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Stabilizer design for dynamic stability control of multimachine power systems Li, Qinghua 1992

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STABILIZER DESIGN FOR DYNAMIC STABILITY CONTROL OFMULTIMACHINE POWER SYSTEMSByQinghua LiB. Sc., Jiangxi Polytechnic University, PRC, 1982A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1991© Qinghua Li, 1991Signature(s) removed to protect privacyIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_____________________Department of /ef1’ (i / nee)’The University of British ColumbiaVancouver, CanadaDate ce )‘ 2 0 /?DE-6 (2188)Signature(s) removed to protect privacyAbstractThe development of new techniques for stabilizer design has been receiving considerableattention of power system industry. In this thesis, several new stabilizer design techniquesare developed for the improvement of dynamic stability of multimachine power systems.Three kinds of dynamic stability problems are dealt with: the low—frequency oscillations, the stability of power systems with wide—range changing operating conditionsand the multi—mode torsional oscillations of a two—machine system. Therefore, differentstabilizer design techniques are developed.First, mathematical models are developed for dynamic studies and stabilizer designof multimachine power systems. Methods are also developed to select the number andthe locations of stabilizers for the entire power system.A new pole-placement technique is developed for a decentralized Power System Stabilizer (PSS) design to control low—frequency oscillations. The computation is economic.The technique is applied to the stabilizer designs of a three—machine system and a nine—machine system. Simulation results show that PSSs thus designed are very effective tocontrol low—frequency oscillations.A direct self—tuning regulator (STR) design method is developed for power systemswith wide-range changing operating conditions. The indirect STR of Clarke based onthe Generalized Predictive Control (GPC) method is improved so that the initial stepcontrol parameters are directly estimated and that the subsequent control parametersare recursively computed. The techniques developed are applied to the STR design of anine—machine system. Comprehensive tests show that the STRs designed can effectively11stabilize the power system with wide—range changing operating conditions while well—designed PSSs fail to do so.Finally, another pole—placement technique is developed for the excitation control ofmulti—mode torsional oscillations of a power system due to the subsynchronous resonanceof a capacitor—compensated transmission line. This is a decentralized linear feedback design. The new technique is applied to the stabilizer design of the Second BenchmarkModel of IEEE, System 2. Test results show that stabilizers designed can effectively control the torsional oscillations of the system over a wide—range of capacitor compensations.111Table of ContentsAbstract iiList of Tables viiiList of Figures ixAcknowledgement xi1 INTRODUCTION 11.1 Motivation 11.2 Objectives of the Thesis 21.3 Outline of the Thesis 32 SYSTEM MODELS FOR STABILIZER DESIGN AND SIMULATION 52.1 Component Models of a Power System 52.1.1 Synchronous Generator 52.1.2 Excitation System 72.1.3 Governor Systems 72.1.4 Power System Stabilizer 102.1.5 Transmission Network 102.2 Complete Model of an M—machine System 112.3 Linear Model for Eigenvalue study and Stabilizer Design 132.4 Discretized Model for Computer Simulation 172.4.1 Trapezoidal Rule of Integration 18iv2.4.2 Obtaining Algebraic Equations from Differential Equations . 192.4.3 Reduced Equations 222.4.4 Complete System Equations for Simulations 243 SELECTION OF NUMBER AND SITES OF STABILIZERS 263.1 Introduction 263.2 A Nine—Machine System under Study 273.3 Participation Factor 303.3.1 Definition of the Participation Factor and Concept 313.3.2 Participation Factors of the Nine—machine System 323.4 Nonlinear Simulations for Open-Loop System 353.4.1 Coherent Groups 353.4.2 Speed Deviation Analysis 383.5 Stabilizer Designs 393.6 Conclusions 414 PSS FOR MULTIMACHINE SYSTEMS WITH LOW-FREQUENCYOSCILLATIONS 434.1 A New Pole—Placement PSS Design Method . . 444.2 PSS Designs Using the New Pole—Placement Method 474.2.1 Algorithm of Solving PSS Parameters by Gauss—Seidel Method 484.2.2 Selection of Eigenvalues for the Closed—Loop System 494.2.3 Design Example 1 — A Three—Machine Power System 494.2.4 Design Example 2 — A Nine-Machine Power System 514.3 Conclusions 56V5 DIRECT MIMO STR FOR MULTIMACHINE SYSTEMS WITH CHANGING OPERATING CONDITIONS 605.1 Review of Self—Tuning Controls 615.1.1 Minimum Variance Regulator (MVR) 615.1.2 Generalized Minimum Variance Control (CMV) . 635.1.3 Pole—Assignment Control (PAC) . 645.1.4 Extended Horizon Control (EHC) . 645.1.5 Generalized Predictive Control (GPC) 665.1.6 Summary of STRs 675.2 A New Direct MIMO STR for Power System 685.2.1 Basic Equations and Long-Range Output Prediction 685.2.2 Control Laws 725.2.3 Recursive Computation of Control Parameters 745.2.4 Direct Estimation of Initial Step Control Parameters 765.2.5 Algorithm of the STR Design 775.3 Example of Design and Simulation Test of the New STR 785.4 Conclusions 816 EXCITATION CONTROL OF SHAFT TORSIONAL OSCILLATIONSOF A MULTIMACHINE SYSTEM 876.1 Introduction 876.2 System 2 of the Second Benchmark Model (SBM) 896.3 Mathematical Model for the System 2 of SBM 896.3.1 Mechanical System 896.3.2 Governor and Turbine 926.3.3 Exciter and Voltage Regulator 92vi6.3.4 Synchronous Generator 946.3.5 Transmission Network 966.3.6 Summary of Mathematical Model 996.4 A Direct Pole—Placement Method for Control Design 996.5 Eigenvalues Analysis of the System without Control 1036.5.1 Natural Torsional Oscillating Modes 1036.5.2 Unstable Mode Eigenvalues 1046.6 Stabilizer Design for the System 2 of SBM 1046.6.1 State Variables for Control Feedback 1056.6.2 Prespecified Eigenvalues 1076.6.3 Feedback Gain Matrices 1076.6.4 Nonlinear Simulation Test 1106.7 Conclusions 1107 CONCLUSIONS 1177.1 New Stabilizer Design Techniques Developed 1177.2 Applications and Conclusions 1187.3 Future Research 119Bibliography 121viiList of Tables2.1 a and b Coefficients. 213.1 Eigenvalues of the Open—Loop System 333.2 Participation Factors of Unstable Modes 343.3 Coherent Groups of a 9—machine System 383.4 Speed Deviation Indices of Machines for the Open—loop System . . . 393.5 System Stability Indices of Various Designs . 414.1 Tuned Parameters of PSSs . . . 514.2 Eigenvalue Comparison 524.3 PSS Parameters of Various Designs . . 544.4 Electromechanical Mode Eigenvalue Comparison 556.1 Torsional Modes 103viiiList of Figures2.1 A Fast Excitation System 72.2 Hydro turbine, Steam turbine, and Governors 92.3 Transfer Function of PSS 103.1 An Initially Unstable Nine—Machine System 283.2 Angular Swings of Machines 1 and 2 363.3 Angular Swings of Machines 4 and 9 363.4 Angular Swings of Machines 5 and 6 . . . . 373.5 Angular Swings of Machines 3, 7 and 8 374.1 A Three—Machine Power System for PSS Design 514.2 Responses to a Short—Circuit near Gi Bus. (a) 584.2 Responses to a Short—Circuit near G1 Bus. (b) 595.1 Schematic Diagram of the Direct MIMO STR 795.2 Responses to Step Changes in Reference Voltage of G3 . . 835.3 Responses to Step Changes in Gate Opening of G3 845.4 Responses to a Short—Circuit near G3 Bus 855.5 Responses to a Short—Circuit and the Removal of the faulted Line . . 855.6 Responses to Step Changes in Gate Opening of G8 866.1 The System 2 of the SBM 906.2 Governor, Turbines and Excitation System 936.3 Individual Machine and Common System Coordinates . . . 97ix6.4 Real—Part Eigenvalue Loci of the Torsional Modes of System without Control 1096.5 Real—Part Eigenvalue Loci of the Torsional Modes of System with Control 1096.6 Responses to a Step Torque to Gi for the System without Control. (a) . 1126.6 Responses to a Step Torque to G1 for the System without Control. (b) . 1136.7 Responses to a Step Torque to G1 for the System with Control. (a) . . . 1146.7 Responses to a Step Torque to Gi for the System with Control. (b) . . . 1156.7 Responses to a Step Torque to Cl for the System with Control. (c) . . . 116xAcknowledgementI would like to thank my research supervisor Dr. Yao—nan Yu for his invaluable guidanceand encouragement throughout the course of this research. I am also indebted for thefinancial support of Grant A3626 from the Natural Science and Engineering Council,Canada.I would like to express my gratitude to the Electrical Engineering Department andthe UBC Computing Centre for computing support.Finally, I would like to thank my parents and my wife, Yunwei, for their supportthroughout my graduate program.xiChapter 1INTRODUCTION1.1 MotivationIncreasing electrical energy demand results in higher transmission voltages, more andlarger generating units, and more complex interconnections in a power system. New devices such as fast—response excitation system, series capacitor compensated transmissionand HVDC transmission are also introduced. As a result, many new problems arise, forexample, dynamic and transient stability, subsynchronous resonance, reliability, security,and voltage instability. A great deal of research is going on to solve these problems. Thisthesis is mainly concerned with the stabilizer design to improve the dynamic stability ofpower systems, especially multimachine power systems.The dynamic stability of a multimachine power system will be considered in themore general context that after a disturbance in the system such as a change in load,a change in voltage regulator reference, or a change in governor reference, generators inthe system must settle down to the synchronous speed. Supplementary stabilizers areusually required for a poorly damped or negatively damped power system to improve itsdynamic stability. Control signals generated by these stabilizers may be applied throughthe excitation loops and/or the governor loops of the generating units that have poorlydamped or negatively damped mechanical modes.One dynamic stability problem is the low-frequency oscillation of a power system,which is usually stabilized by Power System Stabilizer (PSS). PSS has been developed1for power system for many years. However, designing PSSs for multimachine systemsremains a challenging problem because the stabilizers must be decentralized ill structureand only locally measurable signals are used for feedback.There are two prerequisite decisions to be made prior to multimachine PSS design:1) how many stabilizers are required and 2) where they should be located. Obviously,the stabilizers must be located at the most strategic locations so that the number ofstabilizers can be minimized. However, methods of finding the strategic locations requireimprovement.The power system operating conditions are not necessarily constant. Indeed, they areconstantly changing for a large power system. The self—tuning regulator (STR) has beenused in industries other than power system for many years. It may also be beneficial forpower systems to have STR—type stabilizers.There is another type of dynamic stability problem in a power system, the multi—modetorsional oscillations of turbine—generator shaft due to the subsynchronous resonance(SSR) of series—capacitor compensated transmission network. Designing stabilizers fora power system with SSR, especially for a multimachine system, is also a challengingproblem of dynamic stability control. The IEEE SSR Working Group has publishedSystem 2, Benchmark Model II for the study of such phenomena.1.2 Objectives of the ThesisThe main objectives of the thesis are:1. To develop mathematical models for multimachine dynamic stability analyses, stabilizer designs and nonlinear high—order digital simulations..2. To develop a precise technique to determine the number and location of stabilizersfor multimachine system stabilizer design.23. To develop a pole-placement technique for decentralized PSS design in order tostabilize multimachine power systems with low—frequency oscillations.4. To develop an efficient algorithm for a direct multiple-input multiple—output (MIMO)self—tuning stabilizer design in order to control multimachine power systems withwide—range changing operating conditions.5. To develop a direct pole—placement method for a decentralized linear feedbackcontrol design in order to stabilize the multi—mode shaft torsional oscillations of atwo-machine system.1.3 Outline of the ThesisIn Chapter 2, a basic multimachine power system model for dynamic stability studiesis developed. The transmission network equations are related to individual machineequations in d-q coordinates. Based on this model, a linearized model for eigenvalueanalysis and stabilizer designs, and a discretized model for nonlinear digital simulationsare derived.Methods to determine the number and site of stabilizers are developed in Chapter 3.Besides the participation factor method of linear analysis, a speed deviation index methodbased on nonlinear simulation is proposed. Numerical examples are included.For the PSS design of multimachine power systems, a new pole-placement techniquewith less computation than the existing methods is developed in Chapter 4. The designprocedures are illustrated by two example systems: a three-machine system [21] and anine-machine system [3]. The effectiveness of the proposed design method is demonstrated.A new design technique of a direct MIMO self—tuning regulator (STR) for a multimachine power system with wide-range changing operating conditions is developed in3Chapter 5. This is an extension of the principle of Clarke’s indirect SISO GPC [29] withtwo improvements: the direct estimate of initial step control parameters and the recursive computation of subsequent control parameters. A nine—machine system is chosen asan example for the design. The system with designed STRs is thoroughly tested overwide—range changing operating conditions.The problem of multimachine multi-mechanical mode torsional oscillations due tosubsynchronous resonance (SSR) has been posed as System 2 of the Second BenchmarkModel (SBM) by IEEE SSR. Working Group [43]. The system has two nonidenticalmachines and a series—capacitor compensated transmission network. Decentralized linearfeedback stabilizers are designed for the excitation control of torsional oscillations of thesystem in Chapter 6. For the design, a detailed mathematical model of the system isderived and a new direct pole-placement design method is developed. Participationfactors are used to find the most effective variables for feedback. Computer simulationsof the system are carried out to examine the performance of the designed stabilizers.Conclusions and achievements of the thesis are summarized in Chapter 7. Furtherdevelopments of the thesis area are also recommended.4Chapter 2SYSTEM MODELS FOR STABILIZER DESIGN AND SIMULATIONFor stabilizer design and simulation of a multimachine power system, mathematical modeling is desirable. Power system component models are presented in Section 2.1 and acomplete system model is presented in Section 2.2. From the complete system model,a linearized model for eigenvalue analysis and stabilizer design is derived in Section 2.3and a discretized model for computer simulation test in Section Component Models of a Power System2.1.1 Synchronous GeneratorThe torque equations for a synchronous generator may be written as= Tm — Te — Dw) (2.1)S = Wb(W — 1.0) (2.2)wherew : generator rotor speedWb : base speed6 : rotor angleTm, Te : mechanical input and electric output torquesM, B : inertial constant and damping coefficientand the dot over a variable denotes the derivative with respect to time. All values are inper unit, except S in rad, LOb 111 2irf rad/s, and M in second.5Assuming one damper winding per rotor axis, the voltage equations for the generatorrotor windings [1] are= T_[EFD — e’ — x)] (2.3)= -[—eq— id(Xd — Xd) + eq + Tdoeq] (2.4)A do= -[—e + iq(xq — x)] (2.5)qOwhereq transient voltage of a field winding Fe’ : q subtransient voltage of a D damper windinge’ : d subtransient voltage of a Q damper windingXd, x, x d—axis synchronous, transient and subtransient reactancesXq, X’q’ : q—axis synchronous and subtransient reactancesT0, T’0 : armature—op en—circuited d—axis transientand subtransient time constantsT0 : armature—open—circuited q—axis subtransient time constantEFD : field voltage as seen from the armatured, iq : d—axis and q—axis components of armature currentAll values are in per unit, except time constants in seconds.For low—frequency oscillation and dynamic stability studies, the stator armature andthe transmission network are usually described by algebraic equations since, in mostcases, their eigenmode frequencies are very high and the decay is very fast. Therefore,the armature voltage equations may be written asVd ed Ra X= + (2.6)Vq eq Xd Ra6A subscript “k” should be given to signify the k—th generator, but is dropped herefor clarity.2.1.2 Excitation SystemAssuming a fast—response exciter and voltage regulator system, the differential equationfor the excitation system may be written asEFD [KAUE — KA(vt — Vref) — (EFD — EFD0)] (2.7)with the following constraintsEmin EFD Emaxwhere TA denotes a time constant, KA an overall gain, Vt the generator terminal voltage,Vref a reference