Variable Structure Servomechanism Design and Applications to ...

0005-1098/821040385-16503.0010

Automatica, Vol. 18, No. 4, pp. 385--400, 1982.

Pergamon Press Ltd. © 1982 International Federation of Automatic Control

Printed in Great Britain.

Variable Structure Servomechanism Design and Applications to Overspeed Protection Control* KAR-KEUNG D. YOUNGt and HARRY G. KWATNYt

The concepts of variable structure systems and sliding mode are well suited for 'emergency' control system design and lead to valuable insights into the problems of large turbine overspeed protection control. Key Words--Power station control; integrated plant control; nonlinear control systems; servomechanisms; varying structures; singular control

servomechanism with its linear feedback elements replaced by variable structure feedback elements. Our principal motivation for the study of such systems stems from an interest in the control of electric power plants and systems under 'emergency' conditions. However, as will be evident, there are many other areas in which this control strategy has potential application. The notion of 'emergency' control has at its core the implication of the necessity to take 'heroic' control measures in some sense. One obvious situation is that in which there exists a need to utilize the full range of plant control capability, in whatever manner is necessary, in order to maintain stability and to insure that critical process variables are contained within allowable tolerances. From this vantage point, we are led to the consideration of 'cheap' control in the presence of external disturbances, and, in view of the connections between cheap control and variable structure systems (Young, Kokotovic and Utkin, 1977), we arrive at the variable structure servomechanism. The formulation of this variable structure servomechanism is presented in the next section. Its properties are summarized in three key theorems. The first of these completely characterizes the motion of the system in sliding mode and the second establishes the existence of a sliding mode. The third theorem shows that for an arbitrary class of disturbances which can be modeled as a response of a known system of linear time invariant differential equations, the two goals of servomechanism design, namely, regulation and internal stability, are achieved. An illustration of the utility of the developed variable structure servomechanism is made through an application to the problem of turbine overspeed protection control. The dynamical behavior of an electricity generating plant when

Abstraet--A new class of variable structure feedback systems capable of rejecting a persistent disturbance is developed. This type of system which is called the VS servo is structurally similar to linear multivariable servomechanisms. Its behavior on the switching manifold and its internal stability and output regulation properties are examined using the theory of variable structure systems and sliding modes and existing results on high gain feedback. The concepts of the VS servo are particularly well suited for the study of 'emergency' control system design. The design of the VS servo is applied to the problem of overspeed protection control for large steam turbines. This study leads to an understanding of the generic difficulty associated with the application of overspeed protection control to large coal fired power plants and provides a feedback control structure with considerable promise for such applications. In addition, the paper serves as a useful illustration of the flexibility and potential applicability of variable structure designs. The effect of actuator parasitic dynamics is considered. I. INTRODUCTION

IN THIS PAPER, a new class of variable structure feedback systems called the variable structure servomechanism is developed. The major design objectives are those of the linear multivariable servomechanism: (I) to regulate a set of e r r o r variables to zero for all exogenous disturbances belonging to a specified disturbance class (regulation), and (2) to ensure stability of the feedback system and possibly to meet other transient performance requirements (internal stability). In general terms, the variable structure servomechanism is a linear multivariable

*Received 30 March 1981; revised 16 November 1981; revised 8 December 1981. The original version of this.paper was presented at the 8th IFAC Triennial World congress on Control Science and Technology for the Progress of Society, which was held in Kyoto, Japan during August 1981. The published proceedings of this IFAC meeting may be ordered from Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by associate editor V. Utkin under the direction of editor H. Kwakernaak. ?Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, U.S.A. 385

386

K.-K. D. YOUNG and H. G. KWATNY

there is an abrupt isolation of an area caused by a system disturbance can be classified as an 'emergency' condition. This is a situation where fast control action is needed to prevent rapid acceleration of the turbine-generator so that turbine speed does not attain its allowable upper limit at which time the unit would be tripped. Two variable structure servomechanisms are designed for overspeed protection. In the design of these controllers the actuator dynamics are neglected, however, their effects on the overall system behavior are evaluated from the simulated system responses. 2. FORMULATION OF A VARIABLE STRUCTURE SERVOMECHANISM The conception of the variable structure servomechanism which will be examined in the next two sections is essentially motivated by two key factors in the design of control systems for 'emergency' conditions. First, because of the presence of external disturbances a linear multivariable servomechanism formulation is the most suitable. Second, the 'emergency' conditions to which the control system must attend requires use of the full range of control capability in order to maintain stability and to limit excursions of critical variables. This latter factor leads naturally to a cheap control formulation of the feedback law design in the linear multivariable servomechanism design. Formulation of a variable structure servomechanism emerges once it is realized that the cheap control solution is in fact related to a variable structure feedback solution as described in Young, Kokotovic and Utkin (1977). We begin our formulation of this variable structure servomechanism with a brief description of the robust linear multivariable servomechanism which in recent years has been studied extensively by many theoreticians and practitioners. The open loop plant dynamics are modeled by a linear time-invariant system = Ax + Ew + Bu

