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A dynamic test model for power system stability and control studies Dawson, Graham Elliott 1969

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A DYNAMIC TEST MODEL FOR POWER SYSTEM STABILITY AND CONTROL STUDIES by GRAHAM ELLIOTT DAWSON B.A. Sc., University of British Columbia, 1963 M.A. Sc., University of British Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee Acting Head of the Department . . Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1969 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 tha permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment of Ele&f)rica,( En The University of British Columbia Vancouver 8, Canada ABSTRACT A new model for power system transient stability tests lias been developed. It includes a dc motor simulated prime mover with a governor control synthesized by dc booster generator field control, a solid state voltage regulator and exciter, a synchronous machine with a large field time con stant realized by negative resistance in the field circuit, a transmission system with time setting SCR controlled fault and clear sequence switchings, an accurate torque angle de viation transducer (Chapter 2), and analogs to realize con ventional stabilization and nonlinear optimal control (Chapter 5). Three state variable mathematical models of the test model with various degrees of detail are derived, in Chapter 3 Comparisons of results of digital computation and real model tests of a tj'pical power system disturbed by a short circuit are given also in Chapter 3. A parameter sensitivity study is carried out in Chapter 4. Comparisons of digital computa tion of transient stability with a nonlinear optimal control derived in this thesis and power and speed stabilization derived by another colleague of the power group at U.B.C., with the transient stability tests on the test model are given in Chapter 5. iii TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iiLIST OF TABLES v LIST OF ILLUSTRATIONS vi ACKNOWLEDGEMENT . . viii NOMENCLATURE ix 1 . INTRODUCTION ' 1 2. REALIZATION OF THE DYNAMIC TEST MODEL 4 2.1. Modeling Procedure 4 2.2. A Typical Power System 5 2.3. Setting a Typical Power System on the Test Model 6 2.3.1 Determining Base Impedence, Power and Voltage2.3.2 Numerical Values of System and Model Generator Parameters 8 2.3.3 Numerical Values of Model Base Quantities 9 2.3.4 Measurement of Model Generator Parameters 10 2.4. Development of Dynamic Test Model for Power System Simulation II 2.4.1 Governor-Prime Mover ....... n 2.4.2 Regulator-Exciter 14 2.4.3 Transmission Line and Circuit Breakers . 2.8 2.5. Auxiliary Measuring and Control Devices . . 21 2.5.1 Time-Sequence Control 22.5.2 Torque Angle Deviation Measurement . . 23 2.5.3 Forced Excitation Control 2 5 i v Page 3. STATE EQUATION MODELS OF THE DYNAMIC MODEL • AND TEST RESULTS 28 3.1. Seventh Order Synchronous Machine State Equations 30 3.2. State Equations of Controllers and DC Motor-Booster 2 3.3. Initial State of a Power System 3^ 3.4. State Equations of Fifth Order-Synchronous Machine and Controllers 39 3.5. State Equations of Third Order-Synchronous Machine and Controllers 42 3.6. State Equations of Third Order-Synchronous Machine and Regulator-Exciter bub Without Governor 45 3.7. Computation and Test Results of the Dynamic Test Model . . 46 4. PARAMETER SENSITIVITY OF THE TEST MODEL ..... 58 4.1. Sensitivity Equations 55 4.2. Parameter Sensitivity of System Response . . 55 5. NONLINEAR OPTIMAL STABILIZATION OF A POWER SYSTEM AND DYNAMIC MODEL TESTS . . . . . . . g0 5.1. Power System Stabilizing Signal . gC 5.2. Dynamic Optimization and Computational Method. go 5.3*. Computation and Test Results g^ 6. CONCLUSION 10 7 APPENDIX 3A 109 APPENDIX 3B 114 REFERENCES 116 LIST OF TABLES 2.1 Model and Actual System Generator Parameter Values g 5.1 Performance Functions Used During Transient Steps 84 vi LIST OF ILLUSTRATIONS Figure Page 2.1 Power System to be Modeled 4 2.2 DC Motor-Booster Unit 11 2.3 Model Governor-Prime Mover System 15 2.4 Governor-Hydraulic System Transfer Function . 15 2.5 Booster Field Compensation and Current Amplifier Circuit • • 1.6 2.6 Model Regulator-Exciter System 12.7 Regulator-Exciter Transfer Function • • • • 16 2.8 Scheme to Realize Negative Resistance . • • 17 2.9 One Phase Lumped Parameter Equivalent of a Single Circuit 19 2.10 Model JT Unit of Transmission Line .... 20 2.11 One Phase of the Model Transmission System . . 20 2.12 Circuit Breaker and Fault Time-Sequence Control 22 6 2.13 Torque Angle Deviation Measuring Device ... 24 2.14 Forced Excitation Time-Switching Control . . 26 3.1 Dynamic Test Model of One Machine-Infinite Bus System 2 9 3.2 Torque Angle Transient Responses 48 3.3 Governor Actuator Position Transient Responses. 49 3.4 Governor Actuator Feedback Transient Responses. 50 3.5 Governor Gate Position Transient Responses . .51 3.6 Turbine Torque Transient Responses .... 52 3.7 Booster Armature Voltage Transient Responses . 53 3.8 Terminal Voltage Transient Responses .... 54 3.9 Regulator Voltage Transient Responses . . .55 vii LIST OF ILLUSTRATIONS (Cont.) Figure Page 3.10 Field Voltage Transient Responses 56 3.11 Field Current Transient Responses 57 4.1 Parameter Sensitivit}' of Air Gap Flux ... 68 4.2 Parameter Sensitivity of D-Axis Flux .... 69 4.3 Parameter Sensitivity of Q-Axis Flux . . . . 70 4.4 Parameter Sensitivity of Actuator Position . . 71 4.5 Parameter Sensitivity of Actuator Feedback . • • 72 4.6 Parameter Sensitivity of Gate Position ... 73 4.7 Parameter Sensitivity of Turbine Torque ... 74 4.8 Parameter Sensitivity of Booster Armature Voltage 75 4.9 Parameter Sensitivity of Mechanical Speed . . 76 4.10 Parameter Sensitivity of Regulator Voltage . . 77 4.11 Parameter Sensitivity of Field Voltage ... 78 4.12 Parameter Sensitivity of Delta Angle .... 79 5.1 Calculated Transient Responses of Various Performance Functions 87 5.2 Calculated Transient Responses of Various Stabilizing Signals 90 5.3 Transient Responses with Accelerating Power Stabilization  3 5.4 Transient Responses with Speed Deviation Stabilization 96 5.5 Transient Responses with Optimal Control Stabilization 9 5.6 Transient Responses with Composite Stabilization 102 5.7 Transient Responses with Accelerating Power Stabilization and Composite Stabilization . . 10.5 ACKNOWLEDGEMENT I wish to thank Dr. Y. N. Yu, supervisor of this project, for his guidance, interest and encouragement dur ing the course of the research work and writing of this thesis. The development of the solid state voltage regu lator and exciter by Mr. J. Bond and the governor-prime mover system by Mr. R. Siddall is acknowledged and their efforts greatly appreciated. Thanks are due to my colleagues for helpful dis cussions and suggestions, particularly with Mr. N. Thompson concerning optimal control theory and with Mr. C. Siggers concerning power system stabilizing signals. Mr. B. Blackball's interest and efforts in the construction and subsequent testing of the test model were most helpful. Support from the British Columbia Telephone Com pany and the National Research Council of Canada through a Studentship award for 1967-1969 and grant A3626 is gratefully acknowledged. Thanks are also due to Mrs. W. Greig for typing this thesis and Mr. A. HcKenzie for draughting the figures. I am grateful to my wife Beverley for her patient understanding and encouragement throughout my graduate program. NOMENCLATURE Genera1 H Hamiltonian J cost functional J, augmented cost function a], a p d/dt, time derivative operator relative value of parameter q parameter qD intial value of parameter xi a sensitivity coefficient for state variable i with ' 1 respect to parameter qr A prefix denoting a linearized variable A costate variable DC Machines Parameters LaF a coef f icient ;6JmLap being the speed voltage coefficient for the dc motor R dc motor-booster armature resistance including R3 Rgm load setting resistance in the dc motor-booster armature circuit R^JJ booster field resistance T'fB booster field time constant Variables i^ dc motor-booster armature current Ip dc motor field current T^ load torque vfB booster field voltage Vy booster armature voltage v dc motor armature voltage Hi * ' vtdc c'c ino^or-booster armature terminal voltage (j mechanical angular speed of booster generator U)]U mechanical angular speed of dc motor k\ub mechanical angular base speed of dc motor, 188.5 rad/sec Hydraulic Turbine and Governor Parameters G(p) governor transfer function H(p) hydraulic operator transfer function H inertia.constant actuator servomotor time constant Tj^ dashpot relaxation time TQ gate servomotor time constant TW water starting time TN.. mechanical starting time of the unit °^\\ coefficient of net regulation g~ permanent speed droop S temporary speed droop Variables a actuator servomotor position a.p actuator feedback position g gate servomotor position n per unit relative angular speed change t turbine torque output Prime-Mover Governor Model Pa rameters J moment of inertia of test model f KjW | -I- K2 : friction K LaF]FWgLaf , , . ^ — —h-.-,.'