Asymmetric Operation of AC Power Transmission Systems : The Key to Optimizing Power System Reliability and Economics 9782553014031, 2553014031

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Richard

Marceau Abdou-R.

Sana

Donald T.

Asymmetric Operation of AC Power Transmission Systems

The Key to Optimizing Power System Reliability and Economics

McGillis

Presses internationales

P o ly t e c h n i q u e

Asymmetric Operation of AC Power Transmission Systems Richard J. Marceau, Abdou-R. Sana, Donald T. McGillis

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This book may not be duplicated in any way without the express written consent of the publisher. Legal deposit: 1st quarter 2006 Bibliothèque nationale du Québec National Library of Canada

ISBN 2-978-553-01403-1 (printed version) ISBN 2-978-553-01591-5 (pdf version) Printed in Canada

Foreword Asymmetric operation, as a concept, defines the operation of a threeĆphase AC transmission line as three independently operated entities. In this approach, a threeĆphase line can be operated with one or two phases out of service for singleĆline transmission corridors or with one, two or three phases out of service in the case of multipleĆline corridors while preserving threeĆphase symmetrical operation at the extremities of such corridors. The objective of this approach is to maintain the overall integrity of threeĆphase transmission in terms of power delivery, and balanced voltages and currents even in the event of faults on some part of the transmission system resulting in the loss of individual phases. Evidently, compensating equipment must be brought to bear in order to achieve the stated objective, but the consequence is to improve security limits for all but the most extreme contingencies while adding considerable flexibility to normal transmission system operation. Stresses on a transmission system arise from many identifiable sources. First of all, in addition to normal load growth, the conversion of many manufacturing processes to electricity has taken place over the years due to the efficiency of electrical energy in terms of more precise products, less rejections and a cleaner working environment. Secondly, environmental impacts have come under greater scrutiny, with the result that it is very difficult to obtain rightsĆofĆway for new transmission facilities. Finally, the advent of deregulation has introduced the concept of electrical energy as a commodity with associated power transactions which often stress the transmission system to its limits. The challenge, therefore, is to maximize the security region of the system while respecting accepted security practices. As will be seen in this book, asymmetric operation, based on the singleĆphase operation of threeĆphase transmission lines, supported by modern compensating equipment, can achieve this objective while significantly improving the reliability, the environmental impact and the economics of modern AC power transmission. As to how asymmetric operation can be implemented, the required compensating equipment can take the form of passive elements, such as capacitors and reactors with conventional switching equipment, or active elements such as GateĆturnĆoff valves with their inherent voltage and current controls. These alternatives are explained in the following chapters along with the impact of asymmetric operation on system security and the possibility of actual applications. Chapter 1 introduces the modern power system and briefly discusses power system engineering and symmetric and asymmetric operation. Chapter 2 defines asymmetric operation and the objectives of this book. The case of a given transmission line maintaining its original power transfer capacity with only one or two functional phases, while preserving threeĆphase symmetrical systems at its extremities, is considered. The impact of asymmetric operation on power system design and operation is discussed. The implementation strategies of asymmetric operation are briefly described. Finally, the promising strategies are identified and are explored in subsequent chapters.

iv

Foreword

Chapter 3 justifies asymmetric operation of AC power transmission systems through its impact on system reliability. This chapter begins with a short review of probability theory applied to transmission system operation and the specific relationship between asymmetric operation, and system reliability is explored. The viability of asymmetric operation is shown through the analysis of transmission system outage statistics and by a comparative analysis of the reliability of a transmission line under both normal symmetric operation and asymmetric operation. Chapter 4 treats the case of asymmetric operation of a transmission corridor with many parallel threeĆphase lines. When a single phase is open on a given line, the strategy implies the application of shunt and series compensation in the same phases of the sound lines. This compensation strategy restores the equivalent parameters of the initial corridor. The solution to a simultaneous contingency on the three phases of a given line is also shown to be possible by this compensation structure, at no additional cost. Chapter 5 treats the asymmetric operation of a transmission corridor with only one threeĆphase line. This case is shown to be the most complex to compensate. Nevertheless, it is the case most encountered for connecting remote loads, remote generating systems, transmission rings surrounding large metropolitan areas, and in situations where building a second line is either uneconomic, impractical or, for all intents and purposes, impossible. The compensation strategy implies the use of shunt and series elements in the sound phases, once a single phase is no longer operational. The compensating elements may be adjusted for extreme load variations in order to filter the zero and negative sequences resulting from the physical asymmetry. Chapter 6 concerns the preliminary economic analysis of asymmetric operation of AC power transmission systems. The investment cost of asymmetric operation is evaluated according to the number of lines in a corridor and the nature of the compensating devices. The cost of the undelivered energy in normal symmetrical operation is evaluated in terms of the risk of nonĆsupplied energy (the cost of the Loss of Load Expectation: LOLE). Finally, the different costs are compared for each given configuration to determine the most economic option. In Chapter 7, the role of asymmetric operation in enhancing the flexibility of power system operation is delineated in terms of system security, bilateral transactions and a singleĆphase approach to power system analysis in order to replace the threeĆphase culture in selected areas of the network. Chapter 8 concludes this book: the key results are summarized. Appendix A of this book summarizes the underlying concepts which govern the traditional design of symmetrical threeĆphase power transmission systems. These concepts deal with: transmission line parameters, voltageĆcurrent relations, generalized constants, power equations, voltage regulation, power system representation, symmetrical components, stability, reliability concepts, insulation coordination. Appendix B presents simulation results of the asymmetric operation of specific examples of singleĆ and multipleĆline corridors.

Acknowledgement The authors thank the Canadian International Development Agency (CIDA) and its Programme canadien de bourses de la Francophonie (PCBF), the National Sciences and Engineering Research Council (NSERC) of Canada and the Fonds québécois de la recherche sur la nature et les technologies (FQRNT) for their financial assistance in the preparation of this book.

Table of Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Chapter 1ĂĂĂThe Modern Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1ĂĂĂIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Power System Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 System Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 System Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Symmetric Versus Asymmetric Operation . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 3

Chapter 2ĂĂĂIntroduction to Asymmetric Operation of Power Transmission Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Concept of Asymmetric Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3ĂĂĂStrategies of Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Corridor With Multiple Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 SingleĆline Corridor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Reactive Power Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 8 8 9 9 10

Chapter 3ĂĂĂProbabilistic Aspects of Asymmetric Operation . . . . . . . . . . . . . 3.1ĂĂĂIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Binomial Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Two Repairable Identical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Many Repairable Identical Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Gauss Law or Normal Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Application to Asymmetric Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Statistics of Transmission Line Failures . . . . . . . . . . . . . . . . . . . . . . 3.5.2ĂĂĂReliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 11 12 13 14 15 15 16 19 19 19 23

Chapter 4ĂĂĂAsymmetric Operation of a Corridor with Multiple Lines . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2ĂĂĂAsymmetric Operation Using Conventional Compensating Devices: Lossless Line With Lumped Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 28

viii

Table of Contents

4.2.1 Calculation of the Compensating Impedances . . . . . . . . . . . . . . . . . 4.2.2 Calculation of the Installed Reactive Power . . . . . . . . . . . . . . . . . . 4.3ĂĂĂAsymmetric Operation Using Conventional Compensating Devices: Uncoupled Lines With Distributed Parameters . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Calculation of the Compensating Elements . . . . . . . . . . . . . . . . . . . 4.3.2 Effects of the Resistance of the Line . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Asymmetric Operation Using FACTS Controllers . . . . . . . . . . . . . . . . . . . 4.4.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2ĂĂĂSample Case 5: One Series Voltage Source at the Sending End and One Shunt Current Source at the Receiving End . . . . . . . . . . . . . . . 4.4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 29 32 32 37 37 37 41 42 43

Chapter 5ĂĂĂAsymmetric Operation of a Corridor With a Single Line . . . . . 5.1ĂĂĂIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Compensation With Conventional Devices . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Compensation of the Positive Sequence by Series Capacitors . . . . . 5.2.2 Compensation of the Negative Sequence by Passive Elements . . . . 5.2.3 Filtering of the Zero Sequence by Transformers . . . . . . . . . . . . . . . 5.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Compensation With FACTS Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1ĂĂĂCompensation of the PositiveĆsequence Currents and Filtering of the ZeroĆsequence Currents by Series Voltage Sources . . . . . . . . . . 5.3.2ĂĂĂFiltering the NegativeĆsequence Currents by Current Converters . . 5.3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4ĂĂĂPaired and Versatile NegativeĆsequence and ZeroĆsequence Currents Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 47 47 52 54 61 61

Chapter 6ĂĂĂEconomic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2ĂĂĂEstimating Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Cost of a New Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Cost of Reactive Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3ĂĂĂInvestment Costs of Asymmetric Operation Using Conventional and Modern Compensating Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Corridor With One Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Corridor With Two Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Corridor With Three Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4ĂĂĂInvestment Costs of Asymmetric Operation Compared to the Cost of LOLE and the Cost of a New Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1ĂĂĂDetermining the Cost of LOLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2ĂĂĂComparing the Cost of the LOLE, the Investment Costs of Asymmetric Operation and the Investment Costs of a New Line . . 6.5 Effect of the Number of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 69 70

61 65 67 67 68

70 70 71 72 73 73 74 76

Table of Contents

ix

6.6 Scenario Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Representative Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2ĂĂĂFrom the Point of View of the N*1 Criterion . . . . . . . . . . . . . . . . 6.6.3 From the Point of View of the N*1 Criterion and Loading Capacity 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 77 77 81 86

Chapter 7ĂĂĂEnhancing the Flexibility of Power System Operation . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Review of the ThreeĆphase N*1 Criterion . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The SingleĆline Corridor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4Ă Distributed Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5ĂĂĂThe Distribution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 89 90 90 91