(1)

~i, = Z w

(2)

y = Cx + F w + D u

(3)

where x is an n-dimensional plant state vector, w is an q-dimensional vector representing a combined state for the exogeneous disturbance and output reference, y is an r-dimensional measurement output vector and u is an mdimensional control input vector. The objective of this control problem is to regulate an Idimensional regulated error vector e e = C~x + F,w + Dru

(4)

by a feedback controller without directly measuring w such that the closed loop system-excluding the disturbance states w, is stable (internal stability) and e ( t ) ~ O as t-~oo for all initial states (output regulation). It is customarily assumed that ( A , B ) is controllable, (C, A) is observable, the matrices B and C are of full rank and e is readable from y. By this we mean that there exists a matrix H such that e - - H y . In the following discussions, it is also assumed that both the states x and disturbance w are observable through y. Necessary and sufficient conditions for the existence of a solution to the servomechanism problem have been stated in Francis (1977) and Davison (1976). Under these conditions, if the closed loop system exhibits internal stability and output regulation, then the 'steady-state' state ~ and control input ti are linear functions of the disturbance w ,~ =

Ww

(5)

=

Uw.

(6)

The matrices W and U satisfy the algebraic equations AW + E+ BU-

WZ = 0

C , W + DrU + Fr = 0

(7) (8)

and their existence is equivalent to the fulfillment of the rank condition for each tri-=

At(z) rank [ A - t r i C,

B]=n+l.

(9)

Dr

The synthesis procedures of the feedback controller proposed to date fall into two distinct categories. One class is based on the feedback of estimates of the disturbance state and the other on the feedback of states of a dynamic system driven by the regulated output. These two design procedures have been summarized and the structure of the resulting feedback controllers has been characterized in Kwatny and Kalnitsky (1978). Define the motion of plant state and the control input relative to their steady-state values found in (5) and (6) as Ax and Au Ax -- x -- ~

(10)

Au-- u-- ~.

(11)

The feedback control Au = KA~

(12)

Variable structure servomechanism design is formed with the feedback gain matrix K being taken from a state feedback design for the undisturbed, open loop plant with the presumption that the state x is available. The quantity A~ is an estimate of Ax A.~ = .~ - W ~ ,

(13)

where ~ and ~, are estimates of the plant state x and disturbance w, respectively. The composite observer is of the form

L -H2C

BTQB > 0

F/-/,] y Lf/2J

(14)

u = Au + U~i,.

(15)

The matrices HI and H2, by our observability assumption earlier, can be chosen such that ~(t)-*x(t) and ~ ( t ) ~ w ( t ) as t--,o0. The structure of the resulting linear multivariable servomechanism is illustrated in Fig. 1. Since Ax satisfies

Arc = AAx + BAu

(16)

and from our earlier remarks, the design of K is equivalent to the design of the full state feedback control law

Au = KAx

= 1 K * (0)Ax.

(20)

In Young, Kokotovic and Utkin (1977), it is shown that the high gain feedback law (20) is related to a variable structure feedback control law Au(Ax) which is discontinuous on the manifold K*(0)Ax = 0 Aui(Ax)

= ~" Au~(Ax), k'f(0)Ax > 0 ( auT(Ax), k*(0)Ax < 0

(21)

where Aui(Ax) is the ith component of Au(Ax) and k*(0) is the ith row of K*(0). If ideal sliding mode exists on K*(0)Ax = 0, then the motion Ax(t) of the high gain feedback system (16), (20) and that of the variable structure feedback system (16), (21) is identical in the switching manifold K*(0)Ax = 0. If the initial states are on this switching manifold, then the ideal sliding mode in this manifold is optimal with respect to the singular performance index

(17)

for (16). Taking a cheap control formulation in this design, the following quadratic performance index with small penalty on Au is minimized

x = l fo®[AxrQAx + l~2AurRAu]dt

(19)

then the resulting optimal feedback gain (lhz)K*(/~) satisfies the conditions: (i) (K*(0)B) -~ exists and (ii) Re ~,(K*(0)B) 0 and R > 0 are symmetric and ~ is a small scalar parameter. If the weighting matrix

Fm.

