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Modelling of parallel HVDC-AC power systems for dynamic studies and optimal stabilization Metwally, Aly Abdelhameed 1973

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MODELLING OF PARALLEL HVDC-AC POWER SYSTEMS FOR DYNAMIC STUDIES AND OPTIMAL STABILIZATION by ALY ABDELHAMEED METWALLY B.Sc. Aln-Shams University, Cairo 1965 M.A.Sc. University of British Columbia, Vancouver 1970 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 THE UNIVERSITY OF BRITISH COLUMBIA September 1973 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 i t 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada Supervisor: Dr. Y.N. Yu ABSTRACT Due to the fast response, the hvdc lines, when properly con-trolled, are very useful in stabilizing power systems. In this thesis dynamic modelling and linear optimal controls for dc/ac parallel power systems are Investigated. Beginning with a detailed representation of one machine-infinite bus dc/ac parallel power system several reduced order models with very good accuracy are developed. They are derived mainly through eigenvalue analysis but are compared in detail with the high order nonlinear model tests. Linear optimal control schemes, comprising excitation and/or dc reference current control signals, are designed and tested for two different one machine-infinite bus systems. After that a dynamic model for a three-terminal dc line in parallel with a delta-connected ac system is developed and again linear optimal controls are designed and tested. Two systems are studied; a two machine-infinite bus system and a three machine system. For each system two cases are investigated; two rectifiers and one inverter, or one rectifier and two inverters. It is found that in a l l cases the hvdc reference current control provides the most effective means in stabilizing large power systems. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE ON CONTENTS i i i LIST OF TABLES .- v LIST OF ILLUSTRATIONS vi ACKNOWLEDGEMENT ix NOMENCLATURE x 1. INTRODUCTION 1 2. BASIC EQUATIONS OF HVDC POWER SYSTEMS 5 2.1 The HVDC Systems 5 2.2 The Overall System 7 2.3 Synchronous Machine Equations 8 2.4 Converters and DC Control 10 2.5 Transmission Lines ' 11 2.6 AC Harmonic Filters, Shunt Capacitors and Local Load 14 2.7 State Variables 14 3'. DYNAMIC MODELLING AND ORDER REDUCTION 17 3.1 Reduction Techniques 17 3.2 Reduced Models 18 3.3 Initial Conditions for a DC/AC Parallel Power System 21 3.4 System Data 25 3.5 Eigenvalue Analysis 27 3.6 Nonlinear Tests with System Disturbances 31 4. STABILIZATION OF DC/AC PARALLEL SYSTEMS BY LINEAR OPTIMAL CONTROL SIGNALS 36 4.1 Linear Optimal Regulator Problem 36 4.2 Control Signals for an Existing System 37 4.3 Nonlinear Tests 39 4.4 A Fourth Order Model Study 44 4.5 Expanded System 45 i i i Page 5. MULTITERMINAL HVDC SYSTEMS 55 5.1 Operation and Control of Multiterminal DC Lines . . . . 55 5.2 System Equations 56 5.3 Initial Conditions 63 5.4 Numerical Example . . . 64 6. LINEAR OPTIMAL CONTROL FOR MULTITERMINAL DC/AC SYSTEMS . . . . 67 6.1 Two Machine-Infinite Bus System with Two Rectifiers and One Inverter 68 6.2 Two Machine-Infinite Bus System with One Rectifier and Two Inverters 77 6.3 Three Machine System with Two Rectifiers and One Inverter 86 6.4 Three Machine System with One Rectifier and Two Inverters 93 7. CONCLUDING REMARKS 105 REFERENCES 108 iv LIST OF TABLES Table Page I Eigenvalues by State Variable Groupting Techniques . . . 28 II Eigenvalues by Nonlinear Elimination Technique . . . . . 29 III Eigenvalues of the Existing Power System for Various Controls 38 IV Eigenvalues for a 4th Order Model 45 V Eigenvalues for Expanded System . 50 VI Eigenvalues for 2 Machine-Infinite Bus, 2 Rectifiers System 71 VII Eigenvalues for 2 Machine-Infinite Bus, 2 Inverters System 81 VIII Eigenvalues for 3 Machine, 2 Rectifiers System 90 IX Eigenvalues for 3 Machine, 2 Inverters System 99 v LIST OF ILLUSTRATIONS Figure Page 2.1 Bridge Connection for DC Converter 6 2.2 Converter Output DC Voltage 6 2.3 Angle of Delay a 6 2.4 A Typical DC/AC Parallel Power System 8 2.5 Equivalent T-sections for Transmission Lines 13 2.6 Resolving I into d-q Components 13 K 3.1 Eigenvalues for 39th Order Model 30 3.2 Step Change in I , ' 32 r b ref 3.3 Step Change in I 32 3.4 Step Change in P^ . . 33 3.5 Step Change in 33 3.6 Three Phase Fault on AC Line 34 3.7 Three Phase Fault on AC Line 34 4.1 Swing Curves for Existing System 40 4.2 Speed Deviations for Existing System 40 4.3 Terminal Voltage Variations for Existing System . . . . 41 4 .4 Variation of DC Power for Existing System 41 4.5 Variation of AC Power for Existing System 42 4.6 Control Effort for Existing System 42 4.7 Swing Curves vith Varying Limits on u^ , . . 43 4.8 Excitation Control Effort with Varying Limits on Ug . . 43 4.9 Swing Curves for a 4th Order Model 46 4.10 AC System to be Expanded 46 4.11 Swing Curves for Expanded System 52 vi Figure Page 4.12 Speed Deviation for Exapnded System 52 4.13 Terminal Voltage Variations for Expanded System 53 4.14 Variation of DC Power for Expanded System 53 4.15 Variation of AC Power for Expanded System 54 4.16 Control Effort for Expanded System 54 5.1 Three-Terminal DC/AC Power System 57 5.2 Transformation Between Common and Individual Machine Frames . . 57 6.1 Swing Curves for 2 Machine, 2 Rectifier System 73 6.2 Speed Deviation for 2 Machine, 2 Rectifier System . . . . 73 6.3 Terminal Voltage Variations at Bus 1 for 2 Machine, 2 Rectifier System 74 6.4 Terminal Voltage Variations at Bus 2 for 2 Machine, 2 Rectifier System 74 6.5 DC Power Variations for 2 Machine, 2 Rectifier System . . 75 6.6 DC Control Effort for 2 Machine, 2 Rectifier System . . . 76 6.7 Excitation Control Effort for 2 Machine, 2 Rectifier System 76 6.8 Swing Curves for 2 Machine, 2 Inverter System 82 6.9 Speed Deviation for 2 Machine, 2 Inverter System . . . . 82 6.10 Terminal Voltage Variations at Bus 1 for 2 Machine, 2 Inverter System 83 6.11 Terminal Voltage Variations at Bus 2 for 2 Machine, 2 Inverter System 83 6.12 DC Power Variations for 2 Machine, 2 Inverter System . . 84 6.13 DC Control Effort for 2 Machine, 2 Inverter System . . . 85 6.14 Excitation Control Effort for 2 Machine, 2 Inverter System 85 6.15 Swing Curves for 3 Machine, 2 Rectifier System 91 v i i Figure Page 6.16 Speed Deviation for 3 Machine, 2 Rectifier System . . . 91 6.17 Terminal Voltage Variations at Bus 1 for 3 Machine, 2 Rectifier System ., 92 6.18 Terminal Voltage Variations at Bus 2 for 3 Machine, 2 Rectifier System 92 6.19 Terminal Voltage Variations at Bus 3 for 3 Machine, 2 Rectifier System . . . 92 6.20 DC Power Variations for 3 Machine, 2 Rectifier System . 95 6.21 DC Control Effort for 3 Machine, 2 Rectifier System . . 96 6.22 Excitation Control Effort for 3 Machine, 2 Rectifier System 96 6.23 Swing Curves for 3 Machine, 2 Inverter System 100 6.24 Speed Deviation for 3 Machine, 2 Inverter System . . . 100 6.25 Terminal Voltage Variations at Bus 1 for 3 Machine, 2 Inverter System 101 6.26 Terminal Voltage Variations at Bus 2 for 3 Machine, 2 Inverter System 101 6.27 Terminal Voltage Variations at Bus 3 for 3 Machine, 2 Inverter System 101 6.28 DC Power Variations for 3 Machine, 2 Inverter System . 102 6.29 DC Control Effort for 3 Machine, 2 Inverter System . . 103 6.30 Excitation Control Effort for 3 Machine, 2 Inverter System 103 v i i i ACKNOWLEDGEMENT I am deeply grateful to Dr. Y.N. Yu, supervisor of this pro-ject, for his continued interest, inspiring guidance and unfailing patience during the research work and writing of this thesis. Thanks are due to Mr. A. MacKenzie for draughting and Ms. Norma Duggan for typing the thesis. The financial support from the University of British Columbia and the National Research Council of Canada is gratefully acknowledged. I am greatly indebted to my wife Magda for her patience, sac-rifices and encouragement throughout my postgraduate work. ix NOMENCLATURE linearized system matrices ac transmission line coefficients synchronous machine coefficients dc network coefficients common reference frame individual machine frame exciter voltage equivalent ac harmonic fi l t e r admittance local load admittance synchronous machine inertia constant synchronous machine terminal current components direct and quadrature damper windings currents field current shunt capacitor current components local load current components ac harmonic filter current components ac transmission line current components at sending end ac transmission line current components at receiving end dc current components dc reference current rectifier and inverter dc currents voltage regulator gain and time constant rectifier firing circuit gain and time constant inverter firing circuit gain and time constant current margin composite system matrix x mechanical input energy conversion power synchronous machine output power active and reactive powers transmitted over ac line active and reactive power input (output) to rectifier (inverter) weighing matrices armature and field resistances direct and quadrature axis damper windings resistances ac transmission line parameters dc transmission line parameters local load resistance ac harmonic f i l t e r parameters direct and quadrature axis sub transient open circuit time cons tants direct axis transient open circuit time constant optimal control vector optimal dc control signal optimal excitation control signal conventional excitation control signals synchronous machine terminal voltage terminal voltage dq components infinite bus voltage synchronous machine reference voltage rectifier and inverter dc voltages armature leakage reactance direct and quadrature axis mutual reactances between stator and rotor xi direct and quadrature axis damper windings reactances field reactance direct and quadrature axis subtransient reactances direct axis transient reactance direct and quadrature axis synchronous reactances commutation reactance shunt capacitor reactance state vector and its transpose rectifier and inverter delay angles inverter's angle of advance (= TT - ct^.) deionization angle of a valve synchronous machine torque angle angle between terminal voltage and quadrature axis speed deviation power factor angle for converters direct and quadrature axis armature flux linkages direct and quadrature axis damper windings flux linkages field flux linkage synchronous speed ( = 377 rad./sec.) x i i 1. INTRODUCTION The use of direct current (dc) goes back to the early days of electricity. The first electrical power source, the galvanic battery, delivered dc. Indeed the first electric central power station was built in New York in 1882^ and supplied dc at 110 volts. One of the 2 earliest dc transmission systems, the Thury system in the 1880's , was operated with the principle of constant current control which is very similar to modern hvdc systems. However, ac soon superseded dc in generation, transmission and utilization of electricity because of the general flexibility in voltage transformation, high voltage for transmission and low voltage for distribution, and the sturdy structure of the ac motors and their simple operation. But resurrection of hvdc has been witnessed in recent years because of the development of grid-controlled, multi-electrode, mercury-arc valves which are capable of 2 handling large powers at very high voltages . The main problem though, is s t i l l the relative costs of ac and dc equipment. The high i n i t i a l cost of converting stations for dc has to be weighed against the saving in transmission. In general, the longer the transmission distance, the more favourable will be the use of hvdc. From 1950 to 1970 eight dc links had gone into commercial operation in various parts of the world\ with transmission voltages up to 800 kV, rated powers up to 1440 MW, and distances up to 850 mil is. There are many other advantages of hvdc transmission. For underground and submarine cables dc has lower losses and the problem of charging currents is eliminated, thus increasing the current carrying 3 capacity of cables . An hvdc line may serve as an asynchronous link to 1 2 interconnect power networks of different frequencies, as witnessed in Japan^, or networks that would be unstable i f ac links were used for the interconnection, as in the case of the Eel River project"'. The fast res-ponse of dc links can be utilized in improving the performance of the associated ac systems'^ thus suggesting a very powerful tool for stabilizing large power systems. Considerable amount of work has been done in the field of mathematical simulation of hvdc systems. Detailed simulation of con-7-12 verter action on digital computers received a lot of attention . The simulation of ac system impedance and converter transformers were also considered"^'"^. Hybrid computers"'"'' and dc simulators^ were developed for the study. Representation of dc links in large ac systems for load 17-19 flow studies were also reported . For transient stability studies, simpler-models were •obtained-by representing converters en-the basis of the average value of their dc voltages^ Whereas the ac power control after a disturbance is rather slow i f governors are used to control the mechanical input, or lacks continuity and smoothness i f capacitor switching or dynamic resistance braking is used to control the electrical output, the power control over a dc line is quick and smooth by simply controlling the firing angles. People have started looking into the use of dc/ac parallel trans-22—26 mission for improving stability . The basic control scheme is to regu-late the dc power according to the system stability requirement using signals. derived from the frequency differences between the interconnected sys-22 23 tems , the ac voltage drops , a combination of frequency change and 24 its integrals , and power angle magnitude or its derivative and several 25 other schemes . A l l these controls resulted in improving the stability 3 of dc/ac parallel systems including the case of short outage of dc lines. This clearly demonstrates the effectiveness of dc control in stabilizing a power system. As for the gains of a l l the control schemes mentioned above, they were arrived at experimentally. There is also 27 a very recent paper designing a linear optimal control for an hvdc link. All existing hvdc links are essentially restricted to two-terminal lines mainly because of the absence of hvdc circuit breakers. 