Discrete control of an a.c. turbogenerator by output feedback

Transcription

1 Helwan University From the SelectedWorks of Omar H. Abdalla October, 1978 Discrete control of an a.c. turbogenerator by output feedback P. A. W. Walker Omar H. Abdalla Available at:

2 PROCEEDINGS OF THE INSTITUTION OF ELECTRICAL ENGINEERS Control & Science Discrete control of an a.c. turbogenerator by output feedback P.A.W. Walker, B.Eng., Ph.D., C.Eng., M.I.E.E., and O.H. Abdalla, M.Sc.(Eng). Indexing terms: Power system control, Turbogenerators, Optimum control, Feedback, Linear systems Abstract The paper develops a constant-linear-output feedback optimal-control system for an a.c. turbogenerator system, requiring only the measurement of readily available signals from the power system. The control law is derived from the output prediction formulation of linear-system dynamics, releasing the designer from the need for either stateestimation techniques, or the selection of the number of output measurements to be equal to the number of states. The method is applied to a single machine connected to an infinite busbar, and the performance of the controlled machine investigated in a wide range of operating conditions. These include step changes in demand, and a 3-phase fault at the generator terminals, the results being compared to those achievable with full state-feedback optimal control. List of symbols Machine variables = stator voltages in d- and g-axis circuits = terminal voltage = stator currents in d- and q-axis circuits = stator flux linkages of d- and g-axis circuits = field flux linkage = synchronous reactances in d- and q-axis circuits x ad = stator-rotor mutual reactance x fd = self reactance of field winding = field current x V fd = field voltage ; # = v fd r fd r fd = field resistance = input to exciter Q = components of busbar voltage in d- and <?-axes X t,xi = 5 = T m = e = busbar voltage transformer and line reactances rotor angle, rad mechanical torque input to rotor steam power inertia constant H = to = angular frequency of rotor co = angular frequency of infinite busbar T e = electrical torque K d = mechanical-damping-torque coefficient T d = damping-torque coefficient due to damper windings P t, P b real power output at terminals and busbar Qt> Qb reactive power at terminals and busbar r e = exciter time constant T g = governor valve time constant T b = turbine time constant U g = input to governor G v = governor-valve position K v = valve constant tf = short-circuit fault duration Control variables X = (n x 1) state vector U = (m x 1) input-control vector Paper 822C, first received 11th October 1977 and in revised form 3rd July 1978 Dr. Walker and Mr. Abdalla are with the Department of Electrical Engineering & Electronics, University of Liverpool, Brownlow Hill, PO Box 147, Liverpool L69 3BX, England PROC. IEE, Vol. 125, No. 9, OCTOBER 1978 Y = (p x 1) output-measurement vector W = compound output vector 9 = (nx n) state transition matrix A = {n x m) driving matrix C = (p x n) measurement matrix T = sampling period Ip = (P x P) umt matrix 1 Introduction Much importance is currently attached to the application of modern control theory to a.c. turbogenerator systems, to improve system performance and overcome reductions in stability limits, due to the use of larger generating units with lower specific inertia, and also of longer transmission lines at higher voltages. The system equations are usually organised into nonlinear state-vector differential equations, which are linearised about a particular machine operating point, and the resulting linear model used as a basis for developing a multivariable control law based upon a particular optimal control policy. Successful studies using this approach have been described by a number of authors. 