. The power transferred varies as the sine of the angle between the two ends of the line, for small angle differences the sine function can be approximately linear while at large angles the nonlinearity becomes apparent. The key issue is that with large angles then there is excess current while delivering limited power and so protection needs to operate to break the links. To understand the control approach to this power system issue disturbances to the system such as lightning strikes introducing additional energy into the system which could be enough to break the links. There is a kinetic energy term (KE) describing the velocity changes of the generators from their nominal speed there is a spring energy or potential energy (PE) describing the energy stored in the lines. At any point in time the idea is to find a control action to reduce the excess energy due to the disturbance. One option could be for one set of controls being the real power loading at the terminals of the generator corresponding to having a brake on the spring mass system (Fig. 4.1). In the case of a train with nonlinear springs between the carriages the aim is to supress any speed variations or oscillations for the train deviating from its set speed. If carriages 1 and 2 are temporarily above the set speed while 3 and 4 are below the set speed then the disturbance energy is reduced by braking carriages 1 and 2. The immediate effect of the braking is to change the acceleration of that carriage the changes to the spring extension are only as a result of the speed changes. The philosophy is to choose the set of braking commands to give the maximum reduction of the total energy (V) at each instant of time. V = K E + PE © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_4

85

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4 Direct Control of Power System Dynamics

Carriage 1

Carriage 2

Carriage 3

Carriage 4

Fig. 4.1 Dynamics between generators

The rate of reduction in total energy by choice of control (u) ∂V ∂ K E(u) ∂ P E(u) = + ∂u ∂u ∂u Because PE is not a direct function of u then the second term is zero. Consider the generators at angle δi with inertia Ji with input prime mover Pi , load power Li  sin δ −δ with power flow to the other generators j=ad jacentgenerator s (Xii j j ) . Here the Xij represents the reactance of the line. Using a normalizing of inertia by the nominal speed Hi = ω0 Ji .  sin δ −δ 2 K E i = 21 Ji δ˙i and the acceleration satisfies ω0 Ji δ¨i =Pi − L i (u i ) − j (Xii j j ) where the load is approximated as L i = a + bu i . ¨ ∂V = ∂ K∂uE(u) = Ji K δ˙i ∂∂uδi Now examining ∂∂uV for the control u i , ∂u i i i

∂V ∂ δ¨i ∂ K E(u) 1 = = Ji δ˙i = − δ˙i b ∂u i ∂u i ∂u i ω0 So if u i = γ δ˙i this choice will always contribute to the reduction of the system energy. This will translate in the case of the train to braking those carriages above the nominal speed. Braking on carriages 3 and 4 will only increase the total energy so the braking is set to zero. In a real power system this approach can be implemented for any control action that can be immediately affecting acceleration. Thus, a Static Var Compensator in line ij will affect the power transfer on that line. Increasing the shunt reactance on the line will decrease the effective series reactance Xij thus increasing the power transfer in the line. This control will change acceleration of both generators I and j, performing the steps above the recommended control action becomes   u i j = γ δ˙i j sin δi − δ j DC links are another way of affecting power flow and for a link between generators ij the control becomes u DCi j = γ δ˙i j . Thus for any control with a direct effect on acceleration, a set of control actions can be determined to reduce disturbance energy and result in good damping of even nonlinear disturbances.

4.1 Wide Area Control Approach

87

4.1.1 Application Considerations For application to real networks with many generators it is worthwhile to aggregate the sets of generators into groups that oscillate together for major disturbances. In the Australian context a good control of interarea damping is achieved by aggregating the inertias of all generators of the state of Victoria into an equivalent generator and similarly for the other interconnected states. Generally a nonlinear Kalman state estimator would use multiple measurements to update the estimate of the Victorian aggregate generator and the interconnected generators in the other states. This aggregation works well when the line impedance between areas are large compared to the impedances within the state. There is sufficient separation of the group of generators in the northern portion of Queensland to justify two equivalent generators to describe Queensland. When the groups are less well separated then the single equivalent generator becomes less accurate and modifications of the power flow equation are required [1] as discussed in Chap. 3.

4.1.2 Controls Acting Behind a Delay One of the powerful tools to affect damping of generator oscillations is excitation control. Here the magnetic field of the generator is changed which changes the voltage of the generator and hence the power flow. The difference compared with an SVC is that the control operates behind a time constant of about 1 s. So making an instant change of the voltage applied to the excitation, will not result in an instantaneous change of acceleration of the generator. Following the design approach above will not result in good performance. One approach to this is to pretend that an instantaneous change in the voltage or flux of the generator can be made. So the control law for the desired flux can be defined. If a Kalman filter of the system is created, future states based on present measurements can be found. If the future states are known, the desired future fluxes can be computed. The problem then decomposes into finding future fluxes and finding the set of excitation control voltages to achieve the desired set of fluxes. If there is a model of flux y driven by excitation E given as yk+n = c(1)yk+n−1 + c(2)yk+n−2 + · · · c(n)yk + E(k) E(k) can be solved as E(k) = yk+n − c(1)yk+n−1 − c(2)yk+n−2 − · · · c(n)yk Which is a simple form of inverse filtering. This aspect is considered in more detail in Chap. 5.

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4 Direct Control of Power System Dynamics

Fig. 4.2 Magnetization curves of 9 ferromagnetic materials, showing saturation. 1. Sheet steel, 2. Silicon steel, 3. Cast steel, 4. Tungsten steel, 5. Magnet steel, 6. Cast iron, 7. Nickel, 8. Cobalt, 9. Magnetite. Steinmetz, Charles (1917). “Fig. 42” Theory and Calculation of Electric Circuits. McGraw-Hill

If the generator has an output nonlinearity inverse filtering is still achievable. Typically the flux will have a saturation nonlinearity, so knowing the set of fluxes these can be mapped to the equivalent excitation currents (Fig. 4.2). This process of inverse filtering needs to be applied only with data of each particular generator. The desired fluxes are a property of the power system. The same approach can be applied for control of the excitation of wind machines to influence damping of the power system oscillations.

4.2 First Swing Stability of Critical Links In linear oscillatory systems the plot of angle versus speed is a set of circles. For power systems with a sine nonlinearity the trajectories are ellipses with a sharpened point. The curves show the trajectory for the SVC in bang-bang mode U = +1/−1. When the system is in danger of separation then the control must be retained until the danger of separation is passed the PS region shown in Fig. 4.3. This deviates from the conventional wisdom of switching based on velocity [2]. This concept of ensuring focus on critical links using a switching law is adopted in [3] where system wide damping based on velocity is the main focus unless one of the links is in danger of separation. In that case the control is kept on maximum until the separation risk is passed. This first swing control as described in [4] is applied to a simple 2 area system as shown in Fig. 4.4 with risk of separation.

4.2 First Swing Stability of Critical Links

89

Fig. 4.3 Phase plane model of single machine stability with bang-bang control

Fig. 4.4 Simplified model of Qld NSW interconnector

4.2.1 Control Law Based on Remote Measurements This control law is based in the angle across the QNI, δ12 and is given by 4.1 below. ⎧ ⎨ Q max δ12 > η u = Q min δ12 < −η ⎩ ˙ k δ12 othervise

(4.1)

90

4 Direct Control of Power System Dynamics

This control law provided maximum synchronising torque when the QNI angle is greater than a pre-defined threshold η which is a function of the distance to the controlling Unstable Equilibrium Point (UEP). This is based on the work described in [5]. When the QNI angle is less than this threshold a linear control is based on the rate of change of the angle.

4.2.2 Results Both these control laws were tested for a bolted three phase fault occurring in the Hunter Valley area with a North–South power flow of 400 MW. The clearing times obtained for the cases of 1. no control 2. control based on local voltage angle measurements at Sydney West and 3. control based on remote measurements giving an estimate of the QNI angle are shown in Table 4.1. The swing curves for the control using remote measurements case using the critical clearing time of 0.47 s is shown in Fig. 4.5. The actual control signal is shown in Fig. 4.6.

4.3 Nonlinear Dynamic State Estimation of the Reduced System The use of Kalman filters to determine the system state in linear systems is well understood. This section explains how the power system nonlinearities are incorporated into the estimation for power systems. The model for the dynamics of the reduced system based on equivalent machines in each area is of the form [6] when the voltages are assumed to be 1 p.u. Table 4.1 Clearing times (equal NSW and Qld machine inertias assumed)

Case

Critical clearing time-seconds

No control

0.38

Control based on local measurements

0.385

Control based on remote measurements

0.47

4.3 Nonlinear Dynamic State Estimation of the Reduced System

Fig. 4.5 Angle swing under critical link control

Fig. 4.6 QNI angle versus time-control based on remote measurements

91

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4 Direct Control of Power System Dynamics

Ji δ¨i = Pm i −

  n  sin δi j xi j j =1 j = i

(4.2)

When the state x can be expressed in the form T x = [δ1 δ2 · · · δn δ˙1 δ˙2 · · · δ˙n ]

(4.3)

then these equations can be expressed for the reduced system model as xr◦ = f (xr , u)

(4.4)

The linearized equations for this system are of the form ⎞



 n n 1 ⎟ 1 ⎜ δ δ j (4.5) Ji δ¨i = −⎝ + cosδ cosδ ⎠ i i j0 ji0 j =1 j =1 Xi j Xi j j = i j = i ⎛

Which can be expressed as x˙ = Ax + Bu + Gw, y = C x + Du + v

(4.6)

where the noise of the reduced system are the local modes which have been excluded from this model. The term v represents the extent of local modes being visible in the measurements and the term w gives a measure of the local modes affecting system modes during large disturbances. In general the controls are the operations of SVCs or series compensators and these do not often directly affect the measurements thus D = 0 is selected. These noise parameters are considered as uncorrelated zero mean Gaussian white noises which are defined by their covariances as below:   E ww T = Q

(4.7)

  E vv T = R

(4.8)

The Kalman estimator in stationary operation is designed based on the system state and measurement equations considering the process and measurement noises. For this approach, the process covariance Q is a segmented diagonal matrix where the segment related to velocity states is set to unity to excite these states while other segments are zero. Equal disturbance is chosen since there is no prior information of whether the disturbance will perturb any particular machine. The measurement noises from PMUs are independent from each other and have similar characteristics;

4.3 Nonlinear Dynamic State Estimation of the Reduced System

93

therefore, the measurement noise covariance R is set to be a diagonal matrix. The local mode noise from a system disturbance may be known to have a higher probability at certain measurements and in that case the R matrix element is modified. For this work R = σ × I is used with a comparatively low value for σ to represent an equal effect of local modes. 









x˙ = (Ax + Bu) + L(y − y ), y = C x

(4.9)

The Kalman estimator in (3.36) represents the system dynamic model considering the measurement data where x is the estimation of state vector perturbation, and y is the estimation of measurement perturbation. The first part of equation as (Ax + Bu) predicts the evolution of system state and is updated with the correction term (y − y ) through the filter gain L. The Kalman gain L is calculated for the system based on the system and measurement equations and the correspondent noises. 







L = MC T R −1 , M˙ = AM + M A T − MC T R −1 C M + G QG T

(4.10)

where M is the covariance of the state estimation error in steady state and is the only positive solution of the algebraic Riccati equation (4.10) when M˙ = 0. The estimation directly arising from the reduced linear model becomes   x˙ r = Ar x r + Br u + L r y − y , y = Cr x r











(4.11)





where x r is the estimation of reduced system state vector, y is the measurement estimation. As the system converges, L in (4.10) becomes a constant and it is this steady state solution that is used in the nonlinear estimator. The inter-area interactions are observed by phasor measurement units which are installed throughout the system. Visibility of local modes in the data obtained from PMUs is not desirable as it will lead into incorrect identification and estimation of the reduced area-based model. To reduce the effect of local modes on the estimation of equivalent area dynamics, a nonlinear Kalman filter is designed such that area angles and frequencies are estimated based on PMU measurements. The correction term is the same as the linear model but the dynamics are for the nonlinear system. Given the robustness of the Kalman filter the filter properties are retained for a wide range of operating conditions x˙r = Axr + B1 u + B2 Pm + Gw, yr = C xr + v

(4.12)

    x˙r = f x r , u + B.Pm + L y − yr , yr = Cr x r

(4.13)











Equation (4.12) is representing the state of equivalent angles and   estimation frequencies of the reduced areas. Where f x r , u is the nonlinear function representing the identified reduced system model and L k is the gain of the Kalman estimator at operating condition k. 

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4 Direct Control of Power System Dynamics

Fig. 4.7 Identification and estimation of area angles and frequencies

To reduce the effect of particular modes on the measurement, the PMUs are assumed to be installed on non-generator buses. Since the area angles and frequencies represent only the inter-area interactions in the power system, the output of Kalman filter must represent only the inter-area modes while ignoring the local modes. In practice, multiple phasor measurement units are installed in each area to increase the accuracy of measurement. The loss of any PMU may result in inaccurate identification and estimation of area angles and frequencies. To overcome this problem the Kalman gain is calculated for several scenarios considering unavailability and loss of different PMUs so the Kalman estimator will continue to work properly by applying an instantaneous change in the estimator if any PMU is lost. As an improvement, the PMU based estimator of area angles and frequencies can be used to update the estimates in the nonlinear parameter identification described in Chap 3 to reduce the error in the identification process. Therefore, the approach will provide an online estimator which is updated as shown in Fig. 4.7.

4.3.1 Robustness to System Dynamic Loads SVC’s can modulate power flow by adjusting the voltage applied to loads or by shunt modulation of the effective series reactance of lines. Load modulation can be quite effective but when the load has dynamics such as induction motors the question arises whether the control becomes less effective. Several load models are used in dynamic studies and can have a strong influence on the system performance [7, 8]. Here a composite load model is used as in Fig. 4.8 combining large and small motors as well as constant impedance loads. Application of a more complex load model as practiced in WECC [9] could improve the accuracy of the load model; however, identifying the parameters for such a complex model would be impractical for the purpose of this study. The fraction of each load component in the composite load model can vary for every load bus.

4.3 Nonlinear Dynamic State Estimation of the Reduced System

95

Fig. 4.8 Composite load model

Table 4.2 Comparison of first swing stability with composite loads

Critical clearing time (CCT) No wide-area control

With wide-area control

289 ms

342 ms

In order to provide a uniform case, 50% of the load at each load bus is modelled as a constant impedance load while the remaining is equally divided between the two differently sized induction motor loads. The system is simulated with the same scenario as in [5] and the effect of wide-area control with dynamic loads is presented by comparing the Critical Clearing Times (CCTs) in Table 4.2. The CCT is a simple measure of the stability of a power system and presents the longest duration of a fault for which the system remains stable [6]. The system model is as seen in Fig. 4.9. The results in Table 4.2 and Fig. 4.10 show that the wide area control is able to give substantial improvement for first swing and damping stability even for systems with dynamic loads. This does not imply that a more explicit consideration of dynamics could not have yielded greater performance. The tools of Chaps. 5 and 8 offer the potential for designs based on explicit consideration of dynamics.

4.3.2 Transfer Capacity Improvement Although CCT is a widely characterization of first swing stability, Transfer Capacity Improvement (TCI) can show the effectiveness of the control algorithms in increasing the secure transfer capacity of the available transmission lines [10]. The former is an important consideration particularly from the utilities perspective as many lines in the network can be limited below their nominal capacity because of transient stability constraints. Similar to the CCT, the transfer capacity improvement needs to be quantified through a defined set of contingencies. Therefore, in the test system the TCI is expressed as the secure power transfer from area 4 to area 2 following a three phase fault at bus 211 with duration equal to the CCT of the system without widearea control. The simulations are performed by changing the power transfer levels to find the maximum allowable power transfer between the areas and the results are presented in Table 4.3 for the dynamic models in Fig. 4.8. It can be inferred from

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4 Direct Control of Power System Dynamics

Fig. 4.9 The 14-generator simplified Australian system with coherent areas

Table 4.3 that the transfer capacity from area 4 to area 2 is substantially increased from the original 500 MW by applying the wide-area control algorithms on the system SVCs. The results also show that the load model did not significantly affect the transfer capacity improvement performance of the wide-area controllers as the TCI indices were more than 30% for both cases which shows the effectiveness of the proposed wide-area control approach regardless of the load models.

4.3 Nonlinear Dynamic State Estimation of the Reduced System

97

Fig. 4.10 The post-fault trajectory comparison of the generator angles with composite load models a without WAC, b with WAC Table 4.3 Transfer capacity improvement through wide-area control

Maximum transfer capacity (MW)

TCI

No wide-area control 500

–

Wide-area control

680

36%

Wide-area control (composite loads)

650

30%

98

4 Direct Control of Power System Dynamics

4.3.3 Robustness to Time Delays in Controls One of the common concerns for the use of remote measurements in control is the potential for communication delays to cause the controls to perform poorly. The common rule of thumb for control systems is that the delays should be no more than 10% of the period of the shortest oscillation. Here the oscillation frequencies for the interarea modes modelled is no more than 0.5 Hz thus tolerance of delays up to 200 ms is expected. Here a simulation of an unrealistic delay of 40 ms still gives performance indistinguishable from the no delay case. In fact wide area control still delivers benefits for delays up to 200 ms where the CCT remains the same as the no-delay case (Fig. 4.11).

4.3.4 Time Delay Compensation in Kalman Filter One of the concerns about implementing wide area control is the potential delays in communication and the effect of closed loop control. In some studies the robustness of the energy based control to delays has been included [5]. Given that a nonlinear Kalman filter has been formed for the reduced system there is the option to include a compensation for the delays. The concept is that when a measurement arrives with a delay ‘n’ the system model can be used to map the current state estimate to the corresponding ‘n’ step backward state. Now the measurement from that time step can be compared with the best estimate of the state at that time and a correction to the estimate formed. This correction can now be translated forward in time and applied to the current state estimate [11]. The Kalman equations can be separated into prediction and update steps as Prediction Stage M(k + 1) = G T P(k)G + H 1 Rw H 1T ∼

∼

x (k + 1) = G x (k) + H u(k)

(4.14) (4.15)

Update Stage   ∼ ∼ x (k) = x (k) + L(k) y(k) − C x (k)

(4.16)

 −1 P(k) = M(k) − M(k)C T C M(k)C T + Rv C M(k)

(4.17)



where the Kalman gain matrix

4.3 Nonlinear Dynamic State Estimation of the Reduced System

99

Fig. 4.11 Post-fault trajectories of the wide-area controlled system with delay

L(k) = P(k)C T R−1 v , ∼

(4.18)



x (k), x (k) are predicted and updated state vector estimates at time k, and Rw , Rv are the process and measurement noise covariance matrices respectively. As described in [12] the Kalman gain in (4.18) will attain a steady state value after a period of time. The Kalman gain represents a compromise between minimizing the effects of measurement and process noise. In steady state the prediction and update equations are given by (4.19) below neglecting any input. ∼



x (k) = Gx (k − 1)

(4.19)

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4 Direct Control of Power System Dynamics

  ∼ ∼ x (k) = x (k) + L y(k) − C x (k)



(4.20)

These may easily be also be expressed as shown in (4.21) below.   x (k) = Gx (k − 1) + L y(k) − C Gx (k − 1)











→ x (k) = [I − LC]Gx (k − 1) + L y(k)

(4.21)

Note that the operation of the standard Kalman filter assumes that all the elements of the output vector as available at time k. Where there may exist communication delays, especially with large interconnected power system this will not be a valid assumption. Hence there is a need to examine compensation strategies.

4.3.5 Proposed Non-linear Delay Compensation Algorithm Consider the two area system presented in [13] where area two measurements have a constant delay d. Because of the delay the standard Kalman filter does not apply though as reported in [11, 13]. In [13] a linear Kalman filter was used unlike [11, 14] which used a non-linear update. The delay compensation algorithm presented in [11] consisted of three steps, starting at time k − 1 looking at estimating the Kalman update at time k. Step 1: Back project from time k − 1 to time k − d − 1 to get another estimate of the Kalman estimate at time k − d − 1 as per (4.22) (2)

(1)

x (k − d − 1) = G −d x (k − 1) 



(4.22)

Step 2: Perform a standard Kalman update to get an updated estimate for (2) x (k − d) as per (10)



(2)

(2)





x (k − d) = [I − LC]Gx (k − d − 1) + L y(k)

(4.23)

(1)



Step 3: Project forward to x (k) using undelayed measurements as per (4.2.2). This is modelled using a Kalman gain matrix L1 which has rows of zeros in positions corresponding to delayed measurements. (2)





(2)

x (k − d + i) = [I − L 1 C]Gx (k − d + i − 1) + L 1 y(k − d + i)

(4.24)

where 1 ≤ i ≤ d. The algorithm in this work replaces step 1 by a non-linear back projection and step 3 by a non-linear forward projection. In this the angle and velocity state estimates are integrated backwards or forwards in time in time using the modified Euler method [15, p. 277].

4.3 Nonlinear Dynamic State Estimation of the Reduced System

101

In particular step 1 becomes backwards integration of the swing equation given by (4.25) T  x˙ = f (x) = f 1 f 2 f 1 (x) = x 2   f 2 (x) = M −1 p − C i j S − Di j C − Dx 2   p = diag Pmi − E i2 G ii M = diag{Mi } Ci j = ET B E Di j = E T G E D = diag{Di }   S(i, j) = sin δi − δ j = sin δi j   C(i, j) = cos δi − δ j = cos δi j

(4.25)

where E is a vector of internal voltages of aggregated machine groups M i is the inertia of the ith machine group B is the imaginary part of the reduced bus admittance matrix G is the real part of the reduced bus admittance matrix Gii is the conductance to ground at the ith machine bus in the reduced network Di is the damping factor for the ith machine group Pmi is the mechanical power for the ith machine group. ∼ Step 3 becomes a forward integration of the swing equation to find x at each time step from the time of the updated delayed estimate with the update at that time instant being computed in accordance with (4.24).

4.3.6 Implementation and Testing of Algorithm To test the algorithm the same test cases are used as were presented in [11] so as to compare compensation algorithm performance. The algorithm is tested on a five bus system, comprising two aggregated machines at buses 1 and 2 with an SVC at bus 3 in close proximity to a load at bus 5, see Fig. 4.12. This system is stabilized through modulation of the load at bus 5 through operation of SVC in accordance with 4.1. It is assumed that the angle and angular velocity measurements from bus 2 are delayed. To test the effect of these delays the response of the system to a bolted three phase fault at bus 4 is considered with 1. Delays being ignored 2. the compensation algorithm described in [11] and expressed by (4.22)–(4.25)

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Fig. 4.12 Two machine test system

3. the new compensation with the first step replaced by a non-linear back projection. As was done in [11] comparison of the responses for a clearing time of 0.1 s are conducted with the critical delays being determined in each case. Then the critical clearing times are compared for a delay of 100 ms.

4.3.7 Results The responses of the compensated and uncompensated cases for a delay of 0.5 s is shown in Fig. 4.13. A comparison of critical clearing times (CCT) is presented in Table 4.4. The critical delay which is the maximum delay allowable without the system being either first swing unstable or showing negative damping is 0.74 s for the no compensation case. The system response for a delay of 0.74 s is shown in Fig. 4.14a, b.

Fig. 4.13 Angle and velocity response—delay of 0.5 s

Table 4.4 Comparison of critical clearing times

CCT—no delay compensation

CCT—linear delay compensation

CCT—Non-linear delay compensation

0.52

0.49

0.56

4.3 Nonlinear Dynamic State Estimation of the Reduced System

103

Fig. 4.14 Angle and velocity response—delay of 0.74 s

With linear compensation method the critical delay was 1.64 s. With the nonlinear compensation the critical delay is in excess of 7.0 s, well in excess of any reasonably expected delay. A comparison of the critical clearing times for a delay of 100 ms revealed that the critical clearing time was 0.49 s for the linearly compensated case and 0.52 s for the uncompensated case. With the non-linear compensated case the CCT is 0.56 s. The responses of the compensated and uncompensated cases for a clearing time of 0.49 s are shown in Fig. 4.15.

Fig. 4.15 Angle and velocity response—delay of 0.1 s and clearing time of 0.49 s

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4.3.8 Discussion The studies undertaken on this simple system in [4] showed that significant delays could be tolerated with linear compensation. For the more complete IEEE test system this tolerance was also demonstrated. With non-linear compensation the amount of delay which may be tolerated expands well beyond any practical delay. Hence non-linear compensation offers significant benefit. The worsening of critical clearing time observed in [11] is overcome with nonlinear compensation with the critical clearing time extending to 0.56 s. The success of the nonlinear compensation method is due to the fact that close to the critical clearing time the transmission system non-linearities are dominant and so the linear compensation scheme can be expected to not perform well. As was shown in [11] it performed less well than not having any compensation at all. This result indicates that a non-linear approach to delay compensation is worthwhile as even a small improvement in critical clearing time may yield a very large improvement in power transfer capability.

