POIilDK SYSTDM OPTIMIZATIONtrlndlNiirftlfifdrSl

D.P. KOTITARI . J.S. DHILLON

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Power system optimization is intended to introduce the methods of multi-objectiveoptimization in integrated electric power system operation, covering economrc,environmental, security and risk aspects as well. Evolutionary algorithms-which mimicnatural evolutionary principles to constitute random search and optimization proceduresare appended in this new edition to solve generation scheduling problems. writtenin a student-friendly style, the book provides simple and understandable basiccomputational concepts and algorithms used in generation scheduling so that thereaders can develop their own programs in any highlevel programming linguage. Thisclear, logical overview of generation scheduling in electric power systehs plrmits bothstudents and power engineers to understand and apply optimization on a dependablebasis. The book is particularly easy-to-use with sound and consistent terminology andperspective throughout.

This edition presents systematic coverage of local and global optimization techniquessuch as binary- and real-coded genetic algorithms, evolutionary algorithms, particleswarm optimization and differential evolutionary algorithms. The eConomic dispatchproblem presented, considers higher-order nonlinearities and discontinuities in input-output characteristics in fossil fuel burning plants due to valve-point loading, ramp-rate limits and prohibited operating zones. search optimization techniques piesentedare those which participate efficiently in decision making to solve the multiobjectiveoptimization problems. stochastic optimal generation scheduling is also updated in thenew edition. Generalized Z-bus distribution factors (GZBDF) are presented to computethe active and reactive power flow on transmission lines. The interactive decisionmaking methodology based on tuzzy set theory, in order to determine the optimalgeneration allocation to committed generating units, is also discussed.

This book is intended to meet the needs of a diverse range of groups interested inthe application of optimization techniques to power system operation. lt requires onryan elementary knowledge of numerical techniques and matrix operation to understandmost of the topics. lt is designed to serve as a textbook for postgraduate electricalengineering students, as well as a reference for faculty, researchers, and powerengineers interested in the use of optimization as a tool for reliable and secureeconomic operation of power systems.

The book discusses:o Load flow techniques and economic dispatch-both classical and.rigorousi Eionomic Oispatch considering valve-point loading, ramp-rate limits and prohibited

operating zoneso ieal coded genetic algorithms for economic dispatcho Evolutionary

-programming for economic. dispatch.

o Particle swarm optimization for economlc olspalcn. Ditferential evoluiionary algorithm for economic disp.atcho Stochastic multiobjective thermal power dispatch w-ith securityc Generalized Z-bui distribution factors lo compute line flowe Stochastic multiobiective hydrothermal generation.schedulingr Muttiobiective ther'mal powbr dispatch u-sing artificial neural networksc Fuzzy multiobiective generation scheduling. Multiobiective generation scheduling by .e"t.hing *"ight p"fi

FD.P. KOTHARI, Ph.D., is Vice Chancellor of VIT University, !e-!org. Earlier' he wasFrotessor at the centie for Energy studies, lndian Institute of Technology Delhi. He

also served as Director-in-Chargd, llT Delhi (2005), Deputy Director (Administration),iii oLini (2003-oo), Principal, National lqstlrlg.9f.-Tech.nolggv-.1tt"gPql (1se7-1se8)

anO-HeiOl Centre'ior Energy Studies, llT Delhi (1997-1998). Dr. Kothari's fields of

specialization include optimal-hydro-thermal scheduling, unit commitment, maintenancesbheduling, energy cohservati'on, and power quality.and energy systems planning

inO-roOdtiing. A"iecipient of several national awards, Dr. Kothari-who contributedextenJivety to" the spdcialized areas, guided -28 Ph.D.s and 60 M.Tech.s, authorediO Ooot<sbn Power'systems and pubiished 625 research papers in various nationaland international journals.

J,S. DHILLON, Ph.D., is Professor, Department of Electrical and InstrumentationEngineering, sant Longowal Institute. of Engineering and Technology, Longowal, wherefre"also seiied as the-head ol the department (from 2OO2 Io 2005)' Earlier he servedas Assistant Professor (1992-2002) Giani Zail singh college of -Ergineering andiechnofogy, Bathinda anb Lecturer (1987-199-2), Thapar Institute. ol Engineering andfecnnotody, patiala. professor Dhillon has published/pre.sented 91 research papers invarious niiional and international journals/conferences. His research interests includemicroprocei"or applications, multi6bjective thermal.dispatch, hydrothermal scheduling'

neural networks, tuzzy set theory and soft computing applications in power systems'

