A COMPREHENSIVE SIMULATION STUDY OF THE VOLTAGE STABILITY OF A LARGE POWER SYSTEM By WENJIE ZHANG B. Eng., Huazhong University of Science and Technology, PRC, 1982 M. Sc., Taiyuan University of Technology, PRC, 1985 A THESIS SUBMITTED IN PARTiAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1993 © Wenjie Zhang, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Electrical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T1W5 Date: L/46ef / ) 1 Abstract The voltage stability problem has become a growing concern in power system planning and operation. Many large interconnected power systems have experienced voltage insta bilities which involve fast transients and/or slow dynamics. Although load flow related static approaches have been well developed to characterize the system maximum loading limit as the voltage collapse point, the mechanism of how system operation approaches its voltage collapse point and how this collapse point is affected by system dynamics are still obscure. This thesis provides the answers to these two basic questions through the investigation of effects of loads and reactive power controls on system voltage stability by detailed time domain system simulations. The importance of system dynamics in the determination of the 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 appropriately modeled. Secondly, a comprehensive power system simulation program is developed based 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 and backward—substitution procedure for solving the system Jacobian matrix equations are devised. Different system disturbances are simulated, and the exact timing of system changes is implemented. Finally, a 21 bus sample power system is chosen for the voltage stability study. In the case studies, the effects of loads, control devices, and system disturbances on system voltage stability are thoroughly examined. The voltage instability of a power system is a very complicated phenomenon, which, 11 depending on the location, the type, and the severity of a system disturbance, may involve a fast transient voltage instability, or a slow voltage deterioration followed by a sharp collapse. It is closely associated with system reactive power—related controls, and is strongly affected by the load characteristics. The beneficial and detrimental effects of loads and reactive power controls on voltage stability should be carefully analyzed so that the information can be used in voltage stability control designs. 111 Table of Contents Abstract List of Tables vii List of Figures viii Acknowledgements 1 INTRODUCTION 1 1.1 Power System Voltage Stability Problems. 1 1.1.1 Power System Angle Stability 1 1.1.2 Power System Voltage Stability 1.2 1.3 1.4 2 xi Power System Voltage Stability Studies . . . . 1.2.1 Static Voltage Stability Studies 1.2.2 Dynamic Voltage Stability Studies Proposed Thesis Study 1.3.1 System Component Modeling 1.3.2 Dynamic Simulation Techniques 1.3.3 Dynamic Simulation of Voltage Stability . Thesis Structure MODELING OF POWER SYSTEM COMPONENTS 2.1 Typical Power System Load Models 2.1.1 A Composite Bus Load Model iv 10 10 10 2.2 2.3 2.4 3 2.1.2 Induction Motor Load 11 2.1.3 Voltage Dependent PQ Load 14 2.1.4 Constant PQ Load with a Recovery Time Constant 15 Component Models of a Generating Plant 16 2.2.1 Synchronous Generator 16 2.2.2 Field Excitation System 18 2.2.3 Governor and Turbine Systems 19 2.2.4 Power System Stabilizer (PSS) 21 System Voltage Control Devices 22 2.3.1 On Load Transformer Tap Changer 22 2.3.2 Static VAR Compensator (SVC) 23 2.3.3 Generator Rotor Overheat Protection 24 Modeling of Transmission Network 25 SIMULATION TECHNIQUE FOR VOLTAGE STABILITY STUDIES 27 3.1 3.2 3.3 Complete System Equations 28 3.1.1 Machine and Transmission Coordinates 28 3.1.2 Complete System Equations 29 Simultaneous Implicit Integration of System Equations 33 3.2.1 Implicit Trapezoidal Rule 33 3.2.2 Discretization of System Differential Equations 35 Power System Simulation Program 39 3.3.1 Load Flow Calculation 40 3.3.2 Solution of System Jacobian Equations 42 3.3.3 Integration Step Size Control and Exact Timing 46 3.3.4 System Contingencies 49 v 3.3.5 4 . . 51 . DYNAMIC VOLTAGE STABILITY STUDIES 53 4.1 A Sample Power System for Voltage Stability Studies 54 4.1.1 A 21 Bus Sample Power System 54 4.1.2 Basic System Data 55 4.2 4.3 4.4 4.5 5 Flowchart of System Simulation Program The Critical System Load Buses 59 4.2.1 Analysis of the System Operating Condition 4.2.2 Critical System Load Buses . . Effects of System Loads on Voltage Stability 59 60 63 4.3.1 Effect of an Induction Motor Load 64 4.3.2 Effect of Exponential PQ Loads 72 4.3.3 Effect of Persistent PQ Loads 77 Control Effects on System Voltage Stability 81 4.4.1 Effect of transformer tap changing 81 4.4.2 Effect of System VAR Compensation 87 4.4.3 Effect of Generator Rotor Overheat Protection 93 Maximum Loading Limit by Load Flow and Simulation CONCLUSIONS AND REMARKS 98 105 5.1 Conclusions of the Thesis 105 5.2 Future Research Work 110 Bibliography 111 A DERIVATION OF STEADY-STATE MOTOR EQUATIONS 117 vi List of Tables 4.1 Data of Transmission System 55 4.2 Data of Generator PV and Load PQ 56 4.3 Data of Generator Parameters 56 4.4 Data of Field Excitation System 57 4.5 Data of M—H Governor and Hydro Turbine 57 4.6 Data of E—H Governor and Steam Turbine 4.7 Data of PSS’s for Given Operating Condition 4.8 System Power Generation and Consumption 59 4.9 Voltage Profile of the Given System Condition 59 . 4.10 System Voltage Sensitivity Matrix 4.11 Number of Buses Largely Affected by Bus 58 . . . . 58 62 j 63 4.12 Data of an Induction Motor Load 64 4.13 Adjusted System Voltage Profile 65 4.14 Adjusted PSS Parameters 65 4.15 Initial State of the Induction Motor 65 4.16 Data of SVC Parameters 88 vii List of Figures 1.1 Power—Angle and Voltage—Power Curves. 2.1 Classification of a Composite Bus Load 11 2.2 Composite Bus Load Modeling 11 2.3 Constant PQ with a Delayed Recovery 15 2.4 A Fast Excitation System 18 2.5 Hydro Turbine and Governor System 19 2.6 Non-Reheat Steam Turbine and Governor System 2.7 A Power System Stabilizer 21 2.8 A Transformer with OLTC 23 2.9 A TCR with Fixed Capacitor 24 4 . 20 . 2.10 Block Diagram of an SVC with Voltage Control 2.11 Overheat protection characteristic 24 25 . 3.1 Machine and Transmission Network Coordinates 3.2 Block Diagram of Component Interaction 30 3.3 Illustration of Implicit Trapezoidal Rule 34 3.4 An Example of System Jacobian Matrix Equation 44 3.5 Step Size Change for Exact Timing 50 3.6 Overall Flowchart of System Simulation Program 52 4.1 A Sample Power System under Study 54 4.2 A System Bus with Typical Load 64 viii . . . 28 4.3 Motor Response to Step Change in Load Torque 4.4 Some Generator Bus Voltages 4.5 System Voltages at Some Load Buses 68 4.6 Transient P—V and Q—V curves of Motor Terminal Bus 69 4.7 Some Generator Rotor Angles 4.8 Transient Responses of Motor Variables 71 4.9 Voltage Response at a Load Bus 73 4.10 Reactive Power Drawn by the Motor 74 4.11 Voltage Response at a Generator Bus 75 4.12 Rotor Angle of a Generator 76 4.13 Voltage Collapse at Bus 21 78 4.14 Load Power and Reactive Power vs. Load Admittance 78 4.15 Some Other System Load Bus Voltages 79 4.16 Some Generator Bus Voltages 80 4.17 Loads at Some System Buses 80 4.18 Tap Changing Effect on Motor Terminal Voltage 82 4.19 Tap Changing Effect on Motor Torque 83 66 • • 4.20 Tap Changing Effect on Motor P and Q . . 67 70 84 4.21 Motor P—V Curve with and without Tap Changing 84 4.22 Effect of Tap Changing at Other System Load Buses 85 4.23 Motor Bus P—V Curves 86 4.24 Tap Changing Effect on Motor Power and Reactive Power . • 4.25 Effects of Different VAR Compensations 4.26 VAR Compensation of Fixed C Switched in Different Time 88 . . 4.27 VAR Compensation of SVC’s with Different Capacities 4.28 Effects of VAR Compensation of Fixed Capacitor and SVC ix 87 89 90 . . 91 4.29 VAR Compensation of Fixed Capacitor and SVC 92 4.30 Some Generator Excitation Voltages (without Generator ROP’s) 94 4.31 System Load Bus Voltages (without Generator ROP’s) 95 . 4.32 Generator Excitation and Terminal Voltages (with ROP’s) 96 4.33 Effect of Generator Overheat Protections 97 4.34 P—V Curves of Load Bus 21 99 4.35 Motor Load Torque Changes in Simulation Study 100 4.36 P—V Curves of Motor Terminal Bus 101 4.37 Q—V Curves of Motor Terminal Bus 4.38 Some Generator Bus Voltages 4.39 Some System Bus Loads A.1 Equivalent Circuit of An Induction Motor x . . . . . 101 102 103 119 Acknowledgements This thesis is dedicated to my parents, who inspired and encouraged me to take this opportunity of challenge. I wish to express my grateful thanks to my research supervisor Dr. Y.N. Yu for his invaluable guidance, constant encouragement, and great patience during the research work and the accomplishment of this thesis. My appreciation goes to Dr. M.D. Wvong and Dr. H.W. Dommel for their readiness to help and inspiring and helpful discussions. I am also indebted for the financial support from the Natural Science and Engineering Research Council of Canada and the University of British Columbia. Finally and gratefully, I thank my wife, Wei, for her special thoughtfulness, encourage ment, understanding, and patience throughout the difficulties of my graduate program. xi Chapter 1 INTRODUCTION 1.1 1.1.1 Power System Voltage Stability Problems Power System Angle Stability Since the beginning of large power system interconnections, low frequency system oscil lations and transient and dynamic stability problems have developed. The main concern of these stability problems is to keep all synchronous generators in synchronous step by providing them with adequate damping if they oscillate, reducing generation or adding dynamic braking resistance if there is a power surplus, and shedding some loads if there is a power shortage. These problems, which may be classified as the angle stability prob lem, have been thoroughly studied and are well understood. Control means to stabilize the system, such as dynamic resistance braking, force excitation, fast valving, HVDC modulation, power system stabilizer (PSS), and generation tripping and load shedding, have been very well developed. 1.1.2 Power System Voltage Stability As large power system interconnections continued, the power demand kept on increas ing. But the environmental restriction on the building of new transmission lines also increased. There is a tendency in power system planning and operation to load the exist ing generationand transmission equipments as much as possible. This practice, coupled with insufficient and inadequate reactive power supplies for a power system, has caused 1 many system voltage failures in the past, which may or may not also involve an angle instability. Voltage stability has gained a special attention recently [1]. To have a better under standing of this problem, some major system voltage failure events are briefly reviewed as follows. EDF — December 19, 1978 [4] It was a severe cold winter morning in France, and the temperature drop was much greater than anticipated. A rapid rise of power demand caused the increase in several power transfers. It was also marked with the increase in active power losses and espe cially reactive power losses. Several 400 KV lines were overloaded, and system voltages deteriorated very badly. Some EHV/HV tap changers were blocked, and a 5% distribution voltage drop was ordered in some area. The system was stabilized for a while. But due to the overload of the persistent load demand and the loss of many reactive power supports, many major transmission 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. Generations were tripped one after another. At one time, the power deficiency was around 9,000 MW and the full installation capacity was 90 GW. The voltage deteriorated from 400 KV to 300 KV and below but did not collapse completely. The underfrequency relay control did not 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 tripping some 400/225 KV transformers feeding a load area. 2 TEPCO — July 23, 1987 [6] Japan has a 50 Hz system in the north and a 60 Hz system in the south, and the two systems 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 maximum power demand of 39.1 GW in the morning, the demand dropped to 36.5 GW at 12:40 during the lunch hour. But the demand increased again rapidly at a rate of 400 MW per minute from 13:00, much faster than estimated. It was attributed to the air conditioning devices 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 500 KV to 460 KV at 13:15 and further to 370 KV in the western part of the system and to 350 KV in the central part of the system at 13:19. Three substations of 8.168 GW were tripped, and 2.8 million customers were lost. The three substations were brought back to 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 an angle instability, and that the voltage does not necessarily collapse completely. It is also very 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 of all reactive power—related generating, consuming and control components must also be taken into consideration. 3 1.2 Power System Voltage Stability Studies Although some effort has been made to clarify the mechanism of voltage instability and to devise some methods to prevent a system from voltage collapse, most researches have been devoted to the determination of maximum loading limit (MLL) of a power system by using 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 Studies While a power—angle curve for equal—area study has a shape of an inverse V with a maximum power point on top, the voltage—power curve for the MLL study has a V shape with the point of maximum load power towards the right as shown in Figure 1.1. Voltage Generator Power electric power mechanical power Pm Rotor Angle Load Power Figure 1.1: Power—Angle and Voltage—Power Curves It is well known that the upper portion of the voltage—power curve represents a stable operation whereas the lower portion represents the unstable [2] [3]. In—between exists a point 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 stable 4 operating state to the point of voltage collapse. All static methods are essentially related to a Jacobian matrix analysis from the results of a system load flow. System voltage may collapse 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 direct relation between the singularity of the load flow Jacobian and the singularity of the 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 load flow. This method was expanded for voltage stability studies by Tamura et al. [9]. 2. Static Bifurcation Theory Bifurcation theory is concerned with the branching of 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 state stability 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 equilibrium and remain stable until one of the parameters reaches a critical value at which system state branches at a saddle point. This is the very point where the load flow Jacobian matrix becomes singular. When multiple solutions of load flow exist, they correspond to the multiple equilibria of the dynamic system in the neighborhood of the bifurcation point. Therefore, bifurcation analysis can be used to characterize these equilibria and to identify the critical parameters, which are the very important information for system control design. 3. Sensitivity Method Sensitivity in voltage to system parameters near the crit ical state provides very useful information to system operation. It can be used to identify critical system buses and also effective means of controls [37] [58]. Based on the analysis, a variety of indicators of proximity of the system state to the point 5 of voltage collapse can also be defined [11] [14] [15]. Therefore, adequate control can be exerted on the system to keep the system state away from voltage collapse. 1.2.2 Dynamic Voltage Stability Studies Many 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 on the 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 maximum loading limit (MLL) based on load flow analysis may give an upper bound of the voltage stability region, the system may have lost its voltage stability before that limit can be reached due to system dynamics [16]. Therefore, the MLL can only indicate the loading condition at which system voltage collapse may occur. It cannot answer the questions of how the system voltage approaches the collapse point and how this collapse point is affected by the system dynamics. For this sake, a comprehensive system simulation must be resorted to so that the system dynamics can be adequately included in the voltage stability studies. 