Q is chosen such that

Z - H2FJ

FB - H~D 1 + L -H2D J u +

387

1. Block diagram of linear muRivariable servomechanism.

if

J =~

(AxTQAx) dt.

(22)

In other words, the ideal sliding mode coincides with the totally singular optimal arc. For a single input system, this aspect of the singular optimal arc was first discussed in Anderson and Moore (1971). Nonideal sliding on K*(O)Ax = 0 due to finite slope switching with slope /~-1 is O(/~ 2) near optimal with respect to the performance index (22) provided that K*(0)Ax(0)=0. We note that this optimal sliding mode is achieved through the selection of the switching manifold rather than through the design of the discontinuous feedback functions Au~(Ax) and AuT(Ax) in (21). Discussions and formulations of other types of optimal sliding modes can be found in Utkin and Young (1978). In general, the initial state Ax(0) lies off the optimal singular manifold K*(0)Ax =0. The optimal feedback control transfers the state from Ax(0) to some point on the optimal sin-

388

K.-K. D. YOUNGand H. G. KWATNY

gular manifold instantaneously if ~ = 0 and in a time period of O(/~) if /x is sufficiently small. Essentially, this optimal transfer onto the optimal singular manifold is a fast motion. The magnitude of the optimal feedback control is proportional to /x -I during this optimal transfer. The control signal will saturate in this time period if there is a physical limit on its magnitude. As an alternative to the cheap optimal regulator problem formulated as an unconstrained optimal control problem, we can pose an optimal control problem to minimize the singular quadratic index (22) with control magnitude constraints. In the multivariable case, it has been shown in Anderson and Moore (1971) that in some neighborhood of the origin of the state space the optimal trajectory lies in the optimal singular manifold K*(0)Ax = 0 of the unconstrained cheap regulator solution. The total optimal trajectory is composed of nonsingular and singular optimal arcs. Computation and patching of the nonsingular and singular arcs require the solving of a two point boundary value problem. The rationale behind the minimization of a singular quadratic index is that the control designer is willing to use the maximum allowable control magnitudes to minimize the deviations of the states. Due to the complexity of the patching conditions, it may be difficult to compute the required optimal control solution on-line, thus limiting the usefulness of this optimal control formulation. The minimality of the performance index and the implementability of the resulting optimal control solution may have to be compromised in a multivariable control design of the cheap regulator type. An on-line implementable suboptimal cheap control solution is a variable structure feedback control of the type (21) where the optimal singular manifold K*(0)Ax = 0 is chosen as the switching manifold. The discontinuous feedback control Au~(Ax), and Au~.(Ax) are designed so that sliding occurs on the optimal singular manifold. This approach essentially admits the presence of the control magnitude constraints and guarantees an overall asymptotically stable system whose state trajectory also consists of two distinct parts: a motion not necessarily fast that moves onto the optimal singular manifold followed by the optimal singular arc. Clearly, the suboptimality of this variable structure feedback solution is due to the reaching phase: the optimal singular arc is optimal with respect to the index (22) with the initial time zero replaced by the time instant t~ that marks the tFor simplicity, we shall denote from hereon the near optimal feedback gain matrix K*(0) introduced in (20) by G.

beginning of sliding on the optimal singular manifold. So far we have discussed the variable structure feedback solution as a viable alternative to the high gain feedback solution in the design of the feedback control (17) in the linear multivariable servomechanism design. Formation of a variable structure servomechanism is obtained upon replacing the feedback control (12) by its variable structure alternative Aui(Ag)

f k*(0)ai>0 l Au~(A£), k*(0)A~ < 0

(23)

which is e s sentially (21) with Ax substituted by its estimate A~. We emphasize that the switching surfaces depend only on A£ The constitutive equations of this variable structure servomechanism are (1)--(3), (14), (15), and (23). In the above development of a variable structure servomechanism, the variable structure feedback control is exploited to provide an online implementable suboptimal control problem with control magnitude constraints. Thus, our primary motivation to employ variable structure feedback in a control system is not the usual one of reducing system sensitivity to parameter variations and unknown disturbances. Nevertheless, our variable structure servomechanism enjoys these same benefits of variable structure feedback system. Our variable structure servomechanism, albeit its objective is to regulate against disturbances, is conceptually different than the variable structure control systems with the aim of disturbance rejection which were developed by Utkin (1965, 1966) and discussed in Itkis (1976) and Utldn (1977). In these previous works on disturbance rejecting variable structure systems, the dynamics of the disturbance are essentially unknown: only the bounds on the disturbance and its derivatives are used in the design. In contrast, we assume, in the same spirit as in linear multivariable servomechanism problems, that the disturbance can be modeled by a response of a known system of linear time invariant differential equations. We shall call the variable structure servomechanism developed in this section the VS servo. 3. DYNAMICAL BEHAVIOR OF THE VS SERVO