Chapter 8ĂĂĂConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

AppendixĂAĂĂĂUnderlying Concepts of Symmetrical ThreeĆphase System Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Transmission Line Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 VoltageĆcurrent Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 The Generalized Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 The Power Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Voltage Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Representation of a Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Symmetrical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 The PowerĆangle Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 Reliability Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11 Insulation Coordination Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 97 100 105 108 110 115 120 124 126 132 137

AppendixĂBĂĂĂSimulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1ĂĂĂAsymmetric Operation of a Corridor With a Single 120ĂkV, 100Ăkm Transmission Line Using Conventional Compensating Devices . . . . . . . . B.2ĂĂĂAsymmetric Operation of a Corridor With a Single 120ĂkV, 100Ăkm Line Using FACTS Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3ĂĂĂAsymmetric Operation of a Corridor With Two 400ĂkV, 150Ăkm Lines Using Conventional Compensating Devices . . . . . . . . . . . . . . . . . . . . . . . . B.4ĂĂĂAsymmetric Operation of a Corridor With Two 400ĂkV, 150Ăkm Lines Using FACTS Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5ĂĂĂAsymmetric Operation of a Corridor With Three 400ĂkV, 150Ăkm Lines Using Conventional Compensating Devices . . . . . . . . . . . . . . . . . . . . . . . . B.6ĂĂĂAsymmetric Operation of a Corridor With Three 400ĂkV, 150Ăkm Lines Using FACTS Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.7ĂĂĂSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 142 149 156 161 165 169 173

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

List of Figures Figure 1.1 Figure 3.1 Figure 3.2

System operation and system planning . . . . . . . . . . . . . . . . . . . . . A twoĆstate repairable system . . . . . . . . . . . . . . . . . . . . . . . . . . . . The three states of a system consisting of two identical repairable units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.3 Comparison of Weibull distribution for n = 4 and m = 5 and Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.4 Equivalent logical circuits of a threeĆphase line . . . . . . . . . . . . . . Figure 4.1 Simplified representation of a corridor with many lines . . . . . . . . Figure 4.2 Equivalent circuit of one phase (AĆphase) of a corridor of N lines Figure 4.3 Maximum compensation scheme for the asymmetric operation of a corridor with two 400ĂkV, 300Ăkm lossless lines . . . . . . . . . . . . . . Figure 4.4 Minimum compensation scheme for the asymmetric operation of a corridor with two 400ĂkV, 300Ăkm lines . . . . . . . . . . . . . . . . . . . . . Figure 4.5 Arbitrary location of series compensating elements in the lines . . Figure 4.6 Illustration of the compensation of one corridor phase by series voltage and shunt current sources during asymmetric operation . . Figure 4.7 Principle of asymmetric operation with one series voltage source at the sending end and one shunt current source at the receiving end Figure 4.8 Load flow for corridor AĆphase in both symmetric and asymmetric operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.1 Principle of asymmetric operation of a corridor with one threeĆphase line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.2 Asymmetric operation of one line with series capacitors at the sending end for positiveĆsequence compensation . . . . . . . . . . . . . Figure 5.3 Principle of negativeĆsequence current compensation for asymmetric operation of one line using passive LC elements . . . . Figure 5.4 Principle of zeroĆsequence current filtering by two grounding transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.5 T transformer connection for zeroĆsequence current filtering . . . . Figure 5.6 DĆY and Zigzag transformers for zeroĆsequence current filtering . Figure 5.7 ZeroĆsequence filtering by the power transformers . . . . . . . . . . . . Figure 5.8 Asymmetric operation of one 120ĂkV line with one phase open using conventional compensating devices . . . . . . . . . . . . . . . . . . . Figure 5.9 Asymmetric operation of one 735ĂkV line with one phase open using conventional compensating devices . . . . . . . . . . . . . . . . . . . Figure 5.10 Asymmetric operation of one 1Ă200ĂkV line with one phase open using conventional compensating devices . . . . . . . . . . . . . . . . . . . Figure 5.11 Series voltage sources located at the sending end of the line . . . . . Figure 5.12 Shunt current sources at each end of the line for the compensation of negativeĆsequence currents . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 11 13 17 20 27 28 31 32 33 38 41 42 46 48 52 54 55 55 56 57 58 60 62 65

xii

List of Figures

Figure 5.13 Shunt current sources at each end of the line for the compensation of the negativeĆsequence currents . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Figure 6.1 Comparison of costs of symmetric and asymmetric operation for three scenarios according to line length and N*1 criterion . . . . . 80 Figure 6.2 Comparison of costs of symmetric and asymmetric operation for three scenarios according to line length, N*1 and loading capacity 85 Figure A.1 Equivalent
configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Figure A.2 Phasor diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Figure A.3 Distributed parameters of one phase of a transmission line . . . . . . 105 Figure A.4 Adjusted
configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure A.5 The regulation curve for a 315ĂkV singleĆconductor 400Ăkm line . 112 Figure A.6 Typical characteristic of a static compensator . . . . . . . . . . . . . . . . 113 Figure A.7 Application of a static compensator . . . . . . . . . . . . . . . . . . . . . . . . 114 Figure A.8 Section of a sample power system . . . . . . . . . . . . . . . . . . . . . . . . . 116 Figure A.9 Segment of a typical transmission system . . . . . . . . . . . . . . . . . . . 118 Figure A.10 Load flow across a 735ĂkV line . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Figure A.11 Symmetrical components of unbalanced voltages . . . . . . . . . . . . . 120 Figure A.12 One 325Ăkm, 800ĂkV line transmitting 2Ă500 MW to an infinite bus 124 Figure A.13 NegativeĆsequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Figure A.14 ZeroĆsequence network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Figure A.15 Illustration of the transfer impedance . . . . . . . . . . . . . . . . . . . . . . 125 Figure A.16 PowerĆangle curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Figure A.17 Generator feeding into an infinite bus . . . . . . . . . . . . . . . . . . . . . . 127 Figure A.18 Areas Aa and Ad for the equalĆarea method . . . . . . . . . . . . . . . . . . 128 Figure A.19 Two time intervals, each of duration Dt . . . . . . . . . . . . . . . . . . . . . 129 Figure A.20 Examples of swing curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Figure A.21 R/X diagram of the impedance relay . . . . . . . . . . . . . . . . . . . . . . . 132 Figure A.22 Example 1 - singleĆbusbar arrangement substation . . . . . . . . . . . . 136 Figure A.23 Example 2 - breakerĆandĆaĆhalf arrangement of a substation . . . . . 136 Figure A.24 Procedure for insulation coordination . . . . . . . . . . . . . . . . . . . . . . 140 Figure B.1 Asymmetric operation of a corridor with a single 120ĂkV, 100Ăkm line using conventional compensating devices . . . . . . . . . . . . . . . 142 Figure B.2 Simulation results of the asymmetric operation of one 120ĂkV, 100Ăkm line transmitting 50 MW with conventional compensating devices: currents of the source and the load . . . . . . . . . . . . . . . . . . 145 Figure B.3 Simulation results of the asymmetric operation of one 120ĂkV, 100Ăkm line transmitting 50 MW with conventional compensating devices: currents of the sequence filters . . . . . . . . . . . . . . . . . . . . . 146 Figure B.4 Simulation results of the asymmetric operation of one 120ĂkV, 100Ăkm line transmitting 30 MW with conventional compensating devices: currents of the sequence filters . . . . . . . . . . . . . . . . . . . . . 148

List of Figures

Figure B.5 Figure B.6 Figure B.7 Figure B.8 Figure B.9 Figure B.10 Figure B.11 Figure B.12 Figure B.13 Figure B.14 Figure B.15 Figure B.16 Figure B.17 Figure B.18 Figure B.19 Figure B.20 Figure B.21

Compensation of the positive, negative and zero sequences at the sending end of the line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering of the negative and zero sequences at the receiving end of the line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative scheme for the compensation of the positive, negative and zero sequences at the sending end of the line . . . . . . . . . . . . . Simulation results of the asymmetric operation of one 120ĂkV, 100Ăkm line transmitting 50 MW with FACTS controllers: currents of the load and the source . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the asymmetric operation of one 120ĂkV, 100Ăkm line transmitting 50 MW with FACTS controllers: currents of the sequence filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetric operation of a corridor with two 400ĂkV, 150Ăkm lines using conventional compensating devices . . . . . . . . . . . . . . . . . . . Simulation results for the asymmetric operation of a corridor with two 400ĂkV, 150Ăkm lines, using conventional compensating devices: voltages at the source and the load . . . . . . . . . . . . . . . . . Simulation results for the asymmetric operation of a corridor with two 400ĂkV, 150Ăkm lines, using conventional compensating devices: currents of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetric operation of a corridor with two 400ĂkV, 150Ăkm lines using FACTS controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results for the asymmetric operation of a corridor with two 400ĂkV, 150Ăkm lines using FACTS controllers; bus and injected voltages and injected current . . . . . . . . . . . . . . . . . . . . . . Simulation results of the asymmetric operation of two 400ĂkV, 150Ăkm lines using FACTS controllers: system currents . . . . . . . Asymmetrical operation of a corridor with three 400ĂkV, 150Ăkm lines using conventional compensating devices . . . . . . . . . . . . . . . Simulation results for the asymmetrical operation of a corridor with threeĂ400ĂkV, 150Ăkm lines using conventional compensating devices: bus and injected voltage and injected current . . . . . . . . . Simulation results for the asymmetrical operation of a corridor with three 400ĂkV, 150Ăkm lines using conventional compensating devices: currents of the line, the source and the load . . . . . . . . . . . Asymmetrical operation of a corridor with three 400ĂkV, 150Ăkm lines using FACTS controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results for the asymmetric operation of a corridor with three 400ĂkV, 150Ăkm lines using FACTS: voltages in the system, injected voltages and injected current . . . . . . . . . . . . . . . . . . . . . . Simulation results for the asymmetrical operation of a corridor with three 400ĂkV, 150Ăkm lines using FACTS controllers: system currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

