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Linear optical stabilization and representation of multi-machine power systems Moussa, Hamdy Aly Mohammed 1971

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LINEAR OPTIMAL STABILIZATION AND REPRESENTATION OF MULTI-MACHINE POWER SYSTEMS by • HAMDY ALY MOHAMMED MOUSSA B.Sc, Ain Shams University, Egypt, 1965 M.Sc, Ain Shams University, Egypt, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY' in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of the Department of Electrical Engineering THE UNIVERSITY OF"BRITISH COLUMBIA July, 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his 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 Date AlAA^j 3 , 197/ ABSTRACT Linear optimal regulators have been designed for power system stabilization by introducing control signals to voltage regulators and/or governors. A new technique is developed in this thesis to determine the state weighting matrix Q of the regulator performance function with a dominant eigenvalue shift of the closed loop optimal system. The technique is used to investigate the stabilization of a typical one-machine infinite system and a multi-machine system with different stabilization schemes. The objective is to find the best way to stabilize a power system. An optimally sensitive controller is also developed to offset the effects of the changing system operating conditions on the effort of the stabilizing signal. The controller automatically adjusts its gains so that it always provides the system with the optimum stabilizing signal. A new multi-machine state variable formulation, necessary for these studies,' is developed. It requires minimum computations and retains all the parameter information for sensitivity studies. An exact representation of synchronous machines is investigated and test methods are suggested for the determination of exact circuit parameters. ii TABLE OF CONTENTS / • Page ABSTRACT / 11 TABLE OF CONTENTS ..../. ii:L - LIST OF TABLES .' vi LIST OF ILLUSTRATIONS viACKNOWLEDGMENT viiNOMENCLATURE . ix 1. INTRODUCTION 1 2. EXACT EQUIVALENT CIRCUITS AND PARAMETERS OF SYNCHRONOUS MACHINES 4 2.1. d-Axis Exact Equivalent. Circuits ^ 2.2. q-Axis Exact Equivalent Circuits 8 2.3. Circuit Parameters in Terms of Conventional Parameters........ 11 2.4. Extra Tests to Determine T^ and x" • 13 D do 2.4.1. Determination of TQ From a Varying Slip Test. 13 2.4.2. Determination of TQ From Decaying Current Test 14 2.4.3. Determination of xV 16 do 2.5. Laboratory Test Results .' 17 3. STATE VARIABLE EQUATIONS OF MULTI-MACHINE POWER SYSTEMS 19 3.1. Terminal Voltages and Currents : 19 3.2. Nonlinear Machine Equations 21 3.3. Linearized Machine Equations..... 23 3.4. Exciter and Voltage Regulator System 5 3.5. Torque Equations 26 3.6. Governor-hydraulic System 28 iii Page 3.7. State Equations 29 3.8. Multi-Machine System with an Infinite Bus..... 29 3.9. Simplification of Power System Dynamics 31 / / 4. OPTIMAL LINEAR REGULATOR DESIGN WITH DOMINANT EIGENVALUE 'SHIFT 34 4.1. Linear Optimal Regulator Problem. 34.2. Eigenvalue Shift Policy.... 35 4.3. The Shift .. 36 4.4. Determination of Aq 37 4.5. Sensitivity Coefficients X,q 34.6. Algorithm 40 5. OPTIMAL POWER SYSTEM STABILIZATION THROUGH EXCITATION AND/OR GOVERNOR CONTROL '. . 41 5.1. System Data 45.2. Case 1: u£ Control 3 5.3. Case 2a: u_ Control, with Dashpot i 44 G 5.4. Case 2b: u' Control, without pashpot 45 G 5.5. Case 3: u„ Plus u' Control 46 E (j 5.6. Nonlinear Tests v 47 6. OPTIMAL STABILIZATION OF A MULTI-MACHINE SYSTEM. 51 6.1. System Data and Description 56.2. Case 1: One Machine Optimal Excitation Control u„T 54 EI 6.3. Case 2: Multi-Optirtal Controllers u^ 55 6.4. Case 3: Approximated One Machine Optimal Design 57 6.5. Case 4: Subsystems Optima]. Design • 58 Page 6.6. Nonlinear Tests 59 7. OPTIMUM STABILIZATION OF POWER SYSTEMS OVER WIDE RANGE OPERATING CONDITIONS 63 7.1. Optimally Sensitive Linear Regulator Design 63 7.2. Sensitivity Equations of the Linearized Power System.... 66 7.3. Optimally Sensitive Stabilization of a Pox^er System. . 71 8. CONCLUSIONS 8APPENDIX A 4 APPENDIX B ... 86 REFERENCES 90 LIST OF TABLES TABLE PAGE 3-1 Eigenvalues of the Typical One Machine Infinite System of Different Modelling 33 7-1 Controller Gains for u^g and u* at Different Operating Conditions .: 77 7-2 Dominant Eigenvalues of the System with the Different , ... . Controllers 78 vi LIST OF ILLUSTRATIONS FIGURE £AGE 2-1 General d-Axis Circuits.... 7 2-2 Simplified d-Axis Circuit2-3 General q-Axis Equivalent Circuit 9 2-4 q-Axis Circuit, (x ' - X ) . 10 qQ q 2-5 q-Axis Circuit, (XqQ ~ x^) . . . . 12-6 Determination of T^ from Slip Test 4 2-7 Connection for the Decaying Current Test 12- 8 Resolving Decaying Current into Two Components 16 3- 1 Components of V in dq and DQ coordinates 20 3-2 A Typical Exciter-Voltage Regulator System 5 3- 3 A Typical Governor-Hydraulic System 28 4- 1 Algorithm to Determine Q with Dominant Eigenvalue Shift. 40 5- 1 A Typical One-Machine Infinite System 42 5- 2 Nonlinear Test Results 50 6- 1 A Typical Four-Machine Power System 52 6- 2 Nonlinear Tests of the Multi-Machine System 67- 1 Structures of Nominal and Optimally Sensitive Controllers 66 7-2 Speed and Torque Angle Gains for the Controllers 79 7-3 Nonlinear Test Results 80 vii ACKNOWLEDGMENT I wish to express my most grateful thanks and deepest gratitude to Dr. Y.N. Yu, supervisor Of this project, for his continued interest, encouragement and guidance during the research work and xwriting of this thesis. I also wish to thank Dr. E.V. Bohn, Dr. M.S. DaVies and Dr. H.R. Chinn for reading the draft, and for their valuable comments. The proof reading of the final draft by Mr. B. Prior is duly appreciated. Thanks are due to Miss Linda Morris for typing this thesis. The financial support from the National Research Council and the University of British Columbia is gratefully acknowledged. I am grateful to my wife Zainab for her encouragement throughout my graduate program. viii NOMENCLATURE General A system matrix B control matrix Y state vector u control vector Q positive semi-definite symmetric matrix, weighting matrix of Y R positive definite symmetric matrix, weighting matrix of u q vector, diagonal elements of Q K Riccati matrix G closed loop system matrix A=£+jri eigenvalue vector of G A. sensitivity vector of the eigenvalue A. w.r.t. <i x,q J G 1 S sensitivity matrix M composite matrix as defined in (4.18). A,X,V eigenvalue vector, eigenvector matrices of M and M' o subscript denoting initial condition Y time derivative of Y * superscript denoting conjugate ' or T superscripts denoting transpose A prefix denoting a linearized variable [ ] diagonal matrix with elements of each machine 0)^ synchronous angular velocity: 377 rad/s p differential operator ix suffices a,d,q armature a-phase, d-axis, and q-axis windings suffices F,D,Q rotor field, d-axis damper and q-axis damper windings sensitivity of matrix A with respect to parameter q. System parameters (P.U., except as indicated) Y„, network node admittance matrix N Z^ network node impedance matrix Z network node impedance matrix in individual machine m r coordinates r+jx tie-line impedance G+jB terminal load admittance R's,r's winding resistances in ti, and per unit X's,x's self and mutual reactances in ti, and per unit L's self and mutual inductances, H Z armature base ohm, ti n x^x^x^ d-axis synchronous, transient and subtransient reactances x" newly defined open field d-axis subtransient reactance do J r x ,x" q-axis synchronous, and subtransient reactances q q T'T" short circuit d-axis transient and subtransient time a a constants, s T' , T" open circuit d-axis transient and subtransient time do do constants, s d-axis damper winding time constant, s T" open circuit q-axis subtransient time constant, s qo r i . exciter amplifier gain exciter amplifier time constant, s exciter time constant, s governor permanent droop governor temporary droop gate actuator time constant, s dashpot time constant, s hydraulic turbine gate time constant, s water time constant, s inertia constant damping coefficient les (P.U., except as indicated) - . optimal excitation signal, one-machine infinite system conventional excitation control signal optimal governor control signals with and without dashpot one-machine optimal excitation control, multi-machine system multi-machine optimal excitation controls, multi-machine system optimally sensitive excitation control flux linkages, currents, voltages torque angle, radians angular velocity, electrical rad/s exciter regulator voltage gate actuator signal dashpot feedback signal xi g gate movement h hydraulic head t ,t mechanical, electrical torques me v ,i machine voltages and currents in common coordinates n n v ,i machine voltages and currents in individual coordinates m m V ,I voltage and current matrices with diagonal elements mm v and i of each machine mm U =U ,+iU sensitivity matrix of v with respect to <$.' m md mq m v infinite bus voltage o v generator terminal voltage P+jQ generator output power V's,v's applied voltages in V, and per unit U's,u's rotational voltages in V, and per unit I's,i's currents in A, and per unit V ,1 base armature voltage, current n n V^^. jV-^ tV^T, base field, D-winding and Q-winding voltages rJ3 DD QB Ipg, I^gj Iqg base field, D-winding and Q-winding currents. xii 1. INTRODUCTION The stabilization of power systems has become increasingly impor tant because of the increase in the size of power systems, the number of interconnections, the voltage level, the number of large generating units, and the introduction of fast-response excitation systems and dc transmission lines. Much attention has been focussed recently on the application of control signals to the excitation system for stabilization, or stability control, to improve the ability of a power system to return to its synchronous operating equilibrium after 12 3 a disturbance. These signals can be derived from shaft speed ' ' , terminal frequency^'^'^, or terminal power'''8. They are used to off set the voltage regulator reference in the transient period with the object of producing positive damping torques on the synchronous machine shaft In view of the fast development of control theory, more work must be done to explore the possibility of deriving better methods and techniques for power system stabilization. Optimal linear regulators 11 12 are designed and quadratic performance functions are chosen ' There are many problems unsolved. Four of them are mentioned below. The first is that in the optimal state regulator design, the choice of the weighting matrix Q associated with the' performance function is based entirely upon past experience or guessing. Therefore, the designed controller is not necessarily the best. The second problem is the normal controller is designed for only one particular operating condition, and this condition cannot be estimated prior to a disturbance. Can