28—31 Since such breakers are being developed , a sizeable number of pub-lications is already available for multi-terminal hvdc system operation - n 32-41 and control In this thesis the dynamic modelling and linear optimal control design for two-terminal as well as multi-terminal hv dc/ac parallel power systems are investigated. In Chapter 2 the dynamics of a two-terminal hv dc/ac parallel system are explained and the system components are represented mathematically in detail. This detailed model is reduced in order by applying simplifying assumptions and making approxi-mations in Chapter 3. The validity of these assumptions and approxima-tions is tested by eigenvalue analysis and nonlinear disturbance tests. In Chapter 4 optimum control theory is employed to design control signals for linearized models of dc/ac parallel power systems and the nonlinear system responses under disturbance are compared for different control schemes. In Chapter 5 the operation and basic control systems for a tapped three-terminal dc line superimposed on a delta connected ac network are discussed and a dynamic model for the system is constructed-Finally in Chapter 6 linear optimal control schemes are designed and tested for a two machine-infinite bus system as well as a three machine system. Both systems are studied for two modes of operation; two 4 rectifiers and one inverter, or one rectifier and two inverters. 5 2. BASIC EQUATIONS OF HVDC POWER SYSTEMS In this chapter the basic elements of an hvdc system will be described and equations describing such a power system will be given. 2.1 The HVDC Systems The main components of an hvdc system are the dc transmission lines, the ac power supplies, the converters and controls, the smoothing reactors, the ac harmonic filters, and the shunt capacitors. Each con-verter unit consists of six valves in a bridge connection for a six-pulse operation where valves are connected in pairs to each phase terminal, one with the anode and the other with" the cathode as shown in Figure 2.1. The output voltage Figure 2.2, has a ripple six times the main frequency where each valve carries the fu l l dc current for 120°. At least there are always two valves conducting in series. From the above description i t is clear that the ac current contains lots o-f harmonics inherent in the conversion process. Therefore filters are required to reduce the ac voltage and current ripples to an acceptable level. For a six-pulse operation the filters generally com-prise branches tuned to 5th, 7th, 11th and 13th harmonics of the supply frequency. These are connected either to the primary or tertiary wind-42 ings of the converter transformer The dc voltage output of a converter bridge is controlled by changing the firing angie, a, of the valves. In Figure 2,3 valve 3 will take over current from valve 1 at b instead of a. This means that the current will lag behind the voltage and reactive power must be drawn from the supply. For this reason reactive compensation via shunt capa-citors is normally used. In addition, the shunt capacitor banks of the R X Y 1 B 7 3 5 r; x—~XT X 4 6 2 •f Fig. 2.1 Bridge Connection for DC Converter B5 Rl Y3 BS Rl Y'6 B'2 R'4 Y'6 B'2 . Fig. 2.2 Converter Output DC Voltage oHH. J*S IR , . 3Y 5B ' IR Fig. 2.3 Angle of Delay a I 7 ac harmonic filters also supply a portion of the reactive power re-quired. Converter valves should be operated within their current ratings. Constant current control is thus.desirable and is usually em-7 43 ployed for rectifier bridges ' . This is achieved by comparing the actual value of dc current to the desired value and adjusting the firing angle a accordingly. For inverter operation each valve must start firing at an angle 3 (=Tf-a^) in advance of the voltage zero such that after completion of commutation there is s t i l l an angle A available, greater than or equal to A q , the deionization angle of the valve, to prevent com-mutation failure. Since voltage and current are liable to considerable changes, 3 must be large enough to accommodate such changes. On the other hand, the amount of reactive power required by the inverter is directly proportional to g, which means that the smaller the angle 3 the smaller is the reactive power required. These considerations show that a compromise has to be made between these two conflicting require-ments. Such a control scheme is termed constant extinction angle con-trol. In addition to this control a current override control is sup-plemented such that i t allows the inverter to take over current control, at a reduced value of reference current, in case of a failure of the rectifier's current control due to low ac voltage or loss of a converter group. In such a case the rectifier will deliver voltage up to its capacity with a set to a minimum value of about 5°. K 2.2 The Overall System In this thesis optimal stabilizatin of parallel hvdc-ac systems is investigated. For that a dynamic modelling of the system is necessary. The model must be fairly accurate to be able to predict 8 the dynamic performance of the system yet not too complicated computa-tion-wise . Figure 2.4 shows a general layout of a parallel hvdc-ac system. The generator is connected to an infinite system through both ac and dc transmission lines. The generator is equipped with a voltage regulator and has a local resistive l o a d a t the terminal. The dc line has a constant current control at the rectifier end and constant extinction angle control with current override at the inverter end. There are ac harmonic filters and compensating capacitors at the sending end. Fig. 2.4 A Typical DC/AC Parallel Power System 2.3 Synchronous Machine Equations The flux linkages are expressed in terms of per unit reactances instead of per-unit inductances. Since x = to L, where CJ = 377 radians/ e e second, u) will be attached to the p^ equations a l l the time. Also 9 negative reactances instead of negative currents are chosen for the generator armature circuit. Thus one has, for the flux linkages yq -(x + x„ ) X aq la aq • i q -x (x + x„, ) aq aq £kq *d _ ( x a d + x*a> -x ad -x ad x ad a^d (x j + X., ,) X , ad £kd ad x ad ( xad + W Lkd •fd. (2.1) and for the power P = i b , i - ib 1 e d q q d 2 2 . 2 V,_ = V , + V t d q . 4 4 (2.1a) Then Park's equations of a synchronous machine44 and a voltage regulator equation can be arranged in a state variable form to read p6 = Au (2.2a) 0) piK, = to r_, ( i,,) r r f d e fd x , fd ad (2.2b) pE = — [k (V - - v J - E ] x T r ref t x r pi|). , = -to r. , i , , r rkd e kd kd (2.2c) kq "e rkq *kq pip, = to (v, + r i ,) + (to + Aco) ib d e d a d e q (2.2d) p^ = to (v + r i ) - (to + Aio) i b , r r q e v q a q v e ' M 10 Equations (2.2a) are the mechanical equations in state variable form, where 6 is in radians, Aw in rad./sec, P and P in p.u. and H in m e seconds. The second equation of (2.2b) describes the voltage regulator of the control loop. For simplicity only one time constant is included. 2.4 Converters and DC Control Techniques for simulating converter action in detail are 7-12 available . For transient stability studies a simple model may be 20 21 achieved by representing the converters by their average dc voltages ' This method will be used throughout this thesis. The average dc voltages for the rectifier and inverter bridges respectively are V,, = v cos OL. - — x I_. (2.3) R rr t R TT co R T7 3/3 3 _ VT = v cos a_ x I T I TT O I TT C O 1 where a and a are the firing angles of the dc control. R I The constant current control for the rectifier bridge and the constant extinction angle control with current override for the inverter bridge are described as follows. KR . . C O S aR = v~- ( I r e f " V + C O S aRs 1 2 cos a T = -cos A + — (—x I T + e _) (2.4) I o V Q co I cl where C L is the steady state value of O L and e T is the current override Rs J R cl control signal and is restricted to positive values. e T = KT ( k I . - I_) 0 (2.4a) cl I ref I Equations (2.3) describe a continuous relation between the dc voltage, the firing angles, and the dc current. Since a change in the firing angle 11 following a commutation does not affect the average output voltage immediately, not until the end of the normal conduction period, and a change in converter current does not cause an instant change in voltage, time lags must be incorporated into the transfer functions for a tran-sient study. Thus equations (2.3) may be rewritten 3/3 3 T t VT> = V_ C O S ar> X I ' R TT t R TT CO R w 3 y /3 3 _,• VT = v cos a T x I' I Tf O I Ti CO I (2.5) where P*R ' f " XR) P cos a R = ^ [ K R ( I R E F " I.R)/vt + cos a R s - cos a R] p cos aT = ±- [-cos a ; - cos A Q + ( - | ^ + e ^ / v j (2.6a) (2.6b) A single time constant is incorporated into each control loop of the dc system. 2.5 Transmission Lines Both ac and dc lines are represented by equivalent T-sections as shown in Figure 2.5. For long transmission lines more than one section can be used i f more accuracy is required. The dc line is re-presented by to P XR = 3T <VR ~ V c " R V ' P V c = "e X c ( IR " V ( 2 ' 7 ) J_i to P 1 ! ^ ( V I + V c " R V 12 For the ac line let i = i , + j i , v 85 v, + j v d J q d J q (2.8a) The ac line in Park's d-q coordinates may be represented by pi . = — (v - v - r i ,) - (co + Aco) i ,. v qA q qc qA e dA to pi,. = — (v, - v, - r i,.) + (to + Ato) i . r dA x „ d dc dA e qA pi „ = •- (v cos 6 - v - r i ) - ( lo + Au) i r qB x^ o qc qB e dB to PAdB = < v 0 S i n 6 " Vdc - r ldB ) + (u,e + A u ) V (2.8) pv = to x (i + i ) - (to + Ato) v r qc e c qA qB' v e ' dc pv, = to x (i + i ) + (to + Aw) v r dc e c dA dB e qc To relate the dc current to the ac reference frame a second set of axes, d' and q', is defined such that the q' axis coincides con-tinuously with the hypotenuse of the right triangle formed by the two components of the synchronous generator terminal voltage, v^ and v , as shown in Figure 2.6. The angle between the two sets of axes, 6 , is K given by & = arc tan (—) K v (2.9) and the rectifier current .1^  may be resolved into two components, I, K a and Iq, in the ac reference frame. 2/3 sin6 cos6 K K cos6 -sin6 K K. COS<J> R sin<j> R where , A TT R cos<p = — — — R 3/3 v t (2.10) r *l Vt 'A Vr Xl 'U 'B Xr fa) AC T.L R XL XL R [fY-A^A I (b) DC T.L 1 •f Fig. 2.5 Equivalent T-sections for Transmission Lines Fig. 2.6 Resolving I into d-q components R CO 14 2.6 AC Harmonic Filters, Shunt Capacitors and Local Load Four parallel filter branches tuned for 5th, 7th, 11th and 13th harmonics are considered. Each branch is represented by 1 2. P i T {pv. - (w +Ao))v - R... pi,,.. + 2(to + A t i ) ) L , . pi -dfj L f j r d e q fj ^  dfj e f] K qfj [— (to +Au)2 L - . ] i , c . + [L £. pAco + (ai +Ato)R j r.] i ,..} (2.11) c fj e f j J dfj fj ^  e f j J qfj J 2 1 P i c . ~ -z {pv + (to +Aw)v, - 2(a) +Au))L_ pi,,. - R,-. pi r. -qfj L f r q e d e. fj F dfj fj qfj [L.. pAco + (u +Au))R,.]i... - [— (w +Ato) 2 . ] i ..} fj e f j J dfj l c f j e f j J qfj j = 5, 7, 11, 13 Next the shunt capacitors may be represented by pv, = to x i , + (to +Ato) v d e c c d e q (2.12) pv = co x i - (to +Ato) v, q e c cq e d where Xcd xd "^dA " 1df • 1dA ~ Id (2.13) i = i - i . - i . - i - I cq q qA qf q£ q Finally the local load currents are given by — -i v d i \ V. q_ (2.14) 2.7 State Variables Equations (2.2), (2.6), (2.7), (2.8), (2.11) and (2.12) form a complete set of state equations for the system under study. The state variables are 15 ** = (&>^>*fd> V*kd'V*d'V V W IR'II' C° SV C O S Q r 1dA' iqA' 1dB' V'^c'V'^fS'^dfS' ±qf 5 ?? ± q f 5 ' ±df 7 ' P l d f 7' ^ 7 ' P lqf7' idfll» P ± d f l l ' iaflV P i q f l l ! l d f i 3 ' p i d f l 3 ' ±qfl3' p i q f l 3 ' V The system is of the 39th order. The auxilliary algebraic equations are given in (2.1a), (2.4a), (2.5), (2.9), (2.10), (2.13) and (2.14). The current solutions of equations (2.1) are: - -i q —y Y ^kq yaq • *q Xkq ~yaq yq£ % q *d " ydfk y df y dk *d Akd = " ydf y df£ ~ Y d£ • *kd ifd "ydk " y d£ y dk£ (2.15) where — — ^ q I X + X „ , aq £kq yaq ~ \ X aq _ yq^_ x +x„ aq £a A , = (x +x„ ) (x +x„. ) - x 1 aq £a aq £kq aq ydfk Xad ^Afd^Akd^ + x£fdX£kd ydf xad x£fd ydk 1 A 2 xad X£kd ydf£ xad ( x£fd + x£a ) + x£fdX£a yd£ Xad X£a ydk£ Xad ( x£kd + X£a ) + x£kdX£a (2.16) 16 A~ = x JX„ (x. r, + x„. ,) + x.,, x., , (x , + xn ) 2 ad Jla lid Ikd ifd Ikd ad la" These equations form the base of the dynamic modelling of hvdc systems which will be developed in the next chapter. 