149 150 151 154 155 159 156 160 161 163 164 165 167 168 169 171 172

List of Tables Table 3.1 Table 4.1 Table 4.2 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5

Table 5.6 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 6.9

MinimumĂnumber of static compensators required for a given nonĆavailability to guarantee the operation of at least six static compensators with a probability of 99% . . . . . . . . . . . . . . . . . . . . . Total reactive power requirements for the asymmetric operation of two 400ĂkV, 300Ăkm lossless threeĆphase lines equipped to sustain the loss of any three phases in the corridor . . . . . . . . . . . . . . . . . . . Summary of compensation cases of a multipleĆline corridor for asymmetric operation by means of series voltage sources and shunt current sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the cases of compensation of the positive sequence for asymmetric operation of a corridor with a single line using conventional devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated results of the operation of one 120ĂkV line with one phase open, using conventional compensating devices . . . . . . . . . . Calculated results of the operation of one 735ĂkV line with one phase open, using conventional compensating devices . . . . . . . . . . Calculated results of the operation of one 1Ă200ĂkV line with one phase open, using conventional compensating devices . . . . . . . . . . Summary of the compensation scenarios for the asymmetric operation of one line with series voltage sources to eliminate zeroĆsequence currents and to compensate positiveĆsequence currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated results of the asymmetric operation for a single 120ĂkV line with one phase open, using FACTS controllers . . . . . . . . . . . . Transmission line cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investment costs of asymmetric operation for a single 120ĂkV, 100Ăkm line with one phase open, using conventional devices . . . . Investment cost of asymmetric operation for a single 120ĂkV, 100Ăkm line with one phase open, using FACTS devices . . . . . . . . Investment cost of asymmetric operation of two 400ĂkV, 150Ăkm lines, using conventional compensating devices . . . . . . . . . . . . . . . Investment costs of asymmetric operation of two 400ĂkV, 150Ăkm lines, using FACTS devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investment costs of asymmetric operation of three 400ĂkV, 150Ăkm lines, using conventional compensating devices . . . . . . . . . . . . . . . Investment costs of asymmetric operation of three 400ĂkV, 150Ăkm lines, using FACTS devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of the LOLE for four different cases . . . . . . . . . . . . . . . . . . . . Comparing the cost of many options for reducing the risk of nonĆtransmitted energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 31 40 51 57 59 60

64 66 70 70 71 71 71 72 72 74 75

xvi

List of Tables

Table 6.10 Investment cost of asymmetric operation for 735ĂkV, 300Ăkm lines using conventional devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 6.11 Transmission line characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 6.12 Transmission system scenarios for comparison of symmetric vs. asymmetric operation from the point of view of the N*1 criterion Table 6.13 Investment costs of additional equipment permitting asymmetric operation of a single line (loss of a single phase) for different nominal line voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 6.14 Cost associated with symmetric and asymmetric operation for the three scenarios from the point of view of the N*1 criterion . . . . . Table 6.15 Summary of the cost of symmetric and asymmetric operation for three scenarios according to line length and N*1, in M$CA . . . . Table 6.16 Transmission system scenarios for comparison of symmetric vs. asymmetric operation from the point of view of N*1 and identical loading capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 6.17 Cost of unavailability (symmetric operation) . . . . . . . . . . . . . . . . . Table 6.18 Investment costs associated with singleĆline asymmetric operation for different nominal line voltages from the point of view of N*1 and identical loading capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 6.19 Cost associated with symmetric and asymmetric operation for the three scenarios from the point of view of N*1 and identical loading capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 6.20 Summary of the cost of symmetric and asymmetric operation for three scenarios according to line length, N*1 and loading capacity, in M$CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table A.1 Typical values of x and y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table A.2 Typical values of surgeĆimpedance loading (SIL) . . . . . . . . . . . . . . Table A.3 Most probable operating conditions . . . . . . . . . . . . . . . . . . . . . . . . Table A.4 Real and reactive power for different loadings of a 300Ăkm, 735ĂkV line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table A.5 Values of ER , r and PR for the regulation curve . . . . . . . . . . . . . . . Table A.6 SVS rating as a function of the power transfer . . . . . . . . . . . . . . . . Table A.7 Equivalent impedance for different types of fault . . . . . . . . . . . . . . Table A.8 Equivalent impedance of a simplified system . . . . . . . . . . . . . . . . . Table A.9 Power transfer for different types of fault . . . . . . . . . . . . . . . . . . . . Table A.10 Construction of the swing curve for a threeĆphase fault cleared in 6Ăcycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table A.11 Construction of the swing curve for a threeĆphase fault cleared in 9Ăcycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table A.12 Mean evaluation of lost load (MELL) in M$CA . . . . . . . . . . . . . . Table B.1 Calculated results of the compensating currents for the asymmetric operation of one 120ĂkV, 100Ăkm line with phase a" open and using FACTS controllers and pairedĆsequence filtering . . . . . . . . . . . . . .

76 77 78 79 79 81 81 82 83 84 85 100 102 103 110 112 113 123 125 126 131 131 136 152

List of Symbols
z     R S     A B f FACTS Fx Fw fij fx fw G Ip IR I R0 I R1 I R2 IS I S0 I S1 I S2

Probability density function of the Gauss standard law Angle of constant A 0 of the generalized constants of a transmission line Angle of constant B 0 of the generalized constants of a transmission line Angle of constant C 0 of the generalized constants of a transmission line Transmission angle: angle difference between voltages V S and V R Angle of the load voltage V R Angle of the source voltage V S Failure rate of one repairable unit Repair rate of one unit Mean of the normal law Standard deviation of the normal law Pulsation at network frequency Availability of one unit Susceptance general symbol Frequency of the network Flexible Alternating Current Transmission Systems Cumulative probability distribution function: general Cumulative probability distribution function of Weibull distribution Frequency of transition from state i to state j Probability distribution function: general Probability distribution function of Weibull distribution Conductance general symbol Total current of a single phase of a transmission corridor Total line current at the receiving end of a corridor ZeroĆsequence current at the receiving end of a line during symmetric operation DirectĆsequence current at the receiving end of a line during symmetric operation NegativeĆsequence current at the receiving end of a line during symmetric operation Total line current at the sending end of a transmission corridor ZeroĆsequence current at the sending end of a line during symmetric operation DirectĆsequence current at the sending end of a line during symmetric operation NegativeĆsequence current at the sending end of a line during symmetric operation

xviii

L LOLE N N*1 P Pi Pk p pasym psym pw Q q qasym qsym R T U U50 Uaw V X Y Z Z0

List of Symbols

Number of phases out of service Loss of Load Expectation Number of lines in a transmission corridor N minus one" reliability criterion Active power of the load Probability for a system of being in state i Probability of having exactly k units functional within N units Probability, for a system, of being in the functional state Combined probability of successful transmission of a threeĆphase during asymmetric operation Combined probability of successful transmission of a threeĆphase during symmetric operation Probability density of flashover Reactive power of a line Probability, for a system, of being in the outĆofĆservice state Combined probability of nonĆsuccessful transmission of a threeĆphase during asymmetric operation Combined probability of nonĆsuccessful transmission of a threeĆphase during symmetric operation Resistance: general symbol Total power transmitted in a transmission corridor Flashover voltage of a strain of insulators 50% withstand voltage Withstand voltage of a strain of insulators Voltage general symbol Reactance general symbol Admittance general symbol Impedance general symbol Surge impedance of the line

Introduction Existing electric power transmission systems are operated with three physically different systems of conductors, typically referred to as phases", where the alternating voltages and currents in each are offset with respect to each other in order to take advantage of Nicolas Tesla's centuryĆold invention of the threeĆphase AC generator and motor. The threeĆphase power generator is a remarkably robust and economical technology which creates the three sets of voltages and currents, the soĆcalled phases, in threeĆphase transmission lines. This results in a steady torque characteristic, which in turn translates into high generator reliability. The threeĆphase AC motor, which requires an infeed of the three different phases to operate, is also a highly reliable and economical piece of machinery and is a workhorse of industry. Since threeĆphase generators (in power plants) and motors (in industry) are so widely used, they must be connected by means of a threeĆphase highĆvoltage (HV), extraĆhighĆvoltage (EHV) or ultraĆhighĆvoltage (UHV) power transmission system which is symmetrically operated, meaning that if any problem develops on one or more phases of a threeĆphase line, all three phases are taken out of service. This requirement is due to the fact that, if symmetric operation is not enforced, undesirable voltages and currents are generated in the system which are harmful to many types of loads, including rotating machinery. There are exceptions to this rule: for economic reasons, singleĆphase systems equipped with appropriate mitigating measures have often been used in railway electrification and for distributing power at the household level. However, throughout the world, standard practice is to operate bulk power transmission systems symmetrically.