Our study of the properties of VS servo can be divided into three stages. First, we examine the system's behavior on the switching manifold GA~ = K*(0)A~ = 0,t under the assumption that sliding mode occurs in this manifold. Secondly, the motion of the VS servo is shown to remain in a neighborhood of GA~ = 0 and to reach this

Variable structure servomechanism design manifold asymptotically. Finally, we show that regulation of the system error e defined in (4) is achieved for the class of disturbance w defined in (2). The following theorem shows that the behavior sliding mode is governed by an 2n + q - m dimensional system. Theorem 1. Suppose sliding mode occurs on the switching manifold GAI = 0 in the variable structure servomechanism composed of equations (1)-(3), (14), (15), and (23), then the dynamical behavior of this variable structure feedback system in sliding mode is governed by

MtH,- WH IF7 E - H,F Z - H2F

[ .A

Lw-;J

(24)

x - Yc

should not be surprising since in sliding mode variable structure .feedback systems behave like linear feedback systems. From Theorem 1, it is clear that if sliding mode occurs on GA~ = 0, then A~s(t)~0, x(t)--*~(t) and w ( t ) ~ ¢v(t) in this manifold as t ~ ~. In the next theorem, we examine the behavior of the VS servo before sliding mode occurs, that is, the period of reaching to GA~ =0. It is convient to consider the motion of the system in the 2n + q-dimensional space with coordinates A ~ , x - 2 , w - f f . We will show first that the variable structure servomechanism steers the system to a domain ~ of the state space which contains the subspace defined by GAS = 0. Furthermore, it will be shown that ~ shrinks asymptotically to this subspace so that in finite time the trajectory is within an arbitrary bounded neighborhood of the sliding manifold. Theorem 2. Let the l x ( n + q ) vector pT denote the ith row of the m × (n + q) matrix P P =- G[H,- WH2]"

[C

A~, - MAR

B T M T ----0

(26)

and the columns of the n x ( n - m) matrix N span the null space of G G N = 0.

(27)

Proof: See appendix.

It was shown in Young, Kokotovic and Utkin (~xT77) that X ( M A N ) are the transmission zeros ox the triple (G, A, B) which also dictates the behavior of the high gain feedback system (16), (20) in the optimal singular manifold. The asymptotic stability requirement of the cheap control solution guarantees that M A N is a Hurwitz matrix. Design of the composite observer (14) (matrices H1 and /-/2) guarantees the lower diagonal block matrix in (24) to be Hurwitzian. Thus, Theorem 1 shows that sliding mode on GA~ = 0 is asymptotically stable. The system (24) is a linear servomechanism in which the well-known separation property in linear observer feedback systems is exhibited: the matrix G which results from the cheap control formulation is designed independently from the composite observer matrices H, and /-/2. This AUT Vol. 18, No. 4---B

FI[A-H1C t - H2C

(25)

the columns of the n x ( n - m) matrix M T span the null space of B T

389

E - H I F ] -1 Z - H2F J "

(28)

Suppose that a variable structure feedback control Au(Ax) which is discontinuous on the switching surfaces s(Ax) = G A x is designed for the system (16), A~ = A A x + B A u , such that reaching of the manifold defined by G A x = 0 is guaranteed. Let the resulting discontinuous feedback control be defined by the feedback functions A~(.),Au~(.) as in (21) [with G = K*(0)]. If the variable structure control Au/(A~), which is defined by (23) [with G = K*(0)] is employed, the trajectory of the variable structure servomechanism (1)-(3), (14), (23) reaches a domain ~ defined by

after some time tl and remains in ~ for t t> t,. Proof: See appendix.

An important result regarding the possibility of sliding mode on GA~ = 0 is directly obtainable from Theorem 2. Since the estimation error of the composite observer satisfies

d[: ai

rA--,c -

=

t

-

Z-H: FJ"

39o

K.-K. D. YOUNG and H. G. KWATNY

which is asymptotically stable, we have the following corollary:

Corollary 1. For a fixed A > 0 there exists a finite t'(A) such that A~ is confined to the domain

to Theorem 2, then the motion A:~(t) of the VS servo reaches a domain (31) for all t i> t' and can be approximated by (32). Furthermore, for the class of disturbance defined by (2), the system error e(t) of (4) is regulated to zero asymptotically. 4. D E S I G N OF O P T I M A L S W I T C H I N G M A N I F O L D S