17 . 3. DYNAMIC MODELLING AND ORDER REDUCTION The mathematical model constructed in the previous chapter has 39 state variables which describe the system dynamics in great detail Such complexity presents a very difficult problem in digital computation. In this chapter the order reduction of the system model is achieved throu engineering approximations with the aid of eigenvalue analysis. The validity of the low order models is established through tests on the nonlinear models with disturbances. 3.1 Reduction Techniques Numerical techniques for reducing high order systems to low 45-47 order equivalents are available . But the identity of a l l the para-meters will be lost in the process of reduction. In this thesis the •o reduction will be approached by engineering approximations instead. Again there are two approaches for the approximation. The 47 first is called the state variable grouping technique , which may be called linear approximation technique, by which the nonlinear system equations are linearized first and then the system equations order is reduced in possibly several steps. At each step the state variables x are separated into two groups, x^ and y.^* associated^ with large and small time constants respectively. The linearized system equations are written in a partitioned form as follows: x„ A l l A12 A A 21 22 (3.1) Neglecting the small time constants or the fast and short-lived trans-ients by setting equal to zero, is given by 18 = -A, -1 A, (3.2) x 2 22 which is then substituted back into (3.1) to give 1 [ A11 A12 A22 -1 (3.3) The order of the system is thus reduced by the number of state variables in According to this procedure several linear models will be ob-tained, but there is only one high order nonlinear model for the system. technique by which a l l the time derivatives as well as speed deviation terms in the nonlinear differential equations to be eliminated are set equal to zero at each step. This is equivalent to a static representation of the system components for these equations to be eliminated. The resulting algebraic equations are then solved and substituted back into the rest of the nonlinear differential equations before linearization for linear optimal control design or eigenvalue analysis. The advantage of this approach is that for each linearized model there is one original nonlinear model. This is the approach to be followed in this chapter. 3.2 Reduced Models presented. For the procedure, i t is chosen th&t the ac harmonic fi l t e r dynamics shall be eliminated first since i t consists of the largest num-ber of equations. The next group are the ac transmission lines and machine stator windings, and so on. The details are: STEP 1: Neglecting the ac harmonic fi l t e r dynamics of equations (2.11) Another approach may be called the nonlinear approximation In the following the models resulting from reductions will be the model is reduced from the 39th to the 23rd order. The resulting algebraic equations are: 19 "df qf G f B £ -Bf G f v where = I 3 ( R f / + x f/) (3.4) "R f i e fj & Cj.. J J e fj j = 5, 7, 11, 13 (3.5) STEP 2: Neglecting the ac transmission line dynamics the system's order is reduced to 17. Equations (2.8) are replaced by idA _ a i "a2 a3 a4 v sin6 o V a2 a i ~% a3 v cosS o idB a3 a i "a2 Vd V = ~a4 a3 a2 a l V _q Vdc X c (a2-a^) x c (ai+a3) •x c ( a2- a4 } X c ( a l + a 3 } V qc -X c ( a l + a 3 ) x c (a 2-a 4) -x c (a 1 +a 3) X c (a 2-a 4) where "a. 1_ A , 2rx c(x £ - xc) xc ^ " x / + 2 x i x c > r [ r 2 + (x £ - x c ) 2 + x c 2] (x£ ' xc ) + X£2 " 2 Vc } 4r 2 (x £ - x c ) 2 + (r 2 - x/ + Ix^)1 (3.6) (3.7) STEP 3:. Neglecting the transformer voltages and voltages due to speed deviation generated in the armature windings of the synchronous 20 machine a 15th order model is achieved. Equations (2.2d) are replaced by ( 1 + r a ^ q W 2 2 ra ykq ydk ra ykq Ydf ra yaq r y 1 a kq L" ra ydk " ra ydf aJ aq df k ra ydfk. , [ * £ d *kd kq v, v ] d qJ (3.8) STEP 4: Representing the dc transmission line by its static impedance yields a 12th order model. Equations (2.7) become ^ ' k ( VR + V V c - 2 <VR " V X I - XR (3.9) STEP 5: Dynamics of the shunt capacitor of equations (2.12) are neglected to give a 10th order model. The capacitor currents are thus given by (3.10) STEP 6: The damper winding effects of equation (2.2c) are neglected. The resulting model is of the 8th order. The resulting auxilliary i , - V cd = _JL_ q i c V cq d equations are 1 A, >. 4 x ,[r +x„ (x +x„ )] r x ,x0J., ad1 a la aq la a ad Ifd xad x£fd ( xaq + x£ -r x ,x a ad aq [*fd Vd V q ] -x [x 3x n +xn£,(x ,+x„ )] r x (x ,+x„r, aq 1 ad la £fdv ad la 1 a aq ad Ifd (3.11) A, = r (x ,+x.^ .j) + (x +x„ ) [x ,x. + x.,.(x , + x. ) 1 4 a v ad Ifdy v aq la 1 ad la lfdK ad la J 21 STEP 7: The small time constants of equation (2.6a) are neglected and a 6th order model is obtained. The resulting equations are IR= h V = I I I (3.12) STEP 8: Neglecting firing circuits dynamics of equation (2.6b) a 4th order model is obtained and the new algebraic equations are R ref 1 2 cosaT = — (— x I_ + e _) - cosA I v ^/j co I cl o (3.13) STEP 9: Representing the voltage regulator by a static gain and neglec-ting the field flux decay of equation (2.2b) yields a 2nd order model with the new auxilliary equations E _ x ad E = K (V , - v j (3.14) x r ref t The steps thus chosen at this stage are, indeed, a matter of convenience, not because of the order of importance of their eigenvalues-Of course a l l nonlinear differential equations which are not to be e l i -minated are kept intact at each step. 3.3 Initial Conditions of a Parallel AC/DC Power System For this thesis study the i n i t i a l conditions assumed are the synchronous machine's terminal voltage v , the infinite bus voltage V Q , the power transmitted over the ac lines P , and that over the dc line cLC P, , and the minimum extinction angle for the inverter A • Other i n i t i a l dc o conditions, such as the reactive power over the ac line Q and that r ac 22 drawn by the rectifier Q^c» and other synchronous machine i n i t i a l con-ditions are to be determined. Since the active and reactive powers transmitted over the ac line may be given by P ac "ac 1dA XqA XqA . "^dA — — V , d V (3.15) Substituting for i ^ and i from (3.6) into (3.15) and solving for sin6 and cos6 yields sin6 coscl v (a -a0) o 1 I a l a2 "a2 a l ac 2 " a3 ac ~2 ~ a4 Q P ac -~2 + a4 v ac ~2 " a3 v q (3.16) But 2 2 2 2 2 Pac + <*ac = V t <1dA + V > 2 2 2 2 2 2 2 = v t {(a 1 +a2 ) V q + (a 3 +a^ )vfc + 2V Q [ ( a ^ a ^ - a ^ a ^ (vdsin6+vqcos<5) + (a.a.+a0a )(v sin6-v ,cos6)]} i. H £ 5 q d (3.17) Substituting (3.16) into (3.17) and solving for Q gives 3 C Q = a.v 2 + \v [(a 2+a 2)v 2 - a„ v + 2aJ? ] - P 1 ^ac 4 t — i t 1 2 o 3 t 3 acJ ac (3.18) To determine the dc reactive power Q c^ the rectifier voltage V has to be determined fir s t . From (3.12) and (3.13) one has K I =1 = I' = I 1 R I R I * , 2 R cosctT = -cosA + — x — I o ^ co v o (3.19) Substituting (3.19) into (2.5) gives 3/3 3 V T = v cosA -\ x I„ I -no o IT co R (3.20) 23 From (3.9) and (3.20) 3/3 3 V D = v cosA + (2R - - x ) I_ R ir o o TT co R (3.21) But P = — V I dc 3 R R (3.22) Solving (3.21) and (3.22) for V_, yields V P = T^Jr v cosA^ + 1| (-^ v cosA) 2 + 12 P_ (R - ~ x )] 'R 2L TT o ,3/3 TT "O dc 2TT co (3.23) Since cos $ u VR R 3/3 V t (3.24) Thus Q, = P, tan $_ %dc dc R (3.25) Next the synchronous machine i n i t i a l conditions can be deter-mined as follows. The active and reactive powers of the synchronous machine are given by P = P + P . + v 2 (G, + |-) ac dc t f RL ^ = V + Q d c + v t 2 < B f - h c (3.26) Next equations (2.15), (3.8) and (3.11) are combined to give •fd C l C2 C3 -c„ c4 "C5 •c6 "c7 c8 C9 "c6 c10 "c10 ~c8 c l l (3.27) 24 where ~ c l ~ r [x , x + x (x + x. )] a ad &a £fd ad la C2 (x + x. ) [x ,x„ + x„-, (x ^  + x. )] aq la L ad la Ifd v ad la J C3 2 r X j a ad c4 r (x , + x„,.,) (x + x„ ) a v ad ilfd aq la C5 1 r x , (x + x. ) a ad aq la C6 " A4 r (x , + x. £ J) a ad ilfd C7 (x , + X n J-j) (x + x. ) ad ilfd aq >la C8 x , (x + X. ) ad v aq Jta C9 x ,x, + x„.. 3 (x , + x. ) ad Jta ilfd v ad >la C10 r X A a ad _C11_ 2 r + (x , + x„ ) (x + x. ) _ a ad la aq la _ and A. is 4 defined in equation (3.11). Substituting (3.28) P = v , i , + v i and solving for \b. , gives d d q q & r f d 6 fd P + c 6 vfc + (c 7-c 9) v dv q  ( c8 Vd + C10 V (3.29) Substituting (3.27) and (3.29) into Q = v i , - v,i q d d q and solving for v^ gives [ r a P + ( xaq + x£a> Q + v t ' ]  V<*~ « xaq + x * a ) P " r a Q ] Substituting (3.31) into (2.1a) gives (3.30) (3.31) 25 vd ~ ,| o 2 2 2 2 4 <3'32> K + ( xaq + X.a ) J <P +Q > + 2 v t I r a ? + < Xaq + X*a ) (^ + V t Thus v d can be calculated from (3.32), from (3.31) and i ^ f d from (3.29). Finally the torque angle 6 can be determined from (3.6) to read . ( a 2 a 4 " a l 3 3 ) v d " ( a i a 4 + a 2 a 3 ) v q + *1 ±dA + a2 *qA „ „, o = arc tan ; r ;—; r — 3 : ; : (3.33; ( a l a 4 + a 2 a 3 ) v d + ( a 2 a 4 " a l a 3 ) v q " a2 xdA + "l/qA where the ac transmission line currents i , . and i , are determined from dA qA (2.13) after i , and i , i . . , and i I_, I, and I .and i , and i are d q' df qf R d q' cd cq determined from (3.27), (3.4), (3.22), (2.10) and (3.10) respectively. The rectifier and inverter firing angles are determined from (2.5) and (3.13) respectively. 3.4 System Data ^The*sysit-eiiv'sirudied "has 'the "fo'l-lowing data, a l l "in per "unit except for angles in degrees and inertia constant and time constants in seconds. Most data are taken from reference 20. The rest are assumed. Synchronous machine and voltage regulator r 0.005 r,, 0.00055 r. , 0.02 r, 0.04 a fd kd kq x«.a °- 1 x£fd °- 1 XJlkd °' 1 X«,kq °'2 x , 1.0 x 0.7 H 3 sec. k 20.0 T 2.0 sec. ad aq r r AC and DC transmission lines r 0.0784 x„ 0.52 x 5.556 JI c R 0.2792 X. 18.112 X 20.28 x 0.432 L c co 26 AC harmonic filters, local load, and shunt capacitor r f 5 0.065 r f 0.OB92 r f l l 0.062 r f l 3 0.0704 L,_ 0.00862 L,_ 0.00862 L,,,-- 0.00332 L..- 0.00332 r5 f7 f l l fl3 c„ 0.0000325 c,_ 0.00001695 c „ . 0.000017625 c,, _ 0.000012575 f5 f7 f l l fl3 Rj^  6.05 x c 212.0 DC Controllers K_ 30 T 0.002778 sec. KT 30 T_ 0.002778 sec. k 0.9 is. K 1 1 Operating point v. 1.1 v 1.0 P 0.4 P, 0.4 t o ac dc A 5° 6 51.05° i b . , 1.165 ± . 1.0057 o Tfd kd iK 0.947 -0.497 * -0.568 v J 0.565 d kq q d -V 0;944 -E 1:59 V . 1vl»8 P 1.-0 q x ref Q 0.15 P 1.004 1. 0.584 i 0.71 m d q *fd ^ l U °- 1 9 V ° ' 3 1 - ° ' 3 4 i- -0.18 v , 0.71 v 0.82 Q 0.0073 qB dc qc ac I R 0.363 I d 0.331 I 0.225 VR 1.65 V 1.55 VT -1.45 O L 8.25° a T 141.7° c I R I Q, 0.185 i,„ 0.09 i 0.16 i , -0.004 dc dl ql cd i 0.003 -0.012 i „ 0.007 i,,, -0.006 cq df5 qf5 df7 i 0.004 -0.006 i „, 0.004 -0.0045 qf7 d f l l q f l l dfl3 \jfl3 0.003 27 3.5 Eigenvalue Analysis Both state variable grouping technique and nonlinear approxi-mation technique described in section 3.1 are employed to calculate the eigenvalues for the different system models developed in section 3.2. The results are tabulated in Tables I and II respectively. There is not much difference in high order models for the two methods. But the difference becomes evident x^ hen the model order becomes low. This is because of two different ways of making approximations. For example the ac harmonic filters equations (2.11) are replaced by (3.4) in non-linear approximation and the filter currents are given in terms of v^ and v^ only. On the other hand when the same equations (2.11) are linearized and then solved the resulting currents are in terms of Ato, ' I ' J T J S & i> • • • > IT,> C O S O L , .... v, and v due to the presence of pv,, fd kd R R d q d »p.v^  .:and 4>Ato .terms, /in the-.eq.uations. ..The nonlinear tapproximation ...approach seems to be giving more accurate results. It is especially clear for the second order model. While the nonlinear approximation technique gives a pair of imaginary eigenvalues, which is expected since damping is neglected in the synchronous machine equations, the state variable grouping technique yields a conjugate pair of eigenvalues with'positive real parts indicating an unstable system which contradicts the results from higher order models analysis. The eigenvalues for the 39th order model are plotted in the complex plane in Fig. 3.1. It is noticed that a majority of the eigen-values are clustered in the region between -10 and -200 of the real axis. The dominant eigenvalues are found to be these corresponding to 6, Ato, if i ^ and E^ equations. The largest eigenvalues correspond to v., v , v, and v equations. Based on this information the following d q dc qc n & 2nd Order 0.126+J6.76 4 th Order -0.052+j7.1 - 0 . 