direct methods for transient stability studies in power system analysis (lyapunov- energy function)

332 IEEE TRANSACTIONS ON AUMUTIC CONTROL, VOL. ac-16, NO. 4, AUGUST 1971

Direct Methods for Transient Stability Studies in Power System Analysis

JACQUEs L. WILLEMS

Absfruct-This paper deals with recent advances in developing direct methods for studying the transient stability problem of single- machine and multimachine power systems. The paper starts out with the construction of the mathematical model that is usually employed in the analyis of power system transient stability. Com- puter simulation methods are then briefly discussed, and it is in- dicated why accurate direct methods for transient stability investigations would be most welcome. It is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov’s second method. The main purpose of the paper is to give an exposition of the interesting results that have been obtained by applying Lyapunov’s second method to the transient stability problem of single-machine and multimachine power systems. In the final portion of the paper some areas for further research are discussed.

I. INTRODUCTION

T HE PROBLEM of transient stability of power systems becomes increasingly important as t,he size

of the interconnected areas becomes very large. Indeed, the tendency of a system to lose synchronism and dis- integrate, and the resulting possibility of oscillations in the power transfer between interconnected areas is much more prevalent for large systems than it is for relatively small isolated groups. Optimum control and stability investigations are presently used to a large extent in the analysis and design of power systems. A comp1et.e survey of the application of optimum control to power systems wa.s presented at the 1968 Joint Automa.tic Control Conference [l]. It is the aim of this paper to give an exposition of recent results and mbthods concerning the transient stability problem of power systems. The writing of t.his paper is motivated on one hand by a desire t.0 expose this problem to the cont,rol audience a t large, and on the other hand by the timeliness of illustrating an area of application for sophisticated stability analysis techniques to a practica.1 real-world problem. This is done in the hope that such an exposition could. lead to a contribution in bridging the well-publicized gap between t.heory a.nd practice. r\’ot,e that, even in the optimization of t,he steady- stat.e operation of power systems [I], stability crit,eria are important, since they const,itute some of the opthizat,ion constraints.

I n recent y e m a large amount of papers on stability has been published in t,he automatic control literature. The applicat,ion of these ideas and methods to actual

Paper recommended by E. F. Infant,e, Chairman of the IEEE S C S Manuscript received May 29, 1970; revised January 26, 1971.

Stability Theory and Nonlinear Systems Commit.tee. The aut.hor is with the Division of Engineering and Applied

Physics, Harvard University, Cambridge, Mass. 02138, on leave from the University of Ghent, Ghent, Belgium.

systems has, however, rema,ined rather limited. The area of power system stability is an except,ion t,o this rule, since some interesting results have been obtained using t,hese methods. I n particular, much research in control theory has been devot,ed to the problem of the stability of syst,ems 1%-ith a single-variable or multivariable nonlinear feedback element [2]. Interesting results have been obtained for the t,ransient, stabilit,y of power systems by representing a. pon-er system by a nonlinear feedback system, and, in part,icular, by using t.he t.echniques developed for the construction of Lyapunov functions for the proof of the well-known Popov criterion and its generalizations.

Two main approaches can in general be taken t.0 the problem of system stability: one relies on the concept of input-output stability, the other on the concept of dynamic system st.ability in the sense of Lppunov. Although t.he idea of input-out.put st,a.bility appears to be more suited for certain applications, in particular for syst.ems driven by constantly acting disturba,nces, it is not well suited to dea.1 1vit.h the stability problem of power syst.ems. Indeed, the stability problem of pow-er syst,ems arises as follows. A fault occurs somewhere in the system, which dist.urbs the operat,ing conditions. This triggers a sequence of events; the fault is cleared and the system is restored to a healthy post-fault condition. The state of t,he system after fault clearing is in general not the desired equilibrium &ate of the post.-fault syst.em. The question is whether or not the system will converge to this equilibrium statme. This is hence a typical example where asymptotic stability in the sense of 1,yapunov is of prime importance. It should also be emphasized that t,he stability problem as encountered in pon-er systems is not a problem of globd st:abilit,y, but. a problem of estimat,ing the domain of attrac.tion of an equilibrium state of the syst,em. Power syst,ems are never asympt,ot.ically stable in t,he large. The aim of transient. stability studies is hence to compute regions of asymptotic stabilitry of equilibrium solutions. This particularity of the stability problem is responsible for the fact that. the direct methods described in this paper almost exclusively deal with the a.pplication of t,he direct method of Lya.punov. Although it appears t.0 be a fair appraisal that for input-output. st,ability and asymptotic stability in t.he large, Lyapunov methods have been surpassed by operator techniques [3]-[6], they are at, t,he present time the only general procedure available for computing stability regions. The study of the con- struct,ion of Lyapunov functions remains hence a very fruhful area of research.