1 ~ 4 Assuming a priori knowledge of the system parameters, the control philosophy is based on the minimisation of a scalar-quadratic performance index to generate a constant control law, comprising a linear weighting of a state vector target and the measured system-state vector, assuming all the states to be accessible for measurement. One of the first problems to be faced when considering the application of optimal control theory in this field is the difficulty of measuring all the system state variables; for example, rotor angle and field flux linkage. Since measurement difficulties exist even in a micro-machine system, these may well be acerbated in a fullscale plant. Interest therefore attaches to the possibility of basing the controller design on output feedback rather than state feedback. Output, 5 ' 6 or suboptimal, 7 feedback controllers have been designed by omitting the feedback of inaccessible states with a concomitant deterioration in performance, and loss of guarantee that the closedloop control system will any longer be stable. Other studies have been reported 8 in which the number of outputs has been chosen to be equal to the number of states. An approach that may be considered as state feedback with an appropriate transformation of the system states, and one that provides no relaxation in the number of output variables to be measured. As an alternative to the above approaches, and also the use of state-estimation techniques, 9 the authors have adopted an output prediction formulation for the dynamics of a linear, multivariable system. This predicts the output vector in terms of past outputmeasurement vectors and input-control vectors, and completely defines the process in terms of known and measurable vectors, /78/ $1-5/

3 avoiding any reference to the state vector that may contain unmeasurable components. The output prediction formulation provides an effective basis for the application of optimal control techniques, based on either minimum time control, 1 ' n or the use of dynamic programming techniques to minimise a quadratic performance index. 12 For linear systems, these yield a control law comprising a linear weighting of past output measurements, and input-control vectors. The objective of the present paper is to propose a discrete outputfeedback optimal-control strategy appropriate to the requirements of an a'.c. turbogenerator system, based on the assumption that in a fullscale plant many of the system state variables will be either inaccessible, or difficult to measure, due for example to serious contamination by noise. A feature of the proposed output-feedbackcontrol philosophy is that it yields the same optimal-control trajectory as that of state-feedback control when applied to a linear system. This allows a direct comparison to be made of the behaviour of the two types of control. Simulation studies are presented to demonstrate the effectiveness of the control law in a wide range of operating conditions, including step changes in demand, and a 3-phase fault at the generator terminals, the results being compared to those achievable with full-state feedback optimal control. Yk-N+2 *-, ~Y k Y h. x =.../, / m....,. A, 2 Control philosophy 2.1 Process model formulation A linear, time-invariant, multivariable process can be described in a standard, discrete-time, state-space form by and ) Yk-N+l Uk-i U k - 2 -t I m (6) Y k = CX k Using eqns. 1 and 2, the state-prediction equation 1 X k + 1 = F l Z k +F 2 V k can be obtained, (2) (3) _ More compactly the compound output vector (7) Y h Z k (pnx\) = V k {mnx 1) = U h and L = pn + m(n-l) and the derivation of matrices F { and F 2 is discussed in Ref. 1. This equation predicts the state vector at time t {k + 1)7", in terms of the previous N output-measurement vectors and input-control vectors from t = kt, back to t = (k N+ l)t. The minimum number of stagestv is chosen such that N^n/p, rounded up to the nearest integer. Multiplying both sides of eqn. 3 by the measurement matrix C and the second and fourth matrices in eqn. 6 are represented by 9 and 2, respectively. When the system is in the steady state X^, U k - U^ and (4) a (p x pn) = CF X P (pxmn) = CF 2 the output-prediction formulation of the system dynamics that may be used as effectively as eqn. 1 as a basis for the generation of optimalprocess control laws. To facilitate further analysis, eqn. 4 is rewritten Then, subtracting ^ from both sides of eqn. 7, after vertically partitioning p into p 2 and p,, eqn. 5 is augmented by additional rows to form (5) +Qu k (8) H> fe = W k - W u -and i/ fe = U k - This equation completely defines the process in terms of known and measurable vectors, and avoids any reference to the state vector itself, which may have unmeasureable components. It is seen to have an iterative structure identical to that of the state-prediction eqn. 1 and may therefore, with great convenience, be similarly employed to generate appropriate optimal-control laws. 132 PROC. IEE, Vol. 125, No. 9, OCTOBER 1978