voltage, and uE a supplementary excitation control signal, if any. Emax.and Emin. are the constraints for EFD. The block diagram for the excitation system isshown in Fig. 2.1Vref-VtUE___KA EFigure 2.1: A Fast Excitation System2.1.3 Governor SystemsTwo types of governors are considered, a mechanical—hydraulic governor for a hydro—electric plant and an electrical—hydraulic governor for a thermo—electric plant. The differential equations of the hydro turbine and governor system [2] for the hydro—electric7plant are given byG1 = (uG + Wref — w — G2 — crGl)G2 = _G2+si— G— G3 (2.8)— (G3Tm+G0) G3— O.5T —with the following governor speed and opening constraintsG1—G3—9GOmin <(G3 + G0) GOmax.In these equations, Gi represents the output of an actuator, G2 that of a dashpot, G3that of a gate servo, G0 the initial value of gate opening, and Tm the mechanical torqueoutput of the hydro turbine. UG is a supplementary governor control signal, if any.GSmin., GSmax., GOmin. and GOmax. are the gate speed and openmg collstrarnts. Theblock diagram for the governor—turbine system is shown in Fig. 2.2 (a)The differential equations of a non-reheat steam turbine and governor system for thethermo—electric plant are given bya — (UG+WrefW)kgGTsm(2.9)=(G+G0—Tm)with the following governor speed and opening constraintsGSmin <G < GSmaxGOmin <(G + G0) <GOmaxIn these equations, G represents the output of a gate servo and Tm the mechanicaltorque output of the steam turbine. UG is a supplementary governor control signal, if8(I) Go(b) Steam Turbine and GovernorFigure 2.2: Hydro Turbine, Steam Turbine, and Governors(a) Hydro Turbine and GovernorGo9--+W— T— _.fa K KT1— T2-T2UE=—m; + + Xps2For low-frequency oscillation and dynamic stability studies, the transmission network isusually described by algebraic equations since its eigenmode frequency is very high andthe decay is very fast. The equations of a transmission network of an rn—machine powersystem are given by[ID,Q] = [Y][VD,Q] (2.11)any. GSmin., GSmaz., GOmin. and GOmar. are the gate speed and opening constraints.The block diagram for the governor—turbine system is shown in Fig. 2.2 (b)2.1.4 Power System StabilizerA conventional Power System Stabilizer (PSS) is shown in Fig. 2.3. It has two lead—lagcomponents and one reset block. The differential equations for the PSS areXps2UE(2.10)Figure 2.3: Transfer Function of PSS2.1.5 Transmission Network10where [ID,Q] and [VD,Q] respectively includes the D and Q components of all generatorarmature currents and voltagesT[ID,Q] = {ZD1, Q1, D2, iQ2, , Dm, ZQmjT[VD,Q] = [vDl,vQl,vD2,vQ2,.. .,VDm,VQm][Y] is the admittance matrix of the transmission network, which is obtained from theresults of a load flow study by eliminating all non—generator buses[yii] [y12] . . . [Yim][Y] = [y] [Y221 ... [Y2rn1[ymi] [Ym2] [ymm]whereG3 -B,[Yji] =B3 G,and denotes the real component and B3 the imaginary component of the admittanceYij.2.2 Complete Model of an M—machine SystemLet all differential equations for the k-th machine obtained in section 2.1 be expressed as= [F(Yk, dk, qk, Vtk)] (2.12)where k is a vector containing all state variables and dk, and Vtk are non—statevariables. The terminal voltage Vtk may be expressed asVtk = Vdk + Vk or Vtk = + Vk11Let an additional subscript k be added to Eqs. (2.6) to signify the armature voltagesof the k—th generator[ Vdk ] [ e ] + [ Rajc Xqk ] [ dk ] (2.13)Vqk eqk Rak qkThe armature currents for the k—th generator can be obtained from Eq. (2.11)[ Dk ] = [ Z7l(GkvDj — BkvQ,) ] (2.14)1Qk l(BkvD + Gk3vQ)While the armature current and voltage components in Eq. (2.14) are described byD—Q coordinate of the system network, those in Eqs. (2.12) and (2.13) are describedby individual d—q coordinates of the k-th generator. Coordinate transformation must becarried out. Since there are more equations (2.12) and (2.13), which includes dk andthere will be less computation if the currents Dk and iqk of Eqs. (2.14) are transformedinto the and qk• As for the armature voltage, since there are more VD’s and VQ’S inEq. (2.14), it is preferred that only the voltages vdk and Vqk of Eq. (2.13) be transformedto vDk and vQk. As result, Eq. (2.13) and Eq. (2.14) becomes[ sin6k —cos5k ] [ VDk ] [ e ] + [ Raji Xqk ] [ icii ] (2.15)cos sin vQk eqk —Xdk -l?ak qkandsinSk cos45k Zdk = >[l(GkvD— BkvQ3) (2.16)— COS 8k sin ‘5k qk Z_-l(BkvD + Gk3vQ)where 5j, is the rotor angle of the k—th generator and is one of the state variables in kof (2.12). The four algebraic equations of Eqs. (2.15) and (2.16), which are required forthe four non—state variables VDk, vQk, Zdk, and and all differential equations of (2.12)12form a complete mathematical model for the k-th generator as follows[k]= {F(Yk, Zdk, qk, VDk, VQk)]Sfl COS VDk = edk+Rak Xqj; Zdkcos Sk sin 6k vQk eqk —Xdk Rak qk (2.17)Sfl cos— Bk3vQ)—cos6k Sifl5k qk Z,—i(Bk,vD3+ Gk3vQ)For a power system of m machines, there are m sets of Eqs. (2.17). The last equation ofEqs. (2.17) shows the interconnection of all machines. This system model will be usedfor most dynamic stability studies in this thesis.2.3 Linear Model for Eigenvalue study and Stabilizer DesignFor eigenvalue analysis and controller design of a multimachine power system, a linearstate model is used. The model consists of linearized first—order differential equationswith state variables only. These equations are derived from the nonlinear differentialequations obtained in the previous section. The non—state variables VD, VQ, d, and qmust be eliminated.For dynamic stability studies, the effects of rotor D arid Q damper windings, e’q’ ande, are usually ignored. The reason for this is that a damper winding of a synchronousgenerator may be considered as a short—circuited winding of a transformer or an induction motor and the voltages induced in these windings at the low frequencies arenegligibly small. The governor system is also ignored because its dynamic response isusually slow and hence the governor output may be treated as a constant. With theseconsiderations, there are four differential equations remaining for each machine and thelinearized equations may be rewritten in a matrix form as± = Ax + Cy (2.18)13wherex = [AS,LW,LeJ,LEFD1Ty = [/id, jqjTIn Eq. (2.18), the currents ziid and /iq are written separately. They are the interactingvariables of machines of the system. A is a 4 x 4 matrix and C a 4 x 2 matrix. Lx =x(t) — x(O), and x(O) is the given initial operating point of x(t).The multimachine linear model that has been developed for a decentralized optimalstabilizer design [3] can be derived in a slightly different way as follows. For a powersystem with m machines, there are m sets of Eqs. (2.18) and a subscript “k”, k=1, 2,m, may be added to signify the k—th machine. The m sets of equations may also bewritten in a matrix form asA1 0 C 0A2 x2 C2+ . . LZq2 (2.19)0 Am Xm 0 CmAZdmlXZqmNext, the armature voltages and currents are non—state variables and they must beeliminated. Since the effects of the damper winding have been ignored, Eqs. (2.15) forarmature voltages should be modified assin 8k — cos vDk 0 , Rak Xqk dk= eqk + (2.20)cos 8k 5fl 6k VQk 1 dk -‘4ik qk14Linearization of Eqs. (2.20) givesF LVDk 1 F 0 1 F Rak Xqk ] F Zjj 1[Dk] b’k + [Ek] I I = I I eqk + L I I (2.21)L LVQk ] L 1 ] Xdk Rak L /Zqk ]whereF COS SkoVDko + Sill SkoVQko 1 F 8k0 — C05 6k0 1[Dk]= I ] , [E,]= I IL —SZflSkfJVDkO + COS SkOVQkO L cos 6k0 Sill 8k0 ]where a variable with a subscript 0 denotes its initial value and should be treated asknown quantity. Next, linearization of the armature current equations (2.16) gives[ zidk ] [ — BkzvQ) 1[Fk] 8k + [Hk] = I (2.22)qk ZLl(BkLvD + GkLvQ) ]whereF cos5kOZdkO — sin5’kOqkO 1 F Sill 8k0 cos 8k0 1[Fk] = I I and [Ilk] = IL sin5kOZc1kO + cos Skoqko j L — cos 8k0 Sfl 6kQ jEqs. (2.21) and (2.22) respectively can be rewritten asFAVDk1 FAidklI = [Jk]22 I I + [Lk]24 [xk] (2.23)L VQk j L Aqk JandF dk 1I = [Mk]2xm[/VD1, /vQ1, LVD2, ZvQ,.. . , /VDm, /.\Vqm]T + [Nk]24 [xk] (2.24)L /Zqk jThere are m sets of Eqs. (2.24) for m machines and they can be assembled into a matrix15equationZd1 /VD1LZqi /.vQ1M1 N1 0d2N2= LVQ + (2.25)Mm 0 Nm Xmdm VDmZZqm LVQmThere are also m sets of Eqs. (2.23) for m machines and they can be substituted intoEqs. (2.25) to eliminate the voltage vector. With the substitution, Eqs. (2.25) becomesLZqiLZd2x2= [‘92mx4m (2.26)XmqmSubstituting Eq. (2.26) into Eq. (2.19) givesA11 A12 . .. Aim= A21 A22 ... A2m (2.27)Ami Am2 ... Amm XmFor conciseness, Eq. (2.27) is usually written as[rh] = [A][xj (2.28)16where [x] is a 4m x 1 state vector representing the perturbation of the system state fromits chosen operating point. [A] is a 4m x 4rn system matrix depending on the systemparameters as well as the system operating condition. Eq. (2.28) is the linear model ofthe open—loop rn—machine power system, which will be used for eigenvalue analysis andstabilizer design in Chapter 3 and Chapter 4.2.4 Discretized Model for Computer SimulationFor dynamic stability studies of power systems, the dynamic responses of the powersystems with and without supplementary stabilizer are usually investigated by digitalcomputer simulations. The major task of simulations is to simultaneously solve all differential and algebraic equations of Section 2.2. Let the two differential and algebraicequation sets be written in matrix form, respectively as[dY(t)]= [F(Y(t),Z(t))] (2.29)[(Y(t), Z(t))] = [0] (2.30)where Y is a vector containing all state variables and Z is a vector containing all non—state variables (id, Zq, VD, and VQ of all generators).Usually, the differential equatiolls, Eqs. (2.29), are solved using some integrationtechniques including the Runge—Kutta method while the algebraic equations, Eqs. (2.30),are solved by a numerical method. Hence, the equations are solved in two separate loopsand the interface error between the two loops must be dealt with. This is not desirable.A better method is to use the trapezoidal rule of integration to transform Eqs. (2.29)into a set of algebraic equations so that all equations to be solved are algebraic equationsand can be solved simultaneously.172.4.1 Trapezoidal Rule of IntegrationConsider a differential equation of Eq. (2.29)dy(t)_ f[Y(t) Z(t)] (2.31)Since the time responses of y(t) are computed on a digital computer at discrete intervalsof time (step size At ), the numerical solution of Eq. (2.31) at time t may be expressedin integral form asy(t) - y(t - At)= f f[Y(r), Z(r)]dr (2.32)t-tSince the RHS of Eq. (2.32) may equal the trapezoidal area of- At), Z(t - At)] + f[Y(t), Z(t)]}we havey(t) = f[Y(t), Z(t)] + yo (2.33)with Yo known from the solution at the preceding time step,Yo = y(t — At) + f[Y(t — At), Z(t — At)] (2.34)The trapezoidal rule has been discussed in detail in Electromagnetic Transients Program(EMTP) [4]. The error introduced by the trapezoidal rule of integration is negligible fora very small step size At.The following is an example. Consider Eq. (2.4),, 1 ,, . , ,, I II Ieq = -[—eq— Zd(Xd — Xd) + eq + TdOq1doIt may be rewrittell asd ,, , 1 ,, . , ,,— eq) = r[—eq— Zd(Xd—Xd) + eq]do18Thereforey(t) = e—ef[Y(t), Z(t)] = [_e- id(Xd - x) + eq]From the trapezoidal rule Eqs. (2.33) and (2.34), we haveII I At ,, . ,eq— eq = .-[_eq— Zd(Xd — Xd) + eq] + Yo (2.35)doandII I t II . I II IYo = eq — eq + -[—eq — d(Xd — Xd) + eqj (2.36)dowhere Yo is calculated from the values of variables at time = t — At and hence is treatedas known quantities at time = t. Rearranging Eq (2.35) and (2.36) yieldseq = eq — auqid(x — x) + be”qwhereae”q= 2T + Atbe”q = e — e + ae”q[2(e — e)— id(Xd—The benq includes the values of variables at time = t — At. Hence, besides a, b is theknown coefficient at time = t2.4.2 Obtaining Algebraic Equations from Differential EquationsIn this subsection, the differential equations of a power system will be transformed to algebraic equations according to the trapezoidal rule of integration. The resultant a and b coefficients of these algebraic equations obtained are listed in detail in Table 2.1. Note thatthe computation of b coefficients requires the values of some variables at time = t — A t,19which must be updated after every Lit. However, both a and b coefficients are alwaystreated as known quantities at time = t.Applying the trapezoidal rule of integration, the following algebraic equations areobtained.Governor for a hydro—electric plant. From differential equations (2.8 ) wehaveGi = agl(wrcf W G2 + UG) + bgi.G2 = ag2Gl + bg2 (2.37)G3= ag3(G1— G3)+ bg3, GSminGi— G3GSma.Tm = atmiG3 + atm2Go + btm, GOmin (G3 + G0) GOmaxTorque equations of the generator. From the differential equations (2.1) and(2.2) we have= aw(Tm — Ta) + b , T = edid + e’q’iq— ( — X’q’)idiq (2.38)6 = aS(w— 1.0) + b3 (2.39)Power system stabilizer. From the differential equations (2.10), we havex1 = a,81 + b31+ b32 (2.40)tIE = auexps2 + bueExcitation system, F winding and D damper winding. From the differentialequations (2.7), (2.3) and (2.4), we haveEFd = aefd(uE — + V + Vf) + befd, Emin EFD Emaxe’q = aelq[EFD — id(Xd—x1)j + be’q (2.41)eq = eq — aeIlqid(x—x) + benq20Table 2.1: a and b Coefficientsagi= 2T+t bgi = Gi + agl(wref — — G2 — 2aGl + tIG)2Tr6T b92 — — 1)G2— ag2Glag2= 2Tr+t — “—_______ag3— T bg3 = G3 +aga(G1_G3)— t—2Tatml— T+Atatm2=btm = (2atm2 — atmi)G3 + (1 —2atm2)Tm + atm2Goit= 2M+Dt = w + aw(Tm — Te — 2DL.))a3= bs=+a3(w—1.O)2T__________2T-ta81= 2T-f-t b31 = 2T+tXP81 — a31wK(2Ti+a.t) b82 — K(t—2Ta)_______a82= — 2T+It X9i + 2T+tXP82— 2Ti+t bue — t—2T1 2T—Itaue— 2T+t — 2+tXP82 + 2T+taefd— tKA___________— 2TA+t befd = EFD + aejd(UE — + V + Vj— 2(EFD—EFDO))KAaelq 2T0+t be’q = e + ae!q[EFD — 2e — id(Xd — x)}Ltae”q= 2T+tbenq = — e + ae”q[2(e — e’) — — x)]I, Ia’d= 2T+tbe”d = ed + aeud[iq(xq — Xq) — 2ed]21Q damper winding. Finally, from the differential equation (2.5) we havee = ae”diq(xq — x) + be”d (2.42)2.4.3 Reduced EquationsTo save computation time, the number of equations involved in the numerical solutionfor the complete system will be minimized. First, Eqs. (2.38) and (2.39) can be reducedto a single equation by eliminating w,6 = ai(Tm — Te) — (2.43)wherea1 = aab1 = aö(1.O— b) —Te edzd + e’iq — (x’ — x)idiqNext, Eq. (2.39) can be rewritten asw = (1 + 6/a3 — b3/a) (2.44)Substituting Eq. (2.44) into the first equation of (2.37) and then eliminating the variablesGi, G2, and G3 of (2.37) successively, we shall haveTm = amS + bm (2.45)with modified coefficients asam = a13tm1 , bm atmi(a,3b,1+ b) + atm2Go + btm— —agi b’ — agib/as + bgi—a91(b2— UG)a91—aS(a912 + 1) gi — a912 + 1a’— a93 b’ —_________—I + a93 ‘ T9 + a9322Inserting Eq. (2.45) into Eq. (2.43) gives(1— aiam)6 + al[e’d’id + e’iq — (x — X’q’)diq] — aibm + b1 = 0 (2.46)Thus, equations (2.37), (2.38) (2.39) have been reduced to a single equation i.e., Eq. (2.46).Similarly, equations (2.40) and (2.41) can be reduced to a single algebraic equation.First, the w in the first equation of Eq. (2.40) can be replaced with Eq. (2.44). Next,substituting X31 and x2 into the equation of UE, we shall haveUE = a’ue6 + b’j (2.47)whereaueapslaps2ue ab’ue = aue(apslaps2 +a32bi+ b82) + bue— a’uebsInserting Eq. (2.47) into the first equation of Eq. (2.41) and then eliminating EFD ande, we shall haveeq — a218 + a22id + a23vt — b2 = 0 (2.48)with modified coefficients,a21 = aeIqaefda , a2 = ae’q(xd — x) + aeiiq(x — x)a23 = ae’qaefd , b2 = be’q + bei’q + aelq(aefdvref + aefdbue + befd)Hence, all differential equations for each machine can be represented by only threealgebraic equations, namely, Eqs. (2.46), (2.48), and (2.42). They are summarized as(1— aiam)6 + al[eid + eqiq— ( — Xq)dZq — aibm + b1 = 0eq — a216 + a22id +a23v+ V — b2 = 0 (2.49)ed — aelldiq(xq— ) — bend = 023Eqs. (2.49) can be used for any machine. If a different type of governor is used or thereexists a supplementary governor control, only am and bm coefficients will change. For adifferent supplementary excitation control other than PSS, a21 and b2 will change dueto change in a’ue and To take into account the upper and lower limits of variables,a and b coefficients may be modified. For example, once EFD reaches its upper limitEmax, Eqs. (2.40) and the first equation of Eqs. (2.41) should betemporarily excluded.Also, EFD in the second equation of Eqs. (2.41) should be set to Em,j. This is equivalentto setting a21 = a23 = 0 and setting b2 = be’q + be”q + aeiqEma. in Eqs. (2.49). Hence,the form of Eqs. (2.49) holds for various control circuits of any one of machines in amultimachine system. This is very useful in computer programming for the dynamicstability studies where the behavior of each machine with and without various stabilizerswill be simulated.2.4.4 Complete System Equations for SimulationsAs already mentioned in the previous subsection, all differential equations can be transformed into algebraic equations and subsequently be reduced to Eqs. (2.49). Replacingthe differential equations of Eqs. (2.17) with Eqs. (2.49), we have(1—a1ak)Sj, + alk[e’d’kidk + e’iq,.