WIJAFXS: TRdNSIENT STABILITY STUDIES 333

The main purpose of this paper is to review the results that have been obtained for developing direct methods for the study of transient stability of power systems, and to indicate areas for further research. The format of the paper is as follows. I n Section TI the mat,hematical model of a multimachine power system is considered and it is pointed out how the transient stability problem arises. Section 111 deals briefly with simulation, which is the routine procedure used in power system stability analysis. The purpose of this section is mainly to show why accura.te direct methods for stability invest.igations would be most welcome. Classical direct methods are also introduced, mainly to point out that they constitute a particular application of Lyapunov's direct method, and that they are generalized by the Lyapunov approach. Sections N and V are the main sections of the paper; they review the progress that has been made concerning the application of Lyapunov's second method to the power system transient stability problem. These sections also contain some unpublished results. The Lyapunov approach is illustrated by some numerical examples and compared with simulation methods. However, it should be realized that further improvements are necessary to make

where t denotes time, 6i is the angle (in electrical degrees) between the rot.or shaft of t,he ith ma.chine and a sha,ft running at synchronous speed, M i is the inert.ia constant of the ith machine, ai its damping const.ant., Pei the electrical power delivered by the ith machine, and P,, the mechanical power input to it.

In most transient st.ability st.udies t,he mechanical power input is assumed const,ant. This is a sat,isfactory assumpt,ion for most practical cases, since the time constants of t,he governors are usually much larger than t,he duration of the tra,nsient swings. It will be shown, however, that governor action can be taken into a.ccount rather easily at the expense of an increasing complexity. Consider first the case of constant mechanical power input.

The electrical power of the ith machine is the sum of the power dissipated by its short-circuit conducta,nce and the powers delivered to t.he transmission lines connecting the machine under consideration to the other machines of t,he power system:

n

Per = GiEj2 + E,EjYij COS (6, - 6, - ei,) (2) j = 1 j # i

the Lyapunov approach more competitive with direct some of the important topics for further where Ei is t,he internal voltage of t.he it.h machine, Gi is its

research a,re therefore discussed in Section VI. It is hoped short-circuit Conductance, and yij a,nd eij are the modulus that this paper might prompt furt,her research on and the phase angle of the short-circuit t.ransfer admittance this important practical problem. between the ith and the jt.h machines. For most power

systems the transfer conductances G i j = Yij cos eij (i # 11. MATHEUTICAL MODEL FOR A

MULTIMACHINE POWER SYSTEM Consider a power system consisting of n synchronous

machines (or groups of machines). The usual assumpt,ions used in setting up the mathematical model are as follows

1) A synchronous machine is represented by a constant voltage behind its transient reactance; in other words, it is assumed that the flux linkages are constant during the transient period. Flux decay and voltage regulation are hence not taken into consideration. This assumption is valid in most practical cases since the time const,snts involved are much larger than t.hose of the transient phenomena., which are of interest in tra.nsient stabilit,y analysis.

2) Damping power is assumed proportional to slip velocit.y, and is t.hus assumed to be due mainly to mechanical friction and asynchronous torques.

