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A comprehensive simulation study of the voltage stability of a large power system Zhang, Wenjie 1994

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A COMPREHENSIVE SIMULATION STUDY OFTHE VOLTAGE STABILITY OF A LARGE POWER SYSTEMByWENJIE ZHANGB. Eng., Huazhong University of Science and Technology, PRC, 1982M. Sc., Taiyuan University of Technology, PRC, 1985A THESIS SUBMITTED IN PARTiAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© Wenjie Zhang, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Electrical EngineeringThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T1W5Date:L/46ef / ) 1AbstractThe voltage stability problem has become a growing concern in power system planningand operation. Many large interconnected power systems have experienced voltage instabilities which involve fast transients and/or slow dynamics. Although load flow relatedstatic approaches have been well developed to characterize the system maximum loadinglimit as the voltage collapse point, the mechanism of how system operation approachesits voltage collapse point and how this collapse point is affected by system dynamics arestill obscure.This thesis provides the answers to these two basic questions through the investigationof effects of loads and reactive power controls on system voltage stability by detailed timedomain system simulations. The importance of system dynamics in the determination ofthe voltage stability limit is emphasized.Firstly, a multimachine power system with steam and hydro electric generating units,various types of loads, and system reactive power—related control devices is appropriatelymodeled. Secondly, a comprehensive power system simulation program is developedbased on the implicit trapezoidal rule and an integration step size control algorithm.A new variable elimination method for load flow, and a new forward—elimination andbackward—substitution procedure for solving the system Jacobian matrix equations aredevised. Different system disturbances are simulated, and the exact timing of systemchanges is implemented. Finally, a 21 bus sample power system is chosen for the voltagestability study. In the case studies, the effects of loads, control devices, and systemdisturbances on system voltage stability are thoroughly examined.The voltage instability of a power system is a very complicated phenomenon, which,11depending on the location, the type, and the severity of a system disturbance, mayinvolve a fast transient voltage instability, or a slow voltage deterioration followed by asharp collapse. It is closely associated with system reactive power—related controls, andis strongly affected by the load characteristics. The beneficial and detrimental effectsof loads and reactive power controls on voltage stability should be carefully analyzed sothat the information can be used in voltage stability control designs.111Table of ContentsAbstractList of Tables viiList of Figures viiiAcknowledgements xi1 INTRODUCTION1.1 Power System Voltage Stability Problems.1.1.1 Power System Angle Stability1.1.2 Power System Voltage Stability . .1.2 Power System Voltage Stability Studies . .1.2.1 Static Voltage Stability Studies1.2.2 Dynamic Voltage Stability Studies1.3 Proposed Thesis Study1.3.1 System Component Modeling1.3.2 Dynamic Simulation Techniques1.3.3 Dynamic Simulation of Voltage Stability .1.4 Thesis Structure2 MODELING OF POWER SYSTEM COMPONENTS2.1 Typical Power System Load Models11114466677810102.1.1 A Composite Bus Load Model 10iv2.1.22.1.32.1.411141516Induction Motor LoadVoltage Dependent PQ LoadConstant PQ Load with a Recovery Time Constant2.2 Component Models of a Generating Plant2.2.1 Synchronous Generator 162.2.2 Field Excitation System 182.2.3 Governor and Turbine Systems 192.2.4 Power System Stabilizer (PSS) 212.3 System Voltage Control Devices 222.3.1 On Load Transformer Tap Changer 222.3.2 Static VAR Compensator (SVC) 232.3.3 Generator Rotor Overheat Protection 242.4 Modeling of Transmission Network 253 SIMULATION TECHNIQUE FOR VOLTAGE STABILITY STUDIES 273.1 Complete System Equations 283.1.1 Machine and Transmission Coordinates 283.1.2 Complete System Equations 293.2 Simultaneous Implicit Integration of System Equations 333.2.1 Implicit Trapezoidal Rule 333.2.2 Discretization of System Differential Equations 353.3 Power System Simulation Program 393.3.1 Load Flow Calculation 403.3.2 Solution of System Jacobian Equations 423.3.3 Integration Step Size Control and Exact Timing 463.3.4 System Contingencies 49v3.3.5 Flowchart of System Simulation Program . .. 514 DYNAMIC VOLTAGE STABILITY STUDIES 534.1 A Sample Power System for Voltage Stability Studies 544.1.1 A 21 Bus Sample Power System 544.1.2 Basic System Data 554.2 The Critical System Load Buses 594.2.1 Analysis of the System Operating Condition . . 594.2.2 Critical System Load Buses 604.3 Effects of System Loads on Voltage Stability 634.3.1 Effect of an Induction Motor Load 644.3.2 Effect of Exponential PQ Loads 724.3.3 Effect of Persistent PQ Loads 774.4 Control Effects on System Voltage Stability 814.4.1 Effect of transformer tap changing 814.4.2 Effect of System VAR Compensation 874.4.3 Effect of Generator Rotor Overheat Protection 934.5 Maximum Loading Limit by Load Flow and Simulation 985 CONCLUSIONS AND REMARKS 1055.1 Conclusions of the Thesis 1055.2 Future Research Work 110Bibliography 111A DERIVATION OF STEADY-STATE MOTOR EQUATIONS 117viList of Tables4.1 Data of Transmission System 554.2 Data of Generator PV and Load PQ 564.3 Data of Generator Parameters 564.4 Data of Field Excitation System 574.5 Data of M—H Governor and Hydro Turbine 574.6 Data of E—H Governor and Steam Turbine . . 584.7 Data of PSS’s for Given Operating Condition . . . 584.8 System Power Generation and Consumption 594.9 Voltage Profile of the Given System Condition 594.10 System Voltage Sensitivity Matrix 624.11 Number of Buses Largely Affected by Bus j 634.12 Data of an Induction Motor Load 644.13 Adjusted System Voltage Profile 654.14 Adjusted PSS Parameters 654.15 Initial State of the Induction Motor 654.16 Data of SVC Parameters 88viiList of Figures1.1 Power—Angle and Voltage—Power Curves. 42.1 112.2 112.3 152.4 182.5 192.6 202.7 212.8 232.9 242.10 242.11 . 253.1 . . . 283.2 303.3 343.4 443.5 503.6 525464Classification of a Composite Bus LoadComposite Bus Load ModelingConstant PQ with a Delayed RecoveryA Fast Excitation SystemHydro Turbine and Governor SystemNon-Reheat Steam Turbine and Governor System . .A Power System StabilizerA Transformer with OLTCA TCR with Fixed CapacitorBlock Diagram of an SVC with Voltage ControlOverheat protection characteristicMachine and Transmission Network CoordinatesBlock Diagram of Component InteractionIllustration of Implicit Trapezoidal RuleAn Example of System Jacobian Matrix EquationStep Size Change for Exact TimingOverall Flowchart of System Simulation Program4.1 A Sample Power System under Study4.2 A System Bus with Typical Loadviii4.3 Motor Response to Step Change in Load Torque 664.44.54.64.84.94.104.114.124.134.144.154.164.174.184.194.204.214.224.234.244.254.264.274.28• . 676869• . 70717374757678787980808283848485868788899091Some Generator Bus VoltagesSystem Voltages at Some Load BusesTransient P—V and Q—V curves of Motor Terminal Bus4.7 Some Generator Rotor AnglesTransient Responses of Motor VariablesVoltage Response at a Load BusReactive Power Drawn by the MotorVoltage Response at a Generator BusRotor Angle of a GeneratorVoltage Collapse at Bus 21Load Power and Reactive Power vs. Load AdmittanceSome Other System Load Bus VoltagesSome Generator Bus VoltagesLoads at Some System BusesTap Changing Effect on Motor Terminal VoltageTap Changing Effect on Motor TorqueTap Changing Effect on Motor P and QMotor P—V Curve with and without Tap ChangingEffect of Tap Changing at Other System Load BusesMotor Bus P—V CurvesTap Changing Effect on Motor Power and Reactive Power . •Effects of Different VAR CompensationsVAR Compensation of Fixed C Switched in Different Time . .VAR Compensation of SVC’s with Different CapacitiesEffects of VAR Compensation of Fixed Capacitor and SVC . .ix4.29 VAR Compensation of Fixed Capacitor and SVC 924.30 Some Generator Excitation Voltages (without Generator ROP’s) 944.31 System Load Bus Voltages (without Generator ROP’s) . 954.32 Generator Excitation and Terminal Voltages (with ROP’s) 964.33 Effect of Generator Overheat Protections 974.34 P—V Curves of Load Bus 21 994.35 Motor Load Torque Changes in Simulation Study 1004.36 P—V Curves of Motor Terminal Bus 1014.37 Q—V Curves of Motor Terminal Bus . 1014.38 Some Generator Bus Voltages . . 1024.39 Some System Bus Loads . . 103A.1 Equivalent Circuit of An Induction Motor 119xAcknowledgementsThis thesis is dedicated to my parents, who inspired and encouraged me to take thisopportunity of challenge.I wish to express my grateful thanks to my research supervisor Dr. Y.N. Yu for hisinvaluable guidance, constant encouragement, and great patience during the researchwork and the accomplishment of this thesis.My appreciation goes to Dr. M.D. Wvong and Dr. H.W. Dommel for their readinessto help and inspiring and helpful discussions.I am also indebted for the financial support from the Natural Science and EngineeringResearch Council of Canada and the University of British Columbia.Finally and gratefully, I thank my wife, Wei, for her special thoughtfulness, encouragement, understanding, and patience throughout the difficulties of my graduate program.xiChapter 1INTRODUCTION1.1 Power System Voltage Stability Problems1.1.1 Power System Angle StabilitySince the beginning of large power system interconnections, low frequency system oscillations and transient and dynamic stability problems have developed. The main concernof these stability problems is to keep all synchronous generators in synchronous step byproviding them with adequate damping if they oscillate, reducing generation or addingdynamic braking resistance if there is a power surplus, and shedding some loads if thereis a power shortage. These problems, which may be classified as the angle stability problem, have been thoroughly studied and are well understood. Control means to stabilizethe system, such as dynamic resistance braking, force excitation, fast valving, HVDCmodulation, power system stabilizer (PSS), and generation tripping and load shedding,have been very well developed.1.1.2 Power System Voltage StabilityAs large power system interconnections continued, the power demand kept on increasing. But the environmental restriction on the building of new transmission lines alsoincreased. There is a tendency in power system planning and operation to load the existing generationand transmission equipments as much as possible. This practice, coupledwith insufficient and inadequate reactive power supplies for a power system, has caused1many system voltage failures in the past, which may or may not also involve an angleinstability.Voltage stability has gained a special attention recently [1]. To have a better understanding of this problem, some major system voltage failure events are briefly reviewedas follows.EDF — December 19, 1978 [4]It was a severe cold winter morning in France, and the temperature drop was muchgreater than anticipated. A rapid rise of power demand caused the increase in severalpower transfers. It was also marked with the increase in active power losses and especially reactive power losses. Several 400 KV lines were overloaded, and system voltagesdeteriorated very badly.Some EHV/HV tap changers were blocked, and a 5% distribution voltage drop wasordered in some area. The system was stabilized for a while. But due to the overload ofthe persistent load demand and the loss of many reactive power supports, many majortransmission lines were successively lost resulting in island operation of the entire system.EDF — January 12, 1987 [1]It was again a severe cold weather day, and the system was overloaded. Generationswere tripped one after another. At one time, the power deficiency was around 9,000 MWand the full installation capacity was 90 GW. The voltage deteriorated from 400 KV to300 KV and below but did not collapse completely. The underfrequency relay control didnot act because there was no significant frequency deterioration warranting the action.Finally, the voltage profile was restored by shedding 1500 MW of load and by trippingsome 400/225 KV transformers feeding a load area.2TEPCO — July 23, 1987 [6]Japan has a 50 Hz system in the north and a 60 Hz system in the south, and the twosystems are connected by two 300 MW frequency converters at Sakuma and Shinshinano.The Tokyo Electric Power Company, TEPCO, belongs to the 50 Hz system.It was a very hot summer day, much warmer than anticipated. After a maximumpower demand of 39.1 GW in the morning, the demand dropped to 36.5 GW at 12:40during the lunch hour. But the demand increased again rapidly at a rate of 400 MW perminute from 13:00, much faster than estimated. It was attributed to the air conditioningdevices developed by then, which drew more current fast despite voltage deterioration.The power demand at 13:10 was 39.3 GW.Although all shunt capacitances were in service, the system voltage dropped from 500KV to 460 KV at 13:15 and further to 370 KV in the western part of the system and to350 KV in the central part of the system at 13:19. Three substations of 8.168 GW weretripped, and 2.8 million customers were lost. The three substations were brought backto service from 13:23 to 13:35, and about 60% of total load loss was recovered at 13:36,80% at 14:30, 90% at 16:00 and completely recovered at 16:40.To summarize, it is observed that a voltage instability does not necessarily involve anangle instability, and that the voltage does not necessarily collapse completely. It is alsovery important that, for a comprehensive voltage stability study, special types of loads,like those in the TEPCO event, must be adequately modeled, and that the functions ofall reactive power—related generating, consuming and control components must also betaken into consideration.31.2 Power System Voltage Stability StudiesAlthough some effort has been made to clarify the mechanism of voltage instability andto devise some methods to prevent a system from voltage collapse, most researches havebeen devoted to the determination of maximum loading limit (MLL) of a power system byusing the steady state formulation for a static voltage stability analysis. In these studies,only small load variations are considered, and the system dynamics are not included.1.2.1 Static Voltage Stability StudiesWhile a power—angle curve for equal—area study has a shape of an inverse V with amaximum power point on top, the voltage—power curve for the MLL study has a V shapewith the point of maximum load power towards the right as shown in Figure 1.1.Generator Power VoltagePmLoad PowerFigure 1.1: Power—Angle and Voltage—Power CurvesIt is well known that the upper portion of the voltage—power curve represents a stableoperation whereas the lower portion represents the unstable [2] [3]. In—between exists apoint of critical voltage. Several methods have been developed to find this critical point,and a variety of indicators have been defined for the proximity of the system stableelectric powermechanical powerRotor Angle4operating state to the point of voltage collapse. All static methods are essentially relatedto a Jacobian matrix analysis from the results of a system load flow. System voltage maycollapse at the point where the load flow Jacobian becomes singular.1. Load Flow Analysis Venikov et al. [8] found in 1975 that there exists a directrelation between the singularity of the load flow Jacobian and the singularity ofthe system dynamic state Jacobian and the changes in sign of system eigenvalues.Therefore, the stability of a dynamic system may be estimated by means of loadflow. This method was expanded for voltage stability studies by Tamura et al. [9].2. Static Bifurcation Theory Bifurcation theory is concerned with the branchingof static solutions of a dynamic system with a slow change in system parameters.With this technique, Kwatny et al. did a thorough analysis of loss of steady statestability and voltage collapse [10] in 1986. With a slow change in system parameters,the system stable operation determined by load flow will move to a new equilibriumand remain stable until one of the parameters reaches a critical value at whichsystem state branches at a saddle point. This is the very point where the load flowJacobian matrix becomes singular. When multiple solutions of load flow exist, theycorrespond to the multiple equilibria of the dynamic system in the neighborhood ofthe bifurcation point. Therefore, bifurcation analysis can be used to characterizethese equilibria and to identify the critical parameters, which are the very importantinformation for system control design.3. Sensitivity Method Sensitivity in voltage to system parameters near the critical state provides very useful information to system operation. It can be used toidentify critical system buses and also effective means of controls [37] [58]. Basedon the analysis, a variety of indicators of proximity of the system state to the point5of voltage collapse can also be defined [11] [14] [15]. Therefore, adequate control canbe exerted on the system to keep the system state away from voltage collapse.1.2.2 Dynamic Voltage Stability StudiesMany power system voltage failures were triggered by large disturbances of the system.Due to the dynamic interaction among system components and the nonlinear constraints,the dynamics of a voltage instability process is rather complicated. It depends not only onthe stability of generators in the system, the type and location of system contingencies,but