—(Xj’ — X’q’k)dkqk1 — alkbmk + bik 0eqk — a2lk6k+ a22kidk + a23k.Jvk + Vk — b2k = 0edk — ae”dkiqk(xqk— Xqk) — b”dk = 0[ sin6j —cosSk ] [ vDk ] [ edk ] + [ Rai, Xqk ] [ dk ] (2.50)cos Sill VQk eqk Xdk Rak qk5fl 8,, cos dk 1(GkvD — BkvQ)—cos6k 5illk qk l(BkvD + GkvQ)The subscript k is added to the first three equations to signify the k—th generator. Forthe simulation of a m—machine power system, there exists m sets of these equations for24all machines. Together they constitute a complete set of simultaneous equations. Theyare nonlinear algebraic equations and can be solved by the Newton—Raphson method.Once the solutions for 8, e’, e, and armature currents and voltages are obtainedfrom simultaneous equations (2.50) at time = t, they are substituted into the algebraicequations in subsection 2.4.2 to solve for other state variables which were eliminatedduring the equation reductions. For instance, with 6 known, the w is obtained directlyfrom Eq. (2.39). Thereafter, Gi is obtained from the first equation of Eq. (2.37), G2 fromthe second one of Eq. (2.37), and so on. Hence, the solutions for other state variablesinvolve only substitutions. After the solutions for all variables, the lower and upper limitsfor some state variables must be applied and all b coefficients must be updated for thenext computation of Eqs (2.50) at time = t + /t.25Chapter 3SELECTION OF NUMBER AND SITES OF STABILIZERS3.1 IntroductionPower system stabilizers (PSSs) are designed as supplementary control devices in a powersystem for the improvement of its stability. To improve the stability of a large powersystem, all generators larger than a certain capacity are recommended to be equippedwith PSSs. This may not be economic nor effective since not all large generators aresituated at strategic locations. Therefore, in designing stabilizers for a multimachinepower system, two prerequisite decisions must be made: 1) how many stabilizers arerequired and 2) on which machines these should be located.Eigenvector methods ([5], [6]) were commonly used for the stabilizer site selection.Later, the participation factor method was proposed by Perez—Arriaga et al. ([7], [8]) andapplied by Hsu and Chen [9] for one stabilizer siting. Participation factors are calculatedfrom the eigenvectors of a system matrix [A], which may be called the right vectors, andthe eigenvectors of the transpose of the matrix [A], which may be called the left vectors.Hsu and Chen’s work was concerned with the location of only one stabilizer because thepower system under study had only one unstable oscillating mode. For a large powersystem with many unstable modes, further research must be performed.This chapter is aimed at presenting the results of this research on the number andsite selection of stabilizers for an unstable nine—machine power system. Two methods areincluded; the participation factor method and the speed deviation index method. While26the participation factor method is based on a low—order linearized generator model, thespeed deviation index method is based on a high—order nonlinear model. The indexmethod defines a weighted speed deviation index from computer simulation results. Theindex can identify the most unstable generators which must consequently be equippedwith stabilizers. In these simulations, coherently swinging groups of the system arealso identified. The final selection of stabilizer number and sites, however, can only bemade after examining the results of stabilizer design. For the stabilizer design, a pole-placement design technique is developed in the next chapter, but the results of severalstabilizer designs are assessed in this chapter using a system stability index (SSI).Many results of this chapter and the next chapter are published ([it], [12]).3.2 A Nine—Machine System under StudyA nine—machine system [3], Fig. 3.1, is used for this study. The nine—machine systemis chosen because a system with two or three machines is lot large enough to displaythe nature of coherent group behaviour of a large power system. On the other hand,a system with too many machines will be crowded with results and thus it is hard toextract useful information from the results.The system under study comprises nine synchronous generators interconnected bya transmission network. Machines 3 and 9 are hydro—electric plants with mechanical—hydraulic governors while the others are thermal—electric plants with electrical—hydraulicgovernors. An exciter and voltage regulator system of the fast—response type is assumedfor each generator. The multimachine power system is initially unstable. The systemdata are as follows.27GeneratorFigure 3.1: An Initially Unstable Nine—Machine SystemI II II I II IINo. Xd 1’q Xd d xq M D Td Td Tq1 0.70 0.70 0.12 0.098 0.098 25.0 5.0 7.0 0.091 0.4552 0.60 0.60 0.10 0.084 0.084 30.0 5.0 7.0 0.091 0.4553 0.50 0.40 O15 0.070 0.070 20.0 5.0 8.0 0.104 0.5204 1.60 1.60 0.23 0.224 0.224 12.8 5.0 7.0 0.091 0.4555 0.95 0.95 0.15 0.133 0.133 19.8 5.0 7.0 0.091 0.4556 0.95 0.95 0.15 0.133 0.133 19.8 5.0 7.0 0.091 0.4557 1.00 1.00 0.17 0.140 0.140 18.0 5.0 7.0 0.091 0.4558 1.00 1.00 0.17 0.140 0.140 18.0 5.0 7.0 0.091 0.4559 0.39 0.32 0.06 0.055 0.055 32.0 5.0 6.0 0.078 0.390All data are in per unit except M, T, T, and in seconds.28Bus LoadBus Pg (p.u.) V (p.u.) Bus Pload (p.u.) Qload (p.u.)1 (slack) 1.06 11 2.5 0.22 3.0 1.04 12 1.5 0.13 3.5 1.035 13 2.4 0.24 2.0 1.03 14 4.5 0.55 1.0 1.035 15 0.0 0.06 2.5 1.04 16 3.5 0.27 2.5 1.04 17 2.0 0.28 2.0 1.01 18 2.0 0.09 2.0 1.015 19 1.5 0.010 3.5 1.06 20 0.0 0.021 3.5 —1.7Excitation SystemTAO.lOs KA=50.0p.U. For machines 7, 8 and 9TA=0.05 s KA=100.0 p.u. For other machines—0.7 p.u.EFD<0.7 p.u. —0.12 p.u.UE0.12 p.u.Hydro Turbine and Governoru=0.05 p.u. 6=0.25 p.u. T9=0.5 sT0.02 5 Tr=4.8 S T1.6 8—0.1 p.u./sSL0.1 p.u./s 0.0PLPmax—0.15 p.u.UG0.15 p.u.29Steam Turbine and GovernorTsmO.1 S TCHO.4 S K9=20.0 p.U.—0.1 p.u./sSL<0.1 p.u./s 0.0PLPmax—0.15 p.u.UG<0.15 p.u.Transmission LineBus I Bus J R (p.u.) X (p.u.) Bus I Bus J R (p.u.) X (p.u.)2 13 0.0 0.059 13 14 0.024 0.243 13 0.0 0.0135 20 13 0.0096 0.0965 12 0.0 0.29 20 13 0.0096 0.09612 13 0.0068 0.068 14 15 0.007 0.076 16 0.0 0.05 15 4 0.02 0.23216 19 0.03 0.3 15 11 0.007 0.077 16 0.0 0.10 14 10 0.012 0.1216 17 0.00145 0.0145 14 18 0.0204 0.20419 20 0.0106 0.106 14 18 0.0204 0.20420 21 0.0064 0.064 18 9 0.0 0.2821 17 0.0161 0.161 17 18 0.0057 0.05721 14 0.0025 0.025 17 8 0.032 0.3213 14 0.024 0.24 11 1 0.0 0.0323.3 Participation FactorThe participation factor was introduced in [7] for selective modal analysis (SMA). Abrief introduction to the participation factor is given in Section 3.3.1. The calculatedparticipation factors of the nine—machine system are presented in Section Definition of the Participation Factor and ConceptConsider a linear time-invariant system[] = [A][x] (3.1)where [x] is an n state vector and [A] an n x n system matrix. There are n eigenvaluesfor the system, A, i = 1, ..., n. For each eigenvalue , there is an eigenvector [y] of [A]satisfying[A][y} = )j[y], [y] 0 (3.2)It is well—known that both [A] and [AlT have the same eigenvaliies but the differenteigenvectors. Let the eigenvector of [AlT for ) be [uj[A]T[u]= X[u] , [ui] 0 (3.3)or[uJT[A] = [u]TA, [ui] 0 (3.4)Let [y] be called the right eigenvector and [ui] the left eigenvector. Note that there aren entries for each eigenvector and the k—th entry of [y] or [ui] belongs to the lc—th statevariable of [x]. A participation factor is defined in [7] asPki = UkiYkj (3.5)where Yki (uk) is the k—th entry of the i—th right (left) eigenvector [y] ([ui]).Consider the time response of the system (3.1). For an initial condition [x(0)], thesolutions of the [x(t)} are[x(t)] =[u]Tx(O)}ey (3.6)31If [x(0)] = [y], Eq. (3.6) becomes[x(t)] = (3.7)Since [u3] and [yj] are orthogonal for j i [7], we haveT[uj] [y,] = 0.0 for j z (3.8)Therefore, Eq. (3.7) becomes[x(t)] = [uj]T[yj]et[yj]= (Pij + P2i + + pni)e[yi] (3.9)Hence, when only the i—th eigenvalue is excited, Pki, k =1, 2, ..., n measures the relativeparticipation of the k—th state variable in the time response of the i—th eigenvalue mode.Based on this concept, participation factors of a system can be used to find the mostsensitive state variable to the electromechanical eigenvalues of interest.3.3.2 Participation Factors of the Nine—machine SystemBased on the linearized state equation, Eq. (3.1), the system matrix of the nine—machinepower system can be readily obtained by a Fortran program following the linear modelprocedure given in Chapter 2. Each machine is described by a fourth—order model inwhich state variables are arranged in a definite order of M, w, and /.EFD. Thereare 36 state variables in [x] for the nine—machine system and the system matrix [A] isa 36 x 36 matrix. The 36 eigenvalues of the system without supplementary stabilizers(open—loop system) are calculated and listed in Table 3.1. Eigenvalues 11—28 are electromechanical modes. These modes are identified with participation factors. When theparticipation factors of a pair of complex—conjugated eigenvalues have the maximum values for state variables LS and &. of a particular machine, this pair of eigenvalues isclosely associated with the mechanical modes of that machine.32Table 3.1: Eigenvalues of the Open—Loop SystemNumber Eigenvalues1 -17.51782, 3 -9.9615 ±j 11.17004 -15.85145, 6 -10.5035 +j 6.86767, 8 -10.3454 ±j 6.43749, 10 -10.2149 ±j 2.912011, 12 -0.3354 ±j 9.1267 #13, 14 -0.2188 +j 8.6423 #15, 16 -0.0947 ±j 7.2452 #17, 18 -0.0627 +j 7.0242 #19, 20 -0.1970 ±j 6.5043 #21, 22 0.5083 +j 5.323523, 24 0.5260 +j 5.4462 #25, 26 0.2248 +j 6.1273 #27, 28 0.0922 ±j 2.8020 #29, 30 -5.2957 +j 5.235431, 32 -6.1138 ±j 4.057933, 34 -5.8807 +j 4.008035 -4.590836 -2.6307# Denotes electromechanical mode eigenvaluesIt is found from Table 3.1 that there are four unstable eigenmodes, or eigenvalue—pairswith a positive real part (eigenvalues 2 1—28). The participation factors of these unstablemodes are listed in Table 3.2.For the participation factors of each of the first three unstable modes, there are twolargest values corresponding to M and &4. of a particular machine. It is found that theybelong to machines 7, 8, and 3, respectively. This clearly suggests that the machines 7,8 and 3 should be the first three to be designed with PSSs. But it is much less clear forthe fourth unstable mode. Since the participation factors of machines 7, 8 and 9 of thisunstable mode all have the largest value, whether a stabilizer designed on machines 7,33Table 3.2: Participation Factors of Unstable ModesEigenvalues of Participation Factors MachineUnstable Modes Corresponding to State Variables: Number(Eigenvalue No.) 116(i) 11w(i) 11e(i) IIEFD(i) (i)0.05 0.05 0.00 0.00 10.05 0.05 0.00 0.00 20.02 0.02 0.00 -0.00 30.06 0.06 -0.00 0.00 40.5083 ±j 5.3235 -0.00 -0.00 0.01 0.00 5(21 and 22) -0.00 -0.00 0.00 0.00 6I 0.281 10.271 0.11 0.04 7-0.06 -0.05 -0.02 0.01 80.03 0.03 -0.00 0.00 9-0.01 -0.01 0.00 -0.00 1-0.01 -0.01 0.00 -0.00 2-0.01 -0.01 -0.00 -0.00 30.01 0.01 0.00 -0.00 40.5260 ±j 5.4462 0.00 0.00 -0.00 0.00 5(23 and 24) 0.00 0.00 -0.00 0.00 60.05 0.05 -0.02 0.01 710.401 10.391 0.10 0.05 8-0.01 -0.01 0.00 0.00 90.01 0.01 0.00 0.00 10.01 0.01 0.00 0.00 2I 0.441 I 0.441 0.06 0.00 30.02 0.03 -0.00 0.00 40.2248 ±j 6.1273 0.00 0.00 0.00 0.00 5(25 and 26) -0.00 -0.00 0.00 0.00 6-0.00 -0.00 0.00 -0.00 70.00 -0.00 0.00 -0.00 8-0.01 -0.01 0.00 0.00 90.06 0.06 -0.00 -0.00 10.06 0.06 0.01 -0.00 20.02 0.02 0.00 -0.00 30.04 0.04 -0.00 -0.00 40.0922 ±j 2.8020 0.05 0.05 0.01 -0.00 5(27 and 28) 0.05 0.05 0.00 -0.00 60.07 0.07 0.00 -0.00 70.07 0.07 0.01 -0.00 80.07 0.07 -0.00 -0.00 9348, or 9 will effectively improve the fourth unstable mode is uncertain. Also, it is unclearwhether three stabilizers on machines 3, 7, and 8 are sufficient to stabilize the entiresystem or a fourth stabilizer must be designed. Since the participation factor methodis a linear analysis using a low—order linearized generator model, further analysis of thenine—machine system through nonlinear simulations using high—order nonlinear systemmodels probably will provide a more accurate answer.3.4 Nonlinear Simulations for Open-Loop SystemTo find a more accurate answer to the question of how many stabilizers are required andwhere they should be located, comprehensive time—domain simulations using a high—ordernonlinear system model are performed by the computer simulation method described inChapter 2. A three-phase fault is assumed to occur near the terminal bus of eachof the machines in turn for the simulation tests. There are nine machines and ninedifferent short—circuit tests. All time responses of angle, speed, torque, etc. are recorded.Therefore, there are numerous curves which can be plotted. Our primary concerns at themoment, however, are the angular swings and the speed deviations of the nine machines.3.4.1 Coherent GroupsOne of our interests is how these machines behave. Are they swinging coherently orindividually? Typical results found from the short—circuit test at the terminal bus ofgenerator 1 are shown in Figs. 3.2 through 3.5. The generator angular swings for allshort—circuit tests show that there are six coherent groups and the results are summarizedin Table 3.3.Although swing curves from the comprehensive nonlinear simulation tests give a clearpicture of how many coherent groups, they still cannot tell how many stabilizers are really3550.00— ‘. I/ \. / —45.00— / —40.00— —0.00 1.00 2.00 3.00 4.00 5.00Time (second)Figure 3.2: Angular Swings of Machines 1 and 21.00 2.00 3.00 4.00 5,00Time (second)Angular Swings of Machines 4 and 9615264______— U91L00 —6E\/ \R150.00— \ I ,145.00—1’ / —40.00 —35(0)—.130.00 —0.00Figure 3.3:360.00 1.00 2.00 3.00Time (second)Figure 3.4: Angular Swings of Machines 5 and 600a)I006367680.00 1.00 2.00 3.00Time (second)Swings of Machines 3, 7 and 8Figure 3.5: Angular6536I:,.uU——.80.00— S—75.00— /-:: V35.00——4.00 5.004.00 5.0037Table 3.3: Coherent Groups of a 9—machine SystemL Groups 1 2 3 4 5 6Machine No. 1, 2 4, 9 5, 6 3 7 8required for the stabilization of the system and where they should be located. AlthoughHiyama [10] suggested at least one stabilizer for each coherent group of a large powersystem, it remains to be examined.3.4.2 Speed Deviation AnalysisSearching for the most unstable generators in a multimachine system may be helpful indeciding the stabilizer number and sites for the system. Although the stability performances of generators may be compared with each other by directly observing generatorspeed deviations from system simulations, the comparison of a great number of speeddeviation curves of all machines for various simulation tests is very tedious. To avoidthat, a speed deviation index (SDI) is defined for individual machine generator as followsSDI =-J M&4dt (3.10)k1 owhere Aw is the speed deviation of a machine with respect to the system synchronousspeed in the j—th test, M is the inertia constant of the machine used as a weightingfactor, M&4 has the same unit as kinetic energy, which is integrated over time from 0to t, and k is the total number of simulation tests. This means that the SDI is definedby the average value of weighted speed deviations of a machine for all short—circuit testsover the simulation time. The SDI of a machine measures the degree of stability of themachine. The larger the SDI value of a machine is, the less stable the machine.A three—phase short—circuit for 0.12 second is given to each generator terminal ofthe system in turn. There are nine simulation tests for the nine—machine system and38Table 3.4: Speed Deviation Indices of Machines for the Open—loop System( Normalized with Respect to Their Maximum Value)r SDI 1.000 0.6713 0.1028 0.0666 0.0658 0.0630 0.0609 0.0404 0.0308Mc.No. 7 8 3 2 9 1 4 6 5each simulation lasts 5 seconds, i.e., k = 9 and t = 5 in Eq. (3.10). The SDI values ofall machines are calculated for all simulation tests and normalized with respect to theirlargest value. They are listed in Table 3.4 for comparison.Table 3.4 shows that there are three machines of G7, 08 and 03 which have relativelylarger SDI values and another six machines which have relatively smaller SDI values.Therefore, machines 3, 7, and 8 are the first three candidates for stabilizer sites. Theremaining questions are whether the three stabilizers are sufficient to ensure the stabilityof the entire system or a fourth stabilizer is required.3.5 Stabilizer DesignsSo far it has been found from both participation factors and speed deviation indices thatthe first three stabilizers must be designed for machines 3, 7, and 8. Whether threestabilizers on these machines are sufficient or not must be further investigated throughactual stabilizer designs. For this, different combinations of machines for PSS locationsare as follows:39Design No. Machines with Stabilizer1 3782 37813 37824 37845 37856 37867 3789A pole—placement technique developed in Chapter 4 is applied to these power systemstabilizer (PSS) designs for the multimachine system. The design procedure and thedesigned PSS parameters will be presented in detail in the next chapter and are notdescribed here. Our concerns at this moment are the dynamic performances of the nine—machine system with designed stabilizers in order to make the final decision of stabilizernumber and sites. To assess the merit of a PSS design from the nonlinear simulationtests in time—domain, a system stability index (SSI) is introduced, which is defined as= (3.11)where n is the number of machines in the system, i the i—th machine, j the j—th testand k the number of simulation tests. For each PSS design, the speed deviations of allmachines of the power system are integrated over the entire simulation duration, andthe results are added for all