3 3 6 i J l . i l 6th Order -0.052+J7.1 -0.336+jl.ll -362, -35156 8th Order -0.052+j7.1 -0.336+jl.ll -362, -35156 10 th Order -0.15+j7.14 -0.33+jl.l -363, -30532 12th Order -0.15+J7.14 -0.33+jl.l -363, -39685 15th Order -0.15+j7.14 -0.33+jl.l -365 17th Order -0.15+j7.14 -0.33+jl.l -365 23rd Order -0.164+J7.14 -0.33+jl.l -365 39 th Order -0.164+J7.14 -0.33+jl.l -365 360,-360 350,-360 -41.9,-26.1 •360,-360 -41.9,-26.1 -18250+j59136 360,-360 -41.9,-26.1 -6623+J64990 •360,-360 -41.8,-26.2 -4415+J10930 •360,-360 -41.9,-26.3 -i039+j5524 •360,-360 -41.9,-26.3 -6577+J15988 Table I Eigenvalues by state v a r i a b l e grouping technique •185+J761 •20.5+j 327 •122+J785 •15.5+J325 •143+J809 •14.8+j 326 •143+j 811 •14.8+j 326 847+J1892 •73+J1993 •76.7+J1977 •2620+J4267 •59 U +J1216 •50+j 376 •34339,-7018 •61+J1198 •50+j 376 -8.95+j1490 -9.5+J2245 -18+J2191 -23.7+J2947 -55+J4358 -73+J5118 - 92+j 3568 -123+J4335 2nd Order 4th Order 6th Order 8th Order 10th Order 12th Order 15th Order 0.0+J6.76 -0.083+J7.1 -0.083+J7.1 -0.083+J7.1 -0.18+J7.13 -0.18+j 7.13 -0.18+j7.13 -0 .34+j l . l l -0 .34+j l . l l -0 .34+j l . l l -0.33+jl . l -0.33+jl . l -0.33+jl . l 23rd Order -0.164+J7.13 -0.33+jl. l -362,-35156 -362,-35156 -360,-360 -362,-30531 -360,-360 -41.9,-26.15 -362,-31455 -360,-360 -41.9,-26.15 -166910+j495377 -360,-360 -41.9,-26.15 -^49562+j488820 -364 17th Order -0.17+J7.13 -0.33+jl . l -364.5 -365 39th Order -0.164+J714 -0.33+jl. l -365 -360,-360 -41.8,-26.2 -37936+j72148 -360,-360 -41.9,-26.? -6572+J13689 -360,-360 -41.9,-26.3 ^-6577+jl5988 . -185+j760 -20.5+j327 -123+J783 -15.5+J325 -143+J806 -14.9+J326 -1116+J1963 -64.3+J2004 -143+J811 -76.7+J1977 -14.8+J326 -36945,-5154 -55+J1227 -50+J376 -34339,-7018 -8.95+J1490 -61+J119S -50+J376 -9.5+J2245 -18+J2191 -23.7+J2947 -55+J4358 -73+J5118 -92+j3558 -123+J4 335 r o Table II Eigenvalues by nonlinear elimination technique REAL JL X X X X X * -X—X L IMAGINARY 4 -105 -JO4 -103 -W2 -10 -1 10~7 Z O - 2 ' 70" Fig. 3.1 EIGENVALUES FOR 39th ORDER MODEL 31 classification of system models, is suggested: 1. A fourth order model comprising §, Ato, ip^^ and as state variables is the simplest model one can have for system dynamic studies. 2. A sixth order model including cosa and cosaT in addition to the 4th R 1 order states is fairly accurate and can be used for control design including an hvdc system. 3. If one wants more accurate results the 5th harmonic f i l t e r may be included resulting in a tenth order model. • 4. The thirty-ninth order model is accurate but impractical in digital computation. 3.6 Nonlinear Tests with System Disturbances To compare the four suggested models nonlinear system response tests subjected to different types of disturbances are compared. Figs. 3.2 and 3.3 show the change in rotor angle 6 and speed deviation Ato with time for a 25% step change in the dc reference current. For this disturbance i t is seen from Fig. 3.2 that results are very close but the 6th order model gives the closest response to that of the 39th order model. Figures 3.4 and 3.5 show the system's response with a 25% step reduction in mechanical torque input. The 6th and 10th order models give identical response closer to that of the 39th order model than the 4th order model. Figures 3.6 and 3.7 show the system response to a three phase ground fault at the middle of one ac line for 6 cycles followed by isolating the faulted line at both ends and a successful reclosure after 0.4 second after the fault is removed. In this case only the res-ponses of the 39th and 6th order models are plotted. It is noticed that even for such a severe disturbance the 6th order model results are s t i l l SRAD 32 1.0[ 0.8L 0-6 \ 0-4 0.21 39 th ORDER 10th ORDER x xxx 6th ORDER • o « . 4th ORDER a o " — " — i — L . _ i i i i i i 0.5 W Fig. 3-2 STEP CHANCE IN Iref 1.5 t SEC AGO RAD/SEC t SEC 39th ORDER V 10th ORDER v xxxx 6th ORDER . . . . 4th ORDER Fig. 3.3 STEP CHANCE IN Iref SRAD 39 th ORDER 7.0 0.5[ 10 th ORDER xxx x 6th ORDER . • • • *4th ORDER _i i i ' • i i i ' • 0.0 0.5 1.0 15 t SEC Fig. 3.4 STEP CHANGE IN P, m A CO RAD/SEC 2.01 t SEC 39th ORDER 10th ORDER xxxx 6 th ORDER . . . . 4th ORDER -3.0 Fig. 3.5 STEP CHANGE IN P S RAD 35 close to those of the 39th order model. From these test results, suppor-ted also by eigenvalue analysis, i t is decided that from now on in the suc-ceeding chapters only the 6th order model will be used for the control design. 36 4. STABILIZATION OF DC/AC PARALLEL SYSTEMS BY LINEAR OPTIMAL CONTROL SIGNALS In this chapter linear optimal control theory is employed to stabilize dc/ac parallel power systems. Two systems are investigated; the first is to stabilize an existing system, and the second is to stab-ili z e an expanded system by adding a parallel dc link to the existing ac line in order to increase the transmission capacity of the system. Several control schemes are designed and the disturbance test results on nonlinear system models are compared. For a l l designs the sixth order system model, i.e. A6 , Ato, Ait-,, AE , A C O S O L and AcosaT, is fd x R I employed. 4.1 Linear Optimal Regulator Problem The system's linearized equations are written as x = Ax + BU (4.1) It is required to find an optimal control U that minimizes the quad-ratic cost function oo J = Y J (xfc Ox + T/ RU) dt (4.2) o subject to (4.1), where Q and R are positive definite matrices. The 48 required optimal control is given by U = -R_1 Bt Kx (4.3) where the Riccati matrix K is obtained from the solution of the non-linear algebraic matrix equation t -1 t KA + A K - KBR B K + Q = 0 (4.4) Several computation techniques for the solution are available utilizing 49 50 the properties of the state and costate composite system matrix M ' , 37 A M = (A.5) the matrix K may be computed from K X. (4.6) where X. T X. I l l X = (4.7) XII XIV is the eigenvector matrix of M and the eigenvectors X^  and X^ correspond tb the stable eigenvalues of M or the eigenvalues of the controlled system. 4.2 Control Signals for an Existing System The system considered here is the same as described in Chapter 3, which was shown in Figure 2.3. Three different stabilization schemes are investigated; the first with an optimal excitation control u^ on the synchronous machine alone without any stabilization signal on dc, the second with optimal current control u^ on dc but without excitation stabilization on the synchronous machine, and the third with both u^ and Up controls designed together. The control signals designed for the system's linearized model are tested on the original nonlinear system and the system's response to the disturbances and also the system's eigenvalues are compared. For the data given in Chapter 3 the linearized state equations for the system are 38 1 — A6 0.0 1.0 0.0 0.0 0.0 • -52.1 0.0 -112.6 0.0 -92.1 -0.14 0.0 -0.46 0.21 -0.2 AE -0.24 0.0 -6.92 -0.5 -1.11 X Acosct R -278.2 0.0 -8042.0 0.0 -14197 Acosct^ . 5.09 0.0 147.1 0.0 259.7 0.0 -46.7 -0.2 0.92 -10672.0 -164.8 A6 Aw Aifj AE fd A cos a R + BU The B matrices for the three controls are given by t Acosa, (4.8) Ug! B = [0 0 0 k r / T r 0 B = [0 0 0 0 0]' V T R V t ° ] t (4.9) Ug and u^rB 0 0 0 k /T r r 0 0 0 0 V T R V t ° In a l l three cases the matrices Q and R of equation (4.2) are' taken as unit matrices. The system's eigenvalues for a l l cases are listed in Table III and the corresponding control laws are given in (4.10). Table III Eigenvalues of the existing power system for various controls Control Used Eigenvalues no control -0.084 + J7.093 -0.339 + jl.108 -365, -13997 "E -0.728 + j7.154 -0.403, -9.915 -365, -13997 UD -1.512, -43.89 -0.349 + jl.13 -348, -17097 " E A N D % -1.56, -43.89 -0.383, -9.87 -348, -17097 39 UE ~ -5.79 -0.36 -7.32 -1.09 -0.002 -0.025 ~ "D 0.415 1.004 -0.17 -0.0004 -0.319 -0.082 UE -0.159 -0.0017 -0.495 -0.961 0.8xl0~5 -0.0045 _UD_ 0.402 1.005 -0.532 0.008 -0.319 -0.082 [A6 Aw Aii_, AE A C O S O L AcosaT] fd x R I (4.10) It is noticed in Table III that the dominant pair of eigenvalues in a l l three cases have been shifted to the left of the complex plane indicating the stabilizing effect of these controls. The damping ratio for the dominant eigenvalues is improved in the case of u^ , control. For the other cases the dominant pair is decoupled into two real eigenvalues. For Up and u^ controls a l l the eigenvalues are real indicating a non-oscillatory system. 4.3 Nonlinear Tests The system disturbance considered in a l l cases is a three phase to ground fault at the middle of one circuit of the ac transmission line for 0.1 sec. The faulted line is then isolated by disconnecting i t from both ends followed by a successful reclosure at 0.5 sec. after the fault is completely removed. The system's response is summarized in Figures, 4.1 to 4.6. It is noticed from these figures that the excitation con-trol signal Ug by itself does not improve the system's stability effec-tively despite the fact that the control effort reaches maximum a l l the time as shown in Fig. 4.6. The limits set for the control signals are +0.12 p.u. for Ug and +0.3 p.u. for u^. The limits for u g were chosen as +0.12 p.u. after several other values were tried. It is shown in Figure 4.7 that the higher the Ug limits, the higher will be the overshoot in machine I Par P"-42 Fig. 4.5 Variation of AC Power for Existing System u 4 1 NO CONTROL 2 u£ CONTROL 3 uD CONTROL CONTROL Fig. 4.6 Control Effort for Existing System / rad ;.0| Up p.u. 44 swing. The machine goes unstable when u^ , is not restricted. Figure 4 . 8 shows the control efforts for different cases. The dc control limits are so chosen that i t will not overload the dc line or operate i t under very light load. It is found that the stabilization is very effective either with the dc control signal u^ alone or combined with the excitation control signal u^ ,. The angular deviation is very small, and the dc line picks up the load originally carried by the faulty line very quickly. It is also noted that the system's responses with u^ control alone or with Ug and u^ controls are very close except that the terminal voltage changes are smaller in the later case. 4.4 A Fourth Order Model Study Due to the presence of two large eigenvalues on the sixth order model corresponding to the dc firing circuits dynamics the step size in numerical integration is very small thus requiring longer compu-tation time. It was thought that i t would be interesting to find out i f these two small time constants associated with the firing circuits can be neglected in order to save the computation time. The resulting system's linearized equations of the fourth order become A6 n Ato A * f d AE - A6 " - 0 .0 1.0 0 . 0 o .o -• Ato - 5 0 . 3 0 .0 - 6 0 . 4 0 .0 A * f d - 0 . 1 3 5 0 .0 - 0 . 3 4 5 0 .207 AE _ x _ - 0 . 2 1 7 0 .0 - 6 . 3 - 0 . 5 + BU ( 4 . 1 1 ) The matrix B is given by 1 ^ : 6 = [0 .0 0 .0 0 .0 1 0 . 0 ] u D: B = [ 0 . 0 - 5 2 . 5 - 0 . 1 6 3 0 .108 ] 45 " E and u^: B 0.0 0.0 0.0 10.0 0.0 -52.5 -0.163 0.10.8 _ t (4.12) The system's eigenvalues are listed in Table IV and the corresponding control laws are given in (4.13) Table IV Eigenvalues for a fourth order model control used eigenvalues no control -0.084 + j7.093 -0.339 + jl.108 " E -0.728 + J7.154 -0.388, -9.915 -1.39,-51.49 -0.349 +jl.13 u E and u D -1.42, -51.49 -0.35, -9.87 \~ ~-5.78 -0.359 -7.24 -1.09 " ~ A6 %<• 0.444 1.005 -0.023 -0.02 Aco UE -0.109 -0.002 -0.286 -0.957 A^  0.43 1.006 -0.45 0.004 AE fd (4.13) The same nonlinear test with system disturbance is applied to this model and the corresponding swing curves are plotted in Figure 4.9. It is found that there is considerable deviation' between the fourth and sixth, order models and some accuracy is sacrificed by neglecting the two small time constants for faster computation. It is decided that the dc firing circuit dynamics will be retained from now on in the dc system studies for the rest of this thesis. 4.5 Expanded System The system investigated in this section had an ac system con-sisting of a synchronous generator transmitting power to an infinite system over a double circuit ac transmission line, Figure 4.10. The 46 Fig. 4.10 AC System to be Expanded 47 generator is equipped with a voltage regulator and also has a local load at the terminal bus. This system is expanded by adding a dc link in parallel after the generating capacity is doubled. The system was unstable at the outset and a conventional excitation control signal is designed. The ac system's per unit data was as follows: Synchronous machine and voltage regulator: x, 0.973 x 0.55 d q do 7.76 sec. x' 0.19 d x„ 0.04 H 4.63 sec. £a K 130 r T 0.05 sec. r AC transmission line and local load G r -0.034 x £ 0.997 Operating point: 0.249 B £ 0.262 v 1.05 v 1.02 o fd 0.02 1.22 0.952 E 1.34 x v 65.2' 0.45 0.95 0.4 0.81 From the data given the linearized fourth order system's model, equations (2.2a) and (2.2b) were — 1 i A6 Aco Aiji A6 0.0 1.0 0.0 0.0 • Aw -22.4 0.0 -39.7 0.0 A $ f d -0.1 0.0 -0.2 0.15 AE x _ 223.6 0.0 1794.2 -20.0 fd AE X _ i ( 4 . 14 ) She system's eigenvalues were 0.18+J5.16, -10.3+J13.3 which indicates an unstable system. A conventional excitation control 48 signal u^ is designed''"'", u k S e x 1 + T S e . 