an optimal controller be designed to cope with the wide range operating condition? The third problem is the multi-machine dynamics formulation. The problem is not how to obtain a set of state equations but how to avoid the large number of high-order matrix inversions and how to retain all the parameter information for sensitivity investigations. Finally there is the problem of exact representation of synchronous machines and how to determine the circuit parameters from simple field tests. This must be done in order to obtain an accurate evaluation of system dynamic behaviour during and after a disturbance. This thesis provides some answers to the problems mentioned above.. In Chapter 2 the exact equivalent circuits for the synchronous machines are derived from the MKS voltage equations and by the use of per unit systems. Simple field tests to determine the exact machine parameters are then suggested. The multi-machine state equations are derived in chapter 3 by relating the transmission network algebraic equations to individual machine dq coordinates. Detailed representation of excitation and governor systems is presented. The one machine infinite bus system is only a special case of the multi-machine system. Dynamic approxi mation of the formulation is then discussed. A new technique for the design of optimal regulators is developed in Chapter 4. The choice of the state weighting matrix elements of Q of the performance function is related to the movements of the dominant eigenvalues of the closed loop system. The dominant eigenvalues are shifted to the left on the complex plane within the practical limits of the controller. - . The technique is then applied in chapter 5 to stabilize a typical one-machine infinite-bus system. Various stabilization schemes are investigated. Optimal excitation and/or governor controls are compared with conventional excitation control. The objective is to find the best way to stabilize a pox^er system. Some of the stabilization techniques for the one-machine infinite-bus system are further developed for multi-machine system in Chapter 6. Although the one-machine design is more often than not the only case considered, no more difficulty is involved in the formulation or computation for multi-machine systems. Several schemes are investigated, multi-machines x^ith multi optimal controllers or with one optimal controller as compared x^ith multi-machines with individual optimal controllers or an equivalent one-machine with one optimal controller. An answer to the wide range operating condition problem is given in Chapter 7. An optimally sensitive controller is developed which provides stabilization for a power system which departs widely from normal operation conditions. A comparison is then made of the optimally sensitive control design x^ith other nominal designs. 2. EXACT EQUIVALENT CIRCUITS AND PARAMETERS OF SYNCHRONOUS MACHINES For stability studies of large power systems, accurate re presentation of the synchronous machine is required. As pointed out 14 by Canay , the conventional equivalent circuits for synchronous machines do not give accurate computed field voltage and current values. He suggested several circuits and showed good agreement between his test and calculated results. His circuit parameters were calculated from design. Questions arise: How to determine accurate circuit parameters from simple field tests and how to choose the equivalent circuits. The circuits are not unique because of different base volts, base amperes and circuit elements. In this chapter exact equivalent circuits for synchronous machines are derived from voltage equations in MRS units. Some constraints are then imposed so that the equivalent circuits will lead to the simplest form. A systematic procedure is then developed to determine these circuit parameters from simple field tests. 2.1. d-Axis Exact Equivalent Circuits Applying Park's transformation, the d-axis voltage equations of a synchronous machine in MKS unit can be written in the form Vud R R_ X, Xap X _ d ar aD 3 T^a XF XI 2^3 XDF h I, d T D_ . 15 The X-matrix is not symmetric .• Here all X's are reactances of (2.1 single-phase excitation except X^ which is of three-phase excitation. The numerical coefficient 3/2, and hence the asymmetry of the matrix, results from the a,b and c three-phase excitation on the stator and the F or D single-phase excitation on the rotor. The matrix form itself suggests that the per unit reactances must, and voltages and currents may, be defined as follows X „ ; • x dF aF V X DB dD aD V n x_ = X DB FD "FD V FB 3 In 3 In 'Fd = ^Fa-*^ XDd = ^Da-*^ r B UD DF = ^F "FB F V. DB d V R a V FB rF ~ h V, FB FB XD = XD; DB V DB I. DB V DB (2.2 a n h = V^B iD = V^B v. = V./V d d n VF=.VVFB The matrix of x's is not necessarily reciprocal. To make it reciprocal"^, the following constraints must be imposed, 6 —V I = V I = V I 2 n n FB FB DB DB (2.3) resulting in XdF XFd Z 4 ] ' XdD XDd Z 4 } n n n n - 1 XFD/IFB, ^DB. XFD XDF 3 Z I I n n n = 2 XFrT'FB, 2  ^/SB.2 Xd Z ' XF 3 Z 4 ; ' XD • 3 Z 4 ; n n n n n (2.4) =\ _ 2 RF TFB 2 ra Z ' rF 3 Z 4 ; n n n V^B^ ' rD 3 Z 4 ) n n Z = V /I n n n The d-axis voltage equations can be written now, in per unit, as vd"ud = r +px, a r d PXdF PXdD VF pXFd rF+pXF PXFD _ 0 -PXDd PXDF VPXD • A (2.5) One of the general d-axis equivalent circuits corresponding to (2.5) is as shown in Fig.2-1, which reduces to Fig. 2-2, the simplest form, if one sets XFd = XDd = XFD (2-6) Note that x^, x^ and xm are no longer leakage reactances. They are defined as Xd£ Xd XFd' XF£ ~ XF XFd' XM ~ XD " XFD. (2.7) 7 .—V\A-W—r-rrp——-o ro n —OnfT^VV*—I Fig. 2-1 General d-Axis Circuits JJff\ ^VVN . Fig. 2-2 Simplified d-Axis Circuit The following information, although not needed in the determination of parameters from field tests, is useful in design. From (2.6) the current ratios of (2.4) can be determined as follows 1FB 3 XaD 2 X. (2.8) FD •""DB 3 XaF 2 X. (2.9) FD Substituting (2.8) and (2.9) into (2.4) and the results into (2.7) the circuit parameters of Fig. 2-2 can be expressed in terms of winding parameters as follows 8 3 XaF XaD n n TD = 3 faD /F XaD XaF XF£ 2 X_ 4 X_n Z J FD n FD n (2.10) XD£ - 2 XFD4n XFD " Z aDs _ 3 XaF XaD  XFD XDd XFd 2 Z x R n JFD vx = _a = _3 ^ /*AD>2 ra Z ' rF 2 Z ' rD n n TD 3 VjF,2 2 Zn XFD 2.2. q-Axis Exact Equivalent Circuits The q-axis voltage equations for a synchronous machine in MKS unit are as follows r- -i "v - u q q = R a + P X q XaQ I q 0 _2XQa XQ A (2.11) The X-matrix is again not symmetric"'""'. While X^ is a reactance of three-phase excitation, X , X and X are of single-phase excitation. The matrix form suggests the following definitions of per unit reactances, voltages and currents XqQ = XaQ VVn > XQq= ^VW Xq " Xq VVn > XQ = XQ W r = R I /V , r_ = R Inp/Vn_ a a n n Q Q QB OB i =1/1 , i = .I./I_, v = V /V , u = U /V q qn'Q Q QB q q n' q qn (2.12) 9 / 16 To make the x-matrix reciprocal , the following constraint must be imposed / resulting in — VI = V I 2 n n QB QB XqQ ~ XQq " Z h } ' Xq " Z H* M n n n v = i ^Q /pj. 2 = 2 _Q,_QBs2 XQ 3 Z ' ' Q 3 Z 1 ' n n n n The per unit q-axis voltage equation now can be written as r + px px _ i a q qQ v - u = q q 0 (2.13) (2.14) (2.15) pXQq rQ+pXQ, The general q-axis equivalent circuit corresponding to (2.15) is as Fig. 2-3. AAA *ql -OTP Ql '0 AAA 'Q Fig. 2-3 General q-Axis Equivalent Circuits where x = x - x , xr„ = x - x (2.16) qH q qQ QJ6 Q Qq Although x ^ of Fig. 2-3 exactly represents the mutual reactance and x . and x^„ the leakage reactances, mathematically, qJi QJi however, the branch reactance x ^ can be set equal to xq or x^ resulting in two simplified equivalent circuits, Figs. 2-4 and 2-5 respectively. -A/\A—rirpL-Fig. 2-4 q-Axis Circuits (XqQ = V Fig. 3-5 q-Axis Circuit (XqQ = V The parameters of these two circuits can be easily determined from field tests. They can also be expressed in terms of winding parameters: Fig. ,2-4,, xqQ = xq From (2-14) one has I X n aQ Hence XQ " XQq R (3XaQXaQ ^ * K \ Q 3ZnXaQ (2.17) Fig. 2-5, xqQ = xQ From (2.14) one has ""•QB _ 3 XaQ n 2 X, 11 The solutions of the circuit parameters are x ,. = x - / x 2 XaXd(xd Xdo) XdXdo(xd ^c? where and d* I Cxd-xJo)-(x'-xJ) d d d do (xd-Xdo)-(xd-Xd) r„ = F (u T' x ,-x' o do d d 1 rXd"Xd£)2 •D u T" x'-x" o do d d Next, if T^ is separately determined, we have another equation instead of (2.21). The solutions are (2.23) and (2.22) = Xd Xdil XDd Xd Xd£' XFi x ,-xi XDd d d ^Xd Xd£^Xd /n 00-. x_. = ; ri (2.23) d • d 2 1 XDd UoTD " VrD (2'24) xd. = xd- J^-^^a-^Y^- <2'22a) D do The current ratio I^/I of (2.4) can nov7 be determined, but not FB n 1^ /I since there is no way to measure because of the short circuit. DB n D The voltage ratio V„_/V can then be determined from (2.3). r JJ n The q-circuit parameters can be easily determined. For Fig. 2-4 we have q XqQ+XQ£ ° q° Q Q£ qQ / The solutions are „ 2 / XX ^ X ! X ~ ~ X , X~ — 11 i r^ — rritl It ^Z.ZO/ qQ q ' Q£ x -x Q u) T x -x q q o qo q q For Fig. 2-5 we have' x" = x . , (j T" r. = x . , x = x . + x „ (2.27) q qit ' o qo Q qQ q qQ qH The solutions are x . = x" , x• n = x =x -x" , r_ = —^rr (x -xV) (2.28) ql q qQ Q q q 0 % q0 ^ ^ 2.4. Extra Tests to Determine T^ and x'1 . _ D do Two test methods are suggested to determine T^ and one to determine x'' . All methods were tested in the laboratory, do 2.4.1. Determination of T^ from a Varying Slip Test The rotor is driven at various speeds. Positive sequence voltages are applied to the armature winding with the field open. From phase voltage-current ratio equivalent reactances x^Cs) and x^(s) are approximately determined. Replacing r^ by r^/s in Fig.2.2, the imaginary part of the circuit impedance is a function of slip' s as follows x2 x Xd(s) = Xd " ^p-f T X,U2 (2'29) or . j 1 X° (~)2 + -TT (2.30) x,-x,(s) 2 s 2 d C XdDXD . • ' XdD Fig. 2-6 Determination of from Slip Test which can be plotted as Fig.-2-6 for the determination of T^. An accurate value of x^, from open and short circuit tests, must be used for the calculations. ' •2.4.2. Determination of T from Decaying Current Test Fig. 2-7 Connection for the Decaying Current Test 19 Kaminosono and Uyeda's indicial response method is modified to determine T . Since a clear step voltage is hard to obtain, a decaying current is used instead. Apply a constant cur 15 to one phase winding in the d-axis position and then suddenly short circuit the armature terminals with the switch Sw in Fig. 2-7. The rheostat protects the power supply. The voltage equations for Fig. 2-7 in Laplace transform are 0 I 0 r +sL sL -ad dD sL dD r^+sL, "i (s)" a • -Ld LdD i ao -Vs>. -LdD LD - o (2.31) where i is the initial current in the armature winding, ao The solution of I (s) can be written in a convenient form a s+ T' Vs) = —T-^ (s+f)(s+i) 30 1 2 (2.32) where TD " ^Hd5' T1T2 = Vi > Tl+T2 = VTD (2.32a) and TD " VrD > Td = Ld/ra > 4d = LDd/LdLD (2.32b) I (s) of (2.32) can be resolved into two components a . Ia(s) X10 . X20 + 4. 