IgT A2I ~< A,

i = 1. . . . . m

(31)

for all t I> t'. Moreover A->0, t'->oo. Theorem 2 and its corollary shows that sliding mode does not occur on the switching manifold GA:~ = 0 but instead this manifold is reached asymptotically as well as :~(t)-> x(t) and ~,(t)-> w(t) as t->oo. One convenient way to estimate the transient behavior of the VS servo after the motion A:~(t) reaches a neighborhood of GA:~ = 0, such as (31), is to interpret this motion as a nonideal sliding mode and to consider the inability to reach the switching manifold in finite time as the effects of a nonideal switching mechanism. The ideal sliding mode is defined by the solution of the system (24) of Theorem 1. We denote the ideal sliding mode by A:~*(t). From Theorem 1 A:~*(t)satisfies GA:C*(t) = 0 and MA:~* (t) has the same solution as A~,(t) in (24) so that A:~*(t)->0 as t->oo. Using the results of Utkin (1971) on the approximation of ideal sliding mode by nonideal sliding mode, it can be shown that if A:~ is confined to the domain given by (31), then there exists a positive c such that for all t >I t'

] A ~ * ( t ) - A~(t)] 0 as t->oo. Regulation of the error can be demonstrated as follows. Equation (4) can be written in terms of the state variables A:~, x - :~, w - ff as e = C,(x

The design of optimal switching manifolds has been considered in Utkin and Young (1978). We shall first review their results and then discuss the implications of restricting the optimal switching manifold to coincide with an 'outputnulling' linear manifold: if the motion of the system lies entirely on this manifold, then the system output is zero. Following the discussions in Section 2, we note that the design of optimal switching manifolds can be done with the assumption that all the states are available for synthesis of the manifold. Henceforth, the open loop system (16) is to be considered in this design and the switching manifolds are manifold in the Ax-space. Let M be the ( n - m ) x n matrix defined in (26) and V be an m x n matrix whose rows are linearly independent with the rows of M. Define

(33)

(34)

Since the system is stable and in view of (A2) Au(A$)->Aue,->0 as x - ~ , w - f f and A~->0 we can summarize the behavior of the VS servo in the next theorem. Theorem 3. Given that the variable structure feedback control in (23) is designed accordingly

(35)

m

(T-,)TQT-,_ ro,, .........:Q,2]} "-m,,,

(36)

LQ,2 : Q22]} •

Assume that BTQB > 0, (A, B) is stabilizable and (Au, D) is detectable where DTD = Qll - Q~2Q2~QI2

which in view of (8) reduces to e = C r ( x - ~ ) + F , ( w - f v ) + DrAu(AYO.

_ r a i l : A12]} .-r. L~; • i "A~2J}m n-m

- ¢v)

+ DrAu(A.f) + (C,W + F~ + D,U)fv

m

(37)

then from Utkin and Young (1978) the optimal switching manifold with respect to the singular quadratic index (22) is given by GAx = 0 where the matrix G is given by =-R

-|

T -1 T " B2P/[(~22((~12+ AT2Ps) [ra×m]Y.

(38) The matrix P, is the unique positive semidefinite solution of the matrix Riccati equation 1 T 0 = P~(A11-AnQ~-~QI2)+(AI1 -1 Ql2) T TPs - A12Q22 --

P~A1202~A~2Ps + Dr D

(39)

Variable structure servomechanism design B2 = V B

(4O)

Note that this Riccati equation is associated with the optimization of the ( n - m ) - d i m e n sional system

(41)

AYC~= ( A l l - A12H21H1)Ax~ +'A12Au~

and PI is a positive definite matrix given by Pi = S-m(SlnQ22Sln)ln S - m

391

with

(50)

with respect to the performance index (42)

S =--B2R-IBT2.

.Is = ~

Suppose now a specific set of outputs Az is weighed in the singular performance index, that is J =[

(AzTAz) dt

(43)

where

Az= HAx

(44)

and H is a rank m matrix factorizing Q (45)

Q = HTH.

In the following discussions, we shall retain our earlier assumption that B T Q B > 0 which implies that H B is invertible. In Young, Kokotovic and Utkin (1977) it is shown that if the transmission zeros of the triple (H, A, B) all lie in the open left half-complex plane, then the matrix D defined in (37) is a null matrix and the solution of the Riccati equation (39) is Ps = 0. The expression for matrix G becomes G = -R-IBTptH21H

(46)

where the matrix/-/2 is defined in H T -l =- [Hi

H2].