Preface. Preface to the First Edition1. Introduction2. Load Flow Studies3. Economic Load Dispatch of Thermal Generating Units4. Optimal Hydrothermal Scheduling-,5. Mfuftiob;eciive Generation Scheduling^ - . .6. Stocha6tic Multiobiective Generation Scheduling- --7. Evolutionary Progiamming lor Generation Scheduling ̂8. Multiobiectiire Ge-neration Schedulingl Weight Pattern SearchAppendices . lndex

Multi Colour Services 09/2010

POWER SYSTEMOPTIMIZATION

D.P. KOTHARIVice ChancellorVIT University

Velloreand

Former Director-in-ChargeIndian Institute of Technology Delhi

New Delhi

J.S. DHILLONProfessor

Department of Electrical and Instrumentation EngineeringSant Longowal Institute of Engineering and Technology

(Deemed-to-be-University)Longowal, Punjab

New Delhi-1100012011

SECOND EDITION

v

ContentsContentsContentsContentsContents

Preface .................................................................................................................................................. xiPreface to the First Edition .............................................................................................................. xiii

1. Introduction ........................................................................................................................... 1–11

1.1 A Perspective ...................................................................................................................... 11.2 The Components of a Power System ................................................................................ 21.3 Power System and Computers ............................................................................................ 31.4 Planning and Operating Problems ..................................................................................... 4

1.4.1 Resource and Equipment Planning ........................................................................ 41.4.2 Operation Planning ................................................................................................. 51.4.3 Real-Time Operation .............................................................................................. 5

1.5 Artificial Intelligence and Neural Networks ..................................................................... 61.6 Fuzzy Theory in Power Systems ....................................................................................... 61.7 Evolutionary Algorithms .................................................................................................... 7References ................................................................................................................................... 11

2. Load Flow Studies ............................................................................................................. 12–134

2.1 Introduction ....................................................................................................................... 122.2 Network Model Formulation ............................................................................................ 142.3 YBUS Formulation .............................................................................................................. 17

2.3.1 No Mutual Coupling between Transmission Lines ............................................ 172.3.2 Mutual Coupling between Transmission Lines ................................................... 18

2.4 Node Elimination in ZBUS .......................................................................................................................................... 212.5 ZBUS Formulation .............................................................................................................. 23

2.5.1 No Mutual Coupling between Transmission Lines ............................................ 232.5.2 Mutual Coupling between Transmission Lines ................................................... 29

2.6 Load Flow Problem .......................................................................................................... 442.6.1 Slack Bus/Swing Bus/Reference Bus .................................................................. 462.6.2 PQ Bus/Load Bus ................................................................................................. 472.6.3 PV Bus/Generator Bus.......................................................................................... 472.6.4 Voltage-Controlled Buses ..................................................................................... 472.6.5 Limits ..................................................................................................................... 47

vi Contents

2.7 Computation of Line Flows ............................................................................................. 482.8 Modelling of Regulating Transformers ........................................................................... 492.9 Gauss–Seidel Method ....................................................................................................... 532.10 Newton–Raphson Method ................................................................................................ 662.11 Decoupled Newton Method.............................................................................................. 832.12 Fast Decoupled Load Flow (FDLF) ................................................................................ 952.13 Initial Guess for Load Flow .......................................................................................... 1052.14 DC System Model .......................................................................................................... 1092.15 AC–DC Load Flow ........................................................................................................ 1132.16 Conclusion ....................................................................................................................... 117References ................................................................................................................................. 133

3. Economic Load Dispatch of Thermal Generating Units ........................................... 135–248

3.1 Introduction ..................................................................................................................... 1353.2 Generator Operating Cost ............................................................................................... 1363.3 Economic Dispatch Problem on a Bus Bar .................................................................. 137

3.3.1 Limit Constraint Fixing ...................................................................................... 1393.4 Optimal Generation Scheduling ..................................................................................... 1423.5 Economic Dispatch Using Newton–Raphson Method .................................................. 1493.6 Economic Dispatch Using the Approximate Newton–Raphson Method..................... 1543.7 Economic Dispatch Using Efficient Method ............................................................... 1573.8 Classical Method to Calculate Loss Coefficients ......................................................... 1623.9 Loss Coefficient Calculation Using YBUS .................................................................................................... 1723.10 Loss Coefficients Using Sensitivity Factors ................................................................. 176

3.10.1 DC Load Flow .................................................................................................... 1763.10.2 Power Loss in a Line ......................................................................................... 1773.10.3 Generation Shift Distribution (GSD) Factors .................................................... 1783.10.4 Generalized Generation Shift Distribution (GGSD) Factor ............................. 1793.10.5 Derivation of GGDF ........................................................................................... 1793.10.6 Evaluation of B-Coefficients .............................................................................. 181