also on the load characteristics and system controls. Since the system maximumloading limit (MLL) based on load flow analysis may give an upper bound of the voltagestability region, the system may have lost its voltage stability before that limit can bereached due to system dynamics [16]. Therefore, the MLL can only indicate the loadingcondition at which system voltage collapse may occur. It cannot answer the questionsof how the system voltage approaches the collapse point and how this collapse point isaffected by the system dynamics. For this sake, a comprehensive system simulation mustbe resorted to so that the system dynamics can be adequately included in the voltagestability studies.1.3 Proposed Thesis Study1.3.1 System Component ModelingThe major part of this thesis is to investigate the effects of load and system reactivepower components on voltage stability. For that, the system behavior will be simulatedcomprehensively. All important loads and all system components that generate, consumeand control the reactive power of the system will be modeled in detail. Other functions,such as the generator rotor field overheat protection, which may affect the reactive power6of the system, will also be modeled. There are in general four major components to apower system, the generating plants, the transmission network, the system loads, andthe system control devices. All of these components will be modeled.1.3.2 Dynamic Simulation TechniquesA nine—machine pOwer system is chosen for the simulation study. It is felt that thesystem is large enough to display the dynamic interactions among system components,such as generators, loads, transmission network and system controls. The high ordersystem model with inherent system ilonlinearities requires development of an appropriatesimulation technique. The system equations are discretized based on the trapezoidal rule,and Newton—Raphson’s iteration method is used to carry out the solution. To avoid directinversion of a large Jacobian matrix in each integration step, a new technique of solvingthe Jacobian matrix equation is developed.1.3.3 Dynamic Simulation of Voltage StabilityThe main objective of this thesis is to investigate the gelleral effects of special typesof loads and major reactive power—related components on the dynamic behavior of thevoltage stability of a multimachine power system. Several types of loads, control devices,and system disturbances of single or double contingencies will be considered.Since the dynamics of induction motor loads [51], transformer tap changers [17],generator field excitations [37], and reactive power supply deficit [59] play a importantrole in a voltage collapse process, their effects must be examined on a large power systemwith detailed system models so that the dynamic interactions among system componentscan be included. In addition, the persistent PQ load characteristics [54] contributing tothe TEPCO voltage failure should also be investigated. For this reason, the case studiesare designed and carried out to clarify the effects on voltage stability of induction motor7stalling, transformer tap changing, fixed capacitor and SVC compensations, persistentand general static loads and generator overheat protections.1.4 Thesis StructureIn Chapter 2, the power system component models are presented. System bus loads aremodeled by the combination of typical loads. Load dynamics are represented by inductionmotor loads and constant power and reactive power (PQ) loads with delayed recovery.Conventional static loads are modeled as voltage dependent loads. A generating plant ismodeled in detail, including synchronous generator, field excitation system, governor andturbine system, and power system stabilizer. System controls, which have strong effectson system voltage stability, such as on load tap changer of a transformer(OLTC), thestatic VAR compensator(SVC), and generator rotor overheat protection(ROP) are alsomodeled. Finally, the transmission network is represented by the system node voltageequations.Chapter 3 describes the nonlinear time domain simulation technique. The overall system equations can be obtained through a hybrid coordinate system for both generatorsand the transmission system. Based on the trapezoidal rule, system differential equations are discretized to obtain the corresponding difference equations. Newton—Raphson’smethod is then used to solve the system equations. A systematic method is developedto solve the high dimension Jacobian matrix equation without involving a direct inversion of a large matrix. In order to capture both fast transients and slow dynamics of avoltage instability, a variable step size mechanism is implemented. System disturbancesand sudden topological changes are considered and exactly timed.In Chapter 4, a 21 bus power system is presented. Since the system has low frequencyoscillations for the given operating conditions, power system stabilizers are designed.8Next, the most voltage sensitive bus of the system is identified through system voltagesensitivity allalysis. Typical system loads and control actions are considered for the mostvoltage sensitive bus. Case studies are then conducted to clarify the effects of variousloads and controls on system voltage stability. Finally, the results of voltage stabilityfrom load flow and simulation studies are compared to demonstrate the effect of systemdynamics on the system maximum loading limit.Finally, conclusions are drawn and future research projects are suggested in Chapter6.9Chapter 2MODELING OF POWER SYSTEM COMPONENTSFor voltage stability studies, an appropriate power system model is required. Althougha static model, like load flow equations, is adequate for steady state analysis, a moredetailed model, including both load dynamics and reactive power generating and controlcomponents, must be used in a voltage stability study of a system involving dynamics. This is because the voltage instability phenomenon of a power system may involveboth fast transients and slow dynamics [13]. Therefore, all system components involvingtransients and dynamics should be included in the voltage stability study.Typical power system load models are presented in section 2.1, generating plant insection 2.2, system control devices in section 2.3, and, finally, transmission network equations in section 2.4.2.1 Typical Power System Load Models2.1.1 A Composite Bus Load ModelExaminations of major voltage failures show that system loads have significant effects onthe voltage stability of a power system. But, a power system load is usually made upof numerous individual loads with different characteristics, and the information aboutsome individual loads may not be available [18][19]. Therefore, it is almost impossibleto derive an exact model for a power system load. Instead, the power system loadsmay be approximately represented by a few equivalents, e.g., industrial, commercial, and10residential loads [20] as shown in Figure 2.1.load busindustrial loadcommercial loadresidential loadFigure 2.1: Classification of a Composite Bus LoadIt is suggested in [21], for example, that a composite system bus load may be modeledby an equivalent induction motor in parallel with a static load as shown in Figure 2.2,load busdynamic loadstatic loadFigure 2.2: Composite Bus Load Modelingwhere the equivalent induction motor represents the dynamics associated with the majorindustrial load, while the static load represents the voltage dependence of commercialand residential loads. Therefore, the load effect on system performance may be found bystudying the following typical loads.2.1.2 Induction Motor LoadInduction motors constitute the major part of an industrial load. They have a fastresponse to system disturbances to maintain more or less a constant power and drawmore reactive power from the power supply. This feature of quick load pick—up and moresystem bussystem bus11reactive power absorption during a system disturbance is one of the major causes of adynamic voltage instability [22].The basic equations of a three—phase induction motor may be derived from Park’sequations for a synchronous generator. However, there is no field winding, and the dand q axis windings are symmetrical for the motor. There is a slip of rotor windingswith respect to the stator rotating field of the motor. Neglecting the electromagnetictransients (EMT’s) in the stator windings, the induction motor may be described by athird order model as follows.(a) Rotor motion equation:2HTL—TE (2.1)(b) Rotor winding voltage equations:Tè1 = —e—(xo—x’)Iqs+bTse (2.2)T0 eq —eq + (x0 — x ) ‘ds — b T0 S ed(c) Stator winding voltage equations:Vd8 = e+rsIds—xIqs (23)= eq+xIds+rslqs(d) Electromagnetic torque equation:TE = edldS + eIqs (2.4)12whereH : motor inertia constantr : stator resistancerotor open circuit reactanceblocked rotor reactanceT : rotor open circuit time constante, e : rotor internal voltagesVcjs, Vqs : stator terminal voltages‘ds,’qs : stator currentss : motor slipTL : mechanical load torqueTE : electromagnetic torquewb : synchronous speed of the systemIn steady state system condition, both mechanical and electrical transients of the motor are not included. Hence, equations (2.1) and (2.2) are reduced to algebraic equationsto represent the steady state behavior of the motor. In that case, the motor torque,power, and reactive power can be determined byTE fE(s,p)V2Fm = fp(s,p)V2 (25)Qm = fQ(s,p)V2V =where fE, fp and f are functions of motor slip s and motor parameters p. The derivationof equation (2.5) is given in Appendix A.132.1.3 Voltage Dependent PQ LoadSince a voltage instability does not necessarily involve system frequency deterioration,the frequency dependence of commercial and residential loads may be neglected. Thepower and reactive power drawn from a system bus may then be described as a functionof bus voltage. It has been a common practice that a bus load is divided into a constantpower, a constant current, and a collstant impedance load components. This conceptleads directly to a load model of quadratic form [23].P = Po [pp + pi () + Pz (2.6)Q = Qo[qp+qi()+qz()]where Po and Qo are power and reactive power at normal operating voltage Vo, pr,, p, andPz are the coefficients of power portion of constant power, constant current, and constantimpedance loads, and qp, qj and q are those of the reactive power portion of the load.These coefficients must satisfyPp+Pi+Pz = 1qp+qi+qz = 1A more general static PQ load model [24] has exponential forms as follows.F Po()VQ (2.7)Q Qo()Here, again, P0 and Qo are power and reactive power at normal operating voltage V0, anda and characterize the voltage dependence of the load. Equation (2.7) also includesthree special cases, that is, the constant power load with a = = 0, the constant currentload with a = = 1, and the constant impedance load with a = = 2.The appropriate selection of a combination of pr,, Pi, Pz, qp, qi, and q in equation(2.6), and the exponents of a and 3 in equation (2.7) should be made by investigatingthe actual system load.142.1.4 Constant PQ Load with a Recovery Time ConstantThis type of load demands a constant power throllgh self control to maintain the requiredpower despite a system voltage decrease. A typical load power response to a step changein voltage may be in the form as shown in Figure 2.3.(a) voltage step changePo(b) load power responseFigure 2.3: Constant PQ with a Delayed RecoveryThe sudden dip in voltage causes an instant decrease in load power demand followedby a recovery to its normal value P0. The demand is persisting, but involving a timedelay. There is a such kind of load of modern air conditioning device as reported inTEPCO power failure [6]. This type of constant power load, which may be referred toas persistent PQ load, may be modeled by a changing load equivalent admittance witha time delay. The time delay is associated with the time needed for the control devicesto respond, and may also involve a human factor [25].Assuming an exponential recovery in both power and reactive power, this type of PQ15load may be modeled as follows.TGGL = Po—GLV2 (2.8)TBBL = Qo—BLV2where V is the load terminal voltage; GL and BL are the varying equivalent load conductance and susceptance; TG and TB are the corresponding time constants; and P0 andQ are the load power and reactive power at normal operating condition withD — i—I v-210—UIL0V0V (2.9)— 0 i72—2.2 Component Models of a Generating Plant VA generating plant consists of a synchronous generator, a field excitation system, a governor controlled turbine system, and probably a supplementary control, such as a powersystem stabilizer. These components contribute the major dynamics of a power system,which should be adequately modeled in a dynamic system study.2.2.1 Synchronous GeneratorNeglecting electromagnetic transients in armature windings, a synchronous generatormay be represented by a fifth order model as follows.(a) Rotor motion equations:M = TmTeDW (2.10)6 wb(w—1.0)16(b) Rotor winding voltage equations:T0E = —E—(xd—x)Id+EFDTE’ = -E’- (x-x)Id+E +TE (2.11)TE E+(xq_x)Iq(c) Armature winding voltage equations:Vd = E—raId+xIq (2.12)Vr = E’—xId—raIq(d) Electromagnetic torque equation:Te EId + EIq + (x — X)IdIq (2.13)The voltages and currents are described in individual machine d—q coordinates.In the foregoing equations,M : inertia constantD : mechanical damping coefficientw : rotor speed6 : rotor angleTm, Te : mechanical and electromagnetic torquesra : armature resistancecl—axis transient reactancex, x : d— and q—axis sub—transient reactancesT0 : d—axis transient time constantT,, T : d— and q—axis sub—transient time constant17q—axis transient voltaged— q—axis sub—transient voltagesfield voltaged and q components of terminal voltaged and q components of armature currentsynchronous speed of the systemwhere S is in radian, M and time constants are in seconds, and all other variables are inper unit.2.2.2 Field Excitation SystemA fast—response exciter and voltage regulator system shown in Figure 2.4 is chosen forthe study.It may be described by the differential equationTA EFD = —EFD + EFDO + KA (UE + VREF— 4) (2.14)where KA and TA are respectively the equivalent gain and time constant of the excitationsystem. 4 and VREF are the generator terminal voltage and its reference, respectively.E’, E’EFDVd,Id, ‘qVt EFDOUEEFDFigure 2.4: A Fast Excitation System18UE is the supplementary excitation control. EFDO is the initial value of excitation voltageEFD.The excitation voltage has physical limits, which may be called the first limitEmini Eyjj EmaiIt also has a lower operating limits (Emin2, Emax2), which may be called the second limit,determined from the consideration of the generator rotor overheat protection. When agenerator has been excited continuously over the second limit for a prescribed period oftime, relay protection will cramp the excitation voltage to the second limit, or even tripoff the generator.2.2.3 Governor and Turbine Systems(a) Mechanical—Hydraulic Governor and Hydro Turbine SystemThe block diagram of a mechanical—hydraulic governor and turbine system for ahydro—electric plant is shown in Figure 2.5.GoThe corresponding differential equations areTG1TrG27gG30.5 T Tm(00)REFGomaxFigure 2.5: Hydro Turbine and Governor System= —uG1+UG+wREF—..’—G2= G2+6TTrG1= -G3+1= Tm+CgG+G0wG(2.15)19subject to the governor speed and opening constraints byGamin 03 <GmaxG0min (GgG3 + G0) G0maxwhere1/oT, Tr, Tg, TG1,G2G3:WREFUG:overall gain of speed governortransient regulation constanttime constants of actuator, dashpot, servomotor,and water, respectivelyoutputs of servo actuator and dashpot, respectivelygate incremental openinginitial gate openingspeed referencesupplementary governor controlThe corresponding differential equations areTsmG = G+Kg(UG+WREFW)TGHTm = Tm+C9G+G0(b) Electrical-Hydraulic Governor and Steam Turbine SystemThe block diagram of an electrical—hydraulic governor and non—reheat steam turbinefor an steam electric plant is shown in Figure 2.6.COGominFigure 2.6: Non-Reheat Steam Turbine and Governor System(2.16)20subject to governor speed and opening constraintsGamin C < GmaxG0min <(cG+G0)<G0mawhereKg : overall gain of speed governorTsm : servo motor time constantG : valve incremental openingTCH : steam chest time constantIn a multi—machine power system, the power output of a generator is expressed on thesystem base, while the governor output is usually expressed on the individual generatorbase with a full load output as 1 per unit. As a result, an interfacing factor 0g must beintroduced as shown in the Figures.2.2.4 Power System Stabilizer (PSS)For a voltage stability study, the low frequency system oscillation phenomena must beisolated by power system stabilizer applications. A power system stabilizer with twocompensation components and one reset block is shown in Figure 2.7.