(51)

The open loop eigenvalues of system (50) are the transmission zeros of the triple (H, A, B). The existence of right half plane transmission zeros implies that system (50) is not asymptotically stable in open loop. Since there is no state weighted in Js, if asymptotic stability is not required in the resulting optimal feedback system, the trivial solution Ps = 0, hence us = 0 is an optimal control solution and J, = 0 is the optimal cost for this problem. Imposing asymptotic stability of the origin as a constraint on this optimization problem, as it was in Willems (1971), Ps = 0 of (49) is no longer an admissible solution. The optimal singular index has nonzero value. The relationships between the value of the singular quadratic index and transmission zeros are also discussed in Kwakernaak and Sivan (1972). The existence of the solution of (49) that stabilizes the optimal feedback system is proved by Martensson (1971), in which the case of nonnegative definite symmetric weighting matrix of the states was dealt with. The solution Ps is obtained as the limiting solution of a matrix Riccati differential equation - d P = P A , + Ar, P - PAI2(HTH2)-IAI2P dt

(52)

(47)

In this case, the null space of the matrix G coincides with the null space o f the matrix H and the optimal singular arc occurs in the manifold where the outputs are zero A z = H A x = O.

(AuTHTH2Au,) dt.

(48)

If

there exists a transmission zero of in the closed right half plane, the optimal singular arc no longer occurs in the 'output-nulling' manifold (48). A Riccati equation similar to (39) has to be solved to determine G

with A , - A I 1 - A I 2 H ] I H 1 dition

and boundary con-

P(tl) = I ( t I is arbitrary)

(53)

that is Ps = lim P(t).

(54)

(H,A,B)

0 = Ps(An - AI2H21HI) +

- P, An(HIHz)-IAT2P,.

(Au

-

A12H~LHI)TPs

(49)

The resulting matrix G in this case takes the same form of (38) with Ps given by (54). Thus, although only the outputs are weighed in the singular index, the existence of a right half plane transmission zero requires the optimal (stable) singular arc to occur in a manifold different from the 'output-nulling' manifold.

392

K.-K. D. YOUNG and H. G. KWATNY

5. AN APPLICATIONTO TURBINE OVERSPEED PROTECTION CONTROL A problem of current interest to the power system community is that of regulating frequency following a system disturbance which results in the abrupt isolation of an area in which there is insufficient load to comsume all of the prefault generation (Younkins and Johnson, 1981; Schatzmann, 1981; Baldwin and McFadden, 1981; Termuehlen and Gartner, 1981). Because of the consequent mismatch between steam power delivered to the rotating turbine shafts and electrical power delivered from the generator terminals, and in view of the relatively small rotating inertia, the machine speeds will accelerate rapidly. The purpose of the plant overspeed protection control (OPC) system is to bring the machine speed under control before it attains its allowable upper limit at which time the unit would be tripped. It is desired, of course, to avoid the tripping of any plant, so that all units will remain synchronized and it will be possible to restore the system to its original configuration in minimum time. The prevailing operating philosophy of U.S. utilities is to apply this concept to the majority of fossil fueled drum-type plants and once-through type units. OPC systems on oncethrough boilers act to limit speed excursions but the plant is typically tripped upon initiation of OPC action. This is because the dynamic characteristics of once-through systems generally require that they use sufficiently large bypass systems in order to avoid excessive boiler transients following OPC action. Such bypass systems are not typical in the U.S. although they are in Europe. The operation of a typical power plant OPC system will be briefly summarized (Dunlop and Olken, 1980). OPC systems have two functions: (1) identification of a load rejection situation, and (2) execution of control actions. The identification function consists in monitoring shaft speed and acceleration. Should they simultaneously exceed prespecified limits, then OPC is initiated. Control action generally includes rapid closure of turbine intercept valves and/or governing or throttle valves followed by a period in which the valves are maintained shut, a subsequent period during which the valves are reopened, and finally restoration of normal speed governing control. Because of the rapid acceleration of the rotors, attention has naturally focused on the problem of arresting the speed excursion by fast closure of the turbine valves. There has been considerable discussion in the literature pertaining to the use of fast valving in power sys-