3.11 Transmission Loss Coefficients ..................................................................................... 1843.12 Transmission Loss Formula: Function of Generation and Loads ............................... 1883.13 Economic Dispatch Using Exact Loss Formula ........................................................... 1893.14 Economic Dispatch Using Loss Formula Which Is Function of Real and

Reactive Power ............................................................................................................... 1983.15 Economic Dispatch for Active and Reactive Power Balance ...................................... 2033.16 Evaluation of Incremental Transmission Loss .............................................................. 208

3.16.1 Alternative Method to Evaluate Incremental Loss ........................................... 2113.17 Economic Dispatch Based on Penalty Factors ............................................................. 2123.18 Optimal Power Flow Based on Newton Method ......................................................... 218

3.18.1 Limits on Variables ............................................................................................ 2233.18.2 Decoupled Method for Optimal Power Flow ................................................... 233

3.19 Optimal Power Flow Based on Gradient Method ........................................................ 2363.19.1 Inequality Constraints on Control Variables ..................................................... 2393.19.2 Inequality Constraints on Dependent Variables ................................................ 239

References ................................................................................................................................. 245

Contents vii

4. Optimal Hydrothermal Scheduling ............................................................................... 249–324

4.1 Introdcution ..................................................................................................................... 2494.1.1 Classification of Hydro Plants ........................................................................... 2504.1.2 Long-Range Problem .......................................................................................... 2524.1.3 Short-Range Problem.......................................................................................... 253

4.2 Hydro Plant Performance Models ................................................................................. 2534.2.1 Glimn–Kirchmayer Model .................................................................................. 2544.2.2 Hildebrand’s Model ............................................................................................ 2544.2.3 Hamilton–Lamonts’s Model ............................................................................... 2544.2.4 Arvanitidis–Rosing Model .................................................................................. 255

4.3 Short-Range Fixed-Head Hydrothermal Scheduling ..................................................... 2554.3.1 Thermal Model .................................................................................................... 2554.3.2 Hydro Model ....................................................................................................... 2564.3.3 Equality and Inequality Constraints ................................................................... 2564.3.4 Transmission Losses ........................................................................................... 2574.3.5 Discrete Form of Short-Range Fixed-Head Hydrothermal

Scheduling Problem ............................................................................................ 2584.3.6 Initial Guess ........................................................................................................ 2614.3.7 Alternative Method for Initial Guess................................................................. 262

4.4 Newton–Raphson Method for Short-Range Fixed-Head Hydrothermal Scheduling ..... 2674.5 Approximate Newton–Raphson Method for Short-Range Fixed-Head

Hydrothermal Scheduling ............................................................................................... 2724.6 Short-Range Variable-head Hydrothermal Scheduling—Classical Method ................ 282

4.6.1 Thermal Model .................................................................................................... 2844.6.2 Hydro Model ....................................................................................................... 2844.6.3 Reservoir Dynamics ............................................................................................ 2844.6.4 Equality and Inequality Constraints ................................................................... 2854.6.5 Transmission Losses ........................................................................................... 2854.6.6 Discrete Form of Short-Range Variable-Head Hydrothermal

Scheduling Problem ............................................................................................ 2864.6.7 Approximate Newton–Raphson Method for Hydrothermal Generations ........ 2884.6.8 Initial Guess ........................................................................................................ 290

4.7 Approximate Newton–Raphson Method for Short-Range Variable-HeadHydrothermal Scheduling ............................................................................................... 292

4.8 Hydro Plant Modelling for Long-Term Operation ....................................................... 3034.8.1 Hydro Plants on Different Water Streams ........................................................ 3044.8.2 Hydro Plants on the Same Water Stream ......................................................... 3054.8.3 Multi-Chain Hydro Plants .................................................................................. 3074.8.4 Pumped Storage Plants ....................................................................................... 309

4.9 Long-Range Generation Scheduling of Hydrothermal Systems .................................. 3104.9.1 Fuel Cost ............................................................................................................. 3104.9.2 Water Storage Equation...................................................................................... 3114.9.3 Hydro Generation ............................................................................................... 3114.9.4 Power Balance Equation .................................................................................... 3114.9.5 Optimal Control Strategy ................................................................................... 3134.9.6 Direct Root Method ............................................................................................ 315

References ................................................................................................................................. 322

viii Contents

5. Multiobjective Generation Scheduling .......................................................................... 325–390

5.1 Introduction ..................................................................................................................... 3255.2 Multiobjective Optimization—State-of-the-Art ............................................................. 326