4.3.9 Summary A non-linear delay compensation has been presented which allows much greater than expected communication delays to be catered for in applying the control algorithm presented in [5, 14]. It also provides for improvements in critical clearing time which will translate to big improvements in power transfer capability. Work is currently underway to apply this algorithm to much larger systems with multiple time delays. The controllers for power system dynamics can be by HVDC links adjusting the flow between areas, by Static Var Compensators modulating voltages of loads, by batteries directly controlling power injection. These shunt type loads make for simple velocity based control design (Fig. 4.16).

Fig. 4.16 HVDC link controlling flow

4.4 Application

105

4.4 Application For the control of power systems using PMU’s the approach taken has been to align the control action with the control which can help create a fast reduction of kinetic energy. The issue that is still open is to find the control gain for this action. For a control u = −K δ˙ what is the right value of K? In Linear Quadratic Regulators (LQR) it is known that a gain between 0.5 and 2 times nominal will still be stable. The task requires the strongest control that will still be guaranteed stable no matter the size of the disturbance on the system. The basic assumption is that multiple controls have a small influence on convergence and that the convergence time does not become less than one cycle of oscillation. Trouble will emerge when there are multiple controllers with very high gains contending to supress oscillations, where the sum of good controllers can become counterproductive. Another issue is that the control design has been based on ignoring local modes of intermachine oscillations or dynamics of inverters. When there are large numbers of PMU’s then the control can be based on the Kalman model of what the interarea modes are doing. With low number of PMU’s the process is are somewhat vulnerable to the presence of local modes which can upset the design.

References 1. A. Vahidnia, G. Ledwich, Y. Mishra, Correction factors for dynamic state estimation of aggregated generators, in 2015 IEEE Power & Energy Society General Meeting, 26–30 July 2015, Piscataway, NJ, USA (2015), pp. 1–5 2. T.W. Chan, G. Ledwich, E.W. Palmer, is velocity feedback always best for machine stability control? Presented at the AUPEC 2002, Melbourne (2002) 3. T. Li, G. Ledwich, Y. Mishra, J.H. Chow, A. Vahidnia, Wave aspect of power system transient stability—Part II: Control implications. IEEE Trans. Power Syst. 32, 2501–2508 (2017) 4. E. Palmer, A. Vahidnia, G. Ledwich, Delay compensation in the wide area control of SVCs for first swing stabilisation and damping, in Power Engineering Conference (AUPEC), 2016 Australasian Universities, Brisbane (2016), pp. 1–5 5. A. Vahidnia, G. Ledwich, E.W. Palmer, Transient stability improvement through wide-area controlled SVCs. IEEE Trans. Power Syst. 31, 3082-9 (2016) 6. P. Kundur, Power System Stability and Control (McGraw-Hill, New York, 1994) 7. Standard load models for power flow and dynamic performance simulation. IEEE Trans. Power Syst. 10, 1302–1313 (1995) 8. J. Milanovi´c, K. Yamashita, S. Martinez, S. Djoki´c, L. Korunovi´c, International industry practice on power system load modelling, in 2014 IEEE PES General Meeting | Conference & Exposition (2014), pp. 1–1 9. D. Kosterev, A. Meklin, J. Undrill, B. Lesieutre, W. Price, D. Chassin, et al., Load modeling in power system studies: WECC progress update, in 2008 IEEE Power and Energy Society General Meeting—Conversion and Delivery of Electrical Energy in the 21st Century (2008), pp. 1–8 10. J.J. Ford, G. Ledwich, Z.Y. Dong, Efficient and robust model predictive control for first swing transient stability of power systems using flexible AC transmission systems devices. IET Gener. Transm. Distrib. 2, 731–742 (2008)

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11. E. Palmer, G. Ledwich, A. Vahidnia, Delay compensation in the wide area control of SVCs for first swing stabilisation and damping, in 2016 Australasian Universities Power Engineering Conference (AUPEC), 25–28 Sept. 2016, Piscataway, NJ, USA (2016), 5 pp 12. B.D.O. Anderson, Optimal Filtering (Dover Publications, Newburyport, 2012) 13. E. Palmer, G. Ledwich, A. Vahidnia, Non-linear compensation of delays in the wide area control of SVCs for first swing stabilisation and damping, in 2018 Australasian Universities Power Engineering Conference (AUPEC), 27–30 Nov. 2018, Piscataway, NJ, USA (2018), 5 pp 14. E. Palmer, A. Vahidnia, G. Ledwich, Wide area control of SVCs for first swing stabilisation and damping in longitudinal systems, in 2014 Australasian Universities Power Engineering Conference (AUPEC), 28 Sept–1 Oct 2014, Piscataway, NJ, USA (2014), 5 pp 15. R.L. Burden, J.D. Faires, Numerical Analysis (Belmont, CA, 2005)

Chapter 5

Indirect Control of Power System Dynamics

5.1 The Need of Wide Area Damping Control: Inverse Filtering One of the main causes of power system instability is low-damped inter-area oscillations. It is important to damp out low-frequency oscillations effectively and to provide necessary precautions in order to avoid major separations of power systems and cascading blackouts. Development of reliable control strategies to keep the system in a stable condition is a way to prevent system blackouts or brownouts. In the current power system, many of the damping controllers use local signals to perform their task which may not be capable of satisfying the overall defined objective. This is because local signals do not provide comprehensive dynamic information on the whole network. In this situation, global signals can be used by wide area damping controllers (WADC) to improve their performance. The wide-area measurement system provide the option of choosing global signals as stabilizing signals to generate control signal. Global signals obtained through PMUs, such as current, voltage, angle and frequency, contain dynamics of the power system much more accurately. When they are applied in WADC, there is the potential to reduce the frequency and severity of catastrophic failures [1]. The damping of inter-area oscillations focuses on oscillations of coherent areas of the network against each other. This can be improved by taking into consideration the dynamics of other areas of the system. Damping controllers using only local signals that are independent of the conditions of other areas may result in unsatisfactory performance and do not provide required observability and controllability. For example, after a severe perturbation, the inadequate local signal controller may cause low-damped inter-area oscillations that separate a part of the system from the rest of the system. Traditional local signal damping controllers are also unable to damp both local oscillations and inter-area oscillations for all operating conditions. All of these circumstances point to the need of controllers that use global signals or wide-area measurements. Many literature reveals that using global signals from remote areas can improve the damping of inter-area oscillations significantly [2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_5

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5.2 Wide Area Damping Controllers The steps of designing an efficient wide-area damping controller is as follows: 1. Identifying dominant modes of the system 2. Appropriate states of the system are employed in the design of the controller that can influence the dominant modes of the system 3. Implement the controller at locations that have the most impact on the inter-area modes of interest [1]. Traditional devices used for WADCs include power system stabilizer (PSS) and Flexible AC Transmission System (FACTS) devices such as Static VAR Compensators (SVCs). Synchronous generator excitation system can be used to damp out inter-area oscillations if an appropriate auxiliary input is provided. It is more costeffective because the desired performance for the power system is attained by using existing equipment in generation units [1].

5.3 Comparison Between SVCs and Excitation Systems WADCs such as SVCs are fast acting actuators with fast dynamics. They have a small time constant which means that the effect of input changes is instantaneous in their output. There is negligible time delay. However, WADCs such as excitation system are slow acting actuators with slow dynamics. They have a large time constant which means there is significant time delay. In this situation, the inverse filtering technique is needed to cancel the time delay. The Kalman filter needs to provide advanced future predictions of the desired output and then the inverse filtering technique can be used to find the required input. There is output nonlinearity in the excitation system due to flux saturation as shown in Fig. 5.1. This nonlinearity can also be cancelled through the inverse filtering technique.

5.3.1 Excitation System Figure 5.2 illustrates a general block diagram of a synchronous machine and excitation control system. It consists of synchronous machine, exciter, terminal voltage transducer, excitation control elements and in some cases PSS [1]. Figure 5.3 shows the fast static exciter model, which receives the power from an auxiliary generator winding or transformer and rectifier. This model is used in this research because of its simple structure. In this model Vt is generator terminal voltage, Vs is the auxiliary input, E f d represents field voltage, K A is AVR gain and Ta is its time constant [1].

5.3 Comparison Between SVCs and Excitation Systems

Fig. 5.1 Magnetization curve (open-circuit characteristic) of a synchronous machine

Fig. 5.2 General block diagram for synchronous machine excitation control system [1]

Fig. 5.3 Static exciter model [1]

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By controlling the flux for excitation systems, the proposed control approach improves damping of electromechanical oscillations. The nonlinear excitation controller uses an inverse filtering technique.

5.4 Inverse Filtering Inverse filtering, or deconvolution, is a technique which obtains the input of the system using the prior knowledge of the desired output. Inverse filtering has two categories. The first category is when the dynamics of the plant are known and is modelled using the time invariant system. The second category is where the plant dynamics are unknown (blind deconvolution) or partially known [3]. The proposed inverse filtering technique in this chapter is a nonlinear controller that requires knowledge of the dynamics of the actuator and can compensate the delay due to slow dynamics of actuators. In this proposed method, a variable of the actuator is selected to be controlled to accomplish the goal of the controller. In the excitation system of the synchronous generator, the flux of the generator is selected to be controlled. Desired values for the flux variations for the next sampling times are calculated based on a criterion that can damp out inter-area oscillations. The changes of the generator flux for the next sampling times are obtained by using the data acquired from PMUs in a nonlinear Kalman filter, which can estimate area equivalent parameters. By updating the nonlinear Kalman filter, future values for the desired variables can be calculated [1].

5.5 Controller While most of the current controllers in the network are based on linear methods which causes their performance to be limited to a number of presumed operating conditions. The proposed controller is a nonlinear controller and preserves its satisfactory performance over a wide range of operating points. The proposed controller is carried out in a centralized wide area control unit which determines the desired flux for generators in each area. The design of the excitation controller maximizes the reduction rate of the kinetic energy from equilibrium point in the power system. Through the inverse filtering method, a supplementary control input is generated for the excitation system of generators in each area to ensure the local control tracks the required changes in the flux of each area [1].

5.5 Controller

111

5.5.1 Controller Design Steps The steps required to achieve the wide-area damping control using inverse filtering technique are summarized.

5.5.1.1

Obtain a Reduced Model and Estimate Equivalent Area Parameters

The reduced model considers coherent generators in same area to provide equivalent area rotor angles, velocities, inertia and other reduced parameters as shown in Fig. 5.4. The reduced model contains purely inter-area modes while local modes are excluded. It is assumed that at least one PMU is available in each area to provide the needed data. A non-linear Kalman estimator or filter is designed in (5.1) to provide state estimation equation of the equivalent area states based on the non-linear dynamic equation of the multi-machine systems.     ˆ u + L k y − yˆ , Yˆ = C xˆ x˙ = f x,



(5.1)

where xˆ is the of reduced system state vector, yˆ is the measurement  estimation  estimation, f x, ˆ u is the non-linear function representing the identified reduced system model, L is the gain of the Kalman estimator at operating condition k and C is the relationship between the estimated measured data and the reduced states of the system. By using this nonlinear Kalman estimator, the angle and velocity of each coherent area can be estimated. Equations (5.2) and (5.3) represent the states of the reduced order power system. More details of the identification of the reduced model is

Fig. 5.4 Representation of a multi-area system by equivalent generators

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5 Indirect Control of Power System Dynamics

presented in [4]. 

δ˙i = ωˆ i − ωs +

l 

L i j (Y P MU j − Yˆ j )

(5.2)

j=1

⎛ 

ω˙ i = ⎝ Pmi −

 n E  E  sin δˆi − δˆ j  q j qi Xi j

j=1

+

l 

⎞   − Di ωˆ i − ωs ⎠/Ji

 L i+n, j Y P MU j − Yˆ j

(5.3)

j=1

where Y P MU is the PMU measurements, l is the number of measurements, J is inertia, n is the number of coherent groups in the system, δ is the rotor angle of the generator, ωs or δ˙ is synchronous speed, ω˙ s is the acceleration of rotor angle, Pm is the mechanical power of the generator, D is the damping ratio and X i j is the reactance between generator area i and j. E q is the q component of internal voltage of generator.

5.5.1.2

Determine Kinetic Energy of the Whole System Algebrically

The kinetic energy of the whole system is expressed in (5.4). 1 Ji ωi2 2 i=1 n

VK E =

(5.4)

where VK E is the kinetic energy associated with the rotors.

5.5.1.3

Obtain the Desired Flux for the Equivalent Generator in Each Area

The derivative of total kinetic energy in the system can be expressed as a function of generator rotor angle, speed and fluxes as in (5.5). Generator flux is the parameter that is selected to be controlled to maximize the reduction rate of the total kinetic energy in the system. V˙ K E =

n  i=1

Ji δ¨i δ˙i =

n  i=1

(Pmi −

  n   E q j E qi sin δi − δ j j=1

Xi j

− Di (ωi − ωs ))ωi (5.5)

5.5 Controller

113

where V˙ K E is the derivative of the kinetic energy associated with the rotors, δ¨ is the acceleration of rotor angle J is inertia, n is the number of coherent groups in the system, δ is the rotor angle of the generator, ωs or δ˙ is synchronous speed, δ¨ is the acceleration of rotor angle, Pm is the mechanical power of the generator, D is the damping ratio and X i j is the reactance between generator area i and j. E q is the target variable and called the q component of internal voltage of generator which is proportional to the field flux linkage [5]. By differentiating the derivative of total kinetic energy with respect to the flux of the generator, the desired value of the flux variations that maximizes the reduction rate of the total kinetic energy is obtained in (5.6). This flux generates the reference for the local controllers installed for each generator. n  E q j     ∂ V˙ K E sin δi − δ j δ˙i − δ˙ = −  ∂ E qi Xi j j=1     n  sin δi − δ j δ˙i − δ˙ j = KT Xi j j=1

 =  E qi

(5.6)

where K T is a user designed system gain to improve the performance of the controller and  shows the difference between desired value of the flux and its steady state value.

5.5.1.4

Choose an Appropriate System Gain KT for the Desired Flux

The sensitivity analysis using different values of K T is performed and its effect on the damping of the inter-area oscillation is observed. The most effective value of the gain depends on the system.

5.5.1.5

Derive an Moving Average (MA) Model for Inverse Filtering

The overall block diagram of a synchronous generator and excitation system is shown in Fig. 5.5. The power system model is a full order generator and nonlinear dynamics such as flux saturation are considered. Us is the supplementary control input which is added to the excitation system. The following inverse filtering procedure in (5.7)–(5.9) leads to the calculation of the supplementary control input from desired generator flux output. Vxi   = E qi  1 + sTdoi Vxi = E f di −

   b  n  E qi Idi − X di − X di a

(5.7) (5.8)

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Fig. 5.5 Dynamics of synchronous generator and excitation system

E f di = −

K Ai u Si 1 + sTai

(5.9)

Figure 5.6 illustrates the overall concept of the inverse filtering. The continuous transfer function between an input and output of a typical plant in (5.10) is discretised into Autoregressive Moving Average (ARMA) model in (5.11) using sampling frequency significantly larger than two times of the inter-area frequency [1]. For the purpose of this nonlinear inverse filtering controller, (5.11) is modified to an Auto-regressive (AR) model as in (5.12), which is invertible. This in turn is easily transformed to a moving average (MA) model as in (5.13), which has the advantage of being stable and is used for inverse filtering. Y (s) u s (s) q 1 + i=1 bi z −i p H (z) = 1 + i=1 ai z −i H (s) =

H (z) =

1+

u s (k) = 1 +

1 p

p  i=1

Fig. 5.6 Inverse filtering for excitation systems

i=1 ci z

i

(5.10)

(5.11) (5.12)



ci z

i

y(k)

(5.13)

5.5 Controller

115

The AR model is obtained using ‘invfreqz’ function in MATLAB. Based on the order of AR model, the number of required future values for flux, n, can be specified.   + . . . + an E qi Vxik = a0 E qi k k+n

E f dik = Vxik +

p   b   E qi Idik + X di − X di k a

u Sk = b0 E f dik + . . . + bn E f dik+n

(5.14) (5.15) (5.16)

where a0 , . . . , an, b0 , . . . , bn, are coefficients which are obtained after carrying out discretization and inverse filtering and sub-script k represents the time step. In this step, the output of the system is already assigned to a specific value. By inversing the path from output to supplementary control input, the controller can accomplish the control objective of alleviating inter-area fluctuations.

5.5.1.6

Update Kalman Filter to Calculate Desired Flux Values for Next Sampling Steps

The non-linear Kalman filter is used to obtain the estimated parameters such as rotor angle and generator speed for each coherent area. By updating the Kalman filter, the required values for the next time steps are achieved.

5.5.1.7

Track Reference Flux with Actual Flux

After applying u s as control input to the exciter, the actual flux is obtained. The actual flux tracks the desired flux (reference flux). The flux tracking is expected to be perfect if control signal saturation is not experienced. This perfect tracking will prove that the imposed control signal has performed its duty as expected. This validation step verifies the effectiveness of the inverse filtering technique [1].

5.6 Example The proposed controller is applied on a Single Machine Infinite Bus test system and it can be demonstrated that the inverse filtering technique to a small disturbance is effective and valid. A t = 1 s, the load at the generator bus is increased by 0.1pu for 10 cycles. Figure 5.7 shows that the supplementary control input forces the system to track the desired flux accurately. Figure 5.8 shows that the controller is effective on damping the oscillations, which results in a lower kinetic energy for the system.

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Fig. 5.7 Flux tracking for small disturbance

Fig. 5.8 Velocity of generator for small disturbance

5.7 Summary Indirect control of power flow is when the control is filtered through a transfer function before affecting the power flow. Excitation system are one example. Some wind power machines have the rotor dynamics in the process while others are connected with all the power flowing through an inverter (Fig. 5.9).

5.8 Applications For what systems does this inverse filtering work There are three characteristics needed: 1. A filtering technique which can reasonably predict the future dynamics 2. Any transfer function which delays the control signal are radial and not engaged with the main system dynamics. 3. The nonlinearities are well known.

5.8 Applications

117

Fig. 5.9 Wind turbines

In a power system the controllers have a weak influence over the intermachine dynamics. The controllers are unable to supress oscillations in the first half cycle of the oscillation. This means that the phase of the required damping signal is reasonably predictable. For excitation systems, the delay in the control of the field is all contained within the power station, the dynamics of the excitation system are not close coupled with the operation of any other machine Thus in seeking to compensate for the transfer function of the exciter to machine flux local signals can be largely relied on. For the power flow between machines the sine nonlinearity is well known and can be incorporated into the Kalman filter to predict the machine dynamics. For the magnetic field it is a known output nonlinearity so if the transfer function to the MMF is known the correction for magnetic saturation can be applied at the output stage of the inverse filtering. Overall when there is a single element of technology slowing the time domain impact of a control then inverse filtering can be a great solution. When the filtering can be spatially diverse, such as control of PV in an area, the inverse may not be so cleanly defined.

References 1. R. Goldoost-Soloot, Wide Area Damping Control Through Inverse Filtering Technique (School of Electrical Engineering and Computer Science, Queensland University of Technology, Doctor of Philosophy, 2016).

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2. J. Ma, T. Wang, Z. Wang, J.S. Thorp, Adaptive damping control of inter-area oscillations based on federated kalman filter using wide area signals. IEEE Trans. Power Syst. 28, 1627–1635 (2013) 3. A. Saberi, A.A. Stoorvogel, P. Sannuti, Inverse filtering and deconvolution. Int. J. Robust Nonlinear Control 11, 131–156 (2001) 4. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Identification and estimation of equivalent area parameters using synchronised phasor measurements. IET Gener. Transm. Distrib. 8, 697–704 (2014) 5. P.W. Sauer, M. Pai, Power System Dynamics and Stability (Prentice Hall, New Jersey, 1998).

Chapter 6

Inverters Operating in Power System in Weak Grids

6.1 Introduction When connecting inverters to a weak grid, one significant concern is the interaction between the phase locked loops of each inverter and thus with system stability. This chapter examines some of the characteristics of using Linear Quadratic Regulators and state estimators for a more robust connection. For the control of inverter in the grid the standard design is based on a tightly controlled current loop to follow a reference. In more general applications the current reference is set by a voltage control requirement [1]. This works fine for assuming the filtering of the inverter is by a single output inductor. For LCL filtering an additional stabilizing term of capacitor voltage/current is added to the basic tight output current loop [2]. These inverters are typically synchronized to be in-phase with the terminal volts. As discussed in [3] there are many papers in the performance of current source inverters in weak grids many of which including [3] aim for frequency domain compensation of inverters but typically for single inverters in a grid. The issue for real systems is that there can be several inverters in close proximity in a weak grid and the change of the injection current due to its phase locking affects the terminal voltage of all other inverters in the phase locking. This interaction rises with the level of current injection and with line impedance. The approach taken here is to consider Linear Quadratic Regulator based designs LQR [4] but synchronizing to a voltage less affected by the changes of injections. A Kalman filter is used based on the injection current to infer the source voltage for the feeder and the inverter is synchronized using that information. One of the aspects of concern is the progress of the Phase Locked Loop under fault conditions. When a synchronous machine is exporting to the grid and a fault occurs close to the terminals of the machine the exported power falls dramatically while the shaft driving power still continues to operate. Even with fast valving there is little that can be done during the fault to keep input power matching the exported power thus the angle of the generator rises rapidly. If the fault lasts too long the Kinetic energy gained during the fault may not be able to be balanced by the potential energy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_6

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6 Inverters Operating in Power System in Weak Grids

of the link. In a single machine infinite bus case the equal area criterion can be used to assess the stability as a function of fault duration {Kundur, 1994 #74}. For a phase locked loop instability it is entirely up to the designer what change of angle occurs during the fault period. If the magnitude of the terminal voltage falls dramatically its angle will convey very little information about the system Thevenin voltage and low gains should be used in the fault condition to avoid this ramp away. In this chapter the update of the PLL angle is presumed suspended during a major fault. There are different level of modelling that can be used to assess stability of inverters in weak grids. Much of common practise is to use full PSCAD simulations to assess performance of PLL stability. This can give precise evaluation of stability but provides little insight to the ways to improve performance. Phasor level models do not give any representation of line dynamics but can give some indication of the nature of oscillations of PLL and interactions with machine dynamics as discussed in Sect. 6.2. One possibility to reduce the sensitivity of PLL to weak grids is to develop a Kalman filter to infer the angle of the Thevenin source and use this as a reference for current injection. Rather than explicitly model the pulse width modulation in the inverter a common practise is to model the control action averages over a switch cycle {Ledwich, 2002 #8718}. The filter lines and inverter are all modelled as linear items and the majority of dynamics are faithful apart from the impact of switch harmonics. The design and performance of the switch averaged design are discussed in Sects. 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9 and 6.10. Evaluation of the interaction with switching is incorporated from Sect. 6.7 using PSCAD simulations. This set of design and evaluation tools shows how changing the reference voltage can enhance stability in weak grids. A key aspect for improvement is to cease to update the PLL during severe faults or to force the tracking to a pure 50 Hz signal in the Kalman filter.

6.2 Simple Model of Ideal Current Source and Quadrature Voltage Locking The analysis tools and approximations for modelling a set of synchronous machines is well known, as is the process for an inverter connected to an infinite bus. This presentation develops a process to analyse a set of inverters with a set of synchronous machines. The main result is that the stability of the Phase Locked Loop for the inverters dominates the interactions. The inverters do not change system modes but offset the operating point. The inverter modes are decoupled from machine modes and one another. The inverter modes are affected by system strength. This process can help determine setting for control design. The analysis approach starts from the single machine model from [5] which uses a Proportional plus integral controller to adjust the terminal voltage in quadrature

6.2 Simple Model of Ideal Current Source and Quadrature Voltage Locking

121

to the PLL. This assumes that the current control of magnitude is fast and the main dynamic is in alignment with voltage. For a simple system of inverters and synchronous machines the model is built based on the source voltages Vs and the inverter currents II . It is assumed here that the inverter injects a constant sinusoidal current in phase with the terminal voltage VI . 