tem operations. Hydraulic actuated turbine valves, including governor valves, are generally equipped with fast closure capability and reopening typically is accomplished at normal rates. It is accepted that fast valve closures can be accomplished in sufficiently short time to contain the initial speed excursion. Furthermore, studies have indicated that maintaining the turbine valves closed for a matter of a few seconds can restore frequency to near its desired level with acceptable disturbance of the state of the boiler. For instance, in Koyanagi, Komukai and Abe (1977), fast closure of the intercept valves results in a reheat pressure rise which opens safety valves after about 2 s, peaks at about 5 s with pressure 35% above full load pressure and then decays to nominal within approximately 25 s. Nevertheless, operating experience has been far from satisfactory. Although there has been little reporting of OPC operating experience in the open literature, there has been considerable discussion within the U.S. utilities and at least one informal report has been circulated along with a utility recommentation for an industry study. This report refers to a system separation experienced in the U.S. by Union Electric Company and portions of the Illinois Power Company and Associated Electric Cooperative Systems in February 1978 (Union Electric Company, 1978; Kundur, 1981). The separation resulted in a generation-rich island causing frequency to increase. Overspeed protection control (fast valve closure) was initiated in eleven of the fourteen plants. Normal governing action reduced output in the other three plants. Four of the fourteen plants tripped. Three of these were plants in which OPC did act. Furthermore, all three were large (555 MW) coal fired units. All of these experienced oscillations in water level, pressure, firing rate and feedwater rate and were tripped either manually or automatically. The fourth unit to trip did not experience OPC but normal governing action caused a large fluctuation in output and a subsequent trip due to high furnace pressure. It is interesting to examine the ten surviving units, eight of which experienced OPC. Two of these were 555 MW coal units in the same class of the three that tripped. However, on one only, 80 MW was dropped indicating less than normal sensitivity to the frequency excursion. In addition, the boiler controls were on manual. On the second unit, the boiler was also on manual control and the operator took control of the turbine control valves and restored generation to the level existing prior to OPC. Two other units were 480 MW once-through units. In each

Variable structure servomechanism design case, following OPC action, the operator took command of the turbine control valves to regulate throttle pressure. Governor action was cancelled by this procedure. A 300 MW coalfired unit experienced OPC action after which the operator assumed manual control of pressure. Two 125 MW and a 250 MW coal-fired unit all responded well, without intervention, following OPC action. Fast valve closure was not initiated on the remaining two units, a 584 MW plant and a small 356 kW light oil-fired plant. Both units reduced generation in response to manual governor action without indident. In summary, only three of the eleven plants equipped with OPC survived the incident without either manual intervention or a trip. In this section, we consider an application of the VS servo to the problem of overspeed protection. As a result of this study, a tentative explanation of the unsatisfactory behavior of OPC is provided, which is consistent with the above observations as well as other field experiences. It is felt that the VS servo formulation is particularly suitable in the design of OPC, for the following reason. Variable structure systems with sliding mode can be considered as an implementation of high feedback systems (Young, Kokotovic and Utkin, 1977). The motivation of using high feedback gain is often that fast control action is needed due to the occurence of an emergency situation. One obvious situation is that in which there exists a need to utilize the full range of plant control capability, in whatever manner is necessary, in order to maintain stability and to insure that critical process variables are contained within allowable tolerances. This is precisely the concerns in overspeed protection. Through the design of a variable structure servomechanism, we arrive at a possible alternative OPC mechanism. The plant model adopted for the OPC study is presented in Section 5.1. A single loop control

393

system which uses the governor valve control actions, typical of those employed in OPC systems is designed as a variable structure servomechanism. The controlled system behavior is discussed in Section 5.2. One result of the analysis of this single loop control system is a possible explanation of the difficulties experienced in applications of OPC. As a consequence of these observations we formulate a two-input OPC strategy which utilizes the governor valve and a feedforward signal to the feeder stroke as controls. A two-loop variable structure servomechanism is designed and analyzed in Section 5.3. A study of the effects of parasitic actuator dynamics on system responses is also included in Section 5.3 and the results illustrate the inherent capability of variable structure designs for dealing with model uncertainty. 5.1. Plant model description The plant model utilized in this study is illustrated in Fig. 2 with parameter values listed in Table 1. It is a simplified model of the type frequently employed in automatic generation control studies, except that it is linear and incorporates a characterization of drum level dynamics and the associated level control system. This model was chosen, for preliminary studies only, on the basis of two principal virtues. The first is its obvious simplicity. Our primary interest at this stage was to gain experience with the variable structure servomechanism in a context of limited complexity. The second is the fact that the model may contain nonminimum phase zeros (dependTABLE 1. MODEL PARAMETER VALUES

K =0.011 a = 100.0

Ci =200.0 C2= 20.0

• F = 200.0

~r = 4.0

~b= 0.3

k =0.15 K', = 3.0 K, = 10.0 y = 0.005

J = 0.1666

D = 0.01 a* = 0.2

&P---

FIG. 2. Block diagram of boiler model. A~oJco*, feedwater flow; AoJ*/~o*,steam flow; ALIL*, drum level, xl, integrator state--fuel controller; x2, fuel flow A~ol/~o~; x3, pressure APIP*, x4, volume fraction of steam Aala*; x~, integrator state-level controller; x6, turbine mechanical power AP~IP~; x~, frequency Af; u~, valve position; u2, feeder stroke feedforward.

