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AN OPTIMAL CENTRALIZED CONTROLLER WITHNONLINEAR VOLTAGE CONTROLEDGARDO C. MANANSALA a & ARUN G. PHADKE aa Department of Electrical Engineering , Virginia Polytechnic Institute and State UniversityBlacksburg , Virginia 24061, USAPublished online: 14 Feb 2007.

To cite this article: EDGARDO C. MANANSALA & ARUN G. PHADKE (1991) AN OPTIMAL CENTRALIZED CONTROLLER WITHNONLINEAR VOLTAGE CONTROL, Electric Machines & Power Systems, 19:2, 139-156, DOI: 10.1080/07313569108909512

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AN OPTIMAL CENTRALIZED CONTROLLER WITH NONLINEAR VOLTAGE CONTROL

EDGARDO C. MANANSALA and ARUN G. PHADKE

Department of Electrical Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia 2406 1, USA

Abstract

Power system multiswing transient stability during a disturbance may be substantially improved through proper control of power system bus voltages. These bus voltages are normally nonlinear functions of state variables defined for the generators and their control elements. Postfault chan es in the power system may also require a power system operating point different ! rom the prefault value. An optimal centralized controller capable of tracking these changes and controlling the nonlinear elements of the power system state vector is proposed in this paper. The primary role played by voltage control and the need to represent the dynamics of the external system are shown from the results of simulation case studies.

1. Introduction

Utilization of optimal controllers for multimachine power system stabilization has been the subject of much research. The state regulator formulation was normally used to derive the control input signals to either or both the excitation and speed-governing s stems of the individual generators as in Yu and Moussa [I], Fosha and Elgerd [2], and Wilson and Aplevich [3 . This approach has the draw- back of depending too much on the linearization a b out the prefault operating point without an explicit mechanism to track changes in power system operating point and structure. It also assumes that the postfault and prefault conditions on the power system are identical. Thus, the target trajectories for the controller state variables are always set equal to zero.

This paper discusses a centralized control scheme based on real-time measurement of the power system state vector. Complete and accurate system dynamics is included in the controller design. The controller utilizes full-state feedback and measurements performed on the controller output vector. These measurements allow the direct computation of residual terms in the controller state and output equations. Information pertaining to changing power system operating point is included in these residual terms. Inclusion of output residual terms allow the control of nonlinear functions of controller state variables such as power system bus voltage magnitudes. This feature is important since direct control of bus voltages during a disturbance allows indirect control of the electromechanical swings.

Field implementation utilizes synchronized real-time measurements which are processed to compute the required control input signals. Phadke, et.al. [4] proposed a positivesequence phasor measurement technique. This technique which features

Electric Machines and Power Systems. 19: 139-1 56. 1991 Copyright O 1991 by Hemisphere Publishing Corporation

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E. C. MANANSALA AND A. G. PHADKE

Fig. I . Cmtrllized Contml W m e

high speed and high accuracy performs system-wide synchronized measurements at appropriately chosen remote locations in a ower system and transmits the data to a control center via communication links I]. Fig. 1 illustrates this scheme in a general framework. Fig. 2 shows the same centralized scheme of Fig. 1 but with emphasis on the system dynamics assumed by the controller for a typical generator. The elements of the power system state vector are determined through the phasor and field voltage measurements and fast dynamic state estimators. The controller derives the required input signals which are applied to the excitation and speed governing systems of the generators. The test system used in the studies was a 39-Bus loGenerator power system (Fig. 3). Generator 10, the lar est generator, may represent an equivalent machine previously obtained from a co%erency-based dynamic equivalencing program. The other generators may correspond to the retained actual power system units. Implementation of the centralized controller on an actual system can utilize this procedure of reducing the effective size of the -power system to obtain a realistically small number of controller state variables.

The controller state variables correspond to the incremental changes from prefault values in the state variables of the power system components. In the controller problem formulation of this paper, generator rotor angles relative to the angle of the largest generator are used to yield a stable (nonsingular) system matrix. This is always possible provided that linearization is performed about a stable prefault operating point. Use of a stable system (feedback) matrix guarantees the existence of a steady-state solution K, to the Riccati equation (6). Use of the s teadystate solutions to the Riccati and auxiliary differential equations in turn substantially reduces the on-line computation time requirements of the controller. These real-time solutions are used immediately in the control law which simul- taneously minimizes the control input energy and the deviation of the power system state vector (controller output vector) from a target trajectory appropriate for the known disturbance. The target trajectory is based on a desirable postfault equilibrium point and the known immediate postfault value of the power system state vector. An off-line computer program was used to simulate the controller with access to real-time state vector (phasor) measurements from the power system.