„ : model coefficient ^ K K ~ — (l*FJr\ K K + \ / : coefficient for the dc motor 4 1 —R  torque cancellation K Bm load setting resistance in dc motor-booster armature circuit r/ 3 ; simulation coefficient Regulator-Exciter Parameters a-|,a2 constants -used to obtain characteristics of the field voltage limiter regulator gain Tj^p regulator time constant Tp exciter time constant Variables v^ regulator voltage vrej regulator-exciter reference voltage Synchronous Machine Parameters D damping coefficient l- armature resistance a R^ field resistance R^ transformed field resistance 'slope1-ratio of synchronous machine steady state armature short circuit current and field current Tj^-| d-axis damper leakage time constant T' d-axis transient short circuit time constant d T^o d-axis transient open circuit time constant 'T'a'T'q d- and q-axis subtransient short circuit time constant r^do,''"qo C'~ anC' (i~ax^-s subtransient open circuit time constant x mutual reactance between stator and rotor in d-axis ad x , x d- and q-axis synchronous reactances d q x' d-axis transient reactance d x ,x d- and q-axis subtransient reactances d q Variables ifj , i d— and q-axis current "*'fd field current P real power output of the machine Q reactive power output of the machine T energy conversion torque mechanical torque output to the rotor v^,v^ d- and q-axis voltages armature terminal voltage field voltage vr advf | ; a voltage proportional to field voltage RI VFR voltage proportional to field current VDR voltage proportional to d-axis damper winding current V0R voltage proportional to q-axis damper winding current ^d'fq ^~ anc* cl"ax^s flux linkages y field flux linkage V fLsjdV'c. , i f"l,lx proportional to field flux linkage F BI fa f ^D'^Q ^~ and Q-sxis damper winding flux linkages (j^ electrical angular speed CJeo base electrical angular speed, 377 rad/sec 8 torque angle x i :i. i Tran sni i ssion Sy st em Parameters 13 shunt susceptance G shunt conductance r series resistance x series reactance Variables v infinite bus voltage o INTRODUCTION 1 Modern power systems have become so complex in struc ture and so large in size that on-line tests of some control schemes to improve the steady state and transient stability are entirely prohibitive. This is because the experiments are not only costly but also possibly destructive in nature. Therefore, it is desirable to perform tests on small test models which have similar characteristics to the actual system. The development of the test models for large power systems is not new. Robert"*" *"*" in 1950 constructed a micro-machine and rnicroreseaux system. He investigated the electro magnetic and mechanical similarities of the model and real system. The same per unit reactances, equal time constants, similar inertia constants, and torque speed characteristics are used in the modeling. A rotary machine was used to obtain a negative resistance to increase the field time constant. Venikov designed another micromachine in 1952. A commutator machine was used to realize the negative resistance for the field time constant. Three rotors with different saliency, x^ = 0.85? 0.55? 0.40 were used. Flywheel effect was applied to vary the inertia of the micromachine. Adkins' " micromachine was reported in I960 and was used to investigate short circuits, synchronizing and damping torques, swing curves, asynchronous operation and resynchron-ization. The negative resistance of the field circuit was 2 realized by electronic circuits. The microturbine was real ized by a separately excited dc machine with thyratron control. More tests are reported recently in Canada by Roy^*^ using a micromachine and microreseaux and in the U.S.A. by 1.5 Dougherty with a thyristor controlled dc motor as the prime mover for dc transmission tests. A new model for power system dynamic studies has been developed at U.B.C. Preliminary work has been done by J. Bond on the design of a solid state voltage regulator and exciter and by R. Siddall on the simulation of a governor-prime mover. Further development work has been completed by this thesis including additional features and important tests of the com plete system. The model has the following features. 1) It is versatile in that most power systems with con ventional controllers can be simulated on the model on a per unit basis. 2) It has solid state components and analog simulated regulator and exciter with both linear and forced excitation characteristics. 3) It has an analog simulated governor-hydraulic operator in conjunction with a dc motor simulated prime mover. 4) It has a synchronous machine and exciter with a nega tive resistance in the field circuit. The negative resistance is realized by electronic circuits. 5) It has an ac transmission line which at the moment is designed for a special project. 3 6) It has electronic controlled switching' to realize fault, fault cleared and successful or unsuccessful line reclosure at preset times. 7) It has an accurate torque angle deviation -transducer utilizing the zero crossings of the terminal voltage and reference voltage waves. 8) It has a stabilizing signal generator. The thesis also includes the following features. 1) Derivation of state variable equations for the test model with different degrees of details, Chapter 3. 2) Comprehensive comparison of digital computation and model test results of a transient short circuit on the system, Chapter 3. 3) Parameter sensitivity study, Chapter 4. 4) Comprehensive comparison of digital computation and model tests of a "bang-bang" type nonlinear optimal control of the system. It is hoped the test model developed will be useful to the transient stability study of practical power systems. 2. REALIZATION OF THE DYNAMIC TEST MODEL 4 2.1 Modeling Procedure A typical one machine-infinite bus system, Fig. 2.1, is chosen to be modeled for power system dynamics studies. The methods and procedure of modeling are kept general so that they can be extended to any multimachine power system. II I Ll.H L II •-• IIIIIT' GOVERNOR -_ REGULATOR-EXCITER Fig. 2.1. Power System to be Modeled The first step of modeling is to obtain a detailed mathematical description of a typical power system. For ex ample, the synchronous machine is described by seventh order state equations, fifth order for the electrical and second order for the mechanical, but will be approximated • by lower order models after checking with the test results. The second step is to decide what ranges of parameter values of the synchronous machine, the controllers, and the transmission line of actual power systems are to be simulated. 5 The third step is to find the means, circuits and machines, to realize the mathematical model. For example, how is the governor-prime mover system to be simulated by conven tional d--c machines with solid state electronic circuits. The fourth step is the circuit design and construction details. Evidently, it is important to carry through the tests of all the components, subsystems and the complete model. One of the basic problems is how to determine the model para meters with accuracy. The last step is to select a base impedance and a base voltage of the test model for a particular system under in vestigation . 2.2 A Typical Power System The typical power system to be modeled by the dynamic test model is shown schematically in Fig. 2.1. The four main components are the governor-prime mover which supplies power to and maintains a synchronous speed of the system, the syn chronous machine for the electromechanical energy conversion, the regulator-exciter for the machine terminal voltage control, and the transmission system which connects the machine terminal to the infinite bus.. For the modeling of a one machine-infinite bus system, the infinite bus is assumed to be a machine of infinite inertia and zero internal impedance. 2.3 Setting Up a Typical Power System on the Test Model 2.3.1 Determining Base Impedance, Power and Voltage The main objective of the modeling is to achieve the same per unit value of parameters of the actual system on the dynamic test model. In other words, to make the test model per unit impedance equal to the system per unit impedance. Zm,pu = zpu Per unit (2*X) where ^ Zm,pu -per unit Zmb and zm)pU ~ Per unit impedance value of test model, zpu = Per unit impedance value of actual system, Zm = an impedance of the model, Z ^ = model base ohms. Equation (2.1) applies to all system components such as gen erators, transformers, transmission lines and loads. There fore, the model base ohm (lmb) must be selected such that Zmb ~ £m_ ohm (2.2) ^pu This condition must be met in order to obtain a good comparison on the per unit basis between the main parameters of the actual system and that of the test model. On the other hand, to real ize a non-existing impedance (Zjn) on the model, one may apply the well known formula = ZmbZpu - Z/kVmb\ 2 / MVAb \ ohm (2.3) 1<V, / I MVA , , o / \ mb/ 7 Next, the base power of the test model is established from a comparison on the per unit basis of the accelerating torque equation of the test model and that of the actual sys tem. As will be revealed in Section '2.4.1* a simulation parameter CX is related to the inertia constant II(sec) of the actual power system and the moment of inertia J (joule-sec2/ rad2) of the dynamic test model by the equation CX = JL joule-sec/rad2 (2.4) 2H and the base torque of the model is given by Tmb =t*Wmb joule/rad (2.5) where CO ^ (188.5 rad/sec) is the model base mechanical speed. Since the base power is given by Pmb =(Jmb Tmb watt (2.6) substitution of (2.4) and (2.5) into (2.6) gives Pmb = J^mb watt (2-7) 211 Equation (2.7) is the condition to be met for setting the base power of the model. Finally, the model base voltage is given by Vmb = Zmb Pmb <2'8) and equation (2.7) and (2.8) must be satisfied simultaneously. Of course, the model voltage and power bases must be within 8 the rating of the model generator. It is desirable to estab lish a base voltage less than the model voltage rating for two reasons; first the machine saturation effects can be neglected and second, the sustained fault currents in the study will not damage the generator. 2.3.2 Numerical Values of System and Model Generator Parameters Numerical values of generator parameters of an actual power system and those of the model are listed in Table 2.1 side by side on the per unit basis for comparison. The para-Parameter Mode! - System Actual System Measured Synchronous Machine Parameters ra .664-n. .0 42 pu .00247 pu xd 16. 2 1.025 pu .973 pu xq 9.71-n. .614 pu . 