Alt.hough both assumptions are good approximations a.nd satisfa.ctory for most applications, it is sometimes desirable to consider a more exact mathematical model. This will be discussed in Section VI. Furt,hermore, for mat.hematica1 simplicity, only round-rotor machines will be considered in this paper. The motion of t,he ith machine is then described by t.he differential equation

[7 1.

d26,(t) M i -

d t 2 + a i 7 + Pet@) - P,,(t) = 0,

j ) are negligible, and only the transfer susceptances Bij = Y i j sin Bi j have to be taken into consideration. This assumption is also used in the following analysis; the effect of the assumption n<11 be discussed in Sect.ion VI. Thus

Pei = GiE,2 + EiEjBIj sin (Si - S,), n

j = 1 j#i

i = 1, . . . , n.. (3)

If the mechanical power input is assumed constant, then the equilibrium st.at.es of the power system are the solutions of t.he set. of 272 equations

with w i = d&/clt denoting t,he difference betxeen actual angu1a.r machine speed and the synchronous speed. Although t.he equations only contain 211. - 1 unknowns, wl, w2, . . . , w n , 61 - ti,, ii2 - 6,, linP1 - 6 , , t.he set of equations is not overdetermined. Indeed, an equilibrium can occur only if the total mechanical power input equals the t.ot.al power dissipated. Hence

n n

Pmi = Ei2Gi i = 1 i = l

is a. necessary condition for t,he existence of an equilibrium, and thus one of the equations of the set (5) is

i = 1, . . - ,n (1) redundant. The above power syst.em model is hence not

334 IEEE TRANSACTIONS ON AUTObCATIC CONTROL, .AUGUST 1971

of order 2n, but only of order 2n - 1 with state variables

Since the differential equa,tions (l), which govern the behavior of t.he power system are nonlineax, there are two import,ant stability concepts to be discussed, namely local and nonlocal stability of an equilibrium stat.e. The following stability definitions are frequently used in poxer system analysis [SI, [9].

1 ) Dynamic Stabilitg: An equilibrium state of a power system is said t.o be dynamically stable [ lo ] , [ l l ] if under t,he preceeding assumptions t,he state of the power syst,em remains close to the equilibrium state for small initial deviations of the system variables (power input,, load angle, machine speed, etc.) from their equilibrium values. The system does not, lose synchronism for small variations of the variables. This property is equivalent to local stability in the sense of 1,yapunov [ E ] , and can be a.nalyzed by considering t,he linearized power system model and applying classical stability criteria. (Routh, Hurwitz, Kyquist,).

2) Transient Stability: Consider a. dynamically stable equilibrium st,at,e of a power syst.em. If t.he system is perturbed from this state due to a change of the pou-er input, or to a fault, in the system, then the system con- verges to its equilibrium for sufficienblg small values of the deviations. Hou-ever for large perturbat,ions, t.his is not necessarily true. This is a nonlinear phenomenon, and t.he problem of determining whether or not convergence to t.he equilibrium nil1 occur is called t,he transient. &ability problem. Its study requires the determination of est$imates of the domain of att,rac,tion of equilibrium states, which is called the t,ransient stability region.

A very important characterist.ic of transient stability of a power system is it.s crit.ica1 sn-itching time. When a fault occuls in t,he syst.em a t.ransient, phenomenon is start.ed; aft,er clearing the fault, the system will converge to its equilibrium if the system state after fault clearing is mit.hin the trmsient st<abilit,y region of the postiault syst,em. The critical switching t,ime is defined as the maximum allowed fault durat,ion for transient. stabihy.

The remainder of t.his pnper deals with the transient stability problem. For t.he furt,her developnlerlts it. will be useful to write the differential equations of t.he power system in a canonical form. Let,, therefore, 610, 6$, . . . , S n 0 denote the equilibrium load angles for which t.ransieut stability has t,o be investigat,ed; of course t.he angles involve arbitrary constants since only their differences have physical meaning. Define the column vect,ors