IS II





   Ya Yb VS = · VI Y b Y c

(6.1)

This can be re-expressed as VI = −Y c−1 Y b VS + Y c−1 I I = T 1 · VS + Z 2 · I I   I S = Y a − Y b · Y c−1 Y b VS + Y b · Y c−1 I I = Y 1 · VS + N 2 · I I

(6.2) (6.3)

The effective impedance between the synchronous machines is treated as the –imag(Y1(1, 2)) and for small angles this becomes the acceleration term for the machines with an offset based on the real power from the inverters. This is a linearized small signal model for the generators. The correction to the Phase Locked Loop (PLL) is based on an equation ω I = (K p + K i · s) · V q

(6.4)

where Vq is the terminal voltage component in quadrature to the reference angle. If the inverter current is injected at the PLL angles δ I the quadrature component can be approximated as the sum of magnitude terms in quadrature to the reference offset by the reference angle δ I . V q = r eal(Z 2) · Idiag + T 1 · δ M − δ I

(6.5)

where Idiag is a matrix with the inverter injections on the diagonal. And for an real impedance term Z2 starting from the derivative of (6.5) ·

ω˙ I = K p · V q +K i · V q

(6.6)

·

ω˙ I = K i · T 1 · δ M + K p · T 1 · δ M −K i · δ I − K p · Idiag (Z 2r − Dz)· ·

δ I +K i · (Z 2r − Dz) · Idiag · δ I + K i · (Z 2i − Dz) · Idiag

where Z 2r = real(Z2) andZ2 i = imag(Z2), ·

Given that δ23 = ω˙ I the system equations are given by

(6.7)

122

6 Inverters Operating in Power System in Weak Grids

⎡ · ⎤ ⎡ δ 0 I ⎥ ⎢ M ⎢ · ⎥ ⎢ M D ⎢ δM ⎥ ⎢ ⎥=⎢ ⎢ ⎢ δ ⎥ ⎢ 0 0 ⎣ I ⎦ ⎣ · K im · T 1r K pm · T 1r −K i δI .

0 0

0 0

0

I

⎤





+ rZ2. K im · Idiag (Z 2r − Dz) − Dt K pm Idiag (Z 2r − Dz) − Dt + K dm

⎥ ⎥ ⎥ ⎥ ⎦

⎡

⎡ ⎤ ⎤ ⎡ ⎤ δM 0 0 ⎢ · ⎥ ⎢ ⎥ ⎢ ⎥ ⎢δ ⎥ ⎢I ⎥ ⎢ ⎥ M ⎥ + ⎢ ⎥ · N 2 · I + ⎢ 0 ⎥ K im · Z 2i · I ·⎢ diag I ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ δI ⎦ ⎣ 0 ⎦ ⎣0⎦ · I 0 δI .

(6.8)

where M is the matrix of power flow between the synchronous machines linearize for small angles., and Kim and Kpm are matrices with elements of Ki and Kp on the diagonal. M = [y14/J1 −y14/J1; −y14/J2 y14/J2] and D is the 1/J times the damping term. The operating angle of inverters across a line depend on the current being drawn by all inverters on the line. When all inverters are subject to a large transient such as a nearby fault, the transients can cause severe angle changes for all inverters leading to a loss of synchronism for one or more inverters. The eigenvalues of the inverter controls are somewhat dependant on the magnitude of the injection currents. Where diagonal elements Dt are the sum of the terms in a row of Tr and Dz is a diagonal matrix with terms the sum of rows of Z2r. These equations are linearized for small angles . In [5] the analysis shows the sine nonlinearity of the stability equations for the PLL in the sngle machine case. The transients for this system are like those of a single machine infinite bus system, when the angles are high small disturbances can cause the Phase locked loop to lose synchronism. The main difference between synchronous machine dynamics and PLL dynamics is that the designed can choose to reduce the gains if the terminal volts are very low during the application of a fault. So while there can be a significant overshoot in stabilizing synchronous machines the transient for PLL can be limited so there is less chance of losing stability after a fault. For the system in Fig. 6.1 the system response can be found, in general the common impedances between inverters 7 8 9 is strong The oscillations visible in Fig. 6.2 shows the PLL oscillations on top of

Z2

Z1 B1

B2

Z3 B3

B4

Z7

I4

I5 I6

Z4 Z5

Z6

Fig. 6.1 Inverters in series operating with machines

I7 Z8

I8 Z9

I9

Vs2

6.2 Simple Model of Ideal Current Source and Quadrature Voltage Locking

123

Inverter angle 1 2 3 4 5 6

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 16

22

20

18

24

26

28

30

32

34

Fig. 6.2 Inverter angles

intermachine dynamics. In general when the current is injected in weak systems, the angle of the terminal volts increase and the magnitude decreases. During any system fault the high initial inverter angles and reduced voltages make the synchronization to terminal voltages more problematic. In this example a fault is placed at the terminal of Inverter at bus 9 (Figs. 6.3, 6.4 and 6.5). If an offset term is included as in (6.9) when Vq is computed the angles of the inverters can decrease as seen in Fig. 6.6 and the magnitude of the voltage at the inverter increases as seen in Fig. 6.7. This case has rZ2 = 0. mag Inverter V 1.02

1 2

1

3 4 5

0.98

6

0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 16

18

20

22

Fig. 6.3 Inverter terminal volts magnitude

24

26

28

30

32

34

124

6 Inverters Operating in Power System in Weak Grids wi 1

3

2 3 4

2

5 6

1

0

-1

-2

-3 16

18

22

20

24

26

28

30

32

34

Fig. 6.4 PLL frequency transient for fault at I6

mag Inverter V 1 2 3 4 5 6

1.15 1.1 1.05 1 0.95 0.9 0.85

15

20

25

30

35

Fig. 6.5 Inverter terminal volts magnitude: offset synchronization

V q = T 1 · δ M − δ I + imag(T 2) · Idiag

(6.9)

A key consequence if these changes is that the limiting export power increased by 32% when an offset of the PLL operating angle was applied. One item of note is the eigenvectors for the system in (6.8) is that there is a significant decoupling between the modes of machines and the presence of the inverters. The first 4 states are machine states and show no shift due to inverters. The next 6

6.2 Simple Model of Ideal Current Source and Quadrature Voltage Locking

125

Inverter angle 1 2 3 4 5 6

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 16

18

20

22

24

26

28

30

32

34

Fig. 6.6 Inverter angles: offset synchronization

1 f0

0.9

f1 f2

0.8

f3 f4

0.7

f5 f6

0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

12

14

16

Fig. 6.7 Eigenvectors for the first 6 frequencies of Fig. 6.1

states are PLL angles and the next 6 are PLL velocities. There is some engagement of machine states at the PLL frequencies. Note also that each of the PLL modes shows no coupling to other PLL mode. Thus in the simple case with no line dynamics and a Proportional plus integral controller there is seen to be no strong coupling of inverter modes. Changing the offset of the reference angle can improve the ability to synchronize through major faults. To analyse a more sophisticated reference process the line dynamics and filter

126

6 Inverters Operating in Power System in Weak Grids

dynamics need to be incorporated. The next section takes a full state space model approach.

6.3 Estimator Based Controllers of Inverters in Weak Grids (Switch Averaged) In control theory, full state feedback of every energy storage element to the control of each input is necessary to guarantee stability. When not all states are measured then an observer is required to form the state estimates. Combining a Kalman observer with a Linear Quadratic controller achieves the same performance as without disturbance from the “Certainty Equivalence” property [4]. When controlling inverters connected within a weak grid full system parameter knowledge is not available, the control of other inverters is unknown and the resonant interactions with other filters is unknown. Thus a reduced feedback is the first choice for control of each inverter with no knowledge of the external system. The issue then arises of how to create a state estimator of the rest of the connected system. In this chapter the case of a low impedance at the inverter terminals is consider and uses a Kalman estimator based on a Thevenin equivalent of the system supplying the inverter. The source is represented as a 2 state oscillator at 50 Hz and the estimator feedback is based on the terminal voltage simulated in αβ space. This is a form of unknown input observer adjusted to fit the current task [6]. This section treats each inverter as a 2 phase system of LCL filters with independent control of the two axes with no coupling between. The Kalman filter is able to provide the source voltage as it sees it, learning the phase of the 50 Hz source as the filter converges. The reference current and voltage of the inverter is found from this inferred source voltage. Initially the inverter current is set in phase with the estimated source. The weak grid is represented by a high value of the impedance connecting to the source Vs shown in Fig. 6.8 near ix1. The problem is that the inverters who aim to synchronize with the terminal voltage and change the parameters for synchronization, continually change the angle of the injected current. In addition, any neighbouring inverters will be changing their injection and thus the angle of the terminal voltage of other inverters. Oscillations between inverters are the result of this changing reference. With the Kalman model, the speed of inverter tracking can be adjusted and the synchronization point less susceptible to line impedance. The model of current and voltage interactions in each phase for LQR design is x˙ = Ax + Bu where ⎡

⎤ ⎡ ⎤ 0 −1/L1 0 1/L1 A = ⎣ 1/C f 1 0 −1/C f 1 ⎦ B = ⎣ 0 ⎦ 0 1/L2 0 0

(6.10)

6.3 Estimator Based Controllers of Inverters in Weak Grids (Switch Averaged)

127

Fig. 6.8 Inverter connection model

This model of the Pulse Width Modulated inverter is averaged across the switching cycle and the average of the voltage as u. The aim in LQR control to minimize   is given T   T the performance index J = ∫ x − xr e f Q x − xr e f + u Ru dt. This form of control balances the error in the states with the effort in the control {Anderson, 2014 #8667}. Where the input is the voltage from the inverter. This system is converted to discrete time to reflect the sampled nature of the PWM process. The main design choice then is whether to emphasize the error in the output current by a high value in the 3, 3 element of Q or to focus more on voltage tracking by a high value on the 2, 2 element. Intermediate values in Q give a range of harmonic impedance that can be of value in having some general harmonic absorption characteristic. The full system matrix for one phase for both inverters their filters and the lines can be expressed as ⎡

0 ⎢ ⎢ 1/c f ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 A=⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎣0 0

−1/L1 0 1/L2 0 0 0 0 0 0 0

0 −1/c f −R1/L2 0 0 0 0 0 0 0

0 0 0 0 1/c f 2 0 0 0 0 0

0 0 0 −1/L3 0 1/L4 0 0 0 0

0 0 0 0 −1/c f 2 0 0 0 0 1/C x1

0 0 0 0 0 −R2/L4 −Rs1/L X 1 0 1/C x0 0

0 0 0 0 0 0 0 −Rs2/L X 2 −1/C x0 −1/C x1

0 0 0 0 0 0 −1/L X 1 1/L X 2 −1/Rx1 · C x 0

⎤ 0 ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ −1/L4 ⎥ ⎥ ⎥ 0 ⎥ ⎥ −1/L X 2 ⎥ ⎥ ⎦ 0 −1/Rx2 · C x1

128

6 Inverters Operating in Power System in Weak Grids

⎡

1/L1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 B=⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 1/L3 0 0 0 0 0 0

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ 1/L X 1 ⎥ ⎥ ⎥ 0 ⎥ ⎦ 0 0

(6.11)

If the full system matrix is used for the LQR feedback control design, the closed loop eigenvalues improve a little, if remote current information is made available, the difference is small compared with purely local feedback of the filter states and terminal voltage. The results in [7] show that provided the separate inverters are stable and the local control gains are larger than the interaction terms, global stability can be expected. A similar result in [8] where subsystems may or may not be connected, gives conditions on the size of the interconnection terms for global stabilization. For the system under consideration the reference terms for the inverter are derived from the local state estimator. Stability of the system consists of local feedback and local state estimates where the interaction terms between the PLLs (in the model of the network) grow with the size of the current and the impedance of the lines. As the strength of this interaction between subsystems grows because of the angles between the separate inverters also grows, then loss of guaranteed stability is to be expected. Figure 6.9 shows the eigenvalues of the control system (without estimator dynamics) inside the unit circle for discrete stability. In general the local feedback is less stable than full state feedback. The reference value for current is determined by the required output power. The reference voltage of the filter capacitor can be found from the fundamental of the output current and the terminal voltage given by the state estimator. With these references the LQR solution will give the same gains for both phases. With the estimator for 50 Hz the term ω = 2 * pi * 50. The states for this Kalman Filter system for the inverter and one line segment are the states of the 50 Hz oscillator followed by the αβ states of the inductor current and terminal capacitance. ⎡ ⎤ 0 0 0 ω 0 0 0 0 ⎢ 0 ⎢ −ω ⎥ 0 0 0 0 0 0 ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ 0 0 −1/Ls1 0 ⎢ 0 ⎢ 1/Ls1 0 −R/Ls1 ⎥ A=⎢ ⎥B = ⎢ ⎢ 0 ⎢ 0 1/Ls1 ⎥ 0 0 −R/Ls1 0 −1/Ls1 ⎢ ⎢ ⎥ ⎣ 1/C f ⎣ 0 ⎦ 0 0 1/C f 0 −1/(Rp.C f ) 0 0 1/C f 0 0 0 1/C f 0 −1/(Rp.C f ) ⎡

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6.12) where the two inputs are the currents from the 2 phases of the inverter

6.4 Case 1: Model with Unknown 50 Hz Phase and Angle of Terminal Volts

129

reduced feedback o local, + red, * full 1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 6.9 Eigenvalues of system as the feedback goes from full ‘*’ to partial ‘+’ to fully local ‘o’

6.4 Case 1: Model with Unknown 50 Hz Phase and Angle of Terminal Volts Here the model for the estimator only consists of the top left 2 × 2 of A in (6.12). This ensures the model is of some 50 Hz signal so pure 50 Hz signals in the inverter are being tracked. This ensures the model is of some balanced 50 Hz signal, so pure 50 Hz signals are being tracked in the inverter. The LQR design for the inverter has gains 0.4923 −0.0430 4.8651 on the inverter current, capacitor voltage and export current respectively. As seen in Fig. 6.10 the estimate of terminal voltages yh2 approaches the measured yy2 and the states of the oscillator xh2-1 and xh2-2 converged to a balanced set after some transients. Because this model assumes that the terminal volts are fixed and there is no network impedance, the process is sensitive to the changes of the angle of injected current.

130

6 Inverters Operating in Power System in Weak Grids

1.5

Xh2-1 Xh2-2 yy2 yy2q vs vsq

1

yh2 yh2q

0.5 0 -0.5 -1 0.005

0.01

0.02

0.015

0.025 Time

0.03

0.035

0.04

0.045

Fig. 6.10 Stability with phase lock to terminal volts

6.5 Case 2 Broadcast of Reference Angle One approach to the synchronization process is to broadcast the angle of the local source voltage. This completely removes any concern of interactions between converters. One item to consider however is to ensure the voltages at both ends of the line are well managed and this may require an angle offset on the synchronization with the source. The results in Fig. 6.11 shows a short initial transient exciting none of the high frequency transients. 3 vc1 vc2 vcx1 vcx2 vs

2

1

0

-1

-2

-3 0

0.05

0.1

0.15

0.2 Time

0.25

0.3

Fig. 6.11 Performance of inverters synchronized to a broadcast reference angle

0.35

0.4

6.6 Case 3 Projecting to Vs Using Inverter Current

131

6.6 Case 3 Projecting to Vs Using Inverter Current To get a form of projection from the terminal back to the source more of the network as shown in Fig. 6.8 needs to be considered. Now the inverters in αβ space adopt a phase shift from the inferred references. Now the filter finds the first 2 states in (6.12) as the inferred source and from these the reference current and voltage can be formed. Now the system can be stabilized with a high gain on current tracking 2.7488 −0.0835 32.0479 while the state estimator can converge in 2 cycles. The dynamics are clearly superior to Fig. 6.10. Figure 6.12 shows the convergence but there is still some error in the angle of the source voltage because the effect of the other inverter is not considered in the LQR design or in the state estimation (Fig. 6.13). 3

Xh2-1 Xh2-2 yy2

2

yy2q vs vsq

1

yh2 yh2q

0 -1 -2

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Time

Fig. 6.12 Dynamics when more complete Kalman model is used

Offset ref for i2 2

i2o xh1 vcr

1.5

Vs i2 vinv

1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0

0.005

0.01

0.015

0.02

Fig. 6.13 Iref = 0.4 and aligned with inferred source voltage

0.025

0.03

0.035

132

6 Inverters Operating in Power System in Weak Grids

6.7 Offset the Current Reference with Respect to Inferred Source One of the problems of a current reference aligned with the source voltage is the phasor diagram of the line flow which tends to require large values of the inverter voltage as seen in Fig. 6.14. As seen in Figs. 6.11 and 6.12 the magnitude of the inverter voltage and the capacitor voltage is high when current is synchronized to source voltage. This is for the case of a high current reference value (0.4) and one inverter operating. Now if the angle reference is shifted to 55°, Fig. 6.15 shows a much reduced inverter voltage. Now with both inverters Iref1 = 0.4 Iref2 = 0.25 the results for angle offset from Vs = 0 are in Figs. 6.16 and 6.17. offset = 55°, the required voltage are higher than at lower transfer levels (Figs. 6.18 and 6.19). The use of the offset between the inferred Thevenin voltage and the injected current is seen to give better control of voltage at the injection point. Vs I

IX Vt

Fig. 6.14 Unity power injection to terminal volts compared with source reference Offset ref for i2 i2o xh1 vcr Vs i2 vinv

1.5 1 0.5 0 -0.5 -1 -1.5 0

0.005

0.01

0.015

0.02

Fig. 6.15 Using an offset of 55° for current reference

0.025

0.03

0.035

6.7 Offset the Current Reference with Respect to Inferred Source

133

Offset ref for i2 2

i2o xh1 vcr Vs i2 vinv

1.5 1 0.5 0 -0.5 -1 -1.5 -2 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Fig. 6.16 Higher currents needing higher inverter voltages Offset ref for i4 2.5

i4o xh1 vcr Vs i4 vinv2

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0

0.005

0.01

0.02

0.015

0.025

0.03

0.035

Fig. 6.17 Tracking for second inverter offset = 0 Offset ref for i2 2.5

i2o xh1 vcr

2

Vs i2 vinv

1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0.005

0.01

0.015

Fig. 6.18 Tracking with offset = 55° inverter 1

0.02

0.025

0.03

0.035

134

6 Inverters Operating in Power System in Weak Grids Offset ref for i4

3

i4o xh1 vcr

2

Vs i4 vinv2

1 0 -1 -2 -3 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Fig. 6.19 Tracking with offset = 55° inverter 2

6.8 Kalman Filter Based PLL This model makes a more detailed representation of PWM in an inverter context. The control and estimator equations are designed in αβ then implemented in abc form. Pulse centering is used to retain the full range of line to line voltages. Centering refers to shifting the switch transitions in the a b c voltages until they are centered in the switching window. The differences between the voltages are retained in this model but the limitation of depth of modulation is reduced {Ledwich, 2002 #8718}. This centering uses zero sequence voltages and is found as (max(Va, Vb, Vc) − min(Va, Vb, Vc))/2 and is a correction in going to abc because the zero sequence component does not cause any current flow. This analysis in PSCAD is for a single inverter and Kalman estimator. The implementation of the Kalman filter is discussed in 4.3.7 (4.13)–(4.16). The Thevenin voltage source in αβ form as a 50 Hz oscillator of unknown magnitude and phase. As far as the inverter is concerned it is the only source injecting to the node. The model of the line in αβ becomes ⎤⎡ 0 −ω 0 0 0 0 ⎢ ω ⎥⎢ 0 0 0 0 0 ⎢ ⎥⎢ ⎢ ⎥⎢ 0 −1/Ls 0 ⎢ 1/Ls 0 −R/Ls ⎥⎢ x˙ = ⎢ ⎥⎢ ⎢ 0 1/Ls ⎥⎢ 0 −R/Ls 0 −1/Ls ⎢ ⎥⎢ ⎣ 0 ⎦⎣ 0 1/C x 0 −1/(Rp · Ls) 0 0 0 0 1/C x 0 −1/(Rp · Ls) ⎡

V sα V sβ I sα I sβ V tα V tb

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

6.8 Kalman Filter Based PLL

135

⎡

⎤ 0 0 ⎢ 0 0 ⎥ ⎢ ⎥  ⎢ ⎥ 0 ⎥ I lα ⎢ 0 +⎢ ⎥ ⎢ 0 0 ⎥ I lβ ⎢ ⎥ ⎣ −1/C x 0 ⎦ 0 −1/C x

(6.13)

where Vt refers to the terminal voltage of the inverter Which when state noise w is added to the first two states becomes of the form x˙ = Ax + B.Iαβ + Ew. This system can be expressed in discrete time based on the inverter control rate as xk+1 = F xk + G · Iαβ + Dwk ,

yk = C · xk + H n k

(6.14)

where the terminal voltage outputs of the system are the last two states and n k represent measurement noise. The Kalman state estimate based on parameters QF, RF in (4.13)–(4.16) are the process and measurement noise covariance matrices respectively and the estimate is then found as   xˆk+1 = F xˆk + G.Iαβ + L yk − C xˆk

(6.15)

This will identify how the inverter sees the apparent source. One simple solution is to make the inverter track a current which is a scaled version of the inferred source which are the first two states. This scaling sets the magnitude of the current to be injected.

6.9 LQR/Kalman Control in Switching Inverter For a system in the form xk+1 = Axk + B · Uαβ the performance index can be expressed as J=

n  

    x − xr e f Q x − xr e f + Uαβ RUαβ

(6.16)

i=0

where Q is the penalty on state errors to be balanced against the control effort scaled by R. The Ricattii equation can be solved to give a control of the form   Uαβ = −K x − xr e f

(6.17)

Now these quantities are need to be transformed into abc form for implementation in a real inverter

136

6 Inverters Operating in Power System in Weak Grids

The mapping is exemplified in the case of a reference for the filter capacitor voltage presuming the zero sequence is set to zero. V cαβr e f

  0 −1 ˆ ˆ = V tαβ + jωL · Iαβ = V tαβ + ωL · Iαβ 1 0

(6.18)

 Thus using the Clark  transform ignoring the zero sequence for T 0.816 −0.408 −0.408 0 0.707 −0.707 V abcr e f

V abcr e f

=

  0 −1   ˆ ˆ = T V tαβ + j ωL · T · Iαβ = V tαβ + ωL · T · · T · Iabc 1 0 ⎡ ⎤ 0 −1 1 = T  · Vˆ tαβ + ωL · 0.5774⎣ 1 0 −1 ⎦ Iabc (6.19) −1 1 0 

The parameters for this example are Inverter 1 L1 = 0.4 mH Cf = 20 μF L2 = 10 mH DC bus voltage 600 V Inverter 2 L1 = 0.3 mH Cf = 20 μF L2 = 10 mH Line segment 1 R = 0.2 L = 10 mH Line segment 2 R = 0.2 L = 10 mH Source 415 V line-line When the combined set of inverters with local LQR controls and the reduced size Kalman filter is simulated in PSCAD for a reference current as a scaled version of the Kalman oscillator, the tracking is good in Figs. 6.20, 6.21 and 6.22. In Fig. 6.23 the source voltages are presented in kV in αβ form. The transient covers the rising the DC bus voltage and Fig. 6.24 shows the terminal voltage tracking error. Fig. 6.20 Current tracking ‘a’ phase

40

20

0

-20

-40 Ila1 Ilar1

-60

0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

6.9 LQR/Kalman Control in Switching Inverter Fig. 6.21 Current tracking ‘b’ phase

137

60 Ilb1 Ilbr1

40

20

0

-20

-40

0

0.05

0.1

0.15

0.2

0.25

0.3

0.2

0.25

0.3

Time (s)

Fig. 6.22 Current tracking ‘c’ phase

40

20

0

-20 Ilc1 Ilcr1

-40

0

0.05

0.1

0.15 Time (s)

For Q as diagonal 0.001 0.001 10000 R = 0.0001 the LQR gains for this case are 6.2193 0.2862 165.2226 for states I1 Vc I2 (see Fig. 6.8). For Kalman parameters QK as diagonal 10000 and RK as diagonal 30 the Kalman gains LK are LK=[0.0018 -0.0001; .0001 .0018; 0 0; 0 0; 0.0018 -0.0001; .0001 .00018] (Figs. 6.25, 6.26, 6.27, 6.28 and 6.29).

138

6 Inverters Operating in Power System in Weak Grids

Fig. 6.23 Estimated source voltages

400

200

0

-200 xh1 xh2

-400 0

0.05

0.1

0.15

0.2

0.25

0.3

Time (s)

800 xh1 xh2

600 400 200 0 -200 -400 0

0.05

0.1

0.2

0.15

0.25

0.3

Time (s)

Fig. 6.24 Terminal volts tracking error in kV

⎡

0.0018 ⎢ 0.001 ⎢ ⎢ ⎢ 0 LK = ⎢ ⎢ 0 ⎢ ⎣ 0.0018 0.0001

⎤ −0.0001 0.0018 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ −0.0001 ⎦ 0.0018

When switching in L = 0.03 H at t = 0.25 s, the estimators are set for an impedance of L = 0.01 H and now have L = 0.04 H. There is a transient in the estimators at the time of insertion as seen in Figs. 6.30 and 6.31.