5.2.1 Weighting Method .............................................................................................. 3285.2.2 Min-Max Optimum ............................................................................................. 3285.2.3 e-Constraint Method [Haimes, 1977] ................................................................ 3295.2.4 Weighted Min-Max Method [Charalambous, 1989] ......................................... 3305.2.5 Utility Function Method [Rao, 1987] ................................................................ 3305.2.6 Global Criterion Method [Osyczka and Davies, 1984] .................................... 330

5.3 Fuzzy Set Theory in Power Systems ............................................................................ 3335.3.1 Basics of Fuzzy Set Theory ............................................................................... 334

5.4 The Surrogate Worth Trade-Off Approach for Multiobjective ThermalPower Dispatch Problem ................................................................................................ 3375.4.1 Multiobjective Problem Formulation ................................................................. 3375.4.2 The e-Constraint Method.................................................................................... 3395.4.3 The Surrogate Worth Trade-Off (SWT) Function ............................................ 3425.4.4 Utility Function ................................................................................................... 3435.4.5 Test System and Results .................................................................................... 346

5.5 Multiobjective Thermal Power Dispatch Problem—Weighting Method ..................... 3505.5.1 Decision Making ................................................................................................. 3545.5.2 Sample System Study ......................................................................................... 354

5.6 Multiobjective Dispatch for Active and Reactive Power Balance .............................. 3615.6.1 Sample System Study ......................................................................................... 367

5.7 Multiobjective Short-Range Fixed-Head Hydrothermal Scheduling—ApproximateNewton–Raphson Method .............................................................................................. 3695.7.1 Sample System.................................................................................................... 380

References ................................................................................................................................. 387

6. Stochastic Multiobjective Generation Scheduling ....................................................... 391–504

6.1 Introduction ..................................................................................................................... 3916.2 Multiobjective Stochastic Optimal Thermal Power Dispatch—e-Constraint Method ... 393

6.2.1 Stochastic Problem Formulation ........................................................................ 3936.2.2 Algorithm ............................................................................................................ 3966.2.3 Application of the Method ................................................................................. 399

6.3 Multiobjective Stochastic Optimal Thermal Power Dispatch—The SurrogateWorth Trade-Off Method ............................................................................................... 4046.3.1 Multiobjective Optimization Problem Formulation .......................................... 4056.3.2 Solution Procedure .............................................................................................. 4076.3.3 Surrogate Worth Trade-Off Algorithm .............................................................. 4106.3.4 Sample System Study ......................................................................................... 412

6.4 Multiobjective Stochastic Optimal Thermal Power Dispatch—Weighting Method ...... 4176.4.1 Stochastic Multiobjective Optimization Problem Formulation ........................ 4176.4.2 Solution Approach .............................................................................................. 4206.4.3 Decision Making ................................................................................................. 4226.4.4 Results and Discussion ....................................................................................... 423

Contents ix

6.5 Stochastic Economic-Emission Load Dispatch ............................................................. 4276.5.1 Stochastic Economic-Emission Problem Formulation ...................................... 4276.5.2 Solution Approach .............................................................................................. 4296.5.3 Test System and Results .................................................................................... 431

6.6 Multiobjective Optimal Thermal Power Dispatch—Risk/Dispersion Method ............ 4376.6.1 Multiobjective Optimization Problem Formulation .......................................... 4386.6.2 The e-Constraint Method.................................................................................... 4396.6.3 Parameter Sensitivity .......................................................................................... 4406.6.4 Risk Index and Sensitivity Trade-Offs .............................................................. 4416.6.5 Test System and Results .................................................................................... 445

6.7 Stochastic Multiobjective Short-Term Hydrothermal Scheduling ................................ 4506.7.1 Stochastic Multiobjective Optimization Problem Formulation ........................ 4506.7.2 Solution Procedure .............................................................................................. 4576.7.3 Decision Making ................................................................................................. 4646.7.4 Test Systems and Results ................................................................................... 465

6.8 Stochastic Multiobjective Long-Term Hydrothermal Scheduling ................................ 4716.8.1 Stochastic Multiobjective Optimization Problem Formulation ........................ 4736.8.2 Optimal Control Strategy ................................................................................... 4786.8.3 Sample System Study ......................................................................................... 483

6.9 Multiobjective Thermal Power Dispatch Using Artificial Neural Network (ANN) ...... 4886.9.1 Stochastic Economic-Emission Problem Formulation ...................................... 4896.9.2 Membership Functions ....................................................................................... 4916.9.3 Performance Index .............................................................................................. 4936.9.4 Structure of ANN ............................................................................................... 4946.9.5 Backpropagation Algorithm ............................................................................... 4966.9.6 Sample System Study ......................................................................................... 497