4 Of course, this equation can only accurately represent the dynamics of a nonlinear system, such as the a.c. turbogenerator, for relatively small displacements of the system states from a particular operating point. Thus, the matrices and Q must be established by an initial theoretical linearisation of the nonlinear system-state vector differential equation. Alternatively, these matrices might be identified from measurements of inputs and outputs of the system, assuming the system is at some steady-state operating condition, and is only perturbed within an effectively linear operating region. 2.2 Equivalent-performance index The state-feedback optimal-control law may be found by minimising the quadratic-performance index fe=o x fe + 1 = X k + 1 X ss, based on the system dynamics of eqn. 1, and using the dynamic programming procedure. 13 As the number of stages r -*, this leads to an optimal control law of invariant form U k = F s x k (1) Similarly, characterising the system dynamics by eqn. 8, a quadratic performance index r I" 1 Jo = Wfe+i QoWft + i +UkH o u k \ (11) fe= [ J can be defined that, when minimised by the dynamic programming procedure, leads to the invariant control law u k = F Q w k (12) For a given linear system, the two control laws of eqns. 1 and 12 will drive the process from a particular state to a steady-state target Yss ~ CX S s over quite dissimilar control trajectories. This difference is due to the lack of correspondence between the respective weighting matrices in the cost functionals of eqns. 9 and 11. To facilitate a direct comparison of the behaviour of the two control laws, one of the chief objectives of this paper, an alternative performance index to that of eqn. 11 is proposed, from which the output-feedback-control law of eqn. 12 is generated, in which the weighting matrices are derived directly from those used in the stateperformance index of eqn. 9. The resulting control law still takes the form of eqn. 12, but a guarantee is obtained that the control trajectories generated by the two control laws will be identical when applied to a linear system. If the matrix F 2 in eqn. 3 is vertically partitioned into F 3 and y, that equation may be rewritten as x k+l = IF, \F- Uk-N+1 (9) (13) or, subtracting Xss f rom both sides (14) = [F, :F 3 ]. Substituting eqn. 14 into eqn. 9 to eliminate x k + 1, yields the performance index in which Q(LxL) = R(mxL) = S(mxm) = (15) (16) Minimising this equivalent-performance index using the dynamic programming procedure still leads to a control law of the form of eqn. 12, the procedure being summarised in Appendix 8.1. Therefore, the following control philosophy proposed. With a priori knowledge of the system dynamics, the weighting matrices Q s and H s of the state performance index of eqn. 9 may be selected, on the basis of simulation studies, to produce an acceptable process behaviour. The equivalent performance indices Q, R and 5 can then be determined from eqn. 16 for insertion into the output performance index eqn. 15, for computation of the output feedback control law (Appendix 8.1). The efficacy of this control law can be examined, either by means of simulation studies as in the present paper, or by direct application to an online process. Alternatively, if the process dynamics are unknown, the matrices of eqn. 8 might be identified online by least-squares identification, from measurement of the process inputs and outputs, and the performance index of either eqn. 11 or eqn. 15 employed directly to generate the control law, selecting the weighting matrices by trial and error to obtain acceptable control performance. 3 A.C. turbogenerator model A schematic diagram of the open loop a.c. turbogenerator system to be studied is shown in Fig. 1. It comprises a synchronous generator tied, via a step-up transformer and transmission line, to an infinite busbar, and includes a representation of the associated prime mover and field-excitation system. The set of equations used to describe the system are summarised in Appendix 8.2, and the system parameters given in Table 1 are representative 14 of a 37-5 MW generator. A (6 x 1) state vector X, a (2 x 1) input vector U and a (4 x I) output-measurement vector Y are defined as X = [6,5,^/a, " /a>j P.,7' m )' Y = [P u v t,b,e fd Y (17) The output vector is chosen so that its four components, real power output, machine terminal voltage, rotor-angle velocity and field voltage, are readily accessible for measurement in a fullscale plant. The system equations may be written as the following set of first infinite busbar Fig. 1 A.C. turbogenerator system exciter PROC. IEE, Vol. 125, No. 9, OCTOBER