(0WF sT sifl 1÷sTi S2 1+sTi UE1+sT [J 1+sT2s 1+ST2sFigure 2.7: A Power System Stabilizer21The differential equations for the P55 areT3S1 = —81T8thT28S = S2+KT18S (2.17)T23 LIE = — UE +82+ T18 S2where K is the overall gain, T3 is the time constant of the reset block, and T18 and T28are those of the compensation blocks.2.3 System Voltage Control DevicesObviously, system voltage controls have significant effects on system voltage behavior.These control devices may include synchronous condensers, transformer tap changers,and static VAR compensators. In addition, a generator rotor overheat protection maylimit the excitation voltage and hence the reactive power output of the generator, whichmay result in a loss of system voltage control ability. Therefore, It may also be consideredas a voltage control device.2.3.1 On Load Transformer Tap ChangerDistribution transformers are usually equipped with on—load tap changers (OLTC’s). AnOLTC controlled load bus is shown in Figure 2.8. The secondary voltage V is controlledthrough the change of the transformer ratio a which has limits.a = (V5) (2.18)amim aSince a transformer has only limited number of taps, the ratio a must be changed insteps. In addition, time is required for completion of each tap changing. Therefore, thecontrol function may be modeled in discrete way with time delay. That is, for voltage22VloadFigure 2.8: A Transformer with OLTCcontrolled bus i with normal operating voltage °, a tap step of A.aj, and a prescribedvoltage tolerance , the tap changing function may be described bya(k+1) (2.19)where it is a sign function as follows.1 ifV—°<—0 if —V° <—1 if—°>2.3.2 Static VAR Compensator (SVC)Static VAR compensators (SVC’s) have a significant influence on the system voltagebehavior. A thyristor—controlled reactor (TCR) compensator with fixed shunt capacitorsis shown in Figure 2.9 with a block diagram of SVC control circuit in Figure 2.10.The differential equations for the SVC system areTbB1 = —B1+Kb(VREF—4)T2bB —B2 + T1bB + B1 (2.20)BL = B2+BLO.23system buswith constraintsFigure 2.9: A TCR with Fixed CapacitorBsFigure 2.10: Block Diagram of an SVC with Voltage ControlB2min < B < B2maxwhere K,, and Tb are the gail and time constant of voltage regulator. T and T2b aretime constants associated with the thyristor firing system.2.3.3 Generator Rotor Overheat ProtectionA generator rotor overheat protection may be approximately modeled by an excitationreduction of the generator which has been excited continuously over its operating limitor the second excitation limit Emax2 (see in Figure 2.4) for a certain period of time. TheinputVREFB2B2min Bc24timing of excitation reduction depends on the accumulated heat and temperature riseof a field winding dile to actual excitation at a level over Ema2 as in Figure 2.11. Forinstance,Em:1Emax2I TimetxFigllre 2.11: Overheat protection characteristicif the excitation voltage EFD is higher than E continuously over a time period of t,the overheat protection will cramp the excitation voltage immediately to its second limitEmax2.2.4 Modeling of Transmission NetworkNeglecting the electromagnetic transients in the transmission system, the network equation, which describes the relationship between bus voltage and current injection, maybe expressed algebraically with network admittance matrix YN in the following phasorform,[IBUS] = [YN] [VBUS] (2.21)or alternatively in X—Y real number coordinates as[I,y] = [YGBI [Vx,y] (2.22)25In equation (2.22), [V.x,y] and [Ix,y] respectively denote the system bus voltages andcurrent injections in X and Y coordinates as[Vx,y] [Vx1,Vy1,Vx2,Vy2,”,VxN,VyN] T[Ix,yj = [Ix1,Iy1,Ix2,Iy2,”,IxN,IyN] Tand the corresponding matrix YGB has the form of[Y11] [Y12} [Y1][Y21] [Y22]... [Y2N][YGB] = . .[YN1] [YN2] ... [YNN]with-B[)1] =B1where N denotes the total system bus number, and B1 are respectively the real andthe imaginary components of Nj•Hence, the network injection current at system bus i can be expressed as‘Xi = Zi (GVx — BjVyj) (2.23)Ii = Z=1(BijVxj+GijVyj)Power system component models are presented in this chapter. While synchronousgenerators are described in individual machine’s d—q coordinates, other components areexpressed in a coordinate system which rotates at synchronous speed. Coordinate systemtransformation and the overall system presentation are given in Chapter 3.26Chapter 3SIMULATION TECHNIQUE FOR VOLTAGE STABILITY STUDIESThe voltage instability phenomenon of a power system is far more complicated than thatof a conventional transient and dynamic angle instability of the system. They differ innature. Voltage stability depends not only on the stability of synchronous generators, butalso largely on the load characteristics and power and reactive power control dynamicsof the system. For different system operating conditions, voltage instability may involvea fast voltage drop or a slow voltage decline followed by a voltage collapse. In otherwords, a heavily loaded power system may experience either a slow or a fast systemvoltage instability, which depends on the type, the location and the severity of systemdisturbances, the load characteristics and inadequate system controls [13].The stability of a power system can be best assessed by the time responses of systemstate variables to different system disturbances. This requires the solutions of the systemequations which usually constitute a very complicated high order system with manyinherent nonlinearities. Therefore, computer simulation methods must be resorted. Aspart of the thesis work, a comprehensive system simulation program is developed.Following a discussion of coordinate transformation of machines and transmissionnetwork, a complete set of system equations are presented in section 3.1. A simultaneousimplicit integration method based on trapezoidal rule is described in section 3.2. Finally,a flowchart of the system simulation program with some detailed discussions are presentedin section 3.3.273.1 Complete System Equations3.1.1 Machine and Transmission CoordinatesSiflSj COSkCOSSk Sill6kIn Chapter 2, synchronous machines are described in individual machine’s d and q coordinates, while transmission network was initially described in static coordinates but maybe deemed as synchronously rotating X—Y coordinates. To interface the machines withthe transmission network at machine terminals, coordinate transformations of terminalcurrents and voltages are necessary. The relationship between the kth synchronous machine coordinates dk—qk and the common network coordinates X—Y is shown as in Figure3.1.xY/dkFigure 3.1: Machine and Transmission Network CoordinatesTherefore, the voltage and current transformations of the kth generator and the ithtransmission bus may be written as follows.Vdk vxivYi(3.1)28and‘dk SinSk —COS6k ‘Xi (3.2)‘qk COS6k sin6k ‘Yiwhere 6k is the rotor angle of the kth generator.To save the computation of coordinate transformation, a hybrid coordinate systemis preferred [39], by which synchronous machine terminal voltages are transformed intothe common network coordinates, while machine currents remain in individual d and qcoordinates. As a result, for the kth synchronous generator connected to the ith systembus, the generator armature winding voltage equation (2.12) and the network injectioncurrent equation at the ith system bus (2.23) becomesink —cos6k VX = — rak 1qk -[dk (3.3)cos6k sin6k Vy Xdk rak ‘qkandsin5k cosSk ‘dk = Z_1(GijVxj—BijVyj) (3.4)CO56k ‘qkEquation (3.4) shows that the network injection current at a generator bus has beentransformed into the corresponding machine coordinates.3.1.2 Complete System EquationsThe overall internal system behavior is the result of interactions among system components. For a power system with Ng generating plants, N1 system bus loads, and Nsystem control devices, a block diagram of component interaction of the power systemmay be shown as in Figure 3.2.With this block diagram and the coordinate system, equations (3.3) and (3.4), thecomplete system equations can be derived by aggregating all system component equations29bus load I(i=1,2 . Ni)Figure 3.2: Block Diagram of Component Interactiondescribed in Chapter 2, which can be organized into a set of differential and algebraicequations as follows.F(X,X,Y,D) = 0G(X,Y,D) = 0(3.5)(3.6)where X is the vector of system differeiltial variables and Y the vector of system algebraic variables of system voltage and current and their related variables such as machineelectromagnetic torque. X and Y together constitute the system state variable vector,which may be subjected to certain operating constraints. D is the parameter vector ofexternal system changes, such as the changes of system references, load variations, orthe system contingencies. F and G are vector functions which depend on the systemcomponent models and parameters and subject to change due to the changes in systemtopology and/or system operating conditions.The state variables X and Y and their possible constraints of system equations obtained in Chapter 2 may be summarized as follows.generating plant i(i=1,2 Ng)control device i(i=1,2 Nc)301. Typical system loads:(a) Induction motor loadx = [s e e]Ty [Vd Vis ‘ds Iqs TEIT(b) Voltage dependent PQ loady=[Vx Vy Ix Iy]T(c) Persistent PQ loadx=[GL BL]T2. Generating plant:(a) Synchronous generatorx = [w 6 E E1’ E’]Ty = [Vd ‘d ‘q T8]T(b) Field excitation systemx = [EFD]Emini EFD Emaxi(c) Mechanical—hydraulic governor and hydro turbineX[G1G23Tm]Gamin G3 < GmaxG0min (GGG3 + Go) <G0max31(d) Electrical—hydraulic governor and steam turbineX[GTm]TGamin G < GmaxG0mn < (CGG + G0) <G0mar(e) Power system stabilizerx=[Si S2 UE]TUEmjm < UE < UEmax3. System control device(a) On load transformer tap changery=[a]amin a < amax(b) Static VAR compensatorxi[B1 B2]TB2mim < B2 < B2m4. Transmission networky={Vx Vy ‘Xi 11T i=1,2,••,N323.2 Simultaneous Implicit Integration of System Equations3.2.1 Implicit Trapezoidal RuleFor accuracy and numerical stability, simultaneous implicit integration methods are often used to solve the system differential and algebraic equations like (3.5) and (3.6). Inthis method, both differential and algebraic equations are replaced by finite differenceequations through discretizations, and then solved simultaneously by Newton-Raphon’smethod. The simultaneous implicit integration method may be implemented with different algorithms depending on the accuracy of the difference equations used to approximatethe corresponding differential equations. In general, the higher the discretization orderis, the better the accuracy of the result will be, but a more complex algorithm will beinvolved.For power system stability studies, it has been suggested that low order integration would be best in both efficiency and stability [40]. Implicit trapezoidal integrationmethod would then be the most suitable candidate for power system stability study.Firstly, as a single step discretization method, the trapezoidal rule is easy to implement.Secondly, and more importantly, the trapezoidal rule has an order of two and has an essential characteristic of symmetrical A—stability. The latter means that if the differenceequation is symmetrically A—stable, it demonstrates the same stability results as thosedetermined by the solution of the original differential equation [40].The basic idea of implicit trapezoidal rule may be illustrated by Figure 3.3. Tointegrate the differential equation= f@,t) (3.7)xQt) =with known x, the solution of x at t,, the solution of x at tn+1 = t + or can33f(x,t)f(Xn+i tn÷i)cr dAtn -‘tn tn+1Figure 3.3: Illustration of Implicit Trapezoidal Rulebe obtained by integrating equation (3.7) from t to ti as follows.rtn+1Xn+l = Xn + J f(x, t) dt (3.8)tnAs shown in Figure 3.3, the shaded area corresponds to the definite integral in equation(3.8). If the At is small enough, the arc between points a and b may be replaced bythe dashed line ab, and the shaded area may then be approximated by the area of thetrapezoid abed, which results inf(x,t)dt = [f(x,t) + f(x1,t)]+ eLT (3.9)where eLT is the local truncation error introduced in the integration interval (ta, t+1),which can be estimated [42] aseLT = (3.10)for some E (t,t+1).34Equation (3.8) then becomesItnzn+i x + + f(n+i,tn±i)] + eLT (3.11)The evaluation of eLT wilibe presented in section 3.3.3.The trapezoidal integration algorithm for the differential equation (3.7) is obtainedby dropping the error term in equation (3.11).x + ——[f(xn,tn) + f(n+i,tn+i)] (3.12)With x and f(n, tn) known at t = t, a difference equation constant, z, can be definedas— X + f(n,tn) (3.13)Equation (3.12) can then be written in the following standard form.Xn+l = n + f(n+i,tn+i) (3.14)This is the discretization algorithm of implicit trapezoidal integration.3.2.2 Discretization of System Differential EquationsWith the implicit trapezoidal integration rule of equation (3.14), all differential equationsof system components described in Chapter 2 can be discretized and summarized asfollows. For convenience, define h = At / 2.1. Typical system loads:(a) Induction motor load2Hs (i+t) = Zm1t + h(TL —Te(t+) = z + h(—e — (x0 — x’)Iqs +WbTo8eqfiQ+t) (3.15)Zm3t + h[—e + (x0—x’)Ids— wbTOsed](+t)35with the difference equation constantsZmlt = 2Ht+h(TL—TE)tZm2 = T et + h [—e — (x0 — x’) Iqs + Wb T e] tZm3t = + h[—e + (x0 — x’)Id —bTsefit(b) Persistent PQ loadTGGL(t+t) = Zit + h(P0 — (3.16)TBBL(t+t) Zc2t + h(Q0 — BLV2)(+t)with the difference equation constantsZit = TGGLj+h(Po—GLV2)tZc2t = TBBLt+h(Qo—BLVt2. Generating plant:(a) Synchronous generator= Zglt + h(Tm — Te — D) (t+t)= z9 + hwb(w —Zg3t + h[_E — (xd — x)Id + EFDfi(t+t) (3.17)I II= Zg4t + h [_Eq— (xd — Xd) ‘d +Zg5t + h[_E + (xq — X)Iq1(t+t)Zg1t = MLt+h(TmTeDLL))tzq2 =Zg3t =0Et+h[E(xdx)Id+EFD]tZg4ft = T(E’ E+h[E’ - (x-x)Id+E]ItII II IIZg5t = TqoEdt + h[—E + (xq — xq)IqHtMw(t+t)T0E (t+t)T(E’-TE (t+t)with the difference equation constants36(b) Field excitation systemTAEFD(t+t) ZeIt + h[EFD + EFDO + KA (UE + VREF— )](t+t) (3.18)with the difference equation constantZet = TAEFDt + h[EFD + EFD0 + KA(UE + VREF(c) Mechanical—hydrailhic governor and hydro turbineT Gl+t) = Zh1 + h [—u G1 + UG + REF — — G2](t+)(TrG2—6TTrGl)(t+Lt) = Zh2t —(3.19)T9G3(t+t) — ZJ + h (—G3 + Gl)(t+t)(O.5TwTm +TG3)(t+t) — Zh4t+h(Tm +CgG3+Go)(t+)with the difference equation constantsZh1t = TG1+ h[—aGi + UG +WREF —Zh2 = (TrGT rGl)thGZh3t TgGt+h(+Gi)tZh4t = (O.5TwTm + TG3)+ h(Tm + CgG3 + Go)(d) Electrical—hydraulic governor and steam turbineTsm G(t+t) = Z81 + h [G + Kg(UG + REF — w)fi(tt) (3.20)TCHTm(t+t) = Zs2t + h(Tm + CgG+ Go)(t+t)with the difference equation constantsZsit = TsmGt+h[G+Kg(UG+wREFLi.))fitZs2t = TcHTmt+h(Tm+GgG+Go)t37(e) Power system stabilizerT8(S1 +‘)(t+t) = — hSl(t+t)(T283— KTl3Sl)(t+t) Zp2t H- h(—S2 + KCSl)(t+t) (3.21)(T28UE TlSS2)(t+t) = Zp3t+ h(UE+S) +t)with the difference equation constantsz1 = T8(Si+w)t—hSitz2 =z3 = (TUE—Tl)+h(—UE+St3. Control devices(a) Static VAR compensatorTbBl(t+t) = z1 + h[—B1 + Kb(VREF— (322)(T2bB — TlbBl)(+) = z2 + h [—B2 + Blfi(t+t)with the difference equation constantsz1 = TbBlt + h[—B1 + Kb(VREF—z2 = (T2bB — TlbBl)t + h[—B2 + BifitIn the above difference equations, the difference equation constants, z’s, are known attime t and the values of state variables (x, y) at time (t H- t) are to be solved.Finally, with some manipulations of equations (3.15) — (3.22), the overall systemdifference equations may be written in a compact form as= Z + D) (3.23)where /t is the integration step size and D is determined at time t and assumed notto change during (t, t H- z\t). With X, Y, D and H(X,Y, D) known at time t, thedifference equation constant vector Z is defined asZ X + D) (3.24)38Therefore, for a system solution (Xt , Yt) at time t and a chosen integration step sizeAt, equation (3.23) together with discretized system algebraic equation (3.6) forms thesimultaneous implicit integration algorithm based on the trapezoidal rule for the systemequations (3.5) and (3.6)——= 0 (3.25)= 0 (3.26)These are nonlinear algebraic equations and can be solved for Y+) by Newton—Raphson’s method.3.3 Power System Simulation ProgramBased on the simultaneous implicit integration method described in the previous section,a comprehensive power system simulation program is developed in this section for systemvoltage stability studies. The program consists of the following major functions.1. System data input and pre—processing2. Load flow and initial system condition3. Selection of system contingencies4. System state monitoring, recording and control logics5. Topology change and new initial system values6. System equation integration and variable constraints7. Step size control and exact timing8. System data output39The program is very involved since it has to deal with various types of load andreactive power component models, nonlinear constraints and logics, such as generatorrotor overheat protection, etc.. Some details of the simulation program are presented asfollows.3.3.1 Load Flow CalculationFor the dynamic system simulation studies of a power system, a load flow is requiredto determine the initial steady state values of the system. For that, the initial systemgeneration and loading conditions must be specified. With a load flow, all initial systemstate values (X0Y)can be determned.In a load flow, generator buses are usually specified as PV buses, while load busesas PQ buses. However, the power and reactiv power drawn by a dynamic load may bea function of its terminal voltage and other state variables which can not be specifieda priori. For example, the power and reactive power drawn by an induction motor insteady state are determined by equation (2.5), which are functions of not only motorterminal voltage V but also the motor slip s. In order to determine load power andreactive power, the steady state equations of the load must be included in load flow.Since most loads are also nonlinear, their effects must be reflected in the following loadflow Jacobian equation.P(e,v)= [JLFI (3.27)Q(ø,V)where P and zQ are power and reactive power mismatch functions, e and V arerespectively the angle and magnitude of system bus voltage, and LF is the correspondingload flow Jacobian matrix.There are two different methods in dealing with the load flow equations with nonlinearloads, the sequential method and the unified method [38]. By the former, the steady state40equations of the load are solved separately in each iteration. The nonlinear effect of theload is included only in the power mismatch equations, but not in the system Jacobianmatrix, which may reslut in possible convergence problems. By the united method, theidea is to add the load variables, such as motor slip, into the solution vector. It amountsto an extension of a very large load flow Jacobian matrix, which is