1.3 1.3.1 Proposed Thesis Study System Component Modeling The major part of this thesis is to investigate the effects of load and system reactive power components on voltage stability. For that, the system behavior will be simulated comprehensively. All important loads and all system components that generate, consume and 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 power 6 of the system, will also be modeled. There are in general four major components to a power system, the generating plants, the transmission network, the system loads, and the system control devices. All of these components will be modeled. 1.3.2 Dynamic Simulation Techniques A nine—machine pOwer system is chosen for the simulation study. It is felt that the system is large enough to display the dynamic interactions among system components, such as generators, loads, transmission network and system controls. The high order system model with inherent system ilonlinearities requires development of an appropriate simulation 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 direct inversion of a large Jacobian matrix in each integration step, a new technique of solving the Jacobian matrix equation is developed. 1.3.3 Dynamic Simulation of Voltage Stability The main objective of this thesis is to investigate the gelleral effects of special types of loads and major reactive power—related components on the dynamic behavior of the voltage 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 important role in a voltage collapse process, their effects must be examined on a large power system with detailed system models so that the dynamic interactions among system components can be included. In addition, the persistent PQ load characteristics [54] contributing to the TEPCO voltage failure should also be investigated. For this reason, the case studies are designed and carried out to clarify the effects on voltage stability of induction motor 7 stalling, transformer tap changing, fixed capacitor and SVC compensations, persistent and general static loads and generator overheat protections. 1.4 Thesis Structure In Chapter 2, the power system component models are presented. System bus loads are modeled by the combination of typical loads. Load dynamics are represented by induction motor loads and constant power and reactive power (PQ) loads with delayed recovery. Conventional static loads are modeled as voltage dependent loads. A generating plant is modeled in detail, including synchronous generator, field excitation system, governor and turbine system, and power system stabilizer. System controls, which have strong effects on system voltage stability, such as on load tap changer of a transformer(OLTC), the static VAR compensator(SVC), and generator rotor overheat protection(ROP) are also modeled. Finally, the transmission network is represented by the system node voltage equations. Chapter 3 describes the nonlinear time domain simulation technique. The overall sys tem equations can be obtained through a hybrid coordinate system for both generators and the transmission system. Based on the trapezoidal rule, system differential equa tions are discretized to obtain the corresponding difference equations. Newton—Raphson’s method is then used to solve the system equations. A systematic method is developed to solve the high dimension Jacobian matrix equation without involving a direct inver sion of a large matrix. In order to capture both fast transients and slow dynamics of a voltage instability, a variable step size mechanism is implemented. System disturbances and sudden topological changes are considered and exactly timed. In Chapter 4, a 21 bus power system is presented. Since the system has low frequency oscillations for the given operating conditions, power system stabilizers are designed. 8 Next, the most voltage sensitive bus of the system is identified through system voltage sensitivity allalysis. Typical system loads and control actions are considered for the most voltage sensitive bus. Case studies are then conducted to clarify the effects of various loads and controls on system voltage stability. Finally, the results of voltage stability from load flow and simulation studies are compared to demonstrate the effect of system dynamics on the system maximum loading limit. Finally, conclusions are drawn and future research projects are suggested in Chapter 6. 9 Chapter 2 MODELING OF POWER SYSTEM COMPONENTS For voltage stability studies, an appropriate power system model is required. Although a static model, like load flow equations, is adequate for steady state analysis, a more detailed model, including both load dynamics and reactive power generating and control components, must be used in a voltage stability study of a system involving dynam ics. This is because the voltage instability phenomenon of a power system may involve both fast transients and slow dynamics [13]. Therefore, all system components involving transients and dynamics should be included in the voltage stability study. Typical power system load models are presented in section 2.1, generating plant in section 2.2, system control devices in section 2.3, and, finally, transmission network equa tions in section 2.4. 2.1 2.1.1 Typical Power System Load Models A Composite Bus Load Model Examinations of major voltage failures show that system loads have significant effects on the voltage stability of a power system. But, a power system load is usually made up of numerous individual loads with different characteristics, and the information about some individual loads may not be available [18][19]. Therefore, it is almost impossible to derive an exact model for a power system load. Instead, the power system loads may be approximately represented by a few equivalents, e.g., industrial, commercial, and 10 residential loads [20] as shown in Figure 2.1. load bus system bus industrial load commercial load residential load Figure 2.1: Classification of a Composite Bus Load It is suggested in [21], for example, that a composite system bus load may be modeled by an equivalent induction motor in parallel with a static load as shown in Figure 2.2, load bus system bus dynamic load static load Figure 2.2: Composite Bus Load Modeling where the equivalent induction motor represents the dynamics associated with the major industrial load, while the static load represents the voltage dependence of commercial and residential loads. Therefore, the load effect on system performance may be found by studying the following typical loads. 2.1.2 Induction Motor Load Induction motors constitute the major part of an industrial load. They have a fast response to system disturbances to maintain more or less a constant power and draw more reactive power from the power supply. This feature of quick load pick—up and more 11 reactive power absorption during a system disturbance is one of the major causes of a dynamic voltage instability [22]. The basic equations of a three—phase induction motor may be derived from Park’s equations for a synchronous generator. However, there is no field winding, and the d and q axis windings are symmetrical for the motor. There is a slip of rotor windings with respect to the stator rotating field of the motor. Neglecting the electromagnetic transients (EMT’s) in the stator windings, the induction motor may be described by a third order model as follows. (a) Rotor motion equation: (2.1) 2HTL—TE (b) Rotor winding voltage equations: 1 Tè = 0 eq T —e—(xo—x’)Iqs+bTse —eq + (x 0 — x ) ‘ds — b (2.2) 0 S ed T (c) Stator winding voltage equations: 8 Vd = e+rsIds—xIqs = eq+xIds+rslqs (23) (d) Electromagnetic torque equation: TE = edldS + eIqs 12 (2.4) where H : motor inertia constant r : stator resistance rotor open circuit reactance blocked rotor reactance T e, e : rotor open circuit time constant : rotor internal voltages Vcjs, Vqs : stator terminal voltages ‘ds,’qs s : stator currents : motor slip TL : mechanical load torque TE : electromagnetic torque wb : synchronous speed of the system In steady state system condition, both mechanical and electrical transients of the mo tor are not included. Hence, equations (2.1) and (2.2) are reduced to algebraic equations to represent the steady state behavior of the motor. In that case, the motor torque, power, and reactive power can be determined by TE 2 fE(s,p)V Fm 2 = fp(s,p)V m Q 2 = fQ(s,p)V (25) V = where fE, fp and f are functions of motor slip s and motor parameters p. The derivation of equation (2.5) is given in Appendix A. 13 2.1.3 Voltage Dependent PQ Load Since a voltage instability does not necessarily involve system frequency deterioration, the frequency dependence of commercial and residential loads may be neglected. The power and reactive power drawn from a system bus may then be described as a function of bus voltage. It has been a common practice that a bus load is divided into a constant power, a constant current, and a collstant impedance load components. This concept leads directly to a load model of quadratic form [23]. P = Po Q = [pp + pi () + Pz (2.6) Qo[qp+qi()+qz()] where Po and Qo are power and reactive power at normal operating voltage Vo, pr,, p, and Pz are the coefficients of power portion of constant power, constant current, and constant impedance loads, and qp, qj and q are those of the reactive power portion of the load. These coefficients must satisfy Pp+Pi+Pz = 1 qp+qi+qz = 1 A more general static PQ load model [24] has exponential forms as follows. F Po() VQ Q Qo() (2.7) Here, again, P 0 and Qo are power and reactive power at normal operating voltage V , and 0 a and characterize the voltage dependence of the load. Equation (2.7) also includes three special cases, that is, the constant power load with a = load with a = = 0, the constant current = 1, and the constant impedance load with a = The appropriate selection of a combination of pr,, Pi, Pz, qp, = 2. qi, and q in equation (2.6), and the exponents of a and 3 in equation (2.7) should be made by investigating the actual system load. 14 2.1.4 Constant PQ Load with a Recovery Time Constant This type of load demands a constant power throllgh self control to maintain the required power despite a system voltage decrease. A typical load power response to a step change in voltage may be in the form as shown in Figure 2.3. (a) voltage step change Po (b) load power response Figure 2.3: Constant PQ with a Delayed Recovery The sudden dip in voltage causes an instant decrease in load power demand followed by a recovery to its normal value P . The demand is persisting, but involving a time 0 delay. There is a such kind of load of modern air conditioning device as reported in TEPCO power failure [6]. This type of constant power load, which may be referred to as persistent PQ load, may be modeled by a changing load equivalent admittance with a time delay. The time delay is associated with the time needed for the control devices to respond, and may also involve a human factor [25]. Assuming an exponential recovery in both power and reactive power, this type of PQ 15 load may be modeled as follows. TGGL = 2 Po—GLV TBBL = 2 Qo—BLV (2.8) where V is the load terminal voltage; GL and BL are the varying equivalent load con ductance and susceptance; TG and TB are the corresponding time constants; and P 0 and Q are the load power and reactive power at normal operating condition with D 10 v-2 UIL0V 0 i—I — — (2.9) V 0 — i72 — 2.2 Component Models of a Generating Plant V A generating plant consists of a synchronous generator, a field excitation system, a gov ernor controlled turbine system, and probably a supplementary control, such as a power system 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 Generator Neglecting electromagnetic transients in armature windings, a synchronous generator may be represented by a fifth order model as follows. (a) Rotor motion equations: M 6 = TmTeDW wb(w—1.0) 16 (2.10) (b) Rotor winding voltage equations: E 0 T = —E—(xd—x)Id+EFD TE’ = -E’ TE - (x-x)Id+E +TE (2.11) E+(xq_x)Iq (c) Armature winding voltage equations: Vd = E—raId+xIq Vr = E’—xId—raIq (2.12) (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 constant D : mechanical damping coefficient w : rotor speed 6 : rotor angle Tm, Te : mechanical and electromagnetic torques ra : armature resistance cl—axis transient reactance x, x : d— and q—axis sub—transient reactances 0 T : d—axis transient time constant T,, T : d— and q—axis sub—transient time constant 17 q—axis transient voltage d— q—axis sub—transient voltages E’, E’ field voltage EFD Vd, d and q components of terminal voltage Id, ‘q d and q components of armature current synchronous speed of the system where S is in radian, M and time constants are in seconds, and all other variables are in per unit. 2.2.2 Field Excitation System A fast—response exciter and voltage regulator system shown in Figure 2.4 is chosen for the study. Vt EFDO EFD UE Figure 2.4: A Fast Excitation System It may be described by the differential equation TA EFD = —EFD + EFDO + KA (UE + VREF — 4) (2.14) where KA and TA are respectively the equivalent gain and time constant of the excitation system. 4 and VREF are the generator terminal voltage and its reference, respectively. 18 UE is the supplementary excitation control. EFDO is the initial value of excitation voltage EFD. The excitation voltage has physical limits, which may be called the first limit Emini It also has a lower operating limits Eyjj Emai (Emin2, Emax2), which may be called the second limit, determined from the consideration of the generator rotor overheat protection. When a generator has been excited continuously over the second limit for a prescribed period of time, relay protection will cramp the excitation voltage to the second limit, or even trip off the generator. 2.2.3 Governor and Turbine Systems (a) Mechanical—Hydraulic Governor and Hydro Turbine System The block diagram of a mechanical—hydraulic governor and turbine system for a hydro—electric plant is shown in Figure 2.5. Go (0 Gomax 0)REF Figure 2.5: Hydro Turbine and Governor System The corresponding differential equations are 1 TG = 2 = TrG 7gG3 = 0.5 T Tm = —uG1+UG+wREF—..’—G2 +6TTrG 2 G 1 +G 3 -G 1 + Tm+CgG T 0 3 wG G 19 (2.15) subject to the governor speed and opening constraints by Gamin min 0 G 03 <Gmax (GgG3 + G ) 0 max 0 G where overall gain of speed governor 1/o transient regulation constant T, Tr, Tg, T time constants of actuator, dashpot, servomotor, and water, respectively outputs of servo actuator and dashpot, respectively ,G 1 G 2 : gate incremental opening 3 G initial gate opening speed reference WREF UG: supplementary governor control (b) Electrical-Hydraulic Governor and Steam Turbine System The block diagram of an electrical—hydraulic governor and non—reheat steam turbine for an steam electric plant is shown in Figure 2.6. CO Gomin Figure 2.6: Non-Reheat Steam Turbine and Governor System The corresponding differential equations are TsmG = TGHTm = G+Kg(UG+WREFW) G+G 9 Tm+C 0 20 (2.16) subject to governor speed and opening constraints Gamin C < Gmax ) <G 0 min <(cG+G 0 G ma 0 where Kg Tsm : overall gain of speed governor : servo motor time constant G : valve incremental opening TCH : steam chest time constant In a multi—machine power system, the power output of a generator is expressed on the system base, while the governor output is usually expressed on the individual generator base with a full load output as 1 per unit. As a result, an interfacing factor g 0 must be introduced as shown in the Figures. 2.2.4 Power System Stabilizer (PSS) For a voltage stability study, the low frequency system oscillation phenomena must be isolated by power system stabilizer applications. A power system stabilizer with two compensation components and one reset block is shown in Figure 2.7. (0 WF sT 1+sT sifl 1÷sTi [J 1+sT2s S2 Figure 2.7: A Power System Stabilizer 21 1+sTi 1+ST2s UE The differential equations for the P55 are 1 S 3 T = — 1 —8 t 8 h T 28 T 2 S = 2 —S + 1 S 18 KT KS 23 LIE T = — UE +82+ (2.17) 18 S T 2 where K is the overall gain, T 3 is the time constant of the reset block, and T 18 and T 28 are those of the compensation blocks. 2.3 System Voltage Control Devices Obviously, 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 may limit the excitation voltage and hence the reactive power output of the generator, which may result in a loss of system voltage control ability. Therefore, It may also be considered as a voltage control device. 2.3.1 On Load Transformer Tap Changer Distribution transformers are usually equipped with on—load tap changers (OLTC’s). An OLTC controlled load bus is shown in Figure 2.8. The secondary voltage V is controlled through the change of the transformer ratio a which has limits. a amim = ) 5 (V (2.18) a Since a transformer has only limited number of taps, the ratio a must be changed in steps. In addition, time is required for completion of each tap changing. Therefore, the control function may be modeled in discrete way with time delay. That is, for voltage 22 V load Figure 2.8: A Transformer with OLTC controlled bus i with normal operating voltage °, a tap step of A.aj, and a prescribed voltage tolerance , the tap changing function may be described by (2.19) a(k+1) where it is a sign function as follows. 