01, WZ, . . . 1 wn, 61 - 6,, * . . 7 6,-1 - 6,-

z = (to1 (21, 22, ' . ' , x,) x = col (Wl, wg, . . . , wn, 21, ' . . , 2,)

where

2 . 1 1 = 6. - 62, (i = 1,2, . . . , n)

denot,es the difference betu-een the load angle of the ith machine and its equilibrium value. Define t,he diagonal (?a x n ) ma.t,rices

and the matrices

where m = n(n - 1)/2, In, denotes the (n X n) identity matrix, Ojk the ( j X k) zero matrix, D is a.n (m X n) matrix such that 3 = Dz has the components Cl = z1 -

crn = z,-l - z,, and the superscript T denotes transpose. A is an (2n X 2n), B an (2n X m), and C an (m X 2n) matrix. Let f(() be a vect,or-valued function with m components, which is of the diagonal uncoupled type, i.e., t.he i th component of f(3) only depends on the ith component! of 3. The function is defined by

Z Z ~ {2 = ZI - ~ 3 , . . ' , {n-1 = ZI - z,, { n = zz - 24, . . . 2

f i ( r i ) = E,E,B,,[sin (ri + si") - sin {?I where {4 is the ith component of p = D eo1 ({lol . . . , ano)) , and \\-here p and q are the indices of the components of z on which {, is dependent. Then it is easily checked that the set. of differential equations (1) is equivalent t.0 the st.at,e vect,or differential equation

d ~ ( t ) / d t = Ax(t ) - Bf[Cx(t)]. (6)

The study of t.he shbility of the equilibrium state under considerat,ion is now reduced to the study of the stability of t.he null solution of (6). System (6) is of the multivariable Popov type whose st,ability propert.ies have been studied quit.e ext.ensively by cont,rol theorists in the last decade [2].

It is sholvn next that the mat.hematica1 model of t,he poxer syst,em is also of the Popov type, even if t.he as- sumpbion of constant poxver input is dropped. Let, some or all of the machines of the power system have governors that, are very fast, such t.hat. the rela.tive values of governor time constants and t,he t.ransient period do not allow the assumption of const.ant mechanical power input. Let the governors be linear elements. The mechanical pori-er input to the ith machine is

P m i = Pn,,O + pmt

where Pn,p is c.onst,ant, and pmi is related to w i by t,he relation betxeen the Laplace transforms

P m i ( 4 = -Ht(+,(s)

with Hi(s ) the transfer function of the governor, which is assumed rational. Let Ai, bi, and c, be a realization of Hi(s ) , i.e., [13]

B ~ ( s ) = c ~ ~ ( s I - Ai)-lbi. (7)

Then the different,ial equations describing the dynamics of t,he i th machine a.re

WILLEYS: TRANSIENT STABILITY STUDIES 335

&c(t)/dt = At~yi + bia,

&Jdt = W f (8)

M,dw,/dt = i- P J - ciTyj - Pet

where yi is the state of the governor. It is clear tha.t this power sgst.em model can also be reduced t.o the form given by (6) and hence t.0 the Popov type. To keep matt,ers somewhat. simple, in the remainder of t.his paper me- chanicd power input is assumed constant.

The procedure out.lined in t.his section to set up the mathemat,ical model can also be used for a set of n - 1 machines connected to an infinite bus. Then it suffices to let M, + 03, such t.hat w , = 0, 6, = const.ant,. The constant 6, is then usually t,aken as the reference a.ngle.

111. SIXULATIOX AKD CLASSICAL DIRECT ~IETHODS The routine procedure for transient stability a.nalysis of

power syst,ems is simulat,ion on an analog [la-] or a digkal comput,er [15]-[17]; t,he former met.hod is only applicable for low-order syst,ems. In t,his xvay the differential eclua- t.ions of t.he syst,em a.re integrated either by elect.rica1 analogy or numerically. The main a,dvant,age of simulation is t,hat improvements in the mathemat,ical model can be taken into account wit,hout difficu1t.y. Its disadvant,ages are as follows.

1) To find t,he boundary of the transient stability region the equat,ions must be integrat,ed fGr many initial conditions; especially for high-order systems this is a time consuming and expensive task.

2) The procedure must be carried through separately for each numerical example. This is part,icularly import,a.nt, at, the design level where system parameters are fre- quent.ly changed. There one would prefer analyt.ica1 (even conservative) expressions of the stability boundary in t,he system parameters.