0.04 S . Au = - — : — „ ,„ Au (4.15) — — A6 « Aw A * f d = AE X u X A6 Aw Aip AE fd 1 + 0.5Sand is added to the system resulting in 0.0 1.0 0.0 0.0 0.0 -22.4 0.0 -39.7 0.0 0.0 -0.1 0.0 ~0.2 0.15 0.0 223.6 0.0 1794.2 -20.0 2600.0 -1.8 0.0 -3.2 0.0 -2.0 The corresponding eigenvalues become -2.5+J3.5, -4.6, -6.3+J11.9 Thus the system is stabilized. Next the system is expanded by doubling its capacity. With the base MVA of the synchronous machine doubled, the p.u.. ac line and local load data become r -0.068 x^ 1.994 0.1245 B 0.131 The new dc line and harmonic filters' data are R 0.066 x 0.2 G. 0.22xl0~3 ' B, -0.14 co f f It happened that the reactive power can be sufficiently provided by the harmonic filters and no special condenser for power factor correc-tion is needed. The system's new operating point is given as v„ 1.05 v 1.02 P 0.9765 Q 0.0622 t o P 0.504 Q 0.187 P. 0.335 Q, 0.135 ac ^ac dc dc 6 81.15° A 15° v , 0.449 v 0.949 o d q (4.16) 49 1.233 E- 1.388 V ref 1.061 V 1.61 R VT -1.57 cosa_ 0.962 cosaT -0.868 I R 0.312 For the given data the system's linearized equations are A6 0.0 Aw Aifi fd AE A cos a R Acosa, u X 1.0 0.0 0.0 0.0 0.0 0.0 0.0 -119.4 0.0 -173.3 -120.3 0.0 0.0 -0.38 0.15 -0.35 -0.33 0.0 -0.04 183.0 0.0 -1860.0 -20.0 -443.8 134.0 2600.0 2171.1 0.0 -22068.0 0.0 -40020.0 -32172.4 0.0 -17.7 0.0 . 179.5 0.03 0.0 -9.55 0.0 318.2 0.0 -13.9 -105.4 -9.62 [A6 Aa> AiJ;,., AE Acosa„ Acosa,. fd x R I u ] x 0.0 -2.0 (4.17) and the corresponding eigenvalues are -0.87 +j2.4, -4.76, -7.85 + j l l , -363.6, -39762.6 The system is s t i l l stable. However, because of the new situation, the excitation control signal originally designed for the ac system is modified to become 0.09 S Ux " 1 + S Aw (4.18) and the system's eigenvalues for this case are -1.94+J2.04, -2.06, -7.63+J10.8, ~363.6, -39762.6 The system is more stable with the new adjustment. In addition to the above two cases where the system is stabilized by excitation control signals u x and u^ respectively, three new controls are designed and compared with old schemes J 50 1. No modification of the excitation control u and no optimal control x signals. 2. With only the modified excitation control signal u^. 3. An optimal dc current control is designed with excitation control removed. 4. A dc control u^ is designed in conjunction with u^. 5. A dc control is designed in conjunction with the modified exci-tation control u'. x The system's eigenvalues for the last three cases are listed in Table V. Table V Eigenvalues for expanded system Control Eigenvalues -0.93, -38 .5 -10.75 + J13.1 -379, -41071.2 u and u^ x D -1 .06 + jl.5 -55.35 + J66.7 -0.8 -384.5 , -41071.2 u' and u^ x D -0.85 + jl.01 -58.4 + j71 -0.667 -385.2 , -41071.2 The control laws for different cases are given by " V " 2.05 1.13 -13.2 -0.045 -0.133 -0.27 0.0 " ux and u^ UD = 1.37 0.92 -7.73 0.75 -0.14 0.88 19.75 ux a n d "D _UD_ 1.52 0.99 -9.3 0.76 -0.14 0.9 18.8 [A6 Au) Ail),., AE A C O S O L AcosaT u or u'] (4.19) fd x R I x x The nonlinear test of section 4.3 with the same disturbance is applied to this system but with a fault duration reduced to 0.05 sec. and line reclosure at 0.25 sec. after the fault occurrence. The nonlinear system response is summarized in Figures 4.11 to 4.16. Figure 4.11 in-dicates that in the first two cases, where u and u' respectively are applied, the system is unstable despite the fact that the corresponding 51 eigenvalues are stable. This clearly demonstrates the necessity of nonlinear tests and that eigenvalue analysis, although necessary, is not sufficient. Although the dc control signal u^ is more effective than the excitation controls, i t is not sufficient by itself to maintain stability. The system's response with combined excitation and dc controls is much better and the two responses with u or u' controls are very x x close whether the excitation control signal is modified or not. The terminal voltage variations, Figure 4.13, are more pronounced in cases 1 and 2 and the voltage oscillation continues after the line is restored whereas the terminal voltage approaches its steady state value asymp-totically after 0.35 sec. in cases 3,4 and 5. Figures 4.14 and 4.15 show the variation-of transmitted power over the dc and ac lines res-pectively. In cases 1-and 2 the dc link is operating under constant current control and the dc power is nearly constant about, the original value, 0.335 p.u. while the ac power is changing widely after line reclosure. In cases 4 and 5 the dc line picks up the power lost by the ac line and returns to rated value when the line is reclosed. In the mean time the ac line is transmitting the rated power. A similar behav-iour is noted in case 3 up to a point where the ac power starts to drop again and the dc line picks up the difference. The dc control effort in cases 4 and 5 is smaller than case 3 as shown in Figure 4.16. The control stays at limit values, + 0.25, for about 0.3 sec. then decreases rapidly and approaches zero asymptotically. It is concluded from the above discussion that a combined dc and excitation control scheme is the best for the dc/ac parallel system with slightly better results for u^ and than and u^ controls. S RAD < i i l i i i i i i i L. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 t S E C Fig. 4.11 Swing Curves for Expanded System t S E C Fig. 4.12 Speed Deviation for Expanded System 53 0] 02 03 t l 05 OS 07~~ ts 05 Fig. 4.14 Variation of DC Power for Expanded System 54 55 5. MULTITERMINAL HVDC SYSTEMS Al l hvdc systems designed so far have been restricted to the two-terminal operation in which power is transmitted over an hvdc link from system A to system B. However, hvdc need not be so limited and can be readily extended to the multi-terminal operation. In this and the next Chapter the operation and control of a three terminal dc link in conjunction with a delta connected ac system are studied. Two operation modes are considered^ two rectifiers and one inverter or one rectifier and two inverters. The ac system is either a two-machine in-finite system or a three-machine system. The equations describing system dynamics and the determination of the operating point are given in this chapter. 5.1 Ope-rati-on.antd-Conitrnl<,sf ;Multl-fce-rminal..-DC.Lines As partly explained in Chapter 2 the normal operation mode of a two-terminal dc line is to control the currents of both the rec-t i f i e r and inverter with identical current orders but giving the inver-ter a current margin signal of a polarity opposite to the current order. The inverter is normally operating on constant extinction angle control which maintains a minimum safe angle of advance and a maximum dc line voltage while the line current is regulated by the delay angle a R. The current control will be taken over by the inverter only i f the rectifier is unable to provide the ordered current at its minimum delay angle. The same principles of operation can be adopted without change for a tapped dc line with two rectifiers and one inverter. The inverter current order is the algebraic sum of the two rectifiers' current orders plus the current margin. However, the same policy can not be applied 56 to the case of one rectifier and two inverters. If both inverters were on constant extinction angle control there would only be one ratio of power flow between the two receiving stations. For this reason one of the inverters and the rectifier are given specific current orders 37 while the second inverter regulates the dc line voltage .: The difference between these two current orders minus the current margin is applied to the second inverter's current override control. Changing the current order of the first inverter without changing the rectifier' s current order would result in current transfer between the two inverters. An alter-40 native control policy was also proposed where constant extinction angle control is applied at more than one inverter. It is argued that the first control method has the disadvantage that reactive power com-pensation must be provided for constant current operation at a l l but one of the converter stations whereas the second method improves this situation. In the second method the desired power flow can be achieved by adjusting the ac voltages at the inverter stations by means of tap changer control. In this chapter the first control method is adopted. 5.2 System Equations The system to be studied is given in Figure 5.1, consisting of a tapped dc line superimposed on a delta connected ac network. Bus number 3 is an infinite system which will be replaced by a small gen-erator for later studies. There are in general three converter stations where converter station 1 is always operated as a rectifier and that of station 2 always as an inverter. As for converter station 3 i t is operated either as a rectifier or as an inverter depending on which cases are studied. The system is modelled in general as follows; the Fig. 5.2 TRANSFORMATION BETWEEN COMMON AND INDIVIDUAL MACHINE FRAMES 58 synchronous machines are represented by third order models with single time lag voltage regulators. Converter controls are also represented by single time lags. The transmission network is represented by static impedances and the ac harmonic filters are not considered but the con-densers supplying the required reactive power are included. The system state equations are given by p<S. = Aw. , pAu). = - r r ~ ( P . - P .) J 3 3 2Hj mj ej E . ^ rfdj e fdj v x a d j fdj' PE . = [K . (V ,. - v .) - E .] (5.1) v xj T rj refj tj xj v P C O S C L , . = — ^ (I ,. - I„.) v Rj T„.v . v refj Rj' Rj tj 1 1 2 pcosaT. = — — T-cosaT. - cos A . H (— x . I T . •+- e T.)] V X J - 13 oj v t j ^ coj Ij c l j ' j where e _. = K (k I - I ) d j Ij J refj I j y k. = 1 for constant current control (5.2) < 1 for constant extinction angle control where the subscript j indicates the unit number. For a multi-machine power system study a common reference 53 frame (D,Q) must be chosen . This frame is rotating at the synchronous speed. The individual machine angle 6_. is defined as the angle between the quadrature axis q^  of the individual machine with respect to the quadrature axis Q of the common frame. The transformation between the D,Q and the individual d^, q^ frames is given by 59 F . ~ Dj F . L*Qj J cos6. -sinS. J 3 sin<5. cos6 . J 3 -1 f . i - q j -(5.3) f . <- q j -> cos6 . sincS . 3 3 -sinS. cos6. 3 3 • Dj F " L Qj J where f can be either voltage or current. The synchronous machine quantities referred to individual dq frames are given by P = ip,. 1 . - ib •. 1 ej dj q j q j d j v 2 2 4- 2 v . = V,. + V . t j d j q j " c i j C2j C3j •qj - C 2 j C4j _ C 5 j 1dj " C6j " C7j C8j i . q j C9j C I O J _ _ c i o j " C8j C l l j -vdj V . • - q j L*fdj. (5.4) where coefficients c^. to c^.^ are given in equation (3.28) Next, the. local loads and capacitor currents are given by "d£j q£j G£j " B£j Z-3 13 v dj v . q j (5.5) "cdj c q j -v q j 'dj (5.6) and the currents transmitted over line £ connecting busses j and k are 60 given in DQ coordinates by & l l a2l a3l ~ahl ~a2% al£ a4£ a3l XDk£ a3£ ~aH au a2l a H a3£ -a2i a u Dj v_. • Qj VDk (5.7) where to a^  are the transmission line coefficients given in equation (3.7). Substituting (5 .3) into (5.7) yields a2l (a3*COSV+ ( a3£ S l n 6jk-a 4 J t s l n 6 j k ) a 4 £ C O s 6 j k ) ~a2l au - ( a3£ s i n 6jk- ( a 3 £ C O s 6 j k + sa4ros<S.k) a4£ S l n 6 jk ) • Xdk£ ( a3 £ C O s 6 j k ~ - ( a 3 ^ S i n 6 j k + a u a2£ a4£sinSk) a 4 £cos6. k) Xqk£ ( a 3 £ s i n 5 j k + ( a3£ C O s 5jk" ~a2l a u a4£COS<V [v,. V . V „ v , l * " dj qj dk qkJ ( 5 . 8 ) The dc voltages and currents are determined from r v 3=1 J k=l vVRJ = v t j c o s a R j " Koj hi J = ^ 2> 3/3 w 3 T ——• v . cosa x . I T . IT tj Ij TT COJ Ij j = 1, 2, v 61 V_. = V, + 2R. I_ . j = 1, 2, r j = 1, 2, v (5.9) The formulation is rather in general where r is the number of rectifiers, V_. = -V, + 2R. I_. Ij b ] I] v is the number of inverters, and is the voltage at the tapping point. For the three-terminal system with rectifier stations 1 and 3, and inverter station 2, the solution o f (5.9) becomes _ V R l " _ d l ~d2 d3. _ V t l cosa R 1 VI2 "d4 d5 "d6 • Vt2 cosa^2 VR3 d7 "d8 d9 _ Vt3 cosa R 3 XR1 d10 d l l "d12 II2 d l l d13 d14 TR3 "d12 d14 d15 (5.10) On the other hand for rectifier station 1 and inverter stations 2 and 3, the solution becomes VR1 VI2 VI3 = *R1 XI2 XI3 d l "d2 -d. d10 d n d l l d13 d12 _ d14 12 -d 14 15 v t l cosa R 1 v t 2 cosa I 2 V t 3 C O S a I 3 (5.11) where 62 10 d l l d12 d13 d U d15. and R^ 3^3 2R. + 3 RE2 ^3 + 2 R1 (RE2 + *E3> 3 / l T Xcol ^3 3 / 7 r Xcol *E2 3 / l T Xco2 ^3 *E1 *E3 + 2 R2 ( RE1 + 3 h Xco2 *E1 . 3 / * Xco3 ^2 3 / l T Xco3 *E1 ^1 ^2 + 2 R3 ( RE1 + RE2 ) ^2 + ^3 ^3 ^2 hi + ^3 hi hl + hl' j = 1, 2, 3 3 — x IT C O J (5.12) h = *E1 (RE2 + V + ^2 ^3 Finally the current components referred to individual machine frame are given by dj 2/3 T IT dcj sin6 R. cos5R. cos6R. -Sin<5R. cos<pRj j = 1, 2, 3 S i n*Rj (5.13) 63 where§_ . = arc tan (v,./v .) Rj dj qj , TT d C j COSCJ) . = >>-R J 3/3 \ j sin* R. = + W - cos2<j> Rj Of sin $ the positive sign is used for rectifiers and the negative sign Rj for inverters. The terminal voltage components v,. and v . are deter-dj qj mined from i , . = i,„ . + i ,. + V i.,„ + I,, dj d£j cdj L djil dj (5.14) i . = i . . + i .