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Sallam and Om P. Malik The Institution ofEngineeringand Technology PublishedbyTheInstitutionofEngineeringandTechnology,London,UnitedKingdom TheInstitutionofEngineeringandTechnologyisregisteredasaCharityinEngland& Wales(no.211014)andScotland(no.SC038698). †TheInstitutionofEngineeringandTechnology2015 Firstpublished2015 ThispublicationiscopyrightundertheBerneConventionandtheUniversalCopyright Convention.Allrightsreserved.Apartfromanyfairdealingforthepurposesofresearch orprivatestudy,orcriticismorreview,aspermittedundertheCopyright,Designsand PatentsAct1988,thispublicationmaybereproduced,storedortransmitted,inany formorbyanymeans,onlywiththepriorpermissioninwritingofthepublishers,orin thecaseofreprographicreproductioninaccordancewiththetermsoflicencesissued bytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethose termsshouldbesenttothepublisherattheundermentionedaddress: TheInstitutionofEngineeringandTechnology MichaelFaradayHouse SixHillsWay,Stevenage Herts,SG12AY,UnitedKingdom www.theiet.org Whiletheauthorsandpublisherbelievethattheinformationandguidancegiveninthis workarecorrect,allpartiesmustrelyupontheirownskillandjudgementwhenmaking useofthem.Neithertheauthorsnorpublisherassumesanyliabilitytoanyoneforany lossordamagecausedbyanyerrororomissioninthework,whethersuchanerroror omissionistheresultofnegligenceoranyothercause.Anyandallsuchliabilityis disclaimed. 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BritishLibraryCataloguinginPublicationData AcataloguerecordforthisproductisavailablefromtheBritishLibrary ISBN978-1-84919-944-5(hardback) ISBN978-1-84919-945-2(PDF) TypesetinIndiabyMPSLimited PrintedintheUKbyCPIGroup(UK)Ltd,Croydon To our wives Hanzada Sallam and Margareta Malik Contents Preface xiii 1 Power system stability overview 1 1.1 General 1 1.2 Understanding power system stability 1 1.3 Classification of power system stability 3 1.3.1 Small signal stability 3 1.3.2 Transient stability 6 1.4 Need for modelling 8 1.5 Stability margin increase 9 References 10 Part I Modelling 11 2 Modelling of the synchronousmachine 13 2.1 Introduction 13 2.2 Synchronous machine equations 14 2.2.1 Flux linkage equations 14 2.2.2 Voltage equations 15 2.2.3 Torque equation 16 2.3 Park’stransformation 17 2.4 Transformation of synchronous machine equations 18 2.4.1 Transformation of flux linkage equations 18 2.4.2 Transformation of stator voltage equations 19 2.4.3 Transformation of the torque equation 25 2.5 Machine parameters in per unit values 26 2.5.1 Torque and power equations 30 2.6 Synchronous machine equivalent circuits 32 2.7 Fluxlinkage state space model 34 2.7.1 Modelling without saturation 34 2.7.2 Modelling withsaturation 40 2.8 The current state space model 42 References 44 3 Synchronousmachine connected to apower system 47 3.1 Synchronous machine connected toan infinite bus 47 3.1.1 Flux linkage state space model 49 3.1.2 Current state space model 55 viii Power system stability: modelling, analysis and control 3.2 Synchronousmachine connected to an integrated power system 57 3.3 Synchronousmachine parameters in different operating modes 58 3.4 Synchronousmachine-simplified models 62 3.4.1 The classical model 62 3.4.2 The E0 model 64 q 3.5 Excitation system 67 3.5.1 Excitation system modelling 68 3.6 Modelling of prime mover control system 74 3.6.1 Hydraulic turbines 75 3.6.2 Steam turbines 77 References 79 4 Modelling of transformers, transmissionlines andloads 81 4.1 Transformers 81 4.1.1 Modelling of two-winding transformers 81 4.1.2 Modelling of phase-shifting transformers 91 4.2 Transmission lines 93 4.2.1 Voltage and current relationshipof a line 94 4.2.2 Modelling of transmission lines 95 4.3 Loads 97 4.3.1 Static load models 99 4.3.2 Dynamic load models 101 4.4 Remarks onload modelling for stability and power flow studies 103 References 104 Part II Power flow 107 5 Power flow analysis 109 5.1 General concepts 109 5.2 Newton–Raphsonmethod 111 5.2.1 Power flowsolution with polar coordinate system 113 5.2.2 Power flowsolution with rectangular coordinate system 114 5.3 Gauss(cid:2)Seidel method 121 5.4 Decoupling method 123 5.4.1 Fast-decoupled method 125 References 129 6 Optimal power flow 131 6.1 Problem formulation 131 6.2 Problem solution 132 6.3 OPFwith dynamic security constraint 137 References 142 Contents ix Part III Stability analysis 145 7 Small signal stability 147 7.1 Basic concepts 147 7.1.1 Equilibrium points 149 7.1.2 Stability of equilibrium point 150 7.1.3 Phasordiagrams of synchronousmachines 152 7.2 Small signal stability 154 7.2.1 Forced state variable equation 162 7.3 Linearised current state space model of a synchronousgenerator 164 7.4 Linearised flux linkage state space model of a synchronousgenerator 172 7.5 Small signal stability of multi-machine systems 177 References 183 8 Transient stability 185 8.1 Synchronous machine model 186 8.2 Numerical integration techniques 192 8.3 Transient stability assessment of a simple power system 193 8.4 Transient stability analysis of a multi-machine power system 201 References 218 9 Transient energy functionmethods 221 9.1 Definitions of stability concepts 221 9.1.1 Positive definite function 222 9.1.2 Negative definite function 222 9.1.3 Lemma 222 9.1.4 Stability regions 223 9.1.5 Lyapunov function theorem 223 9.2 Stability of single-machine infinite-bussystem 225 9.3 Stability of multi-machine power system 234 9.3.1 Energy balance approach 235 9.3.2 TEFmethod 241 References 250 Part IV Stability enhancementandcontrol 251 10 Artificial intelligence techniques 253 10.1 Artificial neural networks 253 10.2 Neural network topologies 255 10.2.1 Single-layer feed-forwardarchitecture 255 10.2.2 Multi-layer feed-forward architecture 255 10.2.3 Recurrent networks 256 10.2.4 Back-propagation learning algorithm 256 10.3 Fuzzy logic systems 259