SYMMETRIC OPERATION OF POWER TRANSMISSION SYSTEMS When one of the three phases of a power transmission line touches an object, causing the current of that phase to be redirected either to the ground (i.e. through a tree) or to another phase, such shortĆcircuit conditions are generally referred to as faults". Under normal operation, when a fault condition occurs, circuit breakers at both ends of the line interrupt the current in all three phases, effectively removing the line from operation, even though one or two healthy phases remain which could conceivably carry useful power: this means that a problem on 33% of a line automatically deprives the network of 100% of this same line. Furthermore, the presence of three conductors (i.e. phases) on a line rather than one augments the probability of fault by a factor of 3. A final disadvantage is that, after the fault is cleared and the line is switched out of service, the power system operator must then apply remedial measures to the system, such as changing the amount of generation from different power plants, in order to redirect power flows and ensure that the system is capable of sustaining further contingencies, thereby ensuring system reliability and continuity of supply.

xx

Introduction

One therefore concludes that the prevalent operating strategy of the power industry is basically wasteful of expensive transmission equipment and imposes needless stresses on network operation, including both equipment and operators. Symmetrical operation therefore represents the least effective strategy for operating any given threeĆphase transmission investment. In the light of existing pressures on electric power transmission due to both deregulation and greater environmental awareness, symmetrical operation exposes the system to unnecessary risks and results in a waste of potential transmission capacity. This was particularly shown to be true in the Northeastern U. S. and Canada blackout of August 2003, where numerous lines were switched out due to singleĆphase faults.

ASYMMETRIC OPERATION OF POWER TRANSMISSION SYSTEMS Though symmetric operation of power transmission systems was justified over a century ago when the industry and knowledge of power systems were basically in their infancy, simple, effective and economic measures can now be implemented to yield a far more effective and surgical strategy from both a reliability and economic perspective. This operating strategy is called asymmetric operation". Asymmetric operation, as a concept, defines the operation of a threeĆphase transmission line as three independently operated entities. In this approach, a threeĆphase line can be operated with one or two phases out of service for singleĆline transmission corridors, or with one, two, or three phases out of service in the case of multipleĆline corridors while preserving threeĆphase symmetrical operation at the corridor extremities. When faults on individual phases occur, undesirable voltages and currents are contained" within the affected corridor, the latter therefore appearing essentially unaffected from outside. As a final benefit, the implementation of asymmetric operation precludes the application of postĆfault remedial measures by the system operator: since the faulted corridor returns to its preĆfault electrical state, the postĆfault remedial measures are essentially builtĆinto" the strategy! To achieve this, a corridor's postĆfault electrical characteristics are returned to their preĆfault values by introducing the minimum required compensating equipment into the corridor. Such compensating equipment can either be conventional, inexpensive passive devices, such as capacitors and reactors equipped with appropriate switching devices, or more complex and expensive power electronics devices. Both approaches have their strengths and weaknesses and the final choice of each will depend on the particular constraints imposed by the system planner. The consequence is to improve corridor reliability, increase security limits (i.e. the amount of power transferred under normal conditions) and generally add flexibility to transmission system operation. Thus the global efficiency of the transmission system is increased if such an approach is implemented.

Introduction

xxi

EXAMPLES As will be seen in the present book, a technical and economic study consisting of retrofitting a number of existing transmission corridors yields very interesting results. For example, in the case of a corridor composed of two 400ĂkV, 150Ăkm lines, the estimated payback time for the equipment investment permitting asymmetric operation is approximately one year. Comparing this to the cost of a new symmetrically operated 400ĂkV, 150Ăkm line, the equivalent estimated payback period is approximately four times longer. However, the asymmetrically operated twoĆline corridor has far superior reliability and economical performance to that of the three symmetrically operated parallel lines! In the case of a corridor equipped with three 400ĂkV, 150Ăkm lines, the estimated payback time for installing the asymmetric operating strategy is approximately six months. If one has large blocks of power to transmit but space is at a premium, such as in highĆdensity metropolitan areas, preliminary studies show that beyond a certain length, a single asymmetrically operated EHV transmission line has essentially the same or lower cost and provides the same security performance and capacity as two symmetricallyĆoperated transmission lines of the same voltage. Retrofitting singleĆ or multiple line interconnection corridors consequently promises to have significant impact on the economical and reliability performance of such installations.

Chapter 1 The Modern Power System

1.1 INTRODUCTION The modern power system really came into its own in the second half of the twentieth century. During this period, massive power projects were undertaken and rendered feasible and viable with the associated development of AC transmission technology. As examples, one can cite the 800 kV system operating in Canada (Churchill Falls, Manicouagan, James Bay), Brazil (Itaipu), Venezuela (Guri) and the United States (AEPĆNYPA). These major landmarks were based originally on the groundbreaking research into air insulation and corona effects of the 1940s and 1950s and, later, the incorporation of power electronics to provide additional operating flexibility. The success of these major projects has led to some profound conclusions regarding the scope of AC power transmission. On the one hand, we have come to realize that 800ăkV transmission is a practical voltage limit based on field effects since even at this voltage level, the economic power transfer and distance are at least 10ā000 MW and 1ā000Ăkm. On the other hand, DC transmission as an alternative to longĆdistance AC power transmission appears practical only beyond these limits. Thus, the anticipated emergence of DC valve technology has been usurped by AC technology which continues even today to be one of the most promising power transmission technologies.

1.2 POWER SYSTEM ENGINEERING Power system engineering consists essentially of two functions with dissimilar objectives, namely system planning and system operation. While the objective of system planning is the integrity of the network in the long range, the objective of system operation is the efficiency of the network in the short range. As illustrated in Figure 1.1, system planning is an act of creativity based on design criteria and targeted transmission capacity, thereby yielding an appropriate topology; system operation is an act of applying the same design criteria to the present state of the topology, thereby operating the system within an appropriate resulting transmission capacity [36]. The association of these two functions can be mutually beneficial if the ground rules are clear. The symbiotic relationship of system planning and system operation is one example of the relation between science and technology. The pursuit of science is largely analytic whereas technology is speculative or even heuristic. Science answers the question of Why while technology provides the How. Thus science provides the framework for technology which in turn reveals the world [30].

2

Chapter 1

Planning criteria

a) Network topology

Transmission capacity

Planning criteria

b) Network topology

Transmission capacity

Figure 1.1 System operation and system planing: a) System Planning: Capacity + Criteria yield Topology; b) System Operation: Topology + Criteria yield Capacity.

1.2.1 System Planning Power system planning is the art and science of selecting the components of a power system so that their combination satisfies the design criteria. The components could be a transmission line or a substation or the associated equipment. The selection process could be described as bottomĆup" as the components are assembled in some rational order. The design criteria can be implicit (subjective) or explicit (objective) and both types of criteria govern the design. They are the road map on this journey of design. The implicit criteria reflect the user's preferences or constraints with respect to the design. They are subjective in that they are mainly a given and not subjected to analytical scrutiny. They are imposed on the system design and in this respect every system is different: each one has its own culture and history. Such preferences or constraints include purchasing policies, available energy resources, environmental constraints and the geography of the region. In a sense, these subjective criteria reduce the search space for the selection of credible alternative designs for a given project. The explicit criteria are objective in that they can be defined or measured against some standard, whether deterministic or probabilistic. Such criteria relate to the survival of the system in the face of adverse events with which the system must cope (security), the probability of meeting the load (adequacy), the risk of failure of the insulation or the

The Modern Power System

3

availability of individual components and subsystems, all of which contribute individually to the longĆterm reliability of the actual power system. The verification of these criteria in terms of system performance is the stuff of power system analysis. It is the yardstick that leads to the selection of those components that ultimately make up the system design.

1.2.2 System Operation In a system containing thousands of components, it can be understood that the network will always be in a degraded state, not that such a state will necessarily affect system security. Redundancy of components will assure this security at least as a first line of defense but multiple contingencies can occur. In a security region determined on the basis of deterministic criteria, it has been observed that there can exist elements of uncertainty such as the probability of occurrence of adverse events and the accuracy of the analytical methods involved in evaluating their impact. This observation has led to the conclusion that the deterministic evaluation of the security region is rather conservative and that a probabilistic approach may yield a more realistic assessment of system security [36]. System operation is quite different from system planning in that the maintenance of system security is performed on a dayĆtoĆday or even an hourĆtoĆhour basis. There is always a balance to be found between the stresses occurring in the network and the suitable response. The stresses are usually associated with overvoltage, voltage control and stability. The response, in turn, is associated with the corrective actions required in terms of system components strategically located whose characteristics are chosen so that the design will result in a network that is reliable and efficient.

1.3 SYMMETRIC VERSUS ASYMMETRIC OPERATION The modern power system is based on symmetrical operation where the sum of the voltage vectors at each node is zero and the line currents are equal. This design philosophy is the result of Tesla's work on symmetrical vectors where he showed that the summation of N coplanar and concurrent vectors of the same magnitude and regularly spaced in angle is zero [63]. This property applied to polyphase circuits results in an optimal equilibrium. Thus, a perfect polyphase system is ideal and must be symmetrical to be optimal. While many polyphase systems have been tested, the threeĆphase system has been retained over time as the best compromise between economy, simplicity and power transfer capacity. The introduction of a fault on a symmetrical three-phase transmission system results in a type of asymmetric operation with harmful unbalanced voltages and currents that are reflected across the whole network. Such a situation can be withstood in modern power systems for the few cycles of the fault duration until the protection system isolates the fault and restores the system to a state of symmetry and equilibrium. If however, a situation exists in the network whereby one wishes to operate some threeĆphase element in steady state without the full availability of all three phases (i.e. a transmission line with only one or two fully operational phases), equally harmful unbalanced voltages and currents are generated by this type of operation and injected

4

Chapter 1

into the network which must either be inhibited or filtered by means of appropriate measures before they enter the network. It is evident, therefore, that a system policy enabling steady state asymmetric operation of certain threeĆphase elements in the transmission system must respect certain conditions, as follows: 1. The inherent threeĆphase operating symmetry of the network must be respected at the extremities of such threeĆphase elements so as not to transfer any unbalanced voltages and currents into the network during the period of time that this operating mode exists. Symmetrical power delivery at the juncture of asymmetricallyĆ and symmetricallyĆoperated elements therefore represents the boundary conditions to be respected by all elements of the power system. 2. Just as traditional protective measures restore the symmetry and equilibrium of the system after a fault, the corrective measures associated with asymmetric operation in terms of compensating equipment must perform the same function of maintaining the symmetry and equilibrium of power delivery at predefined network locations during this mode of operation. Such predefined locations correspond, for example, to the extremities of a threeĆphase transmission line with only one or two phases fully operational. 3. Such threeĆphase elements, operated asymmetrically, must be capable of sustaining unbalanced operation with no harmful effects. This is the underlying philosophy of successful asymmetric operation which is described in this book and which represents, in a very real sense, a window of opportunity. Indeed, as we shall see further on, the implementation of this philosophy results in a postĆcontingency system which is electrically equivalent to the preĆcontingency system. This observation leads to farĆreaching conclusions regarding system security, system reliability, operational flexibility and the amount of power that can securely and reliably be transmitted in power systems. However, to integrate asymmetric operation effectively within existing power systems requires a sound understanding of the underlying concepts of power system planning practices for symmetrical threeĆphase systems. Such concepts deal with: 1) transmission line parameters, 2) voltageĆcurrent relations, 3) the generalized constants, 4) the power equations, 5) voltage regulation, 6) the representation of a power system, 7) symmetrical components, 8) stability, 9) reliability concepts, 10) insulation coordination. An overview of these issues is presented in Appendix A at the end of this book.