short—circuit simulations to give a SSI value. The SSI valuemeasures the degree of stability of the closed—loop power system for this design. Thesmaller the SSI value of the system is, the more stable the system.Again, a three-phase short—circuit for 0.12 second is assumed to occur near a terminalbus of all generators in turn and the results are recorded for 10 seconds for each short—circuit test. There are nine machines and nine short—circuit test, i.e., n = 9, k = 9 and40Table 3.5: System Stability Indices of Various DesignsDesign No. Machines with PSS SSI of the System1 3 7 8 0.553312 3781 0.547213 3782 0.546804 3784 0.550365 3785 0.550916 3786 0.543827 3789 0.54988t = 10 in Eq. (3.11). The nine simulation tests and calculations of SSI are repeated foreach PSS design. For all simulation tests, the initially unstable nine—machine systembecomes stable for all stabilizer designs. The results of calculated SSI are listed inTable 3.5.The results in Table 3.5 indicate that the three—stabilizer design of G3, G7, and G8 isjust as good as any of four—stabilizer designs. It is concluded that the three stabilizers aresufficient to stabilize the nine—machine system. Some simulation results of the closed—loop system for the three-stabilizer design will be given in the next chapter.3.6 Conclusions1. Both participation factor method of linear analysis and the speed deviation index(SDI) based on nonlinear simulations are helpful in deciding stabilizer number andsites for a multimachine system. Stabilizers should be installed on machines whosespeed state variables have relatively larger participation factors of unstable modesor on machines having relatively larger speed deviation indices.2. Coherent groups can be found from the nonlinear simulation tests, but it is notnecessary to have a stabilizer for each coherent group.413. For the initially unstable nine—machine system, three PSSs on machines 7, 8, and 3are sufficient to ensure the stability of the system although there are four unstablemodes and six coherent groups for the open—loop system.Note that the third conclusion has been verified not only from the results of PSSdesigns in Chapter 4 but also from the results of a self—tuning stabilizer design in Chapter 5.42Chapter 4PSS FOR MULTIMACHINE SYSTEMS WITH LOW-FREQUENCYOSCILLATIONSLow—frequency electro-mechanical mode oscillations between interconnected synchronousgenerators in a large power system stem from the increase in size and in complexity ofpower systems. The oscillating frequencies range approximately from 0.2 to 2.5 Hz [13].To improve damping of low—frequency oscillations, Power System Stabilizers (PSS) aredesigned. The use of PSSs has been popular in power industry [14]. However, thedesign of PSS parameters for a large power system is still a tough task since PSSs areusually installed on some large generators in the system and only local variables of thesegenerators can be readily utilized as feedback signals. Due to the decentralized controlstructure and dynamic interaction between machines [15], much effort has been made inPSS design to coordinate all PSSs of a multimachine system to provide certain dampingfor low—frequency oscillating modes of the entire system ([16]—[18]). Progress also hasbeen made in developing pole—placement methods for determining PSS parameters toyield exact damping to negatively or poorly damped low—frequency oscillating modes([19]—[20]). Although these exact pole—placement techniques are useful for PSS design,they are still complicated and involve much computation. A simpler pole—placementtechnique with less computation is desirable.This chapter presents a new pole—placement design technique for determining PSSparameters of multimachine power systems in order to move unstable or poorly dampedlow—frequency oscillating mode eigenvalues to desired locations on the complex plane.43The mathematical formulation of the design method is concise and systematic and thecomputational requirement is less than the previous methods ([19]—[20]). Two multimachine power systems are used as examples to illustrate the design procedure and to showthe effectiveness of the proposed method.4.1 A New Pole—Placement PSS Design MethodConsider a design of m stabilizers for a multimachine power system. The number ofmachines of the system may be equal to or larger than the number of the stabilizers. Inthe frequency domain, the linear state equation of the power system may be written as[sx(s)] = [A][x(s)] + [B][u(s)] (4.1)where [x] is an n state vector of all machines with the speed variables of m generators inleading positions, [u] is an m control vector, [A] is an n x n system matrix, and [B] is ann x m control matrix. Assuming that the control signals of stabilizers are applied to theexcitation systems of the m machines, Eq. (4.1) may be partitioned asszw(s) A11 A12 /Xw(s) 0= + [u(s)] (4.2)sy(s) A21 A22 y(s) Bwhere [(s)] consists of m speed elements of the m machines for which the stabilizersare designed and [y(s)] the remaining state variables of [x]. [Ba] is part of [B], alsoconsisting of mostly null elements except those associated with the transfer functions ofexcitation systems.The speed variable of the j—th machine, is used as feedback input of the decentralized stabilizer on the j—th machine to produce a control signalui(s) = h(s)(s) (4.3)44whereh(s)=i 2rj ; j=1, 2, ..., m (4.4)and h3(s) is the transfer function of a Power System Stabilizer (PSS), with an additionalreset block, on the j—th machine. The PSS time constant and the reset block timeconstant are usually preselected and treated as known quantities. The other PSSparameters of h,(s) are treated as the unknowns. For the m stabilizers of the powersystem, there are m Eq. (4.3) and they may be assembled into[u(s)] = [H(s)][.(s)] (4.5)where [H] is an rn x m matrix with only diagonal elements h,(.s),j = 1,.. . , m. Therefore,the last term of Eq. (4.2) becomes0 0 sw(s)[u(s)] = [H(s) 0]B B y(s)0 0 &(s)= (4.6)BH(s) 0 y(s)Substituting Eq. (4.6) into Eq. (4.2) gives[sLw(s)] = [A11][Lw(s)] +[A12][y(s)][sy(s)] = [A21 + BH(s)][/.w(s)] +[A22][y(s)]Eliminating [y(s)j from Eq. (4.7) yields[F(s)][/w(s)] = [H(s)][w(s)] (4.8)where[F(s)] = {A12(sI —A22)’B] 1 [sI — A11 — A12(sI —A22)A1] (4.9)45and [F(s)] is an m x m matrix.To find the unknown parameters of hi(s) of the j—th stabilizer, let hi(s) be expressedexplicitly first. Let the j—th speed variable of [&(s)] of Eq. (4.8), L\w(s), be moved tothe first position. Hence, [F(s)] and [H(s)] must be reordered as [F3(s)] and [H3(s)j asfollowsF,11(s) F,12(s) Awi(s) = hi(s) 0 \w3(s) (4.10)F21(s) lj22(s) x3(s) 0 H2(s) x3(s)where hi(s) is separated from other PSS transfer functions and H2(s) is an (m — 1) x(rn— 1) matrix with the remaining rn — 1 stabilizer transfer functions as its diagonalelements. [x(s)] is a vector with m — 1 speed variables of [/w(s)] excluding z\w(s).Eliminating [x(s)] from Eq. (4.10), we shall have the explicit expression of h3(s).hi(s) = F(s) +Fj12(s)[H2s) —.F22(s)]’F1s) (4.11)The process of reordering Eq. (4.8) to obtain Eq. (4.11) must be repeated for j = 1,. . . , mso that rn equations of Eq. (4.11) can be obtained for m stabilizers. Let PSS transferfunctions of Eq. (4.11) be replaced by Eq. (4.4). Then, Eq. (4.11) may be rewritten in ageneral formh(s,K,T) = f(s,K1,T... .,Km,Tm) (412)j=1, 2, ..., mThe remaining problem is to determine PSS time constant T and the gain K,. Thiscan be done in such a way that m pairs of open—loop unstable eigenvalues can be movedexactly to the desired locations for the closed—loop system. For this purpose, replacing sin every j—th equation of Eqs. (4.12) with the j-th desired eigenvalue A, or its conjugate46value ), we haveh1(X,KT) = fi(;\i,K2,TK3... m,Tm)h2(.A,KT) = (\,1.. .Km,Tm)(4.13)fj(\j,Ki,Ti,...,Kj_i,Tj_i,Kj+i,Tj+j,...,Km,Tm)hm(.Am,Km,Tm) = fm(\m,Ki,Ti,K2,...,K _i,Tm_i)Finally, the pole—placement design problem has been reduced to the solution of 2m PSSparameters K, and T3 (j = .. ,m) from the algebraic equations (4.1.3).Two major advantages of the new design method are clear at this moment. Firstly, itis obvious that the mathematical formulation involved for reducing system state equationto algebraic equations (4.13) is concise and systematic. Secondly, since h, is explicitlyexpressed in the j—th equation of Eqs. (4.13) (or h, is separated from the argumentsof fj), a relatively simple method, the Gauss—Seidel method or the fixed point method,can be used to solve for K., and T, j = 1,.. . , ni. But, this cannot be done if allh, j = 1,.. . , m were implicitly expressed in a set of algebraic equation as in [20]. Insuch case, the Newton—Raphson method must be used to find the solution. But, therequired computation of Jacobian matrix at each iteration is very time consuming.4.2 PSS Designs Using the New Pole—Placement MethodThis section discusses how to apply the proposed new pole—placement method to thePSSs designs for power systems. Two multimachine power systems are used as thedesign examples to show the design procedure and the effectiveness of the method.474.2.1 Algorithm of Solving PSS Parameters by Gauss—Seidel MethodIn this subsection, the Gauss—Seidel method is used to solve for the PSS parameters K.and T (j = . . ,m) from Eqs. (4.13). The algorithm is as follows.1. Specify a set of desired new electromechanical mode eigenvalues ) for closed—loopsystem and set K.(O), T,(O) for j = 1, ..., m.Specify error tolerances 6K and ET.Set iteration index I = 1.2. Solve K(I) and T(I), j = 1, ..., m fromh1(),J((I),T1(I))= f)i,K2(I 1),T2(I_ 1),K3(I_ 1),T3(I_ 1),...Km(1 1),Tm(1 1))h2(,K2(1),T2(I))=f,(I),TI),K3(I_ 1),T3(I_ 1),...Km(1 1),Tm(1 1))K(I),T3(I))= f(;\,K1(I) T,...K+1(I_ 1),Tj+(I— 1),...,Km(1 1),Tm(1 1))hm(Am, Km(I), Tm(I))= fm(m,Ki(I),Ti(1),K2(I),...,Kmi(I),Tm_i(I))one by one.3.AreK(I)—K(I—1)<eKandITj(I)—Tj(I—1)I<eT j=1,2,...,rn?4. Stop if both yes;Otherwise set I = I + 1 and goto step 2.484.2.2 Selection of Eigenvalues for the Closed—Loop SystemThe natural frequency w, of an oscillating mode is equal to the absolute value of thecorresponding eigenvalue. For the pole—placement design, new eigenvalue pairs , ))are specified by designating a new damping factor for initially unstable mode eigenvalueswith their natural frequency remaining unchanged. For example, if is the desired newdamping factor, we shall specify= + j/1 — 2wThe purpose of selecting eigenvalues for closed—loop system in this way is to concentrate the control efforts of PSSs on improving the damping of specific low—frequencyoscillating modes.4.2.3 Design Example 1 — A Three—Machine Power SystemTo test the new pole—placement method for PSS design, a three—machine power system,Fig. 4.1, is chosen as the first example system. It consists of three machines and aninfinite busbar. Each machine is equipped with a static exciter. The detailed data of thesystem can be found in ([21], [16]). The linearized system state equation is included in[16]?c=Ax+Bubut with state variables reordered here asx=w2, )3, L61, z62, t53, /eq2, /EFD1, EFD2, 1’FD3]andU=[u1, u2,49—0.039 0.004 0.02 —0.147 0.022 0.046 —0.013 0 0.003 0 0 0—0.034 0.032 —0.028 0.004 —0.149 0.079 —0.0065—0.008 0 0 0 0—0.017 —0.01 —0.017 0.001 0.017 —0.056 —0.003 0 —0.009 0 0 0377. 0 0 0 0 0 0 0 0 0 0 00 377. 0 0 0 0 0 0 0 0 0 00 0 377. 0 0 0 0 0 0 0 0 0A=—3.393 0.754 1.131 —0.266 —0.087 —0.250 —0.922 0.024 0.072 1. 0 01.131 —1.885 0.754 0.121 —1.60 0.460 0.021 —0.21 0.06 0 1. 00 0 —1.131 0.083 0.220 —1.20 —0.002 0.011 —0.197 0 0 1.—309.14 —91.99 —1675 —30.1 24.599 62.051 —60.943 —3.501 —20.194 —20. 0 0—64.47 —51-6.11 —171.91—18.48 106.09 16.99 —12.55 —21.67 —11.41 0 —20. 0—33.93 —46.37 —893.49 —10.1 17 70.1 —6.78 —2.1 —54.4 0 0 —20.T0 0 0 0 0 0 0 0 0 800 0 0B= 0 0 0 0 0 0 0 0 0 0 900 00 0 0 0 0 0 0 0 0 0 0 1000The eigenvalues of the system without PSSs are calculated and listed in the firstcolunm of Table 4.2. There are three pairs of complex—conjugated eigenvalues with lownatural frequency and poor damping. A new damping factor 0.3 is specified for all threepoorly damped modes. The new eigenvalues are as follows:Sites of PSSs Old eigenvalues Specified eigenvalues ()Machine 1 —0.0627 + j7.3692 —2.1800 + j6.9200Machine 2 0.0953 + j7.8364 —2.3700 + j7.5300Machine 3 0.2637 + j4.0915 —1.1700 + j3.720050Infinite Busbar______________0.066610.1782 +jO.7998 0.0926+ jO.6508+ jO.3520_____ __ _______0.1293 +jO.71690.0923+jO.53130.0628 +jO.4745Figure 4.1: A Three—Machine Power System for PSS DesignTable 4.1: Tuned Parameters of PSSsK1 T1 K2 T2 K3 T323.6881 0.1828 34.9859 0.2293 18.4519 0.1599The parameters of the three PSSs are calculated with a Fortran program writtenaccording to the proposed design algorithm and the results are listed in Table 4.1. Theeigenvalues of the closed—loop system with the designed PSSs are computed and listed inthe second column of Table 4.2. As expected, the exact assignment of specified eigenvaluesis achieved by the PSSs designed with the new pole—placement method.4.2.4 Design Example 2 — A Nine—Machine Power SystemThe nine—machine power system described in Chapter 3 is chosen as a second examplefor PSS design with the new pole—placement method. The system configuration is shown51Table 4.2: Eigenvalue ComparisonSystem without PSSs System with PSSs-0.0627 +j 7.3692 -2.1800 ±j 6.9200 *0.0953 ±j 7.8364 -2.3700 +j 7.5300 *0.2637 ±j 4.0915 -1.1700 ±j 3.7200 *-1.5112 -1.3226-3.4305 -3.1685-5.8914 -5.8332-15.1893 -40.1939-17.0519 -42.5035-18.8713 -46.5227-11.7893 ±j 16.0235-13.5454 +j 12.1889-15.5560 ±j 9.4230-0.2083-0.2048_____________________-0.2027*: Exactly Assigned Eigenvaluesin Fig. 3.1 and the system data are given in Chapter 3. There are 36 eigenvalues forthe nine—machine system without supplementary control as listed in Table 3.1. Theelectromechanical modes eigellvalues of the 36 eigenvalues are listed here. in the firstcolumn of Table 4.4. There are four unstable eigenmodes (last four eigenvalue—pairs)and two poorly damped eigenmodes (the third and fourth eigenvalue—pairs) among theelectromechanical modes.It was mentioned in Chapter 3 that either three stabilizers for machines 3, 7, and 8or four stabilizers for the three machines plus a fourth machine must be designed beforethe final decision of stabilizer number and sites. The different combinations of machinesfor PSS locations are listed as follows:52Design No. Machines with PSS1 3782 37813 37824 37845 37856 37867 3789The details of these designs are as follows. In the first example of PSS design in theprevious subsection, a uniform damping C = 0.3 was specified for all mechanical modeeigenvalues. The choice of damping factor here for the nine—machine system is not assimple as that in the first example since the four stabilizers are supposed not only tomove four unstable mode eigenvalue pairs to new places on the complex plane but also toimprove the dampings of two other poorly damped modes. Four stabilizers on machines3, 7, 8, and 9 have been chosen to carry out designs in order to find proper dampingfactors for the four unstable eigenvalues. We began with a uniform damping C = 0.3 forall four initially unstable modes for the design. The four stabilizers designed with the newpole—placement method did provide exact damping factor to the four unstable modes asintended, but failed to improve dampings of the other two poorly damped modes. Thenwe varied the dampings one at a time, but kept the other dampings unchanged andcontinued the design. Finally, we found that the best damping for the entire systemcan be obtained by specifying new non—uniform damping factors 0.8, 0.7, 0.3, and 0.4,respectively, for the four unstable mode eigenvalues. The specified eigenvalues are asfollows:53Locations of PSSs Old Eigenvalues Specified Eigenvalues ( Damping)Machine 7 0.5083±j 5.3235 -4.2782±j 3.2087 (0.8)Machine 8 O.5260+j 5.4462 -3.8301±j 3.9075 (0.7)Machine 3 0.2248±j 6.1273 -1.8394±j 5.8490 (0.3)A fourth machine 0.0922+j 2.8020 -1.1214±j 2.5695 (0.4)For all designs, the reset block time constant Tr of Eq (4.4) is preselected as 5.0 s andone time constant T of PSSs as 0.035 s. To find the unknown parameters K and T3 ofthe PSSs so that the unstable mode eigenvalues can be changed to those specified newvalues, a Fortran program is written according to the proposed pole—placement algorithmfor all designs. The results of all designs are listed in Table 4.3.The eigenvalues of the nine—machine power system with the designed stabilizers(closed—loop system) are then computed. Note that since each PSS has three statevariables, there are 9 more state variables for the power system with three stabilizersand 12 more state variables for the system with four stabilizers. Therefore, there are 45eigenvalues for the closed—loop system with three stabilizers and 48 eigenvalues for allclosed—loop systems with four stabilizers. Only electromechanical mode eigenvalues ofclosed—loop systems for all designs are listed in Table 4.4 for comparison because othereigenvalues have much larger dampings and higher frequencies. The results show notTable 4.3: PSS Parameters of Various DesignsDesign No. T3 K3 T7 K7 T8 K8 T1 0.1029 18.0754 0.1039 28.2710 0.1239 31.3517.2 0.1047 18.0116 0.1033 27.7738 0.1210 32.4351 0.1961 52.29463 0.1047 17.9839 0.1046 28.0538 0.1202 29.4844 0.2093 24.44124 0.1036 17.9782 0.1038 28.2490 0.1238 31.4645 0.2207 43.98475 0.1124 15.9459 0.1030 31.6431 0.1340 37.2103 0.1215 19.57346 0.1050 17.7158 0.0969 29.2139 0.0991 20.9985 0.1652 40.44077 0.1016 18.6228 0.1039 27.8718 0.1197 30.9386 0.1384 36.330754Table 4.4: Electromechanical Mode Eigenvalue ComparisonSystem without PSSs System with PSSs System with PSSs System with PSSs.onMachine378 onMachine378l onMachine3782-0.3354±j 9.1267 -0.3352±j 9.1273 -0.8393±j 7.4043 -0.8846±j 7.3731-0.2188±j 8.6423 -0.2259±j 8.6496 -0.2309±j 8.6538 -0.2252±j 8.6497-0.0947±j 7.2452 -0.1267±j 7.2762 -0 .6384±j 7.9944 -0.2448±j 7.6732-0.0627±j 7.0242 -0.1964±j 6.8179 -0.2902±j 6.1798 -0.3024±j 6.1700-0.1970±j 6.5043 -0.2521±j 6.1470 -0.2131±j 6.8286 -0.2127±j 6.83160.5083±j 5.3235 -4.2782±j 3.2087 * -4.2782±j 3.2087 * -4.2782±j 3.2087 *0.5260±j 5.4462 -3.8301±j 3.9075 * -3.8301±j 3.9075 * -3.8301±j 3.9075 *0.2248±j 6.1273 -1.8394±j 5.8490 * -1.8394±j 5.8490 * -1.8394±j 5.8490 *0.0922±j 2.8020 -0.5275±j 2.7844 -1.1214±j 2.5695 * -1.1214±j 2.5695 *Denotes Exact Assignment of EigenvaluesSystem with PSSs System with PSSs System with PSSs System with PSSson Machine 3 7 8 4 on Machine 3 7 8 5 on Machine 3 7 8 6 on Machine 3 7 8 9-0.3330±j 9.1156 -0.3356±j 9.1252 -0.3353±j 9.1267 -0.3342±j 9.1235-0.2260±j 8.6497 -0.1260±j 7.3206 -0.5298±j 8.0959 -0.2249±j 8.6497-0.8807±j 7.3808 -0.6402±j 7.3419 -0.1838±j 7.2780 -0.1118±j 7.1568-0.1705±j 7.0351 -0.5448±j 6.8192 -0.3232±j 6.6798 -0.8450±j 7.3728-0.1261±j 6.7231 -0.6727±j 6.1768 -0.4222±j 5.9310 -0.2690±j 6.1339-4.2782±j 3.2087 * -4.2782±j 3.2087 * -4.2782±j 3.2087 * -4.2782±j 3.2087 *-3.8301±j 3.9075 * -3.8301±j 3.9075 * -3.8301±j 3.9075 * -3.8301±j 3.9075 *-1.8394±j 5.8490 * -1.8394±j 5.8490 * -1.8394±j 5.8490 * -1.8394±j 5.8490 *-1.1214±j 2.5695 * -1.1214±j 2.5695 * -1.1214±j 2.5695 * -1.1214±j 2.5695 ** Denotes Exact Assignment of Eigenvalues55only that the exact assignments of specified dampings to unstable modes can be achievedfor all designs but also that the dampings of the other two poorly damped modes areimproved.To assess the actual performances of all designs, comprehensive nonlinear simulationtests are performed and system stability indices are calculated as already described inChapter 3. Typical simulation results of the closed—loop system for three-stabilizer designare plotted in Fig. 4.2. The results show how three stabilizers on machines 3, 7, and 8can effectively stabilize the nine—machine system under a severe disturbance. Note thatthe nine—machine system is initially unstable for the same short—circuit test as shown inFigs. 