5 Table 1 SYSTEM PARAMETERS mva = 37-5 mw = 3 p.f. = -8 lagging kv = 11-8 r/min = 3 x d = 2- p.u. x q = 1-86 p.u. x ad = 1-86 p.u. = 2 p.u. x fd R fd = 17 p.u. // = 5-3 MWs/MVA T d = 5 x t = -345 p.u. x, = -125 p.u. e = 1 p.u. T e = 1 s. T a = 1s. b = -5 s. = 1-42 B = C = 1 v u \ Kn KH c) 1 the coefficients are as given in Appendix 8.3. (23) (24) order nonlinear equations, the coefficients are defined in Appendix 8.3: Xi = X 2 X 2 = (X 6 sin Xi K 2 X;=^+^ TB T b Te -K 2 (18) and the outputs Y x, Y 2 may be expressed in terms of these state variables by sinjt, +K 2 sinxj (19) 4 Simulation studies Unless otherwise stated, the a.c. turbogenerator model was assumed to be initially operating at a nominal operating point, taken as an output power of -8 p.u., at -9 lagging power factor, at the generator terminals, with corresponding initial steady-state vectors X S s Y S s = [1,, 1-152, 2-314, -8, -8]' = [-8, 1-152,,2-314]' Uss = [2-314,-563]' (25) 4.1 Open-loop test Open-loop response tests on the nonlinear model were first conducted, for an ultimate 12% change in real power from -8 p.u. to -9 p.u., at the same power factor of -9 lagging. The deviations in the variables from their initial steady-state values, computed by solving eqns. 18 and 19 by the Runge-Kutta procedure, are shown in Fig. 2, and are comparable with those presented in Anderson 1 for a similar machine. The system was driven from the nominal operating conditions defined in eqn. 25 to the new steady-state conditions X ss yss = [1-78,, 1-16, 2-495, -9, -9]' = [-9,1-19,,2-495]' (26) v d = K s sin Xi v Q = K 6 X 3 +K-, cosx, The linearised form of the above equations, x = Ax + Bu y = Cx (2) (21) by a step increase in the input vector to U S s = [2-495,-634]' These open-loop results have proved useful for judging the efficacy of the optimal control laws studied subsequently. 3- OOO6r for small displacements about a particular operating point, is also required for computation of the transition and driving matrices <p and A in eqn. 1 The matrices A, B and C have the form 1 K. (K d + T d )u 2H K 9 2// A K. x ad ' e (22) Fig. 2 Open-loop step response 4.2 Optimal control laws -1 The system was linearised about the nominal operating point and, for a sampling period of -5 seconds, the constant matrices of eqns. 1 and 2 were computed to be 134 PROC. IEE, Vol. 125, No. 9, OCTOBER 1978

6 " (27) conditions are as given in eqns. 25 and 26, respectively. A direct comparison is offered between the step response with the statefeedback optimal-control law of eqn. 1 and the proposed outputfeedback optimal-control law of eqn. 12. Clearly, the two controllers provide very similar responses to this stipulated step change, slight differences being due to the nonlinearity of the a.c. turbogenerator. Fig. 4 shows the transient response with the two control laws to a step change in demanded terminal power factor from -9 lag to -85 lag, at constant real power, i.e. the steady-state target is given by X ss Y ss U ss = [-92,, 1-245, 2-447, -8, -8]' = [-8, 1 148,,2-447]' = [2-447,-563]' (3) In this case, the responses are almost identical, proving that for this transient the system is effectively operating over a linear region. Comparison of these results with the open-loop transient response in Fig. 4, illustrates the substantial improvement affected by the adoption of optimal control, particularly in the terminal voltage. These results also provide direct confirmation of the value of using an equivalent performance index, to generate the output optimalcontrol law, employing the same weighting matrices used to generate the state-feedback-control law. 1 1 A state-feedback optimal controller was first designed by minimising the quadratic performance index of eqn. 9 using the dynamic programming method. 