1 4. 1 s+ — s+ — 1 2 (2.33) and it can be shown that the initial component current ratio i T T -T' 10 = Jl_ 1 D i x T'-T 20 1 D 2 (2.34) From T^, T1 + 1^ of (2.32a) and (2.34), the following solutions are ob tained 16 Td = (i10Tl + i20T2)/iao> TD = TlVTd> TD = <110T2+120Tl)/iao <2'35' T^, T^, i^Q and i^Q are determined from a semilog plot as Fig. 2-8. / The T, value from (35) should be checked with d. • / . (2.36) Fig. 2-8 Resolving Decaying Current into Two Components 2.4.3. Determination of x" „ , , •• ..- , ^0 20 Dalton and Cameron's method to determine x\j is adapted to determine x^' . The rotor remains stationary and the field winding is open-circuited. Single phase Voltage of rated frequency is applied to each of., a pair of stator terminals in turn, leaving - the third terminal open. Three such tests are performed Xi/ith the rotor position fixed throughout the test'. The armature voltage and current and the field voltage are recorded in each test. Let the single-phase reactance X be a function of 6, the angular position of the rotor X = K + M cos 26 (2.38) and let the voltage-current ratio of the three tests be A, B, and C. It can be shown that ^' / v - A + B + C /' K = , , (2.39a) and M = J(B-K) 2 + ^~-2 The open field d-axis subtransient reactance is then given by (2.39b) The plus sign should be used if the largest measured reactance, A, B or C, and the largest measured field voltage occur in the same test. 2.5 Laboratory Test Results The methods thus developed were applied to a small synchronous machine to determine the circuit parameters. From IEEE test code the following d-axis parameters are determined. r = 0.72Q, x, = 16.2n, x' = 2.74Q, x" = 2.42 ^ a a a a T' = 0.27s, T" = 0.027s do do The per unit values can be obtained when the base ohm Z is chosen. n From extra tests the following are determined A TD = 0.049 s (varying slip test) B Tn = 0.055 s (decaying current test) C x^'o = 8.18 Q (adapted Dalton and Cameron) The computed results of d-circuit parameters in ohms are as follows XDd XD£ rF rD A 15.8 0.40 2.75 14.6 0.182 1.66 B 15.5 0.68 2.38 10.9 0.176 1.28 C 15.9 0.33 2.84 15.5 . . 0.184 1.76 / / / The discrepancy in results of B is attributed to the difficulty of resolving the decaying current into components. The field resistance Rp is 70ft and the current and voltage ratios are IFB/In = 0.0625 , VFB/Vn = 24 For the q-axis x = 9.71 ft, x" = 7.2..ft q q are determined by conventional methods and T" = 0.0165 s qo by a decaying current method similar to Fig. 2-7. The computed results of q-axis parameters Fig. 4 x . = 9.71 ft, x0„ = 27.8ft , rA = 6.05 ft qQ QSL Q Fig. 5 xq£ = 7.2 ft , xqQ = 2.51 ft, r^ = 0.407 ft 3. STATE VARIABLE EQUATIONS OF MULTI-MACHINE POWER SYSTEMS21 / In stabilization studies of large interconnected'multi-machine power systems, the system dynamics must be expressed in the 22 state variable form Y = AY + Bu. Laughton suggested a method of building the A matrix from matrix elemination of algebraic and 23 2 A differential equations. Undrill ' proposed to build up the A matrix from individual system submatrices. Undrill's method requires a matrix inversion of mn x mn for m machines each described by n-th order equations. The system parameters are not retained in the final formulation. This is also the case in Laughton's formulation. In this chapter a new multi-machine formulation is proposed. The main objective is to reduce the number of matrix inversions and to keep them of low order. All the system parameters are retained in the final formulation making it convenient for sensitivity and control studies. The synchronous machine parameters are based on an exact equivalent circuit, and can.be determined from field tests as described in chapter 2. 3.1. Terminal Voltages and Currents Let the individual synchronous machine rotating coordinates be d and q and the common rotating coordinates of the complete system be D and Q. Let the terminal voltages and currents of all machines in dq coordinates be a vector v and a vector i and those in DQ coordinates m . m be a vector V and a vector i respectively, and let the phase relation of the k-th machine x^ith respect to the two coordinate systems be as in Fig. 3-1. / / Fig. 3-1 Components of in dq and DQ Coordinates Then we have for the k-th machine VNk e VimV XNk Z > " Xmk and for a group of m machines (3.1) The transmission system is usually considered as a static network in stability and control studies, i.T = YT1v.T N N N (3.2) Substituting (3.2) into (3.3) x<re have v = Z i m mm where and Zm = [e-j6i]ZN[ej6J] m Z = Y (3.3) (3.4) (3.5) (3.6) 21 Note that the highest order matrix inversion required in the formulation is Expanded we have where ~vd' R - X m m _1d / - ( V L qJ X R L . m m J i L qJ ZN = RN + Z m R + jX m J m (3.7) (3.8) Rm(i,3) xm(i,j> R^i.j) -XN(i,j) cos6 . . sin<5.. 13 J (3.9) 6 . . = 6. - 6. (3.9a) 3.2. Nonlinear Machine Equations The synchronous machine equations are as follows, the i-th machine For VF " P^F + rF S where d r d a d e q v =p^ - ri +ibiK q q a q e d 0 = P^D + VD 0 = P^Q + VQ "V "XF XFd XFD ip • *d XdF Xd XdD -:Ld 0) o -XDF XDd - j (3.10) (3.10a) 22 1_ 0) qQ -i q (3.10b) Note that -i^ and' -i^ are used in the synchronous generator equations. Actually all the notations of (3.10) should be given a suffix "i" for the i-th machine, except for p and <i) which are common to all machines. The suffix is dropped for clarity. It is also intended that the same equations be used for the description of multi-machine systems. In such a case all the v's, i's and i|> * s of (3.10), become column vectors, and x's and r's, diagonal matrices. These statements apply also to the rest of the chapter. The current solutions of (3.10a) and (3.10b) have the form -l Y. FF dF DF Fd dd Dd FD dD DD - V ^d (3.11a) and Q qq -- V (3.11b) Note that the solution of currents from (3.10) for individual machines does not involve equations of other machines. The Y matrices of (3.11) are not the inverses of the x matrices of (3.10). If equal per unit mutual reactances are used, the elements of the Y matrices of (3.11a) of individual machines can be determined directly from the d-axis exact equivalent circuit of Fig. 2-2 using the well-known star-mesh relations in network analysis. Substituting i^ and i of (3.11) into (3.7), and the results into v, and v of (3.10), we have d q "*d" = •V "RYdF "RYdd w +X Y —RY X Y e m qq dD m qQ -X Y,_ -0) -X Y,, -RY m dF e m dd qq -X Y -RY n m dD qQJ * [*p; V *q. V V where (3.12) R = Re Z + [r ] m a (3.12a) Substituting !„, in and i of (3.11) into v^, v =0, and v =0 of (3.10), r 1) Q r D Q we have -r Y '\b -F FF ^F rFYFd^d " r Y - TJJ + v F FD VD F P^D = -r Y •\b -D DF VF rDYDd^d " r Y 't D DD VD (3.13) P'^Q = Q Qq \ RQYQQ'*Q Thus the transmission line relation (3.7) at the machine terminals has been included in the nonlinear state form of machine equations (3.12) and (3.13). 3.3. Linearized Machine Equations When equation (3.4) is linearized, it has three terms, Av = Z Ai + jZ [A6.]i - j[A6.]v (3.14) m mmJm.imJim' which can be written as Av = Z Ai + jU A6' m m m m (3.14a) where U = Z I - V m mm m (3.14b) 24 Note that A6, i and V are column vectors and [A<S], I and V are mm m m diagonal matrices. Since Vm * Vd + j V ^ = ^ + j ^ = ^ + ^ m rm the voltage equations v^ and .v of (3.10) can be written as v = pib m ^rm [r ] i + j [to ]i> a m J e rm After linearization and making use of (3.14a), we have PA^ ' m [Z +(r )]Ai -j[co ]AUJ -j |> ] Ato +jU A<5 m a -m J e rm J rm e J m (3.15) (3.16) Expanded-and with the substitution of i, and i from (3.11) we have d q Ail), Aiji -RY dF -RY dd co +X Y -RY,^ X Y e m qq dD m qQ -X Y,_ -to -X Y,, -RY m dF e .EI dd qq -X Y -RY m dD qQ J [A^p, A^d, A^q, A^D,Ai{)Q] + -U mq A6 Aw ^ e Umd "'*d] Equation (3.13), after linearization, becomes P^F = - rF YFp.A^F - rFYpd.A^ - r^-A^ + Avp PA^D = " rD VA*F ~ rDYDd"A*d " VW^D PA*Q = " rQ YQq*AV rQ.VA*Q (3.17a) (3.17b) Equations (3.17a) and (3.17b) are the linearized multi-machine equations 'i in state variable form. 25 3.4. Exciter and Voltage Regulator System Fig. 2 shows the block diagram of a typical exciter voltage regulator system ^6 \ KA- r i VF ) l+TAs 1+TEs Fig. 3-2 A Typical Exciter-Voltage Regulator System The corresponding state equations are PAVF = - Avp 4- |- AvR 1 KA KA PAvR = - - AvR - - Av + - uE A A A Since v Av = v,Av, + v Av t t d d q q then from (3.14a) Av, d R m -X " m + -u mq Av X _ m R mj Ai . q_ U A md. A6 (3.18a) (3.18b) (3.19) (3.20) Substituting Aid and Aiq of a linearized (3.11) into (3.20), the results into (3.19), and the results into (3.18b), we have pAvR = A(7,1)A*F + A(7,2)Ai|jd + A(7,3)A<fq + A(7,4)A^D 1 KA + A(7,5)A.j; - —- Av + A(7,8)A<5 + „ ^ A TA L (3.21) where A(7,l) = MYdF , A(7,2) = MY^ , A(7,4) = MY^ A(7,3) = NYqc. , A(7,5) = NYqQ , KA / A(7,8) = -[«r—KV U , - V, U ), ' AVt q mq K K M =, [-A-](V R + V X ), N = [-^-](V R - V X ) (3.21a) T.v dm q m l.v q m dm At • At KA Note that [- ], V,, and V are diagonal matrices built up from the data T.v d q At of individual machines. So far we have eight state variable sets in the order of (V V'V ^D' V V V 6) 3.5. Torque Equations The linearized torque equation in MKS may be written pA<5 = Aw (3.22) pAw = TCAt - AT - AT ] (3.23) v m J m e D Now if Aue's unit is changed from MKS to per unit, and per unit mechanical torque At and electrical torque At are used in the formulation, me (3.23) becomes w Aw T Aw p n e = -Jl (t At - Dt — -At ) (3.23a) pp J o m o • w e o where pp is the number of pole pairs, w^ the base electrical rad/s, T^ the base torque of the complete system, t T the base operating torque of an individual primemover, and 27 A ATD j Aw D = t T ' w (3.23b) o n o Thus we have .0) Aw pAco = ~- (t At - At - Dt —-) (3.24) e2Homeoto o where 1 W W 9 H = £ j(_°_S.)z/p (3.24a) 2 pp n and is the base power of the system. Note that co co = 120-n- rad/s (3.24b) o n Thus to = 1 if real time is chosen as the base of computation. Other wise all time constants and H must be multiplied by co^. Now since At = A(iKi - i> i.) (3.25) e d q q d and At = g + 1.5h (3.26) m b for a hydraulic system, substituting i^ and i from (3.11) into (3.25) and the results into (3.24), we have pAtoe = A(9,l)AipF + A(9,2)A^d + A(9,3)AiJ) + A(9 ,4) A^D 2 2 to D to t to t + A<9'5>% " 25" VUe + ."^21 8 + !"¥ h (3'27) where 2 to A(9,l) - [*q] YdF 2 A(9,2) = ([Iq] + [^q]Ydd) 2 to A(9,3) =I§ ([^]Yqq+ [ld]) to2 (3.27a) A(9,4) --2JH*q]YdD 2 to A(9,5) = 2i [^d]YqQ 28 (3.27b) The complete state variable sets are / <ipF»^d» iq> iP-Qy i>q> VF» VR» 5> <*>e> a> af> g» h) (3.28) including governor actuator signal a and feedback a^ as in Fig. 3-3. 3.6. Governor-hydraulic System Fig. 3-3 shows the block diagram of a typical governor-hydraulic system I 1+Trs I Fig. 3-3 A Typical Governor-Hydraulic System The corresponding state equations are pa T 3 ~ T af 1_ C a 1 . 