6.9 LQR/Kalman Control in Switching Inverter Fig. 6.25 References for the filter capacitor using equations (8), L = 0.01 H

139

400 VCrefa VCrefb VCrefc

200

0

-200

-400 0

0.05

0.1

0.15

0.2

0.25

0.3

Time (s)

Fig. 6.26 Transient with 4 fold increase in line impedance inverter 1

20 Ial1 Ibe1

10

0

-10

-20

-30 0

Fig. 6.27 Transient with 4 fold increase in both line impedance inverter 2, L = 0.4 H

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

20 Ial2 Ibe2

10

0

-10

-20 0

0.05

0.1

0.15

Time (s)

0.2

0.25

0.3

140 Fig. 6.28 Halving line impedance L = 0.005 H

6 Inverters Operating in Power System in Weak Grids 20 Ial2 Ibe2 10

0

-10

-20

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (s)

Fig. 6.29 Inserting additional impedance

6.9.1 Notes Using PWM signals or switch averaged does not change performance when using LQR/Kalman designs. The capacitors must be line to line in the filter to avoid zero sequence current from the inverter. There is a limit to the exportable current but similar to the usual line impedance considerations.

6.10 Extreme Case of Step Impedance Fig. 6.30 Error in terminal volts in estimator 1

141

800 errVta errVtb

600 400 200 0 -200 -400 -600

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Fig. 6.31 Error in terminal volts in estimator 2

600 errVta2 errVtb2

400

200

0

-200

-400

-600 0

0.1

0.2

0.3

0.4

0.5

Time (s)

6.10 Extreme Case of Step Impedance Here the same step in impedance is used but keep the reference current magnitude constant by correcting for the drop in terminal voltage and hence inferred source signal. This case is extreme since terminal voltage drops to 0.6 p.u in Fig. 6.32. But examining the currents in the first inverter and it is found that the phase of the current stays aligned with the reference in Fig. 6.32, and for the second inverter in Fig. 6.33 (Figs. 6.34, 6.35 and 6.36). With 36° between the source voltage and the inverter capacitor voltages the phasor diagram is significantly stretched.

142

6 Inverters Operating in Power System in Weak Grids

Fig. 6.32 Voltage drop on impedance insertion

600 Va1 Vb1

400

200

0

-200

-400

-600 0

0.2

0.1

0.4

0.3

0.5

Time (s)

20

Ila1 Ilr1

10

0

-10

-20

-30

-40

-50 0

0.1

0.2

0.3

0.4

0.5

Time (s)

Fig. 6.33 Current of first inverter

6.11 Summary of Performance of Inverters The basic problem of many large inverters in weak grids is due to the issue of phase locking. Using a simple phasor model, the interaction between inverters become clearer. This simple analysis indicated that a form of line drop compensation or gain reduction during faults would be helpful. The implementation of this line drop compensation as a state estimator is studied in time domain using LQR with local feedback around each inverter and a simplified Kalman filter estimator for the source Voltage. The process for forming a Kalman filter in discussed in Sect. 4.3.7. This estimator is used to form references for the inverter injection and it is confirmed that better performance is achieved using the inferred source voltage than the terminal voltage. In general the system stability can be improved by lower inverter gains or

6.11 Summary of Performance of Inverters

143

10 Ila2 Ilr2 5

0

-5

-10 0

0.2

0.1

0.3

0.4

0.5

Time (s)

Fig. 6.34 Current of second inverter

300 200 100 0 Vsa1

-100

Vga1 VCrefa1

-200

VCrefa2 Vca1

-300

VCa2

-400 0.18

0.2

0.22

0.24

0.26

0.28

0.3

Time (s)

Fig. 6.35 Step in phase shift of voltages

slower state estimators and is improved with line drop compensation. The synchronization to broadcast angle is a highly desirable mode where possible with a fallback to line drop compensation as fallback.

144

6 Inverters Operating in Power System in Weak Grids Vsa1

300

Vga1 VCrefa1

200

VCrefa2 Vca1

100

VCa2

0

-100

-200

-300

-400 0.255

0.26

0.265

0.27

0.275

0.28

Time (s)

Fig. 6.36 Zoom of phase shift of terminal voltages and capacitor voltages

6.12 Wide Area Control of Power Systems with Inverters and Synchronous Generators With a pure synchronous machine system, the dynamics of groups of synchronous machines can be approximated as flows between equivalent aggregated machines possibly corrected for the distributed voltage support. Local machine oscillations are able to be largely ignored since these higher frequency local modes are typically well damped [9, 10]. When the system includes inverters, the first step in forming an aggregation is to ensure that the “local modes” of the inverters are well damped. The philosophy of the control of the synchronization of inverters is a hierarchy. 1. Broadcast of the angle of the local centre of area of a group of machines and synchronize to that. There will need to be a correction in angle to allow for the angle across local lines, but the overall effect is to remove inter inverter PLL oscillations. 2. The next level of the control options is to use a Kalman filter to infer the source voltage angle using the terminal volts, the line impedance and the injected current. There will come a limit to the gains that can be used and the stability but the sensitivity to other inverter injections is much reduced. 3. The final level is to use a Kalman filter to infer the angle of the terminal volts and to synchronize to that. The speed of convergence can be adjusted to make this less sensitive to PLL interactions but at the cost of responsiveness.

6.12 Wide Area Control of Power Systems …

145

Fig. 6.37 Three area system with inverters

Fig. 6.38 Reduced model

The approach using Linear Quadratic control of the inverter using local measurements makes the system robust against resonant and switch frequency with other inverters. Provided the PLL stability enhancement tools above are applied the inverters become a PQ injection in phase with the local Thevenin voltage or state centre of area. The inverter dynamics are at a different frequency set from the machine dynamics and only act to provide an offset in power flow or system angle. Therefore, the system reduces to the aggregate machine dynamics around a set power flow. Figure 6.37 shows a power system with three areas with a set of inverters in each area (Fig. 6.38). The inverters, if connected to batteries, can modulate their output with a component proportional to area velocities to enhance system damping [11]. For solar farms, the power exported can be reduced but not increased significantly, this modulation can still contribute to system damping. The signal used for area velocity can be formed from the System wide nonlinear Kalman filter as discussed in [12]. The system angle reference and the velocity term can be broadcast to all utility scale inverters.

6.13 Hierarchy of Control Efforts 1. The first level of control is to ensure the PLL of the multiple inverters operate stably.

146

6 Inverters Operating in Power System in Weak Grids

2. The next level is to adjust the angle offset from the reference of the injection to provide good system voltage management. 3. The system wide level of control is to enhance the stability of the interarea oscillations through modulation of the injected power. The control aspect is very much tied to the method of design, and the PWM model of the system with all inverters and generators modelled in detail is not tractable due to the range of eigenvalues and the nonlinearity of the model. Once there is assurance that there is not an adverse interaction between inverters around the inverter switching frequency, then the switch averaged model can be used with the advantage that the major system elements are linear. The drawback is that the model order is huge with every transmission link represented by differential equations. It would be good to have a representation of the low order frequency modes only for the design of the control and then to validate the control as not adversely affecting the other system modes. In [13] groups of synchronous machines are aggregated to one equivalent machine in each area. When there is not a clean separation with a high line impedance between areas it has proven useful to still aggregate system dynamics but to solve for the power flow with voltage sources at some intermediate busses. The simplest form is to have the angle spread of the interarea mode giving the pattern of bus angles and voltage sources but to have only the one differential equation for the area angle. If the bus angles vary linearly between areas then the voltage sources can be determined and then the network can be reduced based on that set of sources. For example when the angles associated with the eigenvector for the low frequency oscillation is normalized such that the centre angles are +1, −1, W = [1 d1 d2 …dn −1] then for where the angles   of the areas are δi , δ j the angles of the areas are at  the case δi − δ j · W/2 + δi + δ j /2. These additional buses at the real generator locations are operating only as voltage props and at each step of simulation can be reduced using standard bus reduction to the buses of the equivalent area machines. The power flow to and from these equivalent machines can be found from this reduced equivalent system. For many locations of renewable generation in Australia the line impedance is high and simply making the inverter current synchronize to the terminal voltage can create problems. As seen in Fig. 6.39, the magnitude of the inverter voltage and thus the capacitor voltage is high when current is synchronized to source voltage. I

Vs IX Vt

Fig. 6.39 Unity power injection to terminal volts compared with source reference

6.13 Hierarchy of Control Efforts

147

This is for the case of a high current reference value to accentuate the difference. In grid control with synchronous machines, the magnitude of source and receiving end voltages would be more equal and the current angle offset from the Vs vector.

6.14 Control Design When the power system is reduced to a set of equivalent machines, the total energy can be represented as the sum of kinetic and potential energy. Because the potential energy derivative is not directly influenced by the available control actions the focus is on kinetic energy 1 2 Ji δ˙i 2    Ji δ¨i = (Vi V j sin δi − δ j /xi j + Psi − PLi ) V =

V˙ =



Ji δ˙i δ¨i

(6.20) (6.21)

where Ps is the input power to the equivalent generator and PL is the load power. The Ps term is taken to incorporate the power to an area from the generators and renewable inverters Ps = PG + PR . ˙ The immediate impact of the PPVi term is found from ∂∂PVRi = δ˙i so to make V˙ negative a good choice is to make the renewable portion of Psi equal to −k δ˙i . So in an area whose speed in a center of area frame is positive the aim is to reduce PV power. In this case, the control action is to reduce Inverter power when its area is at higher velocity compared to the average. Other control actions in the grid will help to support this. Shunt elements will be in phase with velocity of an area, series elements will be controlled based on velocity differences [11, 12]. These controllers can be also expanded to include other elements of the network such as generator excitation systems [14] and costumer-side loads and generation. Using modal decomposition, the system can be approximately represented by a few low frequency modes. In this case the inverter operates like an ideal power source aligned with the equivalent center of area generator. There can be non-ideal effects from generators at the edge of an area which can be mitigated through the proposed aggregation and control approach. Because the power flow with angle characteristic is known extrapolate to nonlinear flow models is feasible.

148

6 Inverters Operating in Power System in Weak Grids delm dot

0.02

1 2

0.015 0.01 0.005

0 -0.005 -0.01 -0.015 -0.02 0

5

10

15

Fig. 6.40 Base case with no inverter modulation

6.15 Inverters to Add Damping by Reducing Output When Velocity in the Area Is High There is an overall reduction of power output when using single sided modulation so there is some frequency drop with this single sided modulation because this 6 machine system is stand-alone (Figs. 6.40 and 6.41). For the three area problem the disturbance primarily drives the oscillation between areas 1 and 3 (Figs. 6.42, 6.43 and 6.44).

6.16 Discussion The design of the controllers for each inverter is based on local filter currents and voltages which is validated using the full PWM simulation in PSCAD. The design of the PLL controllers is based on local Kalman filters designed using state averaging of only local terminal voltage and injected current measurements, which is initially validated from eigenvalues. Where the PLL modes are all well damped then the model reduction to the equivalent area machines is valid. Any intermachine and inverter dynamics are of higher frequency and are presumed to be well damped. The system wide design for SVC’s, series compensators, excitation systems are all based on the area equivalent machines (potentially modified by voltage props).

6.16 Discussion

149 delm dot

0.015 1 2

0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025

0

5

10

15

Fig. 6.41 With inverter reduction for positive velocity delm dot

0.015

m1 m2 m3

0.01

m4 m5 m6

0.005

0

-0.005

-0.01

-0.015 0

5

10

Fig. 6.42 Velocities of all machines: three area problem no modulation

15

150

6 Inverters Operating in Power System in Weak Grids delm dot

0.01

m1 m2 m3 m4 m5 m6

0.005

0

-0.005

-0.01

-0.015 5

0

10

15

Fig. 6.43 Three area problem with modulation to reduce power in area with positive velocity

power towards machines 0.08 m1 m2 m3 m4 m5 m6

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

5

10

15

Fig. 6.44 Power towards machines from inverters

6.17 Summary of Inverter Synchronization The design process proceeds in a series of steps, each step making appropriate approximations.

6.17 Summary of Inverter Synchronization

151

Linear Quadratic control of inverters using local signals, validated at PSCAD level. Kalman filter design for each inverter based on local measurements validated by system level modelling using a state averaged model and finally by PSCAD. The 50 Hz constraint on the identified Thevenin source reduces the potential for interaction at PLL frequencies. System reduction to equivalent machines in each area (possibly corrected by voltage props). Control design for the areas based on kinetic energy function derivatives. The use of a system wide Kalman filter to focus on only oscillations between area aggregate machines and reject local machine and inverter modes. These direct controls are for elements which directly control power flow such as HVDC links and SVC. Other examples are where the output from solar farms affect the power flows. The designs presented here are for a graduated model of inverters affecting the power system. The reader would be familiar with different models for transmission line. When there is a transient lighting strike causing reflections a travelling wave model is used. When examining overvoltages caused by open circuit line a PI model can be used. When incorporated in stability studies a simple series inductor at main frequency is used. For inverters a similar graduation of models is desired for analysis and control design (Figs. 6.45, 6.46, 6.47 and 6.48). The power electronic interfaces of inverters and thyristors can make fast control of power.

6.18 Distributed Versus Lumped Battery Compensators This section considers the control of a multiarea power system using larger sized lumped battery systems or by more distributed batteries of the same total rating. The analysis tools for seeking set of locations with desirable impact is demonstrated on a well structured longitudinal system of 10 machines as well as a more generalized multimachine system.

6.18.1 Introduction to Distributed Controllers Placement of Phasor Units for measuring interarea modes was considered in [15] considering single locations which were not strongly influenced by local modes. In this chapter a set of locations for controllers are considered which have a strong influence in interarea modes and a combined low effect of local modes. Distributed controllers for control a multiarea system are considered but with little consideration of local modes in [16]. Distributed control has been applied to DC microgrids [17] and to distributed reactive control in wind systems [18]. The

152

6 Inverters Operating in Power System in Weak Grids

Fig. 6.45 Transmission lines in power system

topic area of distributed control for load frequency control has been addressed in [19] once again without the impact on local disturbances.

6.18.2 Battery Controller To find the impact of a battery controller at different busses, the result of a short power injection at the bus is considered. The impulse can be considered as establishing a step change of velocity at machines near the connection bus. The linear model for perturbations in the system states in the simple DE are ·       δ = 0 I · δ + 0 u M D δ˙ 1/J δ˙

(6.22)

˙ For a transformation into So an impulse disturbance on u will cause a step in δ. modal form z = Tx starting from a zero perturbation state only the velocity states are impacted. The modal impact is found from the dot product of the impulsive velocity

6.18 Distributed Versus Lumped Battery Compensators

153

Fig. 6.46 Tracking solar panels at Queensland University of Technology

Fig. 6.47 Power electronic controllers for STATCOMs

perturbation with the velocity component of eigenvectors for all oscillation modes. In the simple case M = J\imag(Yr) where Yr is the admittance matrix reduced to the machine busses. From the original network of Fig. 6.1 there are 10 machine busses and 10 extra busses. For

154

6 Inverters Operating in Power System in Weak Grids

Fig. 6.48 Power electronics for small solar power

 Y =

Ya Yb Y b Y c

 (6.23)

The power change can be expressed as a current change at the extra busses 

Im Ix



  Vm Ya Yb . = Vx Y b Y c 

(6.24)

  Eliminating the Vx term gives I m = Y a − Y binv(Y c)Y b V m − Y b · inv(Y c)I x, and for small signals the acceleration of the machine busses satisfies   J · δ¨ = imag Y a − Y b · inv(Y c)Y b δ − Y b · inv(Y c)P x

(6.25)

˙ For a transformation into So an impulse disturbance on u will cause a step in δ. modal form z = Tx. The dot product expresses the velocity impact into a modal impact. The interarea modal impact on some busses can be less strong but also has a strong impact on local modes. If only local mode impacts exceeding 0.1 of the largest are considered, the sum of magnitude of interarea modes over the rms sum of local impacts gives a modal impact factor MIF

6.18 Distributed Versus Lumped Battery Compensators

155

  sqr t interarea modes 2   MIF = sqr t signi f icant local modes 2

(6.26)

6.18.3 Case 1 Line of Generators In the case consider a regular spaced line of generators with equal inertia and the aim is to control the lowest frequency mode of oscillations by one or more batteries placed at any of the busses. The criteria is to achieve good suppression of the interarea mode with low triggering of other modes. If splitting the battery across multiple busses, the rating is also split for a valid comparison (Fig. 6.49). Here the MIF (Modal Impact Factor) for the 10 machine busses is 0.24 0.22 0.18 0.10 0.03 0.03 0.10 0.18 0.22 0.24 And the main line busses (1–10) 0.34 0.36 0.29 0.16 0.05 0.05 0.16 0.29 0.36 0.34. The main busses (11–20) have less local mode and are thus more effective. The control design is based on the feedback design based on LQR. When the pattern of the injections approximately matches the interarea mode with the first and last line busses with 50% of the battery power, the MIF is 1.4, if more spread with 25% at busses 11 12 and 19 20 the MIF = 2.8. When further spread to 6 busses the MIF becomes 6.2 (Figs. 6.50, 6.51, 6.52, 6.53, 6.54 and 6.55). Overall it is seen that using a distributed controller can have a greater impact on the system modes and reduced local modes. The use of a distributed controller has the potential for more of the desired response and less of the local modes being excited. Fig. 6.49 Regular spaced line of generators

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10

11

12

13 14

15

16 17

18 19

20

156

6 Inverters Operating in Power System in Weak Grids Black interarea, RED local; Solid at mc, Dash t backbone 1.8 inter at mc inter at line local at mc local at line ratio line MIF

1.6 1.4

Effectivness

1.2 1 0.8 0.6 0.4 0.2 0 2

1

4

3

5

6

7

8

9

10

bus number

Fig. 6.50 Impact of local modes and interarea for single location controllers

7

distributed 6 distributed 4 distributed 2

6

Effectivness

5

4

3

2

1

0 1

2

3

4

5

6

7

8

9

10

bus number

Fig. 6.51 Controllers spread over 2, 4, and 6 busses

6.18.4 Case 2 Interconnected Generators There are now 16 bus points as seen in Fig. 6.56, that could affect the modes and 9 intermachine modes. The first three are the inverter points, then the machine busses and finally the extra busses. The modal frequencies in rad/sec are 6.11 1.30 1.85 2.81 3.15 3.57 3.993 3.83 3.89. The impact on the local modes is shown in Fig. 6.57, the machine busses are

6.18 Distributed Versus Lumped Battery Compensators Fig. 6.52 System angles for lumped compensator as bus 11

157 Modes

1

0.5

0

-0.5

-1

-1.5

-2

Fig. 6.53 Modal response for lumped controller at bus 11

4

2

0

6

8

10

6

8

10

angles

0.5

0

-0.5

-1

4

2

0

Modes

1

0.5

0

-0.5

-1

-1.5

-2 0

2

4

Fig. 6.54 System angles for distributed over 6 bus

6

8

10

158

6 Inverters Operating in Power System in Weak Grids angles

1

0.5

0

-0.5

-1

0

2

4

6

8

10

Fig. 6.55 Modal response for distributed control over 6 buses

Fig. 6.56 Interconnected machine set

highlighted with a *. It is clear that these machine busses have a strong impact on at least one local mode. The low level of local modes for the non machine busses is a strong component of the low MIF noted in Fig. 6.58 (Fig. 6.59). The inverter busses 5, 8, 13 are almost the same impact as the extra busses 2, 3, 16 while the machine busses are poor choices because of the local modes. In this case the impedances to busses 13 and 14 are almost the same so the injection from bus 15 does not significantly affect the local mode. If the impedances are changed from 0.2 to 0.4, the impact on the local modes rises from 0.16 to 1.12 which makes this bus less attractive (Fig. 6.60). Focusing on the lowest frequency mode M1 then the splitting the inverter over busses 2 and 16 has a MIF of 2.32 and thus a stronger candidate than any single bus cases. Starting the system with an initial state that emphasises Mode 1 gives a response as in Fig. 6.61 for no gain and Fig. 6.62 with batteries active. The control

6.18 Distributed Versus Lumped Battery Compensators

159

Fig. 6.57 Strength of impact on local modes Fig. 6.58 Impact of batteries at all busses

Interarea Modes

2

M1 M2 RMS Local

1.5

1

0.5

0 4

2

6

8

12

10

Modal Impact Factor Gen buses marked with *

5

4

3

2

1

0

0

2

4

6

8

10

12

Fig. 6.59 Modal impact factor: machine busses marked with *

14

16

14

16

160

6 Inverters Operating in Power System in Weak Grids Modal Impact Factor M1 Gen buses marked with *

1.5

1

0.5

0

2

0

4

6

8

12

10

14

16

Fig. 6.60 For mode M1 the modal impact factor for all busses

Modes

5 4 3 2 1 0 -1 -2 -3 -4 0

5

10

15

Fig. 6.61 No control

is just feedback of an approximation of the selected modal velocity. Figures 6.60 and 6.61 show the control is effective in supressing the chosen mode but in Fig. 6.63 shows that more local mode is created by the lumped response.

6.18.5 Summary of Lumped Vs Distributed Batteries Lumped batteries can influence the interarea modes of a system. There can be benefits in selection of the control by considering the impact on local modes. Local modes can cause angle stresses as well as line overloads. Consideration of a strong influence

6.18 Distributed Versus Lumped Battery Compensators

161

Modes

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0

5

10

15

10

15

Fig. 6.62 Control at bus 2 and 16

Modes

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0

5

Fig. 6.63 Control at bus 2 only

over interarea modes coupled with a low impact on local modes can deliver good performance for both local and interarea. For some cases it is clear that distributing the battery installation can significantly reduce the local modes while maintaining good control of interarea. The increased cost of splitting the battery needs to be balanced against the network stresses when installation is in weaker areas of the network.

162

6 Inverters Operating in Power System in Weak Grids

6.19 Distributed Control by Batteries with a Spread of Delays If distributed batteries in customer premises were to be used for control of the damping of power system oscillations batteries by making a power change proportional to the velocity of an area gives the distributed control discussed in the previous section. One of the concerns about implementing such a control over many thousands of batteries would be the delays in the communication system. If the control were implemented using a broadcast system over competing channels of other traffic there could be a sufficient delay as to adversely affect damping. For a system with maximum frequency of interarea oscillations of 2 Hz a delay of 250 mS would mean that by the time it arrived at a given customer premise the signal would be delayed by 180° and would provide negative damping and a delay of 125 mS would be on the margin for negative damping contribution. The system wide Kalman filter can ensure that no higher frequency components are requested of the batteries. Rather than seek to compensate for delays at every customer computer one valid approach is to inform the local controller that if the time stamp on the command if older than 125 mS it is best to discard the command as “stale”. Evaluation of effectiveness requires consideration the component of the signal that would be aligned with the local velocity signal. If there are 1 MW of batteries with a spread of delays the effective battery power that is contributing to damping of the highest mode can be computed. For this 2 Hz case if there is a gaussian spread of delays then the effectiveness will reduce with the average delay and can be summed across the probability density function of the delay distribution (Fig. 6.64).

Fig. 6.64 Effectiveness of a given battery to the highest mode as function of its delay

6.20 Applications

163

6.20 Applications The present process is to use Phase Locked Loops (PLL) to synchronize the current from the inverter to align with terminal voltage. Unfortunately the phase of the terminal voltage is affected by any changes in the inverter output and also from any inverters in electrical proximity. There higher the impedance of the line (weaker system) the greater this effect This means that the measurement process makes the PLL synchronization highly interdependent in weak networks. This effect has required the limiting of output from some inverters in Australia under certain system conditions. The concept behind the Kalman filter identification of the Thevenin voltage is that the local machine equivalent will form the Thevenin voltage. This inferred voltage is less sensitive to the injected current and thus less of an adverse effect in the phase locking. In addition using the Kalman means that the inferred voltage can be forced to be close to 50 Hz thus reducing the propensity of the PLL’s to oscillate at 6– 12 Hz. For inverters on the outskirts of an area the risk is that the Thevenin may be an amalgam of the areas. Further study is required to focus the convergence of the Kalman filter to a useful reference angle. It is clear that synchronizing of the inverter to the Thevenin voltage or the local terminal voltage both have weaknesses. The issue is to determine the precise offset to use as a function of the injected power. The basic approach to wide area controls does not distinguish the effect on local modes. The latter section here demonstrates a process such that the actuator does not have a strong influence on local modes. In practise with lumped controllers the location is determined by other factors but where customer side batteries are used then a lower set of disturbances of local modes can be expected but this will depend of the distribution of these local compensators. A key consideration for distributed batteries is the distribution of delays but by testing for the delays of the packet the local controller can declare packets “stale” and take no action.