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Fig. 2. Generator Subject to Controller

Results from five simulation cases have been studied: Case 1: Controller Inactive

Voltage Control Prioritized Reduced Voltage Control

d Case 4: Increased Angular Control e Case 5: Local Controllers

Case 1 depicts the power system response to the disturbance with no controller acting on the system. In Case 2, controller computations were carried out using an accurately calculated feedback matrix and with generator field voltage control

Fig. 3. 39-Bus 10-Gnuator Tat System

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142 E. C. MANANSALA AND A. G. PHADKE

included in the algorithm. The optimal criterion is determined by the choice of elements for the weighting matrices in the cost function. Here, control of power system bus voltages is given priority over the control of generator field voltages and rotor positions and speeds. Case 3 illustrates, on the one hand, the performance of the centralized controller when control of power system bus voltages is relaxed from that of Case 2. Case 4, on the other hand, shows the effects of increasing rotor angular control over that of Case 2. Finally, in Case 5, individual generator controllers which do not take into account external power system dynamics are simulated. Simulation results are interpreted in terms of controller ability to simultaneously minimize voltage magnitude and rotor oscillations and force the power system state vector to follow a desirable target trajectory for the known disturbance.

2. Controller Equations 2.1 Statement of the Control Problem.

Given the state equation

~ ( t ) = AX(t) + BJ(t) + f(t) - and output equation

Y(t) = CX(t) + J(t) -

find the control law J(t) = g(t) which minimizes the cost function

where

subject to equations (1) and (2). An infinite control time interval is chosen for the controller. R and Q are positive and semi-positive definite weighting matrices in the cost function. _Yd(t) represents the target trajectory for the controller output vector.

2.2 The Control Law.

The solution to the control problem stated above has been derived by Manansala [7] after extending the work done by Rostamkolai [8] and Kirk [9] and defining an infinite control time interval. The variational amroach was used together with Pontryagin's Minimum Principle. The solution is' repeated below. The control law is given by

9, (t) = -[AT - K ~ B R - ' B ~ ] ~ ~ (t) - CT~_yd(t ) + ?QJ(~) + KS f(t)

(6)

and

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OPTIMAL CENTRALIZED CONTROLLER

which is the algebraic Riccati equation.

2.3 Implementation of the Control Law.

Field implementation of the centralized controller requires real-time computation of the state and output residual terms. Assuming a stable prefault equilibrium point and no controller input to the system before the disturbance,

_a = E(_O,_a, t ) (8)

where t is any time before the disturbance. Assuming fast and accurate state estimates, the time derivative of the state vector is given by a nonlinear system of equations based on the known power system component models. Thus,

at any time after the disturbance. The state residual vector is computed as

The value of the output vector _Y(t) is obtained directly from real-time phasor and field voltage measurements. The output residual vector is directly computed as

The feedback matrix A, control matrix B, and output matrix C are computed through a first-order linearization about the prefault operating point of the power system. Their manner of computation has been described in detail by Manansala 171. Field implementation of the controller assumes that these matrices are time- ~nvariant. Therefore, the residual terms capture both the system nonlinearities and the changes caused by the alteration of the network configuration due to the fault.

The error vector g(t) defined in equation (4) is the real-time difference between the output vector and its target trajectory &(t) at time t . Elements of the controller output vector are all directly measurable. The target trajectory for the controller output vector was based on a desirable postfault equilibrium point and the known immediate postfault value of the power system state vector. Field implementation of the controller would require fast on-line loadflow computations corresponding to the postfault s teadystate conditions of the power system. These further require knowledge of changes introduced by the system disturbance. This operating point is determined by solving the loadflow problem in real-time based on the known post fault configuration.

Equation (5) is the closed-loop optimal control law which yields the required input signals to the excitation and speed governing systems. These signals are denoted by the control vector J(t). If linearization was performed about a prefault stable equilibrium oint, the system matrix A would be stable (all eigenvalues have negative real partsf This guarantees the existence of the s teadys ta te solution K,, the controller Kalman gain matrix. This matrix can therefore be computed off-line. The use of a time-invariant solution to the algebraic Riccati equation (7) points to a substantial reduction in real-time computation requirements of the controller. However, Manansala , Rostamkolai [a], and Kirk all indicate that the auxiliary differential equation must be integrated in time. Difficulties are seen at once since future of the residual terms f(t) and g(t) are unknown and the

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1 44 E. C. MANANSALA AND A. G. PHADKE

backward numerical integration would require unacceptable real-time computa-

tional burdens on the controller. By using the approximation g(s) = _O , however, the solution to gs(t) becomes

% (t) = - [ A ~ - K~ BR-'B~]-~{C* Q[& (t) - _g(t)] - I$ . f(t)) (12)

It is noted here that equation (12) is an approximation to the s teadys ta te solution of equation (6) since the vectors _Yd(t) and ~ ( t ) are in general nonzero values of time especially after the occurrence of any disturbance in the power system. This approximation together with the use of s teadystate solutions is important in real-time applications since it eliminates the need for on-line integration of the differential equation (6).