1 —=~ Ato - = u co 1 e T G o a a PS, = -Pg = T a T f co T a r a o a 1 Acoe - — uG a (3.29) 3.7. State Equations There are altogether 13 sets of state variables, (3/28). Ea-ch set is an m-vector for an m-machine system. Equations' (3.17a), (' (3.17b), (3.18a), (3.21), (3.22), (3.27) .and (3.29) are the complete sets of the system state equations. They are assembled into a matrix equation form as Y = A Y + Bu, Y = A(«,F ^ ^q ijijj ^ u = (u£ ua), R (3.30) <5 to a a„ g h), (3.30a) (3.30b) B = K 0 0 0 0 0 0 "A 0 0 0 0 0 0 O'OOOOOO 00 zi Hi. Ta Ta 0 0 (3.30c) and A is given as equation (3.30d) including (3.21a) and (3.27a) as the auxiliary equations. It is obvious that any other type of exciter and governor systems can be easily incorporated with the rest of the state equations. 3.8. Multi-machine System with an Infinite Bus For the study of m machines with an infinite bus, the. matrix equations (3.4) can be partitioned as V m = " z mm Z m°° i m V CO Z . °°m Z oooo i oo (3.31) Linearization of (3.31) can be written as 'rFYFF "rFYFd 0 "rFYFD 0 •RY._, -RY,, w +X' Y -RY.,. XI. 0 dF dd e m qq dD m qQ 'XmYdF "We"XmYdd -Rl -X Y.„ -RY. qq m dD qQ •r Y -r Y D DF D Dd ~rDYDD 0 -r Y • Q Qq 0 0 -VQQ o o -I/TE I/TE A(7,l) A(7,2) A(7,3) A(7,4) "A(7,5) 0 0 0.0 0 0 0 A(9,l) A(9,2) A(9,3) A(9,4) A(9,5) 0 -1/T, mq >q] U md 0 A(7,8) 2H" 2. "0*0 2H 4H. .l/UoTa -,/Ta -l/T -1 *t t' 0 a v t a r a l/T -2/T -l/T 2/T -2/T. (330d) "Av " m "z mm z m°° "Ai " m + i Av 00 Z . °°m Z COCO Ai Z Z mm m°° Z Z cofrl cooo I 0 m V 0 m 0 v / 01 . / A<5 m AS (3 Note that I and V are diagonal matrices with i and v as- diagonal mm m m ° elements respectively. Since for an infinite bus we have Av = 0 00 A6 =0 oo Substituting (3.33) into (3.32) and eliminating Ai^ results in Av. = Z Ai + j U A<5 m m m m m where U = Z I - V , m mm m and Z = Z - Z Z /Z m mm m<x> °°m °°°° (3.33a) (3.33b) (3.14a) (3.34) The linearized state equations of the multi-machine system with and without an infinite bus have exactly the same form. But we have to eliminate the infinite bus when the network impedance matrix is expressed in machine's dq Coordinates, (3.34). '3.9. Simplification of Power System Dynamics For System analysis and design purposes it is usually desirable to simplfy the dynamic description of the system. Numerical approaches 25 26 of approximating high order systems by low order systems are available ' The principle involved is to retain only the dominant eigenvalues of the exact system in the reduced model. The individual system parametric values, however, are completely lost during the process of numerical approximation. The simplification of power system dynamics is different in nature. It is governed mainly by the degree of accuracy of describing the flux linkage variations of the synchronous machine windings. Three different approximations are suggested A: complete description for the system, 7th order syn chronous machine, first order voltage regulator and 4th order governor. B: neglecting damper winding flux linkage variations, i.e. P^D = P *p = 0 C: neglecting damper and armature flux linkage variations P^D = P^Q = 0, and 'P*d = P*q = °» C': The same simplification as in model C, except that the system has no governor representation. The simplification can be easily implemented on the high . order system equations (3.30) using matrix elimination technique. The linearized state form equations of a multi-machine power system with 5th order synchronous machine, model B, with second order voltage regulator and exciter system are given in appendix A. From the numerical example of a typical one machine infinite system, Fig. 5-1, it is found that the dominant eigenvalues differ very little from each other in the different simplification methods. Table 3-1 shows the eigenvalues of the typical one machine infinite system of different modelling. Although there are dynamic couplings among all system state variables, roughly, the model Eigenvalues #1 #2 #3 u A .165+J4.69 -15.2,-3.99 -14.8,-2.24, -1.15,-.034 -847+J3151,-26.1,-12.4 B .229±j4.67 -16.9,-3.76 -15.1,-2.23, -1.15,-.034 -486+11857 / C .234+J4.67 -16.9,-3.77 -15.1,-2.23, -1.15,-.034 C* .178+J4.77 -16.9,-3.68 Table 3-1 Eigenvalues of the Typical One Machine Infinite System of Different Modelling 4 column eigenvalues correspond to the mechanical system, the voltage regulator and excitation system, the governor system, and the synchronous machine armature and damper windings respectively. Here Column #1. gives the dominant eigenvalues. -4. OPTIMAL LINEAR REGULATOR DESIGN WITH DOMINANT EIGENVALUE SHIFT27 Optimal linear regulators have been designed for power 11 12 28 systemstabilization ' and for frequency control . . The performance function J must be chosen in the quadratic form, J = I • /°°(Y'QY + u'Ru)dt (4.1) I o The choice of the weighting matrix Q of (4.1) is entirely left to experience and guessing until satisfactory results are obtained. In this chapter a new method is developed to determine Q in'conjunction with the dominant eigenvalue shift of the closed loop system as far as the practical controllers permit. For the eigen value shift of an n-th order system, it is found that it is sufficient to adjust the n diagonal elements of the Q matrix alone without the need of changing the off-diagonal elements. This also leaves out the change in R elements which decide the relative strength of the different control signals and can be left entirely to economical and practical considerations. 4.1. Linear Optimal Regulator Problem The linear optimal regulator problem may be formulated as follows. Consider the linearized system state equations Y* = AY + Bu . (4.2) Find the optimal control which minimizes the chosen quadratic performance function of (4.1) subject to the system dynamics constraint (4.2). The 29 optimal control is given by 35 u = -R~1B' K Y (4.3) and the Riccati matrix K satisfies the nonlinear matrix algebraic equation KA + A'K - K B R 1B'K = -Q (4.4) With u decided,the closed loop system equations become Y = GY (4.5) where G = A - BR""1B'K (4.6) Thus the eigenvalues of the closed loop system G depend upon the selection of Q for J in (4.1). Consequently the designed optimal controller is not necessarily the best since Q is arbitrarily chosen. On the other hand if Q is adjus ted.constantly and simultaneously with the dominant eigenvalue shift of the closed loop system, the results will be. the best. 4.2. Eigenvalue Shift Policy The shift is restricted to the real part and to the left. Let all the eigenvalues of G be ordered as a vector always with the eigenvalue with the largest real part as the first' element, A^, and the rest in decreasing order of magnitude. A three-point shift policy is established to avoid unnecessary and undesired large change in Aq which may result in impractical controller gains 1. Assign a negative real shift e to the most dominant eigenvalue A only. 2. Keep all negative movements of less dominant eigenvalues, e.g., those having negative real parts up to five or ten 36 times that of X^, within e and damp out all positive movements to the right to avoid their to and fro motion. 3. Relax the movements of the remaining eigenvalues to avoid unusually large controller gains. 4.3. The Shift Let the incremental change in an eigenvalue X^ resulting from the change in the diagonal elements of the weighting matrix Q, written as a vector q, be AX. = X! Aq (4.7) l x,q Z1 since for a conjugate eigenvalue pair A. - X*+1 (4.8) their sensitivity coefficients are also conjugate X. = X* (4.9) x,q l+l,q Therefore the increments AX. = AX* (4.10) I i+l There are, in general, k real eigenvalues and (n-k)/2 conjugate eigenvalue pairs of the n-th order closed loop system G, and only (n+k)/2 independent eigenvalues need to be considered in the shifting process. Let the number be p. Let the p-eigenvaliie vector shift be AX =X,q Aq (4.11) and let them be separated into real and imaginary parts AX = AC + j An (4.12) Then the real part-of AX may be Written .37 • A £ - S • Aq (4.13) where • ' S ^ Real (A,q) (4.,14) 4.4. Determination of Aq Let the number of dominant eigenvalues be m. Since A^ cannot be shifted alone, let a weighted total real shift of the m dominant eigenvalues be z = e^ed) + e2Ac;(2) + ... + emAs(m) (4.15) From (4.13) We have E - 4> * Aq (4.16) where and <|) = (^,...,<!..,...,tj)^' (4.16a) <j>± = $1S(l,i) + B2S(2,i) + ...+emS(m,i) (4.16b) The 3's are positive numbers satisfying the shift policy point two. To make E negative, Aq is moved in the direction of the steepest descent, Aq « -k<f> , k > 0 , . (4.17) The step size k is so determined that it will have a negative shift for the most dominant eigenvalue A^. 4.5. Sensitivity Coefficients A,q 30 Although Chen and Shen gave two algorithms to compute A,q their method requires many computations and large computer storage. A new sensitivity formula for A,q is developed in this section. The computation of A,q and the solution of the Riccati matrix K will be much simplified through an eigenvector matrix X of a composite matrix M; M = A -Q -BR~1B' -A' (4.18) 31 32 The composite matrix M has the following properties ' 1. The 2n eigenvalues of M are symmetrically located with respect to both real and imaginary axes of the complex plane. Let the eigenvalue vector A of M be partitioned as A = [Ar AII]' (4.19) where A has negative real parts and A^ has positive real parts. Then we have AII = "AI (4.20) 2. The eigenvalues with the negative real parts of M are the same eigenvalues of the optimal closed loop system G, i.e. AT = (X.,...^.,...,* )' (4.21) i i l n 3. The solution of the Riccati matrix equation (4.4) is -1 K - XIIXI where X' X, III X II X IV (4.22) (4.22a) is the eigenvector matrix of M, and the first column of the eigenvector matrix X corresponds to the stable eigenvalues A . 39 4. The eigenvector matrix of M' may be Written V = XIV XII ~XIII ~XI (4.23) Let an eigenvector of the stable eigenvalue X_^ of M be X. - (Xl. , xm)' and that of M' be (4.24). Vi = (XIVi . • -XIIIi)' 33 Following Faddeev and Faddeeva , we have (4.25) AX. = ~ V! AM X. l . C. I I l (4.26) where C = V!X. l ii (4.26a) Since in our case We shall have AM = '•-AQ l] AXi = CT Xilll AQ XIi I For the diagonal changes in Q We write where and AX. = X! . Aq l i,q i,q i,ql' i,q2' i,qj i,qn (4.27) (4.28) (4.29) (4.30) Xi,qj-= cTXIIIi.«> XIi^ (4.31) where n Ci = * [XlVi^)XIi(J) " XIIIi^)XTT.i(J)] 3=1 (4 4.6. Algorithm The algorithm for the design of linear optimal regulators with dominant eigenvalue shift is summarized in Fig. 4-1. o AO A AND X OF M CHECK CONTROLLER CAINS Aq Fig. 4-1 Algorithm to Determine Q with Dominant Eigenvalue Shift 1. Start with a small arbitrary Q. 2. Find the eigenvalues A and eigenvectors X of the composite matrix M. 32 3. Calculate K from the stable eigenvectors of X and check the controller gains at each shift. 44. Find Aq from the sensitivity coefficieints A,q. 5. Update Q and repeat the process until a satisfactory eigen value shift is made or until the practical controller's limit is reached. 