1 ifV—°<— 0 if —V° —1 2.3.2 < if—°> Static VAR Compensator (SVC) Static VAR compensators (SVC’s) have a significant influence on the system voltage behavior. A thyristor—controlled reactor (TCR) compensator with fixed shunt capacitors is shown in Figure 2.9 with a block diagram of SVC control circuit in Figure 2.10. The differential equations for the SVC system are 1 TbB = 2+ —B bB T 2 BL —B1+Kb(VREF—4) = bB T 1 +BLO. 2 B 23 1 +B (2.20) system bus input Figure 2.9: A TCR with Fixed Capacitor B2 VREF Bs B2min Bc Figure 2.10: Block Diagram of an SVC with Voltage Control with constraints B2min < B < B2max where K,, and Tb are the gail and time constant of voltage regulator. T and T2b are time constants associated with the thyristor firing system. 2.3.3 Generator Rotor Overheat Protection A generator rotor overheat protection may be approximately modeled by an excitation reduction of the generator which has been excited continuously over its operating limit or the second excitation limit Emax2 (see in Figure 2.4) for a certain period of time. The 24 timing of excitation reduction depends on the accumulated heat and temperature rise of a field winding dile to actual excitation at a level over Ema2 as in Figure 2.11. For instance, Em:1 Emax2 Time I tx Figllre 2.11: Overheat protection characteristic if 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 limit Emax2. 2.4 Modeling of Transmission Network Neglecting the electromagnetic transients in the transmission system, the network equa tion, which describes the relationship between bus voltage and current injection, may be expressed algebraically with network admittance matrix YN in the following phasor form, [IBUS] = [YN] [VBUS] (2.21) or alternatively in X—Y real number coordinates as [I,y] = [YGBI [Vx,y] 25 (2.22) In equation (2.22), [V.x,y] and [Ix,y] respectively denote the system bus voltages and current injections in X and Y coordinates as [Vx,y] [Ix,yj [Vx1,Vy1,Vx2,Vy2,”,VxN,VyN] = [Ix1,Iy1,Ix2,Iy2,”,IxN,IyN] T T and the corresponding matrix YGB has the form of [YGB] ] 11 [Y } 12 [Y ] 21 [Y ] 22 [Y . . ] 1 [YN ] 2 [YN = ] 1 [Y ... ... N] 2 [Y [YNN] with -B [)1] = 1 B where N denotes the total system bus number, and B 1 are respectively the real and the imaginary components of Nj• Hence, the network injection current at system bus i can be expressed as BjVyj) ‘Xi = Zi (GVx Ii = (BijVxj+GijVyj) 1 Z= — (2.23) Power system component models are presented in this chapter. While synchronous generators are described in individual machine’s d—q coordinates, other components are expressed in a coordinate system which rotates at synchronous speed. Coordinate system transformation and the overall system presentation are given in Chapter 3. 26 Chapter 3 SIMULATION TECHNIQUE FOR VOLTAGE STABILITY STUDIES The voltage instability phenomenon of a power system is far more complicated than that of a conventional transient and dynamic angle instability of the system. They differ in nature. Voltage stability depends not only on the stability of synchronous generators, but also largely on the load characteristics and power and reactive power control dynamics of the system. For different system operating conditions, voltage instability may involve a fast voltage drop or a slow voltage decline followed by a voltage collapse. In other words, a heavily loaded power system may experience either a slow or a fast system voltage instability, which depends on the type, the location and the severity of system disturbances, the load characteristics and inadequate system controls [13]. The stability of a power system can be best assessed by the time responses of system state variables to different system disturbances. This requires the solutions of the system equations which usually constitute a very complicated high order system with many inherent nonlinearities. Therefore, computer simulation methods must be resorted. As part of the thesis work, a comprehensive system simulation program is developed. Following a discussion of coordinate transformation of machines and transmission network, a complete set of system equations are presented in section 3.1. A simultaneous implicit 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 presented in section 3.3. 27 3.1 Complete System Equations 3.1.1 Machine and Transmission Coordinates In Chapter 2, synchronous machines are described in individual machine’s d and q coor dinates, while transmission network was initially described in static coordinates but may be deemed as synchronously rotating X—Y coordinates. To interface the machines with the transmission network at machine terminals, coordinate transformations of terminal currents and voltages are necessary. The relationship between the kth synchronous ma chine coordinates dk—qk and the common network coordinates X—Y is shown as in Figure 3.1. Y x /dk Figure 3.1: Machine and Transmission Network Coordinates Therefore, the voltage and current transformations of the kth generator and the ith transmission bus may be written as follows. Vdk SiflSj COSk vxi COSSk k 6 Sill vYi 28 (3.1) and where k 6 ‘dk SinSk k 6 —COS ‘Xi ‘qk k 6 COS k 6 sin ‘Yi (3.2) is the rotor angle of the kth generator. To save the computation of coordinate transformation, a hybrid coordinate system is preferred [39], by which synchronous machine terminal voltages are transformed into the common network coordinates, while machine currents remain in individual d and q coordinates. As a result, for the kth synchronous generator connected to the ith system bus, the generator armature winding voltage equation (2.12) and the network injection current equation at the ith system bus (2.23) become sink k 6 —cos VX k 6 cos k 6 sin Vy = — rak qk 1 -[dk Xdk rak ‘qk (3.3) and k cosSk 5 sin ‘dk k 6 CO5 ‘qk = (GijVxj—BijVyj) 1 Z_ (3.4) Equation (3.4) shows that the network injection current at a generator bus has been transformed into the corresponding machine coordinates. 3.1.2 Complete System Equations The overall internal system behavior is the result of interactions among system com ponents. For a power system with Ng generating plants, N 1 system bus loads, and N system control devices, a block diagram of component interaction of the power system may be shown as in Figure 3.2. With this block diagram and the coordinate system, equations (3.3) and (3.4), the complete system equations can be derived by aggregating all system component equations 29 generating plant i (i=1,2 bus load I Ng) (i=1,2 . Ni) control device i (i=1,2 Nc) Figure 3.2: Block Diagram of Component Interaction described in Chapter 2, which can be organized into a set of differential and algebraic equations as follows. F(X,X,Y,D) = 0 (3.5) G(X,Y,D) = 0 (3.6) where X is the vector of system differeiltial variables and Y the vector of system alge braic variables of system voltage and current and their related variables such as machine electromagnetic torque. X and Y together constitute the system state variable vector, which may be subjected to certain operating constraints. D is the parameter vector of external system changes, such as the changes of system references, load variations, or the system contingencies. F and G are vector functions which depend on the system component models and parameters and subject to change due to the changes in system topology and/or system operating conditions. The state variables X and Y and their possible constraints of system equations ob tained in Chapter 2 may be summarized as follows. 30 1. Typical system loads: (a) Induction motor load x [s e e]T = y Vis [Vd ‘ds Iqs TEIT (b) Voltage dependent PQ load Vy Ix Iy]T y=[Vx (c) Persistent PQ load x=[GL BL]T 2. Generating plant: (a) Synchronous generator x = [w 6 E E ’ E’]T 1 y = [Vd ‘d ]T 8 ‘q T (b) Field excitation system x Emini = [EFD] EFD Emaxi (c) Mechanical—hydraulic governor and hydro turbine Tm] 3 2 G 1 X[G T Gamin min 0 G 3 < Gmax G max 0 (GGG3 + Go) <G 31 (d) Electrical—hydraulic governor and steam turbine X[GTm]T Gamin G < Gmax ) <G 0 mn < (CGG + G 0 G mar 0 (e) Power system stabilizer x=[Si S 2 UE]T UEmjm < UE < UEmax 3. System control device (a) On load transformer tap changer y=[a] amin a < amax (b) Static VAR compensator ]T 2 1 B xi[B 2 < B2m B2mim < B 4. Transmission network y={Vx Vy ‘Xi T 11 32 i=1,2,••,N 3.2 Simultaneous Implicit Integration of System Equations 3.2.1 Implicit Trapezoidal Rule For accuracy and numerical stability, simultaneous implicit integration methods are of ten used to solve the system differential and algebraic equations like (3.5) and (3.6). In this method, both differential and algebraic equations are replaced by finite difference equations through discretizations, and then solved simultaneously by Newton-Raphon’s method. The simultaneous implicit integration method may be implemented with differ ent algorithms depending on the accuracy of the difference equations used to approximate the corresponding differential equations. In general, the higher the discretization order is, the better the accuracy of the result will be, but a more complex algorithm will be involved. For power system stability studies, it has been suggested that low order integra tion would be best in both efficiency and stability [40]. Implicit trapezoidal integration method 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 es sential characteristic of symmetrical A—stability. The latter means that if the difference equation is symmetrically A—stable, it demonstrates the same stability results as those determined by the solution of the original differential equation [40]. The basic idea of implicit trapezoidal rule may be illustrated by Figure 3.3. To integrate the differential equation = xQt) (3.7) f@,t) = with known x, the solution of x at t,, the solution of x at 33 tn+1 = t+ or can f(x,t) f(Xn+i tn÷i) d cr Atn -‘ tn+1 tn Figure 3.3: Illustration of Implicit Trapezoidal Rule be obtained by integrating equation (3.7) from t to 1 rtn+ Xn+l = Xn + J ti as follows. f(x, t) dt (3.8) tn As 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 by the dashed line ab, and the shaded area may then be approximated by the area of the trapezoid abed, which results in f(x,t)dt = ] + eLT ,t 1 f(x [f(x,t) + ) (3.9) where eLT is the local truncation error introduced in the integration interval (ta, t+ ), 1 which can be estimated [42] as eLT for some (3.10) = E (t,t+ ). 1 34 Equation (3.8) then becomes zn+i The evaluation of eLT x + Itn + f(n+i,tn±i)] + (3.11) eLT wilibe presented in section 3.3.3. The trapezoidal integration algorithm for the differential equation (3.7) is obtained by dropping the error term in equation (3.11). x + ——[f(xn,tn) + With and x f(n, tn) known at t = (3.12) f(n+i,tn+i)] t, a difference equation constant, z, can be defined as — X + (3.13) f(n,tn) Equation (3.12) can then be written in the following standard form. Xn+l = n + (3.14) f(n+i,tn+i) This is the discretization algorithm of implicit trapezoidal integration. 3.2.2 Discretization of System Differential Equations With the implicit trapezoidal integration rule of equation (3.14), all differential equations of system components described in Chapter 2 can be discretized and summarized as follows. For convenience, define h = At / 2. 1. Typical system loads: (a) Induction motor load 2Hs (i+t) = Zm1t + h(TL Te(t+) = z + h(—e t 3 Zm 0 + h[—e + (x 35 — — 0 (x — — x’)Iqs +WbTo8eqfiQ+t) x’)Ids — wbTOsed](+t) (3.15) with the difference equation constants Zmlt = 2Ht+h(TL—TE)t 2 Zm = T et + h [—e t 3 Zm = — 0 (x 0 + h[—e + (x x’) Iqs + — Wb T e] t x’)Id —bTsefit — (b) Persistent PQ load TGGL(t+t) = TBBL(t+t) Zit 0 + h(P Zc2t 0 + h(Q — (3.16) )(+t) 2 BLV — with the difference equation constants Zit = )t 2 TGGLj+h(Po—GLV Zc2t = )t 2 TBBLt+h(Qo—BLV 2. Generating plant: (a) Synchronous generator Mw(t+t) = Zglt + h(Tm = 9 z + hwb(w Zg3t + h[_E Zg4t + h [_Eq Zg5t + h[_E + (xq 0 E (t+t) T T(E’ = - TE (t+t) — Te — D) (t+t) — — (xd — I — (xd x)Id + EFDfi(t+t) II — — Xd) ‘d + X)Iq1(t+t) with the difference equation constants MLt+h(TmTeDLL))t Zg1t = 2 zq = t 3 Zg = Et+h[E(xdx)Id+EFD]t 0 T Zg4ft = T(E’ t 5 Zg = TqoEdt + h[—E + (xq II E+h[E’ II - 36 (x-x)Id+E]It II — xq)IqHt (3.17) (b) Field excitation system ZeIt + h[EFD + EFDO + TAEFD(t+t) KA (UE + VREF )](t+t) — (3.18) with the difference equation constant Zet = TAEFDt + h[EFD + EFD0 + KA(UE + VREF (c) Mechanical—hydrailhic governor and hydro turbine 2 (TrG — T Gl+t) = Zh1 TTrGl)(t+Lt) 6 = Zh2t 9G T (t+t) 3 (O.5TwTm +TG )(t+t) 3 — — 1 + UG + + h [—u G REF — — ](t+) 2 G — (3.19) ZJ 3 + Gl)(t+t) + h (—G t+h(Tm 4 Zh 3 +Go)(t+) +CgG with the difference equation constants Zh1t = 1 + h[—aGi + UG TG Zh2 = TTrGl)thG (TrG t 2 — t+h(G TgG + 3 Gi)t Zh3t Zh4t +WREF = (O.5TwTm + TG ) + h(Tm + CgG3 + Go) 3 (d) Electrical—hydraulic governor and steam turbine Tsm G(t+t) = 81 Z + h [G + Kg(UG + TCHTm(t+t) = t 2 Zs + h(Tm + CgG+ Go)(t+t) REF — ) tt w)fi( with the difference equation constants Zsit = TsmGt+h[G+Kg(UG+wREFLi.))fit t 2 Zs = TcHTmt+h(Tm+GgG+Go)t 37 (3.20) (e) Power system stabilizer (S 8 T 1 2 3 28 (T — = +‘)(t+t) Sl)(t+t) 3 KTl UE 28 (T — Zp2t )(t+t) 2 TlSS = hSl(t+t) h(—S + H- 2 Zp3t+ KCSl)(t+t) (3.21) )(+t) 2 h(UE+S with the difference equation constants 1 z = 2 z = 3 z = (Si+w)t—hSit 8 T U (T ) S 8 ) 2 +h(—UE+S E—Tl t 3. Control devices (a) Static VAR compensator bB (T 2 — TbBl(t+t) = TlbBl)(+) = 1 z 1 + Kb(VREF + h[—B 2 z [—B + Blfi(t+t) +h2 — (322) with the difference equation constants 1 + Kb(VREF + h[—B 1 z = TbBlt 2 z = bB (T 2 — TlbBl)t — h[—B + +2 Bifit In the above difference equations, the difference equation constants, z’s, are known at time 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 system difference 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 not to change during (t, t H- z\t). With X, Y, D and H(X,Y, D) known at time t, the difference equation constant vector Z is defined as Z X+ D) 38 (3.24) Therefore, for a system solution (Xt , Yt) at time t and a chosen integration step size At, equation (3.23) together with discretized system algebraic equation (3.6) forms the simultaneous implicit integration algorithm based on the trapezoidal rule for the system equations (3.5) and (3.6) — — These are nonlinear algebraic equations and can be solved for = 0 (3.25) = 0 (3.26) Y+) by Newton— Raphson’s method. 3.3 Power System Simulation Program Based on the simultaneous implicit integration method described in the previous section, a comprehensive power system simulation program is developed in this section for system voltage stability studies. The program consists of the following major functions. 1. System data input and pre—processing 2. Load flow and initial system condition 3. Selection of system contingencies 4. System state monitoring, recording and control logics 5. Topology change and new initial system values 6. System equation integration and variable constraints 7. Step size control and exact timing 8. System data output 39 The program is very involved since it has to deal with various types of load and reactive power component models, nonlinear constraints and logics, such as generator rotor overheat protection, etc.. Some details of the simulation program are presented as follows. 