3. Power System Component Modeling 3.1 Generator Model

The electromechanical behavior of generator "it' is given by

Pmech,i is the time-varying mechanical power input to the generator. 6q, , the center-f-angle is given by

; ,RG

Pei , the electrical real power output of the ith generator is

6,,,i and 6tr,i are the rotor position and terminal angle values of generator "i" with respect to the rotor position of the largest generator, i.e.,

The angular quantities on the right-hand side of the two preceding equations were in turn measured from a synchronous frame of reference. The effective system frequency may drift away from its prefault synchronous value during a disturbance. Thus, the quantity bqc is used in the swing equations. A third-order generator model (including the swing equations) is often used in dynamic stability studies as suggested by Stagg and El-Abiad [lo] and Anderson and Fouad [ll]. Hence,

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OPTIMAL CENTRALIZED CONTROLLER 145

The phasor diagram assumed for the controller and utilized in the simulation studies is shown in Fig. 4.

Fig. 4. Generator P h v a Diagram

3.2 Exatation System Model.

The IEEE Type 1 exciter model with an applied auxiliary input signal was used [12]. It is a fourth-order model typical of many existing excitation systems.

The block diagram representation of this excitation system model is shown i n ~ i ~ . 5. Vsi in equation (22) is the auxiliary input and is an element of the control input vector. Nonlinearities arise from the saturation function S,i which was assumed to vary exponentially as the generator field voltage Efdi and from the ceiling values imposed on the state variable Vri.

3.3 Speed Governing System Model.

The third-order nonreheat mechanical-hydraulic steam turbine type was assumed. An auxiliary signal Psi is defined and applied at the same point as the reference signal PogYOi (or Po i in the time curves for the control input signals to the speed-governing systems). jarameter values including.the time constants are taken from the IEEE Committee Report 131. The block diagram representation of the L speed-governing system model is s own in Fig. 6. Nonlinearities arise from the ceiling values imposed on the state variable Pg,i and its time derivative.

4. Definition of Controller Vectors The controller sate vector is

where

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E. C. MANANSALA AND A. G. PHADKE

Rg. 5. IEEE T y p I Exata l lm Syslem

The largest generator in the system (G10) has a constant field excitation (Efdlo = constant) making equation (19c) invalid for this case. Moreover, although ten rotor an ular values measured with respect to a synchronous frame of reference are de 1 ned for the transient stability program, only nine relative values need to be defined for the load flow solution at any time. This observation is confirmed by the resulting system (feedback) matrix when linearization is performed about the prefault operating point. Use of ten angular values leads to a singular matrix. However, a nonsingular (stable) matrix is obtainable if nine relative angular values are used. A stable feedback matrix is particularly important in ensuring the existence of a steady-state solution to the Riccati equation. Thus, incremental .relative angular values are used for the controller as shown in equations (18) and (19b). The control input signal to the excitation and speed-governing systems are given by

T T T T T u = [I, I, ... Ii ...I9] - (20)

Fig. 6. Mechanical-Hydraulic Nanrchd Stam Tvrbine

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OPTIMAL CENTRALIZED CONTROLLER 147

where

Qi = [VSi

The controller output vector is defined by

The present value of 1 ( t ) at any time defines the power system state itself. Incremental changes in generator terminal voltages are included in the output vector. These quantities are nonlinear functions of the controller state variables. Thus, the output residual vector is nonzero. Here it is noted that other nonlinear functions (such as remote power system bus voltages and generator electrical power outputs) may be defined as elements of the output vector. Incremental field voltages are included in the controller output vector to yield a positive-definite solution to the Riccati equation in the present formulation. The matrix Q was chosen to be a diagonal positive-definite matrix in the simulation case studies presented in this paper. Therefore, the elements of the output vector defined in

the minimum required measurements for the

All of the controller vectors defined in this section apply to the centralized controller of Cases 2, 3, and 4. Vectors and matrices defined for the local controllers of Case 5 are described in the section "Simulation Studies".