5. OPTIMAL POWER SYSTEM STABILIZATION THROUGH EXCITATION AND/OR GOVERNOR CONTROL27 In this chapter the linear optimal regulator design technique developed in the previous chapter is applied to the optimal stabilization of a typical one machine-infinite system, Fig. 5-1. Three different optimal stabilization schemes are investigated, the first with an optimal excitation control u^,, the second with optimal governor controls U-, and u' , with and without the dash-pot, and the third with u plus u^ control. The linear optimal stabilizing signals thus obtained are tested on a high order nonlinear model of the system with detailed description. It is found from the'test results that the optimal controls are more effective than conventional excitation control, that the optimal governor control with the dash-pot removed is just as good as the optimal excitation control, and that the Optimal u„ plus u' control is the best E G way to stabilize a power system. 5.1.' System Data A typical one machine-infinite system as shown in Fig. 5-1 is chosen for this study. The regulator-exciter and governor-hydraulic systems are shown in Pig. 3--2 and Pig. 3-3 respectively. / Fig. 5-1 A Typical One-Machine Infinite System The system data are as follows r X G B V 0 V t P 0 H . D -.034 .997 .249 .262 1.02 1.05 .952 4.63 0 *d x' xd x" xd X q x" q rpll do rj-. 11 qo TD .973 .190 .133 .55 .216 .0436 .0939 .13 KA TA TE h a *t T a T r T g 50 .05 .003 .182 .05 .25 .02 4.8 .50 controller constraints are, exciter amplifier limits (p.u.) 8.83 and -7, dash-pot signal limits + .025 p.u.,-governor gate speed limit .1 p.u./sec, excitation control limits + .12 p.u. and governor control limits + .0.15 p.u. For the design the synchronous machine is described as a third order system with p» <S, to)' as the state variable vector. Thi is done by neglecting the flux linkage variations in the armature and damper windings. 5.2. Case 1: u 'Control E The system has an optimal excitation control u . The time constant T of a solid state exciter is neglected and the voltage E regulator of Fig. 3-2 is approximated as a first order system. For the data given, the per unit linear state equations for the complete system are where • = -.196 1.0 -1.39 -.003 + "o' -50.9 -20. 87.0 -2.4 *VF 1 A 6 0 0 0 1 *6 0 • . Aco -2.94 0 -22.6 -.008 a 0). 0 u = 1000 u^ ex E (5 (5 The optimal control signal u is found as E (-.099Ai|;„ - .004Av„ - .62A6 + O.lAu) The final value of the diagonal elements of Q are (2524 , 0 , 913.6 , 23865) The control T^eighting R is unity. The eigenvalues of the initial svstem without u„ control are E (.178 + J4.77 , -3.68 , -16.9) and the eigenvalues of the final system with u„ control are (-2.07 + J4.9 , -3.85 , -16.7)' Thus the dominant eigenvalue pairs are shifted from (.178+J4.77) to (-2.07 + j4.9) / 5.3. Case 2a: Control, With Dash-pot ^ / The system has an optimal governor control u . The 4-th order governor hydraulic system is as Fig! 3-3 and the voltage regulator is approximated as a first order system. For the data given the per unit linear system state equations for the complete system are Y = AY + Bu where A = Y = [ AUJ F' Av F* A<5, AGO, a, af, g, h]' u = uG B = [0, 0, 0, o, -50, -12.5, 0 > 0]' -.196 1.0 -1. 39 -.003 0 0 0 . '0 -50.9 -20 87. 0 -2.4 0 0 0 0 0 0 0 1 0 0 0 0 -2.94 0 -22. 6 -.008 0 0 38.8 58.2 0 0 0 -.133 -2.5. -50 0 0 0 0 0 -.033 -.625 -12.7 0 0 • 0 0 0 0 2 0 -2 0 0 0 0 0 -4 0 4 -12.5 (5.2) (5.2a) (5.2b) (5.2c) (5.2d) and R is set R (5.2e) The optimal control signal u is found as 45 (.0255A4»F + .0012Av + .126A6 - .0254Aw + .08a - .112 af - . 3g - .4h) The final values of the diagonal elements of Q are (.56, 4.8 .116, 6.8, lO"4 .034, .0019, .52 , 0 ) The eigenvalues of the initial system without u control are (.23 + J4.67, -3.77, -16.9, -.034, -1.149, -2.23, -15) and the eigenvalues of the final system with u„ .control are (-1.35 + j4.9, -4.1, -16.8, -.049, -1.2, -1.6, -15) Thus the most dominant eigenvalue pairs are shifted from (.23+J4.67). to (-1.35 + j4.9) The eigenvalue -0.034, corresponding to a large time constant of the dashpot, has slow response to system disturbance and does not affect the earlier part of system stability. 5.4. Case 2b: ul control, without dashpot The dashpot is removed from Fig. 3-3 for this study. Neglecting the actuator time constant T the governor transfer function can be a ° written as 1/(a + T's) where T' = aT . For the data given the equations g g g for the complete system are 46 = F A<S • Aw g n -.196 -50.9 0 •2.94 0 0 1.0 -20 0 0 0 0 -1.39 87 0 -22.6 0 0 -.003 -2.4 1 -.008 -.1 .21 0 0 0 38.1 -2 0 0 0 % + 0 AvF 0 A<5 0 Aw 0 g -40 h 80 (5.3) and R is set R = 1 (5.3a) The optimal control signal u' is found as (.00628Aib + .0002AV_-+ .0238A6 - .01620Aw r r -.0216g - .1 h) The final values of the diagonal elements of Q are 10~4(0, 0, .0063, 1.83, 31.1, 0) The eigenvalues of the initial system without u' control are (.715 + J4.35, -4.3, -16.8, -.89, -2.8) and the eigenvalues of the final system with u' control are (-3.7 + j4.9, -3.27, -16.8, -1.18, -2.12) Thus the dominant eigenvalue pairs are shifted from (.715 + J4.35) to (-3.7 + j4.9) 5.5. Case 3: u^ Plus M.\ Control The system under study is the same as that of case 2b except it has both u„ and u' control signals. The last term of the system equations becomes [° 1000 00 0 0 ]' [UE] (5.4) 0 0 0 0 -40 80 where R=103[J ;6Q] (5.4a) and is chosen to coordinate the effort of excitation and governor control signals. The optimal u„ and u' control signals are found respectively as (-.047AiJv " -002AV-, - .319A6 + 0.05Aw + . 779g + .78h), r r (.005Aif^ +:0002Av_, + .025A6 - .0127ALO - .045g - .094h) r r The final values of the diagonal elements of Q are (1.42, 0, .0859, 25.8, 82.28, .025) The eigenvalues of the initial system without u„ and u' control are the same as those of case 2b and the eigenvalues of the final system with u^, and u' control are (-4.13 + J5.33, -3.6, -16.79, -.997, -1.66) Thus the dominant eigenvalue pairs are shifted from (.715 + j4.35) to (-4.13 + J5.33) 5.6. Nonlinear Tests All the optimal stabilizing signals thus obtained are tested on the same system of Fig. 5-1 but described by high order nonlinear differential equations with the synchronous machine as a 7-th order system (ij^, ij> , ifjp, ijjp, !pQ> 6, to), excitation and governor systems respectively as Figs. 3-2 and 3-3 with controller constraints. A conventional excitation control as designed in reference 12 .04s Aw (5.5) EC l+.5s using the speed deviation signal is also included for comparison. The system disturbance for the tests is as follows: a three-phase fault occurs at one of the system buses and the faulted line is isolated at 5 cycles followed by a system restoration at 30 cycles. The results are summarized in Fig. 5-2. The system responses for the system with conventional and the optimal excitation controls are displayed on the left column of the figures, and the system responses for the system with the optimal governor, and the optimal governor and excitation controls are displayed on the right column of the figures. • From the results, it is observed that: 1. Although the effort of the optimal excitation control signal u„ is smaller than that of the conventional excitation signal .hi u„^, the system with u„ control is much more stable. 2. The optimal governor control signal u' for the governor with er out dashpot provides more damping for the sytem than that with a dashpot. 3. The optimal excitation and governor signals, u and u', when coordinated, provide the best means for stabilizing a power system, i.e., more damping with less effort than either u^ or u' control. In other words, for the same amount of effort, the optimal u plus u' control has the ability to stabilize the system under more severe fault conditions. 4< 0.003 0.5 0.5 1.0 1.5 2.0 TIME (SECONDS) (e) 2.5 1.0 1.5 2.0 TIME (SECONDS) 20.0 N 15.0 10.04 9 50 CE -3 0 jj -5.0 -10.0 -15.0 II , 10 (.5 2.0 TIME (SECONDS) 0.010 0.020 £0.000 SO. 020 o •> j0.0<10 • a. So.060 --0.060 -\ -a 3.0 1.0 1.5 TIME (SECONDS) (ll 0.5 1.0 1:5 2.0 TIME (SECONDS) (gi 0.5 1.0 1.5 2.0 TIME (SECONDS) (l>) Continued. 1.5 2.0 TIME tSECONDS) (il 1.5 2.0 TIME (SECONDS) 1.0 1.5 2.0 TIME (SECONDS) (i) SO.005 1.0 1.5 2.0 TIME (SECONDS) If I Fig. 5.2 Nonlinear Test Results u = 0, unstabilized system u„n 5 conventional excitation control Ug , optimal excitation control u , optimal governor control with dashpot u' , optimal governor control without dashpot u plus u' control 6. OPTIMAL STABILIZATION OF A MULTI-MACHINE SYSTEM' The optimal linear regulator design technique of determining the weighting matrix Q in conjunction with the dominant eigenvalue shift developed in chapter 4, is applied to the optimal stabilization of a multi-machine system. Two systems are investigated, the first with a one machine optimal controller, u ,- and the second with a multi-machine hi optimal controllers, u^. Each design is given a nonlinear test on the same multi-machine system. It is found that the multi-machine system with a one machine optimal controller u^, designed for the multi machine system is better than a one machine optimal controller, u , designed for the same system but approximated as a one machine-infinite system, and that the multi-machine system with a multi-machine optimal controller, u , is better still than the multi-machine system with the one machine optimal controller, u j, designed for multi-machine system. 6.1. System Data and Description The system under study, Fig. 6-1, is the same as that of reference 12, consisting of one thermo plant (#1), two hydro plants (#2 and #3), and an infinite system equivalent (#4). Fig. 6-1 Typical Four-Machine Power System (Admittances in p.u. on 1000 MVA) The system data are as follox^s Plant r X T H D a d d d q q do do qo D #1 .0019 1.53 .29 .17 1.51 .17 4 .029 .029 .116 2.31 0 #2 .0023 .88 .33 .22 .53 .29 8 .022 .044 .077 . 3.4 0 #3 .0025 .97 .19 .13 .55 .216 7.76 .044 .094 .131 4.63 0 K, T. T R V V A A E F Rtnax Rrnin 13 .21 .15 .129 4.5 0 45 .07 .5 .237 3.5 -3.5 50 .02 .003 .12 8.8 -7 The operating conditions from load flow studies are Plant P (MW) o Q (MVA) 0 Vto(p.u.) 6(deg.) //l 26.5 37 1.04 -10.7 #2 518 -31 1.025 11.8 n 1582 -49.6 1.03 25 #4 410 49.3 1.06 0 For the design each plant is modelled as a fourth-order system (i/jp, Vp, 6, tii), a third-order synchronous machine plus a first-order exciter-regulator system. The linearized system equations are written as • Yl = "All A12 A13 \" + B 'Ul" • Y2 A21 A22 A23 Y2 u2 Y3. .A31 A32 A33. Y3 -U3. (6. For the data given the numerical values of the A and B matrices are 11 13 = -.922 1 -.266 -.009 -2.75 -2.78 -1.36 -.037 0 0 0 1 -4.95 0 -55.5 -.039 .072 0 -.25 .003 .46 0 2.8 -.02 0 0 0 0 .924 0 17.5 .02 12 21 .024 0 -.087 .158 0 1.11 0 0 0 .222 0 8.17 .021 0 .121 003 -1.1 0 -1.62 015 0 0 0 0 -2.43 0 1.37 034 0 004 22 -.21 -1.9 0 •3.1 1 -1. 0 0 -1.6 9.3 0 -56 -.005 -.12 1 .032 A 23 .06 0 .46 .002 -1 0 1.49 -.04 0 0 0 0 .12 0 29.8 -.028 31 33 -.002 0 .083 0 .011 0 .22 0 -6.78 0 -10.1 -.09 A32 = -2.1 0 1.7 -.123 0 0 0 0 0 0 0 0 -1.24 0 .498 -.017 .