3.2 through 3.64.3 Conclusions1. A new pole—placement technique is presented in this chapter for decentralized stabilizer design of multimachine power systems to damp low—frequency oscillations.A concise and systematic mathematical formulation is developed to reduce thesystem state equations to algebraic equations. The parameters of all PSSs in amultimachine power system can be determined simultaneously from the algebraicequations. Since PSS transfer functions are explicitly expressed in our algebraicequations, less computation is required for the determination of PSS parametersthan the existing methods.2. The effectiveness of the new pole-placement design technique has been demonstrated by various PSS designs of the two multimachine power systems. Exactassignment of any number of eigenvalues associated with low—frequency oscillatingmodes to new specified locations can be achieved for all designs.3. Non—uniform damping factors can be assigned to the eigenvalues to be changed.56Assigning a relatively large damping factor to an unstable mechanical mode canalso improve the damping of poorly damped mechanical modes nearby through thedynamic interaction of machines.4. The pole—placement technique in this chapter is general and it can also be appliedto the decentralized stabilizer design of other industries.57CD CD 0 C’, CD C’) 0 (ID0’0 CD c) C))CD C.,) CD 0 CD C’, CD 0 0CD C,, CD C) 0TerminalVoltageof05(p.u.)RPRPPP8TerminalVoltageofG1(p.u.)CD CD C) 0p 8 0TerminalVoltageof 08(p.u.)8TerminalVoltageof03(p.u.)p S g8Y x v io-3CCITime (second)Time (second)CCTime (second)Figure 4.2: Responses to a Short—Circuit near Gi Bus. (b)4.00 —5SM. —C0Y00ta-)I’I’ /I10.00— —9.00— —8.00— —7.00— —6.00— —cm— —2.80— . —0.00— I.2.00— 1 —-4.00—-8.00— —-9.00— - —V0.00 200 4.00 6.00 8.80 10.00 2.00 4.00 6.0014.00 —12.00 —10.008.00 —Y16.00— —10.00— —; 1 if -.8.00-\J-10.00 — —-12.00 — —-14.00 — —0.03 2.02 4.00 8.00 10.80 0.00 2.80 4 6.00 8.00 10.00Time (second)59Chapter 5DIRECT MIMO STR FOR MULTIMACHINE SYSTEMS WITHCHANGING OPERATING CONDITIONSThe conventional Power System Stabilizer (PSS) is designed for power system in normaloperating state. The system equations for PSS design are linearized around the givenoperating conditions. For the PSS design, the data of a power system must be completeand the parameters of PSSs are fixed. The fixed stabilizers work properly for the powersystem only for the given operating conditions. However, the operating conditions ofmany power systems are constantly changing due to the intentional energy managementof the electric plants or unintentional disturbances to the system. If the operating conditions of a power system vary greatly from the given values for which PSSs are designed,the parameters of the fixed PSSs must be retuned. Therefore, it is highly desirable tohave a self—tuning stabilizer for some power systems, which can constantly sample thesystem output, predict the future behavior of the system and automatically self—tune thestabilizer parameters to maintain an optimal performance for power systems.The self—tuning regulator (STR) has been developed since the early 70’s. The minimum variance regulator (MVR) [22] is designed by minimizing the variance of plantoutput. The generalized minimum variance controller (GMV) [23] is generalized by mmimizing both plant output and control. The pole-assignment controller (PAC) ([24]—[26])allows the prescription of closed-loop poles. The extended horizon controller (EHC)([27]—[28]) is designed with a time horizon for output predictor. Finally, the generalizedpredictive control (GPC) [29] is developed with a long—range output predictor and a60control with horizons. A review of the development of these STRs is given in Section 5.1.In Section 5.2, the principle of Clarke’s GPC for indirect single-input—single--output(SISO) STR design is extended to a direct MIMO STR design for power systems. Thereare two improvements: the initial step control parameters are directly estimated withoutsolving the Diophantine equation and estimating the plant parameters, and the subsequent control parameter is calculated recursively. The computational requirement ofGPC is greatly reduced.In Section 5.3, the method of the new direct MIMO STR is applied to the STRdesign for the nine—machine power system described in Chapter 3. A set of three STRsis designed for the system. For each STR, the local generator rotor speed and theterminal voltage are chosen as plant output variables and both governor and exciter ioopsare controlled. The results of the design are evaluated with comprehensive computersimulations for a multimachine system with wide—range changing operating conditions.A comparison between conventional PSSs and the designed STRs is also made.Many results of this chapter are published [39].5.1 Review of Self—Tuning Controls5.1.1 Minimum Variance Regulator (MVR)In Aström’s MVR, the plant is described by a CARMA (Controlled Autoregressive Moving Average) modelAy(t) = q_kBu(t) + e(t) (5.1)where y(t) is the plant output, u(t) is the control signal, e(t) is a white noise with zeromean, t is the sampling instant, and k is the time delay. A and B are polynomials of the61backward—shift operator q’A = 1 +a1q’ + ... + anaq_naB = b0 + b1q + ... + bbq’To find a predictor for output, a Diophantine equation is defined1 q’F + EA (5.2)where E and F are polynomials in q’ of degree k — 1 and na — 1, respectively. Ak—step—ahead predictor for y(t), which can be obtained from Eq. (5.1) and Eq. (5.2), isy(t + k) = Fy(t) + EBu(t) + Ee(t + k) (5.3)Since Ee(t+k) is the noise in the future, uncorrelated with other RHS terms of Eq. (5.3),the optimal predictor for y(t + k) is(1+ kit) = Fy(t) + EBu(t) (5.4)where (t + kit) represents a k—step—ahead optimal predictor from sampled data up tot. For minimizing the variance of plant output, (t + kit) is set to zero. Therefore, acontrol law becomesu(t) = —y(t) (5.5)whereG=EBThe control parameters F and G are directly estimated from Eq. (5.3) without estimating plant parameters A and B. Therefore, MVR is a direct self—tuning controller.Eq. (5.5) shows that polynomial B must have stable roots for u(t) to be bounded.Therefore, the MVR cannot be applied to a plant with unstable B or a non—minimumphase plant.625.1.2 Generalized Minimum Variance Control (GMV)The same plant model as Eq. (5.1) is defined for GMV but an auxiliary output is considered(t) = Py(t) — qRyr(t) + q_kQu(t) (5.6)where F, Q and R are weighting polynomials chosen by the designer and yr(t) is areference output. To find a predictor for (t), a Diophantine equation is definedp = qkF + EA (5.7)The k—step—ahead predictor for (t), which can be obtained from equations (5.1), (5.6)and (5.7), is(t + k) = Fy(t) + (Q + EB)u(t) — Ry(t) + Ee(t + k) (5.8)Since Ee(t + k) is uncorrelated with other RHS terms of Eq. (5.8), the best predictor for(t + k) is(t + kit) Fy(t) + (Q + EB)u(t) - Ryr(t) (5.9)To minimize the variance of the auxiliary output, (t + kit) of GMV must be zero.Therefore, a control law becomesu(t) = Ryr(t) Fy(t) (5.10)GwhereG=Q+EBThe closed—loop equation with the control isRB Gy(t)= BF + AQYr(t — k) + BP + AQe(t) (5.11)The control parameters F and G of this STR are directly estimated from Eq. (5.8).Therefore, GMV is also a direct self—tuning controller.63Eq. (5.11) shows that in order for CMV to deal with an unstable B (non—minimumphase plant), the Q could be taken as a scalar and must be large enough to let theclosed—loop poles approach the zeros of AQ when A is stable; and that in order to handlean unstable A (open—loop unstable plant), the values of P must be large enough. Hence,the GMV can control a non-minimum phase or open—loop unstable plant with carefullychosen Q and P. However, GMV is sensitive to varying delay time k unless Q is large.5.1.3 Pole—Assignment Control (PAC)Eq. (5.11) shows that the poles of the closed—loop system with CMV control are theroots of polynomial BP + AQ. LetBP+AQ=T (5.12)where polynomial T may be prespecified with its roots equal to the desired closed—looppoles. When polynomials P and Q are chosen to satisfy Eq. (5.12), the CMV becomes aPAC.For self—tuning control, plant parameters A and B of Eq. (5.1) may be estimated andP and Q are then calculated from Eq. (5.12). The drawback of PAC is that Eq. (5.12)cannot be solved if A and B have a common factor.5.1.4 Extended Horizon Control (EHC)When the plant time delay k of Eq. (5.1) is uncertain, a more general plant model maybe consideredAy(t) = q’Bu(t) + e(t) (5.13)where time delay k is assumed to be 1. For an actual time delay k, the k— 1 leadingcoefficients of B of Eq. (5.13) are zero.64The following Diophantine equation is defined for EHC1 = q_TF + EA (5.14)where T is a time horizon chosen for the design, which is usually larger than the actualplant time delay or the upper limit of a varying plant time delay.A T—step—ahead predictor for y(t) can be obtained from Eq. (5.13) and Eq. (5.14)y(t + T) = Fy(t) + EBu(t + T — 1) + Ee(t + T) (5.15)The optimal predictor for y(t + T) is(t + Tit) = Fy(t) + EBu(t + T — 1) (5.16)Let polynomial EB (degree = nb + T — 1) be written asEB = h0 + h1q’ + . . + hT_lq_(T_1) (5 17)+q_(T_1)(giq_1 + g2q2 + + gflbq)A reference output yr(t + T) is specified in EHC to satisfy(t + Tit) = yr(t + T) (5.18)Substituting Eqs. (5.17) and (5.18) into Eq. (5.16), and choosing the constant controlu(t) = u(t + 1) = ... = u(t + T — 1) [27], we obtain a control lawu(t)= ET_lh(YT(t + T) — Fy(t) —E1gu(t — i)) (5.19)For this self—tuning control, control parameters F, h1 and g, are directly estimatedfrom Eq. (5.15). Therefore, the EHC is a direct self—tuning controller. Since the timehorizon T is larger than the actual time delay, the roots of B are not included in thecontrol signal of the EHC. Therefore, EHC can control a non—minimum phase plant.However, simulation experience shows that EHC is unstable for an open—loop unstableplant [29].655.1.5 Generalized Predictive Control (GPC)For GPC design, the plant is described by a CARIMA (Controlled Auto—Regressive andIntegrated Moving Average) modelAy(t) =q1Bu(t) + e(t)/ (5 20)A train of Diophantine equations is definedl=q3Fj+EALS j=1,2,...,T (5.21)where the degree of polynomial F is na and T is referred to as a maximum outputhorizon. A set of predictors for output can be obtained from Eq. (5.20) and Eq. (5.21)asy(t+j) = Fy(t) +EBu(t+j —1) +Ee(t+j) j = 1,2,. ..,T (5.22)LetE3B = H3+G, j=1,2,...,TH3 = ho(j) +h1(j)q’ + . .. +h_1(j)q(’) (5.23)G q_(i_1)(g(j)q_1 +g2(j)q + “+g(j)q)Besides the maximum horizon T, a minimum output horizon nl and a control horizonflu are defined for selecting the number of optimal predictors from Eq. (5.22). Theselected optimal predictors are written in matrjx form as[] = [yr] + [H] [us] (5.24)where []=[yr,] = [F] [y(t), y(t — 1),. . . , y(t — na)]+ [G] [ZXu(t — 1), /u(t — 2),.. , Lu(t — nb)]T[us] = [u(t), /u(t + 1),... , ZXu(t + flu —66wherein the j—th rows of matrices [H], [F], and [G] respectively contain the coefficientsof polynomials H,, F3 and G, j = nl, ..., T. A cost function is then defined as followsJ = ([Th - [yr])T([YA] - [yr]) + r[uc]T[uc] (5.25)Minimizing J with respect to [ut] givesu(t) = u(t-l)+ [F]([yr]-[yp]) (5.26)where [F] is the first row of (HTH + rI)_1HT.For the self—tuning control, plant parameters A and B of Eq. (5.20) must be estimatedfirst and F1 and E1 is then calculated from Eq. (5.21). To compute F and E for j = 2,T, a recursive equation is developed. Finally, the control parameters H3 and G,,j = 1, ..., T, are obtained from E3B according to Eq. (5.23). Therefore, the GPC isan indirect self—tuning controller. Moreover, control parameters H3 and G cannot beobtained directly by a recursive equation. The GPC requires heavy computation.5.1.6 Summary of STRsThe MVR requires the least computation but cannot handle the non-minimum phaseproblem (unstable B polynomial). The GMV control can handle the non—minimumphase plant, but it is sensitive to the varying time delay of a plant. The PAC allowsthe prescription of closed-loop poles, but it cannot cope with a common factor whichmay occur in the numerator B and denominator A of the plant transfer function. TheEHC can handle the non—minimum phase and uncertain system time delay, but maystill have difficulty in dealing with the unstable system poles. The GPC is probably thebest method of STR design to handle plants with non—minimum phase, common factor,uncertain time delay, and unstable open—loop dynamics. The only drawback of GPCdesign is the heavy computational requirement.675.2 A New Direct MIMO STR for Power SystemThese STR principles have been applied to pulp mills, chemical processes and otherindustrial processes ([30], [31]). The application of STRs to power systems has also beenstudied ([32]—[37]), involving only a few of machines. But a larger power system with moremachines should be chosen for study since without a reasonable number of machines, thedynamic interactions between machines cannot be thoroughly investigated. In addition, apower system usually has unstable or poorly damped open—loop poles. Also, it is difficultto have the exact information of both system time delay and non—minimum phase whenthe system operation changes over a wide range. Therefore, the principle of GPC is mostattractive for the STR design of power systems. However, the design technique of GPC,especially the computational requirement, must be improved.In this section, the principle of Clarke’s GPC for an indirect SISO STR design isextended to a direct MIMO STR design. A train of modified Diophantine equationsand a set of output predictions are described in subsection 5.2.1. A control law fromminimization of a cost function of weighted optimal predictors and control signals isderived in subsection 5.2.2. A new recursive equation for control parameter computationis developed in subsection 5.2.3 and a method of direct estimation of the initial stepcontrol parameters in subsection 5.2.4. The algorithm of the STR design is summarizedin subsection Basic Equations and Long-Range Output PredictionFor the MIMO STR design, the vector output of a plant is modeled by[A] [y(t)] = q’ [B] [u(t)] + [e(t)] / (5.27)68whereT[y(t)] = [yi(t),y2(t)]T[u(t)] = [ui(1),u2(t)][e(t)] = [ei(t),e2(t)]T[A] = [i + A1q’ + + Anaq][B] = [B0 + B1q’ + + Bflq_nb]In Eq. (5.27), [y(t)] is the plant output vector consisting of rotor speed and terminalvoltage at sampling instant t, [u(t)] the control signal vector of excitation and governorloops, [e(t)] the errors, and [A] and [B] are polynomials of the backward-shift operatorq’ with 2 x 2 matrix coefficients. The integrator 1/Li is introduced to eliminate thestatic errors. The superscript T indicates the transpose of a vector or matrix.For a long—range predictor, an output horizon T (maximum output horizon) is assumed and the following Diophantine equation set similar to the original GPC [29], butin matrix form for MIMO design, may be defined.