5. Simulation Studies The New England 39-Bus lO-Generator test system illustrated in Fig. 3 was

used to evaluate the proposed control algorithm. The simulated disturbance was the symmetrical three-phase trip of line 20 (between buses 15 and 16) without breaker reclosing. Line 20 transported the largest amount of reactive power among all of the active transmission lines before the disturbance. The prefault operating point of the power system was established by increasing all values of bus load and generation levels from the base case by a factor of 1.13. Generator prefault terminal voltages were set equal to their corresponding base case values. From this condition, the system showed significant recovery oscillations without losing stability after application of the disturbance with no controller action. To improve the power system response to the disturbance, the previously designed controller was activated.

An off-line computer rogram was used to simulate the controller with access to real-time state vector phasor) measurements from the power system. Five simulation case studies will ! e explained in this section. Corresponding results will be shown in the next section. Post fault time curves showing the simulated behavior of selected controller output variables and control input signal will be used to evaluate controller performance.

In Case 1 (Controller Inactive), no controller action is applied to the power system after the disturbance. This effect is achievable by setting the weighting matrix Q=O in the cost function.

Diagonal matrices were used for the square weighting matrices R and Q in the cost function of Cases 2, 3, and 4. In all of the three cases, R was 18.18 while Q

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1 48 E. C. MANANSALA AND A. G. PHADKE

was 37.37. The identity matrix was used to measure and limit the control inpui energy, i.e.,

R = I,, (24)

Thus, the requirement of a positive-definite R matrix was met.

In Case 2 (Voltage Control Prioritized), the nonzero elements of the Q matrix are given by:

~ ( k , k ) = lo4

and

The subscript "k" corresponds to the incremental generator terminal voltages while the subscript "i" corresponds to the other elements of the controller output vector including the rotor speeds and relative positions with respect to the rotor position of the largest generator in the system. The controller units used for the voltages and rotor'angles were per unit voltage and radian respectively. Thus, terminal voltage control was given priority over angular control in this case.

In Case 3 (Reduced Voltage Control), all of the nonzero diagonal elements of the weighting matrix Q were set to:

Thus, control of generator terminal voltages is relaxed form that of Case 2.

In Case 4 (Increased Angular Control), all of the nonzero elements of Q were set to:

Thus, the control action for the generator rotor speeds and relative positions was increased from that of Case 2.

In Case 5 (Local Controllers), the control input signals are computed from local measurements and local feedback only. The Riccati equation is solved for each generator and used in individual decentralized controllers. Each controller is described by the following equations:

where

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OPTIMAL CENTRALIZED CONTROLLER 149

The controller matrices Ai,Bi, and Ci are submatrices of the centralized controller matrices A, B, and C. Individual controller vectors are defined as

T T - xi = 1xgi &,i &lT (3'4

where

a. Generator Terminal Vdtages

b. Internal Rota Angla

Fig. 7. Time Cuwa from Case I (Controller Inactive)

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150 E. C. MANANSALA AND A. G. PHADKE

Each controller attempts to minimize the cost function

Thus, there is no attempt to control the terminal voltage of the lar est generator. B .It is further noted that the magnitudes of the nonzero elements o the weighing matrices Qi and Ri in each individual controller are comparable to those used in Case 2. Time curves related to Case 5 are shown in Fig. 12.

a. Generator Terminal Voltages

b. Internal Rota Angla

Fig. 8. Time Cunes from Case 2 (Voltage Contrd Prioritized)

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OPTIMAL CENTRALIZED CONTROLLER

6. Simulation Results This section will state some practical observations based on results of

simulation of controller performance for the cases described in the previous section.

Fig. 7a and Fig. 7b are curves obtained from Case 1 (Controller Inactive). Time curves corresponding to generators 1, 2, 3, and 4 were chosen for analysis since they are the machines closest to the simulated disturbance. Fig. 7a shows the terminal voltage behavior of these generators while Fig. 7b shows the behavior of the rotor speeds and angles relative to the rotor position of the largest generator in the system.