-,07 0 6.37 -.011 -.197 1 -1. 2 003~ -54.4 -20 70.1 -2.37 0 0 0 1 -3.4 -21 -.017 0 36.1 0 0 0 0 0 0 0 0 0 78.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 0 0 The eigenvalues of the unstabilized multi-machine system are -.013 + J7.8 -1.85 + jl.35 (#1) -.018 + j7.4 -1 + J1.3 (#2) + .177 -3.84 , + J3.98 -16.6 (#3) Although there are dynamic couplings among all three plants, roughly, the three column eigenvalues correspond to three plants respectively. Also the the first row eigenvalues of each column correspond to the plant dynamics, 6 and to. 6.2. Case 1: One Machine Optimal Excitation Control u EI Since it is found from the eigenvalue analysis of the unstabilized multi-machine system that plant //3 is unstable, a one-machine optimal excitation control, u„T, is designed for plant #3 in order to stabilize the multi-machine system. In the design, of course, all system dynamics are included. The diagonal elements of the weighting matrix, Q, determined -3 from the dominant eigenvalue shift are the listed Values times' 10 , for R=l. AVF AS Aco plant #1 .011 .018 .348 19.6 plant #2 .023 .536 .284 18.3 plant #3 0 0 .022 .523 u„T for plant #3 are A^p Avp A6 Aco plant #1 .0172 -.0128 .88 -.04 plant #2 -.0345 -.0109 -.28 -.14 plant #3 -.154 -.0066 -.878 .18 The eigenvalues of the final multi-machine system are -1.17 .+ J7.86 -1.77 + jl.36 (#1) -.3 + J7.86 -1.02 + jl.25 (#2) -1.88 + J3.55 -3.6 , -16.6 (#3) Thus the first txro eigenvalues of the last column are shifted from +.177 + J3.98 to -1.88 + j 3.55, indicating great improvement in damping of plant #3. The control signal, u„T, also improves the damping of plants #1 and #2. 6.3. Case 2: Multi-Optimal Controllers u„„ One would expect that a multi-machine system with multi-optimal controllers will be better stabilized than the system with only one optimal controller. This is studied as case 2. The multi-optimal controllers are designed, of course, simultaneously considering all machine dynamics. 56 The diagonal elements of 0 determined from the dominant eigenvalue -3 shift are the listed values times 10 , A<S AGO plant #1 .145 .001 2.64 97.2 plant #2 4.65 3.36 3.11 93.6 plant #3 .1 .0007 4.02 88.2 The weighting matrix elements of three plant controls which give the best results are R = diag (1, 2, 10) of the three control signals are: UEM(#1) Avp A<5 ACQ plant #1 -1.06 -.029 -.639 •S.18 plant #2 -.052 -.0039 -.588 .0313 plant #3 -.073 -.0026 .127 .137 UEM(#2) A^F AvF A<S Ato plant #1 .0084 -.00427 .539 .0218 plant #2 -.069 -.0399 -.826 -.132 plant #3 -.0406 -.00146 -.0097 -.10 EM(#3) AIJJF Avp A 6 AGO plant #1 -.00569 -.0072 .38 -.0225 plant #2 -.00832 -.00369 .0718 -.0516 plant #3 -.1123 -.00497 -.718 .1156 The eigenvalues of the final multi-machine system are -1.01 + J7.64 -1.94 + jl.099 -.448 + J7.89 -1.7 , -2.74 -2.05 + J4.04 -3.03 , -16.65 (#1) (#2) (#3) There is no doubt that a multi-machine system with multi-optimal controllers, u^, is better than the system with a one plant optimal controller, . 6.4. Case 3: Approximated One-Machine Optimal Design For comparison, the u optimal excitation control signal of the same power system as cases 1 and 2, but approximated as a one-machine infinite system as in Chapter 5, is recorded here... The control signal u„ = -.099AiJ; - .004AvT, - .620A6 + . lAw E F F was designed for plant #3 as the one-machine and infinite system. When this signal is applied to plant //3 of the multi-machine system the eigenvalue are .084 + J7.46 -.1 + j7. -1.46 + jl.15 -.63 + jl.51 -2.88 , -15.9 -3.3 + j4.5 (#1) (//2) (#3) When these results are compared with the eigenvalues of the unstabilized multi-machine system, it is found that the u control signal does improve the damping of plant //3, but not much of plant #1 or #2. 58 6.5. Case 4: Subsystems Optimal Design One would be curious to know what would happen if all plants had individual u„ control-designs. This is to say .that all the dynamic b coupling of the three plants, off-diagonal elements of the A matrix in (6-1), will be neglected and the individual optimal controllers are designed from -' —-- -Yl ~ A11Y1 + blUl Y2 " A22Y2 + b2U2 (6.2) Y3 = A33Y3 + b3U3 respectively. Applying the dominant eigenvalue shift technique, the -3 individual weighting 0 matrices are the listed values times 10 ; for R = 1 in each design, A*F A6 Aco plant #1 0 0 2.296 72.88 plant #2 -4 .7x10 .8xl0~2 2.219 69.38 plant #3 .19 0 .549 12.6 The gains of the individual optimal controllers are AiJJF A<5 Aco u, (#1) -.0738 -.0231 -1.059 -.2018 uE (#2) -.0446 -.0182 -.838 -.228 uE(//3) -.075 -.0035 -.455 .071 The eigenvalues of the individual closed loop systems are (#1) (#2) (#3) -.58 + J7.52 -1.1 + jl.16 -.418 + J7.48 -1.868 + jl.25 -1.9 + J4.766 -3.23, -16.6 Next the eigenvalues of the multi-machine system are: With u„(//l) alone -.36 + J7.42 -1.73 + jl +.014 + J7.84 -.93 + jl.4 -.11 + J3.96 -3.86, -16.62 With u^ (#2) alone -.04 + J7.45 -1.89 + jl.29 -.44 + j7.8 -.956 + jl.15 -.097 + j4 -3.84, -16.62 With u„ (#3) alone hi -.064 + J7.46 -1.47 + 11.25 -.079 + J7.83 -.778 + jl.59 -2.33 + J4.08 -3.24, -16.71 With all three uE's -.419 + J7.58 -1.55 + jl.136 -.463 + J7.843 +.169 , -2.25 -2.53 + 74.47 -2.95, -16.7 Although the individual optimal controller provides good damping to the individual plant, the effects on other plants are unpredictable. 6.6. Nonlinear Tests The optimal stabilizing signals thus obtained are tested on the same system of Fig. 6-1 but described by high order nonlinear differential equations including the controller's constraints. The system disturbance for the tests is the same one used in the previous chapter. The test results are summarized in Fig. 6-2. From the results the following is observed 1. - u„ controller, designed for the system approximated as one-E machine infinite system, case #3, provides the required damping to plant #3 but not much to other plants. 2. u^-j. controller, case #1, provides damping to each plant in the system, allowing the controller to stabilize the system for wider fault locations than the case with u„. E 3. u„,, controllers, case #2, provide the best stabilization for EM . the whole system with less effort than the case with u„ or u„T. 4. The simplified subsystems controllers fail to stabilize the system. TERMINAL VOLTRGE (PUJ • I,, DELTA IDEG) SPEED DEVIATION (PU) to 1/1 ; AND 3 1.3 :.S 2.0 TIME ISECON0S) 3.0 0.C8 0.04 -0.00 -0.04 --o.os --0.12 -0.16 ; AND J 'A // ^ / \ 0.5 1.0 1.5 2.0 TIME (SECONDS) ! z.s 3.0 4.0 0.5 1.0 l.S 2.0 Tlt£ '.CECCNDSl i 0.5 —i 1 1— 1.0 l.S 2.0 TIKE (SECONDS) Je2) 2.5 3. 0.5 1.0 1.5 2.0 TIME '.SECONDS J (d3) 2.5 3.0 :.0 1.5 2.0 TIME: 'isEcoros: fe3) 2.S 2 (plant #1) (plant #2) (plant #3) Fig. 6.2 Nonlinear Tests of the Multimachine System (a three-phase fault disturbance) 2. with multi-optimal controllers u£^ 4. with three individual^optimal controllers 1. with one optimal control u^j on plant #3 3. with.up on plant #3; approximated one machine-infinite system 7. OPTIMUM STABILIZATION OF POWER SYSTEMS / OVER WIDE RANGE OPERATING CONDITIONS34 / I Nominal system operating conditions were assumed in chapters 5 and 6 for the design of the optimal stabilizing signals. In real power systems the operating conditions are not constant but subject to the load demands over the system. The question arises: How can we design an optimal controller for the power system sensitive to and good for the wide range of operating condtions? In an attempt to answer this question, an optimally sensitive controller is developed in this chapter. The controller is capable of adjusting its effort in such a manner that optimum system stabilization can always be achieved over the wide range operating conditions.. The sensitive controller thus designed under such conditions is tested on the nonlinear model. The results are compared with that of the system with a nominal controller.. 7.1. Optimally Sensitive Linear Regulator Design Constructing a controller x./hich preserves optimality for a nonlinear control system in spite of its parameter variations has been 3 *5 36 37 the object of several recent publications '" ' . The synthesis of linear optimally sensitive controllers by means of perturbation of the Riccati equation (A.4) is dealt with in this chapter. Let the linearized system equations be 6. Y = A(q) Y + Bu (7.1) where q is a vector contains the m changable parameters of the/system. / / For a quadratic performance function / 1 00 J = — / (Y'QY + u'Ru)dt, (4.1) the optimal control law is u* = -R-1 B' K(q) Y (7.2) where K(q) satisfies the Riccati matrix equation, K(q)A(q) + A'(q)K(q) - K(q)BR_1B*K(q) = -Q (7.3) In conventional regulator design the controller is computed for nominal values of the plant parameters q u = -R_1B'K(q )Y , (7.4) o o for a constant K(q^) • This will be referred to as the nominal optimal control hereafter. But this becomes impractical for system over wide range operating condition^.. It .implies that it is necessary to recompute K for a large number of sets of the plant parameters q, and the implementation of u* under every operating condition. To approximate the control lav; of (7.2), an optimally sensitive control ug is introduced. This control ug tends to track the new optimum of J whenver there is a variation in q. The first order approximation u s is written as m u . = -R~1B,[K(q ) + .E K Aq.]Y (7.5) ' si no i=l q. a. x The Riccati sensitivity matrices K are obtained from the differentiation v of (7.3) w.r.t. q^^ where and K G + G'K = -C, (7.6a) q. q. 1 i i G = A - BR^B'K, / (7.6b) C = KA + A* K (7.6c) 1 q. q. l i The second order approximation ug may be written as i m - m u „ = -R B'[K(q ) + .ZnK Aq. + ^ ?- • K Aq.Aq.]Y (7.7) s2 V*V 1=1 q± 4i 2 ij q±q 4i Hj where the sensitivity matrices K are computed from equations (7.6), ^i and K from q .q . 