obviously inconvenient.In this thesis, a new method is devised to overcome these difficulties, all extra loadvariables, such as motor slip, are eliminated from the load equations first, and then bothpower and reactive power drawn by the load and their derivatives with respect to loadterminal voltage are computed, which may be done numerically if necessary. The loadnonlinearity can then be included in both power mismatch equations and in load flowJacobian matrix without variable extensions.For a static load, the load power and reactive power may be directly expressed asfunctions of the terminal voltage, such as equations (2.6) and (2.7). The calculationsof load power, reactive power and their derivatives with respect to terminal voltage arestraightforward.For a dynamic load, variable elimination may be involved. For example, the steadystate operating condition of an induction motor load is determined by the equations (2.5)which is rewritten as follows.TL=TE = fE(s,p)V2 (3.28)Fm = fp(s,p)V2 (3.29)Qm = fQ(s,p)V2 (3.30)With TL specified, the motor slip s can be eliminated by solving it from equation(3.28) and substituting the result into equations (3.29) and (3.30). Thus, the power Fmand reactive power Qm determined by the terminal voltage V can be included in thepower mismatch equations. The derivatives of power and reactive power with respect to41voltage can be computed by= dfp(:, p)v2 + 2Vfp(s, p) (3.31)dQm = dfQ(s,p)v2 + 2VfQ(s,p) (3.32)where can be obtained by differentiating equation (3.28) with respect to V as= df(:,p) V2 + 2VfE(s,p) (3.33)or=_2fE(s,p)/(d V) (3.34)Therefore, with load flow Jacobian matrix modified by equations (3.31) and (3.32), thenonlinearity of the load has beell fully included in the load flow.3.3.2 Solution of System Jacobian EquationsThe .major task of solving the nonlinear system equations (3.25) aild (3.26) by NewtonRaphson’s method is to solve the corresponding system Jacobian matrix equation= (3.35)where zSR is the function residue vector, S is the system state variable deviation vectorand J the system Jacobian matrix. R has the form of R [hF hIlT with hF forthe function residue vector of all system components but the system network equations,and hI for those of system network equations. hS has the form of hS = [hX hV]Twith hX for the vector of all system non—terminal—voltage variable deviations and hVfor the system terminal—voltage deviations.For a multi—machine power system with nonlinear load and control dynamics, theorder of Jacobian matrix equation (3.35) is usually very high. For instance, it has more42than two hundred variables for a nine machine power system used in this thesis project.Therefore, the direct formulation and solution of the Jacobian matrix equation are bothheavy memory demanding and very time consuming.Although the order of the Jacobian matrix may be reduced by eliminating somesystem variables from equation (3.25) using the existing relations, this amounts to havea new model for the original system. It will be very complicated in dealing with originalcomponent model changes during integration.To save the computation in solving the Jacobian matrix equation (3.35) yet to keepthe original system component models, a new and systematic method is developed inthis thesis. It consists of two steps, a forward elimination and a backward substitution.In the forward elimination step, all system non—terminal—voltage variable deviations areeliminated and the result is used to modify the sub—Jacobian matrix corresponding to thetransmission network equations. All system terminal—voltage deviations are then solved.In the backward substitution step, the terminal-voltage deviations are back substitutedand all non—terminal—voltage variable deviations are then obtained. Both forward elimination and backward substitution are designed in such a way that the system componentsare arranged in a definite order and processed systematically one after another.In this method, the system buses are ordered in such a way that the N9 generator busescome first, followed by N. induction motor buses, and then N3 SVC buses, followed byN1 nonlinear load buses. Any other different types of components can be easily groupedand added in similar way. As an illustration example, a power system with Ng 2generator buses and Nm = 2 motor buses is considered. The system Jacobian matrixequation may then have the form as in Figure 3.4.43AFgj AX,AF2tFmiAFm2Mgi1Ig2MmiMmiFigure 3.4: An Example of System Jacobian Matrix Equationwhere denotes the non—terminal—voltage variable deviation vector for the ith generator, and L\Xmi that for the ith motor. AV and AV are the corresponding terminal—voltage deviations vectors. While /F is the residue function vector of generator or motorequations, /I is residue function vector of the network equations. Finally, A, B, C,and J are corresponding sub—Jacobian matrices.Forward EliminationReferring to Figure 3.4, the residue function vector AF of the ith system component andthe residue current vector AI of the corresponding network equations may be written asAZX + Bz\14 (3.36)= C /X + Jc V (3.37)where is the ith row sub—matrix of J and V is the vector of all system bus terminalvolt age deviations, including44AXm1l3Xm2AVgiAVg2AVm1tVm,Next, the ith component of the non—terminal—voltage deviation vector, can beeliminated by solving it from equation (3.36) and substituting the result into equation(3.37) resulting inzI = JiV — CA’BzXV (3.38)where=—With equation (3.38), is then modified by subtracting G A’B from the ith elementofFinally, when all system components have been processed one by one in the same wayas described above, the sub—Jacobian matrix J, is modified and all system bus voltagedeviations can then be solved through the following matrix equation, which finishes theforward elimination process.= JiV (3.39)where zI’ is the vector with A1 as its elements, and J is the modified matrix of sub—Jacobian matrix J.Backward SubstitutionOnce the system bus voltage deviation V is obtained through forward elimination, thenon—terminal—voltage variable deviation of the ith system component /X can be solvedby substituting Ls4 back into the component equation (3.36), which givesZXX, = A’(zF - BJ’) (3.40)When all system components have been processed one by one in the same substitutionprocedure, the whole system Jacobian matrix equation (3.35) is finally solved.45With this forward elimination and backward substitution procedure, the system Jacobian matrix equation can be solved systematically and efficiently. There are three advantages. Firstly, since the form of all system component equations remains unchanged,the tedious work involved in equation reductions is then avoided. As a result, it is verystraightforward in dealing with changes in component equations. In addition, since allsystem components are handled in the same way, it is very easy to add or remove components to or from the system. Secondly, since system components are processed one ata time, the highest order of the matrix involved in the forward elimination and backwardsubstitution is the highest order among J and A (i = 1, 2, , N) where N is the totalnumber of system components. To reduce the order of the sub-Jacobian matrix J, thesystem buses with constant impedance loads and fixed capacitor ballks can be eliminatedby lumping the corresponding equivalent admittance of the components into the diagonal elements of the respective submatrix of the system admittance matrix YN. Finally,since matrices As and J are usually sparse, triangular factorization technique can beused to save computation associated with forward elimination and backward substitutionprocedures.3.3.3 Integration Step Size Control and Exact TimingPower system equations (3.5) and (3.6) may have wide—ranged values of time constantsassociated with the system dynamics. For example, a generator sub—transient may involvea time constant of a small fraction of a second, while a time constant associated withload dynamics may be several minutes. Small time constants determine the fast systemtransients while large time constants dominate the slow system dynamics. For systemproblems like voltage instability phenomenon involving both fast transients and slowsystem dynamics, a too large integration step size may result in poor accuracy for fasttransients, while a too small step size may result in excessive computation for the slow46dynamics. Therefore, to take care of both small and large time constants of the system,a step size control procedure is imperatively needed for the system integration.In an implicit integration method, the solutions of differential equations are approximated by those of the corresponding difference equations of finite order. For example,the original system differential equation (3.5) is approximated by the difference equation(3.25). The approximation is made by truncating the higher order terms of the difference equations. The error of the approximation introduced in a single integration stepis called the local truncation error eLT. It is shown in [43] that the relationship betweenlocal truncation error and integration step size can be expressed asd1xeLT C dt(1) (3.41)where p is the order of the numerical integration method, for example, p = 2 for integration algorithm based on trapezoidal rule, At is the integration step size, C is a constantwhich depends on the system equation and the integration method being used, and x isthe solution of the system equation.The basic idea of step size control is to keep the local truncation error eLT within thetolerance limit while maximizing the integration step size At.Computation of eLT from equation (3.41) requires information regarding the orderof the method p, the constant C, and the (p + 1)th derivative of x. In this thesis,instead of computing the (p + 1)th derivative of x by extrapolation with a polynomialof degree p + 1, an alternative approach, the step doubling approach, described in [40] isused to estimate the local truncation error. This method involves integrating the systemequations by taking two steps of step size At and reintegrating over the same intervalwith a single step of length 2 At. With these solutions, the local truncation error eLT canbe estimated as follows.Let x be the solution by taking two single steps, xd the solution by taking a double47step, and xt the corresponding true solution which is unknown. We have the followingequationsxt+2eLT8 (3.42)= Xt + eLTd (3.43)where eLT3 and eLTd are local truncation errors of single step and double step integrations,respectively, which, from the equation (3.41), are given byxeLT8 Cdt(P+l)d1xe— C 2Lt4LTd— dt(P+l) ‘Subtracting (3.42) from (3.43), we get= eLTd—2LT5= c /tP+1(2P+1—2) (3.44)= (21— 2)eLT5which gives the local truncation error of single step integrationeLT = (xd —x5)/(21 — 2) (3.45)From equation (3.45), we can see that computation of eLT by step doubling approachavoids the computation of the derivatives and the knowledge of constant C. Only theorder of the integration method needs to be known. On the other hand, step doublingmethod has an obvious disadvantage. It incurs much more computations because x3and xd have to be computed in order to obtain eLT. However, as also indicated in [40],in addition to easy implementation, this method usually gives more reliable and steadyresult. This means that the changes in step size during integration are more consistentwhich is also observed in the simulation studies in this thesis. These benefits are nearlysufficient to compensate the extra computation cost.48With the local truncation error eLT available, the integration step size At can be sochosen that the eLT satisfiesemin eJff emax (3.46)where emin and emax are prescribed lower and upper bounds of local truncation error eLT.If the eLT is within its bounds, the current step size is acceptable and the integrationcontinues with the step size. Otherwise, if the eLT is less than emin or greater than emax,then the step size is doubled or halved. In order to avoid too large or too small step size,At is also bounded by its upper and lower limitsAtmin At <Atmax (3.47)Therefore, whenever there is a system change which causes some system variables tovary sharply, a small integration step size must be chosen for accuracy. After fast systemtransients, the step size will be increased gradually while keeping the local truncationerror within the prescribed limits.Due to the variation in step size during the integration, the simulation may overshootand miss the exact timing of faulting and clearing, tap changing and rotor overheatprotecting, etc.. In the simulation program, the exact timing t of an event, is pinpointedby changing the last step size according toAt new = At0ld — t + tc (3.48)and the new step size is then used to re—integrate the last step as shown in Figure 3.5.3.3.4 System ContingenciesMost system instabilities are caused by system disturbances. Different disturbances occurring at different locations have different impacts on system stability. System disturbances may include system load changes, system network changes, and system generationchanges. There may also be single or double contingencies.49variablesystem changeAtoldttnewI I timet- Atoid tc tFigure 3.5: Step Size Change for Exact Timing1. System load changes(a) Step change in power and reactive power of a nonlinear PQ load.(b) Step change in load torque of an induction motor.(c) Gradual inrease in system load.2. System network changes(a) opening of a transmission line(b) tripping of a transformer.(c) Ground of a system bus.3. System generation changes(a) Tripping of a generator.(b) Switching—in or —out of a capacitor.Whenever there is an sudden change in system topology, system variables Y willchange instantly since the fast electromagnetic transients in transmission network are50neglected. Although the differential variables X remain unchanged, system equation(3.5) and (3.6) must be solved at that time instant in order to find new initial value ofY for the next step integration.3.3.5 Flowchart of System Simulation ProgramThe complete system simulation program may be recapitulated by an overall flowchartas shown in Figure 3.6.51Figure 3.6: Overall Flowchart of System Simulation ProgramnoSOLUTION OF SYSTEM EQUATIONS (X Y)I. EQUATION DISCRETIZ4TION2. DIFFERENCE CONSTANT Z3. JACOBIAN EQUATIONa. FORWARD EUMINATIONb. BACKWARD SUBSTITUIION4. VARIABLE UPDATE52Chapter 4DYNAMIC VOLTAGE STABILITY STUDIESVoltage stability studies have been concentrated on the determination of maximum loading limit (MLL) of a power system based on load flow related analysis. When the systemloading reaches its MLL determined by conventional load flow, the load flow Jacobianwill become singular, which indicates a possible voltage collapse. Such defined MLL maybe referred to as static voltage stability limit. Thus, the distance between current systemloading condition and the MLL can then be used as an index to measure the degree ofsystem voltage stability. However, when the MLL is determined through load flow analysis, system dynamics are not included. As a result, it may not lead to a realistic solutiondue to the harmful or favorable system dynamics. Moreover, even if an exact MLL couldbe found, it can only tell a system loading condition where a possible voltage collapsemay occur. It can not provide any information of how system voltage approaches thecollapse point and how this collapse point is affected by system dynamics.Therefore, the evaluation of system MLL and clarification of voltage collapse mechanism require a detailed system dynamic simulation which may take into account theeffects of all system dynamics on voltage stability. It is believed that the unfavored dynamic interactions among system components play a key role in the process of voltagedeterioration and collapse. With system models described in Chapter 2 and time domainsimulation technique developed in Chapter 3, the effects of load and control dynamicson voltage collapse process can be better investigated by dynamic simulation of a samplepower system.53In this chapter, a 21 bus sample power system with basic system data is presentedin Section 4.1. Critical system load buses are defined and identified in Section 4.2 basedon bus voltage—reactive power sensitivity analysis. Following that, the effects of differentsystem loads on voltage stability are investigated in Section 4.3, and system control effectsin Section 4.4. Finally, the results of system maximum loading limit (MLL) from loadflow and those from dynamic simulation are compared and discussed in Section 4.5.4.1 A Sample Power System for Voltage Stability Studies4.1.1 A 21 Bus Sample Power SystemA sample power system with 21 buses and 23 branches of transmission system is shownin Figure 4.1. The system has 9 generating plants, two of them at buses 4 and 9 arehydro—electric, and the others are steam—electric. There are 11 load buses with a VARcompensation at bus 21. The system is connected to an infinite system through line11 — 1.Figure 4.1: A Sample Power System under Study544.1.2 Basic System DataThe basic system data for the transmission system, the generating plants and a specifiedsystem generation and loading condition are given as follows. The data for typical systemloads and system control devices are given in the subsequent sections. All data are inper unit unless otherwise specified.A. Transmission SystemTable 4.1: Data of Transmission SystemLine Bus I Bus J Resistance(R) Reactance(X)1 2 13 0.0000 0.05902 3 13 0.0000 0.01353 5 12 0.0000 0.29004 12 13 0.0068 0.06805 6 16 0.0000 0.05006 16 19 0.0300 0.30007 7 16 0.0000 0.10008 16 17 0.0015 0.01459 19 20 0.0106 0.106010 20 21 0.0240 0.240011 21 17 0.0161 0.161012 21 14 0.0025 0.025013 13 14 0.0120 0.120014 13 20 0.0048 0.048015 14 15 0.0070 0.070016 15 4 0.0200 0.232017 15 11 0.0070 0.070018 14 10 0.0120 0.120019 14 18 0.0102 0.102020 18 9 0.0000 0.280021 17 18 0.0057 0.057022 17 8 0.0320 0.320023 11 1 0.0000 0.0320The impedances of transformers in the transmission system are combined with thoseof transmission lines. All line capacitances are ignored.55B. Given System Generation and Loading ConditionTable 4.2: Data of Generator PV and Load PQFV Bus FQ BusBus Pgen Vgen Bus Pload Qload2 3.000 1.040 11 2.500 0.2003 3.500 1.035 12 1.500 0.1004 2.000 1.030 13 2.400 0.2005 1.000 1.035 14 4.500 0.5006 2.500 1.040 16 3.500 0.2007 2.500 1.040 17 2.000 0.2008 2.000 1.010 18 2.000 0.0009 2.000 1.015 19 1.500 0.10010 3.500 1.060 21 3.500 —1.700Slack Bus: V1=1.060 Bus 15, 20: No LoadFor different system loading conditions, generator voltages may be adjusted for anormal system voltage profile. The power balance is taken care by the infinite system.C. Generating PlantSynchronous GeneratorsTable 4.3: Data of Generator ParametersBus xd xq M D T0 T, T,,2 0.700 0.700 0.120 0.098 0.098 25.000 0.050 7.000 0.091 0.4553 0.600 0.600 0.100 0.084 0.084 30.000 0.050 7.000 0.091 0.4554 0.500 0.400 0.150 0.070 0.070 20.000 0.050 8.000 0.104 0.5205 1.600 1.600 0.230 0.224 0.224 12.800 0.050 7.000 0.091 0.4556 0.950 0.950 0.150 0.133 0.133 19.800 0.050 7.000 0.091 0.4557 0.950 0.950 0.150 0.133 0.133 19.800 0.050 7.000 0.091 0.4558 1.000 1.000 0.170 0.140 0.140 18.000 0.050 7.000 0.091 0.4559 1.000 1.000 0.170 0.140 0.140 18.000 0.050 7.000 0.091 0.45510 0.390 0.320 0.060 0.055 0.055 32.000 0.050 6.000 0.078 0.39056where M, T0, T, and T are in seconds.Field Excitation SystemsThe excitation systems for the sample power system may be divided into two groups.Group A has smaller time constants and larger gains, while group B has larger timeconstants and smaller gains as shown in Table 4.4. The excitation systems of generatorsat buses 2, 3, 4, 5, 6, and 7 belong to group A, and those at buses 8, 9, and 10 belong togroup B.Table 4.4: Data of Field Excitation SystemGroup TA Emni EmasiA 100.00 0.050 —7.000 7.000B 50.00 0.100 —7.000 7.000where TA is in second.Governor and Turbine Systems(a) Mechanical—Hydraulic (M—H) Governor and Hydro Turbine SystemTable 4.5: Data of M—H Governor and Hydro TurbineT Tr T9 T0.050 0.250 0.020 4.800 0.500 1.600C9 Gsmin Gsmax Gomin GomaxPgen 0.100 0.100 0.000 Pmaxwhere T, Tr, T9 and T are in seconds.