13 As the number of stages r is made increasingly large, the minimisation procedure leads to the constant-state feedbackcontrol law of eqn "s (28) The elements of the weighting matrices Q s and H s, were selected to obtain satisfactory system performance over a wide range of operating conditions, particularly in comparison to that of the system with a conventional voltage regulator and speed governor. The nonzero elements of these matrices were, Qsih O = Qs(6, 6) = 2; S (1, 3) = Q S (3, 1)= 1-684; Q s (3,3)= 124; Qs (5, 5) = IO;H S (1, 1) = -5 and H s (2, 2) = 1-. The output-feedback optimal controller of eqn. 12 was then derived from the state-feedback controller in the manner proposed in Section 2, using the equivalent performance index of eqn. 15, and was found to be Fo i ' Fig. 3 Transient step response state feedback output feedback steady-state target time, s time, s ' , (29) ! i -21» J The dimensions of this control matrix (m x L) are due to selecting the number of stages N in the output prediction eqn. 4 as three rather than the minimum value of two, to contain the magnitude of the elements within similar bounds of those in the state control matrix 4.3 Transient step response Fig. 3 shows the transient step response to a -1 p.u. step change in demanded power, from -8 p.u. to -9 p.u, at constant terminal power factor of -9 lagging, only deviations from the initial steady-state conditions being shown. The initial and target steady-state time, Fig. 4 Transient step response state feedback output feedback steady-state target 4.4 Steady-state stability time. s Fig. 5 shows the steady-state stability limits of the a.c. turbogenerator in the P-Qb plane computed at the infinite busbar. For the purposes of comparison the limits of three configurations are shown, namely, (a) the open-loop system,(z>) the system with statefeedback control and (c) the system with output-feedback control. Both controllers were calculated for the nominal operating point. Fig. 5 confirms the improvement to be gained by introducing state feedback to the open-loop system, a result well documented in the PROC. IEE, Vol. 125, No. 9, OCTOBER

7 literature. 4-9 It is interesting to see that the adoption of outputfeedback control provides a further small improvement in stability. The stability limits were computed by determining when the complex eigenvalues of the discrete representation of the system, i.e. open loop: x fe+, = Qjc k or w k+l =Qw k (31) state feedback: x k+l = [<f + AF s ]x k (32) output feedback: w k+l = [ + QF o ]M> fe (33) lay within the unit circle. At the nominal operating point itself the closed-loop eigenvalues (to four decimal places) were the steady-state stability limits of Fig. 5. The effect of state feedback on the damping coefficient is to increase it by a factor of between two and three besides extending the stability limit from a reactive power of -4 p.u. to -88 p.u. The adoption of output feedback introduces further, though less significant, increases in the damping coefficient at all levels of reactive power. The very sharp cutoff displayed in all three results is probably due to the lack of positive synchronising torque when the steady-state stability boundary is reached at each particular value of reactive power. state feedback [9 + AF S ]: -855 ±/-2875,-6211 ±/-33,-831,-1473 output feedback [Q + QF ]: -855 ±/-2875,-6211 ±/-33, -831, -1473, ±/-4673, i/4742, ±/-416,,,, It is observed that the first six closed-loop eigenvalues of the outputfeedback control system are identical to the six eigenvalues of the state-feedback control system. The sensitivity of the steady-state stability limit, to changes in the nominal operating conditions at which the two controllers were computed, was also studied; e.g., Figs. 5c? and 5e show the limits with state and output controllers computed for an operating power level of -8 p.u, and reactive busbar power of These studies indicate that the stability of the system with both state and output control is relatively insensitive to changes in the system operating point, a conclusion which accords with those of previous investigations with state-feedback control. 