+ I i . + I . qj qAj cqj J qjil qj 5.3 Initial Conditions Given the terminal voltages vtj> j = 1> 2, 3, the generator output powers P^  and 7^, the dc power ^ a n d the minimum extinction angles A^ for inverters the system's operating point is determined as follows: For one rectifier and two inverters operation equation (5.1) gives 2 Xco2 cosa T O = -cosA + — — I 0 (5.15) 12 02 fi Vt2 1 2 cosa = -cosA + — — - — I (5.16) 13 o3 fi v t 3 13 For two rectifiers and one inverter operation, where the rectifier's current ratio is given as x, equation (5.16) is replaced by I R 1 = x I R 3 (5.17) Let the dc power be given as P d c l = -3 VR1 TR1 ( 5 ' 1 8 ) 64 Then the dc voltages, currents and firing angles can be determined by substituting equations (5.15), (5.16) or (5.17) and (5.18) into (5.10) or (5.11). Substituting the results into (5.13) the dc current compo-nents I,, and I . are given in terms of v,. and v .. dj qj dj qj Choosing the common reference frame (D,Q) to coincide with (d^jq^) the angle §^ is equal to zero. From the given powers P^  and P~ one has z 2 P. + c, . v t , + (c 7.-c n.) v,. v . » f - 3 , 6 J t J + Z J - ^ ^-iUL j = l , 2 (5.19) f d j ( C8J Vdj + C10j V Since v,. can be expressed in terms of v . and the terminal voltages dj qj v , substituting (5.4), (5.5), (5.6), (5.8), (5.13) and (5.19) into tj (5.15) v ., j = 1, 2, 3, 6..> S„ and can be solved, qj 1 2 r f d3 5.4 Numerical Example The system studied is basically the same for a l l four cases. The only differences are the operating points and loading conditions. Synchronous machines 1 and 2 are identical of the same capa-city, synchronous machine 3 is a smaller machine which has a capacity of only one third of either of the two. The machines have the following per unit data x" = (0.09, 0.09, 0.54) p.u. x, = (1.28, 1.28, 4.5) p.u. d a x^ = (0.13, 0.13, 0.6) p.u. x q = (0.8, 0.8, 3.8) p.u. : 'x' = (0.18, 0.18, 0.69) p.u. x„ = (0.1, 0.1, 0.35) p.u. T d o = (2.5, 2.5, 2.3) sec. T Q^=(0.018, 0.018, 0.0192) sec T||o = (0.011, 0.011, 0.0117) sec. H = (4.0, 4.0, 1.33) sec. r = (0.0097, 0.0097, 0.1008) p.u. 3. 65 The ac transmission lines have the following constants r = (0.0364, 0.03185, 0.0455) p.u. x £ = (0.905, 0.79, 1.13) p.u. x £ = (16.05, 18.35, 12.85) p.u. Al l dc lines are assumed of equal length and have the same parameters R = 0.0432 p.u. and X = 64.2 p.u. c The commutation reactances are X^Q = (0.3, 0.3, 0.21) p.u. The voltage regulators are assumed identical and also the converter control parameters are assumed similar: K = 20, T = 2 sec, K = 30, T„ = 0.002778 sec, k = 0.9 r r R R The local loads and shunt capacitors for two rectifiers - one inverter case are given by G£ = (0.145, 1.9, 0.17)p.u. B £ = -(0.09, 0.04, 0.02) p.u. i - = (0.088, 0.0, 0.0795) c The operating conditions for the two rectifiers and one inverter system are given by vfc = (1.03, 1.0, 1.02) p.u. P = (0.96, 0.975, 0.325) p.u. Q = (0.213, 0.323, 0.033) p.u. P d c = (0.197, -0.371, 0.186) p.u. Q = (0.093, -0.178, 0.083) p.u. P = (0.393, -0.173, -0.212) p.u. Q C 3. C Q = (0.114, -0.011, 0.011) p.u. 6 = (0.21, -0.515, 0.0) rad. <5_ = (0.553, 0.55, 0.799) rad. i b c , = (1.121, 1.119, 1.026) p.u. K id ipd = (0.883, 0.859, 0.731) p.u. \p = -(0.548, 0.53, 0.756) p.u. v d = (0.541, 0.522, 0.731) p.u. v q = (0.876, 0.853, 0.711) p.u. E = (1.735, 1.864, 1.861) p.u. V , = (1.117, 1.093, 1.113) p.u. P m = (0.969, 0.985, 0.335) p.u. cosct = (0.936, -0.837, 0.933) V = (1.54,-1.49, 1.54) p.u. I J = (0.192, 0.373, 0.181) oc dc 66 For the one rectifier and two inverters system the local loads and shunt capacitors are given by = (0.145, 1.55, 0.5) p.u. = -(0.09, 0.7, 0.2) p.u. =i = (0.13, 0.0,. 0.0) p.u. c and the other operating conditions are v = (1.04, 1.01, 1.0) p.u. P = (0.96, 0.975, 0.322) p.u. Q = (0.134, 0.678, 0.08) p.u. P d c = (0.371, -0.184, -0.177) p.u. Q, = (0.141, -0.069, -0.061) p.u. P = (0.292, -0.136, -0.139) p.u. dc ac Q = (0.046, -0.002, -0.033) p.u. 6 = (0.124, -0.492, 0.0) rad. 6 R = (0.569, 0.457, 0.738) rad. tyfd = (1.109, 1.227, 1.076) p.u. ij>d = (0.883, 0.912, 0.759) p.u. $ = -(0.566, 0.456, 0.7) p.u. v d = (0.561, 0.446, 0.672) p.u. v q = (0.876, 0.906, 0.74) p.u. E_ = (1.659, 2.229, 2.002) p.u. V r g f = (1.123, 1.121, 1.1) p.u. x P = (0.968, 0.988, 0.333) p.u. cosa = (0.9925, -0.9054, -0.9248) m V d c = (1.608, -1.563, -1.564) p.u. I d c = (0.346, 0.1765, 0.1695) p.u. 67 6. LINEAR OPTIMAL CONTROL FOR MULTITERMINAL DC/AC SYSTEMS In this chapter linear optimal control signals are designed for the three-terminal system described in Chapter 5. Two systems are investigated; a two-machine infinite system, and a three-machine system. For each system two modes of operation are studied; two rectifiers and one inverter, and one rectifier and two inverters. Four control schemes are tested for each case. Firstly, optimal excitation control signals, Ug's, are applied to a l l machines. Secondly, optimal dc current control signals, u^'s, are applied to converters regulating dc current. Thirdly, an excitation control signal is applied to the machine connected to the converter station operating on constant extinction angle control and dc current control signals are applied to the rest of the converter stations. Finally a l l excitation and dc current control signals are applied simul-taneously. In designing the linear optimal regulator a performance func-52 tion of the quadratic form must be chosen. Moussa and Yu developed a method for determining the Q matrix with the dominant eigenvalue shift of the closed loop system. This technique was also applied to multi-53 machine stabilization studies . It was found from their results that weights must be given to Au and A<S in order to obtain a very stable system, a limited number of diagonal matrices are tested with varying emphasis on the deviation of rotor angles <5, that of speed Au, and the field flux linkages i>£^' The reason for giving weight on ty^^ is mainly be-cause of the dc lines involved and it was not necessary for ac lines alone. As for the weighing matrix R i t is always taken as a unit matrix. 68 6.1 Two Machine-Infinite Bus System with Two Rectifiers and One Inverter Converter stations 1 and 3 are operated as rectifiers while converter station 2 is operated as an inverter in this case. For the data given in Chapter 5 the linearized state equations for the system are given in eqn. (6.1). The diagonal matrices Q and the nonzero elements of matrices B for different control schemes are given by u E 1,u E 2 ; Q = diag {10, 100, 1, 1, 1, 10, 100, 1, 1, 1, 1} b^ 1 = 10.0, bg 2 = 1 0 , 0 U j ^ . u ^ . Q = diag {10, 1, 100, 1, 1, 10, 1, 100, 1, 1, l ) b_ . = 10485.44, b . . _ = 10588.23 n. D , 1 11, I (.0 . 1 ) . Q = diag {10, 1, 100, 1, 1, 10, 1, 100, 1, 1, 1> b c , = 10485.44, b Q . = 10.0, b . . . = 10588.23 -> > -L " » ^ J . J . , J u E 1,u D l,u E 2,u D 3. Q = diag {10, 1, 10, 1, 1, 10, 1, 10, 1, 1, 1> b. . = 10.0, bK _ = 10485.44, b Q = 10.0, b . . , = 10588.23 H , i D , z y , J i ^ - > ^ and the corresponding control laws are given in eqn. (6.3). The sys-tem's eigenvalues are listed in Table VI. A three-phase ground fault at the middle of one circuit of ac line 1 is used as a disturbance for the nonlinear tests to compare the different control schemes designed. The fault duration is 0.07 sec. followed by isolating the faulted line from both ends. The system responses are shown in Figures 6.1 to 6.7 for the following cases case 0 case 1 case 2 no optimal control whatsoever. optimal excitation controls for both machines. optimal dc controls for rectifier stations 1 and 3. A6\ Aip fdl A * x l • Acosa AS, Aw, Rl Aip fd2 A Ex2 Acosa * Acosa 12 R3 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -26.0 0.0 -114.0 0.0 -123.4 7.225 0.0 -0.72 0.0 -38.8 -0.35 0.0 -1.5 0.45 -1.26 0.142 0.0 0.145 0.0 -0.393 0.405 0.0 -6.58 -0.5 -0.434 -0.369 0.0 -0.87 0.0 -0.145 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 20.98 0.0 22.42 0.0 17.07 -53.9 0.0 -151.4 0.0 43.89 0.11 0.0 0.388 0.0 0.409 0.617 0.0 -0.834 0.0 -1.26 -1.099 0.0 -6.55 -0.5 -2.89 0.0 63.6 0.82 -0.769 501.6 0.0 -18991.0 0.0 -33213.0 177'.0 0.0 5075.5 0.0 -11125.0 16942.0 0.0 68.0 -0.372 0.0 -1.902 0.447 0.988 0.81 -0.501 16.43 0.0 72.54 0.0 133.0 -34.8 0.0 -228.0 0.0 -78.9 -6.43 1657.4 0.0 12689.0 0.0 21752.4 2339.4 0.0 11649.4 0.0 -13772.0 -35626.0 •[A6, Ato, Aip.,, AE , Acosa,,, A6„ Au. Aij i - , - AE „ AcosaT„ Acosa_~] + BU 1 1 fdl xl Rl 2 2 fd2 x2 12 RJ (6. "El -38.4 6.37 -60.4 -2.48 -0.023 21.4 -1.01 -3.53 -0.057 0.285 -0.003 "E2 5.66 3.38 -3.53 -0.057 -0.004 -40.6 6.84 -88.4 -2.93 -1.04 0.015 UDl 3.1 1.01 -0.98 0.75 -0.22 -1.69 -0.22 -5.0 0.75 • -0.101 -0.108 UD3 -0.39 -0.35 0.49 0.79 -0.11 -2.58 -0.97 -1.02 0.67 0.15 -0.195 UD1 2.63 1.05 -3.3 0.41 -0.22 -1.09 -0.26 -2.41 -0.04 -0.086 -0.108 "E2 0.51 0.02 -1.42 0.51 -0.4x10" -4 -0.77 0.12 -9.81 -1.32 -0.004 -0.3x10" •5 "D3 -0.76 -0.31 -1.4 0.53 -0.109 -2.11 -0.998 0.93 -0.003 0.16 -0.195 "El -0.06 0.03 -2.16 -1.04 -0.8x10" -5 -0.005 -0.003 -0.04 -0.005 -0.002 0.1x10" •4 "Dl 2.4 1.02 -1.3 -0.009 -0.22 -0.85 -0.27' -1.12 -0.013 -0.087 -0.108 "E2 0.16 0.016 -0.25 -0.005 -0.1x10" -4 -0.6 0.034 -3.83 -1.11 0.004 0.1x10" -4 -0.93 -0.33 0.037 0.012 -0.11 -1.94 -1.01 2.06 0.01 0.16 -0.19 . [A6 Ao) Aif/ AE . Acosa A6 Au). Aip AE Acosa 0 Acosa Q ] t (6.3) Table VI Eigenvalues for 2 Machine-Infinite Bus, 2 Rectifiers System Control used Eigenvalues no control -53657.4, -15084.8, -177.3, -0.605+jl.35 , -0. 342+J4.77, -0.293+j6.81, -0.261+J1.83 UE1 ' "E2 -53657.4, -15084.8, -177.3, -15.74,-13.5, -7.45+J12.9, -6.16+jlO.74, -0.35, -0.32 "DI ' UD3 -54675, -18423, -162.1, -39.8, -22.95 ,-3.36, -3.16 , -1.17y2.1, -0.5+J1.58 "DI ' "E2 ' "D3 -54675, -18423, -162 , -39.83,-22.96 -6.98,-5.95, -4.39, -3.37, -0.717 + jl.7 " E I » "DI ' "E2 » "D3 -54675, -18423, -162, -39.44,-22.8,-9.6,-9.2, -3.76,-3.4, -2.39,-1.81 72 case 3: case 2 plus excitation control for the machine connected to inverter station 2. case 4: case 1 plus case 2. Figures 6.1 and 6.2 show that machine 1 loses stability but machine 2 remains stable when there is no optimal control signals, case 0. Next when excitation control signals are applied to both mach-ines the stability of machine 1 is getting worse and machine 2 that was originally stable goes out of step, case 1. S t i l l next where optimal dc .controls are applied to rectifiers 1 and 3 the system is stable, case 2. Finally the system is stable and the responses are very close for both cases 3 and 4. The angle and speed deviations are less in case 2 than cases 3.and 4. The changes in terminal voltages are shown in Figures 6.3 and 6.4. It is noted" that' at ac bus- I "the deviations are small for cases 2 and 3 with larger changes for cases 4, 0, and 1 respectively. For machine 2 the terminal voltage deviations, Figure 6.4, are smallest for case 4 followed by cases 2, 1, 3 and 0 respectively. Figures 6.5 (a, b, c) show the changes in power transmitted over the dc transmission network. Consider case 0 fi r s t . The dc power drops for the first 0.07 sec. during the fault because of ac terminal voltages drops, and then returns to rated values. After 0.69 sec. the power carried by dc line 1 drops sharply for 0.1 sec, between approxi-mately 0.69 and 0.79 sec, then continues to drop at a slower rate because of the deterioration of v ^ while the power carried by line 3 increases in a similar fashion because of drops in v 2 with vfc^ remain-ing constant. But the total dc power is decreased. Consider case 1 next. The same pattern is noticed but with sharper drops in P , .. 73 —« 1 1 1 I I I I , 01 0.2 03 Q.4 05 05 07 OS 09 1.0 t sec Fig. 6.4 Terminal Voltage Variations at Bus 2 for 2 Machine, 2 Rectifier System 76 Fig. 6.7 Excitation Control Effort 'for 2 Machine, 2 Rectifier System 77 because of larger drops in v^. For cases. 2, 3 and 4 the dc reference currents are modified by optimal control signals and therefore the total power transmitted over the dc network is significantly increased to absorb some power from the ac system, which may cause large acceleration of the synchronous machines, and consequently resulting in a more stable system. The control efforts are shown in Figures 6.6 and 6.7. It is interesting to notice that the dc control effort at rectifier station 1, IL^> is larger in case 4 than that of cases 2 and 3. This suggests that the presence of an excitation control signal at machine 1 is work-ing against the dc control at the same bus. This may be explained by the fact that the voltage drop at bus 1 is larger in case 4 than cases 2 and 3. It is also noticed that in general the dc control efforts at rectifier station 1 are much larger than the dc control efforts at rectifier station 3 since v ^ is constant and machine 1 is also closer to the fault location. The excitation control efforts are greatly reduced in the presence of dc controls. 6.2 Two Machine-Infinite Bus System with One Rectifier and Two Inverters The study in this section is similar to that of section 6.1 except that converter station 3 is now operated as an inverter instead of a rectifier. So the only change in the cases listed before will be the fact that station 3 is an inverter. The operating point was given in Chapter 5 and the linearized system state equations are A 6 Ai|> f d l A E x l Acosa A 6 , Aa), R l Aifi fd2 AE x2 Acosa • Acosa 12 13. 0.0 1.0 -33 0.0 -0.38 0.0 0.056 0.0 -286 0.0 0.0 0.0 19.4 0.0 0.09 0.0 0.37 0.0 12.6 0.0 726 0.0 0.0 -117.6 -1.58 -6.51 -19722 0.0 22.1 0.36 -0.8 77.9 -13434 0.0 0.0 0.46 -0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -140.6 -1.36 -1.33 -36163 0.0 20.1 0.41 -1.14 152.4 -23864 0.0 0.0 11.3 0.0 0.16 0.0 0.2 0.0 291 0.0 0.0 1.0 50 0.0 0.28 0.0 0.74 0.0 23.9 0.0 1633 0.0 0.0 0.042 0.17 -1.01 4652 0.0 -139 -1.78 -6.24 -212.3 -12456 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.45 -0.5 0.0 0.0 0.0 -31.5 -0.38 0.14 -11241 0.0 . 