394

K.-K. D. YOUNG and H. G. KWATNY

ing on parameter valves) between the input u, and frequency, Af. Furthermore, such zeros appear to be an inherent difficulty associated with plant control and could be problematical for control structures incorporating high gain. It is easily shown that nonminimum phase zeros appear if the pressure control loop is sufficiently sluggish. This is likely to be the case with large pulverized coal plants. The parameter values used in the simulations described below do result in nonminimum phase zeros. 5.2. Governor valve loop The control objectives of the overspeed protection problem is to regulate the system following a large decrease in electrical load, (APrJP*), in such a manner that: (a) in steady-state the frequency error is zero; and (b) the equilibrium state is asymptotically stable and the maximum frequency, pressure and drum level deviations are tolerable. In case 1, only the governor valve is available for control. We denote Al ul = Z~-, and - 1 ~< ut ~< 1.

(55)

The control system is designed using the quadratic penalty function

] = f : {y~ + ~u~ dt

(56)

AP y, =-~-~--/3Af.

(57)

where

The variable structure control can be approximated by the high gain control with saturation* u, = sat [ f i l + l (/3Af --p---g)], AP

~1

(58)

provided that the open-loop zeros of the system ul-~ Yt all lie in the left half plane. Note that the switching plane is defined by Yl = 0. Figure 3 shows the results of three simulation runs. Each curve illustrates the projection of the state trajectory onto the A P I P * - f plane following a 50% load drop. The first curve illustrates the trajectory followed when the valve instantly shuts and remains closed. Note that a tsat(c) = 1 f o r c ~> 1, s a t ( c ) = - 1 f o r c < 1, a n d s a t ( c ) = c for - 1 ~ 0 , 8 > 0 (62)

and R = diag ( r , r2). Figure 4 illustrates trajectories obtained with/3 = 0.1, 7 = 1, ~ = 1, r I = 1 , r 2 = 0.133, and ¢ = 0.5 x 10-5. The control law is as follows? s 1= 0.99926yl - 0.03845 Y2

(63)

s2 = 0.01489yl + 0.387Y2 + p ,

(64)

AI~L

u~ = p ~

(65)

ul = sat u* - - sl

(66)

u2 = satl (--l s2)

(67)

395

control (66) and (67). The corresponding switching p l a n e s are s, = 0 and s2 = 0 as d e f i n e d in (63) and (64). The behavior of frequency following a 40% load drop is observed in Fig. 4(a). Note the fast transient which follows the initial value closure as anticipated from the results of the single input study. The slow transient, which follows the initial capture of the frequency excursion, is, however, dramatically different. The peak excursion of about 1.4 Hz occurs at about 100 s and is smoothly restored to nominal by 350 s. An acceptable maximum pressure deviation of 8% is observed in Fig. 4(b). Until about 310 s the system is in the reaching mode. Sliding occurs on s l = 0 at about 2 s as seen from Fig. 4(b) which shows the governor valve opening. Sliding in the intersection of s~=O and se=O begins at about 310 s which is most clearly evidenced in Fig. 4(e) which illustrates the fuel feedforward input. The actual fuel flow is shown in Fig. 4(c) where the extensive smoothing due to the fuel dynamics is observed. Note also the ultimate zeroing of the frequency error which is one of the requirements of the servomechanism design. The role of the second control is subtle, but crucial. From one viewpoint, it circumvents the difficulty associated with the nominimum phase zeros of the single input case. T h e s e z e r o s contribute to the development of a slow, unstable, oscillatory interaction between the pressure and frequency control loops when 'high' gain feedback is used to control frequency. However, high gain is essential in the initial period to contain the frequency transient. The second control effectively stabilizes the slow transient and plays no role during the frequency-critical initial seconds. Fast response of the fuel is not required and the control does not call for unreasonable rates of fuel change. Our results show, however, that there must be suitable range of fuel control available. For instance, it is not possible to achieve acceptable 0.080.06.',

/~

0.04. The electrical load (APtJP~) is estimated using a reduced order observer. The estimated load is denoted by (A/~t/P*). The variable structure control is implemented through the high gain tsah(b) = 0.3 for b >/0.3, sah(b) = -0.85 for b ~ 0 and si > 0

(A9)

or the relations cri < 0 and sl < 0 .

(AI0)

In view of the remarks following (A8), we can conclude reaching of the domain ~ when the switching manifold G A £ = 0 is employed.