1 J K G + G'K = -C„ (7.8a) q.q, q.q, 2 where C = K G +G'K +KA +A'K + 2 q.q. q. q. q.q. q.q. Mi Hj "i Hi Hi + K A + A' K (7.8b) q.q. q.q. and G = A - BR_1B'K (7.8c) q. q. q. J J 3 Equation (7.8a) is obtained by differentiating (7.6a) with respect to q^. Other matrices of equations (7.6) and (7.8) are computed for q = q^. The procedure can be extended to obtain a controller with higher order approximation by adding more Taylor series terms. Hoxjever, it will become increasingly difficult to implement the high order controller with a large number of changeable parameters. The structures of the nominal controller and the optimally sensitive controller of the first order approximation are shown in Fig. 7-1 a and b respectively. / SYSTEM SYSTEM Y'= A(q)Y+ Bu R B K Y . A(q)Y+ flu -I ' R 8 K. R-'B'K a. Nominal optimal regulator b. Optimally sensitive regulator Fig. 7-1 Structures of Nominal and Optimally Sensitive Controllers The Riccati sensitivity matrices, necessary for the optimal sensitive regulator design, must be computed from the Lyapunov matrix equations of (7.6) and (7.8). A new technique is developed to solve these equations and is given in appendix B. The computational effort is much reduced by the use of the known eigensystem of the closed loop matrix G. 7.2. Sensitivity Equations of the Linearized Power System For the design of optimally sensitive controllers it is necessary to compute the system sensitivity matrices A^. This section deals with the derivations of these sensitivity matrices for a general multi-machine power system. There are in general (4n-l) variables that affect the steady state operating condition for an n machine power system, three terminal conditions for each machine and (n-1) angular differences between net work buses. The operating conditions are expressed in terms of i^, i , v , and S which give the simplest sensitivity expressions. Referring to the multi-machine equations of chapter 3. the deviations of Z of i m (3.5), U of (3.14b), M and N of (3.21a), >, and i|> of (3.10) for m a q varying operating conditions are as follows AZ = jZ [A5] - j[A6]Z (7.9) mm m The real and imaginary parts respectively are and Next, AR = -X [AS] + [A6]X (7.9a) mm m AX = R [LSI - [A6]R (7.9b) mm. m AU = Z AI + AZ I - AV (7.10) m mm mm m using (7.9) and (3.14b), U can be written as m AU = Z AI + jU [A6] - j[A6]U - AV ' (7.11) mmmm m m The real and imaginary parts respectively are AU , = R AI, - X AI - U [A6] + [A6]U -AV,, (7.11a) md m d m q mq mq d and AU = X AI, + R AI + U ,[A6] - [Ao^U , - AV (7.11b) mq m d m q md md q Note that V = V, + jV , and I = I, + jl . They are diagonal matrices mdq mdq with v and i vector elements of each machine as the diagonal matrix mm elements. Next, ' AM = I^—]{V,ARm + V AX + A2AV R - [^.]AV,X + l.v dm qm v dm I dm At t vfc + A 2AV X - [—^HV R} (7.12) v^_ q m 2 q m t Vt KA Vd 2 AN = [~]{VAR - VAX + R'AVRm + I.v q m dm v q m -SL£LlAv v _ r-iL^AV y - tJLA> + Hy^AV X - C-^] AV.X - [-^]AV,R } / (7.13) 2 am v dm I dm/ Vt t vt / -VdVq Vd Note that [—9 ] > [~—]» etc. are matrices consisting of diagonal Vt elements computed from data of individual machines. Finally the armature flux linkage variations from the normal steady state operating conditions are as follows, From (3.10) we have Ai|>, = —(Av + r Ai ) (7.14) d co q a q o From (3.10b) we have x = - -3- Ai ' (7.15) q • a, q In the case of a one machine infinite system, all matrices become scalars and the sensitivity equations (7.9) through (7.13) reduce to AZ = AR + jAX = 0 (7.16) m m J m u» AU = RAi - (X + x )Ai (7.17) md dm q q AU = X Ai, + R Ai - Av (7.18) mq m d m q q KA 2 AM = {(v R - v.v X )(x Ai - r Ai.) 3 wq m d q m/ q q a &J At + (v2X - v v R )Av } (7.19) dm d q m q KA 2 AN = -^r {(v, R' + v.v X )Av ^ 3 d m d q m q A. t -(v2X + v ,v R ) (x Ai - r Ai,)} (7.20) q m d q m q q a d J 69 The system sensitivity matrices A , q=(i,,i ,v )', for a one machine q a q q 7 infinite system of the 5th order synchronous machine model equations of / appendix B are as follows. / 0 . 0 0 0 0 0 0 0 0 0 0 0 -X m 0 0 0 0 0 0 R 0 0 0 0 0 0 0 0 Y' "dF M-' Y' N.. Y'' x^.qq. 0 0 A. (5,6) d 0 0 0 0 0 0 0 0 0 0 2 % 2H 0. 0 0 V c 2H (7.21) 0 0 i df q o o o i dd q 0 w x w x 0 q v' ° ^CY' -Y» ^1 2H dF 2H K dd qqJ 0 0 N. Y' lq qq to r o a 2H qq 0 0 0 0 0 0 0 0 0 0 0 0 Y1 0 0 m •(X +x ) m q' A± (5,6) q o o -x _£L o 0. 0 0 (7.22) v +x i,+2r i q q d a q 2H D J / 70 v 0 •0 0 v dF q 0 0 0 0 0 M Y^M v dd q 0 0 0 0 0 N Y' Vq M 0 to _° Y' 2H qq 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 Av (5,6) q o o to 0 _1 i) o 0 0 0 - V D 2H (7.23) where M id -K. r (v2R - v,v X )/TAV3 A av q m dqmAt N. = K. r (v2X + v.v R )/TAv3 i^ Aaqm dqmAt (7.24) K 2 V = Vt" d M ~VQ - —T~ ra vt and M. = K.x (v2R - v,v X )/T V3 l Aqqm dqm At q . N. = -KAx (v2X + v.v R )/TAV3 l Aqqm dqm At (7.25) and A. (5,6) q KA v U + v.v U , iAVt d m q?. v2 q M = K. (v2X - -v'v R )/T.V3 v Adm dqmAt q N = Kk(y% + vjV X )/TAv3 V A d m d q m At q (7,26) KAvd K Av (5'6) T~v~ .3 ^vd"md ' vdVmq q At T.v H H A (v?U J + v,v U ) A t Although the system sensitivity matrices are derived in terms of the variations Ai^, Ai^, and Av^, it is always possible to relate these variations to another measurable set through a nonsingular transformation. For example, "Aid" = T"1 AP Ai AQ q Av Av L q L t J (7.27) where T = VA - T- XA d ad V + r i q a q Vd —- r vt a v + x i, q q d -(v + X i ) d q q vd -— X vt 1 V V t (7.27a) 7.3. Optimally Sensitive Stabilization of a Power System The one machine infinite system of Fig. 5-1 is chosen for this study. The synchronous machine is described as a 5th order system with ijjp, lf^* t 6, and 00 as the state variables, appendix A. The voltage regulator is approximated as a first order system by neglecting Tp for the solid state exciter system. Nominal system operating conditions are in p.u.; 72 P = .952 , 0 = .015, and v = 1.05 o o to (7.28) The per unit linear state equations for the system at these nominal operating conditions are where A = o Y = AY+Bu 0 Y = AO1 F» *d» V v 6, to) B = (0 0 0 10 0)' data given in chapter 5, u = 1000 u„ E -.660 8.55 0 1 0 44.9 -723 1230 0 59.9 153 -2848 -250 0 -497 418 6736 -368 -20 1125 0 0 0 0 0 5.95 62.7 86.6 0 0 0 -.449 -.954 0 1 0 (7.29) (7.29a) (7.29b) (7.29c) (7.29d) The technique of determining Q developed in chapter 4 is applied to the nominal optimal regulater design of the system. With the weighting factor for control chosen as R = 1, Q is found to be Q = diag.(0 1.55 16.3 0 737.4 19084) (7.30) The Riccati matrix is 1.79 .215 .08 .078 12.2 -2.48 .215 .04 -.003 .008 .863 -.549 .08 -.003 .015 .005 1.1 .114 .078 .008 .005 .004 .6 -.08 12.2 .863 1.1 .6 133 -6.5 2.48 -.549 .114 -.08 -6.5 7.78 The nominal optimal control through excitation is UE0 = (~-078 --008 --005 --004 -'6 -08) Y (7.32) The system sensitivity matrices, of equations (7.21), (7.22), and (7.23), are computed at the nominal operating conditions. Their values are given in equations (7.33), (7.34), and (7.35). To check the computation of A^ matrices, the system matrix A is computed from the linearized equations (A.3d) and from the sensitivity equation A = A + A Aq. A good agreement between both methods is realized over o q a wide range of system operating conditions. 0 0 0 0 0 0 0 0 0 0 -470 0 0 0 0 0 138 0 -.17 2.75 6 0 202 0 0 0 0 0 0 0 0 0 41 0 0 0 (7.33) It 0 0 0 0 0 0 0 0 0 0 -137 -.55 0 0 0 0 -678 -.003 31 -505 -1103 0 1445 0 0 0 0 0 0 0 -7.3 77 .22 0 0 0 -0 0 0 0 0 —i 0 0 0 0 0 377 0 0 0 0 0 0 -1 -27 431 943 0 -140 n 0 0 0 0 0 0 . 0 0 74 0 0 0 (7.34) (7.35) The Riccati sensitivity matrices are obtained by solving the Lyapunov matrix equations (7.6) using the frequency domain technique developed in appendix B. These matrices are K. l. 103x 1 .227 .009 .044 -10.4 -1.85 .227 .049 .002 .01 -.606 -.396 .009 .002 -.002 0 -.833 -.094 .044 .01 0 .002 -.469 -.092 -10.4 -.606 -.833 -.469 -196 7.11 -1.85 -.396 -.094 -.092 7.11 1.32 (7.36) / 75 K. * 10 x 1 1.49 .074 .045 .056 13.4 ,333 .074 -.013 -.003 .002 1.14 .292. .045 -.003 .015 .004 .756 .161 .056 .002 .004 .002 .578 .046 13.4 1.14 .756 .578 112 -3.73 .333 .292 .161 .046 -3.73 -4.54 (7.37) v 103x -.654 -.03 -.022 -.031 13.1 1.51 -.03 .04 -.025 -.004 .836 -.267 -.022 -.025 .013 0 1.23 .242 -.031 -.004 0 -.001 .613 .085 13.1 .836 1.23 .613 241 -6.61 1.51 -.267 .242 .085 -6.61 -.918 (7.38) The first order optimally sensitive excitation control, equation (7.5), is then designed u^ = (-.078 -.008 -.004 -.6 .08) Y + + 10 (Ai, , Ai Av ) d q. q o o o -44 -9.7 -56 -1.5 31 4.2 -.07 -1.9 470 92 •3.7 -2.4 -580 -46 -.34 1.3 -613 -85 (7.39) The control can be expressed in terms of AP , AQ , and Av instead of o o t o Ai, , Ai , and Av d q ' ( o o qQ through the transformation matrix T, equation (7.27), T = .446 1.17 .814 .953 -.895 .399 -.001 .234 .905 The results are UES (7.40) .078 -.008 -. 005 -.004 -.6 .08) Y + " -77 -8 9 -2. 6 -3.3 50 "51 Y AQ ,A v. ) o t o -10 -6 1. 1 -.43 468 73 108 15 1. 4 4.6 -928 -172 (7.41) For comparison the controller gains of the optimal signal ug, b equation (7.2), for different operating conditions are computed and compared - with the resultant gains of the optimally sensitive controller uEg, in table 7-1. The speed and torque angle gains for both signals are plotted in figure 7-2. It is clear that the optimally sensitive controller u^^ gains adjust themselves to cover the wide range operating conditions and to match the absolute optimal controllers u* gains. The dominant eigenvalues for the system with the different controllers at different operating conditions are given in table 7-2. While a reduction of stability of the system is observed when it departs from the nominal operating condition, the optimally sensitive controller u„0 provides better . bb results than the nominal optimal controller uEQ. Although u* provides the best stability, it is impractical to implement as stated before, on the other hand there is no difficulty to implement , it is just as good as u* except for the worst operating condition (P = 1.25, Q = .45, & oo v = 1.05) • system operating conditions i\ =1.05) o P =1.25 0 Q =.45 o P =1.2 0 Q =.34 o P =1.15 o Q =.25 0 P =.952 Nominal Q =.015 0 P =.7 0 Q =-.15 0 P =.'5 0 Q =-.225 0 P =.3 0 Q0=-.256 3 UES 128 160 117 134 107 117 80.1 80.1 56.3 61 44.9 53.9 41.3 49.9 UES -376 -434 -479 -603 -682 -691 -648 00 o • ,<o Uf -148 . -316 -420 -603 -660 -664 -661 T-t X CO a •H cd OC >:• UES U£ -4.7 -5.18 -4.49 -4.77 -4.29 -4.45 -3.57 -3.57 -2.6 -2.75 -1.9 -2.1 -1.2 -1.5 Controller UES UE* -5.18 -4.57 -5.19 -4.88 -5.17 -5 -4.96 -4.96 -4.4 ' -4.3 -3.9 -3.56 -3.15 -2.5 Controller UES -13 -11.9 -10.9 -7.74 -4.5 , -2.5 -.96 ui -16.7 -13.9 -11.9 -7.74 -5.1 -3.8 -2.9 UES -104 -117 -99.5 -107 -94.9 -99 -77.8 -77.8 -55.8 -58.8 -38.3 -45.3 -21.6 -31 . Table 7-1 Controller Gains For u„_ and u* Eb E ' at Different Operating Conditions 78 Table 7-2 Dominant Eigenvalues of the System with the Different Controllers Operating Conditions v . = 1-05 P 0 o o UE0 UES A / i i 1.25 .45 .717+J2.86 -4.8 -16.8 1.49 -4+J4.3 -17.16 .137 -4+J3.8 -17 -2.1 -3.2+J3.1 -16.9 1.2 .34 .56+J3.47 -4.5 -16.9 .467 -3.56+J4.37 -17.15 -1.1 -3.3+J3.9 -17 -2.6 -2.7+J3.7 -16.9 1.15 .25 .44+J3.9 -4.2 -16.9 -.449 -3.1+J4.43 -17.1 -2 -2.8+J4.1 -17 -2.7 -2.5+J4.1 -16.97 Nominal .952 .015 .17+J4.8 -3.6 -16.9 -1.98+J4.99 -2.89 -16.96 -1.98+J4.99 -2.89 -16.96 -1.98+J4.99 -2.89 -16.96 .7 -J.5 .023+J5.2 -3.4 -16.8 -1.39+J5.3 -4.4 -16.6 -1.6+J5.3 -2.7 -16.9 -1.56+J5.3 -2.96 -16.89 .5 -.225 -.02+J5.27 -3.4 -16.8 -1.04+J5.3 -5.57 -16.1 -1.28+J5.4 -2.6 -16.97 -1.22+J5.37 -3.1 -16.8 .3 -.256 -.023+J5.25 -3.4 -16.7 -.66+J5.2 -6.8 -15.56 -.83+J5.3 -2.7 -17 -.83+J5.3 -3.3 -16.7 P0 (PU.) Fig. 7-2 Speed and Torque Angle Gains for the Controllers (1) uEQ (2) uEg (3) u* Both controllers u_n and u„„ are tested on the nonlinear model EO ES of the system on two operating conditions, -Pq = .952, Q = .015, vt =1.05 (Nominal) (7.28) o and P =1.2, Q =.34, v =1.05 (7.42) o ' o t o The system disturbance is the same as in chapter 5. The test results are summarized in Fig. 7-3. While the optimally sensitive controller u„„ maintains system stability for the operating conditions of Eb (7.42), the nominal controller u A fails to do so. 80 0 .0035-0 .0030-a .0025-3 D. 0 0020-g 0 0015-0 0010 -o 0 0005-a n -0 0000-tn -0 0005-•0 0010-0 0015 n 0.5 1^0 1.5 2.0 TIME (SECONDS) 2.5 3.0 lO.O-i 0.05 0.00 0.5 1.0 1.5 2.0 TIME (SECONDS) lb) 2.5 3.0 E?