[I]q_i[j]+[Ej][Aj j=1,2,...,T (5.28)where[F;] = [F(j) + F(j)q’ + ... + F(j)q][Ej = [Eo(j) + Ei(j)q1 + ... + Ej_i(j)q_(j_1)]Here all F(j), .. . , F(j), Eo(j),.. . E_1(j) are 2 x 2 matrices. To simplify the computation of GPC, the Diophantine equation set may be modified as follows. Since [F] = [I]for q1 = 1 according to Eq. (5.28), [Fj] may be separated into two terms[F;] == [I] + [Fo(j) + Fi(j)q’ + . + Fna_i(j)q’1]69Therefore, the Diophantine equation set of Eq. (5.28) can be modified as[I] = q_3([I] + [Fi] z) + [Es] [A] Li j = 1, 2,.. . , T (5.29)where [F3) has one less matrix coefficient than [Fjj of the original OPC.Postmultiplying both sides of Eq. (5.29) by [y(t)], we shall have[y(t)] = q[y(t)] + q [F] z [y(t)] + [E3] [A] [y(t)}= q[y(t)] + q [Fi] /. [y(t)] + [Es] [B] LS [u(t— 1)1 + [Es] [e(t)]The j-step-ahead prediction [y(t + j)J can be obtained from the above equation by shiftingtime t ahead by j steps[y(t + j)] = [y(t)] + [Fj] LS [y(t)] + [Es] [B] L [u(t + j — 1)1 + [Es] [e(t + i1 (5 30)j=1,2,...,T[Es] [B] of Eq. (5.30) is of degree nb +j — 1 and can be separated into a polynomial [G3]for the known controls of the past and a polynomial [H3] for the unknown controls of thefuture,[Ei] [B] = [H3] + [GJwhere[113] = [Ho(j) +H1(j)q’ + + Hj_1(j)q(3’)j[G3] = q1) {G1(j)q’ +G2(j)q + .. . + Gfl(j)q_nb]All Ho(j),.. . ,H3_1(j), Gi(j),.. . , Gb(j) are 2 x 2 matrices. Therefore, Eq. (5.30) may70be written[y(t+j)]= [y(t)]+ [F0(j)zy(t) +F1(j)zy(t — 1) +. + Fna_i(j)iy(t — na + 1)]+ [H0(j)/u(t + j — 1) + Hj(j)Liu(t + j — 2) + . . .H_1(j)L(t)]+ [G1(j)Zu(t — 1) +G2(j)/u(t — 2) + + Gb(j)/.xu(t — nb)]+ [Eo(j) + Ei(j)q’ + • + Eji(j)q(’)j [e(t +j)]j=l,2,...,T(5.31)There are T equations of (5.31) which may be written in matrix form[yr] = [yr] + [H] [us] + [E] [e] (5.32)where [y] is the predicted output of the future, [yr] the output and control of the past,[us] the control to be determined and [e] the errors of the future. Details are[yr] = [y(t+ i)T,(t+2)T,.,y(t+T)T]T[yr,] [C] [y(t)]+[F] [y(t)T,y(t_1)T,...,y(t_na+1)T]+ [G] [u(t — i)T, Au(t — 2)T,.. . , u(t — nb)T]T[uJ = [u(t)T, u(t + i)T, , u(t + T— 1)T]T[e] = [e(t + i)T, e(t +2)T,... , e(t + T)T]Tand[C]= [ C(1) C(2) C(3) •.. C(T)where [C] is a 2T x 2 matrix and C(1), C(2), etc., are 2 x 2 unit matrices. Other details71areF0(1) F1(1) F2(1) •.. F_1(1)F0(2) F1(2) F2(2) Fna_i(2)[F] = F0(3) F1(3) F2(3) Fna_i(3)F0(T) F1(T) F2(T) Fnai(T)G1(1) G2(1) G3(1) •.. Gb(1)G1(2) G2(2) G3(2)•.. Gb(2)[G] = G1(3) G2(3) G3(3) Gb(3)G1(T) G2(T) G3(T) •.. Gb(T)H0(1)I[(2) H0(2)[H] = H2(3) H1(3) H0(3)HT_j(T) HT_2( ) HT_3( ) •.. H(T) j[E] is similar to [H] except that E replaces H. Note that each element of [F], [G] and [H]is 2 x 2 matrix and that the upper—right matrix elements above the diagonal of [H] andthose of [E] are null. The dimensions of [F], [G], and [H] or [E] are 2T x 2na, 2T x 2nb,and 2T x 2T, respectively.5.2.2 Control LawsConsider Eq. (5.32) again. Since the last term of Eq. (5.32) is the disturbance of thefuture, the first two terms on the RHS of Eq. (5.32) correspond to a set of optimal72predictors. Let it be[] = (t + 1It)T, (t + 2It)T,.. . , (t + TIt)T] (5.33)=[yr,] + [H] [zt]Consider a desirable optimal predictor [Y][Y]=(t + n1 t)T, (t + n1 + lt)T,... (t + TIt)T] (534)which is a subset of [p], or [j less the first 2(n1 — 1) rows, where n1 is a minimumoutput horizon, 1 ru T.Next, since our concern is the increment L\u(t) of the present, the increments Lu(t+j)of the future for j run, 1 n < T, may be set to zero. Therefore, [un] of Eq. (5.32)becomes[] = u(t + 1)T. . . , +u(t + n — 1)T] (5.35)and the desired optimal predictor may be written[Y] {th] + [i] [•i] (5.36)where[th] = [a] [y(t)]+ {] {y(t)T,y(t_ 1)T,...,y(_na i)T]T+ [G] [u(t — i)T, u(t — 2)T,. . . , u(t — nb)Tjand[C]= [C] without the first 2(ru1 — 1) rows[F]= [F] without the first 2(ni — 1) rows[G]= [G] without the first 2(rui— 1) rows (5.37)[H]= the first 2ruu columns of [H]without its first 2(ni— 1) rows73To design a long—range optimal predictive control with minimum control effort, thefollowing output reference sequence is chosen[Yr] [yr(t+fli)T,yr(t+fli + 1)T,...,yr(t+T)T] (5.38)where•T[yr(t+j)] = [yri(t+i),yr2(t+j ] 3 =n1,...,TLet a cost function beJ = ([Y}- [y])T [Q] ([Y] - [yr]) + [u]T [R] [] (5.39)where [Q] and [R] are weighting matrices. Substituting Eq. (5.36) into Eq. (5.39), minimizing J with respect to [i], and solving for [] give[]= [-] T [ii] + [R])’ [u]T ([yr] - [i]) (5.40)The control signals of the present correspond to the first two elements of []u1(t) ui(t—1) 1= + [F] [Hj [Q] ([yr]— j) (5.41)u2(t) u2(t— 1)where [F] is the first two rows of ([H] [Q] [H] + [R]) Recursive Computation of Control ParametersIn the original GPC design [29], a recursive equation was developed to calculate parameters [E3], [F3] of the Diophantine equation, but not control parameters. Additionalcalculations of control parameters [H(j)] and [G(j)] ,j = 1,2,..., T were required. It wastime—consuming even for the SISO control design. In our direct MIMO STR, a recursive74algorithm is developed so that the control parameters [F(j)], [G(j)] and [H(j)] are directly computed from [F(j— 1)], [G(j — 1)], and [H(j — 1)] without solving Diophantineequations.The recursive formula may be derived as follows. Let j of Eq. (5.31) be replaced byj — 1 and t of the same equation by t + 1. In other words, let the time t be shifted onestep ahead, we have[y(t + j)] = [y(t + 1)1 + L [F0(j — 1)y(t + 1)]+ [Fi(j 1)y(t) + ... + Fna_1(j — 1)Ly(t — na + 2)]+ H(j—l)/.u(t+j—l)+H_l)Ziu(t+j_2)+...H_2j_l)Zu(t+ 1)]+ [G1(j — 1)zXu(t) + G2(j — 1)Lu(t — 1) ... + G(j — 1)Lu(t — nb + 1)]+ [E0(j — 1) + Ei(j — 1)q + ... + E_2(j — 1)q_U_2)] [e(t + j)](5.42)Rearranging the first two terms on the RFIS of Eq (5.42) gives[y(t + j)] = [I + F0(j — 1)] [y(t + 1)] — [F0(j — 1)y(t)]+ [Fi(j — 1)Ay(t) + ... + F_1(j — 1)y(t — na + 2)]+ H(j—1)u(t+j—1)+H1)/ (t+j_2)+...H_2j_1)Zu(t+ 1)]+ [Gi(j — 1)Lu(t) + G2(j — 1)Lu(t — 1)... + Gb(j — 1)Lu(t — rib + 1)]+ [Eo(j — 1) + Ei(j — 1)q’ + ... + E_2(j — 1)q_U_2)] [e(t + j)](5.43)For j=1, Eq (5.31) may be written as follows[y(t + 1)] = [y(t)]+ [F0(1)L\y(t) +F1(1)zy(t — 1) + ... + Fna_i(1)Ly(t — na + 1)]+ [H0(1)/.u(t)] (5.44)+ [G1(1)zu(t — 1) +G2(1)Lu(t — 2) + ... +G6(1)zu(t — nb)]+ [E0(1)] [e(t + 1)]75Substituting Eq (5.44) into Eq (5.43) and comparing the results with the like terms ofEq (5.31) give a recursive computation formula of the control parameters as follows• I(F0j—1)+1)(1)+÷i(j—1) for i=O,l,...,na—2( (Fo(j—l)+I)F:(l) for i=ria—l• I(F0j—l)+I)G(l)+G11j— ) for i=1,2,...,nb—1G(j) =(Fj—1)+I)G for i=nb(5.45)I H(j—1) for i=O,1,...,j—2=(Fo(j—1)+I)Ho(1)+Gi(j_1) for i=j—1• I E(j—1) for i=O,1,...,j—2=( (Fo(j—1)+I)Eo(1) for i=j—1Note that E parameters are not required for our STR design. The initial step controlparameters, F1(1), H(1), and G(1), can be estimated from system dynamic responses,which is described in detail in the next subsection.5.2.4 Direct Estimation of Initial Step Control ParametersIn our STR design, the control parameters F(1), H(1), and G2(1) of Eq. (5.45) areestimated directly. Since there are two control loops of a power plant, the excitation andthe governor, the i-th row of the [F(1)], [G(1)], and [11(1)] is identified for the i-th loop,i = 1, 2. To that aim, j of Eq. (5.31) is replaced with 1 and t is back shifted one step,76resulting inz [y(t)] = [F0(1)zy(t— 1) +F1(1)LIy(t — 2) + + Fna_i(1)Ly(t — na)]+ [H0(1)zu(t — 1)]+ [Gi(l)L\u(t — 2) +G2(1)L\u(t — 3) + + Gb(l)L\u(t — rib — 1)]+ [e(t)](5.46)Eq. (5.46) consists of measurement and data of zero mean. Eq. (5.46) may be written= X(t — l)TO(t) + e(t) i = 1,2 (5.47)where the data vector X(t — 1) consists of a sequence of system output and control thatare known at time tX(t — i)T = [z(t — 1)T, — 2)T,. . . L\y(t — na)T,Au(t — i)T, Lu(t —2)T,.. . L.u(t — nb — i)T]ande(t)T = the i-th row of [F0(1),... Fna_i(1), Ho(1),.. . G(1), .. . Gni(1)] j = 1,2Note that F0(1),..., Fa_i(1), H0(1), G1(1), ..., Gb(1) are 2 x 2 matrices. For theestimate of the parameter vector e(t) at each sampling, Bierman’s UDU version of RLS(recursive least squares) [38] is employed. When these estimated F0(1),..., Fnai(1),II(1), G1(1), ..., Gb(1) are available, the subsequent control parameters can be computed recursively using Eq. (5.45) in the previous section.5.2.5 Algorithm of the STR DesignThe algorithm of the direct MIMO STR design is summarized as follows1. Read new [y(t)] and [yr(t)] at sampling instant t772. Estimate [F(1)], [H(1)] and [G(1)} by the RLS— Eq. (5.47)3. Compute [F(j)], [H(j)] and [G(j)] for j> 1— Eq. (5.45).4. Decide [F], [a], and [H]— Eq. (5.37).5. Compute [u(t)] and activate it—Eq. (5.41).6. Set t = t + 1, go back to step 1 and repeat the process5.3 Example of Design and Simulation Test of the New STRIn this section, the principle and method of the new direct MIMO STR developed in theprevious section are applied to the STR design for the nine—machine system described inChapter 3. The system is initially unstable. It has been found in Chapters 3 and 4 thatthe machines 3, 7 and 8 are the strategic plants for stabilizer locations. Three STRs willbe designed for these machines.Some details of the STR design are as follows:1. The speed deviation &4. and the terminal voltage Vt of a machine to be equippedwith STR are chosen respectively as the output variables Yi and Y2 of Eq. (5.27).The control signal u1 of the STR is applied to the excitation loop and u2 to thegovernor loop of the machine as Fig. 5.1. A reset block is used to eliminate theeffects of u2 on steady state mechanical torque Tm.2. The weighting matrix [Q] is fixed as a unit matrix [I] while [R] is fixed as O.OO1[Ijafter trial and error. A relatively small [R] provides a better damping to the systemoutput responses while a relatively large [R] provides a smoother control.3. The selection of n1,n and T is discussed in detail in [29]. They are chosen as 3, 3and 5 respectively in our design.78+ I iTGovernor M+ and Turbine____0)wt ‘+ Poweru’2 Generator Vt Systemeter9Ui______________Control Lawlin(COrandVr)Figure 5.1: Schematic Diagram of the Direct MIMO STR4. The sampling period is chosen as 60 ms and both na and nb as 5. Further decreaseof the sampling period or increase of na does not improve the system dynamicresponse significantly.5. A self—adjusting forgetting factor of 0.95 is introduced in the RLS algorithm forthe estimate of control parameters. The forgetting factor allows a discount of someold data in the estimation so that the estimator may adapt itself to fast changesof power system operations. The factor is set to 1 when the estimation errorI — XT(t)01(t is less than 0.001.After the design, the power system with the STRs is given comprehensive simulationtests to determine the merits of STRs. For the simulation tests , nonlinear high order79models are used for each machine, including a 5th order synchronous generator, a 4thorder hydro—plant governor and turbine or a 2nd order steam governor and turbine, anda 1st order fast excitation system. A computer simulation program for the multimachinepower system is written based on the method in Chapter 2. To simulate the self—tuningcontrol loops according to the algorithm of the new STR, a Fortran program is added tothe system simulation program.Mainly three types of simulation tests are performed: successive step changes involtage reference, successive step changes in governor gate opening, and three—phaseshort—circuits at a machine terminal. These tests are repeated for each machine. Forcomparison, similar tests are repeated for the same nine—machine system but equippedwith PSSs designed in Chapter 4. A test of short—circuit and trip—off of the transmissionline is also performed. For all tests, the responses such as speed, voltage, electric power,excitation and governor control signal etc. of all machines are recorded and examined.Numerous curves could be plotted but only typical results are shown as Fig. 5.2 throughFig. 5.6.Fig. 5.2 shows some responses of 03 (machine #3) to the successive step changes inreference voltage of G3 for the system with the designed STRs. To each step change,the STRs adapt very fast to provide damping to the system and the system oscillationssubside very quickly. The terminal voltage, the speed deviation, the excitation and thegovernor control signals are separately plotted.Fig. 5.3 shows the G3—responses to the successive step changes in governor gateopening of G3 for the system with designed STRs. Again, the STRs are very effectiveand the system oscillations are damped very quickly.Fig. 5.4 shows the G3—responses to a three-phase short—circuit for O.12s near 03terminal for the system with the designed STRs. The transient stability of the systemrecovers very fast, which indicates that the designed STRs are very effective. From80Fig. 5.2 to Fig. 5.4 only the responses of 03 are plotted since G3—responses are the mostviolent ones.Fig. 5.5 shows the speed deviation responses of several machines to a three-phaseshort—circuit of a transmission line for O.12s near Bus 14 and between Bus 14 and Bus 21for the system with STRs. The short—circuit is removed by tripping off the faulted line,leaving the system operating in a quite different condition. In spite of the two successivesevere changes, the transient stability of the system recovers although the settling timeis larger than other cases.Fig. 5.6 compares the responses of 08 for the system with STRs, plotted in solid lines,with those for the system with the PSSs designed in the Chapter 4, plotted in dottedlines, to the same successive step changes in governor opening of 08. It is found that thestability recovers for the system with STRs but not for the system with the PSSs. Thisis the case which shows that the STRs are superior to the fixed PSSs.Note that the PSSs presented in Chapter 4, unlike other PSSs, are of a coordinateddesign ensuring a very stable power system. The capability of the PSSs to stabilizea power system is almost as good as the STRs in most cases. But, when the systemoperating conditions change to certain new values as in the case of Fig. 5.6, these fixedPSSs cannot stabilize the system properly.5.4 Conclusions1. Clarke’s principle of indirect SISO STR design [29] is extended to the direct MIMOSTR design in this thesis. Two improved design techniques are developed: thedirect estimate of the initial step control parameters and the recursive computationof the subsequent control parameters. The computational requirements are reduced.These improvements are general and can be very useful for the STR design of other81industries.2. The principle and method of the direct MIMO STR are applied to the STRs designof a nine—machine power system. Both excitation and governor loops are controlledand only three machines require the STRs. Tests of successive step changes involtage reference, successive step changes in governor opening, short—circuits nearthe machine terminal, and short—circuit and trip—off of the transmission lines allshow that the STRs thus designed can effectively stabilize a power system over awide range of operating conditions.3. The results in Fig. 5.6 show that when system operating conditions change to somenew states, the STRs thus designed still can stabilize the system but the fixedPSSs cannot do so even though they are very well designed. Therefore, furtherexploration of STR design is necessary to the benefit of power system stabilitycontrol.4. The STR design also confirms that only three stabilizers on machines 3, 7 and 8are sufficient for the stability control of the nine—machine power system.821.2a. 1.1C.CC5 10 15 20 25 30Time (second)Time (second)0.10C)C015Time (second)0.020C-)ICE-0.020 5 10 15 20 25 30Time (second)(Sampling period=60 ms)Figure 5.2: Responses to Step Changes in Reference Voltage of G383Il)00(Sampling period=60 ms)Figure 5.3: Responses to Step Changes in Gate Opening of G332.520 5 10 15 20Time (second)25 30 35 405020Time (second)00Cl)0.02‘0.01500.010U0.0050C.)00UEV00 5 10 15 20Time (second)25 30 35 4020Time (second)84x1030>.cz 0.1CCCC.)C0-0.1K‘C0000VV -0.C,,10 00’4-0C0. 00VV -0.10 0(Sampling period=60 ms)Figure 5.5: Responses to a Short—Circuit and the Removal of the faulted Line0.54-0C00•0VC,, -Time (second)5 10Time (second)0.0200-0.0210 00Figure 5.4:105Time (second)near 03 Bus5Time (second)Responses to a Short—Circuitxl 04-0C00-5Cd,4-0C00VV-50.C’,5 10Time (second)dO-35Time (second)0 5Time (second)5Time (second)1085000C)3 STR (solid line)2.: jci%:..