Chapter 2 Introduction to Asymmetric Operation of Power Transmission Systems 2.1 CONCEPT OF ASYMMETRIC OPERATION

As a concept, asymmetric operation signifies the operation of a threeĆphase element as three independent singleĆphase entities with appropriate corrective measures to ensure symmetrical power delivery at preĆdefined locations in a network. This concept is not unknown, since it is the operating mode of most distribution systems. Distribution transformers are connected to individual phases of a threeĆphase distribution feeder and a loadĆbalancing act is carried out to achieve more or less equal current in the three phases. When a fault occurs on any phase, this results at most in the loss of the threeĆphase feeder but such an event involves the loss of load only in this feeder and has no effect on the upstream power system. Such an operating strategy simply represents an economic and practical policy for distribution systems. This concept is also applied in DC power transmission systems, where two poles (one negative and one positive) are used for power transmission. Typically, ground return and the remaining sound pole are used to ensure power transmission when one pole is out of service. This asymmetric operating mode ensures DC power transmission reliability and enhances system performance in a majority of DC transmission projects. As for AC transmission system, a new set of constraints appears. First of all, the transmission system has been planned and designed to operate in a symmetrical mode, where the voltages and currents of the three phases are equal in magnitude and equally spaced in angle. One could call this the threeĆphase culture inherited from Nicola Tesla's invention of the threeĆphase induction motor. The transmission system is intimately related to the reliability and stability of the whole power system, from generation to load. The transmission system is, in part, the linchpin between generation and load and as such must ensure the symmetrical operation of the power system for all events within the design criteria. In fact, as seen in Appendix A, the AC transmission system is designed according to this threeĆphase culture. System studies are modelled on this threeĆphase basis, except for transient studies related to overvoltages in the microsecond range. Faults on the AC transmission system, whatever their nature, are cleared in the prescribed time (a few cycles) by the protection system so as to restore the power system to its preĆfault symmetrical state, albeit a degraded one. The restoration process assumes that the system is stable for the design criteria, normally the loss of one threeĆphase circuit element ( N*1) or a threeĆphase fault or both, without loss of load or generation. To this end, enhancements (if any) are incorporated into the system, such as series or shunt compensation or additional transmission. Such enhancements are all designed and specified on a threeĆphase basis to conform to the symmetrical nature of the network.

6

Chapter 2

The introduction of asymmetric operation into this environment encounters severe constraints. The fundamental requirement of symmetry in traditional power systems constrains the application of asymmetric operation to meeting specific objectives where limits must be clearly defined. In such applications, the ultimate objective is to respect the overall symmetrical operation of the system while solving the problem at hand, that is, while permitting local asymmetries forced upon the network by adverse events. In such cases, the underlying concept of asymmetric operation signifies that some localized portion of the network continues to operate normally despite its local asymmetry, provided that power delivery to and from the asymmetric subsystem appears, from the system perspective, to be symmetric. Symmetrical power delivery at the juncture of asymmetrically and symmetrically operated elements therefore represents the boundary condition to be respected by all elements of the power system. Where this is possible, such a mode of operation leads to enhanced system flexibility. For example, consider the case of a transmission corridor consisting of a single line. If this line is subjected to a singleĆphaseĆtoĆground fault resulting in the loss of the faulted phase, only two phases remain available to maintain the preĆcontingency power transfer and angular spread within acceptable limits. In order to profit from this remaining transmission capability, symmetric power delivery to and from this faulted subsystem Ċ which must now operate asymmetrically Ċ can yet be maintained, provided that appropriate measures are implemented within and at both extremities of the line. This simple case illustrates the potential of asymmetric operation in coping more generally with contingencies leading to local asymmetry. In Chapters 4 and 5, we shall see that this is due to the fact that this type of operation integrates the notion of maintaining system integrity in the postĆcontingency state leading to improved system performance from both the reliability and economic perspectives. Indeed, is this not the basic objective of symmetric operation? However, improvement in system performance is often difficult to evaluate unless one can assign a quantitative value to this improvement. Such advantages as simplicity of design or flexibility of operation are mainly subjective. Early on, reservations were expressed regarding the advantages of FACTS controllers [29] in a costĆbenefit analysis. Even the cost of improved reliability is complicated by the probability of occurrence of adverse events, which is heavily dependent on empirical data. It is for these reasons that the evaluation of improved system performance with asymmetric operation should be based on the economic comparison of alternative facilities to achieve the same result. The yardstick of such improvements is the increased secure power transfer, which translates directly into increased revenue for the power company, assuming that such increased power transfer is needed. A solution must meet a specific need: either increased power transfer, increased security limits, or increased reliability. The philosophical basis of asymmetric operation therefore rests on the objective of a symmetrical power system, along with the acceptance of asymmetric islands for meeting specific needs or constraints. The threeĆphase culture is too strongly embedded in the design of the modern power system to be supplanted by asymmetric operation, nor is this necessary in order to achieve specific goals. In fact, the reverse is true.

Introduction to Asymmetric Operation of Power Transmission Systems

7

Asymmetric operation is fundamentally a quest for symmetry, security and reliability through a new way of thinking.

2.2 SYSTEM SECURITY

In the case of existing practice, the N*1 criterion states that the power system must be stable and not suffer the loss of load in the event of the loss of one threeĆphase element, usually a transmission line with or without an associated fault (see Appendix A). Other elements such as power transformers or SVS are usually equipped with enough redundancy to meet this criterion. Following a threeĆphase N*1 event, the transmission system is in a degraded state and remedial measures must be taken to restore the system to a state whereby it remains stable if faced with another N*1 contingency. Such measures normally include reduction of power transmission, or at least some reallocation of generation within the power system. Let us consider a simple example that brings to light the consequences of imposing a threeĆphase N*1 criterion on a power system. If one is to supply an entirely new load by means of a new transmission corridor in steady state, one must first choose the voltage level in such a way as to ensure that a single transmission line can service this load. However, by virtue of the N*1 criterion, the loss of a transmission line or even of one of its sections must not result in the loss of the load. Consequently, the application of this criterion predicates the construction of two transmission lines, each of which has the capacity to fully supply the load. This also signifies that, at best, in steady state, neither line is operated at more than half of its capacity. Additionally, other criteria, such as system reliability, can bring the transfer limits to well under half of the corridor capacity, as defined by the SIL. The threeĆphase N*1 criterion is therefore seen to be highly restrictive. However, it is justified by the fact that any event that occurs, regardless of the number of phases involved, is met with the same threeĆphase response. This criterion has been enforced by most power pools and is generally referred to as a normal contingency. The adoption of the threeĆphase N*1 criterion, along with the threeĆphase fault and normal clearing times, is standard design practice since it has been observed that such a deterministic criterion is sufficient to assure the security of the system. It is for this reason that the security of the transmission system is defined in terms of such a deterministic criterion. There really has been no compelling reason to advance a probabilistic approach to transmission system design, even though the application of the threeĆphaseĂ N*1Ăcriterion contains several questionable probabilistic assumptions, such as peak loading on the system, worst location of the fault and adverse weather conditions. These observations simply underline the fact that symmetrical operation of systems imposes symmetrically oriented security criteria. However, if the system is operated asymmetrically, asymmetrically oriented security criteria can be implemented, thereby increasing security limits since this approach inflicts far less damage to the power system in its response to unexpected events. Indeed, though the system takes on a physical asymmetrically degraded state, it reverts essentially to the same preĆcontingency electrical state in postĆcontingency operation. This is a very significant result, which precludes the necessity of implementing remedial measures such as generation redispatching or reduction in the postĆcontingency state in order to withstand

8

Chapter 2

the next contingency (as in existing threeĆphase security practice). Hence, for a given level of transmission investment, the system can move more power. As we shall see in Chapters 4 and 5, the N*1 criterion can be adapted to asymmetric operation, whereby a singleĆphase response is proposed to an unexpected event of singleĆphase nature, thereby providing greater operational flexibility. Additionally, it is shown how the loss of one or more phases does not result in a reduction of the power transfer or an increase in the angular spread between the terminals, whose voltages must remain the same. In other words, the objective is to achieve a situation whereby nothing has happened in so far as the security of the system is concerned.

2.3

STRATEGIES OF IMPLEMENTATION

Two different strategies have been chosen to implement asymmetric operation for the cases of one line and several lines in a transmission corridor. One strategy is based on the use of conventional devices and the other strategy on FACTS (Flexible AC Transmission Systems). Conventional devices are passive LC elements with mechanical switching devices or synchronous var systems. According to IEEE terms and definitions, FACTS are Alternating current transmission systems incorporating power electronicĆbased and other static controllers to enhance controllability and increase power transfer capability" [29]. A FACTS controller is a power electronicĆbased system and other static equipment that provide control of one or more AC transmission system parameters".