3.3.1 Load Flow Calculation For the dynamic system simulation studies of a power system, a load flow is required to determine the initial steady state values of the system. For that, the initial system generation and loading conditions must be specified. With a load flow, all initial system state values (X ) can be determned. Y 0 In a load flow, generator buses are usually specified as PV buses, while load buses as PQ buses. However, the power and reactiv power drawn by a dynamic load may be a function of its terminal voltage and other state variables which can not be specified a priori. For example, the power and reactive power drawn by an induction motor in steady state are determined by equation (2.5), which are functions of not only motor terminal voltage V but also the motor slip s. In order to determine load power and reactive 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 load flow Jacobian equation. P(e,v) = (3.27) [JLFI Q(ø,V) where P and zQ are power and reactive power mismatch functions, e and V are respectively the angle and magnitude of system bus voltage, and LF is the corresponding load flow Jacobian matrix. There are two different methods in dealing with the load flow equations with nonlinear loads, the sequential method and the unified method [38]. By the former, the steady state 40 equations of the load are solved separately in each iteration. The nonlinear effect of the load is included only in the power mismatch equations, but not in the system Jacobian matrix, which may reslut in possible convergence problems. By the united method, the idea is to add the load variables, such as motor slip, into the solution vector. It amounts to 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 load variables, such as motor slip, are eliminated from the load equations first, and then both power and reactive power drawn by the load and their derivatives with respect to load terminal voltage are computed, which may be done numerically if necessary. The load nonlinearity can then be included in both power mismatch equations and in load flow Jacobian matrix without variable extensions. For a static load, the load power and reactive power may be directly expressed as functions of the terminal voltage, such as equations (2.6) and (2.7). The calculations of load power, reactive power and their derivatives with respect to terminal voltage are straightforward. For a dynamic load, variable elimination may be involved. For example, the steady state operating condition of an induction motor load is determined by the equations (2.5) which is rewritten as follows. TL=TE = fE(s,p)V 2 (3.28) Fm = fp(s,p)V 2 (3.29) m Q 2 = fQ(s,p)V (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 Fm and reactive power Qm determined by the terminal voltage V can be included in the power mismatch equations. The derivatives of power and reactive power with respect to 41 voltage can be computed by = dQm where = dfp(:, p) dfQ(s,p) 2 v + 2Vfp(s, p) (3.31) v2 + 2VfQ(s,p) (3.32) can be obtained by differentiating equation (3.28) with respect to V as = df(:,p) V2 + 2VfE(s,p) (3.33) or = 2 _ f E(s,p)/(d V) (3.34) Therefore, with load flow Jacobian matrix modified by equations (3.31) and (3.32), the nonlinearity of the load has beell fully included in the load flow. 3.3.2 Solution of System Jacobian Equations The .major task of solving the nonlinear system equations (3.25) aild (3.26) by Newton Raphson’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 vector and J the system Jacobian matrix. R has the form of R [hF hIlT with hF for the 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]T with hX for the vector of all system non—terminal—voltage variable deviations and hV for the system terminal—voltage deviations. For a multi—machine power system with nonlinear load and control dynamics, the order of Jacobian matrix equation (3.35) is usually very high. For instance, it has more 42 than 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 both heavy memory demanding and very time consuming. Although the order of the Jacobian matrix may be reduced by eliminating some system variables from equation (3.25) using the existing relations, this amounts to have a new model for the original system. It will be very complicated in dealing with original component model changes during integration. To save the computation in solving the Jacobian matrix equation (3.35) yet to keep the original system component models, a new and systematic method is developed in this 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 are eliminated and the result is used to modify the sub—Jacobian matrix corresponding to the transmission network equations. All system terminal—voltage deviations are then solved. In the backward substitution step, the terminal-voltage deviations are back substituted and all non—terminal—voltage variable deviations are then obtained. Both forward elimi nation and backward substitution are designed in such a way that the system components are arranged in a definite order and processed systematically one after another. 9 generator buses In this method, the system buses are ordered in such a way that the N come first, followed by N. induction motor buses, and then N 3 SVC buses, followed by 1 nonlinear load buses. Any other different types of components can be easily grouped N and added in similar way. As an illustration example, a power system with Ng generator buses and Nm = 2 2 motor buses is considered. The system Jacobian matrix equation may then have the form as in Figure 3.4. 43 AFgj AX, 2 AF tFmi 1 AXm AFm2 l3Xm2 Mgi AVgi 1Ig2 AVg2 Mmi 1 AVm Mmi tVm, Figure 3.4: An Example of System Jacobian Matrix Equation where denotes the non—terminal—voltage variable deviation vector for the ith gener ator, 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 motor equations, /I is residue function vector of the network equations. Finally, A, B, C, and J are corresponding sub—Jacobian matrices. Forward Elimination Referring to Figure 3.4, the residue function vector AF of the ith system component and the residue current vector AI of the corresponding network equations may be written as = where AZX + Bz\14 (3.36) C /X + Jc V (3.37) is the ith row sub—matrix of J and V is the vector of all system bus terminal volt age deviations, including 44 Next, the ith component of the non—terminal—voltage deviation vector, can be eliminated by solving it from equation (3.36) and substituting the result into equation (3.37) resulting in zI = JiV — CA’BzXV (3.38) where = With equation (3.38), — is then modified by subtracting G A’B from the ith element of Finally, when all system components have been processed one by one in the same way as described above, the sub—Jacobian matrix J, is modified and all system bus voltage deviations can then be solved through the following matrix equation, which finishes the forward 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 Substitution Once the system bus voltage deviation V is obtained through forward elimination, the non—terminal—voltage variable deviation of the ith system component /X can be solved by substituting Ls4 back into the component equation (3.36), which gives ZXX, = A’(zF - BJ’) (3.40) When all system components have been processed one by one in the same substitution procedure, the whole system Jacobian matrix equation (3.35) is finally solved. 45 With this forward elimination and backward substitution procedure, the system Ja cobian matrix equation can be solved systematically and efficiently. There are three ad vantages. 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 very straightforward in dealing with changes in component equations. In addition, since all system components are handled in the same way, it is very easy to add or remove com ponents to or from the system. Secondly, since system components are processed one at a time, the highest order of the matrix involved in the forward elimination and backward substitution is the highest order among J and A (i = 1, 2, , N) where N is the total number of system components. To reduce the order of the sub-Jacobian matrix J, the system buses with constant impedance loads and fixed capacitor ballks can be eliminated by lumping the corresponding equivalent admittance of the components into the diago nal elements of the respective submatrix of the system admittance matrix YN. Finally, since matrices As and J are usually sparse, triangular factorization technique can be used to save computation associated with forward elimination and backward substitution procedures. 3.3.3 Integration Step Size Control and Exact Timing Power system equations (3.5) and (3.6) may have wide—ranged values of time constants associated with the system dynamics. For example, a generator sub—transient may involve a time constant of a small fraction of a second, while a time constant associated with load dynamics may be several minutes. Small time constants determine the fast system transients while large time constants dominate the slow system dynamics. For system problems like voltage instability phenomenon involving both fast transients and slow system dynamics, a too large integration step size may result in poor accuracy for fast transients, while a too small step size may result in excessive computation for the slow 46 dynamics. 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 approx imated 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 differ ence equations. The error of the approximation introduced in a single integration step is called the local truncation error eLT. It is shown in [43] that the relationship between local truncation error and integration step size can be expressed as eLT C d x 1 ) 1 dt( where p is the order of the numerical integration method, for example, p (3.41) = 2 for integra tion algorithm based on trapezoidal rule, At is the integration step size, C is a constant which depends on the system equation and the integration method being used, and x is the solution of the system equation. The basic idea of step size control is to keep the local truncation error eLT within the tolerance limit while maximizing the integration step size At. Computation of eLT from equation (3.41) requires information regarding the order of 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 polynomial of degree p + 1, an alternative approach, the step doubling approach, described in [40] is used to estimate the local truncation error. This method involves integrating the system equations by taking two steps of step size At and reintegrating over the same interval with a single step of length 2 At. With these solutions, the local truncation error eLT can be estimated as follows. Let x be the solution by taking two single steps, xd the solution by taking a double 47 step, and xt the corresponding true solution which is unknown. We have the following equations (3.42) eLT 2 xt+ 8 Xt = where 3 eLT and eLTd + (3.43) eLTd are local truncation errors of single step and double step integrations, respectively, which, from the equation (3.41), are given by x 8 eLT C e LTd d x 1 C dt(P+l) — — dt(P+l) 4 2Lt ‘ Subtracting (3.42) from (3.43), we get = eLT 2 eLTd— 5 = c = (21 /tP+1(2P+1 — —2) (3.44) 5 2)eLT which gives the local truncation error of single step integration eLT = (xd — )/(2 5 x 1 — 2) From equation (3.45), we can see that computation of (3.45) eLT by step doubling approach avoids the computation of the derivatives and the knowledge of constant C. Only the order of the integration method needs to be known. On the other hand, step doubling 3 method has an obvious disadvantage. It incurs much more computations because x and 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 steady result. This means that the changes in step size during integration are more consistent which is also observed in the simulation studies in this thesis. These benefits are nearly sufficient to compensate the extra computation cost. 48 With the local truncation error chosen that the eLT eLT satisfies emin where emin and emax available, the integration step size At can be so eJff (3.46) 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 integration continues 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 limits Atmin At <Atmax (3.47) Therefore, whenever there is a system change which causes some system variables to vary sharply, a small integration step size must be chosen for accuracy. After fast system transients, the step size will be increased gradually while keeping the local truncation error within the prescribed limits. Due to the variation in step size during the integration, the simulation may overshoot and miss the exact timing of faulting and clearing, tap changing and rotor overheat protecting, etc.. In the simulation program, the exact timing t of an event, is pinpointed by changing the last step size according to At new = At ld 0 — 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 Contingencies Most system instabilities are caused by system disturbances. Different disturbances oc curring at different locations have different impacts on system stability. System distur bances may include system load changes, system network changes, and system generation changes. There may also be single or double contingencies. 49 variable system change Atold ttnew Figure 3.5: I I t- Atoid tc time t Step Size Change for Exact Timing 1. 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 will change instantly since the fast electromagnetic transients in transmission network are 50 neglected. 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 of Y for the next step integration. 3.3.5 Flowchart of System Simulation Program The complete system simulation program may be recapitulated by an overall flowchart as shown in Figure 3.6. 51 no SOLUTION OF SYSTEM EQUATIONS (X Y) I. EQUATION DISCRETIZ4TION 2. DIFFERENCE CONSTANT Z 3. JACOBIAN EQUATION a. FORWARD EUMINATION b. BACKWARD SUBSTITUIION 4. VARIABLE UPDATE Figure 3.6: Overall Flowchart of System Simulation Program 52 Chapter 4 DYNAMIC VOLTAGE STABILITY STUDIES Voltage stability studies have been concentrated on the determination of maximum load ing limit (MLL) of a power system based on load flow related analysis. When the system loading reaches its MLL determined by conventional load flow, the load flow Jacobian will become singular, which indicates a possible voltage collapse. Such defined MLL may be referred to as static voltage stability limit. Thus, the distance between current system loading condition and the MLL can then be used as an index to measure the degree of system voltage stability. However, when the MLL is determined through load flow analy sis, system dynamics are not included. As a result, it may not lead to a realistic solution due to the harmful or favorable system dynamics. Moreover, even if an exact MLL could be found, it can only tell a system loading condition where a possible voltage collapse may occur. It can not provide any information of how system voltage approaches the collapse point and how this collapse point is affected by system dynamics. Therefore, the evaluation of system MLL and clarification of voltage collapse mech anism require a detailed system dynamic simulation which may take into account the effects of all system dynamics on voltage stability. It is believed that the unfavored dy namic interactions among system components play a key role in the process of voltage deterioration and collapse. With system models described in Chapter 2 and time domain simulation technique developed in Chapter 3, the effects of load and control dynamics on voltage collapse process can be better investigated by dynamic simulation of a sample power system. 53 In this chapter, a 21 bus sample power system with basic system data is presented in Section 4.1. Critical system load buses are defined and identified in Section 4.2 based on bus voltage—reactive power sensitivity analysis. Following that, the effects of different system loads on voltage stability are investigated in Section 4.3, and system control effects in Section 4.4. Finally, the results of system maximum loading limit (MLL) from load flow and those from dynamic simulation are compared and discussed in Section 4.5. 4.1 A Sample Power System for Voltage Stability Studies 4.1.1 A 21 Bus Sample Power System A sample power system with 21 buses and 23 branches of transmission system is shown in Figure 4.1. The system has 9 generating plants, two of them at buses 4 and 9 are hydro—electric, and the others are steam—electric. There are 11 load buses with a VAR compensation at bus 21. The system is connected to an infinite system through line 11 — 1. Figure 4.1: A Sample Power System under Study 54 4.1.2 Basic System Data The basic system data for the transmission system, the generating plants and a specified system generation and loading condition are given as follows. The data for typical system loads and system control devices are given in the subsequent sections. All data are in per unit unless otherwise specified. A. Transmission System Table 4.1: Data of Transmission System Line Bus I Bus J Resistance(R) Reactance(X) 1 2 13 0.0000 0.0590 2 3 13 0.0000 0.0135 3 5 12 0.0000 0.2900 4 12 13 0.0068 0.0680 5 6 16 0.0000 0.0500 6 16 19 0.0300 0.3000 7 7 16 0.0000 0.1000 8 16 17 0.0015 0.0145 9 19 20 0.0106 0.1060 10 20 21 0.0240 0.2400 11 21 17 0.0161 0.1610 12 21 14 0.0025 0.0250 13 13 14 0.0120 0.1200 14 13 20 0.0048 0.0480 15 14 15 0.0070 0.0700 16 15 4 0.0200 0.2320 17 15 11 0.0070 0.0700 18 14 10 0.0120 0.1200 19 14 18 0.0102 0.1020 20 18 9 0.0000 0.2800 21 17 18 0.0057 0.0570 22 17 8 0.0320 0.3200 23 11 1 0.0000 0.0320 The impedances of transformers in the transmission system are combined with those of transmission lines. All line capacitances are ignored. 