55 pu i xd 2.74.n. .173 pu .190 pu .247 sec Adj ustable* 5.00 sec Base Impedance 15.8 .797 Mechanical Parameters Inertia f J - .165 joule-sec^/rad^ 2.67 x 10-3o> + 1>585 / 2 joule-sec/rad H = 4.63 sec II = 4.63 sec •"-Section 2.4.2 Table 2.1 Model and Actual System Generator Parameter Values 9 metric values of the model, in MKS units are determined directly from tests. The base kV of the system generator is 13.9 and the base MVA is 242. To arrive at the per unit values of the model genera tor for a good simulation of the system, emphasis must be placed on the most 'important' parameters such as x<^, x^, x^, II etc. The generator has so many parameters that all of them cannot be simulated closely simultaneously because of so many constraints involved. For the particular per unit set up of Table 2.1 an exact value of H is retained. 2.3.3 Numerical Values of Model Base Quantities From (2.7) and data of Table 2.1 the model power base equals Pmb = J to mb = -( .165)(188.5)2 = 633 watts • (2.9) 2H (2) (4.63) Next if one would like to have an equal per unit value of t d x, in both systems, one shall have from (2.1) zm,pu = xd •= 2.74 = 14.4" ohm (2.10) However, in order to get x.^, x^ and x^ simulated closely simultaneously, the base ohm is chosen as 15.8 ohms. The remaining base values are v , = 100 v line to line volt (2.11) mb and Imb = 3.65 A amp (2.12) The nominal ratings of the dynamic test model generator are P = 1600 watts V = 208 volts I •= 5.5 amps 2.3.4 Measurement of Model Generator Parameters The synchronous machine reactances and time constants are determined by the standard procedures of the IEEE Test 2 1 Code No. 115. ' For example, the value of the direct-axis f transient reactance x^ is obtained from the armature current envelope of a three-phase sudden short circuit, and the direct-axis open-circuit transient time constant T^Q is obtained from the armature voltage envelope when the field winding with excitation is short circuited. The value of the direct-axis synchronous reactance found by the slip test compared favourably with the value obtained from the steady state open circuit and short circuit tests in the linear region. The moment of inertia J of the d-c motor-synchronous machine set is determined from a retardation test and the friction f is obtained from a steady state test utilizing 2 2 the energy conversion torque of the d-c motor. The parameter values thus obtained are presented in Table 2.1. Fig. 2.2. DC Motor-Booster Unit 2.4. Development of the Dynamic Test Model for Pox^er System  Simulation 9 O 2.4.1 Governor-Prime Mover 0 A d-c motor in series with a booster, Fig. 2.2, is used to simulate the prime mover. The d~c motor is given a constant excitation If and the initial armature current i^ and energy conversion torque are set by a load setting resistor Rgm. To obtain an incremental torque as a function of speed deviation of the d-c motor synchronous machine set, the booster receives an excitation which will cause the model to respond similarly to a power system with governor control. Neglecting the time variational but not the speed voltage effect of the d-c motor-booster armature circuit, the armature current becomes AA = i |vtdc -WmLaP IP ' wg Laf vfJ3 \ R I amp {2.13) V RfB 1 + TfpP/ The second term in the bracket is due to constant excitation and the third term is the speed voltage of the booster. The voltage vtdc is the total d-c voltage applied to the d-c motor-booster armature circuit. The resistance R includes two arma ture resistances and the load setting resistor R]3m. The accelerating torque of the d-c motor is Jd(Jm = LaFIFiA - (Ki«m+K2) - TL joule/rad (2.14) dt The first term of the right hand side of the equation is the energy conversion torque, the second term the friction torque experimentally determined, and the last term the load toi^que. For a load deviation, (2.14) becomes JdAGJm = LflF IFAiA ~ K]AoJm - ATL joule/rad (2.15) dt which corresponds to JdAU)m = G(p)l!(p)(-AWm)-ATL (2.16) Substituting (2.13) into (2.14) and (2.15) and comparing the results with (2.16) yields AvfJ = ~ l+TfBp («G(p)H(p)-K4) (-A«m) volt (2.17)-K3 where K3 = LRpIp(J^ Lap amp-sec R Rpg rad and = + (Lap Ip) watt sec^ R rad It becomes evident that if the voltage AV^can be realized according to (2.17), then the d-c motor will have exactly the same torque-speed characteristics as that of a large prime mover with a governor. Dividing through (2.16) by wtj^ gives J_ dn = G(p) 11 (p) (-n) - ATj per unit (2.18) which compares with Hovey's^'^ equation Tm dn - G(p) ll(p) (-n) - AM per unit (2.19) dt Hence = J_ } = 2H sec (2.20) and Tmb =«Wmb . joule/rad (2.21) For the particular model prime mover developed at U.B.C., the parameters are Laplp - 0.8 5 volt-sec/rad GJlTLaf = 73 ohm R^g - 51.6 ohm TfB - 0.5 sec KX = 2.67 x 10""3 K2 - 1.585 joule/sec joule/rad The total resistance R of the armature circuit including the load setting resistance Rgm at no load is R = 28.1 ohm ; vtd = 229 V The K0 and K. constants are 3 4 1 = 23.29 rad = 0.0258 watt sec2 K3 amp-sec " rad The mechanical parameters are presented in Section 2.3.2. The general layout of the model governor-prime mover system is shown in Fig. 2.3. The governor-hydraulic transfer function G(p) li(p) is set up on an analog specially built for this purpose. The model is capable of representing different governor-prime mover configurations. For the governor-hydraulic transfer function shown in Fig. 2.4* the parametric values are TA = 0.0 2 sec CT= 0.06 $ = 0.5 TR = 5.00 sec TG = 0.5 sec Tw = 1.6 sec The booster field compensator circuit (l + T^^P) is also built with analog components. The circuit is given in Fig. 2.5. A current amplifier is included to match the current level between the operational amplifier and the booster field. No difficulties are experienced with the differentiation since the input signal frequency is less than 1 Hz.. A small value of capacitance (C = lOOpf) is used to stabilize the circuit. 2.4.2 Regulator-Exciter2,5 The layout of the model voltage regulator-exciter system is shown in Fig. 2.6. For the studies in this thesis, the transfer function RE(p), Fig. 2.7, of the regulator-PERMANENT DROOP STrP 1+ Tr P GATE SERVO TEMPORARY DROOP AND DASHPOT 1-TWP t 1+.5TWP HYDRAULIC SYSTEM Fig. 2.4. Governor-Hydraulic System Transfer Function 16 Fig. 2.5. Booster Field Compensation and Current Amplifier Circuit VOLTAGE SIGNAL FIELD VOLTAGE A ~U STABILIZING SIGNAL ANALOG COMPUTER r~ CURRENT AMPLIFIER NEGATIVE RESISTOR SYNCHRONOUS MACHINE FIELD Fig. 2.6. Model Regulator-Exciter System VOLTAGE SIGNAL KA UTAP 1+T£P ———o FIELD "* VOLTAGE Fig. 2.7. Regulator-Exciter Transfer Function exciter is patched onto a specially built analog computer. The parameter values of an example are The voltage signal from the analog computer is then amplified to match the voltage and current levels of the synchronous machine field. The model synchronous machine has a relatively large resistance in the field circuit. The resistance value is reduced by a negative resistor^•4 to obtain the desired open circuit field time, constant (T^Q) • The negative resistor is realized by a scheme shown in Fig. 2.8. The idea is to obtain (W100 TA = 0.0 5 sec T£ = 0.035 sec REGULATOR - EXCITER GROUND Fig. 2.8. Scheme to Realize Negative Resistance a voltage v^ which is proportional to current vi = &L H g3 Rsi=1 volt (2.22) so connected that , Vfd + Vi = (Rf + Rs + LfP)i VOlt (2-23) Thus one has v = (Rf + Re - g)i + LP pi volt (2.24) fd S • i For example, to obtain an open field time constant of 5 sec the original resistance, Rf - 69.6 ohms, of the model is reduced to 3.44 ohms. 2.4.3 Transmission Line and Circuit Breakers A transmission line model is built to simulate a 576 mile double-circuit three-phase three-section high voltage transmission line. The distributed parameters of the high voltage transmission line are z = 0.041 + jO.5309 ohm/mile y = J7.88 x 10"6 mho/mile Each section of each phase of each circuit of the line is simulated by a 3T section with lumped parameters. The model section gives the same per unit voltage and per unit current at the ends as that of the real line with distributed para meters. This includes the effect of the shunt reactors (13 5 MVAR at 525 kV) of the real line at each end of the section. One circuit of one phase of the lumped parameter equivalent of Fig. 2.9. One Phase Lumped Parameter Equivalent of a Single Circuit 2 0 .67/ OHM —AAA 7.98juF Fig. 2.10. Model TT Unit of Transmission Line MACHINE TERMINAL TT TT INFINITE BUS TT TRANSMISSION SYSTEM L 77" • 77 - 77 BRAKING RESISTOR FAULT Fig. 2.11. One Phase of the Model Transmission Line the transmission system and shunt reactors is shown in Fig. 2.9. The parametric values of the equivalent model JT unit are given in Fig. 2.10. There are altogether 18 such units. They are built with resistors, iron core reactances and capacitors which are commercially available. A three-phase braking resistor is connected to the machine terminal for special studies. Three-phase relays are used as circuit breakers for line fault, clearing and braking resistance switching. One phase of the model trans mission system is shown in Fig. 2.11. 2.5. Auxiliary Measuring and Control Devices 2.5.1 Time Sequence Control The time and switching sequence control of the trans mission line and braking resistor circuit breakers, and the fault simulation is achieved with integrated circuit logic elements. The control is schematically shown in Fig. 2.12. The crystal clock produces square waves at 16 kHz. The push button sets the flip-flop which allows the square waves to pass through the 2 input NOR gate to the frequency divider which has an output frequency of 1kHz. The 0 and 1 outputs of the 12 bit counter are connected to units contain ing 12 single-pole double-throw time selection switches which in turn are connected to 12 input NOR gates which output a pulse when all the inputs are G. Two units are necessary, one pulse to turn on and a second pulse to turn off the SCR switch, for the fault simulation and braking circuit breaker operation where a CLOSE-OPEN relay action is 9 3.6 V* PUSH BUTTON SET 16 kHz SR FLIP FLOP \R 12 INPUT NOR GATE CRYSTAL CLOCK I 1kHz 2 INPU T NOR GATE HFREQUENCY! DIVIDER IrjFREQUENCY DIVIDER 1kHz \—H2) 12 BIT COUNTER IN VERTER J TIME SEL EC T ION SW/ TCHES 12 INPUT NOR GATE CLOSE V_ r JF OPEN SCR SWITCH I RELAY COIL BRAKING RESISTOR CIRCUIT BREAKER * FAULT SIMULATION J , J I 9 L _ ^ -I - -OPE A/j _C/.OSEj \OPEN i .TRANSMISSION i 5 Z./A/E CIRCUIT a 5 BREAKERS Fig. 2.12. Circuit Breaker and Fault Time-Sequence Control to so desired. For the transmission line circuit breaker simulation an additional unit is incorporated for an OPEN-CLOSE-OPEN relay action to represent an unsuccessful reclosure. The time-sequence control is terminated with a feedback pulse to the set-reset flip-flop 4.096 seconds after initiating the action with the push button. This time interval allows for normal and abnormal time-sequence operation of the fault simulation, transmission line circuit breakers, and braking resistor cir cuit breakers. 