(b) Electrical—Hydraulic (E—H) Governor and Steam Turbine System57Table 4.6: Data of E—H Governor and Steam TnrbineKg Tsm TCH C920.000 0.100 0.400Gsmjn Gsmax Gomin Gomax0.100 0.100 0.000 Pmaxwhere Tsm and TCH are in seconds.In the governor and turbine system data, the interfacing factor a9 has a value of therated power output P96, of the generator. The maximum gate or valve opening Gomaxcorresponds to the maximum power output which may be certain percent over the ratedpower output, for example, Pmax may be 1.2 times of Pgen.Power System StabilizersThe power system for the given operating condition is originally unstable in termof system low frequency oscillations. For voltage stability studies, this conventionalangle stability problem should be removed from the system. This can be done withthe applications of power system stabilizers (PSS’s). For the given system operatingcondition, three power system stabilizers are furnished on the generators at buses 4, 8,and 9 to provide a supplementary excitation control to quench the osillations [44]. Thecorresponding PSS parameters are given below. For different system loading conditions,these parameters may have to be adjusted so as to give the best damping performance.Table 4.7: Data of PSS’s for Given Operating ConditionGenerator Bus K0 2’4 18.057 5.000 0.103 0.0358 28.271 5.000 0.104 0.0359 31.352 5.000 0.124 0.03558whereT3,T1, and T2 are in seconds.4.2 The Critical System Load Buses4.2.1 Analysis of the System Operating ConditionFor the given system operating condition, load flow studies show that the system is highlypower stressed. A large amount of power is transferred from some remote generatingplants, for instance, at buses 4, 5, 8, 9, and 10, and to some remote loads, such as loadsat buses 14, 19, and 21. The long distant power transmission results in a heavy reactivepower loss in the transmission lines and transformers. As a matter of fact, all the reactivepower supply from generators is consumed in the transmission system, as indicated inTable 4.8. As a result, the large capacity reactive power compensation at bus 21 becomescritical for the system to maintain a nearly normal voltage profile as shown in Table 4.9.Table 4.8: System Power Generation and ConsumptionSlack Bus Generators Compensations Loads LossesPower (F) 1.875 22.000 0.000 23.4000 0.475Reactive 0.969 7.104 1.700 1.500 8.273PowerTable 4.9: Voltage Profile of the Given System ConditionV1 V2 V3 V4 V5 V6 V71.060 1.040 1.035 1.030 1.035 1.040 1.040V8 V9 V10 V1 V12 V13 V141.010 1.015 1.060 1.032 1.008 1.021 0.981V15 V16 V17 V18 V19 V20 V210.994 0.984 0.969 0.956 0.967 0.999 1.00459As a result, the system operation becomes voltage vulnerable due to its critical dependence on the heavy reactive power compensation. This means that system voltagemay collapse due to system disturbances that reduce the compensation.4.2.2 Critical System Load BusesCritical load buses of the system are identified and used to investigate the effects of loadand control dynamics on system voltage stability. The critical system load bus is definedas the bus that is most voltage sensitive among all system load buses. This suggests thatthe load increase at a critical load bus will have a larger influence on the overall systemvoltage profile than those at other load buses. More specifically, a load increase at thecritical load bus will cause large voltage drops at most system load buses. Therefore,the system voltage profile is more sensitive to the load variations at the critical load busthan those at other system load buses.For a given system generation and loading condition, the steady state system operation can be characterized by the following nonlinear load flow equations.P (O,V) = 0 (4.1)Q (ø,V) = 0 (4.2)where P and Q refer to the power and reactive power mismatch equations, respectively.0 and V represents the angles and magnitudes of system bus voltages, respectively.The system perturbation equations can be obtained by linearizing the load flow equations (4.1) and (4.2) around the system operating point, which gives= Pe k0 + P v (4.3)= e0 + Qv V (4.4)60which may be written in a single matrix equation with the load flow Jacobian matrix asJLF (e,V) =QeQvand V are respectively the angle and magilitude deviations of system bus voltagesdue to the bus power and reactive power perturbations of iP and iQ. The Jacobianmatrix JLF (0, V) is essentially a sensitivity matrix, and the corresponding bus voltagesensitivities can be used to identify the most critical system load bus.Since there is relatively strong coupling between reactive power and voltage magnitudein power system, the voltage—reactive power sensitivity [iV/iQ] is a reasonable indexto describe the effects of system loading perturbations on the voltage magnitude. Sincethe effects of changes in the real power injections on the voltage magnitude is usuallyvery small, the relationship between bus reactive power perturbation iQ and the busvoltage deviation iV can be obtained as follows.Let iP = 0 in Equation (4.3) and solve for i0, which gives= —P1PZkV (4.5)Substituting i€ in Equation (4.5) into Equation (4.4) gives= (Qv — Q P1 P) iV (4.6)oriv = SiQ (4.7)whereS= [Qv — Qe P1 Pv1’ (4.8)and S may be referred to as system voltage and reactive power sensitivity matrix.61In Equation (4.7), the element S of matrix S has a value equal to the voltagedeviation at load bus i, due to one per unit change in reactive power L\Q 1 atload bus j,, assuming no load changes at other load buses. From this point, a criticalsystem load bus is defined as the bus which, when having a load increase, will cause largevoltage drops at most system buses. On the other hand, a load bus which, when havinga load increase, will influence the voltages of only its own or/and a few adjacent buses,is not critical to the overall system voltage profile.For the given system operating condition of the sample power system, the systemvoltage and reactive power sensitivity matrix S calculated from equation (4.8) is shownin Table 4.10. The values in the jth column of the matrix are the voltage deviations atall system load buses due to the per unit change in reactive power at load bus j. Thelarger the value is, the larger is the influence of the bus j on the corresponding buses.If 1 percent of voltage deviation due to 1 per unit reactive power change is taken as thethreshold of large influence, the number of buses whose voltages are largely affected bythe change of reactive power at bus j, (j = 11 ,21) are given in Table 4.11.Table 4.10: System Voltage Sensitivity MatrixBus i --- ---- ---- ---- —- --- 4Qii Ql2 Q13 Q1d ‘QlS Qi6 ‘Qii Qi8 Ql9 Q2O Q2l11 —0.026 —0.001 —0.001 —0.005 —0.014 —0.001 —0.002 —0.003 —0.002 —0.002 —0.00412 —0.001 —0.063 —0.008 —0.003 —0.002 —0.001 —0.002 —0.002 —0.005 —0.006 —0.00413 —0.001 —0.008 —0.010 —0.004 —0.002 —0.001 —0.002 —0.003 —0.006 —0.007 —0.00414 —0.005 —0.003 —0.004 —0.030 —0.016 —0.007 —0.010 —0.016 —0.011 —0.012 —0.02215 —0.014 —0.002 —0.002 —0.016 —0.045 —0.004 —0.005 —0.008 —0.006 —0.006 —0.01216 —0.001 —0.001 —0.001 —0.007 —0.004 —0.026 —0.023 —0.016 —0.011 —0.006 —0.00817 —0.001 —0.002 —0.002 —0.009 —0.005 —0.022 —0.032 —0.023 —0.011 —0.007 —0.01118 —0.002 —0.002 —0.002 —0.016 —0.008 —0.016 —0.023 —0.055 —0.011 —0.008 —0.01419 —0.002 —0.005 —0.006 —0.011 —0.006 —0.012 —0.012 —0.011 —0.110 —0.031 —0.01620 —0.002 —0.006 —0.007 —0.012 —0.006 —0.006 —0.007 —0.008 —0.031 —0.038 ‘—0.01721 —0.004 —0.004 —0.005 —0.023 —0.012 —0.008 —0.011 —0.015 —0.016 —0.018 —0.03562Table 4.11: Number of Buses Largely Affected by Bus j11 12 13 14 15Bus j 16 17 18 19 20 21No.ofBuses 2 1 1 6 4 4 6 6 7 4 7Largely AffectedAccording to Table 4.11, the system load buses may be classified into three groupsin terms of voltage sensitivity, the strong bus group, buses 11, 12, and 13, which are lesssystem voltage sensitive, the weak bus group, buses 14, 17, 18, 19, and 21 with the mostcritical buses 19 and 21, which have the largest effects on system voltage profile, and thethird group, buses 15, 16, and 20, which have the effects in between.Since the load and control effects on the system voltage stability can be clearly demonstrated at the system critical buses, and the system voltage profile is largely dependenton the reactive power compensation at bus 21, the load bus 21 is then chosen for subséquent voltage stability studies. Thus, the load at bus 21 will be substituted by a typicalload for each case study. The effect of reactive power controls will also be examined atbus 21.4.3 Effects of System Loads on Voltage StabilityThe effects of the system bus loads described in Chapter 2 on the system voltage stabilityare studied in this section. In the study, a system bus load is represented by a particulartype of load, a transformer with on load tap changer, and a distribution link connected tothe transmission system bus as in Figure 4.2. The distribution link impedance is includedin the transformer model. When the effect of a typical load itself is investigated, the tapchanging may not be considered, that is, t = 0. An induction motor load, an exponentialform static load and a persistent PQ load are included in this part of study.63system busFigure 4.2: A System Bus with Typical Load4.3.1 Effect of an Induction Motor LoadThe basic data of an equivalent induction motor is as follows:Table 4.12: Data of an Induction Motor LoadH T TL6.700 0.010 4.209 0.142 1.600 2.4where H and T are in seconds, and the motor load torque TL may subject to change indifferent case studies.With the induction motor load connected at bus 21, the system will have a lowvoltage profile due to the large reactive power consumed by the motor. To have a systemoperation with a fairly normal voltage profile, some of the generator voltage referencesare raised to a higher level and the parameters of the PSS’s are adjusted to give thebest damping to the system unstable mechanical modes. The system adjustments are asfollows.64Table 4.13: Adjusted System Voltage ProfileVi V2 V3 V4 V5 V6 V71.060 1.040 1.035 1.050 1.035 1.050 1.050V8 V9 V10 Vu V12 V13 V141.050 1.050 1.060 1.026 1.004 1.017 0.960V15 V16 V17 V18 V19 V20 V210.984 0.988 0.972 0.960 0.956 0.982 0.966Table 4.14: Adjusted PSS ParametersCase Study 1: An Induction Motor Near Critical StateIn this case study, the induction motor with a constant load torque TL = 2.4 isconsidered. The motor is connected to bus 21. Other system loads are assumed to beconstant impedance loads. Transformer tap changing effect is not considered. The initialsystem voltage profile is shown in the Table 4.13 and the motor related variables aregiven in Table 4.15.Table 4.15: Initial State of the Induction MotorTerminal Voltage Motor Slip Power Reactive Power0.965 0.024 2.480 1.404Generator Bus It0 T T15 T284 37.573 5.000 0.080 0.0358 34.056 5.000 0.128 0.0359 26.251 5.000 0.121 0.03565.Figure 4.3: Motor Response to Step Change in Load TorqueA step load torque increase of 0.23 is then applied to the motor as a system disturbance. System responses recorded in the figures show a voltage instability phenomenonof slow system dynamics followed by a sudden voltage collapse.2.82.62.42.2CID00h__-L—;— ————.0 20 40 60 80 100 120 140 160 180Time (sec)0 20 40 60 80 100 120 140 160 180Time (sec):,(a)(b)(c)(d)120 140 160 1800 20 40 60 80 100Time (sec)20.9— 080• 0.70.60.50 20 40 60 80 100 120 140 160 180Time (sec)66Figure 4.3 shows the dynaniic behavior of the induction motor. Following the stepincrease in load torque at t = 5 second as in Figure 4.3a, there is a transient period ofabout 10 seconds. Following the disturbance, the motor slip begins to increase accordingto the rotor motion equation (2.1). Both motor power and reactive power drawn fromthe system increase accordingly to pick up the load. As a result, the larger motor currentcauses extra voltage drops in the transmission system, leading to a decrease of motorterminal voltage and other system bus voltages as well, as shown in Figure 4.3d andFigure 4.5. During this transient, generator bus voltages are maintained at the samelevel as those in the pre—disturbance system condition by generator excitation controlsas shown in Figure 4.4.1.0651::1.06“C41 1.0554’ 1.05t11.045. 1.04C1.0351.031.025‘ 1.02Time (sec)Figure 4.4: Some Generator Bus VoltagesAfter the transient, the motor slip càntinues to increase gradually, and the systemexperiences a rather slow system dynamics, in which the system load bus voltages remainfairly normal, and the system frequency is fixed at 60 Hz, as shown in Figure 4.5 andFigure 4.7. During this slow dynamics, the electromagnetic torque developed by the67motor is slightly less than the motor load torque. Motor slip begins to approach itscritical value over which the motor will start stalling. At around t = 105 second, themotor slip reaches its critical value 8c 0.038 with a maximum motor torque TE 2.627,which is still less than the load torque TL 2.63. After that, the motor slip continues toincrease while the motor developed torque begins to drop. The motor starts stalling. Themotor reactive power begins to increase. This causes the motor terminal voltage to dipfurther, which in turn reduces the motor torque. This interacting dynamics continuesuntil a rapid change occurs at about t = 170 second. Due to the fast decreases inmotor developed torque shown in Figure 4.3a, the motor slip increase rapidly as shownin Figure 4.3b. Although the motor power drops quickly, its reactive power goes uprapidly as shown in Figure 4.3c. It is this rapid increase of the motor reactive powerdemand that causes a sharp drop in motor terminal voltage as shown in Figure 4.3d.Other system load buses have similar phenomena, and the results are shown in Figure4.5, in which only voltages at load buses 13, 16, and 21 are shown for the sake of clarity.1.050.950.90.850.80.750.70.650.60 20 40 60 80 100 120 140 160Time (sec)Figure 4.5: System Voltages at Some Load Buses18068At about t = 175 second, the motor reactive power begins to decrease but the motorterminal voltage continues to drop. This means that the system operation has reachedthe lower part of Q—V curve of motor terminal bus as shown in Figure 4.6. The systemloses voltage control at motor terminal bus after t = 175 second. Figure 4.6 also showsthat the maximum motor power limit occurs at t = 165 second, while that of reactivepower occurs at t = 175 second. This suggests that although the motor operates on thelower part of the P—V curve after t = 165 second, the motor terminal voltage still can becontrolled if there is a sufficient reactive power compensation near the motor terminalbus.0.950.90.85• 0.80.750.70.654 4.5Figure 4.6: Transient P—V and Q—V curves of Motor Terminal BusSince all system loads except the induction motor load are constant impedance loads,the large system voltage drops after t = 175 second cause a large load reduction to theentire system. This load reduction happens so fast that the governors and turbines arenot in time to respond so as to reduce the mechanical inputs to the generators. As aresult, all generators are speeded up and ultimately pulled out of synchronous operationPmV0 0.5 1 1.5 2 2.5 3 3.5Motor Power Pm & Reactive Power Qm69successively, which causes a complete system collapse as shown in Figure 4.7 where therotor angles of generators at buses 2, 6, 8, and 10 are recorded. Other generators havesimilar angular instabilities.00—°-———¶ 2.5 66(‘1—b———2——-cs--——— 6ioI 1.5—CC,0 20 40 60 80 100 120 140 160 180Time (sec)Figure 4.7: Some Generator Rotor AnglesIn summary, the results of this case study show that an induction motor load couldcause a slOw voltage deterioration followed by a sudden voltage collapse when the motoroperates near its critical condition. The stable operation of an induction motor dependson its terminal voltage and its critical slip.When a motor operates over its critical state, both motor developed torque andpower will drop, but it will draw more reactive power from the system, which will, inturn, aggravate the motor terminal voltage deterioration. If this interacting process isunchecked, a sharp voltage collapse will occur. In addition, from Figure 4.6, we can seethat although system operates at the unstable branch of P—V curve of motor terminalbus, the system has not yet lost its voltage control until the maximum reactive power isreached. If sufficient Var compensations were added at motor terminal bus before this70reactive power limit is reached, the system voltage would have been controlled, and themotor would move back to stable operation. Therefore, for the system with inductionmotor loads, the system voltage stability can not be judged by P—V curve alone.Case Study 2: Transient Stability with Induction Motor Load4.541! 