55 B. Given System Generation and Loading Condition Table 4.2: Data of Generator PV and Load PQ FV Bus FQ Bus Bus Pgen Vgen Bus Pload Qload 2 3.000 1.040 11 2.500 0.200 3 3.500 1.035 12 1.500 0.100 4 2.000 1.030 13 2.400 0.200 5 1.000 1.035 14 4.500 0.500 6 2.500 1.040 16 3.500 0.200 7 2.500 1.040 17 2.000 0.200 8 2.000 1.010 18 2.000 0.000 9 2.000 1.015 19 1.500 0.100 10 3.500 1.060 21 3.500 —1.700 =1.060 1 Slack Bus: V Bus 15, 20: No Load For different system loading conditions, generator voltages may be adjusted for a normal system voltage profile. The power balance is taken care by the infinite system. C. Generating Plant Synchronous Generators Table 4.3: Data of Generator Parameters Bus xd xq M D 0 T T, T,, 2 0.700 0.700 0.120 0.098 0.098 25.000 0.050 7.000 0.091 0.455 3 0.600 0.600 0.100 0.084 0.084 30.000 0.050 7.000 0.091 0.455 4 0.500 0.400 0.150 0.070 0.070 20.000 0.050 8.000 0.104 0.520 5 1.600 1.600 0.230 0.224 0.224 12.800 0.050 7.000 0.091 0.455 6 0.950 0.950 0.150 0.133 0.133 19.800 0.050 7.000 0.091 0.455 7 0.950 0.950 0.150 0.133 0.133 19.800 0.050 7.000 0.091 0.455 8 1.000 1.000 0.170 0.140 0.140 18.000 0.050 7.000 0.091 0.455 9 1.000 1.000 0.170 0.140 0.140 18.000 0.050 7.000 0.091 0.455 10 0.390 0.320 0.060 0.055 0.055 32.000 0.050 6.000 0.078 0.390 56 where M, T , T, 0 and T are in seconds. Field Excitation Systems The 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 time constants and smaller gains as shown in Table 4.4. The excitation systems of generators at buses 2, 3, 4, 5, 6, and 7 belong to group A, and those at buses 8, 9, and 10 belong to group B. Table 4.4: Data of Field Excitation System Group where TA TA Emni Emasi A 100.00 0.050 —7.000 7.000 B 50.00 0.100 —7.000 7.000 is in second. Governor and Turbine Systems (a) Mechanical—Hydraulic (M—H) Governor and Hydro Turbine System Table 4.5: Data of M—H Governor and Hydro Turbine Tr 9 T T 1.600 0.050 0.250 0.020 4.800 0.500 9 C Gsmin Gsmax Gomin Gomax 0.100 0.000 Pmax 0.100 Pgen where T T, Tr, T 9 and T are in seconds. (b) Electrical—Hydraulic (E—H) Governor and Steam Turbine System 57 Table 4.6: Data of E—H Governor and Steam Tnrbine Kg Tsm TCH 20.000 0.100 0.400 Gsmjn Gsmax Gomin Gomax 0.100 0.000 Pmax 0.100 where Tsm and TCH 9 C are in seconds. In the governor and turbine system data, the interfacing factor a 9 has a value of the rated power output 96 P , of the generator. The maximum gate or valve opening Gomax corresponds to the maximum power output which may be certain percent over the rated power output, for example, Pmax may be 1.2 times of Pgen. Power System Stabilizers The power system for the given operating condition is originally unstable in term of system low frequency oscillations. For voltage stability studies, this conventional angle stability problem should be removed from the system. This can be done with the applications of power system stabilizers (PSS’s). For the given system operating condition, 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]. The corresponding 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 Condition Generator Bus 0 K 2’ 4 18.057 5.000 0.103 0.035 8 28.271 5.000 0.104 0.035 9 31.352 5.000 0.124 0.035 58 , 3 whereT 4.2 4.2.1 , 1 T and 2 T are in seconds. The Critical System Load Buses Analysis of the System Operating Condition For the given system operating condition, load flow studies show that the system is highly power stressed. A large amount of power is transferred from some remote generating plants, for instance, at buses 4, 5, 8, 9, and 10, and to some remote loads, such as loads at buses 14, 19, and 21. The long distant power transmission results in a heavy reactive power loss in the transmission lines and transformers. As a matter of fact, all the reactive power supply from generators is consumed in the transmission system, as indicated in Table 4.8. As a result, the large capacity reactive power compensation at bus 21 becomes critical for the system to maintain a nearly normal voltage profile as shown in Table 4.9. Table 4.8: System Power Generation and Consumption Power (F) Reactive Slack Bus Generators Compensations Loads Losses 1.875 22.000 0.000 23.4000 0.475 0.969 7.104 1.700 1.500 8.273 Power Table 4.9: Voltage Profile of the Given System Condition 1 V 2 V 3 V 4 V 5 V 6 V 7 V 1.060 1.040 1.035 1.030 1.035 1.040 1.040 8 V 9 V 10 V 1 V 12 V 13 V 14 V 1.010 1.015 1.060 1.032 1.008 1.021 0.981 15 V 16 V 17 V 18 V 19 V 20 V 21 V 0.994 0.984 0.969 0.956 0.967 0.999 1.004 59 As a result, the system operation becomes voltage vulnerable due to its critical de pendence on the heavy reactive power compensation. This means that system voltage may collapse due to system disturbances that reduce the compensation. 4.2.2 Critical System Load Buses Critical load buses of the system are identified and used to investigate the effects of load and control dynamics on system voltage stability. The critical system load bus is defined as the bus that is most voltage sensitive among all system load buses. This suggests that the load increase at a critical load bus will have a larger influence on the overall system voltage profile than those at other load buses. More specifically, a load increase at the critical 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 bus than those at other system load buses. For a given system generation and loading condition, the steady state system opera tion can be characterized by the following nonlinear load flow equations. where P and P (O,V) = 0 (4.1) Q (ø,V) = 0 (4.2) 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 equa tions (4.1) and (4.2) around the system operating point, which gives = = v (4.3) Qv V (4.4) Pe k0 + P e0 + 60 which may be written in a single matrix equation with the load flow Jacobian matrix as JLF (e,V) = QeQv and V are respectively the angle and magilitude deviations of system bus voltages due to the bus power and reactive power perturbations of iP and iQ. The Jacobian matrix JLF (0, V) is essentially a sensitivity matrix, and the corresponding bus voltage sensitivities can be used to identify the most critical system load bus. Since there is relatively strong coupling between reactive power and voltage magnitude in power system, the voltage—reactive power sensitivity [iV/iQ] is a reasonable index to describe the effects of system loading perturbations on the voltage magnitude. Since the effects of changes in the real power injections on the voltage magnitude is usually very small, the relationship between bus reactive power perturbation iQ and the bus voltage deviation iV can be obtained as follows. Let iP = 0 in Equation (4.3) and solve for i0, which gives PZkV 1 —P = (4.5) Substituting i€ in Equation (4.5) into Equation (4.4) gives = (Qv — Q 1 P) iV P (4.6) or iv = SiQ (4.7) 1 Pv1’ P (4.8) where S = [Qv — Qe and S may be referred to as system voltage and reactive power sensitivity matrix. 61 In Equation (4.7), the element S of matrix S has a value equal to the voltage deviation load bus at load bus i, due to one per unit change in reactive power L\Q j,, 1 at assuming no load changes at other load buses. From this point, a critical system load bus is defined as the bus which, when having a load increase, will cause large voltage drops at most system buses. On the other hand, a load bus which, when having a 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 system voltage and reactive power sensitivity matrix S calculated from equation (4.8) is shown in Table 4.10. The values in the jth column of the matrix are the voltage deviations at all system load buses due to the per unit change in reactive power at load bus larger the value is, the larger is the influence of the bus j j. The on the corresponding buses. If 1 percent of voltage deviation due to 1 per unit reactive power change is taken as the threshold of large influence, the number of buses whose voltages are largely affected by the change of reactive power at bus j, (j = 11 ,21) are given in Table 4.11. Table 4.10: System Voltage Sensitivity Matrix Bus i ---- —- Q13 Q1d ‘QlS --- Ql2 6 Qi ‘Qii Qi8 Ql9 Q2O l 2 Q 11 —0.026 —0.001 —0.001 —0.005 —0.014 —0.001 —0.002 —0.003 —0.002 —0.002 —0.004 12 —0.001 —0.063 —0.008 —0.003 —0.002 —0.001 —0.002 —0.002 —0.005 —0.006 —0.004 13 —0.001 —0.008 —0.010 —0.004 —0.002 —0.001 —0.002 —0.003 —0.006 —0.007 —0.004 14 —0.005 —0.003 —0.004 —0.030 —0.016 —0.007 —0.010 —0.016 —0.011 —0.012 —0.022 15 —0.014 —0.002 —0.002 —0.016 —0.045 —0.004 —0.005 —0.008 —0.006 —0.006 —0.012 16 —0.001 —0.001 —0.001 —0.007 —0.004 —0.026 —0.023 —0.016 —0.011 —0.006 —0.008 17 —0.001 —0.002 —0.002 —0.009 —0.005 —0.022 —0.032 —0.023 —0.011 —0.007 —0.011 18 —0.002 —0.002 —0.002 —0.016 —0.008 —0.016 —0.023 —0.055 —0.011 —0.008 —0.014 19 —0.002 —0.005 —0.006 —0.011 —0.006 —0.012 —0.012 —0.011 —0.110 —0.031 —0.016 20 —0.002 —0.006 —0.007 —0.012 —0.006 —0.006 —0.007 —0.008 —0.031 —0.038 ‘—0.017 —0.012 —0.008 —0.011 —0.015 —0.016 —0.018 —0.035 —0.004 ---- 4 ---- Qii 21 --- —0.004 —0.005 —0.023 62 Table 4.11: Number of Buses Largely Affected by Bus Bus j No.ofBuses j 11 12 13 14 15 16 17 18 19 20 21 2 1 1 6 4 4 6 6 7 4 7 Largely Affected According to Table 4.11, the system load buses may be classified into three groups in terms of voltage sensitivity, the strong bus group, buses 11, 12, and 13, which are less system voltage sensitive, the weak bus group, buses 14, 17, 18, 19, and 21 with the most critical buses 19 and 21, which have the largest effects on system voltage profile, and the third 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 demon strated at the system critical buses, and the system voltage profile is largely dependent on the reactive power compensation at bus 21, the load bus 21 is then chosen for subsé quent voltage stability studies. Thus, the load at bus 21 will be substituted by a typical load for each case study. The effect of reactive power controls will also be examined at bus 21. 4.3 Effects of System Loads on Voltage Stability The effects of the system bus loads described in Chapter 2 on the system voltage stability are studied in this section. In the study, a system bus load is represented by a particular type of load, a transformer with on load tap changer, and a distribution link connected to the transmission system bus as in Figure 4.2. The distribution link impedance is included in the transformer model. When the effect of a typical load itself is investigated, the tap changing may not be considered, that is, t = 0. An induction motor load, an exponential form static load and a persistent PQ load are included in this part of study. 63 system bus Figure 4.2: A System Bus with Typical Load Effect of an Induction Motor Load 4.3.1 The basic data of an equivalent induction motor is as follows: Table 4.12: Data of an Induction Motor Load H 6.700 where H and T 0.010 0.142 4.209 T TL 1.600 2.4 are in seconds, and the motor load torque TL may subject to change in different case studies. With the induction motor load connected at bus 21, the system will have a low voltage profile due to the large reactive power consumed by the motor. To have a system operation with a fairly normal voltage profile, some of the generator voltage references are raised to a higher level and the parameters of the PSS’s are adjusted to give the best damping to the system unstable mechanical modes. The system adjustments are as follows. 64 Table 4.13: Adjusted System Voltage Profile Vi 2 V 3 V 4 V 5 V 6 V 7 V 1.060 1.040 1.035 1.050 1.035 1.050 1.050 8 V 9 V 10 V 12 V 13 V 14 V 1.050 1.050 1.060 1.026 1.004 1.017 0.960 15 V 16 V 17 V 18 V 19 V 20 V 21 V 0.984 0.988 0.972 0.960 0.956 0.982 0.966 Vu Table 4.14: Adjusted PSS Parameters Generator Bus 0 It T 15 T 28 T 4 37.573 5.000 0.080 0.035 8 34.056 5.000 0.128 0.035 9 26.251 5.000 0.121 0.035 Case Study 1: An Induction Motor Near Critical State In this case study, the induction motor with a constant load torque TL = 2.4 is considered. The motor is connected to bus 21. Other system loads are assumed to be constant impedance loads. Transformer tap changing effect is not considered. The initial system voltage profile is shown in the Table 4.13 and the motor related variables are given in Table 4.15. Table 4.15: Initial State of the Induction Motor Terminal Voltage Motor Slip Power Reactive Power 0.965 0.024 2.480 1.404 65 2.8 2.6 __-L—;— 2.4 2.2 (a) ————. h 0 20 40 60 80 100 120 140 160 180 Time (sec) CID (b) 0 0 0 20 40 60 80 100 120 140 160 180 Time (sec) :, (c) . 0 20 40 60 80 100 120 140 160 180 Time (sec) 2 0.9 — 0 08 (d) • 0.7 0.6 0.5 0 20 40 60 80 100 120 140 160 180 Time (sec) Figure 4.3: Motor Response to Step Change in Load Torque A step load torque increase of 0.23 is then applied to the motor as a system distur bance. System responses recorded in the figures show a voltage instability phenomenon of slow system dynamics followed by a sudden voltage collapse. 66 Figure 4.3 shows the dynaniic behavior of the induction motor. Following the step increase in load torque at t = 5 second as in Figure 4.3a, there is a transient period of about 10 seconds. Following the disturbance, the motor slip begins to increase according to the rotor motion equation (2.1). Both motor power and reactive power drawn from the system increase accordingly to pick up the load. As a result, the larger motor current causes extra voltage drops in the transmission system, leading to a decrease of motor terminal voltage and other system bus voltages as well, as shown in Figure 4.3d and Figure 4.5. During this transient, generator bus voltages are maintained at the same level as those in the pre—disturbance system condition by generator excitation controls as shown in Figure 4.4. 1.065 1:: 1.06 “C 41 1.055 4’ 1.05 t1 1.045 . C 1.04 1.035 1.03 1.025 ‘ 1.02 Time (sec) Figure 4.4: Some Generator Bus Voltages After the transient, the motor slip càntinues to increase gradually, and the system experiences a rather slow system dynamics, in which the system load bus voltages remain fairly normal, and the system frequency is fixed at 60 Hz, as shown in Figure 4.5 and Figure 4.7. During this slow dynamics, the electromagnetic torque developed by the 67 motor is slightly less than the motor load torque. Motor slip begins to approach its critical value over which the motor will start stalling. At around t motor slip reaches its critical value = 105 second, the 0.038 with a maximum motor torque TE c 8 which is still less than the load torque TL 2.627, 2.63. After that, the motor slip continues to increase while the motor developed torque begins to drop. The motor starts stalling. The motor reactive power begins to increase. This causes the motor terminal voltage to dip further, which in turn reduces the motor torque. This interacting dynamics continues until a rapid change occurs at about t = 170 second. Due to the fast decreases in motor developed torque shown in Figure 4.3a, the motor slip increase rapidly as shown in Figure 4.3b. Although the motor power drops quickly, its reactive power goes up rapidly as shown in Figure 4.3c. It is this rapid increase of the motor reactive power demand 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 Figure 4.5, in which only voltages at load buses 13, 16, and 21 are shown for the sake of clarity. 1.05 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0 20 40 60 80 100 120 140 160 Time (sec) Figure 4.5: System Voltages at Some Load Buses 68 180 At about t = 175 second, the motor reactive power begins to decrease but the motor terminal voltage continues to drop. This means that the system operation has reached the lower part of Q—V curve of motor terminal bus as shown in Figure 4.6. The system loses voltage control at motor terminal bus after t = 175 second. Figure 4.6 also shows that the maximum motor power limit occurs at t = 165 second, while that of reactive power occurs at t = 175 second. This suggests that although the motor operates on the lower part of the P—V curve after t 165 second, the motor terminal voltage still can be = controlled if there is a sufficient reactive power compensation near the motor terminal bus. 0.95 PmV 0.9 0.85 • 0.8 0.75 0.7 0.65 0 0.5 1 1.5 2 2.5 3 Motor Power Pm & Reactive Power 3.5 4 4.5 Qm Figure 4.6: Transient P—V and Q—V curves of Motor Terminal Bus Since 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 the entire system. This load reduction happens so fast that the governors and turbines are not in time to respond so as to reduce the mechanical inputs to the generators. As a result, all generators are speeded up and ultimately pulled out of synchronous operation 69 successively, which causes a complete system collapse as shown in Figure 4.7 where the rotor angles of generators at buses 2, 6, 8, and 10 are recorded. Other generators have similar angular instabilities. 00 ¶ (‘1 —°-——— 2.5 66 —b——— ——-cs--——— 6io 2 I 1.5 —C C, 0 20 40 60 80 100 120 140 160 180 Time (sec) Figure 4.7: Some Generator Rotor Angles In summary, the results of this case study show that an induction motor load could cause a slOw voltage deterioration followed by a sudden voltage collapse when the motor operates near its critical condition. The stable operation of an induction motor depends on its terminal voltage and its critical slip. When a motor operates over its critical state, both motor developed torque and power will drop, but it will draw more reactive power from the system, which will, in turn, aggravate the motor terminal voltage deterioration. If this interacting process is unchecked, a sharp voltage collapse will occur. In addition, from Figure 4.6, we can see that although system operates at the unstable branch of P—V curve of motor terminal bus, the system has not yet lost its voltage control until the maximum reactive power is reached. If sufficient Var compensations were added at motor terminal bus before this 70 reactive power limit is reached, the system voltage would have been controlled, and the motor would move back to stable operation. Therefore, for the system with induction motor loads, the system voltage stability can not be judged by P—V curve alone. Case Study 2: Transient Stability with Induction Motor Load 4.5 4 1! 