References 1. A.A.A. Radwan, Y.A.I. Mohamed, Power synchronization control for grid-connected currentsource inverter-based photovoltaic systems. IEEE Trans. Energy Convers. 31, 1023–1036 (2016) 2. L. Poh Chiang, D.G. Holmes, Analysis of multiloop control strategies for LC/CL/LCL-filtered voltage-source and current-source inverters. IEEE Trans. Ind. Appl. 41, 644–654 (2005) 3. X. Chen, Y. Zhang, S. Wang, J. Chen, C. Gong, Impedance-phased dynamic control method for grid-connected inverters in a weak grid. IEEE Trans. Power Electron. 32, 274–283 (2017) 4. B.D.O. Anderson, J.B. Moore, Optimal Control Linear Quadratic Methods (Dover Publications, 2014) 5. Q. Hu, L. Fu, F. Ma, F. Ji, Large signal synchronizing instability of PLL-based VSC connected to weak AC grid. IEEE Trans. Power Syst. 34, 3220–3229 (2019)

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6 Inverters Operating in Power System in Weak Grids

6. Z. Kemin, R. Zhang, W. Wei, On the design of unknown input observers and fault detection filters, in 2006 6th World Congress on Intelligent Control and Automation, 2006, pp. 5638–5642 7. M. Ikeda, D. Siljak, Decentralized stabilization of linear time-varying systems. IEEE Trans. Autom. Control 25, 106–107 (1980) 8. D.D. Siljak, Stability of large-scale systems under structural perturbations. IEEE Trans. Syst. Man Cybern. SMC-2, 657–663 (1972) 9. P. Kundur, Power System Stability and Control (McGraw-Hill, New York, 1994) 10. A. Pai, Energy Function Analysis for Power System Stability (Springer, 1989) 11. A. Vahidnia, G. Ledwich, E. Palmer, A. Ghosh, Wide-area control through aggregation of power systems. IET Gener. Transm. Distrib. 9, 1292–1300 (2015) 12. A. Vahidnia, G. Ledwich, E.W. Palmer, Transient stability improvement through wide-area controlled SVCs. IEEE Trans. Power Syst. 31, 3082–3089 (2016) 13. A. Vahidnia, G. Ledwich, Y. Mishra, Correction factors for dynamic state estimation of aggregated generators, in 2015 IEEE Power & Energy Society General Meeting, 2015, pp. 1–5 14. R. Goldoost-Soloot, Y. Mishra, G. Ledwich, Wide-area damping control for inter-area oscillations using inverse filtering technique. IET Gener. Transm. Distrib. 9, 1534–1543 (2015) 15. E.W. Palmer, G. Ledwich, Optimal placement of angle transducers in power systems. IEEE Trans. Power Syst. 11, 788–793 (1996) 16. E. Vlahakis, L. Dritsas, G. Halikias, Distributed LQR design for a class of large-scale multi-area power systems. Energies 12, 2664 (28 pp.) 17. F. Perez, G. Damm, P. Ribeiro, F. Lamnabhi-Lagarrigue, L. Galai-Dol, A nonlinear distributed control strategy for a DC microgrid using hybrid energy storage for voltage stability, in 2019 IEEE 58th Conference on Decision and Control (CDC), 11–13 Dec 2019, Piscataway, NJ, USA, 2019, pp. 5168–5173 18. B. Raouf, A. Akbarimajd, A. Dejamkhooy, S. Seyed Shenava, Robust distributed control of reactive power in a hybrid wind-diesel power system with STATCOM. Int. Trans. Electr. Energy Syst. 29, e2780 (18 pp.) 19. W. Lei, W. Chuan, W. Kun, W. He, L. Zhaoyu, Y. Wenwu, Distributed load frequency control for multi-area power systems, in 2019 Tenth International Conference on Intelligent Control and Information Processing (ICICIP), 14–19 Dec 2019, Piscataway, NJ, USA, 2019, pp. 100–105

Chapter 7

Travelling Waves

7.1 Electromechanical Wave Propagation In the train model introduced in Chap. 2 if there is a sudden disturbance on Carriage 1 the jolt will travel down the train in much the same way a continuous spring would have propagating waves. In a similar way the sets of generators in a power system have nonlinear springs but still the jolt at one generator will propagate down the power system. Ideally a long spring would carry any frequency disturbance down the train. The discrete case of distinct carriages means that only low frequency transients will propagate well down the train (Fig. 7.1). Electromechanical wave propagation phenomenon is observed using Wide Area Measurement System (WAMS) in the U.S. eastern interconnection and western power (WECC) systems at 400–600 mile/s [1–3]—This traveling wave phenomenon is characterized by a delayed oscillation of the generator rotor angles and load bus voltage phase angles traveling away from the disturbance initiation point. The speed of propagation is dependent on the per unit area impedance and per unit area inertia of the system. Continuum modelling of the power system using distributed parameters gives global system properties and aids in the studying of this dynamic phenomenon [4–7]. If not properly attenuated by control devices, this traveling wave can negatively influence the reliable power transfer limits of the network [5, 6] cause a large number of generators to lose synchronism, and even lead to widespread blackouts [8]. Due to its potential to threaten power system stability, many control strategies have been investigated to mitigate this propagating disturbance [9, 9]. Static Var Compensator (SVC) provides dynamic voltage support and shunt reactive compensation [10, 11] and is frequently used to enhance power system stability [12, 13]. In existing literature on traveling wave phenomenon, there has not yet been any investigation conducted on the frequency range characteristics of discrete power systems with different levels of discretization and on designing SVC controllers to attenuate the electromechanical wave propagation based on the transmission line analogy and the load modulation effect. Part one of this chapter is developed to provide further insight to these aspects by analyzing uniformly and non-uniformly discretized power © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_7

165

166

7 Travelling Waves

Carriage 1

Carriage 2

Carriage 3

Carriage 4

Fig. 7.1 Train analogy for a set of synchronous machines

systems and the simplified Australian power system model IEEE test system. In part two, further case studies illustrate the key insights and control implications that arise from the traveling wave phenomenon. The investigations and concepts introduced here can assist in improving the design of control strategies to enhance power system transient stability considering the spatial aspects.

7.2 Attenuation of Electromechanical Waves Using SVC There are electromagnetic voltage and current waves traveling at the speed of light in open air transmission lines. In power systems, there are electromechanical velocity and power flow waves that travel a much slower speed and are analogous to traveling electromagnetic waves in transmission lines [9]. Velocity or angular frequency ω wave is analogous to electromagnetic voltage wave v while electromechanical power flow wave P is analogous to electromagnetic current wave i [4]. In transmission lines, the reflection of these waves at discontinuities, i.e. open circuit or junctions, can be characterized by a reflection coefficient based on the characteristic impedance [9]. According to the analogy, it is demonstrated that velocity and power waves should behave similarly to voltage and current, respectively, at a discontinuity [4] and should have reflection and transmission coefficients as in (7.1–7.3) [14]. The characteristic impedance and termination are defined in Table 7.1. Rω =

Z − Z0 Z + Z0

(7.1)

Table 7.1 Analogy between electromagnetic and electromechanical waves Types of waves

Variables

(a) Electromagnetic (Transmission line)

Voltage (v)

(b) Electromechanical (Power Systems)

Current (i)

Characteristic impedance/characteristic termination  + Z 0 = CL 00 = VI 0+ 

Rotor velocity (ω)

Power flow (p)

Z0 = ω+ P+

[9]

0

 ωs  2h

1 V 2b



=

7.2 Attenuation of Electromechanical Waves Using SVC

R P = −Rω =

167

Z0 − Z Z + Z0

T =1+ R

(7.2) (7.3)

In a transmission line system, a resistor can absorb energy and reduce wave reflection and transmission. A characteristic impedance termination can eliminate reflection entirely. Based on this analogy, in a power system, a characteristic termination or control device with constant angular frequency to power absorption ratio Δω/ΔP will reduce traveling pulse size by absorbing energy and reducing wave reflection and transmission [9]. The Δω/ΔP characteristic can be utilized for the attenuation of the electromechanical waves. Direct load modulation such as switched loads, can be one of the effective ways. Moreover, modern batteries and demand management has a potential to emulate the Δω/ΔP characteristic and achieve desired load modulation. However, in this chapter, the load modulation effect of SVC is used to reduce wave reflection and transmission. For a distortionless system, an additional loss element will cause the forward traveling wave to be attenuated as in (7.4), where λ is the attenuation constant. In the artificial test power system, this additional loss element is created by a SVC which modulates a frequency dependent load. p = W + (x − ct)exp(−λx)

(7.4)

The SVC controls the load voltage which directly modulates load power, assuming a constant impedance load. The change in load power absorption (P) is proportional to the change in velocity or local angular frequency (ω) by a constant K in (7.5). P = K ω

(7.5)

K is a product of the individual gains of each step as shown in (7.6) and (7.7). k

η

β

ω → B → V → P

(7.6)

K =k×η×β

(7.7)

where ω is change in generator velocity; B is change in SVC shunt susceptance; V is change in load bus voltage; and P is change in load power absorption. Psvc = K δ˙i represents power to the load as modulated by the SVC as in (7.8). This K is modelled in the controlled artificial test system as an extra damping term added to the frequency dependent load where SVC is located as in (7.9). δ¨i =

1 (Pm − Pe − Di δ˙i − Psvc ) Ji

(7.8)

168

7 Travelling Waves

δ¨i =

1 (Pm − Pe − (Di + K )δ˙i ) Ji

(7.9)

The procedure to obtain K involves placing a SVC with susceptance size of jα at the chosen load bus i at steady state condition and observing how much the voltage at the chosen load bus Vx varies when this SVC is added as in (7.10) p128 [15]. The change in power P is then calculated according to (7.11). jα is a shunt term added to the appropriate location on the diagonal of the M submatrix of the bus admittance matrix. Vx = Vxnew − Vx = −(M + M)−1 L V A − Vx

(7.10)

P = Pnew − P = (Vx + Vx )2 G − Vx 2 G

(7.11)

˙ The control The SVC control action is proportional control based on ω or δ. law is: ⎧ ⎨ E k δ˙ > E BSV C = −E k δ˙ < −E (7.12) ⎩ ˙ k δ other wise ˙ Bsvc where Bsvc is the control law based on local generator velocity measurement δ. is SVC susceptance value. E is the upper limit and −E is the lower limit. Since local rotor angle variations is strongly aligned with local voltage angle variations, the SVC control action can also use local bus voltage angular frequency instead of nearby generator velocity.

7.3 Frequency Range and Modal Analysis There is close association between the discretization level of a region, and its frequency range and modal analysis results, as demonstrated in simulation results of this chapter. Frequency range is dependent on the discretization level of the inertia and impedance elements. Frequency range of a uniform region is inversely proportional to square root of the average inertia Have (Inertia of each discrete generator) and the average impedance Z ave (Impedance between neighbouring discrete generators) as in (7.13). A system region with smaller inertia machines and smaller impedances between the machines has a higher frequency range.  Fr equency_Range ∝

1 Have Z ave

(7.13)

7.3 Frequency Range and Modal Analysis

169

The frequency range of a particular uniformly discretized region and the wave phase speed of that region [5] are strongly related. For example, a region with a low frequency range has a slower velocity of propagation. The frequency range of a uniformly discretized region is equal to the oscillation mode with the highest frequency in that region. This means that any wave frequency higher than the highest frequency of a uniformly discretized region will not propagate into or past this region. Furthermore, it is possible to display the traveling wave phenomenon using oscillation modes in time domain. The angle and velocity states of the system can be expressed in terms of modes through the use of the inverse T-matrix [16] as in (7.14). x = T −1 z

(7.14)

where x contains the angle and velocity states of generators, and z contains the position and velocity states of system modes [16]. When a limited number of modes are used for prediction, then the resulting inferred angle and velocity state approximation and traveling wave display are less accurate. The low frequency modes are coherent in phase at the traveling wave initiating time and the time when the wave reaches the termination. A chosen controller must be able to react quickly to track the incident wave [9]. Therefore, it is important to know the frequency range or the frequency of the highest oscillation mode in a particular region, because the shape of the traveling wave can impact the effectiveness of a controller in that region. A controller can usually function well in a low frequency range region and track the slowly changing velocity wave shape. Sharp velocity wave shape in high frequency range region can cause problems, since the controller may not react quick enough to track the wave and be less effective in attenuating the wave or removing kinetic energy. Special care needs to be taken when designing controllers, which are located in high frequency range regions, in order to ensure its robustness.

7.3.1 Forward Wave Analysis An important goal of this chapter is the investigation of the forward electromechanical wave propagation for realistic discrete systems in time and frequency domains. In a longitudinal distortionless power system, where the speed of propagation is independent of frequency, the classical swing equation of generator rotor can be transformed into a wave equation, consists of distributed parameters and characterizes the propagation of electromechanical waves ξ in terms of rotor angle δ, rotor velocity ω or power p as in (7.15) [4], where v is propagation velocity [5] 2 ∂ 2ξ 2∂ ξ = v , ξ = δ, ω, or p ∂x2 ∂t 2

(7.15)

170

7 Travelling Waves

The solution of this second order linear differential equation shows that there are both forward W + () and backward W − () traveling wave components present as in (7.16) [4]. ξ = W + (x − vt) + W − (x + vt)

(7.16)

According to the transmission line analogy, by ending each test power system with a characteristic termination, or a device with constant velocity and power ratio as listed in Table 7.1 (b), the reflections are eliminated and there is only forward traveling wave present as in (7.17). ξ = W + (x − vt)

(7.17)

The frequency domain investigations are accomplished through analyzing the properties of the phase and magnitude plots of transfer functions of angular frequencies (velocities) between two chosen machines. The velocity transfer function is a ratio of velocity between two different machines, say a and b, and can be derived by dividing the transfer functions of the form ωi / p1 as shown in (7.18), where p1 is the input power perturbation at 1st machine. p1 excites ω i , which is the angular frequency of ith machine. The phase and the magnitude of this transfer function can be expressed as in (7.19) and (7.20), respectively. (ωb / p1)/(ωa / p1) = ωb /ωa = ω(b − a)

(7.18)

ph(ωb /ωa ) = ph(ωb / p1) − ph(ωa / p1)

(7.19)

mag(ωb /ωa ) = mag(ωb / p1)/mag(ωa / p1)

(7.20)

If a forward traveling wave is present in a power system, a phase delay between the velocities of two machines is expected. A delay function, x(t − t0 ), in time domain can be represented by X (s)e−st0 in frequency domain using Laplace transform as in (7.21). The delay process, e−st0 , can be described as Y(s), where the magnitude is 1 and phase is ϕ as in (7.22). x(t − t0 )

Laplace trans f or m

←→

X (s)e−st0

Y (s) = e−st0 = e jϕ

(7.21) (7.22)

For examining the frequency responses, s = jω, which makes e jϕ = e− jωt0 and thus, jϕ = −jωt 0 . Therefore, the phase, ϕ, and hence the delay value, t0 , are related as in (7.23). Here, the phase angle decreases linearly with frequency. A sharper slope, or larger t0 , means a longer delay time. This angular frequency ω is equivalent to the linear phase region or frequency range of the magnitude plot, which is the frequency range with sufficient signal magnitude.

7.3 Frequency Range and Modal Analysis

171

ϕ = −ωt0

(7.23)

The characteristic termination at the end of each test system is implemented by adjusting the frequency dependent load Pload = Dload δ˙e near the end machine as in (7.24). δ¨e =

1 (Pm − Pe − De δ˙e − Pload ) Je

(7.24)

where De is the damping ratio, Je is the rotor inertia constant, Pm is mechanical power, Pe is electrical power,δ¨e is acceleration and δ˙e is velocity of the end machine. The total frequency dependent power is P = (De + Dload )δ˙e . Since there is no reverse traveling wave, the power wave of the load is positive only P + = P and the velocity wave is positive only ω+ = ω = δ˙e . The characteristic termination is defined as in Table 7.1. In this special case, it is derived as in (7.25). Z0 =

ω+ δ˙e = + P P

(7.25)

The frequency dependent power at the end machine and the characteristic termination are derived as in (7.26) and (7.27). P=

1 δ˙e = (De + Dload )δ˙e Z0

(7.26)

1 (De + Dload )

(7.27)

Z0 =

7.4 Test Systems and Simulations The electromechanical wave propagation phenomenon concept is investigated in MATLAB in two artificial discrete test systems for time and frequency domain responses and in the real simplified Australian power system IEEE test system [17] for frequency domain response and in PSSE for time domain response. The uncontrolled systems are firstly examined and a SVC controller is later added to show its ability in attenuating the traveling wave in time domain. The two artificial test systems are the uniformly discretized 10 machine system and the non-uniformly discretized 15 machine system. Both test systems have identical and uniform line impedances of ZL of 0.01 + 0.1 and connection impedances of Zcm of 0.001 + 0.01, but with either uniform or non-uniform inertia. They are terminated with a characteristic termination boundary condition based on the inertia of the last machine. This termination is analogous to impedance matching in transmission lines

172

7 Travelling Waves

and eliminates wave reflections [9]. It makes each artificial system equivalent to an infinite length system with only forward traveling waves present. These test system are algebraically linearized for frequency investigations. An idealized continuum system has infinitesimal elements such as continuous transmission, generation and load per unit length [4–6]. An idealized longitudinal continuum system would have infinite frequency range. Hence, these two artificial test power systems with finite frequency range are considered as discrete or coarsely discretized. When a more finely discretized 100 machine equivalent systems is created, which have the same per kilometer parameter value and total distance across the system with 10 times more machines, the frequency range is 10 times wider and the output traveling wave shape is less distorted. This shows that as the discretization becomes finer, the finite frequency range becomes higher and closer to that of the ideal continuum. In other words, the traveling wave phenomenon is only valid to a finite frequency range in discrete systems. The lumpier the system, the lower the frequency range. Inside the frequency range, the system behaves like traveling waves.

7.5 Artificial Test Systems The uniformly discretized system as shown in Fig. 7.2. consists of 10 machines with uniform inertia constant H of 6 s and negligible general damping. The damping at the last machine is modified to model the characteristic termination Z 0 = 1.62. The generation is uniformly 1 everywhere and the loading is uniformly 1 everywhere except 0 at the first load and 2 at the last load [0 1 1 …0.1 1 2], which results in a left to right power flow. The non-uniformly discretized system as shown in Fig. 7.3. consists of 15 Fig. 7.2 Uniformly discretized 10 machine system

Fig. 7.3 Non-uniformly discretized 15 machine system

7.5 Artificial Test Systems

173

machines in three sections: 5 machines low inertia section (H = 1.2 s, D = 0.014), 5 machines high inertia section (H = 6 s, D = 0.07), and 5 machines low inertia section (H = 1.2 s, D = 0.014) ended with a characteristic termination Z 0 = 3.62. The inertia and damping are five times higher in the high inertia section. The generation and loading pattern is similar to the previous system, which results in a left to right power flow. A square shaped input pulse of 0.1 s is initiated at left-most machine’s (machine 1 s) power input for each test system and the output velocity pulse travels from left to right across the system as shown in Fig. 7.2. Simulation results indicate that the delay value t0 , which is calculated using frequency domain characteristics, is commensurate to the measured velocity pulse delay time across each section in both test systems in time domain as listed in Tables 7.2 and 7.3. There is minor discrepancy between measured and calculated values due to discretization. For the uniformly discretized system, there are identical delay value, linear phase region and uniform frequency range of 26 rad/s across each half of the system as shown in Fig. 7.4. The velocity disturbance is shown traveling across the system in time domain in Fig. 7.4. There is no change to pulse shape or travel speed. There is negligible decrease in magnitude as there is negligible damping in the system. In Figs. 7.5 and 6.7, the velocities are originally centered at 0, but an offset value is provided to make traveling waves more visible in the plots. In Fig. 7.5, the undamped system is ended with a characteristic termination, which should ideally eliminate all reflections. However, there are still reflections present in the system. An explanation of this imperfect reflection elimination is that the characteristic termination Z 0 = ω+ /P + is derived based on the wave equation model, distributed parameters, and without damping. This termination is imperfect Table 7.2 Delay in uniformly discretized system Section of 5 machines

Time domain

Frequency domain

Measured delay time (s)

Calculated t 0 delay value

First half

0.33

10.4 rad/(26 rad/s) = 0.4 s

Second half

0.33

10.1 rad/(25 rad/s) = 0.4 s

Table 7.3 Delay in non-uniformly discretized system Section of 5 Machines

Time domain

Frequency domain

Measured delay time (s)

Calculated t_0 Delay value

1. Low Inertia (Finely discretized)

0.15

12.20 rad/(60 rad/s) = 0.20 s

2. High Inertia (Coarsely discretized)

0.34

11.9 rad/(26 rad/s) = 0.46 s

3. Low Inertia (Finely discretized)

0.15

11.5 rad/(60 rad/s) = 0.19 s

174

7 Travelling Waves

Phase (deg)

Magnitude (dB)

0

0 5-1

-200

10-6

-20

-400 -600 -800 0

-40

10

20

30

40

0

10

30

20

40

Freq (rad/s) 26rad/s 25rad/s

Freq (rad/s) 596 ° = 10.4rad 581 ° = 10.1rad

Fig. 7.4 Phase and magnitude plots for uniformly discretized system

1

1 2 3 4 5 6 7 8 9 10

Velocity (rad/s)

0

-1

-2

-3

-4

-5 1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Fig. 7.5 Offset velocity plot of uniformly discretized 10 machine system

in performance in the discrete power system model, which has lumped parameters [9]. In a discrete power test system, the traveling wave power in the line between machines ‘i’ and ‘i + 1” is observed to be more closely related to velocity average (ω i +ωi+1 )/2 of the neighbouring machines Z 0 = ωav + /P + . Although this relationship could provide a more perfect termination, the value of ωi+1 is impractical to measure. For a non-uniformly discretized system with low, high and low inertia regions (LHL system), there are varying delay value, linear phase region and frequency range across each of the different inertia regions in frequency domain as shown in Fig. 7.5. The sharp slope (larger delay value) and narrow frequency range (26 rad/s) of the high inertia region in frequency domain corresponds to a longer delay time across that region in time domain. The flat slope (smaller delay value) and wide frequency

7.5 Artificial Test Systems

175

range (60 rad/s) of the low inertia region in frequency domain corresponds to shorter delay time across that region in time domain. These results imply that in the real system, a region with larger inertia, which is more coarsely discretized, has slower traveling time in time domain, a larger delay value, and smaller linear phase region or frequency range in frequency domain. The frequency range observations of each region in Fig. 7.5. is consistent with the mode of the highest frequency in that region when modal analysis is conducted on that segment of the system. The presence of high frequency local modes is most visible in magnitude plots in Fig. 7.6 The region closer to the left end of the system has the most amounts of local mode oscillations. The local modes attenuate and do not propagate too far away from the disturbance injection point, which is the left-most machine in these two test systems. The LHL system has the most attenuation of high frequency oscillations compared to uniform system. The bandwidth or the frequency range of the high inertia coarsely discretized region in this LHL system defines the actual frequency of the anticipated disturbance that propagates to the rest of the system, since frequencies higher than this frequency range are attenuated along the way. Figure 7.7 shows the disturbance traveling across the damped system in time domain. There is change of pulse shape and travel speed. The pulse is tall and narrow in Sect. 7.1 and short and wide in Sects. 7.2 and 7.3. The travel speed of the pulse is fast in Sects. 7.1 and 7.3 at same speed and slow in Sect. 7.2. There is significant decrease of pulse size going from Sects. 7.1 and 7.2 but no noticeable decrease of pulse size later on. There is no reflection at the end of the system due to the characteristic termination and system damping. There is significant reflection in Sect. 7.1 and some reflection in Sect. 7.2 due to the change of inertia, a discontinuity that the traveling pulse encounters. In the undamped system, there is the same pattern of pulse size decrease, shape change across the regions and delay times, but the system has much more reflections at the end. The effect of the proposed k δ˙ SVC controller on the attenuation of the traveling pulse is investigated by placing a SVC at various locations in the uniform and nonuniform inertia systems. The susceptance of the SVC is ±j20 Siemens and the gain k Phase (deg)

Magnitude (dB) 20

0 5-1 10-6 15-11

-200

0

-400 -20

-600 -800

0

20

40

60

80

Freq (rad/s) 697° = 12.2rad 679.9° = 11.9rad 658.5° = 11.5rad

-40

0

20

40

60

Freq (rad/s) 60rad/s 26rad/s 60rad/s

Fig. 7.6 Phase and magnitude plots for non-uniformly discretized system

80

176

7 Travelling Waves

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

Velocity (rad/s)

-5

-10

-15

-20

1

1.5

2.5

2

3

Time (s)

Fig. 7.7 Offset velocity plot of non-uniformly discretized 15 machine system

of 8 is used for testing in this chapter. Velocity pulse magnitudes at the last machine are measured in each controlled system and compared with that in the uncontrolled system for percentage of attenuation. The machines at the beginning and end of the systems are not considered as control locations because of the irregular result at boundaries of these coarsely discretized test systems. Simulation result shows that this SVC controller successfully attenuates the traveling pulse in both the uniform and non-uniform inertia systems. In the uniform system, the attenuation achieved is same at around 5.6% no matter where the controller is placed as shown in Fig. 7.8. In the non-uniform system, the attenuation is significantly different when the controller is placed in different inertia region. A controller in the first low inertia section achieves an attenuation of less than 1.0% and the last low inertia section achieved less than 9.98%. The most attenuation is achieved by the high inertia section, with 11.6% at machine 8 or 9 being the highest as shown in Fig. 7.9. The effect of a single SVC at fourth machine in the uniform system on frequency domain magnitude plots is investigated as shown in Fig. 7.10. The susceptance of the SVC is ±j20 Siemens. The unchanging constants are calculated for the controlled

5.2

5.8

6.1

5.8

5.6

3

4

5

6

7

% Attenuation 5.3

8

Fig. 7.8 Attenuation of velocity pulse at the last machine caused by a single svc located at machines 3–8 in the uniformly discretized system

7.5 Artificial Test Systems Fig. 7.9 Attenuation of velocity pulse at the last machine caused by a single svc located at machines 3–13 in the non-uniformly discretized system

177 % Attenuation 10.5 11.6 11.6 10.7 9.49 7.62 9.98 6.4 0.8 0.7 0.9 3

4

5

6

7

8

9

10

11

3.2

3.2

3.2

3.2

5

6

7

8

12

13

Fig. 7.10 Uniform system with SVC at fourth machine

Fig. 7.11 Average attenuation in dB magnitude between controlled and uncontrolled uniform system transfer functions ω(a)/ p1 for selected machines

SVC 2.1

2.3

3.2

2

3

4

Attenuation in Magnitude

bus of the test system as η = 0.0105 and β = 6.5933 when a full size SVC is placed there and measurements on voltage and power flow changes are made. Then gain of the SVC is chosen as k = 8. This gives an overall constant of K = 0.55. Transfer function of the form ω(a)/ p1 is obtained at the 2nd–8th machines in the system for both uncontrolled system and SVC controlled system when a disturbance occurs at the leftmost machine. The average difference in magnitude attenuation over all frequency range in dB is plotted in Fig. 7.11. in a location based manner. It is shown that there is the least difference in attenuation for the machines prior to the SVC. From the SVC location at 4th machine onwards to 8th machine, there is much larger difference in attenuation. This shows the effectiveness of the SVC attenuation in frequency domain.