32.5200.50.40.3I:2Time (sec)6Time (sec)Time (sec)02.521.510.52 4 6Time (sec)0.900.60.52.51.510Time (sec) Time (sec)Figure 4.8: Transient Responses of Motor Variables0 2 4 6 2 4 671In this case study, an induction motor load is connected at system bus 21 through adistribution link. All other system loads are assumed to be constant impedance loads.The initial system condition is the same as that in case study 1. But the system losesa line between buses 14 and 21 at t = 0.5 second. The system responses shown in theFigure 4.8 demonstrate a transient voltage instability of the system.Figure 4.8 shows the motor responses to the system disturbance. At the instant of thedisturbance, the motor internal voltages (not shown) and motor slip remain unchanged.The motor terminal voltage and current change instantly. The disturbance causes themotor current to change in such a way that the motor developed torque, and hence themotor power and reactive power dip suddenly. The motor terminal voltage goes up abit. Following that, motor slip begins to increase quickly until the critical slip is reachedat t = 1.2 second. The motor developed torque has reached the maximum which is,however, still less than the load torque as shown in Figure 4.8b. After that, the motorstarts stalling. Both motor torque and power decrease quickly. The motor reactive powerdemand increases rapidly causing motor terminal voltage collapse within 4 seconds. Theresults of this case study show that a system with induction motor loads may involvea transient voltage instability when the operation of the induction motor is upset bysystem disturbances. More reactive power demand of an induction motor against voltagedecline adds an strict constraint on system voltage stability.4.3.2 Effect of Exponential PQ LoadsThe effect of a general static PQ load of the exponential form on system voltage stabilityis examined, which includes three special cases: the constant impedance, the constantcurrent, and the constant power loads.72Case Study 3: Constant Power, Current, or Impedance LoadThe initial system condition is the same as that in case study 1. An inductionmotor with an initial load torque of 2.4 is again connected to system bus 21 through adistribution link. An step load torque increase of 0.23 is applied to the motor as a systemdisturbance. All other system bus loads are assumed to be exponential PQ loads. Theeffects of three different types of the loads, the constant impedance, the constant current,and the constant power loads are examined and compared. Tap changing effects are notconsidered in this study.0.950.90.850.8—c— constant power0.75—ci--——— constant currentconstant impedance0.70 20 40 60 80 100 120 140 160 180 200Time (sec)Figure 4.9: Voltage Response at a Load BusComparison of the results for the three special types of PQ loads reveals some interesting points. A constant impedance load (a = /3 2) is more voltage dependentthan a constant current load (a= /3 = 1), while a constant power load (a = /3 = 0) isnot dependent on its terminal voltage at all. Following the disturbance at t = 5 second,the induction motor will draw more power and reactive power from the system rapidly,73causing voltage drops at all system buses. In responding to the voltage drops, the constant impedance loads will draw less power and reactive from the system, thus having afavorable effect to halt the further decline of system voltage. As a result, the motor canmaintain stable operation at a higher terminal voltage for more than 170 seconds. On theother hand, the constant power load will draw the same power and reactive power fromthe system despite the voltage decreases, which aggravates the system voltage decline.The induction motor starts stalling at about t = 25 second, much earlier than that inthe case of constant impedance loads. The effect of a constant current load is betweenthe two with motor stalling at about t = 125 second, as shown in Figure 4.9.0000Time (sec)Figure 4.10: Reactive Power Drawn by the MotorIn all three cases, bus voltages collapse sharply when the induction motor startsstalling, which is, however, largely affected by the load characteristics. The more thedependence of a load on its terminal voltage, the better the damping effect it will haveon system voltage stability. This damping effect is more crucial to the voltage stability ofa power system where critical induction motor loads are supplied, as in this case study.74The motor reactive power response is shown in Figure 4.10, and the voltage response ofa representative generator is shown in Figure 4.11.1.0511.051.049__________________________________1.0481.0471.0461.0451.01.043 —c— constant powerconstant current1.042_ _constant impedance1.0410 20 40 60 80 100 120 140 200Time (sec)Figure 4.11: Voltage Response at a Generator BusIt is also observed that with constant impedance loads, the system involves the generator angle instability, while with constant current or constant power loads, the systemremains stable in term of system frequency although the stalling of the induction motorcauses a voltage collapse at load bus 21, which is clearly demonstrated in Figure 4.12.In the case of constant impedance loads, large voltage drops due to system disturbancereduce the system load, especially the real power load, causing generators to speed upfollowing the disturbance. System remains stable both in angle and in voltage duringthe slow system dynamics. At about t = 105 second after the disturbance, the motorbegins to stall, resulting in a sharp voltage collapse as shown in Figure 4.9. Both motorpower and other system constant impedance loads drop rapidly. The process is so fastthat the governors can not respond in time. The imbalance of system real power drivesthe generators eventually out of step of the synchronism. On the other hand, in the case160 18075of constant power loads, system load is not affected by the sharp voltage drop due to themotor’s stalling. Only the real power of the induction motor load is lost. The generatorsspeed up in response to this load reduction but remain stable at a higher rotor anglesthan those before the disturbance, as shown in Figure 4.12.41I180 200Time (sec)Figure 4.12: Rotor Angle of a GeneratorThis case study demonstrates the effects of a static voltage dependent load on systemstability. Constant impedance loads draw less power and reactive power when voltagedecreases, which has a favored damping effect on system voltage stability. On the otherhand, the load reduction may tip over the real power balance of the system causing apossible generator angle instability, especially in the cases where rapid voltage collapsemay involve. In contrast, constant power loads have an somewhat opposite effect onsystem stability. Although the constant demand of power and reactive power may aggravate the system voltage decline, which may cause system voltage collapse, it may helpto maintain system real power balance when system voltage drops rapidly as in this casestudy, thus enhancing the system transient stability. This conclusion is drawn from the0 20 40 60 80 100 120 140 16076consideration of load only. The conclusion may be opposite if the loss of system powergenerations is also involved.4.3.3 Effect of Persistent PQ LoadsAs described in Chapter 2, a persistent PQ load demands constant power and reactivepower despite a voltage decline, but involving an inherent time delay constant. This kindof load characteristics is very important to system voltage stability since the insistingdemand of constant power and reactive power may cause system collapse, especially whenthe system loadability is reduced due to system disturbances.Case Study 4: Effects of Persistent PQ loads on Voltage StabilityTo study the effect of this kind of load on system voltage stability, all system loadsof the sample system are considered as persistent PQ loads, modeled as changing equivalent admittances and with same recovery time delay of 10 seconds. The same initialsystem condition as in the previous case studies is obtained by replacing the corresponding induction motor load at bus 21 with a persistent PQ load having an initial loadingof 2.48 + jl.40. The steady state system operation is then disturbed by a loss of a linebetween buses 14 and 21 at t = 5 second. It is anticipated that this system disturbancewould greatly reduce the loadability of system bus 21 because the load now must be supplied through the relatively remote buses 17 and 20. The simulation lasts 1000 seconds,and system responses are as follows.Figure 4.13 records the voltage response at system bus 21, which shows voltage instability. Upon the system disturbance, system voltages dip immediately, causing theinstant load power and reactive power load drops as seen in Figure 4.14.770.90.80.70.60.50.4o 0.30.20.1I2 4 6 8 10 2 4 6 8 10Load Admittance at Bus #21 Load Admittance at Bus #21Figure 4.14: Load Power and Reactive Power vs. Load AdmittanceAfter that, the system load admittance begins to increase according to equation (2.8).The load power and reactive power recover gradually while system voltages continue todecline. This process continues until about t = 139 second at which load power and400 500 600Time (sec)Figure 4.13: Voltage Collapse at Bus 2178reactive power at bus 21 have reached their corresponding maximum values as shown inFigure 4.14. But, the maximum load power and reactive power at bus 21 are still lessthan the pre—disturbance values. As a result, the load admittance continue increasing.Over that point, the load power and reactive power begin to drop monotonically despitethe further increase of the load admittance. The bus voltage goes all the way down asshown in Figure 4.13, which indicates that system has lost voltage control at bus 21.Time (sec)Figure 4.15: Some Other System Load Bus VoltagesSome other load and generator bus voltages of the system are shown in Figure 4.15and Figure 4.16. The results indicate that the system survives the transient stability andvoltages at system load buses other than bus 21 remain a fairly high level. Although thevoltage collapses at bus 21, which causes loss of load at that bus, the other system busloads have recovered to their pre—disturbance levels, some of which are shown in Figure4.17. The persistent loads demand constant power and reactive power, which exceed thesystem loadability at bus 21, causing a voltage collapse at that bus, but maintains thesystem real power balance, avoiding a system angle instability.791.0554 1.5rl)-.1.4800001.05441.0451.041.0351.030 100 200 300 400 500 600 700 800 900 1000Time (sec)Figure 4.16: Some Generator Bus Voltages4.52444.484.464.44“041:: 3.5-..c 3.480. 3.4603.440 500 1000 0Time (sec)500Time (sec)1000110 500 1000Time (sec)Some System Buses0 500 1000Time (sec)Figure 4.17: Loads at804.4 Control Effects on System Voltage StabilityAll reactive power related system components, such as controls, compensations, and constraints, have large impacts on system voltage stability. These system components mayimprove or deteriorate system voltage by supporting or restricting the system reactivepower supply. The effects of system reactive power components on system voltage stability are examined in this section. Among them are the on load tap changing of adistribution trallsformer, the reactive power compensation with fixed capacitor banks orwith SVC’s, and the rotor overheat protection of a generator.4.4.1 Effect of transformer tap changingSystem distribution transformers are equipped with on load tap changers (OLTC’s) tomaintain normal load side voltages by changing the taps. The dynamics of the OLTCis usually slow comparing with those of other system components, such as generatorsand motors. Therefore, the effect of an OLTC on system voltage stability may not beconsidered for fast transient system conditions. But, it must be considered when a slowsystem dynamics is involved, especially when a system is operating near its critical state.To study the tap changing effect on system voltage stability, an induction motor loadis assumed at load bus 21, and all other system bus loads are assumed to be constantimpedance loads. All loads are connected to the system buses through transformer links,and the transformers are equipped with on load tap changers. It is further assumed thateach tap of a transformer is 0.025, the time delay of each tap changing is 10 seconds,and the voltage deviation tolerance is + 2 percent from the normal voltage. The systemdisturbance is simulated by a step load torque increase to the motor with an initial loadtorque of 2.4.81Case Study 5: Tap Changing at Critical Motor Load BusIn this particular study, the tap changing is only assumed at the transformer link ofthe motor load bus. The effects of other transformer tap changings are not considered.A step load torque increase of 0.23 is applied to the motor at t = 5second.To demonstrate the transformer tap changing effect, the result of case study 1 is usedfor comparison. As shown in Figure 4.18, if the transformer tap changing effect is notconsidered (as in case study 1), the system voltage will collapse at about t = 170 seconddue to the motor’s stalling. Otherwise, the system voltage remains stable if the tapchanging effect is considered.2Figure 4.18: TapFollowing the step change in motor load torque at 1 5 second, both motor power andreactive power increase along with the increase of the motor slip, which causes the motorterminal voltage to decline due to the extra voltage drops in the transmission system.At I = 7.13 second, the motor terminal voltage drops below the specified lower limit andthe tap changing is initiated. The tap changing of moving up one tap is completed at820 20 40 60 80 100 120 140 160 180Time (sec)Changing Effect on Motor Terminal Voltaget = 17.13 second. This tap changing raises the motor terminal voltage as shown in Figure4.18. This voltage rise increases the motor developed torque, which becomes larger thanthe load torque as shown in Figure 4.19.2.6502.602.552.502.4502.4Figure 4.19:At the instant of tap changing, the motor slip remains unchanged. Both motor powerand reactive power jump suddenly. After that, motor slip begins to decrease, whichresults in a large motor reactive power decrease as shown in Figure 4.20. This reductionof motor reactive power helps to halt the system voltage from declining, which stabilizesthe motor operation with a higher motor load torque of TL = 2.63.The effect of the tap changing on the P—V curve of the motor terminal bus is shownin Figure 4.21, where the loading limit (the nose of the P—V curve) of the motor terminalbus is appreciably extended by the transformer tap changing.This study shows that the transformer tap changing is helpful to voltage stabilityfor a load with negative reactive power—voltage characteristics, such as induction motorload.0 5 10 15 20 25 30 35 40 50Time (sec)Tap Changing Effect on Motor Torque4583I2.82.62.42.22—0-———— motor Q—g---—--- motor P1.81.61.45 10 15 20 25 30 35 40 45Time (sec)Figure 4.20: Tap Changing Effect on Motor P and Q50IFigure 4.21:The load side voltage increase due to the tap changing will reduce the load reactivepower, which, in turn, enhences the system voltages. This effect of a transformer tapchanging is especially crucial when the system operates near its critical state.2.55 2.6 2.65 2.7 2.75Power Drawn by the motorMotor P—V Curve with and without Tap Changing84Case Study 6: Tap Changing at Other System Load BusesIn this case study, it is assumed that there are transformer tap changings for allsystem load transformers except for that of the induction motor load. The initial systemcondition is the same as that in case study 5. But a smaller motor load torque increaseof 0.22 is assumed at t = 5 second.