2.5.2 Torque Angle Deviation Measurement For power system dynamic studies, it is evidently very important to have an accurate torque angle deviation signal which can be constantly monitored and used for control pur poses. To this end, a continuous voltage signal proportional to torque angle deviation is obtained by the scheme shown in Fig. 2.13. Two a-c voltage signals? one from the infinite bus and one from an a-c tachometer coupled to the generator shaft, are fed into separate comparators. The output square wave signals are connected to monostables. This eliminates inter mittent switching caused by ringing at the comparator outputs. The time delay between the pulses from the monostables is pro portional to the phase shift of the two signals. The a-c tachometer monostable pulse sets the set-reset flip flop to allow pulses from clock 1 through the NOR gate to the 9 bit counter 1. The frequency of clock 1 is 30.72 kHz so that 512 pulses will be counted if the phase shift is 360 degrees at AC TACHOME TER COMPARATOR COMPARATOR 9 BIT COUNTER 2 9 SWITCHES 9 INPUT NOR GATE RESET OUN TER | 7 9BIT COUN JKFLIP FLOPS JL OUTPUT BUFFER JK FLIP FLOPS] TOGGLE INFINITE BUS 60 Hz DIGITAL TO ANALOG CONVERTER Fig. 2.13. Torque Angle Deviation Measuring Device .to •fa. 60 Hz. The infinite bus monostable pulse lowers the output level of the set-reset flip-flop to stop pulses from clock 1. At the same time, the falling edge of the set-reset flip-flop toggles a flip-flop to allow pulses from clock 2 into both 9~ bit counter 1 and 2. This scheme allows the steady state out put analog signal to be reduced to zero by setting the 9 double-throw single-pole switches. The complement of the binary value set is added to counter 1 at approximately 2.5 x 106.bits per second by clock 2. The end of the addition is detected by an output pulse from a 9-input NOR gate which initiates the following events; first the flip-flop is toggled to shut off the pulses from clock 2, second a pulse from the NOR gate toggles the output buffer to receive the contents of counter 1, and third a monostable is triggered. The monostabl provides a delayed pulse so the contents of counter 1, before being reset, can be transfered to the output buffer. The out put buffer is converted to a continuous voltage signal by a digital to analog converter with a range 0 to -10 volts. The sampling rate is 60 times a second. 2.5.3 Forced Excitation Control Provision is made for the forced excitation control to provide a stabilizing signal for the power system during and after a system disturbance; Chapter 5. The stabilizing signal is an open-loop bang-bang voltage signal summed with the ter minal and reference voltage to provide a total voltage error signal, Fig. 2.6. The block diagram of Fig. 2.14 shows a ©-> SR FLIP R FLOP [ INVERTER NOR GATE JL I 1 DECADE COUNTER I 10 100 1000 IL THUMBWHEEL SV/ITCHES INVERTER r~ NOR GATE EMERGENCY 6i STOP o 3.5 V INVERTER LONG PULSE MONO SHORT PULSE MONO r REPEATED 8 TIMES I. AAA THUMBWHEEL SWITCHES INVERTER 1 X NOR GATE J Fig. 2.14. Forced Excitation Time-Switching Control NEGATIVE VOLTAGE I TRANSISTOR SWITCH TRANSISTOR SWITCH POSITIVE VOLTAGE scheme to realize the switching times and the number of switchings of the stabilizing signal. The events of the forced excitation control are started at the instant of fault by a signal from push button set of the time-sequence control, Fig. 2.12. This signal resets a set-reset flip-flop allowing lkliz square waves from the frequency divider, Fig. 2.12, to be counted by the decade counter. Ten units consisting of thumbwheel switches, in verters and NOR gates set the times of switching. At the instant the events are initiated, the complementary level from the set-reset flip-flop is fed into an inverter whose output performs two functions. First, two monostables are triggered through an inverter. One monostable with a long pulse estab lishes the initial polarity of the forced excitation and the second shorter pulse toggles the initial condition into flip-flop 2. Second, flip-flop 1 is toggled which allows the NOR gates for negative and positive voltages to be enabled. The time setting units toggle flip-flop 2 to produce a bang-bang type of forced excitation voltage signal. The last timing unit output feeds back to the set-reset flip-flop to stop the timing sequence. At the same time, the high level output of flip-flop 1 clamps the negative and positive voltages to a zero value. 3. STATE EQUATION MODELS OF THE DYNAMIC MODEL AND TEST RESULTS One of the most important decisions to be made in power system dynamics studies is how much of the detail of the synchronous machine should be described by equations. The results of a mathematical model is a compromise between computation time and accuracy and is to be decided from direct comparison of computation and test, not by arbitrary choice. It is the objective of this chapter to investigate and to compare the computation results of synchronous machine models of different degrees of details with those from machine tests. Another important point in modeling is that the machine must be described by equations with measureable parameters. There is much more freedom in the mathematical manipulation of equations than the methods available in ob taining reliable parameter values directly from tests. For example, the synchronous machine reactances can be determined with much better accuracy than the leakage reactances. A third point in modeling is that since most optimal control theory, computation and nonlinear stability analysis techniques are developed from system equations in the state variable form, it is desirable to model the system equations as such to allow the application of the theory and obtain the solution by known computational methods. In this chapter a one machine-infinite bus system model schematically shown in Fig. 3.1 will be developed. The state equation representation of the regulator-exciter is Rp 'A + Bm SUMMING AMPLIFIER MACHINE TERMINAL AAA-W g INFINITE BUS G —— —- B Fig. 3.1. Dynamic Test Model of One Machine-Infinite Bus System to based on Fig. 2.1, and that of the governor-hydraulic system on Fig. 2.4 incorporating the state equation of the d-c motor-booster of Fig. 2.2. 3.1. Seventh Order Synchronous Machine State Equations Park's equations for a synchronous machine in d-q 3 1 coordinates ' are: vd = P^d ~WetFq ~ V-d (3-1) vq - PH>q + We^ " raiq (3.2) Vd = 1+ Tmp xad_ vfd _ (1+ Tdp) (1+Tdp) _xd id (l+Tdop)(l+Td'oP) WEORF (l+Tdop)(l+Tdop) 40eo (3.3) H'q = - 1 + Tq-£ _JEfl iq (3.41 + Tq0P CJeo which can be rearranged into the following state variable form P<Pd = vd + WeH>q + raid (3'5) P^q = vq ~ We^d + Vq (3'6) pVF =RvF - vFR (3.7) P*D = ~VDR (3'8) PVQ^-VQR (3'9where Vp^, id and v^ are solved from id V DR xd(Tdo + TD) -xd(Tdo-Tc^eo xd(? it tt , xd Tdo Tdo xd Tdo xdTD do xdo xdT xd ]do - CJ, eo tt xd Tdo do x!i T" d Jdo 'do xd ldo x, T d c v" T" T' xd TdoJdo ^d 9 D (3.10) and iq and VQR from eo •Weo(xq-xq) Lq xq Iqo it " xq Tqo Q (3.11) The derivation of (3.5) through (3.11) is presented in Appen dix 3A. For the study of the machine-infinite bus system as shown in Fig. 3.1, the d- and q-axis machine terminal voltage can be expressed in terms of the infinite bus voltage and the torque angle between synchronous machine q-axis and infinite bus voltage and the machine d- and q-axis current as follows. k-^r+k^x -(knx-k2r) vd q k^x-k r k^r+k0x + kl k2 •k2 kx vQ sinS v cos S o (3.13) where k± = (1-xB+rG)/((1-xB+rG)2 + (xG+rB)2) (3.12) k2 = (xG +rB) /((1-xB+rG) 2+ (xG+rB)2) To complete the description of the synchronous machine dynamics, two additional equations are obtained as follows. From the equilibrium equation for torque Pwm = J <Ti ~ DcJm " Te) (3.13) where the three phase electrical torque is Te - 3 P°*es (<Pd iq - *q id) (3.14) From the relation between electrical torque angle and mechanical speed PS = poles u - to (3.15) 2m GO Equations (3.5) through (3.9), (3.13) and (3.15) are the seventh order synchronous machine equations in the state variable form and (3.10), (3.11), (3.12) and (3.14) are the auxiliary equations. 3.2. State Equations of Controllers and DC Motor-Booster The transfer function of the regulator-exciter, Fig. 2.7, in state equation form is " - ~ VR + |*>ref-vt) IRE !RE for the regulator and Pvfd , X. f(vR) _ i Yfd (3.17) for the synchronous machine field voltage. The character istics of the field voltage limiter are approximated by f(vR) = ai tanh.(a2vR) (3.18) where the values of a^ and a2 are determined from a least squares criterion. The terminal voltage is Vt = /vd + Vq <3.19) The transfer function of the governor-hydraulic system, Fig. 2.4* in the state equation are as follows. For the actuator position pa = JT a -._! af - , (3.20) . " TA TA TA for the actuator feedback position paf = -UJL a - STV+ TA af ~ 6_&">m (3.21) TA Tr TA • TA for the gate position oo- = JL a - _i_ S (3.22) PC> TG TG and for the turbine torque output pt - - a + Tg + TW g - , ,1 .„ t (3.23) . 5TG . 5TGTw . 