7.6 IEEE Benchmark System (Simplified Australian System) Figure 7.12 shows the simplified Australian power system model IEEE test system consisting of 14 generators and 59 buses [17]. This model is used to investigate the time and frequency domain responses of traveling wave phenomenon in a real

178

7 Travelling Waves

QLD

SVC

SA NSW

VIC Fig. 7.12 Simplified 14 generator, 59 bus Australian system [17]

7.6 IEEE Benchmark System (Simplified Australian System)

179

Table 7.4 Observed frequency range of each state in Australia for a disturbance in QLD (QLD case) State

Machine plot ID

Area inertia (s)

Inertia size

Observed frequency signal for a disturbance in QLD (rad/s)

SA (rad/s)

QLD

8–11

202

Low

20

5

NSW

1–5

440

High

5

5

VIC

6–7

177

Low

5

5

SA

12–14

113

Low

5

20

Table 7.5 Generator numbering for simplified Australian system Bus #

Bus code

State

Plot ID

101

HPS_1

NSW-Snowy

1

201

BPS_2

NSW

2

202

EPS_2

NSW

203

VPS_2

NSW

204

MPS_2

301 302

Bus #

Bus code

State

Plot ID

401

TPS_4

QLD

8

402

CPS_4

QLD

9

3

403

SPS_4

QLD

10

4

404

GPS_4

QLD

11

NSW

5

501

NPS_5

SA

12

LPS_3

VIC

6

502

TPS_5

SA

13

YPS_3

VIC

7

503

PPS_5

SA

14

power system. It is shown by existing data that the Australian system is primarily a longitudinal system. It consists of a low inertia QLD region, a high inertia NSW region, a low inertia VIC region and a low inertia SA region as shown in Table 7.4. The generator numbering is color coded according to the state as listed in Table 7.5. Therefore, the frequency characteristic of Australia in terms of frequency range is expected to be somewhat similar to the LHL inertia non-uniformly discretized system, which has a wide, narrow, and wide frequency range. Minor discrepancies are also expected since the Australian system has an open termination, which means there are reflections and both forward and backward traveling waves present. Two test scenarios are conducted to test this frequency range expectation in MATLAB. Firstly, the system is given an input power disturbance at the northernmost generator of the system (the QLD case). Then, the system is given input power disturbance at the southern-most generator of the system (the SA case). For frequency domain investigations, the simplified Australian system is linearized using the forward-difference approximation numerical method (Burden 2005). The observed frequency range of each geographical state of Australia for each test scenario is listed in Table 7.4. The frequency range is derived as the meaningful range of phase angle for individual transfer functions ω(a)/ p1 in which the signal magnitude is sufficient (Fig. 7.13). Simulation results in MATLAB show that the simplified Australian system exhibits some frequency characteristics of the LHL system when a fault is applied at

180

Fig. 7.13 Australian interconnected system and the estimated distances

7 Travelling Waves

7.6 IEEE Benchmark System (Simplified Australian System)

181

either edge of the system. The first two States give the same result as the LHL system with observed frequency range of wide (20 rad/s), and narrow (5 rad/s). The observed frequency range has become so limited through the high inertia section that there is no signal at high frequency present in the next section (VIC, SA) upon which to appraise the real frequency range. When the disturbance signal is in SA, the frequency range of that section is clearly seen to be wide (20 rad/s) as expected. Similar to the LHL system, the frequency range of the high inertia region in the simplified Australian system defines the actual frequency of the anticipated disturbance that propagates to the rest of the system, since higher frequencies are attenuated along the way. Simulations are performed using PSSE software to demonstrate the presence of the electromechanical wave propagation phenomenon in the simplified Australian system for the QLD case in time domain. A 0.25 s fault is applied in northern QLD at bus 406 for a heavy loading condition. All SVCs in the system are set to voltage control. Simulation results show that the Australian system exhibits traveling wave phenomenon primarily in lower frequencies and there are groups of coherent generators oscillating together, such as SA generators. There is high magnitude high frequency oscillation in QLD and low magnitude low frequency oscillation in the other states at a delayed time based on distance from the fault as shown in angle and velocity plots in Figs. 7.14 and 7.15. The estimated wave phase speed from border buses is 3182.6 km/s for QLD, 1566.4 km/s for NSW, 5854.8 km/s for VIC and 6065.6 km/s for SA as listed in Table 7.6. The distance for each simplified state is listed in inset table of Fig. 7.12. This result shows that the wave speed is fast in low inertia QLD, VIC and SA regions, and slow in high inertia NSW region. The Australian result confirms to a

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

3.5 3

Angle (Radians)

2.5 2 1.5 1 0.5 0 -0.5 6

8

10

12

Time (sec)

Fig. 7.14 Angle response of Australian system in PSSE

14

16

18

182

7 Travelling Waves

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

Velocity (Radians/s)

400

300

200

100

0

-100

6

7

8

9

10

11

Time (sec)

Fig. 7.15 Velocity response of Australian system in PSSE

Table 7.6 Estimated wave phase speed for Australian system Travel region

Estimated travel time from PSSE angle plot of border buses

Estimated travel speed in time domain

QLD (416–406)

7.269–6.935 = 0.334 s

1063 km/0.334 s = 3182.6 km/s

NSW (102–416)

8.203–7.269 = 0.934 s

1463 km/0.934 s = 1566.4 km/s

VIC (315–102)

8.327–8.203 = 0.124 s

726 km/0.124 s = 5854.8 km/s

SA (501–315)

8.449–8.327 = 0.122 s

740 km/0.122 s = 6065.6 km/s

certain degree the relationship between frequency range and travel speed as shown in (7.13). The efficacy of the proposed k δ˙ SVC controller to attenuate the traveling pulse in the longitudinal Australian power system is investigated further in PSSE. The same heavy loading condition and fault location as previous simulation is used here. For the uncontrolled system, all SVCs are initially set as fixed shunts, except the three SVCs at buses 412 (QLD), 205 (NSW) and 216 (NSW), which are set to voltage control to stabilize the system. For the controlled system, a single SVC of capacity ±400 MVar is set to k δ˙ control at bus 216 in NSW, while the other SVCs are the same configuration as the uncontrolled system. A 0.26 s fault at bus 406 (in QLD) is applied in all tests. The load at buses adjacent to the controlled SVC is approximately 3000 MW. The angle and velocity plots look very similar to the previous results in Figs. 7.11 and 7.12. The velocity pulse at last machine 12 in SA in the uncontrolled system is compared with that in the controlled system. The proposed controller is successful in attenuating the traveling pulse by 11.49%.

7.6 IEEE Benchmark System (Simplified Australian System)

183

Fig. 7.16 Attenuation of velocity pulse at last machine (12) when a single SVC is located in bus 216 in NSW

Therefore, the proposed SVC controller is a good control strategy for mitigating the effects of electromechanical wave propagation in the Australian power system. Figure 7.16 shows the velocity pulse at machine 12 for the uncontrolled system and proposed SVC controlled system. An enlarged inset plot is shown for a better visual comparison of the peak of the waves. The SVC peaked at 400 MVar. This caused a 0.05 pu change in voltage which resulted in a power change of 1911 MW.

7.7 Summary of Travelling Waves This chapter shows in detail the finite approximation of the electromechanical wave propagation phenomenon and its attenuation in realistic discrete power systems. The artificial test systems are ended with characteristic termination so that only forward traveling waves of generator rotor angles and velocities are present. A delay value derived from frequency results can approximately represent this forward wave’s time delay. In these discrete systems, the waves exist only to a certain frequency range. The frequency range of a uniformly discretized region is a function of average inertia and average impedance and is equal to the mode with the highest frequency. Modes can also be used to display the traveling wave phenomenon. The frequency of the anticipated disturbance that propagates to the rest of the system is determined by the most coarsely discretized region which has the lowest frequency range. The simplified Australian system exhibits traveling wave phenomenon and some characteristics of

184

7 Travelling Waves

the non-uniform discretized system when a fault is applied at either edge of the system. The load modulation effect of SVC is very attractive for some systems where there are numerous units already installed in the network for voltage control purposes. Simulation results show that the proposed SVC controller can successfully attenuate the electromechanical waves in longitudinal power systems. This chapter shows another aspect to the initiation of transients in power systems and the control based on avoiding reflections. Further details of the control aspects of travelling waves can be found in [18].

7.8 Applications One potential advantage arising from the travelling wave concept is that some of the more distant controllers can anticipate the arrival of the wave and potentially give greater attenuation to the disturbance. One of the main contributions from knowing of the travelling wave effect is to note the doubling of the wave at an open circuit. This indicates that the machines at the end of a long line of generators is more at risk of travelling waves in a longitudinal system. For meshed systems the concept allows us to be less concerned of the remote effects from the travelling wave as it attenuates with distance.

References 1. Current, Event Replay of 26.02.2008 Florida blackout caused by substation short circuit fault near Miami (Current, 2013) 2. Y. Liu, A US-wide power systems frequency monitoring network. IEEE PES Power Syst. Conf. Expos. 2006, 159–166 (2006) 3. J.N. Bank, R.M. Gardner, S.S. Tsai, K.S. Kook, Y. Liu, Visualization of wide-area frequency measurement information. IEEE Power Eng. Soc. Gen. Meet. 2007, 1–8 (2007) 4. R. L. Burden and J. D. Faires, Numerical Analysis. Thomson Brooks/Cole, 2005. 5. A. Semlyen, Analysis of disturbance propagation in power systems based on a homogeneoue dynamic model. IEEE Trans. Power Appar. Syst. PAS-93, 676–684 (1974) 6. J.S. Thorp, C.E. Seyler, A.G. Phadke, Electromechanical wave propagation in large electric power systems. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 45, 614–622 (1998) 7. M. Parashar, J.S. Thorp, C.E. Seyler, Continuum modeling of electromechanical dynamics in large-scale power systems. IEEE Trans. Circuits Syst. I Regul. Pap. 51, 1848–1858 (2004) 8. U. Rudez, R. Mihalic, Understanding the electromechanical wave propagation speed, in 2013 IREP Symposium Bulk Power System Dynamics and Control—IX Optimization, Security and Control of the Emerging Power Grid (2013), pp. 1–8 9. M. Ali, J. Buisson, Y. Phulpin, Improved control strategy to mitigate electromechanical wave propagation using PSS, in Melecon 2010–2010 15th IEEE Mediterranean Electrotechnical Conference (2010), pp. 35–40 10. B.C. Lesieutre, E. Scholtz, G.C. Verghese, Impedance matching controllers to extinguish electromechanical waves in power networks, in Proceedings of the International Conference on Control Applications, vol. 1 (2002), pp. 25–30

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11. J.H. Chow, A. Chakrabortty, M. Arcak, B. Bhargava, A. Salazar, Synchronized phasor data based energy function analysis of power transfer paths, in 2006 IEEE Power Engineering Society General Meeting (2006), 5 pp. 12. A. Chakrabortty, J.H. Chow, A. Salazar, Interarea model estimation for radial power system transfer paths with intermediate voltage control using synchronized phasor measurements. IEEE Trans. Power Syst. 24, 1318–1326 (2009) 13. R. Goldoost-Soloot, Y. Mishra, G. Ledwich, Utilizing wide-area signals for off-center SVCs to damp interarea oscillations, in Proceedings of the 2013 IEEE Power and Energy Society General Meeting (PES) (Vancouver, British Columbia, Canada, 2013) 14. K. Uhlen, L. Vanfretti, M.M. Oliveira, A.B. Leirbukt, V.H. Aarstrand, J.O. Gjerde, Wide-area power oscillation damper implementation and testing in the norwegian transmission network, in 2012 IEEE Power and Energy Society General Meeting (2012), pp. 1–7 15. N. Ida, Transients on Transmission Lines, in Engineering Electromagnetics, N. ed. by Ida (Springer International Publishing, Cham, 2015), pp. 833–868 16. S. Ramar, S. Kuruseelan, Power System Analysis (PHI Learning, New Delhi) 17. C. Zhang, G. Ledwich, A new approach to identify modes of the power system based on T-matrix, in 2003 Sixth International Conference on Advances in Power System Control, Operation and Management ASDCOM 2003 (2003), pp. 496–501 18. M. Gibbard, D. Vowles, Simplified 14-Generator Model of the SE Australian Power System (The University of Adelaide, South Australia, 2010). 19. T. Li, G. Ledwich, Y. Mishra, J.H. Chow, A. Vahidnia, Wave aspect of power system transient stability—Part II: control implications. IEEE Trans. Power Syst. 32, 2501–2508 (2017)

Chapter 8

Identification of Dynamics of Inverters and Loads in Power Systems

When adding new inverters to an area the interaction with existing inverters needs to be considered. When the number of existing inverters is limited and perfect models are available then simulation of the new system with all inverters is feasible and control designs able to be refined. However when existing models are not well specified an identification route is highly desirable. Note however that there is not a clear input and output when the inverters operate with load variations as in Fig. 6.1. If one measurement as the machine angles and the other as inverter current angles is considered there is a problem of identification under closed loop as seen in Fig. 8.1. The process in Moyano [1], Parveen et al. [2] identifies a model of each of the measurements in terms of past signals and then separates the residual white noise. Taking these as system inputs the models of transfer function of the feedforward and feedback systems become available. Fig. 8.2 shows the low frequency moderately damped machine mode and Fig. 8.3 shows multiple resonance corresponding to the PLL modes. The system poles are common to all transfer functions identified. The stability of the discrete time system is summarized by having all poles indicated by a ‘*’ well inside the unit circle. The zeros close to poles indicate a cancelling of the effect of those poles. The transfer function b11(z)/a(z) indicates the effect of disturbances on the machine angles and has zeros shown in blue. And b22(z)/a(z) indicates the machine angle noise disturbing the current injection angle with zeros shown in green in Fig. 8.4. As expected, the b11(z)/a(z) term is dominated by the machine poles while the other transfer function b22(z)/a(z) summarises the low damping of the PLL in the case considered. When a stabilizing term is applied to the system the high frequency poles are more stable (Fig. 8.5).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_8

187

188

8 Identification of Dynamics of Inverters …

angI

w1 +

Σ

B(s)/A(s) Angle of I inverter affecting angle of V terrminal

w2 +

+

angV

Σ

D(s)/C(s) Angle of Terminal V changes affecting angle of I

Fig. 8.1 Identification under closed loop

Magnitude (dB)

b11/a

20 0 -20

Phase (deg)

2160 1440 720 0 -720 10 1

10 2

Frequency (rad/s)

Fig. 8.2 Transfer function of machine part

8.1 Summary of Inverter Identification The Inclusion of voltage corrected current controlled inverters can accomplish many of the features of a desired power system. Energy storage is needed in the system at a subset of inverters to achieve frequency regulation. One of the present drawbacks of current control is stability in weak systems, this is overcome by a Kalman based line drop correction to the PLL. Protection concerns for inverter fed systems can be addressed in most cases by adoption of admittance relays (Dewadasa 2010, p. 8725). Designing real controllers needs system models with a graduated level of system modelling and this presentation shows a means of system identification from normal operation.

8.1 Summary of Inverter Identification

189

Magnitude (dB)

b22/a 5 0 -5 -10

Phase (deg)

-15 225 180 135 90 45 10 2

10 1

Frequency (rad/s)

Fig. 8.3 Transfer function of inverter angles as perturbed by machine angles b11 blue, b22 Green

1

0.5

0

-0.5

-1

-1

-0.5

0

0.5

Fig. 8.4 Identified poles and zeros of feedforward and feedback portions

1

190

8 Identification of Dynamics of Inverters … b11 blue, b22 Green 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

-0.5

0

0.5

1

Fig. 8.5 Identified Stability as affected control changes

Fig. 8.6 Load representation based on voltage and power

8.2 Identification of Load Dynamics Using PMUs and SVCs There is often less uncertainty in modelling if a known disturbance signal is added into the control of an actuator. In this case identification of the dynamics of a load with dynamic and motor components is considered in response to the voltage changes caused by an SVC. In transmission systems the line impedances are largely inductive so when reactive power is injected by an SVC it causes primarily voltage changes. If the load were purely constant impedance, then the voltage rise would cause an instantaneous power change to the load. If there were induction motors, then the voltage rise would cause the motor speed to rise but the power to the load would settle close to the original value after a transient period. This implies that for best design of control using an SVC, the dynamics of the load should be considered such as the model in Fig. 8.6. The traditional static load model is used in [3] to represent the power and voltage relation as below.

8.2 Identification of Load Dynamics Using PMUs …

191

V Ref

Fig. 8.7 SVC block diagram with added noise signal

+

|V|

Σ

SVC

V SVC

Noise

Ps = P0 (V /V0 )np

(8.1)

For identification of load dynamics a low level zero-mean uniform white noise is added to the SVC control as shown in Fig. 8.7. The characteristics of the noise signal allows for an efficient identification of load dynamics similar to the application in [4, p. 690] as the same amount of energy can be injected to the system without having a large peak disturbance. The system response is measured using a PMU giving a high accuracy even for these low-level signals. In [5] a simulation was performed on a system with a single machine and SVC and load characteristics similar to an induction motor as in Fig. 8.6 and (8.1) with parameters np = 1.5, T p1 = 1, T p2 = 0.5.

(8.2)

The simulation runs for 15 s to collect input-output data pairs, and the estimated first order transfer function, which is given below, shows a 95.84% fit to the given data which gives confidence in the modelling process 3.008

s + 1.0788 0.5s + 1.0795

(8.3)

In order to examine the accuracy of this identification approach, further simulations are performed on the four machine two area system [6] with various dynamic loading configuration including induction machines [7, p. 691] as shown in Fig. 8.8. Case 1: Simple First Order Dynamic Load Model In the first case, the modelling is applied to a dynamic load with known parameters. It is expected to derive the model parameters with a high degree of accuracy and close resemblance to the known parameters. For this purpose, the following model transfer function was used to represent the dynamic load: H (s) = 0.98087 1.275s+1 0.2s+1

(8.4)

192

8 Identification of Dynamics of Inverters … #7 #1

#5

#8

#9

#6

#10

#11

#3

G1

G3

Load

SVC

Load

#2

#4

G2

G4

Area-1

Area-2

Fig. 8.8 Four machine two area test system

After the implementation of the modelling, the following transfer function was obtained which shows 99.58% match to the measured load data based on the Root Mean Square Error (RMSE) percentage criteria. H (s) = 0.9809 1.2311s+1 0.1999s+1

(8.5)

Also, for further verification, the step response of the estimated model was compared against that of the actual load model by applying a 1% step increase in load bus voltage. The result of the test is shown in Fig. 8.9, which clearly shows an almost perfect match between the responses of the actual and estimated dynamic load models. This confirms the accuracy and effectiveness of the identification method. Case 2: Large Induction Machines (IM) with Constant Impedance load In this case, the load composition is varied by using a large number of 2250 HP induction motors (IM) and a constant impedance load. Each load type contributes to 50% of the total load at the load bus. The implementation of the identification method 1.01 Actual Load Power Response Estimated 1st Order Load Power Response

1

Power (pu)

0.99 0.98 0.97 0.96 0.95 0

1

2

3

4

5

6

Time (S)

Fig. 8.9 First order dynamic load step response comparison (case 1)

7

8

9

10

8.2 Identification of Load Dynamics Using PMUs …

193

yields an accuracy of 99.31%. The estimated transfer function is given below. H (s) =

0.9797s+4.692 s+4.784

(8.6)

The step response test is carried out by the introduction of a 1% step change in the load bus voltage and as a result, a close match in power responses of the actual load and estimated load model is observed which is shown in Fig. 8.10. Case 3: Composite load with Multiple Induction Machines and Constant Impedance In order to introduce more complexity, the load composition is varied to include two types of induction machines namely 500 HP and 2250 HP motors in combination with the constant impedance load. The load configuration is arranged such that each IM type contributes 25% of the total load and the remaining 50% is attributed to the constant impedance load. The utilization of such load composition is inspired from the fact that induction motor loads constitute a significant portion of loads in power systems and the inclusion of a diverse range of IMs is essential in dynamic studies. Based on the RMSE criteria, the identification yields to 99.54% accuracy and once again, the result of step response test is a proof of the effectiveness of the load modelling method which is shown in Fig. 8.11. The estimated transfer function is given below. H (s) =

0.9868s+3.019 s+3.059

(8.7)

Also, it should be noted that in this scenario, the dynamic interaction between load sub-compositions are considered and it does not affect the performance of the identification approach. Therefore, the modelling procedure can consistently yield to an overall aggregate dynamic load model regardless of load composition and sub-constituents. 0.974 Actual Load Power Response Estimated 1st Order Load Power Response

Power (pu)

0.972 0.97 0.968 0.966 0.964 0.962 0

1

2

3

4

5

6

7

8

Time (S)

Fig. 8.10 IMs and constant impedance load step response comparison (case 2 )

9

10

194

8 Identification of Dynamics of Inverters … 0.972 Actual Load Power Response Estimated 1st Order Load Power Response

0.97

Power (pu)

0.968 0.966 0.964 0.962 0.96 0

1

2

3

4

5

6

7

8

9

10

Time (S)

Fig. 8.11 Composite load step response comparison (case 3)

Variation in load composition can result in change of overall system dynamics. As a consequence, the system oscillatory modes are impacted. The impacts are often in the form of variation in mode frequencies or intensity of oscillatory modes at specific frequencies. These impacts are observable through performing spectrum analysis on a system’s generator velocity deviations. This effect is demonstrated for several load compositions which range from constant impedance load, simple firstorder dynamic load to numerous combinations of IM of varying sizes and constant impedance loads. IM loads of different sizes are chosen and used in combination as load sub-composition since IM loads are often a major constituent of power system loads and can impact modes and their intensity. Also, the effect of IM loads on modes is proportional to their power and inertia ratings and their portion of load contribution. The spectrum analysis of generator velocity deviations as excited by random noise for several cases with constant impedance, first-order dynamic load and combination of various IMs are illustrated in Figs. 8.12, 8.13, 8.14, 8.15. From the results of spectrum analysis, it can be inferred that the variation in load composition and dynamics have an impact on the oscillatory mode frequencies and intensity. While the main oscillatory mode of the system (about 0.65 Hz) is present in all the demonstrated cases, lower and higher frequency modes are changed as load composition is varied. Furthermore, generator G2 seems to be the most impacted based on the results of spectrum analysis of all demonstrated cases. This is due to the fact that generator G2 is the closest to the load bus and therefore is the most effected of all. Therefore, the spectrum analysis of generator G2 provides a more detailed insight for this analysis. A comparative spectrum analysis results for generator G2 is illustrated in Fig. 8.15 which represents the results for several load compositions superimposed on the same graph. This clearly demonstrates the variation in mode frequencies and their intensity as a consequence of variation in load composition and associated dynamics.