‘-IITime (sec)Figure 4.22: Effect of Tap Changing at Other System Load BusesFigure 4.22 shows that the system voltage will remain stable if no tap changing effectsare considered while the system loses voltage stability in the case where the tap changingeffects are considered for all load buses but the motor load bus.Following the system disturbance, system voltage decreases as the motor picks up itsload. Since all system loads except the motor load are constant impedance loads, thisvoltage drop will cause a system load reduction, which has a damping effect to voltagedeterioration. If no transformer tap changings are involved, the system will sustain atransient condition, and remain stable at a fairly normal voltage as shown in Figure 4.22.85In the case where the tap changings are considered oniy at load buses with constantimpedance loads, the voltage drops at buses 14, 19, 18, and 17 successively initiate and,after 10 second time delay, activate the tap changers at the corresponding load buses.These tap changings raise the load side bus voltages of transformers, which restores someof the load power and reactive power. More current are then drawn from the transmissionsystem, causing a further voltage decline at system side buses of transformers. As a result,the voltage drop at system bus 21 causes the induction motor to stall at about t = 55second. Following that, a rapid increase in motor reactive power demand leads to asharp voltage collapse. Figure 4.23 shows the P—V curve of the motor load bus. Thedash—line curve shows the case where no tap changings are considered, and the motorremains stable operation at a fairly normal voltage, while the solid—line curve shows thetap changing effects which cause system bus voltage drops, upsetting the motor stableoperation. The effects of tap changings on motor power and reactive power is furthershown in Figure 4.24.0.9600.922O.90.880.860.942.6 2.65 2.7Motor PowerFigure 4.23: Motor Bus P—V Curves86Time (sec)Figure 4.24: Tap Chauging Effect ou Motor Power and Reactive PowerThis case study shows that a transformer tap changing has a detrimental effect onsystem voltage stability for the system loads with positive reactive power—voltage characteristics, such as an exponential form PQ load with positive exponents. The voltage risedue to a transformer tap changing will increase the load power and reactive power. This,in turn, aggravates the system bus voltage deterioration, which may cause a possiblevoltage instability as shown in this case study. This effect of a transformer tap changingwill become salient when the system is operating near its critical state.4.4.2 Effect of System VAR CompensationFor a heavily loaded power system, effective reactive power support is crucial to maintainsystem voltage stability. To study the effects of reactive power control devices on systemvoltage stability, a fixed capacitor compensation and an SVC are considered and compared fof different system operating conditions, which may involve fast system transientsand slow system dynamics. The fixed capacitor is modeled as a constant susceptance B.87The parameters of the SVC’s control circuit are given in Table 4.16.Table 4.16: Data of SVC ParametersCKB TB T1B T2B100 0.15 1.0 10.0Case Study 7: VAR Compensation and Fast System TransientsIn this case study, the additinal VAR compensations are provided to bus 21. Bothfixed capacitor compensation and SVC’s with two different capacities are considered.The fixed capacitor susceptance B is 0.5. SVC capacity may be represented by the limitof B2 as shown in Figure 2.10. The SVC of larger capacity has a value of B2max 0.35,while that of smaller capacity B2max 0.25. An induction motor with initial load torqueof 2.4 is connected to system bus 21 through a distribution link. Other system loads areassumed to be constant impedance loads. The system disturbance is simulated by a lineopening between buses 14 and 21 at t = 0.5 second. No tap changings are considered.1.050.950.90.850.80.750.70.650.6Time (sec)Figure 4.25: Effects of Different VAR Compensations0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 588Figure 4.25 shows voltage responses at load bus 21 with different VAR compensations.After the system disturbance, the voltage at bus 21 drops quickly due to the large currentof motor in order to pick up the load, which now has to be supplied from relatively remotesystem buses 17 and 20. If there is no additional reactive power support at bus 21, thelow voltage will cause the motor to stall, and the large reactive power drawn by thestalling motor will result in a voltage collapse within 4 seconds, as shown by Curve 1.The situations will be different if a fixed capacitor is available at bus 21, in whichCurve 2 shows the case where the capacitor is switched in the system at t = 1.5 second,while Curve 3 that at t = 2.5 second. Although the capacitor is switched in very quickly, itstill can not halt the voltage deterioration and collapse in both cases. Curve 2 and Curve 3also show that the sooner the capacitor is switched in, the better the compensation willpresent. This is because the reactive power supplied by a capacitor depends on the voltagelevel, as shown in Figure 4.26. Since the voltage drops quickly in transient conditions,the fixed capacitor compensation is not effective due to the switch time delay.0-c-)CJD0Figure 4.26: VAR Compensation2 2.5 3Time (sec)of Fixed C Switched in Different Time89Curve 4 and Curve 5 in Figure 4.25 show the cases where the fixed capacitor at bus21 is replaced by SVC’s. The results demonstrate that system voltage can be effectivelystabilized by an SVC of sufficient capacity (Curve 4), but not so for the system with anSVC of limited capacity (Curve 5). In the latter case, the SVC cannot supply enoughreactive power compensation because it has reached its limit, and behaves like a fixedcapacitor thereafter. As shown by Curve 5, the bus voltage has been controlled at a fairlyhigh level for about 4 seconds, but ultimately collapses.Figure 4.27 further shows the reactive power compensations of both SVC’s. It clearlyshows that an SVC can provide reactive power very quickly upon voltage drops, but anSVC must have sufficient capacity in order to stabilize a system voltage.0.350.3Cl 0.250.2?‘J)0J________SVC with larger capacity0.05 — — —— SVC with limited capacity-0.050 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time (sec)Figure 4.27: VAR Compensation of SVC’s with Different CapacitiesCase Study 8: VAR Compensation and Slow System DynamicsIn case study 7, system VAR compensation in fast transient system condition isinvestigated. This case study will show the effects of VAR compensations for a system90condition in slow system dynamics. For this, all system loads are modeled as persistentPQ loads with a time constant of 10 seconds. The induction motor load in case study 7is replaced by a persistent PQ load with the same initial loading. VAR compensationsare considered again at bus 1. The system disturbance is simulated by a loss of linebetween buses 14 and 21 at t = 5 second.Time (sec)Figure 4.28: Effects of VAR Compensation ofFigure 4.28 shows the voltage responses at, bus 21. Upon the system disturbance,the voltage drops suddenly, and then declines monotonically as the load recovers. Ifthere is no additional VAR compensation available at bus 21, the voltage will ultimatelycollapse as shown by Curve 1. Comparatively, Curve 2 shows the case where an effectiveSVC is available at bus 21. The voltage recovers very quickly in about 10 seconds andremains stable at its normal value. The other two dash—line curves, Curve 3 and Curve4, show the situations where a fixed capacitor at bus 21 is switched in at 10 secondsand 20 seconds respectively after the disturbance. The voltage is stably controlled forboth cases. However, there will be more than required reactive power compensation to0.940.92O.9C0.880.860.840 10 20 30 40 50 60 70 80 90 100Fixed Capacitor and SVC91the system along with the voltage recovery. As a result, the voltage will have a valuehigher than its pre—disturbance value, and some of capacitor compensations may haveto be switched out of the system. This can be very critical when a system has majorload with negative reactive power—voltage characteristics, that is, the load demands morereactive power when its terminal voltage goes down. There is no such problem with SVCcompensation, since the reactive power supply is controlled according to the voltagedeviations. When voltage is higher than its specified value, VAR compensation of SVCwill decrease accordingly. The different VAR compensations are shown in Figure 4.29.E4For a heavily loaded power system, an effective reactive power compensation is crucial to system voltage stability since system disturbances changes the power flow in thetransmission network, which may cause an extra reactive power loss and a lower systemvoltage. Depending on the system controls and load characteristics, the system mayinvolve fast transients and/or slow dynamics. For system involving fast transients, thesystem voltage may drop quickly. In such a case, fixed capacitor compensations are not0.350.30.25 4++curve 1 (SVC)—— curve2 (fixed C)—— curve3capacitor+>7stchedin10 20 30 40 50 60Figure 4.29:70 80 90Time (sec)VAR Compensation of Fixed Capacitor and SVC10092effective and SVC’s with sufficient capacities should be used. For the system involvingslow dynamics, the fixed capacitor compensation can be used to sllpport system voltage,but it may cause the voltage to overshoot its normal value because of its positive feedback control characteristics. Therefore, only SVC’s with ample capacities can effectivelystabilize the voltage of a power system.4.4.3 Effect of Generator Rotor Overheat ProtectionGenerators are equipped with rotor overheat protection to prevent the field willding fromoverheating. The current in the field winding is determined by the winding resistanceand the excitation voltage which is controlled by the generator excitation system. Whensystem disturbances cause an extra system reactive power loss, the generator terminal voltages will decrease. To maintain normal terminal voltages, generator excitationvoltages must be automatically increased, which will result in larger currents in rotorwindings. When the accumulated heat is over the permissive limit, overheat protectionswill reduce the excitation voltage to a lower level or even trip the generator off the system.Since this protection limits the generator reactive power output or even trips off the generator, it will have a significant impact on the voltage stability of the entire power system.Case Study 9: Effect of Rotor Overheat ProtectionIn this case study, the effect of generator rotor overheat protection (ROP) on systemvoltage stability is investigated. The ROP is usually realized by the generator excitationreduction to its continuous operating limit or the second limit (see in Figure 2.4) whenthe generator has been operated at a higher excitation level over a prescribed period oftime.In case study 4, the effects of persistent loads on system voltage stability are discussed93without considering the generator ROP. The attempt of persistent PQ loads to recoverloads to the pre—disturbance level by increasing their equivalent admittances pushes thesystem over the post—disturbance loading limit of bus 21, which results in a voltagecollapse at that bus as shown in Figure 4.13.In this case study, all system loads are again modeled as persistent PQ loads withrecovery time delay of 10 seconds, the same as in case study 4, but the effect of generatorROP on system voltage stability is considered. The pre—disturbance loading at bus 21in case study 4 is adjusted from 2.48 + jl.40 to 2.40 + jl.36. The system disturbanceremains unchanged, and is again simulated by a loss of line between buses IA and 21,which occurs at t = 5 second.02.92.83.23.1CC2.9500Time (sec)10003.63.5j 3.43.302.82.72.62.5500Time (sec)10000 500 1000 0 500 1000Time (sec) Time (sec)Figure 4.30: Some Generator Excitation Voltages (without Generator ROP’s)Two different cases are examined. In the first case, no generator ROP’s are included.94Generators could operate continuously at higher excitations, as shown in Figure 4.30where excitation voltages of generators at buses 3, 6, 7, and 8 are recorded. This meansthat the generators could have the ability to maintain their terminal voltages so thatlarge reactive power could be supplied to the system. As a result, the system load busvoltages remain stable at a fairly higher values as shown in Figure 4.31.1.2I__________________________________________________________________________0.80.6 load bus #2100 100 200 300 400 500 600 700 800 900 1000Time (sec)Figure 4.31: System Load Bus Voltages (without Generator ROP’s)In the second case, the continuous operating limit, or the second limit, of excitationvoltage for each generator is assumed to be 10 percent over the corresponding pre—disturbance level. If a generator has been operated continuously at a higher excitationthan its second limit over 100 seconds, the rotor overheat protection will cramp thegenerator excitation to this limit.Simulations show that after system transients, the excitation voltage of the generatorat bus 6 exceeds its second limit at t 142 second and is cramped at t = 242 secondsince it has operated continuously at the higher excitation over 100 seconds.951.0598CSC0.95C0.90.851.041.020.9800C8.960.941.05C98SC00.95so20.90.853.153.13.05S8 3c3o 2.9502.92.852.82.75C982SC0C0SC0.8Time (see)(a)(b)(c)0Generator at Bus #3 Tripped Off1001.1200 300 400 508Time (see)600 700 800100 200 300 400 500 600Time (see)Figure 4.32: Generator Excitation and Terminal Voltages (with ROP’s)As a result, this generator loses its voltage control ability and its terminal voltagebegins to decline as shown in Figure 4.32a. Immediately after the excitation reduction96of the generator at bus 6, the reactive power burden is transferred to other generators.As shown in Figure 4.32b and Figure 4.32c, generators at buses 3 and 7 increase theirexcitation rapidly to pick up the system reactive power load.Consequently, the excitation voltage of generator at bus 3 remains over its secondlimit from t 457 second and then is reduced to the second limit after 100 seconds.The loss of reactive power control of generator at bus 3 aggravates the system voltagedeterioration. This causes the generator at bus 7 to increase its excitation more quickly.At t = 576 second, the excitation voltage exceeds its second limit and is cramped at= 676 second. The loss of voltage controls of generator at bus 7 causes large voltagedrops at system load buses as shown in Figure 4.33. Although other generators attemptto increase excitations to supply more reactive power, the rapid voltage collapse initiatesa system transient instability with generators tripped off consecutively due to their lossesof synchronism.Erl)rjUFigure 4.33: EffectThis case study shows the important effect of generator rotor overheat protection on100 200 300 400 500 600 700 800Time (see)of Generator Overheat Protections97system voltage stability. Under heavy system loading conditions, generators are usuallyoperated at high excitation levels in order to maintain a normal system voltage profile.When a system disturbance causes a further system reactive power loss due to the increased the electrical distance between generators and loads, some generators may beoperated with overexcitation, such as generator at bus 6 in this case study. Once theexcitation of the overexcited generator is reduced by its rotor overheat protection, someof its reactive power supply must be transferred to other machines which may be againsuccessively protected, such as generators at buses 3 and 7. The successive losses ofvoltage control at generator buses will then aggravate the system reactive power shortage. System voltage may therefore collapse if there are no other reactive power controlsavailable to the system. This study also suggests that for a heavily loaded system, localload reactive power compensation should be effectively used for system voltage controlrather than heavily counting on the reactive power supply from remote generators.4.5 Maximum Loading Limit by Load Flow and SimulationIn previous sections, various effects of loads and controls on voltage stability are examined. Due to the system disturbances, the system operation may approach its criticalstate dynamically, which is affected by load and control characteristics. Since the systemcritical state is usually characterized by the system MLL which is crucial to voltage stability analysis and control, methods based on the traditional load flow analysis and thedynamic simulation as proposed in this thesis are compared in this section.Case Study 10: System Loading Limit with Persistent PQ loadsIn this case study, all system loads are modeled as constant PQ loads for load flowstudy except that a variable admittance load with initial loading of P+jQ = 2.48 +jl.4098—aSaais assumed for bus 21. Persistent PQ loads with the same time constant of 10 secondsare used for the simulation study. The initial and final system loading conditions, andload power factors are the same for both studies. A loss of the line between buses 14 and21 is assumed as the system disturbance.Load flow studies are made for both normal and post—disturbance system conditionsby increasing load admittance at bus 21. The system loading limits of bus 21 are observedas P + jQ = 5.77 + j3.26 for the normal system condition, and P + jQ = 2.94 + jl.67for the post—disturbance system condition. For the simulation study, the disturbanceis applied to the system at 1 5 second, and the simulation lasts 1000 seconds. P—Vcurves of bus 21 of both load flow and simulation studies are shown in Figure 4.34.1.20.80.60.40.20.5 3.51.5 2 2.5Power drawn from system bus 21Figure 4.34: P—V Curves of Load Bus 21Figure 4.34 shows the results of P—V curves of various studies: Curve 1 from load flowfor the normal system conditioll, Curve 2 also from load flow but for the post—disturbancesystem condition, and Curve 3 from simulation for the system being disturbed. Fromload flow studies, although the disturbance greatly reduces the loadability of load bus 21,99the system loading still lies within its limit, which leads to a conclusion that the post—disturbance system voltage remains stable. However, for the system being disturbed,Curve 3 of Figure 4.34 shows that the system loading limit is less than