(dashed0 5 10 15 20 25Time (second)30STR (solid flne)APSS (dashed line)(5 10 15 20 25 30Time (second)1.04001.02____________________________________________________at 10.9802Time (second)STR (solid line)PSS (dashed line)5 10 15 20 25 300 5 10 - 15 20 25 30Time (second)(Sampling period=60 ms)Figure 5.6: Responses to Step Changes in Gate Opening of G886Chapter 6EXCITATION CONTROL OF SHAFT TORSIONAL OSCILLATIONS OFA MULTIMACHINE SYSTEM6.1 IntroductionSeries capacitor compensation is used in transmission lines to increase HVAC transmissioncapacity of power systems. When electric power is transmitted from a thermal—electricplant over this kind of capacitor—compensated line, series resonance of the line and thegenerator at subsynchronous frequency may occur. The subsynchronous resonance (SSR)may excite the torsional oscillations of a turbine-generator mechanical system and evencause shaft damage and system interruption. Therefore, stabilizer design to suppress thetorsional oscillations of a power system with SSR is an important power system stabilitycontrol problem.Since the shaft damage caused by SSR at the Mohave power plant [41], two BenchmarkModels have been recommended by IEEE SSR Working Group for SSR studies ([42], [43]).Extensive analysis of SSR has been made, tests and countermeasures for one—machineinfinite—bus system, such as the First Benchmark Model (FBM) and the system 1 ofthe Second Benchmark Model (SBM), have been proposed , and some countermeasureshave been implemented ([44]— [46]). PSS control of SSR with filter is also studied [47].However, a stabilizer for the system 2 of SBM, which has two nonidentical machines withone series—capacitor compensated line, has not yet been developed. This chapter designsa stabilizer through the excitation loop of a generator to damp the multi—mode torsional87oscillations of the system 2 of the SBM.For SSR analysis and stabilizer design for the system 2 of the SBM, a mathematicalmodel must be developed. This model is different from the conventional model for low—frequency oscillation study which models a turbine—generator system as a single massspring system and does not recognize the torsional oscillations between various stages ofturbines , generator and exciter. For SSR study, however, each of all rotating massesmust be modeled by two first—order differential equations because these rotating massesmay oscillate with respect to each other when they are excited by SSR. In addition,the damper windings of a synchronous generator are usually ignored in the conventionalmodel for low—frequency oscillation study. But, these damper windings must be includedin the model for SSR study because of their effects at high frequencies. Also, the statorarmature winding of a generator and the transmission network, which are described byalgebraic equations for low—frequency oscillation study, must be remodeled by differentialequations for SSR study to find the electric resonance of the capacitor—compensatedtransmission line and the generator. Since the transmissioll network and generators areusually described in different coordillate systems, coordinate transformation is necessaryfor SSR analysis and stabilizer design of the system 2 of the SBM.Unlike conventional PSS design using only one state variable of a generator as thefeedback input of a PSS, more state variables must be used as the feedback to controlthe multi—mode torsional oscillations of a turbine—generator mechanical system. A linearcombination of the local measurable variables is used as feedback in this chapter todesign stabilizers for the system 2 of the SBM. To determine which state variables aremost effective for the feedback, the participation factor method is used. To determinefeedback gains of the decentralized stabilizers, a new direct pole-placement method isdeveloped. With this method, exact pole—placement can be achieved and the feedbackgains of stabilizers can be obtained directly without iteration. The results of computer88simulation tests for the systems with and without the designed stabilizers show that thestabilizers thus designed can effectively damp out all torsional oscillations of the systemover a wide range of capacitor compensation.Many results of this chapter are published [50].6.2 System 2 of the Second Benchmark Model (SBM)The electrical circuit of system 2 of the SBM is shown in Fig. 6.1 (a) [43]. There aretwo generator units 01 and 02 connected to a common bus through transformers. Thetransmission line between the common bus and bus 1 is series—capacitor—compensated.R and X1, respectively, denote the resistance and reactance of a transformer, RL andXL the resistance and reactance of a transmission line, and Xc the series capacitancevarying from 10% to 90% of XL1.The mechanical system of unit 01 is presented in Fig. 6.1 (b) [43]. There are fourrotating masses of unit G1; the high—pressure turbine (HP), lower—pressure turbine (LP),the generator (GEN) and the exciter (EX), all on one shaft. Each rotating mass and shaftconstitutes a torsional mass—spring system. The mechanical system of unit G2 is similarto that of 01 except for no rotating exciter.6.3 Mathematical Model for the System 2 of SBM6.3.1 Mechanical SystemAs previously mentioned, for SSR study, each rotating mass of the mass—spring system of a turbine—generator—exciter set should be modeled by two first—order differentialequations. Differential equations for high-pressure turbine, low—pressure turbine, thegenerator, and the exciter of the first turbine—generator—exciter set in Fig. 6.1 (b) can be89RL1 XL1 X(a) Electrical System(b) Mechanical System of GiFigure 6.1: The System 2 of the SBMRti XtiR Xt2RL2 XL2\Infinite BusCommon Bus90written as follows [14]1WH1 , [TH1 — DH1LIJH1 — KHL1(OH1— OL1)].IVIH1°H1 = — 1.0)L1 = [T — DL1WL1 + KHL1(OH1— OL1) — KLG1(OL1 —°L1 = Wb(WL1 — 1.0) (6.1)= M1[Tei — D01w1 + KLG1(OL1—— KGx1(61—— 1.0)1= [—Tl—Dxlwxl+KGxl(6l—0xl)]1’1X1OX1 — 1.0)where the w’s are the rotor speeds in per unit with a base value equal to w = 27rf rad/s,the 0’s are the rotor angles, 6 is the generator rotor angle, the M’s are inertial constantsfor rotating masses, the K’s are the shaft stiffnesses, the D’s are dampings, and theT’s are the torques applied to masses. Subscripts H, L, G and X, respectively, identifythe high— and low—pressure turbines, the generator, and the exciter. Note that turbinetorques TH and TL are also state variables and their differential equations will be givenas Eqs. (6.4) and (6.5) in Section 6.3.2. The generator electric torque output Te is not astate variable but can be replaced with Eq. (6.14) in Section 6.3.4.Differential equations for the mass—spring system of the second turbine—generator setare almost the same as those for G1 except that the last two equations and KGX constantof Eqs. (6.1) should be deleted and the subscript 1 should be replaced by 2.916.3.2 Governor and TurbineA two-time-constant governor is assumed in Fig. 6.2 (a) where a denotes the speed relayposition and g the governor opening [48]. The differential equations for the governor areKG 1a= 1 (‘ref — ,) — —a (6.2)1+ISR ‘SR1 g—gog = —a— (6.3)TSM TSMwhere g0 denotes an initial gate opening.There are two time constants of the steam turbine [48]: TCH for the steam chest andTRH for the reheater and/or the cross-over. The transfer function of the steam turbineis shown in Fig. 6.2 (b). The differential equations for the steam turbine are• FH 1TH = —g— —TH (6.4)ICH ICH• FL 1TL = TH— —TL (6.5)rHIRH ‘RHThe fractions FH and FL are defined asFH+FL=1 (6.6)6.3.3 Exciter and Voltage RegulatorA two—time—constant excitation system is chosen in Fig. 6.2 (c) where yR denotes avoltage regulator output and EFD a generator internal voltage [49]. Two differentialequations can be written for the excitation system1 IAyR = (VrefVt)VR+UE (6.7)‘A ‘A ‘A• 1 1EFD = — -EFD (6.8)1E IE92(a) A Two-Time-Constant Governor(b) A Two-Time-Constant Steam Turbine__ __KA VR 1 IEVf-Vtl+sTA j l+sTEUEt(c) Excitation SystemFigure 6.2: Governor, Turbine and Excitation SystemTH93where uE is the supplementary control signal of the stabilizer to be designed and Vref areference voltage. Vt denotes the generator terminal voltage and can be written asVt = Jv + v (6.9)where Vd and Vq are the d and q components of the Vt and will be described in Section Synchronous GeneratorFor SSR study, it is more convenient to use the generator currents as state variables tomodel a synchronous generator ([14]). The generator model can be obtained from Park’svoltage equations. In Park’s equations, the variables of a generator are described bythe individual d—q coordinate of the generator. For each generator of system 2 of SBM,besides d and q armature windings on the stator, there are four windings on the rotor: adamper winding D and a field winding F on the d axis and damper windings Q and Son the q axis. Therefore, the Park’s equation may be written as followsVd = TaZd+d/WbW/)qVq = —raiq + q/b + dVF = rFiF + /-‘F/Wbo = rDiD+bD/wb (6.10)0 = rQiQ+bQ/wb0 = rsis + bs/wbwhere V denotes a voltage, i a current, )L’ a flux linkage, w a speed, and r a resistance, allin per unit. Subscripts d, q, F, D, Q, and S identify the respective windings. To use thegenerator currents as state variables, the flux linkages of Eqs. (6.10) can be substituted94with currents by using the following equationsXd Xmd Xmd= Xmd XF Xmd (6.11)Xmd Xmd XDXq Xmq Xmq= Xmq XQ Zmq (6.12)Xmq Xmq XSand Eqs. (6.10) becomes1 .. .. . .xdzd + XmdZF + XmdiD) = w(_xqzq + XmqiQ + xmqzs) + raid + Vd1 .. .. . .Xqiq + XmqZQ + xmqzs) = —(—xdzd + XmdZF + XmdiD) + TaZq + Vq1 .Xmdid + XFZF + XmdiD) TZ + VF (6.13)Wb1FXmdid + XmdZF + XDZD) = TDZD1 ..Xmqiq + XQZQ + Xmqis) = —rQZQ1 .. ..Xmqiq + XmqiQ + XSZS) = rszswhere xd, Xq, XF, XD, xq, and xs are the reactances of the respective windings, Xmd isthe mutual reactance of windings on d axis, and Xmq is the mutual reactance of windingson q axis. Note that the generator terminal voltage Vd and Vq are not state variables butcan be eliminated by using Eq. (6.22) in Section 6.3.5.The generator electric torque Te of Eq. (6.1) is not a state variable , but can now bereplaced by generator currents:Te = Zd?IdZq?I3q=(Xq— Xd)idiq + XmdZfZq + XmdqD—XmqQd—Xmqsd (6.14)956.3.5 Transmission NetworkIn individual machine coordinates dk—qk, the terminal voltages of the k-th machine canbe expressed in the sum of the voltages across the transformers and the common busvoltages [51]Vql = R1 X1 q1+Zql+Vcom.ql (6.15)Vdl X1 R1 Zdl -‘b Zdl VcomdlVq2 = R2 X2 Zq2+q2+Vcom.q2 (6.16)Vd2 —X2 R Zd2 d2 Vcomd2where R denotes a transformer resistance, X its reactance, wb the base speed, and Vcom.dand Vcom.q respectively the d and q components of the common bus voltage in individualmachine coordinates. Now, the two individual coordinates must be interfaced by usinga common system coordinate, D—Q coordinate. The position of the infinite bus voltageis chosen as the D axis of the common coordinate which is also the reference axis of therotor angles of both machines. The relationship between the individual d—q coordinateof the k—machine and the common D—Q coordinate is shown in Fig. 6.3. Therefore, theEqs. (6.15) and (6.16) can be rewritten asVql = R1 X1 qi+zqi+ [Ti]Vcom.D (6.17)Vdl X1 R1 c11 d1 VcomQ[ Vq2 ] = [ R2 X2 ] [ ] + [ Zq2 ] + [T2} [ Vcom.D ] (6.18)Vd2 —Xt2 Rt2 Zd2 d2 VcomQwherecos 6 sin 6 cos 82 sin 62[T1}=sin 8 — cos 8 sin 62 — cos 62Vcom.D and Vcom.Q, respectively, are the D and Q components of the common bus voltagein the D—Q coordinate and 6 and 62 are the rotor angles of Gi and G2.96Q axisaxisVqkVQ ///// VD D axisVdkdk axiSFigure 6.3: Individual Machine and Common System CoordinatesIn common D—Q coordinate, the common bus voltages can be expressed in terms ofthe transmission line, capacitor compensation, and infinite bus voltages as follows:[VcomD][RL _XL][ILD] [i] [D] [VOD] (6.19)VcomQ XL RL ILQ ‘LQ EcQ VOQwhere RL denotes the total line resistance and XL its total reactance. ‘LD and ILQ,respectively, are the D and Q components of the line currents; ECD and ECQ the components of the voltage across the capacitor; and VOD and VOQ the components of the infinitebus voltage, all in the common D-Q coordinate.Next, the D—Q components of the transmission line current may be expressed in termsof the d—q components of individual machine currents asILD=[T1] (6.20)ILQ di d297and their derivatives consist of four terms[1LD] d61 [zi ] + [T1] [‘] + [] [ Z2] + [T2] [2] (6.21)-TLQ 1d1 1d1 d2 d2where— sin 6 cos 6 — sin 62 COS 62[T19 = [T} =cos 5 sin 6 ‘52 sin 62Substituting Equations (6.19), (6.20) and (6.21) into Eq. (6.17), the terminal voltagesof Cl in d—q coordinate becomeVql R1 + RL X1 + XLLl q1+XL + X1 Zq1Vdl —(X1 + XLW1) R1 + RL d1 Wb Zd1FeD VOD XL 012 S12 q2+ [T1 +[T1] +—ECQ VOQ b 12 012 d2C12RL +S12XLw2 —S12RL +C12XLW2+ (6.22)S12RL — C12XLW2 C12RL +S12XLW2whereS12 = sin(61— 62) 012 = cos(61 — 62)521 = sin(62 — 61) 012 = cos(62— 61)Similar equations can be written for the terminal voltage of 02 by simply interchanging the subscripts 1 and 2 of Eq. (6.22).Finally, the differential equations for the voltages across the capacitor compensationcan be written from the current relationECD 0 1 BeD ‘LD= +WbXC (6.23)ECQ 1 0 ECQ ‘LQ98where transmission line currents‘LD and ‘LQ are not state variables, but may be replacedby Eq. (6.20).6.3.6 Summary of Mathematical ModelThe complete system model for the SSR study includes 40 nonlinear differential equationsas derived in the previous sections. All 40 state variables may be summarized as[X]T= {L’H1 °H1 L1 0L1 -‘i 8 a1 giTH1 TL1 WH2 °H2 WL2 °L2 ‘2 82 a2 g2TH2 TL2 di q1 ZF1 D1 Qi ZS1 d2 q2ZF2 D2 ZQ2 S2 ECD ECQ EFD1 EFD2 VR1 VR2jThe nonlinear differential equations of the state variables are used for simulationtests. For the control design in the following sections, these nonlinear equations mustbe linearized with respect to a set of initial values of the state variables. The linearizedsystem state equations will be given as Eq. (6.24) in the next section but the detailsof the linearized equations are not included. All data of these differential equations areavailable in the references ([43],[51j).6.4 A Direct Pole—Placement Method for Control DesignIterative pole—placement method has been applied to excitation control design for theFirst Benchmark Model which has only one machine [52]. For the control design of thesystem 2 of the SBM, which has two machines, a new direct pole—placement method isdeveloped in this thesis. This method does not require any iterations and the resultantstabilizers use only local state variables as feedback signals for decentralized control.99The linearized state equation of a multimachine power system with excitation controlu(t) may be written±=Ax+Bu (6.24)where x is the n x 1 state vector and u is the k x 1 control vector. A and B, respectively,are the n x n system matrix and the n x k control matrix. Assume that there are munstable eigenvalues to be replaced and that state variables used as control feedbacksignals have been chosen. Assume further that columns and rows of the system matrixcorresponding to the feedback state variables have been moved to the front position. WehaveXI X1U’x11 = [A] x11 + [B] (6.25)U2XIII X111where X1 contains m local feedback state variables chosen for u1, the excitation controlof the first generator set, and X11 contains m state variables for u2, the control of thesecond generator set, i.e.,K 0 X1= (6.26)U2 0 K11 X11where K1 is a 1 x m gain matrix of u1 and Ku that of u2. Thus, Eq. (6.25) may berewritten as(I XIK1 0 0= A+B x11 (6.27)0 1(11 0XIII XIIINext, let the unstable mode eigenvalues be shifted to desired values )q, i = 1,2,. .. , m,and let the corresponding eigenvectors be P, i = 1,2,. . . , m. For the i-th new eigenvalues,100we shall haveK1 0 0A + B [F,] = [F]) (6.28)0 K11 0orK1 0 0[Fe] = [A — )I]’[B] [F] (6.29)0 K11 0LetK1 0 0— [F] = (6.30)0 K11 0 ,t3where c, and /3 are constants. Since the value of an eigenvector is not unique, cj ( or/3) can be chosen by trial and error, but not zero, and both of them are related to P2.Substituting Eq. (6.30) into Eq. (6.29), we have[F2] = [A — ),I]’[Bj (6.31)where [Pg] is an n x 1 vector. For prescribed eigenvalues i = 1, 2,... , m, the corresponding eigenvectors F, can be solved one by one from Eq. (6.31) and all eigenvectorsthus obtained can be written in a matrix form as[F] = [P1 F2 ... F,... Fm] (6.32)where [F] is an n x m matrix.By using Eq. (6.30) and Eq. (6.32), we haveK1 0 0 a1 a2...am— [F] = (6.33)0 K11 0 /32 ... /3m101Let [F] be partitioned asF’[F] = p11 (6.34)F”where both F, and P11 are rn x m matrices. Then, Eq. (6.33) becomesF’K1 0 0 a1 a2... am—p11 = (6.35)0 K,, 0 /3i ...F”Therefore, the control gains [K,] and [K,,] can be solved from[K,] = —[a’ a2 •. am][F,j (6.36)[K,,] _[3, 2 •.. /3m][1ii]’ (6.37)No iterations are required.The algorithm of the pole—placement method may be summarized as follows:1. Specify a set of desired new mechanical mode eigenvalues ) for closed—loop systemand select nonzero elements a and /3, i = 1, ..., m.2. Let X, include m feedback state variables chosen for u, and X,, m variables for u2.3. Rearrange the columns and rows of system matrix, if necessary, to form the equationEq. (6.25).4. Calculate P from Eq. (6.31), for i = 1, ..., m.5. Form [P] according to Eq. (6.32).6. Pick out F, and F,, from Eq. (6.34).7. Obtain K1 and K,1 from Eq. (6.36) and Eq. (6.37), respectively.102Table 6.1: Torsional ModesMode Frequency of 01 Frequency of 021 24.65 Hz 24.65 Hz2 32.39 Hz 44.9 Hz3 51.10 Hz —6.5 Eigenvalues Analysis of the System without ControlTo apply the pole—placement controller design method to the system 2 of the SBM, theeigenvalues of the linearized system model are analyzed in this section to find the unstablemode eigenvalues. Since the unstable eigenvalues are usually associated with the torsionaloscillation frequencies of mechanical systems, the torsional modes of mechanical systemsare discussed first.6.5.1 Natural Torsional Oscillating ModesThe natural torsional modes of the turbine—generator—exciter mass-spring system maybe found by the eigenvalues