2.3.1 Corridor With Multiple Lines The case of several parallel lines in a corridor is that most commonly encountered and has greatest potential for power system application. It is also easier to resolve than the case of a single line. Chapter 4 examines this case in great detail and the economic analysis is found in Chapter 6. To illustrate the basic strategy, let us consider the case of the loss of one phase on one of such parallel lines. Considering the use of conventional devices, the positiveĆsequence impedance can be corrected to the prefault corridor value by series capacitors installed in the phase of each nonĆfaulted line in the corridor corresponding to the phase that has been switched out of service on the faulted line. The voltages at the sending and the receiving ends can be adjusted by shunt capacitors whose rating is equal to that of the capacitance of one line to compensate for the charging of the phase that has been lost. It may be that there is a voltage drop at the receiving end due to the increased current and recourse may have to be made to a compensating device to provide additional voltage support. Compensation of the residual negativeĆ and zeroĆsequence currents is not necessary if it can be demonstrated that these values are small enough to be acceptable. If one considers the use of FACTS controllers, series voltage sources and shunt current sources can be placed in various locations to compensate for the unbalances resulting from the loss of one phase of one line. There are several combinations of these controllers that can be used to balance the system. Usually, some form of optimization process may be necessary to arrive at the best solution.

Introduction to Asymmetric Operation of Power Transmission Systems

9

2.3.2 SingleĆline Corridor The case of a corridor consisting of a single threeĆphase transmission line is the most difficult to resolve since it involves compensating for the positiveĆ, negativeĆ and zeroĆsequence currents that result from the loss of one phase of the threeĆphase line. It is the case most encountered for connecting remote loads, remote generating systems, transmission rings surrounding large metropolitan areas, and in all situations where building a second line is either uneconomical, impractical or, for all intents and purposes, impossible. Chapter 5 examines this case in great detail and the economic analysis is found in Chapter 6. To illustrate the basic strategy while employing conventional devices, first, series capacitors are used for the compensation of the positive sequence and are usually assumed to be located at the midĆpoint of the line for best results. Compensation of the negative sequence is then provided by connecting a reactor and a capacitor between the three phases at each end of the line. This interphase connection creates a low impedance path for the negativeĆsequence currents. In a sense, this solution is an extension of the principle of load compensation. There should be no need to change the values of the L and C elements for different loading conditions within certain limits. Finally, zeroĆsequence currents are filtered through the use of grounding transformers located at each end of the line. Existing power transformers at each end of the line can serve this purpose, although they may have to be reinforced for this duty. It can be seen that the use of conventional devices for the asymmetric operation of one line is somewhat complicated but possible. With respect to the use of FACTS controllers, the positiveĆsequence and zeroĆsequence currents can be compensated by series voltage sources located at the sending end, the receiving end or the midĆpoint of the line. The negativeĆsequence currents can be compensated by shunt current sources located at each end of the line. Evidently, this solution is simpler but likely more costly.

2.4 REACTIVE POWER MANAGEMENT The role of reactive power is crucial to realizing the prescribed performance of a transmission system, whether in the symmetrical or asymmetrical mode of operation. Reactive power, in fact, defines the operating state or conversely, the operating state defines the reactive power requirements. For example, surplus reactive power is automatically translated into overvoltage and a deficit of reactive power is translated into undervoltage. Consider, for example, the case of power flow through a pure reactance. If it is desired to have the same voltage magnitude at each terminal and no reactive power supply from each end, then the power flow will be zero since sin d = 0 and cosĂdĂ=Ă1 (see Appendix A). In the case of a transmission line, the line capacitance, or more precisely the reactive power that is generated by the line capacitance, makes the power transfer possible. The most interesting value of the power transfer is the surge impedance loading (SIL) at which the line capacitance compensates exactly for the reactive power losses in the line. Most EHV and UHV systems operate at the SIL for these reasons. This loading provides

10

Chapter 2

a benchmark against which to evaluate feasible alternatives such as voltage level and number of circuits. For example, loading below the SIL will result in a surplus of reactive power, just as loading above the SIL will result in a deficit. Since there is usually a normal range of loadings about the SIL, there must be some control of the voltage level to prevent loadingĆrelated reactive power imbalances from being translated into overvoltage or undervoltage, as the case may be. The static compensator has been found to be the most useful device for this purpose. In fact, the static compensator operates as a voltage source within its rating. If the voltage at a given point is higher than the desired value, reactive power will be absorbed by the compensator, and if the voltage is lower than the desired value, reactive power will be supplied by the compensator. Such devices have been in service in most major utilities for 25 years and can be considered as conventional devices even though they are based on power electronics. It may be that the static compensator, combined with conventional passive elements, is the ideal solution to successful asymmetric operation.

2.5 SUMMARY In this chapter, asymmetric operation and the objectives of this book have been defined. Asymmetric operation is the operation of a threeĆphase element as three independent singleĆphase entities with appropriate corrective measures to ensure symmetrical power delivery at preĆdefined locations in a network. Two distinct groups of strategies are identified, one for corridors with a single line, and the other for corridors with multiple lines. All these strategies for implementing asymmetric operation are seen to be based on reactive power compensation. The advantages of asymmetric operation are briefly seen to be either increased operational flexibility, increased power transfer, increased security limits, or increased reliability.

Chapter 3ă Probabilistic Aspects of Asymmetric Operation 3.1 INTRODUCTION Probability analysis is an essential ingredient in power system planning and operation. There are many processes where the application of probability methods is a determining factor, such as in decision making, risk management, choice of alternatives, constraints in terms of probability of occurrence and especially reliability analysis. The application of probability concepts in power system analysis is based mainly on the Binomial Law and the Poisson, Weibull and Gauss distributions. The viability of asymmetric operation is intimately linked to probability concepts and in particular, the concept of risk. For example, if the risk of losing a threeĆphase line is small or zero, it is not necessary to consider asymmetric operation. On the other hand, if the risk of losing such a line is high, then asymmetric operation can be applied to reduce this risk. In the following sections, some sample applications of probability concepts are discussed first, and the relationship to asymmetric operation is then described.

3.2 THE BINOMIAL LAW 3.2.1 Definitions Figure 3.1 Shows the states of a system consisting of one repairable unit: the functional state, or up" state, and the outĆofĆservice state, or down" state. 1 m

State 1: Unit is functional (up" state) l

2

State 2: Unit is out of service (down" state) Figure 3.1 A twoĆstate repairable system.

One considers that this device is out of service at a given rate designated by l and that it is repaired at a given rate designated by m. Thus, the mean time spent in the functional state, typically referred to as up time" (UT), is: UT + 1ńl

(3.1)

and the mean time spent in the outĆofĆservice state, typically referred to as down time" (DT), is: DT + 1ńm

(3.2)

12

Chapter 3

One defines the availability A, as the ratio of the mean time in the functional state (UT) to the total operation time for a given period (total time" or TT = DT + UT): 1ńl m A + UT + + TT l)m 1ńl ) 1ńm

(3.3)

whereas the nonĆavailability (A) is defined as the ratio of the mean time spent in theĆoutĆofĆservice state (DT) to the total time for the same given period (TT): 1ńm A + DT + + l TT l)m 1ńl ) 1ńm The probability p that the device is in the functional state is obtained by: m p+A+ l)m

(3.4)

(3.5)

and the probability q that the device is in the outĆofĆservice state is given by: q+A+

l +1*p l)m

(3.6)

One defines the frequency of transition from one state (initial state) to another as the product of the probability of being in the initial state and the transition rate from the initial state to the next state. For example, the transition frequency from the functional state to the outĆofĆservice state for one device with two states is given by: f 12 + pl +

lm l)m

(3.7)

and the frequency of transition from the outĆofĆservice state to the functional state is given by: f 21 + qm +

lm l)m

(3.8)

In steady state, the frequency of transition from one state to the other is the same.

3.2.2 Two Repairable Identical Units Consider a system of two identical and repairable units, each having a failure rate of l and a repair rate of m. The probability of being in the functional state for each unit is p and the probability of being in the outĆofĆservice state is q. For such a system, there are three possible states, as follows: ćă The two units are functional (no unit is out of service); ćă Only one unit is functional (one unit is out of service); ćă No unit is functional (two units are out of service). Figure 3.2 illustrates the three possible states and the transition rates of one state to another of such a system.

Probabilistic Aspects of Asymmetric Operation

1 m

State 1: two units are functional 2l

2
m

3

13

l

State 2: only one unit is functional State 3: no unit is functional

Figure 3.2ĂĂĂThe three states of a system consisting of two identical repairable units. Let Pi be the probability that the system is in state i (i = 1, 2 or 3). The sum of the probabilities is equal to unity. The transition frequencies from one state to another are constants in stationary steady state and are given by: F 12 + P 1 @ (2l) + P2 @ m + F 21 F 23 + P 2 @ l + P3 @ (2m) + F 32 (3.9) P1 ) P2 ) P3 + 1 Resolving the preceding equations gives the following probabilities: m2 + p2 P1 + (l ) m) 2 lm P2 + 2 + 2pq (l ) m) 2 l2 P3 + + q2 (l ) m) 2

(3.10)

One notices that the probabilities of the three states are given by the terms of the expansion of the expression: (p + q)2.