Figs. 8a, 8b, 9a, and 9b are from the simulation results of Case 2 (Voltage Control Prioritized) for the same generators. Figs. 8a and 8b show the terminal voltage and relative rotor position behavior of there machines when voltage control is emphasized by the centralized controller. Figs. 9a and 9b show the control input signals to the excitation and speed-governing systems.

a. Excitation System Input

b. Speed-Covernlng System Input

Fig. 9. Some Control Input Signals from Case 2

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152 E. C. MANANSALA AND A. G. PHADKE

Figs. 10 and 11 directly compare the power system response to the disturbance using the centralized controllers of Case 3 (Reduced Voltage Control) and Case 4 (Increased Angular Control) from the behavior of some selected controller output variables. Fig. 10 is a comparison of relative rotor position behavior while Fig. 11 is a comparison of generator terminal voltage behavior.

b. Cue 3 ( R e d u d Vdtagc Contrd)

c. Cur 4 ( I d Angular Cmtml)

Finally, Fig. 12 shows the unstable behavior of some output variables corresponding to the generators dosest to the disturbance (GI to G4) when the control input signals to the individual generators are not coordinated as in Case 5 (Local Controllers). Fig. 12a shows the terminal voltages of the first four generators. Fig. 12b shows the relative rotor positions of the same set of generators with respect to the rotor position of the largest generator in the power system. Fig. 12c gives the rotor speeds with respect to a stationary frame of reference.

Some comments will now be made regarding the time curves which have just been presented fiom the simulation studies.

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OPTIMAL CENTRALIZED CONTROLLER

Ftg. I I . Ei?& dQ oo TmninJ V d l l p r

Case 2 (Voltage Control Prioritized) provided the best power system response compared to the last three cases. The power system approached the postfault steady-state condition after about five seconds. There were minimal voltage magnitude oscillations as expected since the generator terminal voltages were given the priority in the controller. Machine rotor oscillations were damped out substantially within the first second of operation.

Case 3 (Reduced Voltage Control), the controller performed much more poorly than in Case 2. Power system response was characterized by larger amplitude and more persistent oscillations during the earlier stages of the time interval of interest (5 seconds). The reduced control on the generator terminal voltages caused these quantities to move away from their desired trajectories.

Case 4 (Increased Angular Control) reveals the ractical impossibility of making all output variables follow the target trapctory Lponential , first-order). With more weight assigned to the angles, the generator terminal voltages did not closely follow their corresponding target trajectories for most of the time interval

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E. C. MANANSALA AND A. G. PHADKE

"A I

e. Rota SprQ

unaer stuay. st^, controller pertormance in this respect is better than in Case 3. The angles themselves followed their target trajectories more closely. Increasing the elements of CJ from that given in Case 1 did not significantly improve controller performance.

In Case 5 (Local Controllers), the voltages were controlled very poorly. The rotor angles and speeds failed to stabilize to s teadys ta te values. In general, the combined performance of the individual controllers was very poor.

7. Conclusions This paper has dealt with the evaluation of the performance of the centralized

controller under various conditions. Some very important questions have been answered.

Results of Case 2 indicate that the proposed controller utilizing excitation system and speed-governin control performs excellently as long as the minimum requirements of positive de i niteness of the Kalman gain matrix and linear closed- loop stability of the feedback matrix are satisfied. The use of s teadvs ta te solutions

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OPTIMAL CENTRALIZED CONTROLLER 155

K, and g, with the approximation of equation (33b) substantially ;educes the real- time computational requirements of the controller.

Cases 3 and 4 show that some care must be exercised in the choice of elements for the Q matrix. Use of small values (Case 3) reduces the effectiveness of the controller and fails to damp out oscillations in a reasonably short amount of time. Use of large values above a certain level (Case 4) does not necessarily improve controller performance and only makes the computation of the solution to the Riccati equation more difficult.

Case 5 shows that in spite of the presence of residual terms in the state and output equations (which are included in the control law) and coordinated trajectories, individual controllers which derive generator control input signals from local measurements only fail to perform satisfactorily as a group. Some mechanism to account for the behavior of the external system must be included in the design of the controller.

Finally, the controller equations are sufficiently general to be applicable to other types of problems. In particular, the controller output vector may contain other nonlinear elements such as bus voltage magnitudes at other buses, and generator electrical power output. Other types of excitation and speed-overning systems that may be connected to individual generators can also be represented in the design of the controller.

8. Acknowledgement The work reported in this paper has been supported by a grant from the

National Science Foundation under Research Grant Number ECS-8715315.

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11. P. M. Anderson and A. A. Fouad, Power System Control and Stability, The Iowa State University Press, 1977.

12. IEEE Committee Report, "Computer Representation of Excitation Systems", IEEE Tramactiona on Power Apparatw and Systems, PAS-87, pp. 1460-1464, June 1968.

13. IEEE Committee Report, "Dynamic Models for Steam and Hydro Turbines in Power System Studies", IEEE Tramactions on Power Apparatus and Systems, PAS-92, pp. 1904-1915, November/December 1973.

Manuscript received in final form September 1 1, 1990 Request reprints from Dr. Arun G. Phadke

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