5TW The booster armature voltage state equation from (2.13) is pvB = _ ^g Laf vfB - 1 vB (3.24) RfBTfB TfB where the booster field voltage signal of Fig. 2.2 is computed by vfB = X (t_) _ fe^R ^ + ^£fi pk) K3 \ 21 \ K3 K3/ 2 K3 V2' _TfB f2^- P(^) (3.25) The state equation of the mechanical speed of the test model of (3.13) becomes Nm\ = LaFjF Lk _ *1 (j^m) ^2 _ £e (3.26) P \ 2 / 2J J \ 2 / 2J 2J where the d-c motor torque of (3.13) is computed by T± = LaFiFiA (3.27) the d-c motor and booster armature current by iA = Kdc" vm - VB) / R (3.28) and the motor armature voltage by vm 2 LaF ip p (3.29) The state variable tJ m instead of U) is chosen because the ~r m tachometer output voltage is 94.25 volts at synchronous speed (188.5 rad/sec). The damping coefficient determined experimentally is D = KJL + K2 W~ (3.30) m Finally, the electrical torque angle state equation from (3.15) becomes pS = 4 _ WEO (3.31) 3.3. Initial State of a Power System The initial states of a power system, vd, Vq, i^, iq, vQ and S, are determined from the operating conditions, i.e., the real power P, the reactive power Q and the voltage at the machine terminal from the following nonlinear algebraic equations. P = vd + vq ^ (3.32) 2 ^ vq id - vd iq (3.33vt = Vd + vq (3.34) vd = "Vd + xq iq(3'35> vd = k-^v^sinS + rid-xi ) + k2(v0cosS + xid+ri ) (3.36) v q - k1(vQcosS + xid+ri )-k2 (vQsinS + rid ~xiq) (3-37) For the particular cases studied in this thesis, these equa-tions are solved by the method of Fletcher and Powell. An initial estimate of the solution of iq, vd, Vq and id is obtained from the closed form solutions3•4 which neglects armature resistance and that of vQ and 8 from the transmission configuration of Fig. 3.1. 1q = • — (3.38) 4 (Pxq)2 + (v| + Q)2 vd = xq iq (3.39) vq = H " Vd (3.40) . 2 q Q + Xn iri , id = v 9 q (3.4D 'b = \/[>d-r(id-vd G+VqB) + x(iq-vdB-vq G)]2 27 + l>q-r(iq-vd B~vqG) " x(id~vdG+vq B) ] (3.42) S = arctan vd-r(id-vd G+vqB)+x(iq-vdB-vqG)  Vq-r(iq-Vd B-v 6)-x(id-vdG+vqB) (3.43) The initial value of the field voltage, required for the integration of (3.17), is determined from (3.1) and (3.3), (vq + Xd Ad + Vaiq) Ri x ad (3.44) where W = & e eo, and Rf ^ad R f \/ra + X /(ra + xd x ) 'slope' (3.45) The 'slope' is determined by the slope of the steady state short circuit characteristic of the synchronous machine, relating RMS armature phase current and d-c field current. The derivation of the expression for is presented in Appendix 3B. xad The initial values of ^p, tpd and are determined from (3A. Tdo _Tdo 1 - xd U)eo 0 0 0 0 -x. eo (3.46) The following conditions are applied DI T 0 Tdo = ° V 0 T Td TD 0 VDR = 0 VQR ~ 0 (3.47) Since VpR = Vp during steady state, the initial regulator voltage is VR arctanh a2 \ ax (3,48) and the reference voltage is established from Vref = (VR + KA V / \ (3'49) The initial values of the governor-hydraulic prime mover are a = 0 (3.50) af = 0 (3.5D g - 0 (3.52) t-0 (3.53The initial value of the booster voltage of (3.24) is vB = 0 (3.54) since v^g = 0, (3.25). The initial real power of the dynamic test model is set by the load setting resistor in the d-c motor-booster armature circuit. The value of resist ance is established from (3.28), (3.20) and (3.54) R = Kdc " um LaF if) / lA (3.55) where iA is obtained from the steady state torque equation and is equal to LA = (Kl + K2 + Vd ^ ~ «Pq id) / LaF % (3'56) The first two terms on the right hand side of (3.56) are torque terms due to friction and the last two terms are the torque of the synchronous machine. The mechanical synchron ous speed of the test model is tOm - 188.5 rad/sec (3.57) 39 3.4. State Equations of Fifth Order-Synchronous Machine  and Controllers The fifth order-synchronous machine state equations are obtained by neglecting the damper winding effects of the seventh order model. By removing (3.8) and (3.9) and setting DI 0 , T do 0 , T" = 0 , T% 0 , T" = 0 qo (3.58) A = 1, B = 0, (3.59) and T = T, Xd Tdo TD = 0 (3.60) in (3A.2), Park's equations reduce to P^d = vd + (Jeij> + rai p<uq = Vq - weyd + raiq P^F = VF ~ VFR (3.61) (3.62) (3.63) where (3.10) is replaced by FR ^d xd "^eo (xd~xd) xd ido xd Jdo eo xd V, Yd (3.64) and (3.11) becomes eo ^q (3.65) Further elimination of vd, Vq, id, iq and Vp-^ in (3.6l) through (3.63) using (3.12), (3.64) and (3.65) results in the following state equations. PVF = - xd VF + ^eoKl-Xd) Yd xd Tdo xd (3.66) + (ra+XdXq)tsl°pe' Vfd P^cl = klr + k2x + ra ^F " klr + k2x + ra ^eo^d xd ldo xd + klX " k2r Weo«Pq + 4 * ° ^ 4>q + klvo sin8 + k2Vo cpsS x 2 q (3.67) P<fq = klX " k2r VF " klX " k2r Weo " 4-° ^m ^d xd Tdo xd 2 ~ klr + k2x + ra ^eo^q " klv0 sin8 + klvo cosS Xq (3.68) The state equations (3.26) and (3.31) for machine dynamics, (3.16) and (3.17) for regulator-exciter, (3.20) through (3.23) for governor-hydraulic system, (3.24) for booster armature volt age, and hence the auxiliary equations of them remain unchanged except the electrical torque T of (3.26) and (3.14) now equal to Te - ecoeo U - JL\ - ~-V WF (3.69) \xd xq/ xd Tdo the d- and q- axis voltages of (3.19) from 3.61), (3.62), (3.64) and (3.65) now equal to vd = P^d ~ we4>q " • irai~ VP + £iL_^eo Vd (3.70) xd ido xd vq = PVq + weVd + ra »eo Vq (3.71) xq These equations consist of the complete set of state and auxiliary equations for the power system with a fifth order-synchronous machine and controllers. The initial conditions for the state equations of the fifth ordei—synchronous and controllers are the same as pre sented in Section 3.3. The variables id* iq, vd, vq, v^. and ipd are calculated from the solution of the state variables at each integration step. 1 VF - 00 eo yd (3.72) a. d xd Tdo 1 q ~ ^ Vq (3.73) vd ~ (kir+k2x) id ~ (k^x-k2r)i.q + ^lvo s*-n ^ + ko v cos S * o (3.74) vq = (k1x-k2r)id + (k1r+k2x)iq + k, vQ cos 8 - k2v0 sinS (3.75) (3.19) 4--fd r 2 4- x 2 1 a xq xd VF (xd"Xd) ^eoVd^ (r2+xdxq) 'slope' VTdoxd t xd (3.76) 3-5. State Equations of Third Order-Synchronous Machine and Controllers The third order synchronous machine state equations are obtained from the fifth order model with two more assump tions. First, the speed voltage effects due to speed variation in (3.61) and (3.62) are negligible; w e<pd ~ weo(|;d , wecpq ~ u>eoH>q (3.77) Second, the induced voltage effects due to the change of flux linkages are much smaller than the speed voltages; p<rd < < "eoYq ., P^q << Vd (3.78) As a result, the two state equations (3.61) and (3.62) reduce to algebraic equations and may be written 1 0 0 1 0 OJ •U) 0 e ^d + r 0 a 0 r. (3.79) where vd and v can be eliminated using (3.12) and id and i . q using (3.64) and (3.65). Finally, ty^ and <f are expressed as 4.3 Vq 0) eo o + x_(xn + xn) ox o J ,1 xq xd ^do •r0 (x0+xd) +rQx0 1 2 mi x,, T do V, Vt rokl-k2(xo+Xq) r k +k (x +x ) o 2 1 o ql rQk24k1(xQ+xd) rokl-K2(xo+xq) t xd x. vQ sin 8 vQ cos S (3.80) where A=C02O r2 + (xQ+xq) (xQ+x-d) xd x and ^o = klr + k2x + ra cQ = klX - k2r (3.81) After eliminating V^, (3.63) reduces to 1 / xd - ( Xd-Xd) ( rg+Xo( x0+xq ))\ifF + ( r!+xdxq) ' slope' xdTdo^ o +(xo+Xq)(xo+xd) \/ra+xq R. fd + (xd~xd)(rokl-K2(xo+xq^ vo sinS + ( xd-xd ) ( rQk 2+l<1 ( x0+xq ) ) vQ cos 8 r^+Uo+x )(xD+xd) ^o+(xo+xq)(xo+xd) (3.82) The other two synchronous machine dynamic state equations (3.26) and (3.31) and the auxiliary equation (3.69) remain unchanged. 44 The exciter state equations (3.16) and (3.17) and the terminal equation (3.19) remain the same. However, with the conditions (3.77) and (3.78), (3.70) reduces to "d " -"e<A " _J^_ f" *d (3_83) xdTdo xd and (3.71) to vq = "eo^q 4 ra ^eo V q Xq (3.84) The state equations describing the governor-hydraulic prime mover (3.20) to (3.23) and the d-c motor-booster state equation (3.24) remain unchanged. In summary (3.82), (3. 26), (3.31), (3.16), (3.17), (3.20) to (3.23) and (3.24) form the state equations of the third order-synchronous machine with controllers, and the auxiliary equations are (3.28), (3.29), (3.69), (3.85), (3.86), (3.18), (3.19), (3.83), (3.84) and (3.25). The method of evaluating the initial conditions is described in Section 3.3. The vari ables .^d, <p , , 1^, v^, Vq, v^. and i^ are calculated from the solution of state variables at each step as follows 2 ' m = . rp + x0 (x0+xq) (j;p + xd( rpkj-k 2 (x0+xq+xq) ) s±n S weoTdo(r^ + Cxo+xq)C^+xd) weo + Xd (rpkz-t-^Cxo+Xq)) cos S (3.85) 4 5 = "q^o^^o^o^^ VF + xq(rok2+k2(xo+xd>> sin 5 weo xd Tdo "eo + xq(r0k1-k2(x0+xd)) CQsS (3.86) ^eo iH - - ^eo <Pd (3.87) *d?do xd iq = - ^eo <pq (3.88) . xq vd = " Weo ^q " raid (3.89) vq = "eo <Pd " raiq (3.90vt - 7Vd + Vq (3'91) ±fci= \/ra + xq / xd ~ (xd-xd) "eoVd^j (3.92) (r|+xdxq)slope \xdTdo xd / 3.6. State Equations of Third Order-Synchronous Machine  and Regulator-Exciter but without Governor For the study of electrical transients of the system, the governor-hydraulic prime mover dynamics are replaced with a constant torque input. The equations (3.8 2) and (3.31) de scribing the synchronous machine and (3.16) and (3.17) describ ing the regulator-exciter remain unchanged. The state equation for mechanical speed (3.26) becomes 46 i 2 ./ 2Jwmb J 2 J 2 (3.93) J \xd xq/ ^do The auxiliary equations are (3.85), (3.86), (3.18), (3.19), (3.83) and (3.84). The initial conditions are evaluated by the techniques described in Section 3.3 and the variables Yd, Yq, id} lq> vq> vt and ±fd (3.85) through (3.92). 