3 2.5 2 0 2 6 Time (sec) Time (sec) 0.5 0.9 0.4 0 0.3 I: 0.6 0.5 0 2 4 6 4 6 Time (sec) Time (sec) 2.5 2.5 2 1.5 1 1.5 0.5 0 2 4 10 6 Time (sec) 2 Time (sec) Figure 4.8: Transient Responses of Motor Variables 71 In this case study, an induction motor load is connected at system bus 21 through a distribution 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 loses a line between buses 14 and 21 at t = 0.5 second. The system responses shown in the Figure 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 the disturbance, the motor internal voltages (not shown) and motor slip remain unchanged. The motor terminal voltage and current change instantly. The disturbance causes the motor current to change in such a way that the motor developed torque, and hence the motor power and reactive power dip suddenly. The motor terminal voltage goes up a bit. Following that, motor slip begins to increase quickly until the critical slip is reached at 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 motor starts stalling. Both motor torque and power decrease quickly. The motor reactive power demand increases rapidly causing motor terminal voltage collapse within 4 seconds. The results of this case study show that a system with induction motor loads may involve a transient voltage instability when the operation of the induction motor is upset by system disturbances. More reactive power demand of an induction motor against voltage decline adds an strict constraint on system voltage stability. 4.3.2 Effect of Exponential PQ Loads The effect of a general static PQ load of the exponential form on system voltage stability is examined, which includes three special cases: the constant impedance, the constant current, and the constant power loads. 72 Case Study 3: Constant Power, Current, or Impedance Load The initial system condition is the same as that in case study 1. An induction motor with an initial load torque of 2.4 is again connected to system bus 21 through a distribution link. An step load torque increase of 0.23 is applied to the motor as a system disturbance. All other system bus loads are assumed to be exponential PQ loads. The effects 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 not considered in this study. 0.95 0.9 0.85 0.8 —c— 0.75 —ci--——— 0.7 0 20 constant power constant current constant impedance 40 60 80 100 140 120 160 180 200 Time (sec) Figure 4.9: Voltage Response at a Load Bus Comparison of the results for the three special types of PQ loads reveals some in teresting points. A constant impedance load (a = /3 2) is more voltage dependent 1), while a constant power load (a = /3 not dependent on its terminal voltage at all. Following the disturbance at t = 5 second, than a constant current load (a = /3 = = 0) is the induction motor will draw more power and reactive power from the system rapidly, 73 causing voltage drops at all system buses. In responding to the voltage drops, the con stant impedance loads will draw less power and reactive from the system, thus having a favorable effect to halt the further decline of system voltage. As a result, the motor can maintain stable operation at a higher terminal voltage for more than 170 seconds. On the other hand, the constant power load will draw the same power and reactive power from the 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 in the case of constant impedance loads. The effect of a constant current load is between the two with motor stalling at about t = 125 second, as shown in Figure 4.9. 0 0 0 0 Time (sec) Figure 4.10: Reactive Power Drawn by the Motor In all three cases, bus voltages collapse sharply when the induction motor starts stalling, which is, however, largely affected by the load characteristics. The more the dependence of a load on its terminal voltage, the better the damping effect it will have on system voltage stability. This damping effect is more crucial to the voltage stability of a power system where critical induction motor loads are supplied, as in this case study. 74 The motor reactive power response is shown in Figure 4.10, and the voltage response of a representative generator is shown in Figure 4.11. 1.051 1.05 1.049 1.048 1.047 1.046 1.045 1.0 —c— 1.043 1.042 1.041 0 20 40 constant power constant current constant impedance 60 100 80 120 140 160 180 200 Time (sec) Figure 4.11: Voltage Response at a Generator Bus It is also observed that with constant impedance loads, the system involves the gen erator angle instability, while with constant current or constant power loads, the system remains stable in term of system frequency although the stalling of the induction motor causes 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 disturbance reduce the system load, especially the real power load, causing generators to speed up following the disturbance. System remains stable both in angle and in voltage during the slow system dynamics. At about t = 105 second after the disturbance, the motor begins to stall, resulting in a sharp voltage collapse as shown in Figure 4.9. Both motor power and other system constant impedance loads drop rapidly. The process is so fast that the governors can not respond in time. The imbalance of system real power drives the generators eventually out of step of the synchronism. On the other hand, in the case 75 of constant power loads, system load is not affected by the sharp voltage drop due to the motor’s stalling. Only the real power of the induction motor load is lost. The generators speed up in response to this load reduction but remain stable at a higher rotor angles than those before the disturbance, as shown in Figure 4.12. 41 I 0 20 40 60 100 80 120 140 160 180 200 Time (sec) Figure 4.12: Rotor Angle of a Generator This case study demonstrates the effects of a static voltage dependent load on system stability. Constant impedance loads draw less power and reactive power when voltage decreases, which has a favored damping effect on system voltage stability. On the other hand, the load reduction may tip over the real power balance of the system causing a possible generator angle instability, especially in the cases where rapid voltage collapse may involve. In contrast, constant power loads have an somewhat opposite effect on system stability. Although the constant demand of power and reactive power may aggra vate the system voltage decline, which may cause system voltage collapse, it may help to maintain system real power balance when system voltage drops rapidly as in this case study, thus enhancing the system transient stability. This conclusion is drawn from the 76 consideration of load only. The conclusion may be opposite if the loss of system power generations is also involved. 4.3.3 Effect of Persistent PQ Loads As described in Chapter 2, a persistent PQ load demands constant power and reactive power despite a voltage decline, but involving an inherent time delay constant. This kind of load characteristics is very important to system voltage stability since the insisting demand of constant power and reactive power may cause system collapse, especially when the system loadability is reduced due to system disturbances. Case Study 4: Effects of Persistent PQ loads on Voltage Stability To study the effect of this kind of load on system voltage stability, all system loads of the sample system are considered as persistent PQ loads, modeled as changing equiv alent admittances and with same recovery time delay of 10 seconds. The same initial system condition as in the previous case studies is obtained by replacing the correspond ing induction motor load at bus 21 with a persistent PQ load having an initial loading of 2.48 + jl.40. The steady state system operation is then disturbed by a loss of a line between buses 14 and 21 at t = 5 second. It is anticipated that this system disturbance would greatly reduce the loadability of system bus 21 because the load now must be sup plied 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 in stability. Upon the system disturbance, system voltages dip immediately, causing the instant load power and reactive power load drops as seen in Figure 4.14. 77 0.9 0.8 0.7 0.6 0.5 0.4 o 0.3 0.2 0.1 400 500 600 Time (sec) Figure 4.13: Voltage Collapse at Bus 21 I 2 4 6 8 10 2 4 6 8 10 Load Admittance at Bus #21 Load Admittance at Bus #21 Figure 4.14: Load Power and Reactive Power vs. Load Admittance After 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 to decline. This process continues until about t 78 = 139 second at which load power and reactive power at bus 21 have reached their corresponding maximum values as shown in Figure 4.14. But, the maximum load power and reactive power at bus 21 are still less than 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 despite the further increase of the load admittance. The bus voltage goes all the way down as shown 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 Voltages Some other load and generator bus voltages of the system are shown in Figure 4.15 and Figure 4.16. The results indicate that the system survives the transient stability and voltages at system load buses other than bus 21 remain a fairly high level. Although the voltage collapses at bus 21, which causes loss of load at that bus, the other system bus loads have recovered to their pre—disturbance levels, some of which are shown in Figure 4.17. The persistent loads demand constant power and reactive power, which exceed the system loadability at bus 21, causing a voltage collapse at that bus, but maintains the system real power balance, avoiding a system angle instability. 79 1.055 00 1.05 44 1.045 1.04 1.035 1.03 0 100 200 300 400 500 600 700 800 900 1000 Time (sec) Figure 4.16: Some Generator Bus Voltages 4.52 “0 41:: 44 3.5 c 3.48 -.. 4.48 0 . 3.46 4.46 0 4.44 4 rl) 0 500 Time (sec) 1000 3.44 0 500 Time (sec) 1000 0 500 Time (sec) 1000 1.5 -. 1.48 0 11 0 0 500 Time (sec) 1000 Figure 4.17: Loads at Some System Buses 80 4.4 Control Effects on System Voltage Stability All reactive power related system components, such as controls, compensations, and con straints, have large impacts on system voltage stability. These system components may improve or deteriorate system voltage by supporting or restricting the system reactive power supply. The effects of system reactive power components on system voltage sta bility are examined in this section. Among them are the on load tap changing of a distribution trallsformer, the reactive power compensation with fixed capacitor banks or with SVC’s, and the rotor overheat protection of a generator. 4.4.1 Effect of transformer tap changing System distribution transformers are equipped with on load tap changers (OLTC’s) to maintain normal load side voltages by changing the taps. The dynamics of the OLTC is usually slow comparing with those of other system components, such as generators and motors. Therefore, the effect of an OLTC on system voltage stability may not be considered for fast transient system conditions. But, it must be considered when a slow system 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 load is assumed at load bus 21, and all other system bus loads are assumed to be constant impedance 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 that each 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 system disturbance is simulated by a step load torque increase to the motor with an initial load torque of 2.4. 81 Case Study 5: Tap Changing at Critical Motor Load Bus In this particular study, the tap changing is only assumed at the transformer link of the 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 used for comparison. As shown in Figure 4.18, if the transformer tap changing effect is not considered (as in case study 1), the system voltage will collapse at about t = 170 second due to the motor’s stalling. Otherwise, the system voltage remains stable if the tap changing effect is considered. 2 0 20 40 60 80 100 120 140 160 180 Time (sec) Figure 4.18: Tap Changing Effect on Motor Terminal Voltage Following the step change in motor load torque at 1 5 second, both motor power and reactive power increase along with the increase of the motor slip, which causes the motor terminal 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 and the tap changing is initiated. The tap changing of moving up one tap is completed at 82 t = 17.13 second. This tap changing raises the motor terminal voltage as shown in Figure 4.18. This voltage rise increases the motor developed torque, which becomes larger than the load torque as shown in Figure 4.19. 2.65 0 2.6 0 2.55 2.5 0 2.45 0 2.4 0 5 10 15 20 25 30 35 40 45 50 Time (sec) Figure 4.19: Tap Changing Effect on Motor Torque At the instant of tap changing, the motor slip remains unchanged. Both motor power and reactive power jump suddenly. After that, motor slip begins to decrease, which results in a large motor reactive power decrease as shown in Figure 4.20. This reduction of motor reactive power helps to halt the system voltage from declining, which stabilizes the 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 shown in Figure 4.21, where the loading limit (the nose of the P—V curve) of the motor terminal bus is appreciably extended by the transformer tap changing. This study shows that the transformer tap changing is helpful to voltage stability for a load with negative reactive power—voltage characteristics, such as induction motor load. 83 2.8 2.6 I 2.4 —0-———— motor Q 2.2 —g---—--- motor P 2 1.8 1.6 1.4 5 10 15 20 25 30 35 40 45 50 Time (sec) Figure 4.20: Tap Changing Effect on Motor P and Q I 2.55 2.6 2.65 2.7 2.75 Power Drawn by the motor Figure 4.21: Motor P—V Curve with and without Tap Changing The load side voltage increase due to the tap changing will reduce the load reactive power, which, in turn, enhences the system voltages. This effect of a transformer tap changing is especially crucial when the system operates near its critical state. 84 Case Study 6: Tap Changing at Other System Load Buses In this case study, it is assumed that there are transformer tap changings for all system load transformers except for that of the induction motor load. The initial system condition is the same as that in case study 5. But a smaller motor load torque increase of 0.22 is assumed at t = 5 second. ‘-I I Time (sec) Figure 4.22: Effect of Tap Changing at Other System Load Buses Figure 4.22 shows that the system voltage will remain stable if no tap changing effects are considered while the system loses voltage stability in the case where the tap changing effects are considered for all load buses but the motor load bus. Following the system disturbance, system voltage decreases as the motor picks up its load. Since all system loads except the motor load are constant impedance loads, this voltage drop will cause a system load reduction, which has a damping effect to voltage deterioration. If no transformer tap changings are involved, the system will sustain a transient condition, and remain stable at a fairly normal voltage as shown in Figure 4.22. 85 In the case where the tap changings are considered oniy at load buses with constant impedance 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 some of the load power and reactive power. More current are then drawn from the transmission system, 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 = 55 second. Following that, a rapid increase in motor reactive power demand leads to a sharp voltage collapse. Figure 4.23 shows the P—V curve of the motor load bus. The dash—line curve shows the case where no tap changings are considered, and the motor remains stable operation at a fairly normal voltage, while the solid—line curve shows the tap changing effects which cause system bus voltage drops, upsetting the motor stable operation. The effects of tap changings on motor power and reactive power is further shown in Figure 4.24. 0.96 0.94 0 0.92 2 O.9 0.88 0.86 2.6 2.65 2.7 Motor Power Figure 4.23: Motor Bus P—V Curves 86 Time (sec) Figure 4.24: Tap Chauging Effect ou Motor Power and Reactive Power This case study shows that a transformer tap changing has a detrimental effect on system voltage stability for the system loads with positive reactive power—voltage charac teristics, such as an exponential form PQ load with positive exponents. The voltage rise due 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 possible voltage instability as shown in this case study. This effect of a transformer tap changing will become salient when the system is operating near its critical state. 