9 oos o -IO -OB -O6 -O-4-2 OOO reactive power p.u. Fig. 6 Damping coefficient of an a.c. turbogenerator as a function of reactive busbar power P b = -8 p.u. open-loop system state-feedback control output-feedback control 1 O8 O-6 8. O-4 2 2O Fig. 7 3-phase fault at generator terminals -12 -O8-6 -O4 reactive -O2 r, P-u- OO t f = 15 s. state feedback control output feedback control steady state target Fig. 5 Steady-state stability limits (a) open-loop system (b) state-feedback control Q = b -83 p.u. (c) output-feedback control Q b = 83 p.u. (d) state-feedback control Q b = -536 p.u. (e) output-feedback control Q b = -536 p.u. 4.5 Damping characteristics The value of K d to make the system oscillate with constant amplitude at the operating point gives a measure of the damping in the system, corresponding to the undamped natural frequency for the given operating condition. An examination of the system damping may provide a useful basis of comparison for the relative stability of different control configurations in the feedback loop. It is therefore necessary to determine the amount of damping K d 5 required to cancel out that in the system due to the load, damper windings and controller. Accordingly, for different levels of leading reactive power, the value of K d was progressively reduced, until one or more eigenvalues of eqns. 31, 32 and 33 describing the dynamics of the three configurations of interest here reached the unit circle.. The variation of K d for a power level of -8 p.u with reactive power is shown in Fig. 6, and may be interpreted in association with Fig. 8 Fault studies with output feedback control tf = O15s. (a) P b = -8 p.u.; (b)p b = -8 p.u.; = -266 p.u. = -536 p.u. 136 PROC. 1EE, Vol. 125, No. 9, OCTOBER 1978

8 4.6 Fault studies Fig. 7 shows the system response to a 3-phase short circuit of -15 s. duration at the generator terminals with state feedback, and also output-feedback controllers. Even with this severe disturbance, the behaviour of the a.c. turbogenerator shows little significant difference between the two types of control, the rotor angle swing being damped out in about 2 s. It is noted that the first positive swing of the rotor angle is almost the same for the two controllers, although the first negative swing is somewhat greater with the output-feedback controller. Further fault studies concentrated on a 3-phase short circuit at the high-voltage side of the transformer. The results are shown in Fig. 8 for a wide range of operating conditions, only the rotor angle and terminal voltage being illustrated. With P t = 8p.u. and Q b = -266, rotor angle swings are damped out in less than 2 s. after fault application, the linear controller again clearly providing satisfactory control over the nonlinear system. From Fig. 5 it is clear that at an operating point of P t = -8, Q b = -536, the open loop a.c turbo-generator is dynamically unstable. By contrast, employing output feedback, a fault at this operating point now leads to the rotor-angle swing being damped out in 2-5 s. 5 Conclusion Reference has been made to the difficulties encountered when designing and implementing an online optimal control law for an a.c. turbogenerator that relied on the measurement of all the system-state variables. From the simulation results presented in this paper, it seems that such difficulties may be largely overcome by the use of linear-discrete output-feedback control, requiring only.the measurement of readily available signals such as terminal power and terminal voltage. The control law is derived from the output prediction formulation of the system dynamics, thereby avoiding either the need for state estimation techniques, or forcing the output matrix to be square, by selecting the number of outputs equal to the number of states. Perhaps it should be emphasised that the method of deriving the weighting matrices in the performance index based on the compound output vector was employed for convenience to allow a meaningful comparison of state, and output feedback control laws to be made. It is available for use when a theoretical model of the process is available in terms of states. When this is 1 not the case and the process dynamics are determined by online least square identification, from measurements of inputs and outputs, the weighting matrices must be selected directly to obtain acceptable control performance. The steady state and transient performance with output-feedback optimal control is seen to be very similar to that with state-feedback control. The improvement in the steady-state stability limits may be noted among the similar characteristics displayed, compared with the open loop system, and particularly the relative insensitivity of these limits to changes in the nominal operating point, for which the control laws are designed. This provides considerable encouragement for the further study of the application of optimal-control concepts, based on output feedback and developed for linear systems, being applied in this essentially nonlinear environment. 