3.7. Computation and Test Results of the Dynamic Test Model In this Section, computation results of various state variable models of Section 3.4, 3.5 and 3.6 and results from actual tests are summarized and a comparison is made in order to verify the mathematical models. A transient test is carried out on the power system described in Chapter 2 simulated on the dynamic test model. The system has a three-phase fault before the fault line of a double circuit transmission system is isolated at 5 cycles. The fault is cleared and the system restored after 30 cycles. The test responses of the torque angle 8, governor actuator position a, governor actuator feedback af, gate tr, turbine torque t, booster armature voltage vg, terminal voltage vt, regulator voltage v^ and field current iP{j are recorded on a Visicorder. The results are plotted along with the computa tion results of the three different state variable models. The curves in Fig. 3.2 through 3.11 are identified as follows: 1. third order machine with exciter, 2. third order machine v/ith exciter and governor, 3. fifth order machine with exciter and governor, 4. dynamic model test results. Hamming's numerical integration method3"^ is used for computation with an integration step size of 0.000 25 seconds for the fifth order machine which includes pf^ and pVq and 0.005 seconds for the other two models. The computation results of all three models are very close except the terminal voltage response of the fifth order machine model, curve 3, Fig. 3.8. The fifth order model predicts the voltage spikes due to line switching at fault cleaz^ed and system restored which is substantiated by actual test results. It is also observed that both the third and fifth order models with gov ernor, curves 2 and 3, Fig. 3.2, are slightly more unstable than the third order ivdthout governor, curve 1, Fig. 3.2. This phenomena is observed also from direct model tests. A close correlation between computational and test model results is observed except for the regulator voltage, Fig. 3.9, and the field voltage, Fig. 3.10, at the instant the system is restored. This may be attributed to the imper fect mathematical model. For example, it does not include transmission line switching, since it is described by steady state equations. However, the prevailing frequency of oscil lation is the same. Fig. 3.3. Governor Actuator Position Transient Responses o 1.5n TIME (SEC) Fig. 3.5. Governor Gate Position Transient Responses 1—1 Fig. 3.6. Turbine Torque Transient Responses 3n TIME (SEC) Fig. 3.7. Booster Armature Voltage Transient Responses OJ 100 -i cc 30 S 20 H 10 0 fau/f cleared \system restored 0 I—I—I—I—I—I—I—I—1—I—I—I I I I 0^5 l!o 1.5 2.0 TIME (SEC) i—i—i—j—i—i—i—i—]—i—i—i—i—i—i—i ' ' i 2.5 3.0 3.5 4.0 Fig. 3.8. Terminal Voltage Transient Responses Fig. 3.9. Regulator Voltage Transient Responses Fig. 3.11. Field Current Transient Responses 4. PARAMETER SENSITIVITY OF THE TEST MODEL Sensitivity analysis^" J ^.2 is applied in this chap ter to investigate the effect of model parameters on system response. As shown in Section 2.3.3, only important model parameters, not all of them, can be matched simultaneously with actual system values (Table 2.1). The model base imped ance chosen is a compromise. Sensitivity analysis is applied to see whether this approach is justified. The controllers are included in the investigation. The state equations for the synchronous machine and controllers are given in Section 3.4. 4.1. Sensitivity Equation The state variable equation is written as xi ~' %(x>qr) > qr = q / q0 (4.i) where qr is the relative value of the parameter, q the true value and qQ a constant equal to initial value of the para meter. The sensitivity equation for small parameter pertur bations becomes n S *fi. x, _ + qQ b£i xi>a„ = f£i . xL, + Q iii (4.2) K J- ox, d q where x,^ = cbq, (4>3) dqr 59 The right hand side of the sensitivity equation (4.2) consists of two terms; the first term includes the state variable sensitivity coefficients only and the second term depends upon parameters explicitly. The two terms will be identified as fl. (x,x q ) and f2. (x) respectively, where n fli a <x>xi, „ ) = S Mi x. n (4.4) i>^r k,qr k=l T k,qr and f2i>qr(x) = qQ Mi U>5) q When the results are applied to system equations of Section 3.4, the terms independent of parameters are fH Xd v + Weo (xd~xd) x. „ flvF,qr = " r^y ^r^r Xltfd>qr (4.6) + (ra 4 xHxa) 'slope' + 2 , SL_3 vfd^r 7 r2 + x2, R a q r f 1 , ro ro "eox + xo weox _ Vd^r x^ X^F^r ^ Vd^r + 7Z + 4 em x + 4(j)q x^^qr + k1vocos8 sinU^.) 2 Q ^ I** n q> *r "2" k0v sinScos (x ) 2 ° S'q- (4.7) 60 fl _ x o d do xo ueo x xd 4 wm x - 4 ip, x - ro ueo x 2 rd^r d «m,<Ir x Vqr 2 q k-,v cosSsin (x ) - k,v sinScos (x ) x ° 6>qn 1 ° s>q„ (4.8) fl 2 (L aF F JR Ha>qr 2 - LaFxF xv 2JR - i^eo /^ _ jW«pd x + V x J ^5 xqj\ ^q>qr q Yd,qr/ Jxd^d^ I F rVqr q ^F^rj (4.9) = 4 x 6J _21> 1 x K A TRE R' r TREvt _1 _ (vd(r0-ra) +v xQ)x xdTdo 4 vF,qr (0. f (vd(r0-ra) + vqXQ) + ^ Cvdx0-vq(r0-ra)) rVqr + Cvd^ ki v0cos^~'s'2Vo sin^) + vq( ~'c2VoCOS ^ ~ k] VoS"'"n ^ ^ 'S,q, (4.11) 61 2, pi = _ _! x + alA2 sech (a,vR) x /, -, ,-> \ fSd,qr TE ^fd^r "TT 2 VR'qr (4"12) fla,q - ~ Z. *a,q " JL xa - _2_ x (4.13) TA Mr TA fqr TA -f'qr fl = -<rj x - &Tr -I- TA x - 2_ x (4.14) af>qr TA a'qr TrTA afqr TA ^^r flff n = xa a " _1_ x q (4.15) g'qr TG 3,qr TQ g,qr t, q = ~ . 1 xa q + TG + TW x - 1 , x. (4.16) • 5TG .5TGTW S,qr >5Tw t,qr flv a = - 2* <*>gLaf 1 x + (2*«R _ 2K4_\ OgLaf x B>qr K3 RffiTfB 2 'qr V K3 K3 / RffiTfB Sn,qr 2«TfB "gLaf 1 fl, a + Tfpf2o(ctR - 2K4\<JgLaf fl K3 RfBTfB 2 ' V \ K3 K3/RfBTfB %i,4r 2 1 X TfB VB>qr (4.17) The parameters involved in the sensitivity investiga-' t txon are ra, xd, Xq, xd, Tdo and J of the synchronous machine, KA, T^JJ- and T^ of the regulator-exciter and S, T, T^, TR, TG and T^ of the governor-hydraulic operator. For the armature resistance r . (4.5) yields f2... = raT slope' (2 x/r^+x2, - V^+XdX2)\v (4.18) T^T V a q ~l^r~~ td 62 f2 = ra w - ra(l>eo o> (4.19) UJ . r- —-r;:rr— T P T > D rVi rn ~"n;rr- ' F dicio xd f2 = - raueo <tf (4.20) iu r — Tq T q,1 a xr for the d-axis synchronous reactance f2 x - - VF • + ^M2 Vd + y^slopet V^Xd x^T^ Rf r2+ x2 for the q-axis synchronous reactance x 'slope' / / 2. 2 x (r +X|x„)\ /. f2„ := - Xoueo ID ' (4.23) Vxq ~x^~ ^q f2 = roweo <p (4.24) u; x T'q Tq, q Xq f2 = AXq Cv.f? + v_(f2 ~ a' eo y _)) (4.25) f2 = - 3weo vd ^q (4.26) (j x ~7 Ti) q Jxq 2 f2v x = Vffi/^ ~ --4 !^£al.V2» x (4.27) VB,Xq q 1 K3 K3 ,.RfBTfJ %,xq 7 2 63 for the d-axis transient reactance f2v vt = Xd; VF - xdfto HJd (4.28) F'xd xdTdo. xd f, « _ ro + roweo ip , (4.29) t2y, v» —i—— F —i— d Yd,xd xdTdo xd f2 , = - xo Vr- + V'eo VH (4.30) f2 , = - xo Vp + xoweo VH " "VXd Wd~o xd~ raS ^ ' " ^,xd x^Tdo * xj: f2„ k. - - ^d_ (vd(f2^ + *F - -Wd q" VV (4.31) f2 , = 3Ueo Vd Vq -__J? "VFVq (4.32) *2* f2y xt = xdTfB - Hi ]«3iaf " (4.33) VB>xd \ Ko K3 ;RfBTfB -f>*d for the d-axis open-circuit transient time constant F2VF'Tio = ^ (4.34) *d> do xd ldQ (4.36) f 2 y ry, f R» do KATdo (v,(f2 , + V d (1), 1 ' • rp • Ya> do xdldo " TREvt 5T" V + W ao (4.37) f2u f ""TV-(4,38m rP » Jx ,X , 1 7'Mo d do T. = TdoTfBf^ " £^L\^AF_ ^2 f (4.39) 13,1 do y K3 K3/RfBTfB — ° f2v m. and for the moment of inertia J ±A = I (vt - 2LapiF f£m - vB) (4.40) f2 = fl n - LaFiF iA + S + 3CJeo /l - lV <p ^m^ J 2~ 2J 2J J lxd xq] q 2 __J__S_ VF cu (4.41) JxdTdo q 'm = - 94.25 (4.42) 2 f2 , =-__J_ "gLaf t + £V Vaf ^m VB,J K3Tm RffiTfB K3Tm RfBTfB 2 +JTfBff^.^\^f. f2fi) j (4.43) V K3 K3 / fB fB -f > Next for the gain of the regulator-exciter, (4.5) yields f2 " Kef ~ V <4-44) R A LRE 6 5 for the regulator time constant T^ f\ TPE = ^ " (V"f"Vt' U-45) ' KL XRE IRE and for the exciter time constant T^ f2v T = JL vf, - fl tanh (a2vR) (4.46) vfd> E TE 1Q To Finally, for the transient droop S of the governor-hydraulic operator, (4.5) yields ••= <im » 94.25 (4.47 2 2 f 2 ' = -^g_ a - ap - 2% ^m (4.48) f' TA TA TA 2 for the permanent droop f2ajVr = - «L a (4.49) TA =-^ia (4.50) af»v TA for the actuator servomotor time constant TA f2 B 1 a + 1_ af + 2_„ *_Sn (4-51) A TA TA TA 2 f2 T =^S a + 1_ ap + 2S (4.52) af> A TA TA TA 2 66 for the dashpot damping time constant T R f2- _ = _1 ap (4.53) for the gate servomotor time constant Tg f2 m - - _1_ (a-g) (4.54) G TG f2. r„ = _2_ (a-g) (4.55) ' G TG and for the water starting time constant Tw f2t T = - _2_ (g-t) (4.56) ^ W TW The equations, (4.6) through (4.56), are used for the computation. 4.2. Parameter Sensitivity of System Response Parameter sensitivity of system response is investi gated in this section. There are fifteen parameters and twelve state variables involved. From each sensitivity curve the max imum and minimum are found. A convenient criteria is set that any parameter whose sensitivity curve maximum and minimum for a state variable are less than one-seventh of those of all the sensitivity curves will be considered insensitive. The results can be summarized in three categories. One, parameters are very sensitive in the beginning and remain sensi-67 tive for the rest of the time period. Two, parameters are very-sensitive in the beginning but not towards the end. Three, parameters are insensitive in the beginning but become sensi tive towards the end. In the first category, it is noted that the sensitivity curvesH'p^ (Fig. 4.1), Vd Y (Fig. 4.2), vRjX (Fig. 4.10), q '"q q and v.pd x (Fig. 4.11), all with respect to x„, have their ' q maximum during the fault and fault cleared period and remain sensitive after the system has been restored. A similar result is observed for the parameter Tdo except for typ <j>» (Fig. 4.1), 5 do which increases during the first 0.2 seconds and remains large throughout the remaining period. In the second category, it is observed that Vd x' ' d (Fig. 4.2), ipn xi (Fig. 4.3) and vR K (Fig. 4.10) are large H, D K,IVA only during the fault period ("Co.08 seconds). In the third category, it is noticed that the moment of inertia J is insensitive during the fault and fault cleared period but becomes very sensitive towards the end. t It is found, in general, from this study that xq, Tdo and J are the most sensitive parameters and that to a certain extent xd, K^, and TRE but not r&, xrf, T^ and the governor-hydraulic operator* parameters. RIR GRP FLUX IWEBERS) 0 50 100 150 200 250 300 350 I i • i ' 1 I I I 1 I 1 I I I I 1 I—I 1—I—I !—I—I—I 1—!—1 1—I—I 1—I—J—I PRRRMETER SENSITIVITY -400 -200 0 200 400 89 D-RXIS FLUX (WEBERS) -0.00 0.05 0.10 0.15 0.20 i i i • i i i i i i—i—i—i—i—i—i—i—i—i—i—I PRRRMETER SENSITIVITY -0.2 -0.1 -0.0 0.1 0.2 69 Q-RXIS FLUX (V/EBERS) -0.10 -0.08 -0.06 -0.04 -0.02 -0.00 0.02 BOOSTER RRMRTURE VOLTRGE (VOLTS) -3-2-10 1 2 3 I i i i i 1 i i i i I i i i i I i i i i !