8.3 Load Adaptive Wide-Area Controlled SVCs

195

10 -8

3

Velocity Change PSD (pu)

Gen 1 2.5

Gen 2 Gen 3

2

Gen 4

1.5 1 0.5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz)

Fig. 8.12 Spectrum analysis of generator velocity deviation for constant impedance load

10

2.5

-7

Velocity Change PSD (pu)

Gen 1 2

Gen 2 Gen 3 Gen 4

1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz) Fig. 8.13 Spectrum analysis of generator velocity deviation for first order dynamic load (case 1)

8.3 Load Adaptive Wide-Area Controlled SVCs As explained in Chap. 4, the kinetic energy-based control is formulated so the SVC can modulate the load and provide damping for low frequency modes through maximizing the reduction of kinetic energy. This can be achieved by modulating the load in the vicinity of the SVC so that the change in load power Pli is equal to k δ˙i . If the load dynamics are not considered, the SVC voltage and load power will have a nearlinear relationship and therefore, the SVS controller can be summarized as u = k δ˙i as shown in Fig. 8.16. The effectiveness of the wide-area controller in damping the post-fault oscillations depends on the active-power modulation capability of the SVC

196

8 Identification of Dynamics of Inverters … -8

10

1.5

Gen 1

Velocity Change PSD (pu)

Gen 2 Gen 3

1

Gen 4

0.5

0 0

0.5

1

1.5

2

3

2.5

4

3.5

5

4.5

Frequency (Hz)

Fig. 8.14 Spectrum analysis of generator velocity deviation for the composite load 10

Velocity Change PSD (pu)

1.2

-7

Constant Z Dynamic Load Small IM + Constant Z Large IM + Constant Z Small IM + Large IM + Constant Z Small IM + Large IM

1 10

0.8

-8

1.5

0.6 1

0.4 0.5

0.2 0

0 0.2

0

0.5

1

1.5

2

0.4

0.6

2.5

0.8

3

1

3.5

1.2

1.4

4

1.8

1.6

2

4.5

Frequency (Hz)

Fig. 8.15 Spectrum analysis of generator G2 velocity deviation for different load compositions Fig. 8.16 SVC controller

5

8.3 Load Adaptive Wide-Area Controlled SVCs

197

Fig. 8.17 Load, SVC and PMU location

PMU

SVC

Load

through reactive power modulation. In reality, the loads in power systems are not static and the SVC controller should consider the dynamics of the loads. The identification of load dynamics enables a better control design for the SVCs through compensation of phase shift and delays associated with the dynamic loads in the SVC wide-area controller. This process can be applied in repeated intervals to better represent the changing nature of the load dynamics. Using the identified model, an inverse filtering approach similar to the method explained in Chap. 5 for the excitation system can be applied to compensate for the load dynamics in the control design. In this section, a direct approach is presented for developing the load-compensated SVC controller which can adaptively modify the controller based on the identified load dynamics. In this approach, dynamic moving average (MA) model of the load is obtained through the SVC and PMUs. In this method a similar noise signal as shown in Fig. 8.7 is applied to the SVC to excite the load dynamic. This will cause small voltage fluctuations in the load terminal voltage and since the load power is also a function of load voltage, any voltage fluctuation will result in active power fluctuations. The load power response to voltage fluctuations is then measured by a PMU over a specific time windows and serves as the raw data for the load modelling process. A block diagram of Load, SVC and PMU locations is illustrated in Fig. 8.17. The estimation of a Moving Average (MA) load model begins by processing of the measured load power and voltage data which are obtained by PMU measurements. This process involves resampling and filtering the measured data in order to extract inter-area dynamics within 0.1–1 Hz frequencies. Then, a suitable MA model order must be chosen based on the required input samples that result in accurate estimation of the output. The MA model structure is given as below which defines the relationship between the load power and the SVC bus voltage: M A(m) : yt = b0 u t + b1 u t+1 + b2 u t+2 + b3 u t+3 + · · · + bm u t+m

(8.8)

where m, t, b, u and y are the order of MA model, sample number, model coefficient, input and output values respectively. The parameters of the MA model shown in (8.8) can be represented in a more compact form by the use of vector notation as followings:

198

8 Identification of Dynamics of Inverters …

β = [b0 b1 b2 b3 . . . bn ]T

(8.9)

T  ϕ = u t u t+1 u t+2 u t+3 . . . u t+m

(8.10)

Therefore, the MA model can be written as: M A(m) : yt = ϕ T β

(8.11)

The SVC controller relies on Wide-Area Measurement Systems (WAMS) using PMUs in order to acquire equivalent area and velocities and to obtain load power and voltage data to identify the Moving Average dynamic load model. The controller aims to damp inter-area oscillations by further enhancing the control algorithm explained in Chap. 4. This needs to be translated into the required load voltage variations that result in the desired control objective. This effectively modulates the area load power by varying load voltage through the SVC based on load dynamics represented by the MA load model; which improves the performance of the wide-area controlled SVCs. The utilization of the MA load model for load modulation is expressed in (8.12), where the required load voltage variation is expressed as a function of desired load power response. By substitution of the kinetic energy based control law into (8.12), the desired controller expression (8.13) is obtained defines the SVC switching action. Vlik = b0 Plik + b1 Plik+1 + b2 Plik+2 + · · · + bn Plik+n

(8.12)

u ik = b0 k δ˙ik + b1 k δ˙ik+1 + b2 k δ˙ik+2 + · · · + bn k δ˙ik+n

(8.13)

The Moving Average (MA) model in (8.13) uses the present and estimated future angle and velocity inputs to improve the control performance. In order to obtain predicted future input values a predictor-corrector or Kalman estimator as described in Chap. 4 can be employed to estimate future values of equivalent area angles and velocities using the PMUs in the system. The block diagram of the adaptive Wide-Area Controller is shown in Fig. 8.18.

8.4 Implementation of Adaptive Wide-Area Controller The adaptive Wide-Area Control method is evaluated on the four machine two-area system with the addition of an SVC placed at Area-1 as Fig. 8.8. A composite load model comprised of induction motor (IM) and constant impedance loads are used in order to introduce complex dynamics. During the load modelling process, it is assumed that the system is at its nominal operating point and at a state of equilibrium.

8.4 Implementation of Adaptive Wide-Area Controller Wide-Area Measurement

199

Wide-Area Control Inter-Area Dynamics Estimator

u

uk = b0 K δ k + b1 K δ k +1 + b2 K δ k + 2 + ... + bn K δ k + n

WAMS

PMU

MA Model Estimator

Fig. 8.18 Adaptive wide-area controller block diagram

In order to implement the Moving Average (MA) load modeling method, the SVC is set to inject a noise signal in the form of a uniform random number for excitation of load dynamics at 0.05 s intervals (20 Hz). This induces small variations in load bus voltage and directly results in variation of load consumed power which are measured by a PMU. These variations result in the raw power and voltage data streams that are used for load modelling which are shown in Fig. 8.19. The load model is then estimated by processing the load power and voltage data and applying the MA load modelling method that can provide the best fit to the measured load data. In this implementation, a composite load model with small and large equivalent induction motors (IM) in combination with a constant impedance load is chosen. The load composition is selected to have a dominant component of dynamic loads by including 30% small, 30% large induction motor (IM) and 40% constant impedance loads. By applying the load modelling method in the studied case, the following third-order MA load

Power (pu)

0.97 0.965 0.96 0

1

2

3

4

5

6

7

8

9

10

Voltage (pu)

0.98 0.975 0.97 0

1

2

3

4

5

6

7

Time (s)

Fig. 8.19 Load power and voltage response to injected noise via SVC

8

9

10

200

8 Identification of Dynamics of Inverters …

model is obtained which results in 95.52% accuracy based on the RMSE criterion: Vlk = − 19.87Plk + 59.89Plk+1 − 61.29Plk+2 + 22.53Plk+3

(8.14)

Therefore, the control action of the SVC based on load dynamics can be expressed as: u k = − 19.87k δ˙k + 59.89k δ˙k+1 − 61.29k δ˙k+2 + 22.53k δ˙k+3

(8.15)

To evaluate the effectiveness of the this adaptive control method, a 120 ms selfclearing three-phase short-circuit fault is applied on the tie-line closer to bus 7 and the effect of controllers in damping post fault inter-area oscillations for the cases with and without wide-area control (WAC) and with load dynamics considerations are compared as shown in Figs. 8.20 and 8.21. The obtained results show that both wide-area control methods exhibit superior performance in damping the oscillation of the system when compared to the case without wide-area control. However, the adaptive load based control method has a further advantage since the load modulation effect of the SVC is enhanced by involving load dynamics in the damping controller implementation. In order to further examine the performance of this load adaptive control method on larger system, simulations have been performed on the 14-generator simplified Australian test system as shown in Fig. 4.9. The loads in the system are assumed to be composite loads as Fig. 4.8 including 50% induction machines of various size. The system was simulated with a three-phase short-circuit fault applied on bus 308 in area 3, and the stability characteristics of the system is compared for the cases without WAC, with WAC and with Load Adaptive WAC controllers. The generator angles for the three compared cases ae illustrated in Fig. 8.22 and the critical clearing time (CCT) are compared in Table 8.1. The results show that the load adaptive wide-area controller can further improve the transient stability of the system by significantly increasing the CCT. The generator angle trajectories in Fig. 8.22 also show the improvement in damping of the post fault oscillations through both WAC methods while the load adaptive WAC shows a superior performance.

8.5 Identification of a Set of Thevenin Impedances from Residential PV Inverters There is a growing concern in distribution systems of the voltage rise caused by PVs in a feeder. Much of the research has proposed droop controllers to give reactive power from an inverter when the voltage deviates. A contribution to reactive power

8.5 Identification of a Set of Thevenin …

201

20 Without WAC

COI Area Angle (deg)

With WAC 15

With WAC and MA load model

10

5

0 0

5

10

15

20

(a)

COI Area Angle (deg)

0

Without WAC With WAC With WAC and MA load model

-5 -10 -15 -20 -25 0

5

10

15

20

Time (S)

(b) Fig. 8.20 Angles of area 1 and area 2

can impair the real power capability of the inverter and increase the losses. If the same gain is used for all customers on the feeder a much higher burden would be placed on customers towards the end of the feeder. If the gain used in the droop control is adjusted based on the distance along the feeder a more even spread of effort can be implemented. It has been found that this spread can be achieved if the droop gain is reduced with any increase of Thevenin impedance of the supply as seen by any given inverter [4, p. 8708]. The actual Thevenin impedance will be affected by connections of sections of a line to different feeders at different times. Thus an offline study to compute the Thevenin impedance at each node would not yield a good solution. This section proposes the inferring of the system impedance by seeing the amount of change of terminal voltage of the inverter as a function of the injected reactive power. If there were no customer load changes and no changes from the adjacent inverters this would be a simple exercise. It is known from the cases in Chap. 5 and for distribution in Chap. 9 that the changes in loading by customers are approximately white noise so this would not bias the evaluation of impedance from a given inverter.

202

8 Identification of Dynamics of Inverters … 10

COI Area Velocity (pu)

2

-3

Area 1 Without WAC Area 1 With WAC

1

Area 1 With WAC and MA load model

0

-1

-2 5

0

10

COI Area Velocity (pu)

2

10

15

20

(a)

-3

Area 2 Without WAC Area 2 With WAC Area 2 With WAC and MA load model

1 0 -1 -2 0

5

10

15

20

Time (S)

(b) Fig. 8.21 Velocities of area 1 and area 2

It is necessary to ensure that the changes in the injected signal are orthogonal to changes from other inverters or from customer loads. If each source were a pseudo random sequence or selected uniquely from a Walsh sequence then there is confidence that there would be no long term bias in the estimates [8]. Uniform probing is where the disturbance signal is based on a uniformly distributed random variable. The advantage is that the peak of the disturbance is kept low while still giving sufficient variation to permit identification. Gaussian variables can have occasional very large disturbances and thus more potential impact on other customers.

8.5.1 Example of Identification of Thevenin Impedance in Distribution For the 33 bus example [9] in Fig. 8.23 uniform probing was used and impedances found as in Fig. 8.24.

8.5 Identification of a Set of Thevenin …

203

COI Angle (deg)

150 100 50 0 -50

(a)

COI Angle (deg)

-100 150 100 50 0 -50

(b)

COI Angle (deg)

-100 150 100 50 0 -50

(c)

-100 5

0

10

15

Time (s) 101 401

201 402

202 403

203 404

204 501

301 502

302 501

Fig. 8.22 Generator rotor angles for case a without WAC, b with WAC c with load adaptive WAC

Table 8.1 Comparison of CCT

Critical clearing time (CCT) No wide-area control

Wide-area control (without load dynamics)

Wide-area control (with load dynamics)

185 ms

225 ms

292 ms

The main aspect of which to be careful is when finding the Thevenin voltage change at a given point of the network without direct use of a PMU. The change of angle should be measured with respect to a local timing reference so that the Thevenin voltage and Thevenin impedance are both identifiable. The results of this section show that on-line identification from small probing signals is feasible provided that none of the sources of variations are correlated.

204

8 Identification of Dynamics of Inverters …

Fig. 8.23 33 bus network with 3 laterals

8.6 Applications of Thevenin Identification The key concept is that the power from inverters within a feeder affect the angle of the connection bus. Also the angle of the voltage at the connection point will in the short term affect the power from the inverters. The noise process is the key issue. The task requires to have an inverter noise from PV in a feeder that is independent of the noise in the Thevenin voltage of the connection point. Thus when there is a strong influence from other PV on the connection point angle the concern of independence should be raised. If customer load perturbations dominate over the PV perturbations then there is an expectation that the identification will work. Note that if there is LQR control over inverters with remote angle references than the concern that other inverters will affect any new inverter diminish significantly and there is no real need to consider the other inverters in the proximity of any proposed connection point. The identification process will then be able to confirm sufficient independence of the phase locking of any inverters in an area.

8.6 Applications of Thevenin Identification

205

Fig. 8.24 Inferred thevenin impedance as function of location

References 1. C.F. Moyano, G. Ledwich, Load modelling: induction motor, in Electric Power Systems in Transition (Nova Science Publisher Inc, 2010) 2. T. Parveen, G. Ledwich, E. Palmer, Model of induction motor changes to power system disturbances, in Australasian Universities Power Engineering Conference AUPEC 2006 (Melbourne, VIC, Australia, 2006) 3. J.V. Milanovic, I.A. Hiskens, Effects of load dynamics on power system damping. IEEE Trans. Power Syst. 10, 1022–1028 (1995) 4. A. Raghami, G. Ledwich, Y. Mishra., Improved Reactive Power Sharing Among Customers’ Inverters Using Online Thévenin Estimates. IEEE Trans. Power Syst. 34, 4168–4176 (2019) 5. K. Q. Hua, A. Vahidnia, Y. Mishra, G. Ledwich, PMU Measurement Based Dynamic Load Modeling Using SVC Devices in Online Environment (2015) pp. 1–5 6. P. Kundur, Power System Stability and Control (New York: McGraw-Hill, 1994) 7. Pekarek, Steven, et al., Analysis of Electric Machinery and Drive Systems. United Kingdom, Wiley, 2013.

206

8 Identification of Dynamics of Inverters …

8. C. Zhang, G. Ledwich, A new approach to identify modes of the power system based on T-matrix, in 2003 Sixth International Conference on Advances in Power System Control, Operation and Management ASDCOM 2003 (2003), pp. 496–501 9. M. E. Baran and F. F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Trans. Power Deliv, 4, no. 2, pp. 1401–1407, April 1989, 10.1109/61. 25627

Chapter 9

Phasors for Distribution

9.1 Introduction Phasor measurements have mainly been applied for transmission systems partly because of cost. There is a continuing reduction in the cost and distribution applications are growing. One of the developing issues is the impact of solar panels in the distribution network causing voltage rises and the potential for battery use to expand. If all customers simultaneously exported power from their battery that would have even more potential for overvoltage problems. The growth in these Distributed Energy Resources (DER) is driving a greater need to observe the flows and voltages in distribution. It has been proposed that estimation of the state of voltages on a distribution line would be a lower cost path rather than extensive measurements and communication. This is examined in Sects. 9.2, 9.3, 9.4, 9.5. Another issue for distribution is the extent to which distribution loads can contribute to overall system performance particularly system angle control. This is examined in Sect. 9.6.

9.2 Distribution State Estimation Transmission State Estimation for an n bus network needs 2n measurement to learn 2n unknowns [1–3]. In transmission there are limited numbers of busses and the assets are much more expensive so having enough measurements is not a problem. For distribution, the numbers of busses is huge while the measurements are sparse. The task is tackled by use of pseudo measurements, intelligent approximations from other operational considerations. For solution, these pseudo measurements are assigned a large uncertainty. The most common DSE is designed based on the weighted least square (WLS) method, as detailed in [4]. For observability of an n bus network still needs 2n measurements real and pseudo. The WLS solution follows the path of transmission estimators, taking a set of measurements at one time instant and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_9

207

208

9 Phasors for Distribution

forming the best estimate of the state at that time. The next time step is treated as a separate problem. The main limitation of WLS-based works is requiring extensive real measurements with fast communication platforms, which is not cost-effective in distribution networks. There is a high use of Kalman filters in power application and given a model of the propagation of system states can track system states in time. WLS as a snapshot algorithm and high resolution measured data is applied in [5, 6], but it requires a high number of measurement devices. This makes it a costly solution. In [7], it is shown that Kalman filter as the most common time series forecastingaided state estimator (FASE) [8] has a better performance compared with WLS for distribution networks. Two decoupled FASE algorithms are reported for estimating voltage magnitude and voltage angle independently in [9]. These algorithms are not attractive because of high computational cost and ignoring the dependencies between magnitude and angle estimation noises. A complex formulation for the state estimator can potentially improve the accuracy of estimation [10]. This chapter shows a complex number Kalman filter solution that in the first instance required complex number measurements and thus are PMU based. The next section shows how the recursive implementation of Kalman can make use of scalar measurements.

9.3 Forecast Aided Complex State Estimator The system equations are based on phasor relations of voltage current and impedances. Implicitly the system is assumed in electrical steady state. Dynamic equations are used to show the progression of electrical loads. A key development is to show an appropriate model for customer loads in the short term.

9.3.1 Augmented Complex Kalman Filter A general linear state-space model is given as [11]: xi = Fi−1 xi−1 + wi

yi = Hi xi + n i

(9.1)

where xi and yi are the complex state and measurement vector of the system at discrete time i, respectively. Fi−1 and Hi denote state transition and observation matrices, and (wi ) and (n i ) represent white noises. In a strictly linear distribution network model with complex states and measurement vectors, (1) can be rewritten in a so-called augmented complex vectors format as [10]:

9.3 Forecast Aided Complex State Estimator

209

a a xia = Fi−1 xi−1 + wia

yia = Hia xia + n ia

(9.2)

where (.)a represents the augmented complex vectors;        xi yi Fi 0 Hi 0 a a a , y , F , H , wia = = = i i i xi∗ yi∗ 0 Fi∗ 0 Hi∗     wi ni a and n i = , = wi∗ n i∗ 

xia =

with (.)∗ as the complex conjugate operation. In the state-space model, it is assumed that the state and measurement noises are uncorrelated with zero mean [12]. The augmented state and measurement noise covariance matrices Q ia and Ria can be calculated using (9.3):     Q CQ Q ia = E wia wia H = C∗  ∗  Q Q   R CR Ria = E via via H = C∗R  ∗R

(9.3)

where E(.) and (.) H are the mean and transpose-complex conjugate operators, respectively.  Q and  R are the covariance matrices, and C R and C Q are pseudocovariance matrices for complex state and measurement noises. After modelling a linear system in a complex state-space format, and introducing state and measurement noises, ACKF can be formulated through the following three steps: State and covariance matrix initialization: xˆ0|0 = E(x0 )       a P0|0 = E (x0a − E x0a )(x0a − E x0a ) H

(9.4)

Updating states and covariance matrix: xˆi|i−1 = Fi−1 xˆi−1|i−1 a a a aH Pi|i−1 = Fi−1 Pi−1|i−1 Fi−1 + Q ia

Updating states based on measurements:

(9.5)

210

9 Phasors for Distribution

 −1 a a G ia = Pi|i−1 Hia H Hia Pi|i−1 Hia H + Ria    ∗ xˆi|i = xˆi|i−1 + G i11 yi − Hi xˆi|i−1 + G i12 yi∗ − Hi∗ xˆi|i−1  a  a = I − G ia Hia Pi|i−1 Pi|i

(9.6)



where x i|i shows a posteriori state at time i giving observation including at time i, P a denotes the augmented state covariance matrix, and G a is the Kalman gain. G ia is a square matrix covering gains on the original and augmented state. If the state is of size ‘m’ then the augmented state is ‘2 m’ and the P is 2 m by 2 m. In practise the steady state solution for the Kalman filter gain can be computed off line. Also since it is easy to find the conjugate state from the real, the equations to propagate the augmented state are reduced to size ‘m’.

9.3.2 Implementation To apply this theory to the power system problem F and H are defined as in (9.1) and (9.2). Note that in Chap. 3 that the major load centers act as the integral of white noise. The changes in load in the short term are independent of past history. Measurements at the distribution level show this same pattern. The measured data of Newmarket suburb, Queensland, Australia with temporal resolution of one minute and for a period of seven days is shown in Figs. 9.1 and 9.2. This measurement set was used to give a pseudo measurement of load currents by using the energy bill to apportion total feeder flow among the customers. Thus if the difference is white, the model in (9.1) gives the injected current approximately as the sum of previous current plus a white noise term. In other words if the state is the set of injected current from customers, F is the identity matrix. The next step is to define the measurement matrix H. In order to define the measurement vector and the observation matrix, the direct power flow algorithm [13] is employed. In this method, the relation between injected Fig. 9.1 One step difference of injected current

9.3 Forecast Aided Complex State Estimator

211

Fig. 9.2 Correlations of one step difference currents

currents (i in j ), branch currents (i branch ) and bus voltages (vin j ) is formulated as: 

i branch = B I BC ∗ i in j vin j = vr e f − DL F ∗ i in j

(9.7)

where B I BC is the Bus-Injection to Branch-Current matrix, DL F is the Direct Load Flow matrix and vr e f is the voltage of the reference bus [13]. The measured vector contains a combination of the calculated pseudo data and the available real measurements. The measurement matrix includes, pseudo measured a subset of injected currents (I pin j ), a limited number of measured bus voltages a (V m in j ) and branch currents (I m abranch ) as: ⎤ a I pin ji ⎥ ⎢ yi = ⎣ I m abranch i ⎦ a V m in ji ⎡

(9.8)

Based on (9.7), the measurement matrix H is given as ⎤ I(m−1) H = ⎣ B I BC m ⎦ −DL F m ⎡

(9.9)

where B I BC m and DL F m contain rows of B I BC and DL F matrices for corresponding measured branch currents and bus voltages. In this section all elements of y are phasor measurements describing the evolution of the load current phasors. Because of the direct load flow approach [13] the phasors are all taken with respect to the source voltage. The use of PMU’s would provide correct phasor current measurements.

212

9 Phasors for Distribution

9.3.3 Weighted Least Squares (WLS) Formulation The WLS approach to power system SE is well-known (e.g. [14]) and can be modified to provide a comparison with the ACKF approach. The conventional approach to WLS is to solve the nonlinear function involving the Jacobian of the power system. In this case, the direct load flow approach was used which relates the measurement vector y to the state vector x with a linear function H. As such as linear WLS can be adopted to solve the problem at one time instant. From (9.2), the WLS approach can be formulated from the linear function, such that state estimate is given by,  −1 T −1 H R (y − v1 ) x = H T R −1 H



(9.10)

for a measurement v1 . This assumes the observational errors are uncorrelated and the R matrix is diagonal (i.e. error variance matrix). This formulation is used as a comparison to the ACKF approach.