3) For large int.erconnected systems extremely large computer systems are necessary to a,ccurately represent the high voltage grid. Because of t,he large inert*ias involved and the long ties, swings must, be carried out for several seconds or more. This indeed amounts to a lot of expensive computer t,ime.

Therefore, especia,lly at the design level of a power system or a part of it, one prefers to have a direct method for stability analysis a t intermediate st,eps to be able t,o study transient stability without. solving the syst,em equations, and to obtain analytical expressions of the st.a.bilit,y b0unda1-37 in terms of the system parametes, or at least est.imates of this bounday. Simulation would then ma.inly be used to check the transient stability properties of the find design.

A mcthod that. is closely related to simulation is phase- pla.ne or state-space a n d p i s [18]-[20]. This procedure is only feasible for second-order systems; its main merit is that it can lead to some classical direct. methods such as the equal-area and energy-integral criteria, which are dealt svit,h in the next section.

It. should be emphasized t,ha.t direct, methods do not

Fig. 1. Equal area criteria.

exclude simulation. The general approach used when direct. methods are applied is as follows. The t,ransient st.abilit,y region of the post.-fault poxer system or a.n estimate of it is comput,ed by means of T. direct method. Then the syst,em is simulated on a computer, and by integrating t,he syst,em differential equations during fault conditions the post-fa.ult, initial state is obtained. It. is checked It-hether or not. tha.t state lies n-ithin t,he computed stability domain. For other applications t,he critical swit,ching t,ime is computed by integrat.ing t,he equations of t.he faulted system until its t.raject,ory reaches t,he boundary of the stability region. Hence the usual direct met,hod applications combine simulation and direct analysis. The direct. met,hod only subst.itut,es for simulation as far as the comput,at,ion of the transient stability region of the post-fault syst.em is concerned. However t.his is the most important, and by far the most. time consuming part. of a. st.abilit,y analysis by computer simulation.

Power system engineers have been using direct methods for t,ransient, stability investigations long before Lya- punov's direct, method beca.me popular in system theory research. The early direct methods use the energy concept [48] of which the idea of a Lppunov function is a generalization. It is mainly for single-machine systems (that is a system consisting of a single machine connected to a busbar of infinite capacity) with const,ant. mechanical power input that, a. direct method called the equa.l-area or energy-integral criterion has been very useful [9], [21], [22]. In Fig. 1 the electrical pou-er output of t.he machine is plotted versus the load angle. The criterion states t,hat an initial load angle 6 0 of the machine and an initial speed deviation wo = [dS(t) /~Zt]~ from t.he synchronous speed give rise to a stable t'rajectory if

M w 0 2 / 2 + Area(ABC) 6 kea(CDE). (9)

This is the energy-int,egral criterion. If the init.ia1 speed equals the synchronous speed, then we obt,ain the equal- area criterion

Area(ABC) 6 Area(CDE). ( 10)

This crit,erion is very simple and is very useful for obt,aining a quick estimate of the sta,bility region. I ts main dra,n-- backs are as follows: 1) t.he criterion does not show any improvement of the stabilit,y region due t,o damping torques, although it is valid for any nonnegative value of the damping constant; 2) the method cannot be generalized to t.ake t.he effect of fast, speed governors into account.

d26(t) dS(t) 111 -

at2 dt + a - + Pe(t) - P,(t) = 0. (11)

If t,he mechanical power input is const,ant, t,hen t.he system is of the second order; if the effect) of fast governors is taken into considerat,ion, t,hen the system order is higher. I n any case the system contains a single non- 1inearit.y; it. is t,he nonlinear relationship between the electrical pou-er output of the machine and its load angle. For round-rot or machines this relationship is sinusoidal. The system hence belongs to the t.ype considered by Popov p2] of systems 11-it.11 a. nonlinear feedback element.