References ................................................................................................................................. 500

7. Evolutionary Programming for Generation Scheduling ............................................ 505–593

7.1 Introduction ..................................................................................................................... 5057.1.1 Coding ................................................................................................................. 506

7.2 Fitness Function .............................................................................................................. 5087.3 Genetic Algorithm Operators ......................................................................................... 510

7.3.1 Reproduction ....................................................................................................... 5107.3.2 Competition and Selection ................................................................................. 5127.3.3 Crossover Operator ............................................................................................. 5137.3.4 Mutation .............................................................................................................. 514

7.4 Random Number Generation ......................................................................................... 5157.5 Economic Dispatch Problem .......................................................................................... 5187.6 Genetic Algorithm Solution Methodology .................................................................... 520

7.6.1 Encoding and Decoding ..................................................................................... 5207.6.2 Calculation for Generation and Transmission Losses ...................................... 5217.6.3 Fitness Function and Parent Selection .............................................................. 522

7.7 Genetic Algorithm Solution Based on Real Power Search ......................................... 5297.7.1 Encoding and Decoding ..................................................................................... 5297.7.2 Fitness Function and Parent Selection .............................................................. 530

7.8 Economic Dispatch with Valve Point Loading ............................................................ 536

x Contents

7.9 Economic Dispatch with Ramp Rate Limits and Prohibited Operating Zones .......... 5377.9.1 Prohibited Operating Zone ................................................................................. 5387.9.2 Ramp Rate Limit ................................................................................................ 539

7.10 Evolutionary Search Method for Economic Dispatch .................................................. 5407.10.1 Evolutionary Search Optimization Method ....................................................... 541

7.11 Evolutionary Programming for Economic Dispatch–I ................................................. 5477.12 Evolutionary Programming for Economic Dispatch–II ................................................ 5577.13 Particle Swarm Optimization for Economic Dispatch .................................................. 5637.14 Anti-Predatory Particle Swarm Optimization ................................................................ 5717.15 Differential Evolution for Economic Dispatch ............................................................. 5737.16 Real Coded Genetic Algorithm...................................................................................... 583References ................................................................................................................................. 590

8. Multiobjective Generation Scheduling: Weight Pattern Search .............................. 594–662

8.1 Introduction ..................................................................................................................... 5948.2 Economic Emission Load Dispatch ............................................................................... 595

8.2.1 Multiobjective Optimization Problem Formulation .......................................... 5958.2.2 Operating Limits ................................................................................................. 5988.2.3 Membership Functions of Objectives ................................................................ 5998.2.4 Weightage Pattern Search: Evolutionary Search Method ................................ 6008.2.5 Weightage Pattern Search: Genetic Algorithm ................................................. 6018.2.6 Test Systems and Results ................................................................................... 6038.2.7 Comparison of Results ....................................................................................... 614

8.3 Multiobjective Secure Load Dispatch............................................................................ 6178.3.1 Power Flow on Transmission Lines .................................................................. 6188.3.2 Generalized Z-Bus Distribution Factors (GZBDF) ........................................... 6208.3.3 Multiobjective Optimization Problem Formulation .......................................... 6218.3.4 Solution Procedure .............................................................................................. 6238.3.5 Sample Study System ......................................................................................... 6248.3.6 Results and Discussion ....................................................................................... 626

8.4 Stochastic Multiobjective Optimal Generation Allocation ........................................... 6338.4.1 Stochastic Multiobjective Optimization Problem Formulation ........................ 6358.4.2 Solution Approach .............................................................................................. 6398.4.3 Test System and Results .................................................................................... 641

8.5 Fuzzy Multiobjective Secure Generation Scheduling ................................................... 6488.5.1 Fuzzy Multiobjective Optimization Problem Formulation ............................... 6488.5.2 Generation Search with Genetic Algorithm ...................................................... 6528.5.3 Results and Discussions ..................................................................................... 654

References ................................................................................................................................. 659

Appendix A: Evaluation of the Expected Values of Functions ......................................... 663–670

Appendix B: Evaluation of a Coefficient of a Generator Output ...................................... 671–673Appendix C: Kuhn–Tucker Theorem ................................................................................... 674–675Appendix D: Newton–Raphson Method .............................................................................. 676–779

Appendix E: Gauss Elimination Method ............................................................................. 680–685Appendix F: Primal-Dual Interior Point Method ................................................................ 686–702

Index ............................................................................................................................................... 703–