—i—i—i—i—I—i—i—i—i—I PRRRMETER SENSITIVITY -30 -20 -10 0 10 20 30 MECH SPEED / 2 (RRD/SEC) 0 20 40 60 80 100 11111 • 11111 • • i • i • • • 11 • 111111111111111111111 PRRRMETER SENSITIVITY -10 -5 0 5 10 15 91 REGULATOR VOLTRGE (VOLTS) -20 0 - 20 40 60 i i • i i i i i • • i  i i i i i i i i 1.1.. i i i 1.1 i i i * i i i i 1.1 PRRRMETER SENSITIVITY w Lt FIELD VOLTRGE tVOLTS) -15 -10 -5 0 5 10 15 DELTR RNGLE IRRD) 61 5. NONLINEAR OPTIMAL STABILIZATION OF A POWER SYSTEM AND DYNAMIC MODEL TESTS 5.1. Power Sysbem Stabilizing Signal The main purposes of a stabilizing signal are to provide additional synchronizing torque during the first torque angle swing after a transient disturbance and to pro vide damping torque for subsequent oscillations. Ellis and others^ • 1 > 5 . 2, 5. 3 were able to obtain effective stabilization using speed, accelerating power and frequency deviation sig nals. The theoretical basis is found from a linear model by deMello and Concordia."'*4 These are the conventional stabil ization techniques. Jones^'^ did a bang-bang control test on a model power system. In this Chapter, optimal control theory is applied to a power system described by nonlinear state equations including field voltage and stabilizing signal limits. The power system is shown in Fig. 3.1. 5.2. Dynamic Optimization and Computational Method The problem being considered is to find an optimum control uJ,;'(t) which minimizes the performance functional J - 0(x(tf)) +j F(x) dt (5.D t o subject to the dynamical constraints or system equations x = f(x,u) (5.2) The variational calculus method of Lagrange Multipliers is used to form an augmented functional 81 5.6 Ja =#(x(tf)) + J (-H(x,A, u) + A1 x)dt (5.3) co where H is the Hamiltonian H (x,A , u) - -F(x) + AT f(x,u) (5.4) and A. is a time-varying costate variable vector. The condi-5.7 tions which must be satisfied at the optimum are x = f(x, u) x(tQ) = Xq state equations (5.5) A= -Hx(x,A , u) costate equations (5.6) 0 = H (x, A , u) gradient condition (5.7) A(t^) = (x(t^)) transversality condition (5.8) Equations (5.5) through (5.8) represent a nonlinear two-point boundary-value problem (TPBVP). For the solution of the TPBVP, the gradient^*^ and r 7 Newton-Raphson with Ricatti TransformationJ'' methods are tested first on a simplified third-order power system. There is no difficulty with the computation. The method of Newton-Raphson with Ricatti transformation is then applied to the fifth-order system using the gradient method to start. Num erical instability in the solution of the Ricatti equation is experienced. The solution of the TPBVP is finally obtained with the gradient method. It takes fifty iterations or less. Three different performance functions are chosen and tested for each step of the line fault, fault cleared (one circuit of a double circuit removed), and line restored (successful reclosure of circuit). The general form of the performance function from (5.1) is 0(x(tf)) = Wi^J + W2 <aS)2 (5'9} F(x) = w3 {&QJmj2 + w4 (&S)2 + w5^&wmj2 (5.10) The first two steps are of fixed time established by the circuit breaker settings. In these two steps, (f) (x(t^) ) is set to zero. Studies are made of alternatively consider ing the weighting factors , or alone. The same pro cedures are repeated for the final step but including W-^ and ^2' ^° emPnasize small deviations of speed and torque angle at the final time and V/^ are set equal to ten, W-^= ^ * The final time of the system restored step is estimated from a stabilizing signal study (~1.7 seconds) anticipating a shorter settling time (1.25 seconds). The Hamiltonian is H = -F(x) + A-^pYp) + A2(p ^Dl) + \3 (pS) 2 + (pvR + uj + A5 (pvfd) (5.11) * TRF/ ' 83 where pSJp, PVR are S'iven by (3.82) and (3.16) respectivel}' The Hamiltonian is maximized or the performance functional is minimized with respect to u if the condition u = umax sSn *4 is satisfied. This is a "bang-bang" type of optimal control. The solution of u is obtained indirectly by optimizing the 1 r g unconstrained value u related by Box's transformation as follows u - Umax sin (u) (5.12) In the gradient method, the correction applied to u at each point is Au - -k Hu (5.13) It is rather difficult to choose k"*'^. In this computation the following value of k~*'^ is chosen k = Sl (5.14) HT H dt ii u where Sl is a step size constraint. 5.3. Computation and Test Results Computation results from an IBM 36O Model 67 and test results from the dynamic test model are summarized in this Section as follows. 8 4 For the computation, investigation of the various performance functions outlined in Section 5.2 reveals that for the fault step, the three performance functions yield the same optimal control and for the fault cleared and system restored steps, the optimal control is the same for F = t&^nxj and F = (AS)2 but different for F = Thus, a single trajectory for the system variables is obtained during the fault step, two trajectories during the fault cleared step, and two trajectories from each of the previous two during the system restored step. These results are pre sented in Fig. 5.1 with the performance functions summarized in Table 5.1. Curve Fig.5.1. Fault Fault Cleared Svstem Restored 1 10^«mj 2 + 10 (AS )2 + J^wmj2 dt 2 io^waj2 + io (&S)2 + jpm)2dt 3 10^umj2 + 10 (AS)2 + j^^j2 JL 4 10^«mj2 + 10 (AS)2 + j^j 2 dt Table 5.1. Performance Functions Used During Transient Steps 85 Comparisons are then made between the system responses with optimal control signals and those without. It is observed that the system responses with optimal control signals are much more damped than those without. Once the. control is removed after 1.25 seconds the system oscillates with a reduced mag nitude. It is observed also that all the system responses with optimal control signals stay very close. The speed de viation, torque angle and field voltage responses are shown in Fig. 5.1a, b, and c respectively. Comparison is then made between conventional speed deviation and accelerating power stabilizing signal responses and one of the optimal controls (curve 2> Fig. 5.1) responses. In general, it seems that the optimal control yields a better damping effect, speed deviation in Fig. 5.2a and torque angle Fig. 5.2b, than the stabilizing signals except for the field voltage in Fig. 5.2c because of the nature of the forced excitation. Power stabilization test results obtained directly from the dynamic test model are plotted along with computation results in Fig. 5.3a, 5.3b, and 5.3c for comparison. All system responses are close. Similar comparisons of computation and test results are made in Fig. 5.4a, b and c for system responses with a speed stabilizing signal. In both cases steady state oscil lations are observed as experienced in practice. However, because of the very nature of the steady state oscillations 86 it is difficult to realize the same initial condition for computation on the test model. Comparison is then made between test and computation of system responses with optimal control. They agree with each other very well, Fig. 5.5a and 5.5b, except the switching transient disturbance in the field voltage Fig. 5.5c which is observed also in Fig. 3.8. The last comparisons of test and computation results are carried out on system responses with optimal control for the first 1.25 seconds and power stabilizing signal for the remaining period, Fig. 5.6a, b, and c. Note that the compari son of the first 1.25 seconds is made in Fig. 5.5a, b and c. It is observed that the overall responses of the test and computation are very close. In Fig. 5.7, system responses with optimal control and power stabilizing signal are plotted along with system responses with power stabilizing signal alone. It is interest ing to note that the composite signal yields the best overall system response and this is realized to a lesser degree on the test model. From the comparisons made above it is concluded the dynamic test model can be used to perform complicated power-system tests, such as stabilizing signal control, and used to check computational predictions. MECHRNICRL SPEED DEVIRTION (RRD/SEC) ^8 DELTA ANGLE (RAD) -0.25 0 0.25 0.50 0.75 1.00 1.25 1.50 38 69 MECHRNICRL SPEED DEVIATION (RRD/SEC) i i no * o »—* rv) Fig. 5.2 (b) Torque Angle Calculated Transient Responses of Various Stabilizing Signals CO I— _J Q LU CE > a _i LU speed deviation Fig. -j 1 1 1 1 ^ 1 1 1 1 1 1 1 1 1 j 1 r 0.25 0.50 0.75 1.00 TIME (SEC) (c) Field Voltage 5.2 Calculated Transient Responses of Various Stabilizing Signals 1.25 to 2n TIME (SEC) (a) Mechanical Speed Deviation Fig. 5.3 Transient Responses with Accelerating Power Stabilization 1.50 -i calculated .test model sr. i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i i | 1 <~ 1.5 2.0 2.5 3.0 3.5 TIME (SEC) i—]—i—i—i—i—|—r 0.5 1.0 (c) Field Voltage g. 5.3 Transient Responses with Accelerating Power Stabilization MECHRNICRL SPEED DEVIATION 1RRD/SEC) 1.50 -i TIME (SEC) (b) Torque Angle Fig. 5.4 Transient Responses with Speed Deviation Stabilization ^ 15-i CO f— _J LU CD cn > o I LU 4.0 (c) Field Voltage Fig. 5.4 Transient Responses with Speed Deviation Stabilization o co 2 CJ LU CO \ CD cr cn 1 rz: o CE UJ Q Q LU LU CL IO 0 ^ 1 CJ-1 CE CJ LU -2 no stabilization fault cleared system restore —i -T 1 1 j 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 r 0 0.25 0.50 0.75 1 .00 TIME (SEC) (a) Mechanical Speed Deviation Fig. 5-5 Transient Responses with Optimal Control Stabilization T 1 r 1.25 vO o cr LU I LD rz. CE cr 1.50 -i 1.25H 1 .00 H 0.75 H 0.50 H 0.25-•0.25 J^.calcula ted 1.25 TIME (SEC) (b) Torque Angle Fig. 5.5 Transient Responses with Optimal Control Stabilization o o TIME (SEC) •(c) Field Voltage Fig. 5.5 Transient Responses with Optimal Control Stabilization g MECHRMICRL SPEED DEVIATION (RRD/5EC) i i f\j >-* O f\j ZOT no stabilization TIMF (SEC) (c) Field Voltage Fig. 5.6. Transient Responses with Composite Stabilization 2n TIME (SEC) (a) Mechanical Speed Deviation Fig. 5.. 7 Transient Responses with Accelerating Power Stabilization M and Composite Stabilization p o cr or LU I CD CC cr i LU Q 1.50 1.25-1.00-0.75 0.50 H 0.25 -0.25 0 0.5 1.0 1.5 2.0. 