-0.05 y-o.io -0.15 —i 1 —i 1— 0.5 1.0 1.5 2.0 TIME (SECBNOS1 (c) 0.10 _ 0.05 0.5 1.0 1.5 2.0 TIME ISECONDS) Id) .3.0 0.5 1.0 1.5 2.0 TIME ISECONDS) 1*) Fig. 7-3 Nonlinear Test Results (1) Nominal operating conditions, U^Q or uEg (2) Nominal optimal control uEQ, for P = 1.2 (3) Optimally sensitive control uEg, for P = 1.2 8. CONCLUSIONS An exact representation of synchronous machines is presented / and a step by step derivation of the exact equivalent circuit given in Chapter 2. It is found that an extra test with the IEEE test code is needed to determine the d-circuit synchronous machine parameters. Three different methods are suggested, a varying slip test or a decaying current test to determine the D-damper time constant T^,o'r an adaptation of Dalton and Cameron's method to determine the newly defined open field d-axis subtransient reactance x1.1 . No extra test is needed to determine do the q-circuit parameters. All three methods gave close results in laboratory tests. A new multi-machine state variable formulation is presented in Chapter 3. The largest matrix inversion is the nodal admittance matrix Y . All system parameters are retained in the final formulation, convenient for sensitivity studies. Systems with an infinite bus are also considered. The results have the same form as that of multi-machine systems without an infinite bus. Dynamic simplification of power systems is discussed. It is found from a numerical example that conventional simplification in power system engineering retains the most dominant eigenvalues of the system. A new technique for the design of optimal linear regulators is developed in Chapter 4. The Weighting matrix Q of the regulator per formance function is determined in conjunction with the dominant eigen value shift of the closed loop system. The eigenvalue sensitivities of the optimal closed loop system with respect to the Q elements are expressed in terms of the same eigenvector matrix of the composite matrix M of equation (4.18), which is required for computing the Riccati matrix K. ^ / / Applying the technique developed in Chapter 4, the optimal stabilization of a one machine infinite system is investigated in Chapter 5. Three different methods of stabilization are considered, through excitation, through the governor, or through both as compared with the conventional stabilization through excitation control. It is found that optimal stabilization through excitation is more effective than conventional excitation stabilization, that optimal stabilization through a governor without dashpot is better than that through a governor With a dashpot, and that optimal stabilization through both excitation and governor without dashpot is the best of all. In Chapter 6, the stabilization of multi-machine systems is investigated again using the technique developed in Chapter 4. Several cases are considered. It is found that a multi-machine system with multi-machine optimal controller uE^, is better than the multi-machine system with only one optimal controller, u„T, which is in turn better than the multi-machine system with the approximated one machine infinite system controller Ug. It is also found that although the individual optimal controller designs are effective in providing damping to individual machines, their effects on other machines are unpredictable. Therefore the dynamic coupling of the multi-machine system must always be included in optimal controller design. The optimal controllers in Chapters 5 and 6 are all for nominal system operating conditions. Since the operating conditions of a real system change from time to time, the controllers so far designed are not adequate for varying operating conditions. In an attempt to face this challenge an optimally sensitive controller is designed in Chapter 7. It is found that the newly developed optimally sensitive controller can adjust itself to stabilize a power system over a wide range of operating conditions and the optimum stabilization is always achieved. A new method to solve the Lyapunov type matrix equation necessary for the design is also developed. Although the techniques have been tested on the detailed non linear mathematical model of the systems, it is highly desirable to implement them on a real poxver system. Other problems remain to be solved. One is to develop test methods to determine exact parameters of synchronous machines with additional rotor circuits. Another problem is how to obtain better approximate representation for system loads and infinite systems for power system dynamic studies. Finally there is the challenging problem of nonlinear optimal stabilization, which needs more investigation to make it practical. 84 APPENDIX A MULTI-MACHINE STATE FORM EQUATIONS FOR 5th ORDER SYNCHRONOUS MACHINE MODEL / ( For a 5th order synchronous machine model, the damper flux linkage variations are neglected, i.e. pAtf»D =0, pA^ = 0, (A.l) (A. 2) implementing (A.l) and (A.l) into (3.30) and elimenating Aij^ and AIJJQ from the results, system equations become, Y* = AY + Bu, Y * A(*_ * v v. F rd Tq F R u = u E» B = [0 0 0 0 K. 6 co)', 0 0]', (A. 3) (A.3a) (A.3b) (A.3c) A A = "rFYFF "RYdF m dF MY' dF 0 A(7,l) -r Y' 0 F Fd -RY* co +X Y' dd em qq. -co -X Y' -RY' e m dd qq MY' dd NY' 0 0 A(7,2) A(7,3) I 0 0 -1 T„ 0 0 0 0 0 1_ TE TA 0 -U mq md 0 -[*d] A(5,6) 0 0 o o 2H (A. 3d) /' 85 where K A(5,6) = h—"](V,U - VII) i.v d mq q md / At / / ' "2 ' A(75l) = - ^§ ^ Y-p / 2 CO A(7,2) = - -r£ i|. (Y' - Y» ) 2H q dd qq 2 A(7,3) --^{*d(Yjd- Y^) + W YFF YFF YFD YDD YDF % *d/xFXd YdF " YdF " YdD YDD YDF = " % XdF/xF Xd (A*3e) YFd " YFd YFD YDD YDd." % XFd/xFXd Ydd Ydd YdD YDD YDd wo/xd Y1 = Y - Y . Y"J Y. = co /x qq qq qQ QQ Qq o q M and N are as given in (3.21a). The governor equations can be easily incorporated into (A.3) if required. APPENDIX B FREQUENCY DOMAIN SOLUTION OF LYAPUNOV MATRIX EQUATION A new method for solving the Lyapunov matrix equation in the frequency domain is proposed. The highest matrix order used in the computation is the same as the system matrix and no matrix inversion.is required. Two algorithms are given,-the first uses the Leverrier algorithm and the second uses the eigensystem of the system matrix. The equation is usually of the form ATK + KA = -Q (B.l) where A is the system matrix, K the matrix to be solved and Q a positive semi-definite symmetric matrix. • Equation (B.l) consists of essentially n(n+l)/2 linear equations for an n-order sytem. The equation can be expanded as Nk = q (B.2) and solved directly. Since for a stable system T K = Ja eA t Q eAt dt (B.3) which has finite value, the integral can be approximated as a series 38 39 summation and evaluated iteratively ' . Transformation approaches are also reported^^. Solutions are obtained after (B.l) is reduced to a special form. In what follows, the method of frequency domain solution of (B.l) will be presented. Applying Parseval's theorem (B.3) becomes K = ~~r f ,:P(s)ds (B.4) 2ITJ -j« 87 where t(s) = (-si - A1)""1 Q(sl - A)""1 (B.5) K can thus be evaluated from the residue theorem. Let (si - A)"1 = R(s)/g(s) (B.6) where R(s) = Is11 1 + R.s" 2 + .. . + R.sn 1-1 + ... + R . 1 l n-1 g(s) = det(sl - A) = s11 - hnsn ^ - ... -h.s11 ^ - ... - h to 1 l n i = 1, 2, n <B'7> The matrix coefficients R_^ of the adjoint matrix polynomial R(s) and the scalar coefficients h. of the characteristic equation g(s) can be 33 determined simultaneously by.Leverrier's algorithm , h. = ~ trace [A.], R. = A. - h.I ii 1 I ' I l i A = A , A. = AR. n 1 i i-i (B.8) (B.9) substituting (B.6) into (B.5) gives g(-s) g(s) which can be written as n C. n D. i 1=1 i where C. and D. are residue matrices of F(s) in the left and right half ii complex planes respectively. It is assumed that are distinct. Let g'(s) = d • g(s)/ds. Then 'RT(-X.) R(X.) or c. = <s-x.) • F(S)|S==Xi =^rrx Qp^ CB.iD 1 "1 1 RT(-X ) Q R(X.) C = 1— r-ir- (B.12) 1 2X. IT (XT-XT)" Applying the residue theorem one has n K = EC. (B.13) i=l 1 Since C±+1 = C* (B.U) for conjugate-pair roots, X , = X*. For a system with m conjugate pair i+1 i roots and I real roots, m n K = 2 E Real C„ + E C. (B.15) • i 2r-l . o .-i J i=l j=2m+l J The residue matrices C_^ can be computed also from the eigenvalues and eigenvectors of the system. Since R/ v n R(X.) —T^Y = E , w \ N (B.16) g(s) j=1 g'(Xj)(s-Xj) 43 and Morgan has shown that R(A.) x,vT = -nf. (B.17) J J g (Xj) T where and v^ are the normalized j-th eigenvectors of A and A respectively, equation (11) may be written as n RT(X.) R(X±) °i = g^XjX-x"^) Q g^TxT)" T nvX = - E -r^- Q X. V. J=1'3 i =-VA.XTQx.vT- <B'18) 1 11 where /' A. = diag[X. + X. , X. + X_, ..., X. + X ]• / (B.19) 1 l 1 i 2 l n / and X, V are eigenvector matrices of columns of X_^ and v_^, respectively; 90 REFERENCES 1. H.M. Ellis, J.E. Hardy, A.L. Blythe and J.W. Skooglund, "Dynamic Stability of the Peace River System", IEEE Transactions, Vol. PAS-85, pp. 586-600, June 1966. 1 2. P.L. Dandeno, A.N. Karas, K.R. McClymont and W. Watson, "Effect of High-Speed Rectifier Excitation Systems on Generator Stability Limits", IEEE Transactions, Vol. PAS-87, pp. 190-201, January 1968. 3. O.W. Hanson, C.J. Goodwin, and P.L. Dandeno, "Influence of Excitation and Speed Control Parameters in Stabilizing Intersystem Oscillations", ' IEEE Transactions, Vol. PAS-87, pp. 1306-1313, May 1968. 4. F.R. Schleif, G.E. Martin, and R.R. Angell, "Damping of System Oscillations with a Hydrogenerating Unit", IEEE Transactions, Vol. PAS-86, pp. 438-442, April 1967. 5. F.R. Schleif, H.D. Humlins, G.E. Martin, and E.E. Hattan, "Excitation Control to Improve Powerline Stability", IEEE Transactions, Vol. PAS-87, pp. 1426-1434, June 1968. 6. F.R. Schleif, H.D. Hunkins, E.E. Hattan, and W.B. Gish, "Control of Rotating Exciters for Power System Damping-Pilot Applications and Experience", IEEE Transactions, Vol. PAS-88, pp. 1259-1266, August 1969. 7. R.M. Shier, and A.L. Blythe, "Field Tests of Dynamic Stability Using a Stabilizing Signal and Computer Program Verification", IEEE Transactions, Vol. PAS-87, pp. 315-322, February 1968. 8. R.T. Byerly, F.W. Keay, and J.W. Skooglund, "Damping of Power Oscillations in Salient Pole Machines With Static Exciters", IEEE Transactions, Vol. PAS-89, pp. 1009-1021, July/August 1970 91 9. F.P. deMello and C. Concordia, "Concepts of Synchronous Machine Stability as Affected by Excitation Control", IEEE Transactions, Vol. PAS-88, pp. 316-329, April 1969. / 10. P.C. Krause and J.N. Towle, "Synchronous Machine Damping by;Excitation Control with Direct and Quadrature Axis Field Windings", IEEE Transactions, Vol. PAS-88, pp. 1266-1274, August 1969. 11. Y.N. Yu, K. Vongsuriya, and L.N. 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