the load demand,and the system voltage at bus 21 collapses. This concludes that load flow study usuallygives the upper bound of a system maximum loading limit.Case Study 11: System Loading Limit with An Induction Motor LoadThe system loading conditions of this case study are the same as those of case study10 except that the load at bus 21 is replaced by an induction motor load. The initial loadtorque TL for both load flow and simulation studies is 2.0. Load flow study is carried outto find the system loading limit by increasing the motor load torque until the load flowsolution disappears, and the result is shown as Curve 1 of Figures 4.36 and 4.37. Theresults show that the system reaches its loading limit at a motor load torque TL 2.47(Pm = 2.56) at which the load flow diverges.I__Time (sec)Figure 4.35: Motor Load Torque Changes in Simulation Study1000.960.9402000.860.920.90.882.2Motor PowerFigure 4.36: P—V Curves of Motor Terminal Bus0.750.652 2.5Motor Reactive PowerFigure 4.37: Q—V Curves of Motor Terminal BusFor simulation study, it is further assumed that the motor load torque has a stepchange of 0.23 at t = 5 second, and then a gradual increase of 0.02/sec until it reachesTL = 2.6 at t 23 second as shown in Figure 4.35. The F—V and Q—V curves are shown2.3 2.4 2.5 2.6 2.7 2.80.950.85curve1curve 2curve 3curve41 1.5 3 3.5101in the Figure 4.36 and Figure 4.37, respectively, as Curves 2, 3, and 4.The results show that the voltage control due to the transformer tap changing atmotor load bus can increase the system loading limit, but is also affected by the timedelay of the tap changing. The simulations show that the motor load torque of TL = 2.6 isstill within system loading limit if the time delay of tap changing is less than 10 seconds.Due to the slow recovery of system persistent PQ load and the slow increase in motorload torque after the step change, there is enough time for the transformer tap changerto respond before the voltage collapses.1.049Curve 2 of Figure 4.36 and Figure 4.37 shows the case that the tap changing time delayis 10 seconds. The tap changes at t = 15.77 second and t = 25.77 second raise the motorterminal voltage, and extend the motor bus loading limit. After the second tap change,the motor developed torque becomes larger than the load torque of TL = 2.6. The motorslip begins to decrease, and hence the reactive power drops accordingly. Although thefinal generator voltages shown in Figure 4.38 are below their normal levels, the motor50 50Time (sec) Time (sec)Figure 4.38: Some Generator Bus Voltages10200remains stable at the normal voltage with the load torque of 2.6 and other bus loadstotally recovered as shown in Figure 4.39.rl)0000Time (sec) Time (sec)ITime (sec)30Time (see)Figure 4.39: Some System Bus LoadsCurve 3 of Figure 4.36 and Figure 4.37 shows the case that the time delay of tapchanging is set to 15 seconds. The tap changer responses at t 20.77, which increasesthe system loading limit. But this limit is again exceeded at t 21 second due to systemload recovery. As a result, the motor power begins to decrease, while its reactive powerincreases despite voltage drop. Finally, before the second tap change could occur atI 35.77 second, the system has lost voltage stability at motor terminal bus at I = 30.7second. After that, both motor power and reactive power drops along with the voltagecollapse. For comparison, the case where no tap changer is considered is also presentedby Curve 4 of Figure 4.36 and Figure 4.37.103This case study demollstrates that although the system loading limit in steady statecan be determined from load flow related static methods, the loading limit for systembeing disturbed must be evaluated by detailed system simulations. Moreover, the voltagecontrol effect of transformer tap changing, if fast enough, can effectively increase the MLLof a system being disturbed to the extent even larger than the MLL of steady state fromload flow.104Chapter 5CONCLUSIONS AND REMARKS5.1 Conclusions of the ThesisThis thesis project is mainly concentrated on the analysis of voltage stability of a powersystem through time domain simulation techniques. Better understandings of systemvoltage instability phenomenon are gained through close examinations of the effects ofload and control components on system voltage stability.A 21 bus sample power system is chosen for the simulation studies. Steam andhydro—electric generating units, various types of loads, and many reactive power controldevices are modeled with emphasis on the dynamic behaviors of system loads and reactivepower related components. System critical buses are defined and identified from voltage—reactive power sensitivity analysis for the voltage stability study.A comprehensive time domain simulation program is developed based on the implicitTrapezoidal integration rule and the step doubling integration step size control algorithm.A new variable elimination method is devised for some dynamic load to include the related nonlinearities in load flow iterations so that the variable extension and convergenceproblems can be avoided. A new two—step procedure is also developed for efficient andsystematic solution of high order system Jacobian matrix equations.The effects of various types of loads and reactive power controls on the voltage stabilityare thoroughly examined through designed case studies so that the dynamic voltagebehavior of a power system in various operating conditions can be clearly demonstrated105and clarified.From case stndies in this thesis, the conclusions are drawn as follows.1. Voltage instability of a dynamic power system is a very complicated phenomenon.It may involve a fast transient voltage instability and/or a slow voltage decline followed by a sudden collapse, depending on the system operating conditions, systemload and control dynamics, and types, locations, and seventies of system disturbances.2. Induction motor loads, which constitute the major part of industrial loads, mayhave great influences on system voltage stability due to its more or less constantpower and negative Q—V characteristics which means that the motor will draw•more reactive power when its terminal voltage decreases.When a system voltage drops, which causes a reduction of motor developedtorque, the motor will pick up the load very quickly by increasing its slip. Morecurrent will be drawn from the system, which causes further voltage decrease. Depending on the magnitude of a disturbance, this interaction between the motor andthe supply system may experience either a slow system dynamics which drives themotor towards its critical state for a long time before it starts stalling, or a fastsystem transients which upsets the motor’s stable operation so quickly and causesthe motor stalling in a few seconds. During the motor stalling, the reactive powerdemand increases very quickly, which causes the system voltage collapse.Unlike system angle instability which is caused by the generator power imbalance, the voltage instability caused by the loss of motor stable operation can notbe judged by the power imbalance alone. It also involves the reactive power equilibrium of the system. This means that the loss of motor power due to the motor106stalling does not necessarily result in a voltage instability since the motor terminalvoltage can be controlled before the motor Q—V characteristics becomes positive.3. When a disturbance causes system voltage drop, a constant impedance load willdraw less power and reactive power from the supply system than that of a constantpower load, which has a favored damping effect to halt the further voltage decline.This damping effect is crucial to system voltage stability when there are inductionmotors operating near their critical state. Constant power load, which is not voltagesensitive, is therefore a stiff system load. This load characteristics is harmful tosystem voltage stability because it will draw the same power and reactive powerfrom the supply system despite voltage decline.Although a voltage dependent load has a favored effect on system voltage stability, it has different impacts on system angle stability. On the one hand, whena disturbance causes a system power supply shortage, such as loss of a generator,voltage sensitive loads will draw less power due to voltage decrease. This loadreduction is helpful to balance system power so as to stabilize the generator operation. On the other hand, when a disturbance causes system voltage collapse dueto, for example, the loss of a motor stable operation, the rapid decrease of loadsdue to voltage collapse may upset the system power balance causing a transientsystem angle instability.4. A persistent PQ load may demand constant power and reactive power but involving a time delayed recovery. The attempt to maintain pre—disturbance load levelby increasing load equivalent admittance despite voltage decline may cause someparticular system load bus exceeding its post—disturbance loading limit, causingvoltage instability at that bus.If the post—disturbance system operating condition is such that the voltage107collapsed bus has a relatively short electrical distance with other system buses, theloss of voltage control at that particular bus may spread out to the other parts ofthe system causing a complete system voltage collapse. On the other hand, if thevoltage collapsed bus is far away electrically from the rest of the system, it mayhas little impact on the other bus voltages, and then the rest part of the systemmay have a chance to maintain both voltage and angle stability.5. A transformer tap changing may have either beneficial or detrimental effect onsystem voltage stability depending on its location and load characteristics. Itseffect is crucial when a system operates near its critical state.The effect of tap changing at step—up transformer of a generator is alwaysbeneficial to system voltage stability since it raises the transmission voltage, andhence reduces the network current and the corresponding reactive power loss.The tap changing of distribution transformer at a system load bus has differenteffects on system voltage stability depending on load characteristics. For a load withpositive Q—V characteristics, such as an exponential form PQ load with positiveexponents, tap changing which raises the load side voltage, will result in morereactive power drawn from the system causing system side voltage to decline further.This may push the system over its loading limit causing a possible voltage collapseif the system has operated near its critical state. On the other hand, for a loadwith negative Q—V characteristics, such as an induction motor load, the voltageincrease by changing the transformer tap will reduce the reactive power drawn fromthe system so as to stabilize the system side voltage.6. Effective system VAR control and adequate compensation is very important tomaintain system voltage stability. The effectiveness of a VAR compensation depends on the types and locations of VAR devices, the system operating conditions,108the VAR control speed, and the VAR capacities.Since the VAR compensation with a fixed capacitor are directly proportional tothe square of the bus voltage regardless of load demands, it may not be effectivein most system situations involving fast voltage drop. This also suggests that aheavy fixed capacitor compensated system may be vulnerable to voltage instability.For a system operating condition involving slow voltage decline, fixed capacitorscompensation can be used to support system voltage, but some of the capacitorsshould be switched out of the system to avoid over—compensation when systemvoltage is back to normal. This is especially crucial when the bus load has anegative Q—V characteristic.The VAR compensation with an SVC of sufficient capacity is very effective tostabilize the voltage of a power system. It is effective in both fast transient andslow dynamic operating conditions due to the fast response of the negative feedbackcontrol. However, there is a dynamic interaction between an SVC voltage controland a generator excitation control, which may cause a system oscillation. Thisobservation suggests that the SVC voltage control must be coordinated with thepower system stabilizer(P S S) design.7. In a heavily loaded system, a generator rotor overheat protection may limit its rotorwinding current by cramping the excitation voltage. As a result, it will reducethe reactive power supply to the system, and that reactive power burden mustbe transferred to other generators which could be also protected. The successivelosses of voltage controls at generator buses will aggravate system reactive powershortage, which may cause a possible system voltage collapse. This also suggeststhat an effective system wide VAR compensation should be effectively designed,and a heavy dependence of reactive power supply from remote generators could109result in voltage instability due to the possible generator rotor overheat protection.8. Case studies in this thesis demonstrate the importance of dynamic effects of system loads and control devices on system voltage stability. Although the MLLdetermined by load flow for power system in steady state is generally larger thanthe MLL determined by simulation for a system being disturbed, the voltage control effect of transformer tap changing, if fast enough, can increase the MLL of aninduction motor load even larger than that determined from load flow.5.2 Future Research WorkThe following aspects are suggested for future research work.1. Although the effects of individual system typical loads on system voltage stabilityare closely examined with a sample power system, the real system bus load is farmore complicated. It may be a combination of these typical loads or more, whichrequires further investigation. The study may involve more detailed modeling of busload, estimation of some other unknown loads, and identification of load parameters.2. 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Sutanto, “Differeilt types of voltage instability,” IEEE/PESSummer Meeting, 93 SM 518-1 PWRS, Vancouver, July 18—22, 1993.116Appendix ADERIVATION OF STEADY-STATE MOTOR EQUATIONSIn this appendix, the steady-state induction motor power equation (2.5) is derived frommotor equivalent circuit which can be obtained from a L’ model for the motor.Similar to the Park’s equations for synchronous machines, the voltage equations of athree phase symmetrical induction motor can be described in a d—q frame of referencerotating at synchronous speed Wb as(a) Stator winding voltage equations:Vds = rsIds—qs+1qs (A.1)rsIqsds+—ds (A.2)(b) Rotor winding voltage equations:Vdr = —(1 wr)qr + ±qr (A.3)= rrIqr+(l—wr)dr+dr (A.4)In the foregoing equations, r’s are winding resistances, wr is motor speed, Vd’s andVa’s are winding voltage d—q components with V = Vqr 0 for the rotor, Id’s and Iq’Sare winding current d—q components, and L”s are corresponding winding flux linkages, inWebers per second, which have the following relations.ds = (X13 +Xm)Ids+Xmldr (A.5)qs = (X18 + Xm) ‘qs + Xm Iqr (A.6)117= (Xir + Xm) ‘dr + Xm I (A. 7)qr = (Xir + Xm) Iqr + Xm Iqs (A.8)where X1. and Xir are respectively the stator and rotor winding leakage reactances, andXm is the magnetizing reactance.For steady state conditions, the winding transients are not included. The corresponding steady state voltage equations can then be obtained by dropping the derivative termsfrom equations (A.1) — (A.4), which gives(a) Stator winding voltage equations:Vd5 r5 Icts—/‘qs (A.9)TIq + ds (A.1O)(b). Rotor winding voltage equations:o rrIdr—(1—wr)qr (A.11)o = rrIqr+(1—wr)bdr (A.12)The steady state equations of an induction motor can also be expressed in phasors.For this, substitute equations (A.5) — (A.8) into equations (A.9)— (A.12), and definemotor slip s = 1— wr, terminal voltage V = Vds + j stator current I I +and rotor current ‘r = ‘dr + ‘qr which, through some manipulations, givesV = (rs+Xis)Is+jXm(Is+Ir) (A.13)o = (+Xir)Ir+jXm(Is+Ir) (A.14)Equations (A.13) and (A.14) lead to the well—known equivalent circuit of an inductionmotor as shown in Figure A.1.Defining Z = r5 + jX18, Zr rr + j Xir, Zm j Xm, Ra iEi r, and Za =Zr + Ra = + j Xir, the impedance Z as seen from the motor terminal becomes118Z=Zs+ZmHZa (A.15)where the symbol “ “ means “parallel with.”rs Xis rr XirAIs1-srrSFigure A.1: Equivalent Circuit of An Induction MotorThe voltages Vm across XM, and Va across as shown in Figure A.1 can beexpressed asVm = (ZmHZa) (A.16)VaRa (A.17)Substituting equation (A.16) into (A.17) givesVamV (A.18)The torque and power equation (2.5) can be derived as follows.(a) Air gap power FEFE = Re[Va1]= Re[Va (Va/Ra)*] (A 19)= Re{Va2/Ra]V 2/j— a /1La119/V4JrVrn Xm(b) Motor developed torque TETE=(A.20)(c) Power drawn by the motor FmPm = Re[VI]= Re[V (V/Z)*] (A.21)= V2/Re[Z*](d) Reactive power drawn by the motor QmQm = Im[VI]= Im[V(V/Z)*] (A.22)= V2/Im[Z*]The above equations can be expanded and re—organized in such a way that the motortorque and powers can be expressed explicitly in terms of motor slip .s, the terminalvoltage V, and the motor parameters p with p = {r8, rr, Xis, Xir, Xm}.Leta1 = rsrra2 = X15 X1,. + Xis Xm + Xir Xma3 = rr(Xis+Xm)a4 = rs(Xir+Xm)a5 = rrX= rs(Xjr+Xm)2= rrX2M= r5120Cl = (Xir+Xm)aiC2—ra3d1 = a+ad2 = 2rsrrXmd3 = a+athe motor torque and power can then be expressed as follows.a5 sTE = V2 = fE(s,p)V2 (A.23)d1s2H-d.s+3b1s2 + b2s + b3 V2 = fp(s,p)V2 (A.24)Fm dis+ds+d3Q — c1s2 +c2 2 = fQ(s,p)V2 (A.25)d1s2 + d2s + d3Equations (A.24) and (A.25) show variable impedance characteristics of an inductionmotor in steady state operating conditions. The power and reactive power of an inductionmotor depend on both terminal voltage V and motor slip s.121

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