of Eqs (6.1) without damping D’s and forcing torque T’s.There are usually m— 1 torsional modes for a rn—mass-spring system. These modesare numbered sequentially according to the values of their frequencies. Mode 1 has thelowest torsional oscillating frequency and mode rn— 1 has the highest one. For thesystem 2 of the SBM, the torsional modes of mechanical systems of two machines (Cland 02) were given in [43], and they are also confirmed by our own calculation. Theseresults are listed in Table 6.1. There are three torsional modes for the mechanical systemof 01 and two modes for that of 02.1036.5.2 Unstable Mode EigenvaluesThe two generators of system 2 of SBM are assumed operating on full load at a powerfactor of 0.9. A 40th—order system model derived in previous sections and linearizedaround the operating conditions is used for eigenvalue analyses. For each given capacitorcompensation ratio Xc/XL1,the system model has a set of forty eigenvalues. The capacitor compensation ratio is assumed to vary from 0.05 to 0.9 in 0.05 increments. Theeigenvalues of the system for all these compensations are calculated and examined. It isfound that a low—frequency mode (MO), the mode 1 of Gi (Mu), and the mode 1 of G2(M12) are unstable, or nearly unstable. The real—part eigenvalue loci of these unstablemodes are plotted in Fig. 6.4. The worst situation occurs around a compensation ratio of0.70. For this compensation ratio, the unstable eigenvalues and their undamped naturalfrequencies areEigenvalues Natural Frequencies f (Hz) Torsional Modes0.9847 +j 155.77 24.79 Mu0.0243 ±j 155.77 24.79 M120.3232 ±j6.9752 1.11 MOTherefore, the system state equation based on the compensation ratio of 0.7 is singledout for the excitation control design.6.6 Stabilizer Design for the System 2 of SBMThe objective of the stabilizer design is to place these unstable mode eigenvalues tothe desired location using local measurable state variables. First, participation factorsof these eigenvalues will be calculated to decide the most effective state variables asfeedback input of stabilizers. Then, the state variables will be transformed to somemeasurable variable for the controller design. Finally, since the gain matrix in the design104is not unique, a gain matrix suitable for the system stability over wide—range capacitorcompensation will be sought.6.6.1 State Variables for Control FeedbackTo move the six unstable eigenvalues to prespecifled locations by the pole—placementdesign method, it is necessary to choose six state variables for [X1] and another six for[Xii]. Participation factor analysis method is used to determine which of the forty statevariables are most sensitive to these unstable modes and should be chosen as the excitation control feedback. It is found that the following state variables are most influentialand may be used for feedback design[Xi]T [Lid1 LZqi M1 L\W1 /&WL1 1iFi][X11]T [L.id2 q2 2 L\W2 ‘L2 F2]The state variables d and q, the d and q components of the armature current, however,are not directly measurable. They must be replaced by measurable variables. In thisdesign, the electric power output Fe and the generator armature current t are chosen.Hence, the desired feedback variables are[y1]T= [IFei LSzLl AS1 Aw1 AWL1 AZF1][y11]T= [APe2 Ai2 2 AW2 ALL’L2 AZF2]They require the following linear transformation= [T] (6.38)where T is a 12 x 12 transformation matrix. Twelve linear algebraic equations are requiredto form the T matrix. In addition to eight identity equations for the AS’s, the Aw’s, the105w’s, and the L.iF’s, the remaining four equations arePci = ZjV + ZqiVqiPe2 = d2’Vd2 + Zq2Vq2 (6.39)ti= ‘v/Iii + iit2 + i2To eliminate Vdi and Vqi of Eqs. (6.39), Eq. (6.22) may be modified as follows. At thepoint of system operation, the derivatives of the currents of Eq (6.22), which constructsa very small portion of the voltage across the transformers and the transmission line, canbe neglected. The [T1j[ECD ECQ]T representing the voltage across Xc of the capacitorcompensation will vanish if the total line reactance XL of Eq (6.22) is replaced by X =XL— Xc. Therefore, Eq (6.22) becomesvql R1 + RL X1 + Xw1 Zqi VOD=vdi —(X1 + Xwi) Rn + RL di VQQC12RL +S12X2 —S12RL +C12Xw2+ (6.40)S12RL — C12Xw2 C12RL + Sl2XNote that besides di, q1 there are state variables 6 and of both machines on the RHSof Eqs. (6.40). Now, Vdi and Vql of Eqs. (6.39) can be replaced with Eq (6.40). Similarrelations can be found for Vd2 and Vq2. Finally, from the results of the linearization ofEqs. (6.39), the transformation matrix [TI may be completed.Using Eq. (6.38), Eq. (6.25) can be transformed into= [A] ] + [u’] (6.41)XIII X111106wherer—i T 0 T’ 0 T 0[Aj = [A] [B = [B]01 0 I 01Now the control vector is chosen asu1 K1 0 Y1= (6.42)u2 0 i(jNote that this is a linear similarity transformation and thus the eigenvalues of the systemmatrix A are the same as those of the original system matrix A. The control designmethod of Section 6.4 still can be applied to the transformed system by simply replacingA and B matrices by A and B, respectively.6.6.2 Prespecified EigenvaluesAs already mentioned in Section 6.5.2, there are six unstable eigenvalues among theforty eigenvalues for the system without control. The six eigenvalues consist of threecomplex conjugate pairs. It is intended to change their damping ratio but not theirnatural frequencies so that unnecessary control efforts can be avoided. The prespecifiedeigenvalues areSpecified Eigenvalues Old Unstable Eigenvalues—3.4270 ± j155.73 0.9847 ± j155.77—3.L155±j155.74 0.0243±j155.77—0.8728 + j6.9279 0.3232 ± j6.97526.6.3 Feedback Gain MatricesSince the feedback gain matrices are not unique, it depends on the choice of the a and/3 constants. The non—zero elements a and /3, = 1, 2,. . . , m, are all assumed to be1071.00 to begin with. The control gains are directly determined and exact pole—placementfor those prespecified eigenvalues is obtained without the need of iteration. However, forstable operation over wide—range capacitor compensation ratio from 0.05 to 0.9, the cand /3 values of 1.00 chosen for the 0.7 capacitor compensation design should be adjusted.The final values are— 1.00 1.00 1.00 1.00 1.00 1.00/3 — 1.04 1.04 0.94 0.94 1.00 1.00and the control gains become[K1] = [1.803 0.198 0.573— 98.0 125.9 — 2.516][KH] = [2.109 — 7.933 8.242 — 8.80 105.8 — 1.503]The eigenvalues of the six unstable mode have been moved to exact new locations—3.4270 ±j 155.73 (f = 24.79 Hz)—3.1155 ±j 155.74 (f = 24.79 Hz)—0.8728 + 6.9279 (f = 1.11 Hz)Although the control design is for a particular capacitor compensation ratio of 0.7, eigenvalues of the system with the control gains are recalculated for compensation ratio from0.05 to 0.90 in 0.05 steps. All eigenmodes are stable. Finally, the real—part eigenvalueloci of the low frequency mode (MO), the mode 1 of 01 (Mu), and the mode 1 of G2(M12) for the system with stabilizer, are plotted in Fig. 6.5. Although the worst dampings of Mu and M12 for the system without stabilizers occur at compensation between0.6 and 0.8 as shown in Fig. 6.4, the best dampings of both Mu and M12 for the systemwith stabilizer occur at 0.7 as shown in Fig. 6.5, and this is exactly what the system isdesigned for.108120.80.60.4Mu—0.20.2 0.4 0.6 08Compensation Ratio Xc/XL1Figure 6.4: Real—Part Eigenvalue Loci of the Torsional Modes of System without Control0M12—2—3—40.2 0.4Compensation0.6 0.8Ratio Xc/XL1Figure 6.5: Real—Part Eigenvalue Loci of the Torsional Modes of System with Control109)6.6.4 Nonlinear Simulation TestNonlinear simulations are performed to test the excitation control design in this Chapter.The forty nonlinear differential equations including the governor opening and voltageregulator constraints are used for the computer simulations. Two types of disturbancesare simulated: a 10% step torque applied to generator 1 or 2, and a 20% pulsed torqueapplied to generator 1 or 2 for 0.2 second. All results indicate that the system is stablefor the system with control but unstable for the system without control. Since thecontrol is designed for a given state of operation, the system responses to the step torquedisturbance are more severe than those to the pulsed torque disturbance. Since thesimulation results of the step torque applied to generator 1 or 2 are similar, only theresults of the step torque applied to generator 1 are presented in this section. Fig. 6.6shows the responses to this step torque for the system without control and Fig. 6.7 showsthose for the system with control. Multimode oscillations appear in all cases, but thedecay is reasonably fast for the system with control. There are high and low frequencycomponents in control signal UE. While the low frequency component in UE providesdamping to the low frequency oscillations of mode 0, like conventional PSS, and thehigh frequency components produce effective damping to the high frequency torsionaloscillations of mode 1 of both generators.6.7 Conclusions1. A mathematical model has been derived for system 2 of the SBM for the SSRstudies.2. A new direct pole—placement method is developed for the stabilizer design of multimachine SSR systems. Only local output signals of individual machines are usedas the feedback input.1103. The method has been successfully applied to a stabilizer design for excitation control of torsional oscillations of system 2 of the SBM. Effective feedback signals canbe found from participation factor analysis and all unstable mode eigenvalues ofthe system can be shifted directly to exact new positions without iteration.4. The stabilizers thus designed can effectively damp out multi—mode torsional oscillations of the system over a wide range of capacitor compensations although thestabilizers are designed for a particular degree of compensation.111-4 -4o 0b-4-0o—C.)o .C.)TilL: Shaft torque between high and low pressure turbinesTLG: Shaft torque between low pressure turbine and generator(a) Responses of Generator 1 (Gi)Figure 6.6: Responses to a Step Torque to G1 for the System without Control. (a)Time (second) Time (second)Time (second) Time (second)112Shaft torque between high and low pressure turbinesShaft torque between low pressure turbine and generator(b) Responses of Generator 2 (G2)Figure 6.6: Responses to a Step Torque to Gi for the System without Control. (b)4200z-20105.0•-505Time (second)5Time (second)0 5 0 5Time (second) Time (second)TilL:TLG:113‘4-04.000 5Time (second)0 5Time (second)z‘S0.381.1bI :Time (second)Shaft torque between high and low pressure turbinesShaft torque between low pressure turbine and generator(a) Responses of Generator 1 (Gi)Figure 6.7: Responses to a Step Torque to Gi for the System with Control. (a)Time (second)THL:TLG:51140.34a)I00Shaft torque between high and low pressure turbinesShaft torque between low pressure turbine and generator—0.335o 0.3250.320.3150Time (second)5 0 5Time (second)84.50Time (second)5 0 5THL:TLG:Time (second)(b) Responses of Generator 2 (G2)Figure 6.7: Responses to a Step Torque to Gi for the System with Control. (b)115t4-0C)0C)0.05-0.050Time (second)Time (second)(c) Excitation Control of Gi and G2Figure 6.7: Responses to a Step Torque to Gi for the System with Control. (c)50 5116Chapter 7CONCLUSIONS7.1 New Stabilizer Design Techniques DevelopedThree types of stabilizers for dynamic stability control of multimachine power systemsare developed in this thesis: the coordinated and decentralized PSS in Chapter 4, thedirect MIMO STR in Chapter 5, and the decentralized linear feedback stabilizer in Chapter 6. A number of new design techniques for these stabilizers are developed. They aresummarized as follows:1. Mathematical models for multimachine dynamic stability studies and for high ordernonlinear simulations are developed in Chapter 2 and Chapter 6.2. Participation factors for linear analysis and speed deviation indices (SDI) fromnonlinear simulations are used in Chapter 3 for the selection of number and sitesof stabilizers. The effectiveness of this method is confirmed by the PSS design inChapter 4 and the STR design in Chapter 5.3. A new pole—placement technique is developed in Chapter 4 for decentralized PowerSystem Stabilizer (PSS) design for multimachine power systems with low—frequencyoscillations. The PSS transfer functions are explicitly expressed in the final equations for the PSS parameter design. The computation required is much less thanthe existing methods.1174. A direct self—tuning regulator (STR) for a multimachine system with wide—rangechanging operating conditions is developed in Chapter 5. Clarke’s indirect STR ofGPC method is improved so that the initial step control parameters are directlyestimated and that the subsequent control parameters are recursively computed.The computational requirement of the original CPC is reduced.5. A direct pole—placement design technique for decentralized linear feedback stabilizers is developed in Chapter 6 for the stabilization of multimachine multi—modetorsional oscillations. This method is applied to the excitation control design forSystem 2 of Benchmark II of IEEE. For the design, a mathematical model for thesystem is also developed.7.2 Applications and ConclusionsBecause of the nature of different types of stabilizers for different kinds of dynamicstability problems, conclusions have been drawn for each topic at the end of each chapter.The final conclusions of applying these new stabilizer design techniques to power systemsmay be drawn as follows.1. Both participation factor method of linear analysis and the speed deviation index(SDI) based on nonlinear simulations are helpful in deciding stabilizer number required for a multimachine system and the sites of stabilizer installation. Stabilizersshould be installed on machines having relatively larger participation factors of unstable modes or on machines having relatively larger speed deviation indices. Forthe initially unstable nine-machine system, three stabilizers on machines 7, 8, and3 are sufficient to ensure the stability of the system although there are four unstablemodes and six coherent groups for the open—loop system.1182. The effectiveness of the new pole—placement PSS design technique has been demonstrated by various PSS designs of the two multimachine power systems. Exact assignment of any number of eigenvalues of low—frequency oscillating modes to newspecified locations can be achieved for all designs. Non—uniform damping factorscan be assigned to the eigenvalues to be changed. Assigning a relatively large damping factor to an unstable mechanical mode can also improve the damping of poorlydamped mechanical modes nearby through the dynamic interaction of machines.3. The principle and method of the direct MIMO STR developed are applied to theSTRs design of a nine—machine power system. Comprehensive simulation resultsshow that the STRs thus designed can effectively stabilize a power system over awide range of changing operating conditions while the stabilizers with fixed parameters may fail-to do so. Therefore, further exploration of STR design is necessaryto the benefit of power system stability control.4. The new direct pole-placement method is successfully applied to an excitation control design to damp torsional oscillations of system 2 of the SBM. Effective feedbacksignals can be found from participation factor analysis and all unstable mode eigenvalues of the system can be shifted directly to exact new positions without iteration.The stabilizers thus designed can effectively damp out multi—torsional—mode sub-synchronous oscillations of the system over a wide range of capacitor compensationsalthough the stabilizers are designed for a particular degree of compensation.7.3 Future ResearchFurther research shall be done on both analyses and applications. For example, althoughpoles can be shifted to the desired locations with the developed pole—placement method, itneeds a method to analytically decide where poles should be shifted for optimal dynamic119stability control. As for the self—tuning control of Chapter 5, the output and controlhorizons id and nu are chosen so far by users and the analytical relationship betweenthese chosen horizons and stability conditions are still unclear. The question is howto select the horizons under defined stability conditions. There are also the modelingproblems for more complex systems with HVDC, SVC, etc. Furthermore, there aremany transient stability control problems such as generator tripping, load shedding, etc.They are beyond the scope of this thesis but should be coordinated with the dynamicstability control of a power system.Although the stabilizers presented in the thesis prove very effective for dynamic stability control of power systems from computer simulations, more work remains to bedone to implement them in real power systems. These include instrumentation, dataacquisition, and communication, especially for self—tuning stabilizers.120Bibliography[1] T.J. Hammons and D.J. Winning, “Comparisons of synchronous—machine modelsin the study of the transient behaviour of electrical power systems,” Proc. TEE 118(10), pp 1442—1458, 1971.[2] L.M. Hovey and L. A. Bateman, “Speed regulation tests on hydro station supplyingan isolated load,” IEEE Trans.on PAS, pp.364—3’Tl, Oct. 1962.[3] Yao-nan Yu, W.Y. 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