3.2.3 Many Repairable Identical Units In a similar manner, if a system is composed of N identical repairable units, each unit having two states with the corresponding probabilities p and q respectively, then there exist N + 1 possible states (N, N*1, ..., 0) and the probability for the system to be in a given state is given by the appropriate term of the expansion of the expression: (pă+ăq) N. The probability Pk of having exactly k units in service within N units is given by the following equation: N N*k P k + k Ăp kĂ(1 * p) (3.11)

ǒǓ

where

ǒNkǓ + k!Ă(NN!* k)! is the coefficient of

expansion of (pă+ăq) N.

p kĂ(1 * p)

N*k

in the binomial

14

Chapter 3

Example 3.1 Static Compensators To operate an electric power transmission corridor, it has been determined that six static compensators must be in service at all times. One needs to determine, therefore, the number of static compensators to be installed (the six and the spares) in order to meet a given reliability criterion during system operation, such as at least 99% availability of the static compensators. The static compensators have the following characteristics: ć Failure rate: one failure per month; ć Mean repair time: one day and a half Thus, the failure rate is lĂ=Ă12 failures/year and the mean yearly down time is 18Ădays. For each static compensator, the nonĆavailability A and the probabilities of being functional p or failed q are given by the following: A + 18 + 0.05 + q 365 (3.12) p + 1 * q + 0.95 Under these conditions, one must install eight static compensators (six with two spares) to guarantee that at least six static compensators are in service with a probability of at least 99%. Table 3.1 gives a summary of the results for different values of nonĆavailability. For a nonĆavailability of 0.1 (2 failures per month), one must install at least three spare static compensators, and for a nonĆavailability of 0.15 (3 failures per month), one would need four spare units. As a result, one can estimate the cost of nonĆavailability in terms of spare units required. Table 3.1ĂĂĂMinimum number of static compensators required for a given nonĆavailability to guarantee the operation of at least six static compensators with a probability of 99% NonĆavailability ratio A

Number of installed static compensators and probability of at least 6 in service 6

7

8

9

0.05

0.7351

0.9556

0.9942

0.9994

0.10

0.5314

0.8503

0.9619

0.9917

0.15

0.3771

0.7166

0.8948

0.9660

Number of spare static compensators

0

1

2

3

3.3 THE POISSON DISTRIBUTION A process for which the mean number of events per unit time or space is constant and the events are independent can be described by a Poisson process. If Xt is a random variable representing the actual number of events per interval of time t and l the expected

Probabilistic Aspects of Asymmetric Operation

15

number of events per interval unit, then the probability distribution function of a Poisson process is given by the following: P(X t + n) +

(lt) n -lt e ĄąąĄn + 0, 1, 2, AAA n!

(3.13)

The Poisson distribution is used in a variety of stochastic processes in power systems analysis, such as determining the expected number of failures of transmission equipment (lines, generators, transformers, etc.) per time interval.

Example 3.2 735ĂkV Power Transformers Assume that one wants to determine the probability of having, in the present year, the number of major failures of 735ĂkV power transformers based on a total of 600 units. Observations during a certain period have shown 100 major failures of 735ĂkV power transformers over a total of 4Ă000 transformerĆyears. Based on this data, the statistical failure rate is 2.5%. Thus on a population of 600 transformers, the failure rate is l = 15 expected failures per year. The probability of having exactly 15 major failures in this year (t = 1) is given by: (15 @ 1) 15 -15 (3.14) P(X 1 + 15) + Ă e + 0.10 15! Thus, even if the expected number of major failures is 15, the probability of having exactly 15 major failures is only 10%. Also of interest is the probability of having 15 or less major failures in this year. This probability is given by: P + e -15

15

ȍ 15n!n

(3.15)

n+0

This value of P is approximately 50%, which corresponds to the midĆpoint of the Poisson distribution as one would expect. One can thus schedule the planning of redundancy and maintenance accordingly.

3.4 THE WEIBULL DISTRIBUTION 3.4.1 The Gauss Law or Normal Law A random variable obeys the Gauss law or normal law if its probability distribution function is given by the following: f x(x) +

2

1 e -(x*m) 2s 2 ĄąąĄĄĄ-R v x v R,ĂĂ s u 0 Ǹ2p s

(3.16)

where m is the mean and s is the standard deviation. The probability density function (cumulative distribution) is given by: x

F x(x) +

1 Ǹ2p s

ŕe -R

(t*m)2 2s 2

-

dtĄąąĄ-R v x v R,ĂĂ s u 0

(3.17)

16

Chapter 3

It is difficult to evaluate the integral in equation (3.17) between two arbitrary numbers a and b. To overcome this difficulty, the probability density function is given in special tables using the GaussĆcentred law or standard normal law with m = 0 and s = 1. The standard law has the following probability distribution and density functions, respectively: z2 f z(z) + 1 e - 2 ąąąąąĄĄĄ-R v z v R Ǹ2p

z

F z(z) + 1 Ǹ2p

ŕ e dtĄąąĄĄĄ-R v z v R

(3.18)

2

-t2

-R

Passing from equation (3.17) to equation (3.18) is done by a change of variable z + (x*m)ńs in equation (3.17). Thus, the probability density function is evaluated through the standard normal law as follows: x*m (3.19) F x(x) + F z s

ǒ

Ǔ

and the probability for variable X to be between two given numbers a and b is given by: P(a v X v b) + F x(b) * F x(a) + F z

ǒb *s mǓ * F ǒa *s mǓ z

(3.20)

3.4.2 The Weibull Distribution The probability distribution function of a random variable according to the Weibull distribution is: b x b-1 -ǒ x Ǔ b (3.21) Ǔ e a ĄąąĄĄĄa, b u 0,Ă x u 0 f x(x) + a ǒa and the probability density function is given by: x

F x(x) +

ŕ f (t)dt + 1 * e x

-ǒax Ǔ

b

ĄąąĄĄĄa, b u 0,Ă x u 0

(3.22)

-R

An alternative representation of the Weibull distribution used in power transmission systems is given by:

ǒ Ǔǒ Ǔ

xǓ m-1 ln 1 1 ǒ f w(x) + m n 1)n 2 2

ǒ1)nxǓ m

ąąĄĄn u 0,Ą m u 0,Ą x w -n

(3.23)

and the corresponding probability density function is: x

F w(x) +

ŕ f (t)dt + 1 * 0.5 w

ǒ1)nxǓ m

ĄąąĄn u 0,Ą m u 0,Ą x w -n

(3.24)

-R

For some value of parameters m and n, the probability density function of the normalcentred law and the probability density function of the Weibull law are approximately

Probabilistic Aspects of Asymmetric Operation

17

the same within a given interval of interest, say -4Ă≤ĂxĂ≤Ă-1 (see Figure 3.3). n is termed the truncation point and is determined so as to have Fw (x)Ă=ĂFz (x)Ă=Ă0 for xĂ=Ă-4. m is determined so that the probabilities for xĂ=Ă-1 are the same according to the normal standard law and the Weibull law [2]. In other words: F w(-4) + F z(-4) + 0 (3.25) F w(-1) + F z(-1) + 0.16 The solution of equation (3.25) gives nĂ=Ă4 and m = 5 and Figure 3.3 shows a comparison of the probability density function of the two distributions with nĂ=Ă4 and mĂ=Ă5. 1.0 Fw Fz

0.8 Fw Fz

0.6

fw 0.4 fz 0.2

fz

0.0 -4

-3

-2

-1

0

fw

x

1

2

3

4

Figure 3.3ĂĂĂComparison of Weibull distribution for nĂ=Ă4 and mĂ=Ă5 and normal distribution.

Example 3.3 Parallel Insulator Strings Assume that it is desired to determine the critical withstand voltage U50_N and standard deviation s N of N identical strings of insulators in parallel if the critical withstand voltage U50 and standard deviation s of one string of insulators are known. In this case, the normal standard law can be replaced by the Weibull alternative distribution with parameters n = 4 and m = 5, so that there is no need to use the probability tables. For one string of insulators, the probability of flashover corresponding to a voltage impulse U is given by the Weibull distribution as: U * U50 x+ s (3.26) 5 x p (x) + P(X + x) + F (x) + 1 * 0.5 ǒ1)4Ǔ w

w

where s is the standard deviation of the normal law and U50 is the flashover voltage with 50% probability. As pw (x) = 0 for x = -4, the Weibull distribution gives the absolute withstand voltage Uaw as: U aw + U 50 * 4s

(3.27)

For N strings of insulators in parallel, if one defines U50_N as the flashover voltage with 50% probability and s N as the standard deviation of the normal law, then the

18

Chapter 3

probability of flashover corresponding to a voltage impulse U for the set of N strings of insulators is: U * U50_N xȀ + sN (3.28) 5 ǒ 1)xȀ4 Ǔ p w(xȀ) + 1 * 0.5 The absolute withstand voltage for the set of N strings of insulators UawN is given for xȀă=ă-4 and is: U awN + U 50_N * 4s N

(3.29)

To relate s N to s, one determines the probability of flashover of the set of N parallel identical strings of insulators in terms of the probability of flashover of one string of insulators (given by equation (3.26)) and then compares to equation (3.28). The probability of flashover of the set of N strings of insulators pw ( xȀ) in terms of the probability of flashover of one string of insulators is: p w(xȀ) + 1 * q w(xȀ) + 1 * 0.5 Nǒ1)4Ǔ x

5

(3.30)

where qw ( xȀ) is the probability of withstand of N identical strings of insulators in parallel and is given by: q w(xȀ) + [ q w(x) ] + 0.5 Nǒ1)4Ǔ N

x

5

(3.31)

where qw (x) is the probability of withstand of one string of insulators given as: q w(x) + 1 * p w(x) + 0.5ǒ1)4Ǔ x

5

(3.32)

Comparing equations (3.30) and (3.28) and using the fact that the absolute withstand voltage is the same for N strings of insulators as for one string of insulators (UawNă =ăUaw ), one can relate s N to s as follows: s N + 5s (3.33) ǸN Thus, if for one string of insulators the critical withstand voltage U50 and the standard deviation s of the normal law are known, one can determine the critical withstand voltage U50_N and the standard deviation s NĂof the normal law for N identical strings of insulators in parallel subjected to the same impulse voltage, using the Weibull distribution. For a 735ĂkV line, one has: U50 = 1Ă600 kV, s = 100 kV and U-4s = 1Ă200 kV. For a 333Ăkm line there are 1Ă024 towers with 1Ă024 insulator strings in parallel. Thus, one has after calculation: N = 1Ă024, s N = 100ń 5Ǹ1024 = 25 kV and U50_NĂ = U-4sĂ + 4s N orĂ1Ă300ĂkV. The ability to carry out this exercise is based on the fact that the Weibull distribution has a truncation point or an absolute withstand.