2.5 TIME (SEC) 3.0 3.5 (b) Torque Angle Fig. 5.7 Transient Responses with Accelerating Power Stabilization r'* ' and Composite Stabilization o ON 1 07 • CONCLUSION A new model.for power system transient stability and con trol tests has been developed capable of -representing any conventional one machine infinite bus system in detail. The model system has a synchronous machine with an adjustable field time constant, a solid state regulator-exciter system and an adapted dc motor prime mover with typical speed governor characteristics. The model can be used to investigate power system dynamics over a wide range of operating conditions. A general fifth order state variable model for a synchronous machine has been derived, from Park's equations. It has been shown that parameters in this state variable model can be determined directly, from experimental machine- tests. This is an improvement over existing models which are based on parameters not directly measurable. Three variations on this general model have been shown to be useful in predicting the dynamic behaviour of a synchronous machine and interconnected systems. From the comparison of computation and model test results, it has been found that the first order synchronous machine (plPcO and the second order dynamics (pS,pCJj are sufficient for most studies except subtransient and switching phenomenon which has not been included in this study. A parameter sensitivity study has been-carried out. It has been found that x , T, and J are the most sensitive narameters q' do but not r , x-, , T^, and the governor hydraulic operator parameters. a Q ii The usefulness of the model has been demonstrated with the study of the stabilizing signal. A "bang-bang" type nonlinear optimal control signal for a fault-fault cleared-system restored power system has been treated as a two-point-boundary-value problem in this thesis. A gradient method has been applied to obtain the switching times of the control. 'The control signal thus obtained has been implemented on the test model. The model test results have been shown to agree favourably with those obtained from computation. Thus, it has proven that the test model provides a convenient means to check the design. The test model has also been used to test conventional speed and accelerating power'stabilising signals. The experimental results have compared favourably with computed values. It has been demonstrated experimentally that the nonlinear optimal control provides better system damping than conventional signals. The principal contri bution of the thesis is the development of a compatible test model and higher order state variable synchronous machine models for power system dynamic studies. For future studies it is felt that 1) the mathematical model could be improved, if necessary, to include subtransient and switching phenomenon, 2) comparison of alternative schemes of prime mover simulation should be made, 3) the prototype test model should be developed to include a universal transmission system since the present one is of special purpose, 4) and the model developed could be multiplied for multi-ma chine s t ud i e s. 109 APPENDIX 3A Park's Equations in the State Variable Form Equation (3.3) is expanded to obtain d R ^eo(l+TdoP) Weo (1+T^p)^ 1+TcP + TDp \ Vweo (l+Tdop) "eo (l+Tdo^> / Xdxd where vp R xad vfd t RF t Tdo - T *D1 Tdo - T" xdo it Tdo " TD1 T^o " Tdo Tdo (Td + Tdo) t II Td Td t " Tdo ~ Tdo _ it , II t it t it TD - ~Tdo (Td + Td - Tdo) + Td TH Tdo ~ Tdo Terms in (3A.1) are combined to obtain Yd = J_(]llF (1+Tcp) xdid + xdid 1 + Tdo p - xd!d eo + _L_ |SvF - Tnpxdid eo 1 + Td> (3A.1) (3A.2) 110 which can be written Yd = -xdj-d + VFR + ^DR (3A.3) Weo Ueo ^eo where vpR Rvp +(Tdo - Tc) xd pi P1d (3A.4) 1 + Tdo P V°R = SvF - TD pxdid (3A.5) 1 + Tdo P From (3A.4) one has VF = -i- (VFR + P (Tdo vFR - xd (Tdo-Tc)id)) (3A.6) and when compared with Rvp = vFR + pYp (3A.7) the results yield ^F - Tdo VFR ~ xd <Tdo - Tc) ^ <3A.8) From (-3A.5) one has 0 = vDR-SvF + p (Tdo vDR + TDxdid) (3A.9) and when compared with 0 - vDR + P^D (3A.10) the results yield = Tdo vDR + TDxdid (3A.11) and v —v — Sv DR DR F Ill Equations (3A.3), (3A.8) and (3A.ll) can be written in matrix form ^d T do 1 -xd(Tdo - Tc) 0 -x d to eo 0 eo eo xd^D T do VFR (3.A.12) .» id and VDR are VFR xd(Tdo + TD) -xd(Tdo-Tc)Weo xd(Tdo-Tc) xd Tdo Tdo xd Tdo xd Tdo Tdo id 1 1 ^d xd Tdo *3 "II xd Tdo VDR " *dTD weo xd Tn xdTo. _ x3 Ido Tdo xd Tdo xJl Td'0 Tdo_ (3.10) A similar approach can be used to obtain state equa tions in the q-axis. The flux linkages due to the armature winding and the damper winding are obtained by rewriting (3.4) as follows Vq = - (T+TqP) Xgiq + xqiq _ xqi( (3A.13) (l+Tq0p)Weo ueo We eo 112 When (3A.13) is compared with (3A.14) eo eo the result yields P x q(Tqo Tn) if (3A.15) 1+T qo Equation (3A.15) can be further written 0 VQR +P(Tao v0P- -xn(Tno-Tq)in) qo VQR ~xq<- x qo_J-q-'xq^ (3A.16) and when it is compared to 0 - v QR + PVQ (3A.17) the result yields p. := Tqo (VQR - (xq-xq) *q) (3A.18) Equations (3A.15) and (3A.18) can be written in matrix form a q eo eo -(xq xq) Tq'0 T"qo RQR (3A.19) and the solutions of iq and VQR are 113 tt), eo 1 xqiqo -Weo(xq-xq) Q (3.11) Therefore Park's equations can be vs'ritten in the state variable form as follows P^q = vq - + ra±q (3.5) (3.6) Rv, FR (3.7) P^D = ~ VDR (3.8) (3.9) 114 APPENDIX 3B Determination of Effective Voltage Ratio Rf/xad The value of Rf in (3.3) is in fact an effective xad voltage ratio of the synchronous machine and can be determined from a short circuit test and machine parameters. For a three phase steady state short circuit of the synchronous machine Park's equations become "eoY'q - -raid <3B.l) "eoTd = raiq (3B.2^eo^d = fad vfd ~ xdld (3B.3) Rf weoVq = -xqiq (3B.4) Combining (3B.1) and(3B.3) yields 1q ~ £a id (3B.5) Xq Substituting (3B.2) and (3B.5) into (3B.3) gives id = *q xad vfd (3B.6) ra+xdxq Rf Substituting id into (3B.5) yields 115 i = . ra fad vfd (3B.7) ra + xdxq Rf Thus the a-phase short circuit current equals i = /i2 + = /rf + x^ x_, v 2 2 a + xq xad vfd (3B.8) ra + xd xq Rf When (3B.8) is compared to the test results obtained from the steady state short circuit test, one has 'slope' = i / /v^ j (3B.9) Thus RF = Rf yjrl + x2 (3>45)  cad (ra + xd xq) 'slope' 116 REFERENCES R. Robert, "Micromachine and microreseaux: study of the problems of transient stability by the use of models similar electromechanically to existing machines and systems," C.I.G.R.E., vol. Ill, 1950. V. A. Venikov, "Representation of electrical phen omena on physical models as applied to power system, C.I.G.R.E., vol. Ill, 1952. B. Adkins, "Micromachine studies at Imperial College Electrical Times, July I960. J. Roy, "Effects of synchronous machine parameters on dynamic and transient stability," Paper presented to C.E.A. Winnipeg Meeting, March 1967." J. J. Dougherty and V. Caleca, "The EEI ac/dc trans mission model," IEEE Transactions, vol. PAS-87, pp. 504-512, February 1968. "Test procedures for synchronous machines," IEEE publication No. 115, March 1965. Y. N, Yu and G. E. Dawson, "Modeling a four-electric machine system on analog using parameters directly determined from tests," IEEE Transactions, vol. PAS-87, pp. 632-641, March 1968. R. G. Siddall, "A prime mover-governor test model for large power systems," U.B.C. MASc. Thesis, January 1968. L. M. Hovey, "Optimum Adjustment of Governors in Hydro Generating Stations," Engineering Institute of Canada Journal, pp. 64-71, Nove. I960. J. A. Bond, "A solid state voltage regulator and exciter for a large po\vrer system test model," U.B.C. MASc. Thesis, July 1967. 117 Chapter 3 3.1 R. H. Park, "Two reaction theory of synchronous machines, generalized method of analysis," AIEE Transactions, vol. 48, pp. 716-730, July 1929. 3.2 K. Vongsuriya, "The application of Lyapunov func tion to power system stability analysis and control," U.B.C. PhD Thesis, February 1968. 3.3 R. Fletcher and M. J. D. Powell, "A rapidly conver gent descent method for minimization," The Computer Journal, vol. 6, pp. 163-168, 1963. 3.4 Y. N. Yu and K. Vongsuriya, "Steady state stability limits of a regulated synchronous machine connected to an infinite system," IEEE Transactions, vol. PAS-85, PP. 759-767, July 1966. Chapter 4 4.1 4.2 R. Tomovic, Sensitivity analysis of dynamic systems. New York: McGraw-Hill, 1963. A. P. Sage, Optimum Systems Control, Englewood Cliffs, N.J.: Prentice Hall, 1968, Chapter 12. Chapter 5 5.1 5.2 5.3 5.4 P. L. Dandenc et al., "Effect of high-speed rectifier excitation system on generator stability limits," IEEE Transactions, vol. PAS-87, pp. 190-201, January 1968. R. H. Schier and A. L. Blythe, "Field tests of dynamic stability using a stabilizing signal and computer program verification," IEEE Transactions, vol. PAS-87, pp. 315-322, February 1968. F. R. Schlief et al., "Control of rotating exciters for power system damping-pilot applications and ex perience," IEEE Transaction Paper 69 TP 155-PWR. F. P. deMello and C. Concordia, "Concepts of syn chronous machine stability as affected by excitation control," IEEE Transactions, vol. PAS-88, pp. 316-329, April 1969. 118 G. A. Jones, "Transient stability of a synchronous generator under conditions of bang-bang excitation scheduling," IEEEE Transactions, vol. PAS-84, pp. 114-121, February 1965. A. P. Sage, Optimum systems control, Englewood Cliffs, N.J.: Prentice Hall, 1968. C. II. Schley, Jr. and I. Lee, "Optimal Control computation by the Newton-Raphson method and the Riccati transformation," IEEE Transactions, vol. AC-12, pp. 139-144, April 1967. M. J. Box, "A comparison of several current optimi zation methods, and the use of transformations in constrained problems," The Computer Journal, vol. 9, pp. 67, 1966. J. W. Sutherland, "The synthesis of optimal control lers for a class of aerodynamical systems, and the numerical solution of nonlinear optimal control problems," U.B.C. PhD Thesis, May 1967. Y. N. Yu and C. Siggers, "Stabilization and optimal control signals for a power system." Accepted as transaction paper for IEEE 1970 Summer Power Meeting. 

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