4.4.2 Effect of System VAR Compensation For a heavily loaded power system, effective reactive power support is crucial to maintain system voltage stability. To study the effects of reactive power control devices on system voltage stability, a fixed capacitor compensation and an SVC are considered and com pared fof different system operating conditions, which may involve fast system transients and slow system dynamics. The fixed capacitor is modeled as a constant susceptance B. 87 The parameters of the SVC’s control circuit are given in Table 4.16. Table 4.16: Data of SVC Parameters KB TB T1B T2B 100 0.15 1.0 10.0 Case Study 7: VAR Compensation and Fast System Transients In this case study, the additinal VAR compensations are provided to bus 21. Both fixed 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 limit of B 2 as shown in Figure 2.10. The SVC of larger capacity has a value of B max 2 while that of smaller capacity B max 2 0.35, 0.25. An induction motor with initial load torque of 2.4 is connected to system bus 21 through a distribution link. Other system loads are assumed to be constant impedance loads. The system disturbance is simulated by a line opening between buses 14 and 21 at t = 0.5 second. No tap changings are considered. 1.05 0.95 0.9 0.85 0.8 0.75 C 0.7 0.65 0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (sec) Figure 4.25: Effects of Different VAR Compensations 88 5 Figure 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 current of motor in order to pick up the load, which now has to be supplied from relatively remote system buses 17 and 20. If there is no additional reactive power support at bus 21, the low voltage will cause the motor to stall, and the large reactive power drawn by the stalling 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 which Curve 2 shows the case where the capacitor is switched in the system at t while Curve 3 that at t = = 1.5 second, 2.5 second. Although the capacitor is switched in very quickly, it still can not halt the voltage deterioration and collapse in both cases. Curve 2 and Curve 3 also show that the sooner the capacitor is switched in, the better the compensation will present. This is because the reactive power supplied by a capacitor depends on the voltage level, 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-) CJD 0 2.5 2 3 Time (sec) Figure 4.26: VAR Compensation of Fixed C Switched in Different Time 89 Curve 4 and Curve 5 in Figure 4.25 show the cases where the fixed capacitor at bus 21 is replaced by SVC’s. The results demonstrate that system voltage can be effectively stabilized by an SVC of sufficient capacity (Curve 4), but not so for the system with an SVC of limited capacity (Curve 5). In the latter case, the SVC cannot supply enough reactive power compensation because it has reached its limit, and behaves like a fixed capacitor thereafter. As shown by Curve 5, the bus voltage has been controlled at a fairly high level for about 4 seconds, but ultimately collapses. Figure 4.27 further shows the reactive power compensations of both SVC’s. It clearly shows that an SVC can provide reactive power very quickly upon voltage drops, but an SVC must have sufficient capacity in order to stabilize a system voltage. 0.35 0.3 Cl 0.25 0.2 ?‘J) 0J SVC with larger capacity 0.05 -0.05 — 0 0.5 1 — 1.5 — — SVC with limited capacity 2 2.5 3 3.5 4 4.5 5 Time (sec) Figure 4.27: VAR Compensation of SVC’s with Different Capacities Case Study 8: VAR Compensation and Slow System Dynamics In case study 7, system VAR compensation in fast transient system condition is investigated. This case study will show the effects of VAR compensations for a system 90 condition in slow system dynamics. For this, all system loads are modeled as persistent PQ loads with a time constant of 10 seconds. The induction motor load in case study 7 is replaced by a persistent PQ load with the same initial loading. VAR compensations are considered again at bus between buses 14 and 21 at t 1. The system disturbance is simulated by a loss of line = 5 second. 0.94 0.92 O.9 C 0.88 0.86 0.84 0 10 20 30 50 40 60 70 80 90 100 Time (sec) Figure 4.28: Effects of VAR Compensation of Fixed Capacitor and SVC Figure 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. If there is no additional VAR compensation available at bus 21, the voltage will ultimately collapse as shown by Curve 1. Comparatively, Curve 2 shows the case where an effective SVC is available at bus 21. The voltage recovers very quickly in about 10 seconds and remains stable at its normal value. The other two dash—line curves, Curve 3 and Curve 4, show the situations where a fixed capacitor at bus 21 is switched in at 10 seconds and 20 seconds respectively after the disturbance. The voltage is stably controlled for both cases. However, there will be more than required reactive power compensation to 91 the system along with the voltage recovery. As a result, the voltage will have a value higher than its pre—disturbance value, and some of capacitor compensations may have to be switched out of the system. This can be very critical when a system has major load with negative reactive power—voltage characteristics, that is, the load demands more reactive power when its terminal voltage goes down. There is no such problem with SVC compensation, since the reactive power supply is controlled according to the voltage deviations. When voltage is higher than its specified value, VAR compensation of SVC will decrease accordingly. The different VAR compensations are shown in Figure 4.29. 0.35 0.3 4 0.25 + — — E — — curve 1 (SVC) curve2 (fixed C) curve3 + 4 capacitor +>7stchedin 10 20 30 40 50 60 70 80 90 100 Time (sec) Figure 4.29: VAR Compensation of Fixed Capacitor and SVC For a heavily loaded power system, an effective reactive power compensation is cru cial to system voltage stability since system disturbances changes the power flow in the transmission network, which may cause an extra reactive power loss and a lower system voltage. Depending on the system controls and load characteristics, the system may involve fast transients and/or slow dynamics. For system involving fast transients, the system voltage may drop quickly. In such a case, fixed capacitor compensations are not 92 effective and SVC’s with sufficient capacities should be used. For the system involving slow 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 feed back control characteristics. Therefore, only SVC’s with ample capacities can effectively stabilize the voltage of a power system. 4.4.3 Effect of Generator Rotor Overheat Protection Generators are equipped with rotor overheat protection to prevent the field willding from overheating. The current in the field winding is determined by the winding resistance and the excitation voltage which is controlled by the generator excitation system. When system disturbances cause an extra system reactive power loss, the generator termi nal voltages will decrease. To maintain normal terminal voltages, generator excitation voltages must be automatically increased, which will result in larger currents in rotor windings. When the accumulated heat is over the permissive limit, overheat protections will 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 gen erator, it will have a significant impact on the voltage stability of the entire power system. Case Study 9: Effect of Rotor Overheat Protection In this case study, the effect of generator rotor overheat protection (ROP) on system voltage stability is investigated. The ROP is usually realized by the generator excitation reduction to its continuous operating limit or the second limit (see in Figure 2.4) when the generator has been operated at a higher excitation level over a prescribed period of time. In case study 4, the effects of persistent loads on system voltage stability are discussed 93 without considering the generator ROP. The attempt of persistent PQ loads to recover loads to the pre—disturbance level by increasing their equivalent admittances pushes the system over the post—disturbance loading limit of bus 21, which results in a voltage collapse at that bus as shown in Figure 4.13. In this case study, all system loads are again modeled as persistent PQ loads with recovery time delay of 10 seconds, the same as in case study 4, but the effect of generator ROP on system voltage stability is considered. The pre—disturbance loading at bus 21 in case study 4 is adjusted from 2.48 + jl.40 to 2.40 + jl.36. The system disturbance remains unchanged, and is again simulated by a loss of line between buses IA and 21, which occurs at t = 5 second. 3.6 3.5 2.9 j 2.8 3.4 3.3 0 500 Time (sec) 1000 0 500 Time (sec) 1000 0 500 Time (sec) 1000 3.2 2.8 3.1 2.7 2.6 C C 2.5 2.9 0 500 Time (sec) 1000 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. 94 Generators could operate continuously at higher excitations, as shown in Figure 4.30 7, where excitation voltages of generators at buses 3, 6, and 8 are recorded. This means that the generators could have the ability to maintain their terminal voltages so that As a result, the system load bus large reactive power could be supplied to the system. voltages remain stable at a fairly higher values as shown in Figure 4.31. 1.2 I 0.8 load bus #21 0.6 0 100 0 300 200 400 700 600 500 800 900 1000 Time (sec) Figure the In 4.31: System Load Bus Voltages (without Generator ROP’s) the second case, continuous operating voltage for each generator is assumed to disturbance level. than its second If over generator excitation to at bus since it 6 be 10 or the second percent over the limit, of excitation corresponding pre— a generator has been operated continuously at a higher excitation limit Simulations show limit, 100 seconds, the rotor overheat protection will cramp the this limit. that after system transients, exceeds its second limit at t 142 the second excitation voltage of and is cramped at has operated continuously at the higher excitation over 95 100 t the = seconds. generator 242 second 1.05 98 C S C (a) 0.95 C 0.9 0.85 0.8 Time (see) 1.04 3.15 3.1 1.02 3.05 S 8 c3 3 o 2.95 (b) 0 2.9 0.98 00 2.85 C 2.8 8.96 Generator at Bus #3 Tripped Off 2.75 0.94 0 100 200 300 400 508 600 700 800 Time (see) 1.1 C 1.05 98 C 98 2S S C 0 C (c) C 0 0 0.95 so 2 S C 0.9 0.85 100 200 300 400 500 600 Time (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 voltage begins to decline as shown in Figure 4.32a. Immediately after the excitation reduction 96 of 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 their excitation rapidly to pick up the system reactive power load. Consequently, the excitation voltage of generator at bus 3 remains over its second limit 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 voltage deterioration. 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 voltage drops at system load buses as shown in Figure 4.33. Although other generators attempt to increase excitations to supply more reactive power, the rapid voltage collapse initiates a system transient instability with generators tripped off consecutively due to their losses of synchronism. E rl) rj U 100 200 400 300 500 600 700 800 Time (see) Figure 4.33: Effect of Generator Overheat Protections This case study shows the important effect of generator rotor overheat protection on 97 system voltage stability. Under heavy system loading conditions, generators are usually operated 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 in creased the electrical distance between generators and loads, some generators may be operated with overexcitation, such as generator at bus 6 in this case study. Once the excitation of the overexcited generator is reduced by its rotor overheat protection, some of its reactive power supply must be transferred to other machines which may be again successively protected, such as generators at buses 3 and 7. The successive losses of voltage control at generator buses will then aggravate the system reactive power short age. System voltage may therefore collapse if there are no other reactive power controls available to the system. This study also suggests that for a heavily loaded system, local load reactive power compensation should be effectively used for system voltage control rather than heavily counting on the reactive power supply from remote generators. 4.5 Maximum Loading Limit by Load Flow and Simulation In previous sections, various effects of loads and controls on voltage stability are exam ined. Due to the system disturbances, the system operation may approach its critical state dynamically, which is affected by load and control characteristics. Since the system critical state is usually characterized by the system MLL which is crucial to voltage sta bility analysis and control, methods based on the traditional load flow analysis and the dynamic simulation as proposed in this thesis are compared in this section. Case Study 10: System Loading Limit with Persistent PQ loads In this case study, all system loads are modeled as constant PQ loads for load flow study except that a variable admittance load with initial loading of P+jQ 98 = 2.48 +jl.40 is assumed for bus 21. Persistent PQ loads with the same time constant of 10 seconds are used for the simulation study. The initial and final system loading conditions, and load power factors are the same for both studies. A loss of the line between buses 14 and 21 is assumed as the system disturbance. Load flow studies are made for both normal and post—disturbance system conditions by increasing load admittance at bus 21. The system loading limits of bus 21 are observed as P + jQ = 5.77 + j3.26 for the normal system condition, and P + jQ = 2.94 + jl.67 for the post—disturbance system condition. For the simulation study, the disturbance is applied to the system at 1 5 second, and the simulation lasts 1000 seconds. P—V curves of bus 21 of both load flow and simulation studies are shown in Figure 4.34. 1.2 — a S 0.8 0.6 a a 0.4 0.2 0.5 1.5 2 2.5 3.5 Power drawn from system bus 21 Figure 4.34: P—V Curves of Load Bus 21 Figure 4.34 shows the results of P—V curves of various studies: Curve 1 from load flow for the normal system conditioll, Curve 2 also from load flow but for the post—disturbance system condition, and Curve 3 from simulation for the system being disturbed. From load flow studies, although the disturbance greatly reduces the loadability of load bus 21, 99 the 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 usually gives the upper bound of a system maximum loading limit. Case Study 11: System Loading Limit with An Induction Motor Load The system loading conditions of this case study are the same as those of case study 10 except that the load at bus 21 is replaced by an induction motor load. The initial load torque TL for both load flow and simulation studies is 2.0. Load flow study is carried out to find the system loading limit by increasing the motor load torque until the load flow solution disappears, and the result is shown as Curve 1 of Figures 4.36 and 4.37. The results show that the system reaches its loading limit at a motor load torque TL (Pm = 2.56) at which the load flow diverges. I__ Time (sec) Figure 4.35: Motor Load Torque Changes in Simulation Study 100 2.47 0.96 0.94 0 0.92 2 0.9 0 0.88 0.86 2.3 2.2 2.4 2.6 2.5 2.8 2.7 Motor Power Figure 4.36: P—V Curves of Motor Terminal Bus 0.95 0 0.85 1 curve 2 0.75 0.65 curve 3 1 1.5 2 2.5 3 3.5 Motor Reactive Power Figure 4.37: Q—V Curves of Motor Terminal Bus For simulation study, it is further assumed that the motor load torque has a step change of 0.23 at t TL = 2.6 at t = 5 second, and then a gradual increase of 0.02/sec until it reaches 23 second as shown in Figure 4.35. The F—V and Q—V curves are shown 101 in 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 at motor load bus can increase the system loading limit, but is also affected by the time delay of the tap changing. The simulations show that the motor load torque of TL = 2.6 is still 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 motor load torque after the step change, there is enough time for the transformer tap changer to respond before the voltage collapses. 