Probabilistic Aspects of Asymmetric Operation

19

3.5 APPLICATION TO ASYMMETRIC OPERATION In this section, the advantage of asymmetric operation with respect to transmission lines is shown by probabilistic arguments. First, the statistics of transmission line failures are examined and then the reliability of transmission lines in asymmetric operation and in symmetric operation are calculated and compared.

3.5.1 Statistics of Transmission Line Failures The failures of transmission lines are caused by events such as bad weather, equipment faults and human error. The results of such causes are usually phaseĆtoĆground faults (more than 95%), some phaseĆtoĆphase faults (due essentially to galloping) and very rarely a threeĆphase fault (caused essentially by exceptional events such as a forest fire). The statistics of transmission line faults show that the percentage of singleĆphase faults increases with the nominal voltage and the dimensions of the line: from 60% for 220ĂkV lines, to 97% for 735ĂkV lines [10, 52]. Single Pole Reclosure (SPR) has been adopted by many companies in order to increase the performance of the transmission system, even if the threeĆphase fault was the design criterion of the system. The number of successful automatic reclosures decreases with the increase of the voltage level from 72% at 330 kV to only 52% at 765ĂkV [10, 52]. This is due in part, to the fact that the effect of the fault cannot always be eliminated by automatic reclosure, especially in extraĆhighĆvoltage (EHV) systems. For voltage levels below 300 kV, SPR is relatively successful because lightning and the associated insulation flashover are the principal causes of failure. For EHV systems, the insulation level is very high and results in a smaller number of failures caused by lightning compared to the number of failures caused by line terminal equipment. As the equipment of EHV systems is singleĆphase, the failures in EHV systems can also be classified in the category of phaseĆtoĆground faults. SPR however, cannot be applied to faulted equipment. In any case, it is not possible to apply SPR when the lines are equipped with shunt reactors as is the case for the 735ĂkV system of HydroĆQuébec. The fact that most transmission line faults are singleĆphase is a motivation for studying asymmetric operation, even if the practice of SPR results in a success rate of about 50%. The threeĆphase fault criterion has been adopted because it is simple to define and verify, and because it is an umbrella that covers a large number of contingencies. However it is wrong to give too much physical significance to this criterion, which is essentially a concept (statistics reveal that the threeĆphase fault has only a 1% probability of occurence). It is more appropriate to consider asymmetric operation to supplement the threeĆphase fault criterion in the majority of cases where this criterion does not apply. Indeed, if the power system is capable of responding to singleĆphase contingencies with a singleĆphase or asymmetric response strategy, the threeĆphase criterion is clearly no longer appropriate. 3.5.2 Reliability Analysis The most frequently used reliability index in transmission system planning is the Loss of Load Expectation (LOLE). The LOLE is the expected mean of the energy not supplied due to the network configuration and the failure of its components [1]. Another form of the LOLE that is currently used is the expected mean of the nonĆtransmitted

20

Chapter 3

energy. This form of the LOLE is used in the following sections for the reliability analysis of a multipleĆline transmission corridor in symmetric operation and in asymmetric operation. In view of the fact that risk, in general terms, is obtained by weighing the consequences of an event by its probability of occurrence, this form of the LOLE is also a measure of the risk of transmitting power. This measure of risk depicts the potential of asymmetric operation [55]. Figure 3.4 shows the logic circuits for the reliability analysis of a threeĆphase transmission line under symmetric or asymmetric operation. In the symmetrical approach, events leading to the loss of any one phase result in the loss of all three phases, hence, the equivalent logical circuit in this case presents the three phases in series. The corollary to this is that if the power system reacts to any contingency with a threeĆphase strategy, such a threeĆphase strategy therefore dictates the use of a threeĆphase criterion for the planning and operation of power transmission systems. In the asymmetrical approach, the three phases function independently and thus, the equivalent logical circuit presents the three phases in parallel. It also follows that if the system can be operated asymmetrically, that singleĆphase criteria need only be used to plan and operate the transmission system. a)

b)

p

p

p

A

B

C

p

A

p

B

p

C

AƐBƐC p sym + p 3

AƒBƒC p asym + 1 * (1 * p)

3

Figure 3.4ĂĂĂEquivalent logical circuits of a threeĆphase line: a) symmetric operation; b) asymmetric operation. The probability of successful transmission of power between the sending and the receiving ends of the line is different in each case and the objective is to calculate and compare the LOLE in each case. For this purpose, consider a corridor of N threeĆphase lines transmitting a total power value of power transmission of T where the probability of successful transmission of each phase of each threeĆphase line is p and the probability of failure of each single phase is q = 1 ć p. The usual practice in the industry is to consider the probability of success of a threeĆphase line as the combined probability of transmission of each phase within the three according to the total exposure of the line to the environment. But this probability can be separated and the contribution of each phase defined. As asymmetric operation is a perĆphase approach, the normal symmetric operation must also converted in per phase so as to make the calculation and the analysis on the same basis. Thus, in the following

Probabilistic Aspects of Asymmetric Operation

21

sections, p is the probability of successful transmission of each phase of a threeĆphase line as it is subjected to its own exposure to events. Expected mean nonĆtransmitted power according to symmetric operation. In normal symmetric operation, the combined probability of successful transmission psym and the combined probability of unsuccessful transmission of one threeĆphase line are given respectively as follows: p sym + p 3

(3.34)

q sym + 1 * p 3

(3.35)

where p is the probability of successful transmission of each phase of the line. Thus, the probability of unsuccessful transmission of k threeĆphase lines within N follows a binomial distribution of parameters (N, qsym ) and is given by: Q sym +

ǒNkǓĂq

k symĂ

ǒ1*q symǓN*k +

N! ǒ1*p 3Ǔkp3(N*k ) k!(N*k )!

(3.36)

The transmission capacity loss caused by the total failure of k lines within N is k ā T. N The loss of load expectation of this event under symmetrical operation LOLEsym (k,N) is the product of the lost transmission capacity and the probability of losing that transmission capacity, as follows:

ǒǓ

N*k N LOLE sym(k, N) + k Ă TĂ k Ăq ksymĂǒ1 * qsymǓ N k N! ąąąąąą+ TĂ k Ă Ă ǒ1 * p 3Ǔ Ăp 3(N*k ) N k!Ă( N * k )!

(3.37)

One can also consider LOLEsym (k,N) as the contribution of this particular event to the overall LOLE of symmetrical operation. The overall loss of load expectation of symmetrical operation LOLEsym is therefore obtained by the summation of the individual contribution of each event and is given by the following: N

LOLE sym + ȍ LOLE sym(k, N) k+0 N

k N! Ă ǒ1 * p3Ǔ Ăp 3(N*k ) + Tǒ1 * p 3Ǔ ąąąă+ T ȍ k Ă N k!(N * k )!

(3.38)

k+0

Expected mean nonĆtransmitted power according to asymmetric operation. During asymmetric operation, the combined successful transmission probability pasym and the combined unsuccessful transmission probability qasym of one threeĆphase line results from a binomial distribution on the threeĆphases with parameter (3, p) and (3, q) respectively. Thus one has for each line, the following equations: p asym + 1*(1*p) q asym + (1*p)

3

3

(3.39) (3.40)

22

Chapter 3

Under these conditions, the probability of nonĆtransmission of k threeĆphase lines within N, Qasym , is also given by a binomial distribution with parameters (N, qasym ): Q asym +

ǒNkǓĂq

k asymĂ

ǒ1*q asymǓN*k +

ƪ

N! (1 * p) 3kĂ 1*(1*p) 3 k!(N*k )!

ƫ

N*k

(3.41)

The loss of load expectation of this event under asymmetrical operation (i.e. LOLEasym (k,N)) is given by the following equation:

ǒǓ

N*k N LOLE asym(ā k, N ) + k TĂ k Ăq kasymĂǒ1 * q asymǓ N

ƪ

3k 3 N! ąąąąąąĂ+ T k Ă Ă (1 * p) Ă 1 * (1 * p) N k!( N * k )!

ƫ

N*k

(3.42)

The overall loss of load expectation of asymmetrical operation LOLEasym is therefore given by: N

LOLE asym + ȍ LOLE asym(k, N ) k+0 N

N! Ă (1*p) 3kĂƪ1*(1*p)3ƫ ąąąą+ ȍ k Ă T N k!(N*k )!

N*k

3 Ă +ă TĂ(1*p)

(3.43)

k+0

Benefit of asymmetric operation.ĂĂThe difference between the loss of load expectation of symmetric operation and asymmetric operation, DLOLE, indicates the benefit of asymmetric operation, which can then be evaluated at the energy generation cost:

ƪ

ƫ

3 DLOLE + LOLE sym*LOLE asym + ǒ1*p 3ǓĂ*Ă(1*p) ĂTĂ + 3pā(1 * p)āT (3.44)

Equation (3.44) shows that DLOLE is always greater than zero (DLOLE > 0) for 0ă