9 OKONGWU, E.H., WILSON, W.J., and ANDERSON, J.H.: 'Optimal state feedback control of a microalternator using an observer'. IEEE Summer Power Meeting.1977, Paper F77, pp SANDOZ, D.J., and WALKER, P.A.W.: 'Output-prediction equation with minimum-time control of linear multivariable systems', Proc. IEE, 1973, 12, (3), pp WALKER, P.A.W., and DUNNE, M.D.: 'Direct formation of a time optimal controller by least squires identification', IEEE Trans., 1977, AC22, pp SANDOZ, D.J.: 'Optimal control of linear multivariable systems based on discrete output feedback',proc. IEE, 1973, 12, (11), pp TOU, J.H.: 'Modern control theory', (McGraw-Hill, 1964). 14 SHACKSHAFT, G.: 'General-purpose turbo-alternator model', Proc. IEE, 1963, 11, (4) pp Appendixes 8.1 Output feedback optimal control algorithm The equivalent multi-stage performance index of eqn. 15 is minimised using the dynamic programming procedure. This leads to the following recursive algorithm to evaluate the constant-control matrix of the output-feedback control law of eqn. 12. and F k -j = -M-jiik-j (34) / (35) Mfe-y = S +Q t ojq (36) starting at the last stage with / = and a =, the equations iterate backwards with increasing; to determine the control matrix F fe _y. As the number of stages / is made increasingly large, the recurrence matrix F k.j converges to a constant matrix F o. 8.2 A.C. turbogenerator system equations Synchronous generator The equations of the generator are based on Park's equations, and in their derivation, apart from the original assumptions made by Park, the following additional assumptions have been made: (i)the effect of change of speed, and the rate of change of flux linkage in the stator voltage expressions, is negligible. (ii) line and stator resistances, and the effect of transients in the transmission lines, are negligible. (iii) No magnetic saturation. (iv) The effect of damper windings can be accounted for by adjustment of the damping coefficient T d in the mechanical equation of motion. v d = - = R fd i fd + \ (38) (39) (4) 6 Acknowledgment The authors are grateful to Prof. J.H. Leek for facilities provided in the Department of Electrical Engineering & Electronics. Mr. Abdalla is grateful to the Egyptian Government for financial support. 7 References 1 ANDERSON, J.H.: "The control of a synchronous machine using optimal control theory', Proc. IEEE, 1971, 59, pp MOUSSA, H.A.M., and YU, Y.N.: 'Optimal power system stabilisation through excitation and/or governor control', IEEE Trans., 1972, PAS-91, pp ANDERSON, J.H., and RAINA, V.M.: 'Power system excitation and governor design using optimal control theory', Int. J. Control, 1974, 2, pp NEWTON, M.E., and HOGG, B.W.: 'Optimal control of a micro-alternator system', IEEE PES, 1976, Winter Meeting, Paper F DAVISON, E.J., and RAU, N.S.: "The optimal output feedback control of a synchronous machine', IEEE Trans., 1971, PAS-9, pp DE SARKAR, A.K., and RAU, N.D.: 'Stabilization of a synchronous machine through output feedback control', ibid., 1973, PAS-92, pp ADCOCK, B.S., and BROWN, W.A.: 'Suboptimal control of voltage and power for a large turbogenerator'. Proceedings of IFAC Congress, Melbourne, Feb. 1977, pp RAINA, V.M., ANDERSON, J.H., WILSON, W.J., and QUINTANA, V.H.: 'Optimal output feedback control of power systems with high-speed excitation system', IEEE Trans., 1976, PAS-95, pp PROC. IEE, Vol. 125, No. 9, OCTOBER 1978 ~x d i d 4>q = -Xgi q ^fd~ x fdhd v] = v\ + v\ Transmission system v d v q = e.sin5 x e i Q = e. cos 5 +x e i d x e = Prime mover G v = U g G v ; (41) (42) (43) (44) (45) (46) (47) (48) (49) 137

9 P s = k v G v (5) -r f x dl oi x I f d X d i _ x adu o r f e x f d X d i m Tb Excitation system r s x m T b = ju e -je fd. -5<E fd <5 (51) (52) Xg 'e Xql x d -e K n = x d i Xe 'X ad X d l 'Xf d +K 2 cos2x,) 8.3 Constants in a.c. turbogenerator Defining K* = - sin i V 2 x ad i _ ' i v = -Aa Sin Xd - x d ; x dl - x d + x e Xfd x d i = x d +x e ; x Q i = x q +x e constants are K l2 K x sin v d K 5, +K 2 K x = ex ad Xfd x dl = eo.(x d -Xg) K 6 V Q 138 PROC. IEE, Vol. 125, No. 9, OCTOBER 1978