1.049 50 50 Time (sec) Time (sec) Figure 4.38: Some Generator Bus Voltages Curve 2 of Figure 4.36 and Figure 4.37 shows the case that the tap changing time delay is 10 seconds. The tap changes at t = 15.77 second and t = 25.77 second raise the motor terminal 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 motor slip begins to decrease, and hence the reactive power drops accordingly. Although the final generator voltages shown in Figure 4.38 are below their normal levels, the motor 102 remains stable at the normal voltage with the load torque of 2.6 and other bus loads totally recovered as shown in Figure 4.39. rl) 0 0 0 0 Time (sec) Time (sec) I 0 0 Time (sec) 30 Time (see) Figure 4.39: Some System Bus Loads Curve 3 of Figure 4.36 and Figure 4.37 shows the case that the time delay of tap changing is set to 15 seconds. The tap changer responses at t the system loading limit. But this limit is again exceeded at t 20.77, which increases 21 second due to system load recovery. As a result, the motor power begins to decrease, while its reactive power increases despite voltage drop. Finally, before the second tap change could occur at I 35.77 second, the system has lost voltage stability at motor terminal bus at I = 30.7 second. After that, both motor power and reactive power drops along with the voltage collapse. For comparison, the case where no tap changer is considered is also presented by Curve 4 of Figure 4.36 and Figure 4.37. 103 This case study demollstrates that although the system loading limit in steady state can be determined from load flow related static methods, the loading limit for system being disturbed must be evaluated by detailed system simulations. Moreover, the voltage control effect of transformer tap changing, if fast enough, can effectively increase the MLL of a system being disturbed to the extent even larger than the MLL of steady state from load flow. 104 Chapter 5 CONCLUSIONS AND REMARKS 5.1 Conclusions of the Thesis This thesis project is mainly concentrated on the analysis of voltage stability of a power system through time domain simulation techniques. Better understandings of system voltage instability phenomenon are gained through close examinations of the effects of load and control components on system voltage stability. A 21 bus sample power system is chosen for the simulation studies. Steam and hydro—electric generating units, various types of loads, and many reactive power control devices are modeled with emphasis on the dynamic behaviors of system loads and reactive power 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 implicit Trapezoidal 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 re lated nonlinearities in load flow iterations so that the variable extension and convergence problems can be avoided. A new two—step procedure is also developed for efficient and systematic solution of high order system Jacobian matrix equations. The effects of various types of loads and reactive power controls on the voltage stability are thoroughly examined through designed case studies so that the dynamic voltage behavior of a power system in various operating conditions can be clearly demonstrated 105 and 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 fol lowed by a sudden collapse, depending on the system operating conditions, system load and control dynamics, and types, locations, and seventies of system distur bances. 2. Induction motor loads, which constitute the major part of industrial loads, may have great influences on system voltage stability due to its more or less constant power 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 developed torque, the motor will pick up the load very quickly by increasing its slip. More current will be drawn from the system, which causes further voltage decrease. De pending on the magnitude of a disturbance, this interaction between the motor and the supply system may experience either a slow system dynamics which drives the motor towards its critical state for a long time before it starts stalling, or a fast system transients which upsets the motor’s stable operation so quickly and causes the motor stalling in a few seconds. During the motor stalling, the reactive power demand increases very quickly, which causes the system voltage collapse. Unlike system angle instability which is caused by the generator power imbal ance, the voltage instability caused by the loss of motor stable operation can not be judged by the power imbalance alone. It also involves the reactive power equi librium of the system. This means that the loss of motor power due to the motor 106 stalling does not necessarily result in a voltage instability since the motor terminal voltage can be controlled before the motor Q—V characteristics becomes positive. 3. When a disturbance causes system voltage drop, a constant impedance load will draw less power and reactive power from the supply system than that of a constant power 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 induction motors operating near their critical state. Constant power load, which is not voltage sensitive, is therefore a stiff system load. This load characteristics is harmful to system voltage stability because it will draw the same power and reactive power from the supply system despite voltage decline. Although a voltage dependent load has a favored effect on system voltage sta bility, it has different impacts on system angle stability. On the one hand, when a 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 load reduction is helpful to balance system power so as to stabilize the generator oper ation. On the other hand, when a disturbance causes system voltage collapse due to, for example, the loss of a motor stable operation, the rapid decrease of loads due to voltage collapse may upset the system power balance causing a transient system angle instability. 4. A persistent PQ load may demand constant power and reactive power but involv ing a time delayed recovery. The attempt to maintain pre—disturbance load level by increasing load equivalent admittance despite voltage decline may cause some particular system load bus exceeding its post—disturbance loading limit, causing voltage instability at that bus. If the post—disturbance system operating condition is such that the voltage 107 collapsed bus has a relatively short electrical distance with other system buses, the loss of voltage control at that particular bus may spread out to the other parts of the system causing a complete system voltage collapse. On the other hand, if the voltage collapsed bus is far away electrically from the rest of the system, it may has little impact on the other bus voltages, and then the rest part of the system may have a chance to maintain both voltage and angle stability. 5. A transformer tap changing may have either beneficial or detrimental effect on system voltage stability depending on its location and load characteristics. Its effect is crucial when a system operates near its critical state. The effect of tap changing at step—up transformer of a generator is always beneficial to system voltage stability since it raises the transmission voltage, and hence reduces the network current and the corresponding reactive power loss. The tap changing of distribution transformer at a system load bus has different effects on system voltage stability depending on load characteristics. For a load with positive Q—V characteristics, such as an exponential form PQ load with positive exponents, tap changing which raises the load side voltage, will result in more reactive 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 collapse if the system has operated near its critical state. On the other hand, for a load with negative Q—V characteristics, such as an induction motor load, the voltage increase by changing the transformer tap will reduce the reactive power drawn from the system so as to stabilize the system side voltage. 6. Effective system VAR control and adequate compensation is very important to maintain system voltage stability. The effectiveness of a VAR compensation de pends on the types and locations of VAR devices, the system operating conditions, 108 the VAR control speed, and the VAR capacities. Since the VAR compensation with a fixed capacitor are directly proportional to the square of the bus voltage regardless of load demands, it may not be effective in most system situations involving fast voltage drop. This also suggests that a heavy fixed capacitor compensated system may be vulnerable to voltage instability. For a system operating condition involving slow voltage decline, fixed capacitors compensation can be used to support system voltage, but some of the capacitors should be switched out of the system to avoid over—compensation when system voltage is back to normal. This is especially crucial when the bus load has a negative Q—V characteristic. The VAR compensation with an SVC of sufficient capacity is very effective to stabilize the voltage of a power system. It is effective in both fast transient and slow dynamic operating conditions due to the fast response of the negative feedback control. However, there is a dynamic interaction between an SVC voltage control and a generator excitation control, which may cause a system oscillation. This observation suggests that the SVC voltage control must be coordinated with the power system stabilizer(P S S) design. 7. In a heavily loaded system, a generator rotor overheat protection may limit its rotor winding current by cramping the excitation voltage. As a result, it will reduce the reactive power supply to the system, and that reactive power burden must be transferred to other generators which could be also protected. The successive losses of voltage controls at generator buses will aggravate system reactive power shortage, which may cause a possible system voltage collapse. This also suggests that an effective system wide VAR compensation should be effectively designed, and a heavy dependence of reactive power supply from remote generators could 109 result in voltage instability due to the possible generator rotor overheat protection. 8. Case studies in this thesis demonstrate the importance of dynamic effects of sys tem loads and control devices on system voltage stability. Although the MLL determined by load flow for power system in steady state is generally larger than the MLL determined by simulation for a system being disturbed, the voltage con trol effect of transformer tap changing, if fast enough, can increase the MLL of an induction motor load even larger than that determined from load flow. 5.2 Future Research Work The following aspects are suggested for future research work. 1. Although the effects of individual system typical loads on system voltage stability are closely examined with a sample power system, the real system bus load is far more complicated. It may be a combination of these typical loads or more, which requires further investigation. The study may involve more detailed modeling of bus load, estimation of some other unknown loads, and identification of load parameters. 2. 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Sutanto, “Differeilt types of voltage instability,” IEEE/PES Summer Meeting, 93 SM 518-1 PWRS, Vancouver, July 18—22, 1993. 116 Appendix A DERIVATION OF STEADY-STATE MOTOR EQUATIONS In this appendix, the steady-state induction motor power equation (2.5) is derived from motor 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 a three phase symmetrical induction motor can be described in a d—q frame of reference rotating at synchronous speed Wb as (a) Stator winding voltage equations: = Vds qs 1 rsIds—qs+ (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 and Va’s are winding voltage d—q components with V = Vqr 0 for the rotor, Id’s and Iq’S are winding current d—q components, and L”s are corresponding winding flux linkages, in Webers per second, which have the following relations. ds = 13 +Xm)Ids+Xmldr (X (A.5) qs = 18 + Xm) ‘qs + Xm Iqr (X (A.6) 117 qr = (Xir + Xm) ‘dr + Xm I (A. 7) = (Xir + Xm) Iqr + Xm Iqs (A.8) where X . and Xir are respectively the stator and rotor winding leakage reactances, and 1 Xm is the magnetizing reactance. For steady state conditions, the winding transients are not included. The correspond ing steady state voltage equations can then be obtained by dropping the derivative terms from equations (A.1) (A.4), which gives — (a) Stator winding voltage equations: 5 Icts r 5 Vd — TIq + /‘qs (A.9) ds (A.1O) (b). Rotor winding voltage equations: o o = rrIdr—(1—wr)qr (A.11) 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) motor slip s = 1 — — (A.8) into equations (A.9) wr, terminal voltage V = Vds + j and rotor current ‘r = ‘dr + ‘qr V = o = — (A.12), and define stator current I I + which, through some manipulations, gives (rs+Xis)Is+jXm(Is+Ir) (A.13) (+Xir)Ir+jXm(Is+Ir) (A.14) Equations (A.13) and (A.14) lead to the well—known equivalent circuit of an induction motor as shown in Figure A.1. Defining Z = r 5 + Zr + Ra = 18 X j, Zr rr + j Xir, Zm j Xm, Ra iEi r, and Za = + j Xir, the impedance Z as seen from the motor terminal becomes 118 (A.15) Z=Zs+ZmHZa where the symbol “ “ means “parallel with.” rs Xis Xir rr A 4 Is Jr / 1-s Xm Vrn V S rr Figure A.1: Equivalent Circuit of An Induction Motor The voltages Vm across XM, and Va across as shown in Figure A.1 can be expressed as Vm (ZmHZa) = VaRa (A.16) (A.17) Substituting equation (A.16) into (A.17) gives VamV (A.18) The torque and power equation (2.5) can be derived as follows. (a) Air gap power FE FE = Re[Va1] = Re[Va (Va/Ra)*] = /Ra] 2 Re{Va — Va 2/j /1La 119 (A 19) (b) Motor developed torque TE TE (A.20) = (c) Power drawn by the motor Fm Pm = Re[VI] = Re[V (V/Z)*] = /Re[Z*] 2 V (d) Reactive power drawn by the motor m Q (A.21) m Q = Im[VI] = Im[V(V/Z)*] = (A.22) /Im[Z*] 2 V The above equations can be expanded and re—organized in such a way that the motor torque and powers can be expressed explicitly in terms of motor slip .s, the terminal voltage V, and the motor parameters p with p = {r , rr, Xis, Xir, Xm}. 8 Let 1 a = rsrr 2 a ,. + Xis Xm + Xir Xm 1 15 X = X 3 = rr(Xis+Xm) a 4 = rs(Xir+Xm) a 5 a = rrX = 2 rs(Xjr+Xm) = rrX M 2 = r 5 120 Cl = 2 C — (Xir+Xm)ai ra3 1 d = a+a 2 d = 2rsrrXm 2 3 d = a+a the motor torque and power can then be expressed as follows. TE = Fm Q — as 5 2 V H s 1 d . 2 3 s+d -d = 2 fE(s,p)V (A.23) b +b 2 s 1 s+b 2 3 2 V d +d + dis s 2 3 = 2 fp(s,p)V (A.24) c +c 2 s 1 2 3 2+d s 1 d s+d 2 = 2 fQ(s,p)V (A.25) 2 Equations (A.24) and (A.25) show variable impedance characteristics of an induction motor in steady state operating conditions. The power and reactive power of an induction motor depend on both terminal voltage V and motor slip s. 121
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A comprehensive simulation study of the voltage stability of a large power system Zhang, Wenjie 1994
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Title | A comprehensive simulation study of the voltage stability of a large power system |
Creator |
Zhang, Wenjie |
Date Issued | 1994 |
Description | The voltage stability problem has become a growing concern in power system planning and operation. Many large interconnected power systems have experienced voltage insta- bilities which involve fast transients and/or slow dynamics. Although load flow related static approaches have been well developed to characterize the system maximum loading limit as the voltage collapse point, the mechanism of how system operation approaches its voltage collapse point and how this collapse point is affected by system dynamics are still obscure. This thesis provides the answers to these two basic questions through the investigation of effects of loads and reactive power controls on system voltage stability by detailed time domain system simulations. The importance of system dynamics in the determination of the 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 appropriately modeled. Secondly, a comprehensive power system simulation program is developed based 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 and backward—substitution procedure for solving the system Jacobian matrix equations are devised. Different system disturbances are simulated, and the exact timing of system changes is implemented. Finally, a 21 bus sample power system is chosen for the voltage stability study. In the case studies, the effects of loads, control devices, and system disturbances on system voltage stability are thoroughly examined. The voltage instability of a power system is a very complicated phenomenon, which, depending on the location, the type, and the severity of a system disturbance, may involve a fast transient voltage instability, or a slow voltage deterioration followed by a sharp collapse. It is closely associated with system reactive power—related controls, and is strongly affected by the load characteristics. The beneficial and detrimental effects of loads and reactive power controls on voltage stability should be carefully analyzed so that the information can be used in voltage stability control designs. |
Extent | 1566493 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0064839 |
URI | http://hdl.handle.net/2429/6908 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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