0.7 1.01 -1.91 -46.8 15930 0.0 -62.3 -0.84 0.82 -16153 0.0 -67.9 -0.7 0.49' 4.45 -35443 A 6 . A I D , A^  fdl AE: , xl Acosa. A 6 R Aw, Rl Alp fd2 A Ex2 Acosa Acosa + BU 12 I3-1 (6 4) oo 79 The weighing matrices Q and the corresponding nonzero elements of matrices B are V L ^ and ^ ; Q » diag. {100, 10, 1, 1, 1, 100, 10, 1, 1, 1, 1} b 4 , l = 1 0 ' b9,2 = 1 0 and u D 3 . Q = diag. {100, 10, 1000, 1, 1, 100, 10, 1000, 1, 1, 1} b. . = 10383.8, b.. 0 = 10799.1 ->, J - 1-L, 2. Qo. D) ^ , 1 ^ 2 and . Q = diag. {100, 10, 1, 1, 1, 100, 10, 1, 1, 1, 1} b 5 ± = 10383.8, b 9 2 = 10, b u 3 = 10799.1 "El* "Dl' UE2 a n d UD3- Q = d l a g ' 1 > 1 0 ' l j 1' 1 0 ' 1» 1 0 ' 1' 1 } b. ' = 10, b_ „ = 10383.8, b. . = 10, b.. . = 10799.1 H , J - D , y , J ±±,4 The control laws are given in equation (6.6) and the system's eigen-values are listed in Table VII. The same disturbance described in section 6.1 is applied to this system for the nonlinear tests. The system responses are summari-zed in Figures 6.8 to 6.14. Figures 6.8 and 6.9 show the angle and speed deviations res-pectively. The system is more stable without optimal control, case 0, than with excitation control, case 1. Cases 2 and 3 give the smallest angular deviations and i t is also close for case 4. The terminal voltage variations at bus 1, Figure 6.10, are largest in case 1 followed by cases 4, 0, 2 and 3 in that order. At bus 2 a l l the voltage deviations are small. In Figures 6.12 (a, b, c) the variations of power transmitted over the dc network are shown. In cases 0 and 1 with the dc network operating on constant current control the dc power goes back to original values immediately after the faulted line is isolated and then decreases " E l -13 1.6 -25.6 -1.75 -0.004 12.15 -0.66 -12 -0.34 0.52 0.0013 ~ "E2 -4.4 2.74 -14.7 -0.34 -0.01 -12 1.11 -23.3 -1.67 -0.36 -0.0016 UD1 10.8 3.1 -0.63 1.99 -0.22 -3.25 -0.25 -11.6 0.27 -0.12 0.102 " D3 0.17 0.57 1.28 -2.81 0.11 10.4 3.1 7.76 -0.49 -0.93 -0.2 UD1 9.04 3.21 -1.39 0.06 -0.21 -1.17 -0.45 -0.72 0.0003 -0.025 0.102 "E2 0.21 0.02 -1.08 0.21 0.3xl0~6 -0.41 0.01 -1.3 -1.0 0.0026 -O.SxlO - 5 UD3 = 1.73 0.48 0.42 -0.08 0.11 8.48 3.24 -1.75 -0.009 -1.0 -0.2 " E l -0.15 0.03 -2.16 -1.05 -0.5xl0~6 0.18 -0.003 -0.006 -0.005 -0.005 -0.15xl0~ 4 "Dl 2.43 1.06 -1.07 -0.0005 -0.21 -0.08 -0.04 -1.01 0.005 -0.12 0.1 "E2 0.09 0.009 -0.24 -0.005 0.5xl0~5 -0.2 0.01 -1.32 -1.01 -0.0014 -0.2xl0~ 4 0.52 0.07 0.49 -0.02 0.11 1.94 1.07 -0.72 -0.02 -0.48 -0.19 •[A61 A W 1 A * f d l AEx^ Acosa -KJL A6 2 Ao 2 A*fd2 A E x 2 Acosaj 2 Acosa.^] t (6.6) co o Table VII Eigenvalues for 2 Machine-Infinite Bus, 2 Inverters System Control used Eigenvalues no control -55443, -16056, -155, -0.785+j1.49, -0.405+J1.82, -0.201+J4.76, -0.168+J6.52 u E l a n d "E2 -55443, -16056, -155, -10.75,-9.91, -3.66+J7.91, -2.92, 2.15+J6.23, -2.11 UD1 a n d "D3 -56428, -19286, -191, -94.45,-48.4, -3.16, -1.82 +j2.2, -1.81, -0.78 + jl.41 UD1'UE2&UD3 -56428, -19286, -191, -93.9,-46.8, -9.64, -3.26, -3.18, -1.59, -0.874+J1.68 UE1'UD1>UE2 and u D 3 -56428, -19286, -149, -32.3, -14.38, -9.55, -9.49, -4.18, -3.39, -2.66, -2.17 82 t sec Fig. 6.8 Swing Curves for 2 Machine, 2 Inverter System Au rad/sec '3 t sec J.O 0.6^  ^ ^ ^ I r r . Fig. 6.9 Speed Deviation for 2 Machine, 2 Inverter System 7.7 7.05 7-0 0.35 .9 0.85 0.8 v., A " 83 2 0 . . • •• *• 0.0 0.1 02 0.3 04 0-5 06 07 0-8 0.9 LO t sec Fig. 6.10 Terminal Voltage Variations at Bus 1 for 2 Machine, 2 Inverter Svstem 7.7 7.05 7.0 0-95 0.90 0.85 0.80 vt2 A"-0-7 0-2 03 0.4 05 0.6 0.7 08 0.9 7.0 t sec Fig. 6.11 Terminal Voltage Variations at Bus 2 for 2 Machine, 2 Inverter System I I I I I I I I I I l_ 0.0 0-7 0.2 0-3 0.4 0.5 0-6 0.7 0-8 0.9 1.0 0.0 0.7 0.2 0.3 0-4 0.5 0.6 0.7 0-8 0.9 10 (c) " f sec Fig. 6.12 DC Power Variations for 2 Machine. 2 Inverter Svsfem 0.15 0.05 -0.05 -0.1 -0.151 CASE 2 CASE 3 x xCASE 4 Fig. 6.13 DC Control Effort for 2 Machine, 2 Inverter System uF p.u. 0.151 0.05 -0.05 -0.151 Fig. 6.14 Excitation Control Effort for 2 Machine, 2 Inverter System CASE 1 CASE 3 x xCASE 4 86 because of the ac voltage drop at the rectifier bus. In cases 2, 3 and 4 where the dc reference currents are modified by optimal control sig-nals, the rectifier power is increased even with the fault on. The power is even increased further after the fault is removed and stays almost constant at about 0.48 p.u. then starts to go down at 0.7 sec. The control efforts are shown in Figures 6.13 and 6.14. Again i t is seen that the excitation control efforts, Figure 6.14, are greatly reduced in the presence of dc controls. It is also noticed that the dc control at converter station 3 jumps to high positive values for the fault duration and then changes sign after the faulted line is iso-lated. The reason for this is that during the fault inverter 3 draws more dc power since inverter 2 is incapable of absorbing this power due to the ac voltage drop.' After the faulted line is removed inverter 2 starts to pick up more dc power to compensate for the ac line power loss. 6.3 Three Machine System with Two Rectifiers and One Inverter The system studied here is the same as that of section 6.1 except that the infinite bus 3 is replaced by a small synchronous machine having one third the rating of either machines 1 or 2. The sys-tem's operating point is the same as section 6.1. The system's linearized state equations are given in eqn. (6.7). A l l 0.0 1.0 0.0 0.0 0.0 0.0 0.0 ,0.0 Au^ -23.2 0.0 -99 0.0 -105 13.4 0.0 7.85 A * f d l -0.36 0.0 -1.23 0.45 -0.9 0.24 0.0 0.35 0.63 0.0 -7.19 -0-5 -1.28 -0.52 0.0 -1.5 Acosa^ -14690 0.0 -14935 0.0 -27366 1069 0.0 9644 ti2 tm 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 27.6 0.0 44.4 0.0 43.9 -44.1 0.0 -141 A*fd2 0.11 0.0 0.7 0.0 0.82 -0.26 0.0 -1.67 A*x2 1.03 0.0 -1.28 0.0 -1.95 -1.14 0.0 -7.21 Acosaj2 2.0 0.0 129 0.0 211 -18.45 0.0 -175 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Au^ 49.6 0.0 42.8 0.0 42.3 29.8 0.0 -9.89 A*fd3 0.52 0.0 0.97 0.0 1.12 0.5 0.0 0.27 A*x3 1.27 0.0 -3.13 0.0 -4.45 -0.76 0.0 -3.33 Acoaa^ 1718 0.0 3593 0.0 8926 36.9 0.0 2264 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -52.6 9.77 0.0 10.7 0.0 31.2 0.0 -0.57 0.125 0.0 0.21 0.0 0.35 0.0 0.03 -0.11 0.0 -0.55 0.0 -0.1 0.0 -11826 400 0.0 3849 0.0 13269 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20.7 . 16.4 0.0 14.75 0.0 15.8 0.48 0.77 0.15 0.0 0.24 0.0 0.25 0.5 -3.03 0.11 0.0 -0.49 0.0 -0.46 0.0 -103 16.45 0.0 49.3 0.0 140 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -86 -79.4 0.0 -129 0.0 -170 0.0 -1.27 -1.02 0.0 -2.25 0.49 -2.73 0.0 0.82 -0.51 0.0 -2.89 -0.5 -3.3 0.0 -11034 -17554 0.0 -8268 0.0 -25443 A5, Ai|> fdl AE , xl Acosa A52 Au. Rl Ai(i fd2 A Ex2 Acosa AS, Au. 12 A^ fd3 AE , x3 L A c o s o R 3 j + BU (6.7) oo —i 88 The matrices Q and the nonzero elements of B are given by ^.u^.^.-Q = diag {10,. 100, 1, 1, 1, 10, 100, 1, 1, 1, 10, 100, 1, 1,1} b 4 , l * 1 0 ' b9,2 " 1 0 ' b14,3 = 1 0 "DI* UD3 : Q = d i a g { 1 0 ' 1» 1 0 0> 1» 1> 10> 1» 1 0 0> 1» X> 1 0» 1' 1 0 0 » 1' 1 } b 5 j l = 10484.6, b 1 5 > 2 = 10587.4 ( 6 > g ) " D l ' ^ ' ^ : Q = d i a g { 1 0 ' 1» 1 0 0 ' 1' 1' 1 0 ' 1» 1 0 0 ' l j 1» 1 0 ' l j 1 0 0 ' 1' 1 }  b 5 , l = 1 0 4 8 4- 6> b 9 2 = 1 0 , b15 3 = 1 0 5 8 7 ' 4 " E l ^ l ' ^ ' ^ E a ' ^ : Q = d ± a g { 1 0 ' 1 0 , 10-' 1 > 1 > 1 0 > 1 0 ' 1 0> 1» 1» 1 0 ' 1 0» 10, 1, 1} b 4 , l = 1 0 ' b5,2 = 1 0 4 8 4-6» b 9 j 3 = 10, b U ) 4 = 10, b 1 5 ) 5 = 10587.4 The corresponding control laws are given in (6.9). The system's eigenvalues for different controls are listed in Table VIII. The same disturbance applied to the nonlinear tests of section 6.1 is applied to this three machine system. Figures 6.15 to 6.22 show the system responses for different control schemes. Again four cases are studied as listed in section 6.1 except that in cases land 4 excitation controls are applied to a l l three machines in this case. Since this three machine system is unstable without optimal controls as indicated by the eigenvalues in Table VIII, the nonlinear response for this case is not shown. • The angle and speed deviation curves, Figures 6.15 and 6.16 respectively, show that in case 1 machine 1 is unstable and that in general the variations in angle and speed deviations are the largest in this case. Case 2 results in the smallest deviations followed by cases 3 and 4 in that order. Figures 6.17, 6.18 and 6.19 show the ac terminal voltages variations. At bus 1, Figure 6.17, the voltage changes most in case 1 " U E l " ~37.4 6.37 -63.4 -2.53 -0.023 21.3 -1.04 -0.26 -0.002 0.19 17.3 -1.19 10.7 0.17 0.002 "E2 26.3 5.24 0.69 -0.002 -0.016 -23.9 7.81 -129 -3.49 -0.4 2.26 2.36 -9.19 -0.19 -0.015 UE3 25 2.88 8.5 0.17 -0.015 40 0.6 -16.. 5 -0.19 0.06 -62.5 4.89 -69.7 -2.74 -0.03 " D I 7.14 1.26 0.47 0.19 -0.21 -3.94 0.64 -21.6 -0.61 -0.14 1.08 0.05 -0.54 1.09 -0.09 "D3 1.26 0.13 0.65 -0.34 -0.09 -1.67 0.31 -8.73 -0.17 -0.12 3.82 1.03 0.3 ' -1.62 -0.25 . " D I 6.24 1.23 -0.001 0.19 -0.21 -2.77 0.49 -15.2 -0.32 -0.13 0.69 0.05 -0.62 1.1 -0.09 UE2 2.83 0.13 1.88 0.03 -0.0003 -2.91 0.61 -20.6 -1.6 -0.03 1.11 0.03 0.25 0.13 -0.0001 "D3 0.89 0.12 0.44 -0.3 -0.09 -1.11 0.25 -6.05 -0113 -0.12 3.64 1.03 0.22 -1.61 -0.25 " E I a -0.73 0.007 -2.41 -1.05 0.2xl0~4 0.78 -0.12 2.92 0.08 0.005 -0.14 -0.006 0.04 0.001 0.8x10" -5 " D I 6.99 3.37 0.17 0.03 -0.22 -5.32 0.48 -21.3 -0.43 -0.16 1.93 0.001 0.27 0.008 -0.09 "E2 9.87 0.44 3.16 0.08 -0.0004 -11.8 2.17 -48.3 -2.21 -0.13 4.13 0.1 0.9 0.02 -0.0002 UE3 0.13 0.02 -0.01 0.001 0.8xl0~5 0.34 -0.001 0.69 0.02 -0.003 -0.43 0.02 -2.48 -1.07 0.3x10' -5 _"l>3_ 2.32 0.14 0.92 0.01 -0.09 -1.92 0.28 -9.06 -0.18 -0.16 3.1 3.19 -0.63 0.004 -0.26 .[Afi1 A* f < J 1 AExx Aco Sc R 1 Afi 2 A* f d 2 AE^ Acosa^ A« 3 A ^ A* f d 3 AE^ Acosa^]' (6.9 Table VIII Eigenvalues for 3 Machine, 2 Rectifier System. Control used Eigenvalues no control -37332, -15211, -372, -0.585 + , -0.526 + , -0.501 + , -0.483 + , -0.314 + , +0.88 xlO - 3 jl.95 jl.16 jl.58 J8.92 J6.88 UE1 , UE2 , UE3 -37332, -15211, -372, -18.5, -15.4, -13.26, -0.9 + , -7.05 + , -6.1 + , -0.35, -0.34, -0.32 J14.9 J14.2 jlO.26 UD1> ^3 -38764, -18557, -342, -70.25, -38.24, -3.28, -3.18, -2.8 + , -1.44, -1.2 + , -0.526, -0.4 + J5.82 jl.98 jl.68 " D l ' ^ ' V , -38764, -18557, -342, -70.2, -38.2, -9.57, -4.27, -4.19 + , -3.25, -3.2, -1.19 + , -0.41 + J6.88 jl.97 jl.67 "E3' "D3 -38764, -18556, -341, -225, -122, -13.06, -9.6, -9.57, -6.1 + , -1.73 + , -1.13, -1.1, -1.0 J9.89 jO.07 O 2.0, Srad 91 7.5 i.ol -x X' £12 0.51 H3 0-0 J 1 1 L I I I I i „j 0-8 0.9 10 t sec 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1.0 S23 Fig. 6.15 Swing Curves for 3 Machine, 2 Rectifier System ACJ rad/sec AOJI ^Aco3 <ACJ2 '.6 0.7 0.8 0.9 J L 1.0 2_ _ x— 3 ^ Aul 2. - - ^ Fig. 6.16 Speed Deviation for 3 Machine, 2 Rectifier System I 1 I I 1 21 I _i I i ^ 00 '•0.1 02 03 04 05 0-6 07 08 09 LO t sec Fig. 6.17 Terminal Voltage Variations at Bus 1 for 3 Machine, 2 Rectifier System I 1 1 1 1 i i i i i 0-/ 02 03 04 05 0.6 0.7 0-8 0.9 J.O t sec Fig. 6.19 Terminal Voltage Variations at Bus 3 for 3 Machine, 2 Rectifier System 93 followed by cases 4, 3 and 2 respectively. The largest changes in termin-al voltage at bus 2, Figure 6.18, are also observed in case 1 followed by cases 2, 3 and 4 respectively. At bus 3, Figure 6.19, the order is changed to 1, 4, 2 and 3. The dc power changes are shown in Figure 6.20 (a, b, c). In case 1, where no dc controls are applied, the dc power returns to pre-fault values after fault removal because of the constant current control and then starts to deteriorate after 0.45 sec. because of ac voltage drops. In cases 2, 3 and 4 the dc power transmitted over line 1 reaches its maximum value after fault removal and stays there for 0.75, 0.63 and 0.52 sec. for cases 2, 3 and 4 respectively. Then they a l l go down rapidly at first and then gradually. The total energy transmitted over the dc network is maximum in case 2 followed by cases 4 and 3 respectively. The control efforts are shown in Figure 6.21 and 6.22. The required excitation control efforts are maximum in case 1 and are greatly reduced in case 4. The dc control signal u ^ is largest in case 2 fol -lowed by cases 3 and 4 while u ^ is largest in case 3 followed by cases 4 and 2. 6.4 Three Machine System with One Rectifier and Two Inverters Similar to section 6.3 again four cases are studied with the exception that converter station 3 is operated as an inverter instead of a rectifier for a l l cases. Converter station 1 is operated as a rectifier and is controlling the dc current. Inverter station 2 is controlling the dc voltage while station 3 is operated under constant current control. The system's operating point is the same as that of section 6.2. The linearized state equations are given by (6.10). The weighing matrices Q and control laws for different control schemes are 0.0 1.0 0.0 0.0 0.0 0.0 0.0 All)^  -28.2 0.0 -101 0.0 -119 18 0.0 A * f d l -0.36 0.0 -1.29 0.45 -1 0.24 0.0 * E x l 0.28 0.0 -6.94 -0.5 -1.88 -0.24 0.0 Acosa^ -1381 0.0 -16638 0.0 -32176 747 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 Ato2 ' 26.3 0.0 44.1 0.0 49.9 -40.4 0.0 **fd2 0.1 0.0 0.61 0.0 0.74 -0.21 0.0 A E x2 0.64 0.0 -1.06 0.0 -1.47 -0.71 0.0 A C O S O J 2 1.66 0.0 124 0.0 213 -14.