9.3.4 Network Layers Even with the use of efficient implementation of Kalman filters the computational cost for an ‘n’ bus system, will still be of the order of n3 . The means to reduce this computational cost is to break the network into several layers. If there are ‘m’ layers with equal bus numbers and low computational overhead of co-ordinating the layers the expected cost would be m. (n/m)3 . Consider a 23-bus Australian distribution network, the unbalanced 23-bus distribution network is shown in Fig. 9.3. The DSE- MACKF algorithm is formed for this network by deploying B I BC and DL F matrices. In order to decrease the simulation time, the network is divided into one main area and three subareas. Subarea 3 has ten buses and imposes a higher computation time compared with other subareas. Hence, by adding another estimation layer, this subarea itself is divided into two subareas, where bus 17 aggregates the loading of subarea 4 in the second estimation layer. Figure 9.4 illustrates the proposed algorithm through the three layers. In the first layer, the current of main branches and the voltages at boundaries between the main area and subareas are estimated. In the second layer, by incorporating results for bus 2, 10 and 11 from the first layer, the state estimation is conducted in parallel for three independent subareas. Finally, based on the estimated state of bus 17, the states in layer 3 are estimated. The three layer approach has negligible loss of accuracy as seen in Table 9.1. As expected the computation cost has significantly improved (Tables 9.1 and 9.2).

9.3 Forecast Aided Complex State Estimator

213

Fig. 9.3 A MV/LV unbalanced distribution network

3

2 3

1

MV/LV

4 4

5 5

6 6

7 Area 1

7

9 8

2

9

8

Area 2 10 11 12 13 14

10

11

12 13

14

15 16 Area 3 20 19

21 22

18

23

20 19 18 17 21 22 23 Area 4 Fig. 9.4 A three-layer state estimation representation

2

1

MV/LV

9 8

2 8

9

Layer 1 10 11

10

3

12 13 14

2 3 4

4 5

5

11 12 13

11

10 15

14

Layer 2

6

16

6 7

7

20 19

17

18

21 22

23

20 19 18 17 21 22 23

Layer 3

214

9 Phasors for Distribution

Table 9.1 Magnitude voltage error in case study One layer ACKF

Three layers ACKF

AMVE (%)

0.18

0.17

MMVE (%)

0.57

0.44

Time (s)

271

68

Table 9.2 Voltage magnitude error and computational time in case study One layer ACKF

Three layers ACKF

Bus 10

Bus 23

Bus 10

Bus 23

AMVE (%)

0.06

0.13

0.06

0.12

MMVE (%)

0.43

0.9

0.32

0.6

9.3.5 Scalar Measurements For scalar measurements, the measurement matrix, H needs to be changed into the ∗ . and the case where the augmented form. Consider the relationship Re(z) = z+z 2 magnitude of voltage is measured and thus the portion of (9.2) becomes  DL F a DL F − x Re(v) = − 2 2 

(9.11)

The voltage angle in distribution does not change significantly and Re(vi ) is an approximation of |vi |. A better approximation is given as |vi | = [−

DL F i − DL F i /2]x a /cos(θi ) 2

(9.12)

where θi is the angle at bus ‘i’ with respect to the source voltage angle. Because the Kalman filter is recursive, the angle of the voltage can be taken from the previous iteration. The angle converges within 2–3 steps of the algorithm. Scalar current measurements can be similarly approximated when the angle of the current is not large. If it is known that the angle of current is closer to 90 degrees then a better starting point is to approximate the magnitude of current from the imaginary ∗ and including the correction term part of the current. Im(z) = z−z 2 |Ii | = [−B I BC i /2B I BC i /2]x a /sin(θi )

(9.13)

9.4 Results The IEEE 34-bus test feeder shown in Fig. 9.5 is used to validate the scalar estimator. The load for each bus was generated using a complex-valued Wiener process with

9.4 Results

215

22 21 20

1

2

3

4

5

6

802 806 808 812 814 (1) (2) (3) (4) (5)

7

858 (14)

824 (8)

816 (7)

14

800

Measurements Magnitude Voltage Branch Power Flow (P, Q) (.) Parentheses show bus numbers used in this simulation

832 (13)

9

846 (21) 844 (20)

862 (19)

842 (17)

19

17 23

8

850 (6)

848 (22)

16

24

18

834 860 836 840 (23) (16) (24) (18)

15

25

888 (15)

890 (25)

13

852 (12) 12

828 (9)

10

11

854 (11)

830 (10)

Fig. 9.5 Modified IEEE 34-bus test feeder

a noise term based on a normal distribution with a non-zero mean. The real and imaginary components are considered as independent Wiener processes. The purpose of the non-zero mean is to reflect the inherent different loadings on each bus. The results consider a single simulation set and a set of simulation to evaluate statistical performance. The purpose of the single simulation set is to show the timeseries performance of the ACKF estimator compared with the actual load flow and with the WLS estimator. For each simulation set, a total of 200-time steps was generated. The methodology consisted of the following: 1. Generate complex-valued current injection for each bus location as described 2. Initialize the ACKF state estimator 3. At each time step, solve the load flow and find angles of for the scalar measurements 4. At each time step, update the ACKF state estimator using the available measurement points 5. Calculate the estimated state of the network.

216

9 Phasors for Distribution

The full simulation set repeats the simulation set 100 times for statistical relevance. The purpose of the full simulation set is to demonstrate the performance over a statistically relevant sample size, for determining accuracy of estimated error variance. The full simulation set performance was assessed using two metrics: mean absolute error (MAE) and the normalized mean error variance (NMEV). MAE was calculated across all the simulations for both bus voltage and branch current magnitude, as per (9.14), M AE =

 1 m 1 n  xi − x i  j=1 n i=1 m

(9.14)

where n is the number of timesteps per simulation, and m is the total number of simulations. x are the parameters of interest which are the bus voltage and branch current magnitudes. In the results, the voltage magnitude MAE and branch current magnitude MAE is denoted as VMMAE and BMMAE, respectively. Over the full simulation set, the actual error variance of the state estimator was calculated using the NMEV for both bus voltage magnitude and branch current magnitude, as per (9.15). M EV =

m n 2 1  1  ei − μ j m j=1 n i=1

N M EV =

M E V − min(M E V ) max(M E V ) − min(M E V )

(9.15)

where ej is the error at timestep i for bus voltage magnitude. The branch flow and voltage measurements are as shown in Fig. 9.6. These results show that the error increases for busses further from the real measurements, Because the WLS only takes data from one time point the estimate is significantly worst than the Kalman approach which refines the estimate at each time step (Fig. 9.7). Repeated runs show that the variance calculated from the Ricatti equation in (9.6) is well matched with the achieved result in Fig. 9.8.

9.5 Summary of Distribution State Estimation Phasors can be used for state estimators in distribution by use of pseudo and real measurements. The Kalman filter shows better performance than least squares because of the recursive update of Kalman compared to the estimate based on a single time step in Least Squares. Scalar measurements can be incorporated into the Kalman process through the time iteration process.

1.5

Actual ACKF

0

100

Branch 9 P (MVA)

200

Q (MVA)

1.2 1.0 Actual

0.8

ACKF

100

0.5

Actual ACKF

100 Actual

0.1

ACKF

0

1.5

100

200

Actual ACKF

0

100

200

2.0 Actual

1.5

ACKF

0 1.5

0.5

200

0.2

0.0

2.0

200

1.0

0

2.5

Q (MVA)

1.0

Q (MVA)

2.0

0

Branch 18 P (MVA)

217

Q (MVA)

Branch 5 P (MVA)

Branch 1 P (MVA)

9.5 Summary of Distribution State Estimation

100

200

Actual

1.0

ACKF

0

100

200 Actual ACKF

0.0

0

100

200

Timestep

Fig. 9.6 Comparison of real and reactive power flow of actual and estimated for selected branches

Voltage Mag (pu)

0.20 0.10 ACKF

0.00

WLS

0

5

10

15

20

25

Bus Number

Voltage Angle (deg)

1.00 0.50 ACKF

0.00

WLS

0

5

10

15

20

25

Bus Number

Fig. 9.7 Comparison of voltage error between ACKF and WLS approach

In general a phasor measurement is superior for computation but usually at a higher cost.

218

9 Phasors for Distribution

Normalized Error Variance

1

ACKF Actual

0.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

Bus Number Fig. 9.8 Normalized average error variance for magnitude bus voltage

9.5.1 Contribution of Distribution Loads and Batteries to System Stability While Assisting Local Constraints Distributed inverters can make a substantial contribution to voltage control in distribution {Perera, 2012 #8719}. The contribution can be for magnitude and balance at MV and LV. The consideration here is for an impact on dynamic stability. The initial study is for a simple two machine system with a distribution system on bus 2 as in Fig. 9.9. The distribution system is as for {Perera, 2012 #8719} as shown in Fig. 9.10. The nature of the distribution system is that the angle changes at the transmission connection point of bus 2 will been seen in the angle changes at each inverter point. The Fig. 9.10 shows the voltage support towards the ends of the LV feeders to gives good voltage support where the effect is strongest. In the study of this section five locations are used with independent real and reactive control of each phase subject to inverter rating.

Fig. 9.9 System area model showing connection point of distribution load

9.5 Summary of Distribution State Estimation

Tap Change Transformer

219

MV busses

Fixed Tap

LV loads DStatcom

Fig. 9.10 Model of distribution system with inverters

9.5.2 Voltage Controllers When there are voltage controllers, the choice is between central controllers and distributed. For centralized control the voltage levels at each point on each line need to be communicated to the controller, an optimized control calculated and the reactive injections communicated to each controller. Distributed controllers aim to make decisions on the level of the local voltages and determine the reactive injection only based on the local voltage magnitude. For conventional generators the droop system enables sharing of voltage control between different generators generally in proportion to rating. The droop setting is the gain which determines the level of reactive injection proportional to the voltage error from a reference. In Fig. 9.11 this is the steady state droop line at the top of the graph. When inverters are used there is a much faster response than conventional generation. And the fast control with high gain is prone to instability. The solution shown in Fig. 9.11 is a low gain instantaneous droop line and then the injected reactive

Fig. 9.11 Integral to droop line control

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9 Phasors for Distribution

power integrates until the steady droop line is reached. This gives a system with low transient gain and high steady state gain. Note this is not an integral controller which would aim for every voltage to reach the reference rather it integrates to reach the droop line.

9.5.3 Centralized Control of Real Power In this case consider identical inverters in the distribution network modulating the battery power based on the rate of change of angle at bus 2 of Fig. 9.10. Consider also that there are non critical loads such as water heaters which can contribute to load modulation in response to the same signal. Consider initially where the there is no control of batteries or loads. In Fig. 9.12 a significant transient in power flow is seen and in Fig. 9.13 the system angles drop significantly while the extra load is applied. p12 -38 -38.2 -38.4 -38.6 -38.8 -39 -39.2 -39.4 -39.6 -39.8

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Fig. 9.12 Power flow on the line between the busses in relation to a step load change on Bus 2 from 2.5 to 4 s. (no control)

9.5 Summary of Distribution State Estimation

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source ang 40

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Fig. 9.13 Variation of bus angles during transient. (no control)

9.5.4 Load Modulation Only If only load modulation is used and control the loads proportional to a broadcast value of the Kalman version of rate of change of angle there is a significant damping of the oscillations in Fig. 9.14 compared to Fig. 9.12. Similarly the angle drop in Fig. 9.15 during the load step is now 7 deg compared with 16 deg with no control (Figs. 9.16, 9.17).

9.5.5 Battery and Load Modulation As seen in Fig. 9.18, when the batteries and the loads are employed then the nett angle change under load steps in further reduced to 3 degrees (Figs. 9.19, 9.20).

9.5.6 Using Local Angle Measurement The control of batteries for wide area control based on the rate of change of angle has positive impact on frequency control and transient stability. The question arises whether the local bus angle is a good signal to use for the control of the batteries

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-38.4

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Fig. 9.14 Power flow between generators (load modulation, Broadcast)

source ang 35 30 25 20 15 10 5 0 -5 -10

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Fig. 9.15 Angles of generators (load modulation, Broadcast)

to avoid the need for communication signals to be broadcast to customer battery controllers within areas. The difficulty is that the local bus angle responds to all local load changes and more particularly from power flow changes from the use of batteries. The other issue is that there are continual changes in customer load which

9.5 Summary of Distribution State Estimation

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Power to cct 39.6 39.4 39.2 39 38.8 38.6 38.4 38.2 38 3

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Fig. 9.16 Power flow to distribution circuit (load modulation, Broadcast)

1.02 1.01

V at Dstatcom

1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0

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Fig. 9.17 Voltage at injection points (load modulation, Broadcast)

will cause variation of the angle at each load node. System operators are familiar with the continual load perturbations causing voltage changes but when the R/X of distribution lines is near unity the angle changes are of the same level as the voltage changes. The approach taken in this section is to use a Kalman filter tuned to the expected modes of the transmission system implemented as a discrete time controller. To represent the distributed batteries there are 5 locations of inverter systems in the distribution network which have a real power as well as a reactive injection control capacity. The reactive component of the inverter is used as an integral to droop line

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Fig. 9.18 Angles of generators under battery and load modulation (Broadcast)

source ang 40 30 20 10 0 -10

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Fig. 9.19 Voltages at injection points under battery and load modulation (Broadcast)

1.05 1.04

V at Dstatcom

1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 3

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controller of local voltage. The real component will be based on the rate of change of angle at the terminals. To reduce the sensitivity to angle changes the Kalman filter is applied to each phase of the terminals of the inverter. The simple version of the system model considered only the second order dynamics of the two area system. The state estimate x follows

˙ x = Ax + Bu + Gw,



y = Cx + v

(9.16)

In discrete time the Kalman equations can be expressed





x (k + 1) = F x (k) + H u(k) + Gw(k) − L ∗ (y (k) − y(k)),





y (k) = C x (k) + v(k) (9.17)

where L is the Kalman gain and y(k) contains the measurement of local bus angle for the appropriate phase. When applied directly the unmodelled component of the

9.5 Summary of Distribution State Estimation

225

1.1 1.08 1.06

MV level

1.04 1.02 1 0.98 0.96 0.94 0.92

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Fig. 9.20 Bus voltage magnitude and the inverter connection points. (Battery and Load modulation broadcast)

different bus angles gave an offset to both the angle and velocity estimates. A pseudo measurement of bus velocity was provided from one-step differencing, thus y(k) also has a second element containing the pseudo “rate of change of angle” measurement. This did not correct that DC offset problem. A corrective term based on the integral of error helped overcome these offset errors.   sum(k + 1) = sum(k) + y (k) − y(k)

(9.18)

  x (k + 1) = F x (k) + H u(k) + Gw(k) − L ∗ y (k) − y(k) + K ∗ sum(k)











y (k) = C x (k) + v(k)

(9.19)

Figure 9.21 shows the tracking of the bus angles at the 5 locations and 3 phases. With comparitively low gains on the kalman filter in Fig. 9.22 there is some tracking error visible in the angle velocity estimates. Zooming in on one phase in Fig. 9.23 shows reasonable tracking at all measurement points for the bus angles with an offset at higher levels of battery injection. The Kalman estimates for distribution angles do not track well as seen in Fig. 9.23 Figure 9.24 shows a small variation in the power flow to the circuit corresponding to the angle and loading changes at the transmission level. The low level of achieved influence was due to the numerical stability issues of control based on measurements

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angles at capbus 4 3 2 1 0 -1

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Fig. 9.21 Kalman estimates of bus angles

angles at capbus 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3

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Fig. 9.22 Zoomed estimates of bus angles

of angles within the distribution network while modulating the loads which changes these angles (Fig. 9.25). The slower task of voltage control can be well handled under droop control but with the load variation and with the issue of the controller injection upsetting the state estimate local angle measurement by PMU does not seem feasible for distribution contribution to dynamic stability.

9.6 Applications

227 vel at capbus 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 3

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Fig. 9.23 Angular velocity estimates transmission bus ‘ + ’ distribution busses ‘−’

Power to cct 38.78 38.77 38.76 38.75 38.74 38.73

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Fig. 9.24 Variation in power flow to the distribution system

9.6 Applications Distribution state estimation seeks to overcome the lack of measurements in many distribution applications by the use of pseudo measurements. The use of dynamic Kalman filters which establish a continuity of load state between time instants makes a step jump in the performance over snapshot estimates which operate for only one time instant. The process is highly dependant on the stochastic model of the loads is the short term. Where there is a substantial penetration of PV that is non uniform along a feeder, this can give rise to an inferior state estimate.

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Fig. 9.25 Angle variations at distribution points under random walk load

When battery systems become more prevalent in distribution then once again the designer needs to be wary of the stochastic model of distribution loads. There is likely to be some impact of the use of batteries correlated to bulk energy markets or to local congestion. If it is not possible to correlate these effect out of the Kalman process a much higher level of measurements will be needed to ensure that DER does still keep the voltages and loadings within the design specifications. The use of load and battery control for voltage management is well supported for distribution voltage control and the integral to droop line is well suited for simultaneous control. For contributions to system wide phenomena such as frequency control and damping there are issue in using local PMU measurements at each load. The limits are due to noise in the angle measurements as well as control actions interfering the estimates. A PMU measurement at higher voltage points in the network could provide a suitable signal for angle control but as seen in earlier chapters for good wide area control with Kalman estimates of aggregated machine groups there are benefits for broadcast control action over a large area of distribution. One of the issues is the latency of a broadcast of a control reference to millions of customers. Current management of data packets need one separate packet to each customer. Improvements as discussed in [15] broadcast one packet which splits at nodes to achieve the very low required latency [16]. With compatible communications and with a transparent reward mechanism strong customer contribution to system events can be achieved through broadcast of the required control action.

References 1. F.C. Schweppe, J. Wildes, Power system static-state estimation, Part I: exact model. IEEE Trans. Power Appar. Syst. PAS-89, 120–125 (1970) 2. F.C. Schweppe, D.B. Rom, Power system static-state estimation, Part II: approximate model. IEEE Trans. Power Appar. Syst. PAS-89, 125–130 (1970)

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3. F.C. Schweppe, Power system static-state estimation, Part III: implementation. IEEE Trans. Power Appar. Syst. PAS-89, 130–135 (1970) 4. M.E. Baran, A.W. Kelley, State estimation for real-time monitoring of distribution systems. IEEE Trans. Power Syst. 9, 1601–1609 (1994) 5. M. Pau, P.A. Pegoraro, S. Sulis, Efficient branch-current-based distribution system state estimation including synchronized measurements. IEEE Trans. Instrum. Meas. 62, 2419–2429 (2013) 6. A.S. Meliopoulos, G. Cokkinides, R. Huang, E. Farantatos, S. Choi, Y. Lee et al., Smart grid technologies for autonomous operation and control. IEEE Trans. Smart Grid 2, 1–10 (2011) 7. S. Sarri, M. Paolone, R. Cherkaoui, A. Borghetti, F. Napolitano, C.A. Nucci, State estimation of active distribution networks: comparison between WLS and iterated Kalman-filter algorithm integrating PMUs, in 2012 3rd IEEE PES International Conference and Exhibition on Innovative Smart Grid Technologies (ISGT Europe) (2012), pp. 1–8 8. K. Nishiya, J. Hasegawa, T. Koike, Dynamic state estimation including anomaly detection and identification for power systems, in IEE Proceedings C (Generation, Transmission and Distribution) (1982), pp. 192–198 9. A. Bahgat, M. Sakr, A. El-Shafei, Two level dynamic state estimator for electric power systems based on nonlinear transformation, in IEE Proceedings C (Generation, Transmission and Distribution) (1989), pp. 15–23 10. S.L. Goh, D.P. Mandic, An augmented extended Kalman filter algorithm for complex-valued recurrent neural networks. Neural Comput. 19, 1039–1055 (2007) 11. F.S. Cattivelli, A.H. Sayed, Diffusion strategies for distributed Kalman filtering and smoothing. IEEE Trans. Autom. Control 55, 2069–2084 (2010) 12. B. Picinbono, Second-order complex random vectors and normal distributions. IEEE Trans. Signal Process. 44, 2637–2640 (1996) 13. J.-H. Teng, A direct approach for distribution system load flow solutions. Power Deliv., IEEE Trans. On 18, 882–887 (2003) 14. IEEE_Task_force_for_dynamic_performance, Standard load models for power flow and dynamic performance simulation. IEEE Trans. Power Syst. 10 (1995) 15. L. Xiaohui, T. Yu-Chu, L. Gerard; M. Yateendra, H. Xiaoqing, Z. Chunjie, “Constrained Optimization of Multicast Routing for Wide Area Control of Smart Grid” IEEE Transactions on Smart Grid, 10(4), 3801–3809 (2019) 16. Y. Ding, T. Yu-Chu, L. Xiaohui, M. Yateendra, L. Gerard, Z. Chunjie, “Constrained Broadcast With Minimized Latency in Neighborhood Area Networks of Smart Grid” IEEE Transactions on Industrial Informatics, 16(1) 309–318 (2020)

Chapter 10

Conclusions

The high accuracy of PMU measurements makes fast estimation of the state and the dynamic response of a power system readily available. The motor content of loads can be identified as well as interarea dynamics. Nonlinear state estimators can tell us whether a section of the network has critical issues. Nonlinear control design based on Kinetic Energy can provide both first swing improvements as well as overall nonlinear damping control. Inverters are rising in importance in the power system dynamics and Chap. 6 shows how these can be incorporated into the overall control philosophy. The expected reduction of conventional generation raises concerns about system control with significant levels of inverters but with system state estimates available, fast command to customer loads and batteries can be a substantial impact on the control of critical faults in the power system. The learning of system properties such as travelling waves and inverter aggregate dynamics are one side effect of the availability of Phasor measurements in the system. Much of the work presented here is in the context of transmission systems and wide area control. The area of most rapid development is in the area of distribution. Extrapolation of the tools in Chap. 9 will enable customer batteries and demand management to be deployed with benefit to the overall system. There are currently not large numbers of PMU ’s in distribution but the benefits can lie in enhancing Distribution State Estimation and characterizing the dynamics of distribution loads. This will enable the system benefits to be gained without compromise of distribution voltages or flow limits.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5_10

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Index

C Closed, 3, 6, 16, 18, 31, 98, 128, 187, 188

D Dynamics, 6, 10, 27, 55, 56, 86, 114, 131, 187, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206

E Energy, 3, 4, 7–9, 27, 33, 34, 37, 39, 41, 42, 50–52, 55, 58, 61, 63, 65, 77, 85, 86, 98, 105, 110, 112, 113, 115, 126, 147, 151, 167, 169, 210, 228 Estimation, 7, 30, 31, 60–62, 207, 231

L Linear, 4, 8, 30, 54, 63, 67, 71, 82, 85, 88, 90, 93, 100, 102–104, 110, 111, 115, 146, 152, 170, 173, 174, 208, 209, 212 Linear Quadratic Regulators (LQR), 9, 105, 119, 126, 128, 129, 131, 135, 137, 140, 142, 204 Load, 1, 3, 4, 6–9, 13–21, 23–28, 32, 33, 35– 37, 40, 41, 44, 52, 55, 57, 59–61, 64, 66, 76–78, 80, 82, 86, 94–97, 101, 115, 147, 152, 165, 167, 168, 171, 172, 182, 184, 187, 190, 197–201, 204, 210–212, 215, 227 Lyapunov, 4

N Nonlinear, 1, 4, 6–8, 10, 27, 30, 31, 51, 85– 87, 93, 94, 98, 104, 110, 111, 113, 114, 145, 147, 165, 212, 231

I Identification, 1, 3, 6, 7, 9, 10, 20, 23–28, 30, 44, 64, 66, 81, 82, 93, 94, 111, 163, 187, 188, 190, 191, 202–204 Inverter, 8–10, 24, 26, 82, 116, 119–121, 123, 126–129, 131–135, 140–149, 151, 156, 158, 163, 187–189, 200, 201, 204, 231

P Phasor Measurement Unit (PMU), 1, 2, 4, 6, 10, 16, 18, 25, 26, 28–30, 32, 64, 65, 75, 93, 94, 105, 111, 112, 191, 197, 199, 203, 208, 231

K Kalman, 8–10, 30, 61

S Small signal, 4

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Ledwich and A. Vahidnia, Phasors for Measurement and Control, Power Systems, https://doi.org/10.1007/978-3-030-67040-5

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