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 AtOj 45.9 0.0 52.3 0.0 73.4 34.3 0.0 A *fd3 0.49 0.0 1.03 0.0 1.41 0.51 0.0 A E x3 0.87 0.0 -2.45 0.0 -3.17 -0.36 0.0 AcosOj 3 -1400 0.0 -6195 0.0 -14464 -379 0.0 0.0 0.0 0.0 0.0 9.58 0.0. -48.1 10.1 0.37 0.0 -0.54 0.12 -1.56 0.0 0.08 -0.04 8155 0.0 -11526 635 0.0 0.0 0.0 0.0 -126 0.0 46.6 14.1 -1.58 0.45 0.85 0.11 -6.69 -0.5 -2.14 0.06 -164 0.0 -59.1 13.1 0.0 0.0 0.0 0.0 5.76 0.0 -97.6 -80.3 0.46 0.0 -1.33 -1 -2.79 0.0 0.23 -0.51 -4604 0.0 14619 1778 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 8.91 0.0 0.22 0.0 -0.73 0.0 4497 0.0 0.0 0.0 11.7 0.0 0.24 0.0 -0.64 0.0 59.8 0.0 0.0 -148 0.0 0.0 -2.51 0.49 -3.58 -0.5 9940 0.0 0.0 -31.4 Aio^  -0.34 * * fd l -0.04 A E x l -10085 A cos 0.0 A62 -25.7 Au2 -0.21 -0.05 i E x 2 -121 Acosaj 2 0.0 A6 3 98.1 Au 3 1.97 •A*fd3 -4.83 A E x3 -21231 AcosOj 3 + BU 0-0 0.1 0.2 0.3 0.4 0-5 0.6 0.7 0.8 0.9 1.0 t sec 0.3L 0-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 • . . ( c ) ' sec Fie. 6.20. DC Power Variations for 3 Machine, 2 Rectifier System 96 Fig. 6.21 DC Control Effort for 3 Machine, 2 Rectifier System Fig. 6.22 Excitation Control Effort for 3 Machine, 2 Rectifier System 97 given in equations (6.11) and (6.12) respectively. The system's eigen-values are listed in Table IX and i t is seen that without optimal con-trols the system is nearly unstable. "El' u ^ , U g 3 . Q «= diag (10, 100, 1, 1, 1, 10, 100, 1, 1, 1, 10, 100, 1, 1, 1> b 4 , l = 1 0 ' b 9 , 2 = 1 ° ' b 1 4 , 3 = 1 ° "Dl' "D3 •' Q = d l a g 1 0 ° ' 1' 1 > 1 0 ' 1» 1 0 0 ' 1' 1' 1 0 ' 1' 1 0 0 ' 1' 1 } b 5 f l - 10384, b 1 5 f 2 - 10799 u D 1,u E 2,u D 3 ; Q = diag {10, 1, 100, 1, 1, 10, 1, 100, 1, 1, 10, 1, 100, 1, 1) (6.11) b 5 , l = 1 0 3 8 4 ' b9,2 " 10> b15, 3 = 1 0 7 9 9 "El' "D1» UE2.' Q = diag {10, 1, 100, 1, 1, 10, 1, 100, 1, 1, 10, 1, 100, U E 3 ' ^ 3 ; 1, 1) b 4 , l = 1 0 ' b5, 2 = l ° 3 6 4 - b9,,3 = 1 ° ' b14,4 = 1 ° ' b 1 5 > 5 - 10799 The test results of nonlinear system responses to the same disturbance used for the three previous systems are summarized in Figures 6.23 to 6.30. Again case 2 is the most stable, as indicated by angle and speed deviations shown in Figures 6.23 and 6.24, followed closely by cases 3 and 4 respectively. Case 1 is the least stable case proving once more that excitation control signals by themselves are not suffi-cient to effectively stabilize such a system. Figures 6.25 to 6.27 show that ac terminal voltages variations are largest in case 1 and smallest in case 2 with cases 3 and 4 in between. " E l " -40.4 6.06 -57.7 -2.43 -0.02 26.2 -0.4 -3.92 -0.05 0.13 15.8 -0.49 5.33 0.08 0.4x10" -4 "E2 20.8 4.35 -2.18 -0.05 -0.006 -28.2 7.81 -107 -3.19 -0.55 11.5 1.24 -10.7 -0.19 -0.004 " E ? 29.2 3.22 4.59 0.08 -0.01 35.5 1.82 -15.7 -0.19 -0.64 -61.4 5.87 -93.5 -3.14 0.02 UD1 8.72 1.51 0.04 -0.03 -0.2 -2.76 0.87 -24 -0.91 -0.27 -0.57 0.4 -8.8 -0.24 0.1 UD3 1.12 0.13 0.02 0.09 0.11 1.01 0.38 1.67 0.2 0.15 -3.13 -0.89 -1.78 1.74 -0.29 " D I 7.37 1.41 -0.13 -0.04 -0.2 -1.76 0.62 -15.6 -0.36 -0.25 -0.61 0.34 -7.17 0.19 0.1 "E2. m 3.52 0.31 0.41 0.05 -0.0003 -1.74 0.77 -20.2 -1.59 -0.05 0.11 0.2 -3.95 -0.83 0.7x10" •5 UD3 1.16 0.11 0.12 0.13 0.11 0.66 0.35 -0.4 0.007 0.15 -3.02 -0.91 -1.66 1.65 -0.29 "El -0.06 0,07 -8.43 -1.27 0.4xl0~5 0.04 -0.05 0.5 0.01 0.005 0.03 -0.01 0.19 0.005 0.2x10" -5 " D I 6.94 1.42 -0.33 0.004 -0.2 -1.88 0.61 -15.6 -0.36 -0.24 -0.21 0.31 -5.38 -0.12 0.1 "E2 3.64 0.29 0.26 0.01 -0.0003 -1.55 0.74 -19.5 -1.56 -0.06 -0.19 0.19 -3.24 -0.09 -0.3x10" -4 "E3 1.63 0.11 0.14 0.005 -0.0001 -0.25 0.25 -3.8 -0.09 -0.02 -0.43 0.2 -10.4 -1.37 -0.5x10" -5 "D3 0.43 0.17 -0.25 0.002 0.11 0.23 0.39 -1.48 -0.03 0.16 -2.07 -0.93 -0.88 -0.005 -0.29 •[t6x Au^ fi*fdl AE^ Acosa R 1 A6,, Au>2 W{A2 AE^ Acoso I 2 A*3 Aw3 Ai)i f d 3 AE x 3 Acosa I 3] C (6.12) Table IX Eigenvalues for 3 Machine, 2 Inverter System Control used Eigenvalues no control -39956, -13146, -366, -0.641 + , -0.608 + , -0.564 + , -0.458 + , -0.245 + , -0.527 xlO" 7 + jl.52 jl.93 J9.05 jl.38 J7.1 j0.68xl0~3 -39956, -13146, -366, -17.74, -16.21, -13.26, -8.12 + , -8.03 + , -6.07 + , -0.355, -0.343, -0.316 J13.8 J15.26 jlO.48 "Dl* "D3 -41301, -16972, -317, -60.7, -38.85, -3.29, -3.21, -3.03 + , -1.72, -1.1 + , -0.478, -0.443 + J4.59 J2.2 jl.59 "DI'^'VJ -41301, -16972, -317, -607, -36.86, -9.32, -4.32 + , -4.19, -3.26 + , -1.25 + , -0.442 + J5.65 jO.054 j 2 . l l jl.59 "E3' "D3 -41301, -16972, -317, -60.7, -36.86, -9.39, -7.93, -6.55 + , -5.71, -4.43 + , -4.42, -4.11, -3.19 J0.9 J5.73 VO VO 100 Fig. 6.24 Speed Deviation for'3 Machine, 2 Inverter System 101 I I I I I I I I 1 k~-0.7 0.2 03 04 0.5 0.5 07 Qc? 0.9 1.0 t sec Fig. 6.26 Terminal Voltage Variations at Bus 2 for 3 Machine, 2 Inverter System . , , i i 1 1 " 1 «~-01 02 03. 04 05 06 07 OS 0.9 10 t sec Fig. 6.27 Terminal Voltage Variations at Bus 3 for 3 Machine, 2 Inverter System 103 0.75 0-05 -0.051 x xCASE 3 _ CASE 4 Fig. 6.29 DC Control Effort for 3 Machine, 2 Inverter System 0-75 0.7 0.05 0.0 •005 -0.1 -0.75 u £ p.u. x 1 X b \ I ,\\. \ UE1 UE3 * _/ u_ * £7 *• \ 0-3 ^ y^-0-4 — S S \ ~ " f > s *UE2 1.5 0.7 ' A)-8 - - , U - ' -v/ JE2 0JuPr> 7.0 ^ E 3 t sec UE1 CASE 1 CASE 3 CASE 4 x x Fig. 6.30 Excitation Control Effort for 3 Machine, 2 Inverter System 104 Figure 6.28 shows the dc power variations. In case 1 the power returns to rated values immediately after fault removal and starts to go down at 0.45 sec. Cases 2, 3 and 4 are very similar. In a l l three cases the power transmitted over the dc network is greatly increased after fault removal and stays at a value of about 0.47 p.u. then goes down. The dc energy transmission is maximum in case 2 followed by cases 3 and 4 respectively. Control efforts are shown in Figures 6.29 and 6.30. The con-trol changes have a pattern similar to previous systems. For the four systems studied in this chapter the nonlinear test results indicate strongly that optimizing the dc reference currents is the most suitable for controlling the dc/ac parallel systems. The dc network in such cases acts as a variable load at the machine terminals with the ability to adjust itself according to the system stability requirements. Optimal excitation controls, on the other hand, are slower and hence far less effective than dc controls. In certain cases where the excitation control signals cause the ac terminal voltages to drop, the power transmitted over the dc network is reduced accordingly, since the dc system is operating under constant current control, and the sys-tem is less stable. It is noted that applying combinations of optimal dc and excitation controls also yield stable systems with responses close to the case with dc optimal control alone. 105 7. CONCLUDING REMARKS The hvdc system dynamics are described in Chapter 2. For a simple two-terminal dc/ac parallel system i t takes a 39th order model to describe the system dynamics in detail including filters, reactive power condensers, ac tie lines, dc tie line and controls, synchronous machine and control loops. Dynamic models for the stability and control studies are devel-oped in Chapter 3. Despite the accuracy of the 39th order model i t is practically impossible for digital computation due to the tremendous time required because of the small step size. Several reduced models are found in this thesis which are accurate enough for practical purposes. The most suitable one seems to be the 6th order model consisting of a 3rd order synchronous machine, a 1st order voltage regulator, and two dc firing circuit controls. In Chapter 4 linear optimal controls are designed for two different single machine-infinite bus systems. It is found that a control signal modifying the dc reference'current is the most effective one in stabilizing a strong dc system. As for the excitation control i t is found that there is very limited effect on system stabilization. When the two controls, excitation and dc, are applied simultaneously an increase of dc control effort is noted. For a weak ac system, by adding a parallel dc link in expansion, i t is found that the best results are achieved by using a dc control signal in addition to the existing excitation con-trol designed for the ac system, but the system response s t i l l can be improved slightly by modifying the excitation control signal with the system expansion. The investigation of optimal control for multi-terminal dc 106 systems begins in Chapter 5. The system equations for a tapped three-terminal dc line superimposed on a three bus ac system are derived. In Chapter 6 two multi-terminal systems are investigated, each with two different modes of operation. The two machine-infinite bus system was stable but the three machine system was unstable when there were no controls. Four different control schemes are tested for each system and i t is found that the application of optimal control signals to modify dc reference currents at the current controlling converter stations is the most effective scheme, for a l l systems studied. On the other hand the application of optimal excitation controls is the least effective scheme and in some cases i t was even worse than not having controls. A combination of dc and excitation controls does stabilize the systems studied but is less effective than the dc control by its e l f . It is also evidenced that the dc control effort is increased rather than decreased in the presence of excitation control signals, suggesting that the two kinds of controls may work against each other. There are a few points deserving special discussion. In choosing the weighing matrix Q for the linear optimal control design for an ac system more emphasis must be placed on speed deviation"'1. This result i s also confirmed in this thesis when designing excitation controls alone. But when dc controls were being designed for a dc/ac parallel system i t i s found that more emphasis should be placed on the field flux linkage. This can be explained as follows. Since the basic function of the dc control is to pick up some power from the ac.system which may cause large acceleration of the synchronous machines in order to achieve stability and since the dc control is regulating dc current, then i t is preferable to keep the change in ac terminal voltages, which are directly related to 107 field flux linkages, to a minimum and hence a larger weighing factor must be assigned to field flux linkage deviations. Another reason for high computer costs, in addition to small step size because of the large eigenvalues, is the iterative solution of the system's auxilliary nonlinear algebraic equations four times for each integration step. One way of overcoming this difficulty is linear-ization of these algebraic equations but the effect of this lineariza-tion on the accuracy of results has to be investigated. As for future work the present study may be extended to the cases of dc line disturbances such as loss of one pole in a bipolar sys-tem, and the possibility of using suboptimal controls by neglecting the smaller gains in the control laws. The switching transients over transmission lines might have some effect on the control performance of dc/ae. parallel systems'-with small time-., constants--in-the- control ioop. Finally, the implementation of such optimal controls in actual systems is of course a challenging problem to utilities. 108 REFERENCES 1. E.W. Kimbark, "Direct Current Transmission, Vol. I", Wiley-Inter-science, N.Y., 1971. 2. C. Adamson and N.G. Hingorani, "High Voltage Direct Current Power Transmission", Garraway Limited, London, England, 1960. 3. C. 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Park, "Two Reaction Theory of Synchronous Machines: I -Generalized Method of Analysis", AIEE Trans., Vol. 48, July 1929, pp. 716. 45. E.J. Davison, "A Method of Simplifying Linear Dynamic Systems", IEEE Trans, on Automatic Control, Vol. AC-11, No. 1, January 1966, pp. 93. 46. A. Kuppurajulu- and S. Elangovan, "System Analysis by Simplified Models", ibid., Vol. AC-15, No. 2, April 1970, pp. 234. 47. A. Kuppurajulu and S. Elangovan, "Simplified Power System Models for Dynamic Stability Studies", IEEE Trans, on Power Apparatus and Systems, Vol. PAS-90, No. 1, January/February 1971, pp. 11. 48. M. Athans and P.L. Falb, "Optimal Control", McGraw H i l l , New York, 1966. 49. J.E. Potter, "Matrix Quadratic Solutions", SIAM, Journal of Applied Math., Vol. 14, No. 3, May 1966, pp. 496. 50. CE. Fosha Jr. and 0.1. Elgard, "Optimum Linear Control of the Multi-Variable Megawatt-Frequency Control Problem", JAAC 1969, pp. 471. 51. F.P. DeMello and C. Concordia, "Concepts of Synchronous Machine Sta-bility as Affected by Excitation Control", IEEE Trans, on Power Apparatus and Systems, Vol. PAS-88, No. 4, April 1969, pp. 316. 52. H.A.M. Moussa and Y.N. Yu, "Optimal Power System Stabilization Through Excitation and/or Governor Control", ibid., Vol. PAS-91, No. 3, May/June 1972, pp. 1166. 53. Y.N. Yu and H.A.M. Moussa, "Optimal Stabilization of a Multi-Machine System", ibid., VoL PAS-91, No. 3, May/June 1972, pp. 1174. 

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