Various techniques can be used for constructing Lya- punov functions for this system. For lox-order systems trial-and-error procedures yield useful Lyapunov funr- tions; such techniques Irere used in the early papers on the application of Lyapunov methods t.o transient power system analysis by Gless [23], and Fallide and Pate1 [24]. In this way results u-ere obtained equivalent to the energT integral crit,erion. More general results were obtained by Dharma Rao [ 2 3 ] , [%I. It. is, hon-ever, clear that, trial-and-error methods are only feasible for l o ~ r - order systems. Another approach is the application of Zubov's method [27], u-hich theoret.ically could lead t.0

the exact doma.in of attraction of an equilibrium st.ate. Hon-ever this met,hod is also limited to low-order systems, and requires a great deal of computation. The same resu1t.s can be obtained more easily by ot.her techniques.

Very interest.ing Lyapunov functions we generated by observing that the system is actually of t.he nonlinear feedback system type considered by Popov. The stability of such systems has been extensively studied in the automatic control area. One of the outst,anding results that have been obtained for t.his type of sJ-stem is 6he celebrat,ed Popov criterion [2], xhich gives a sufficient, condition for the asympt.otic stability i n the large of nonlinear feedback systems if the characterist,ic of the single nonlinear element sat,isfies the condit,ion

uf(4 3 0 for all u. Various techniques have been proposed in control theory t.o const.ruct Lyapunov functions t,hat demonstrat*e the validity of t,he Popov criterion. Alt,hough the non1inearit.y involved in the poxer system model does not. sat.isfy the above inequality for all values of t.he argument u., the same Lyapunov functions can neverthe- less be used t.0 compute t,ransient, st.abilit,y regions for the p o w r syst.em. This idea. was used in [ Z S ] and [29]. In the latter paper IZalman's procedure for comput.ing Lyapunov

u = -a and u = b are hyperplanes in the state space of the system. The technique used in most papers to derive' stabilit,y regions from t,he Lyapunov functions is to find the largest region bounded by a surface V(x) = constant, which lies between t,hose two hyperplanes, where V(x) denot,es the Lyapunov funct,ion under consideration [24]-[26], [29]. This method w-as proposed by Weissen- berger [30], and by Walker and AIcClamroch [31]. How- ever, larger &ability regions often can be obtained by means of the same Lyapunov funct,ions, using a technique proposed in [ Z S ] and [32] for second-order systems and later generalized t.0 syst.ems of any order [33]. The basic idea is to compute a region bounded by a surface V(x) = constant., and by parts of the hyperplanes u = -a and u = b, such t,hat all trajectories passing through those parts of t.he hyperpla.nes are directed inwa.rd to t.he enclosed region. To illustrnt.e the improvement that can be obtnined consider the stability diagram for a system of the second order of Fig. 2 with normalized parameters M = 1, P, = sin6, P, = 0.S67, a = 0.5. Curve A is t,he boundary of t.he stability region obtained by means of the classical energy-integal crit.erion. Curve C and curve B are obt,ained [2S] by mearls of the Lyapunov functions derived from the proof of the Popov criterion; the boundaries are obtnined respectively by meam of the t.echniques of the open and t.he closed Lyapunov cont,ours.

An import,ant pra.ct>icnl problem is the determination of the load mgle range t,hat guarantees transient. stability for the case that the initia.1 speed equals the synchronous speed. This is an important pract.ica1 c,ase since it occurs for inst.ance when one of t.he feeders connecting the machine to t,he busbar is t>ripped or when the mechanical power input, suddenly changes. If governor action is not considered, then the equal-area criterion discussed in Section IV yields a sufficient condit,ion for synchronization. For systems with damping torque, this criterion can be generalized by means of the Lyapunov method. The obtained sufficient condition for transient, &ability [34 J is (Fig. 1)

Area(A3GCA) 6 Area(CDEFC), (12)

where So is the initial load angle, and where BG and FE are parallel such that tan fl = a2/M.

The single-machine problem is not only important for t,he study of the stability properties of an isolat.ed machine tied to an infinite bus; it is also important because oft,en a stabilit,y property of a multimachine syst,em can be analyzed approximately by considering separately each machine of the system while the set of ( n - 1) other

WILLEMS: TRANSIEST ST.4BILITF STUDIES 337

this section; for sinlplicity, constant, power input is assumed, ahhough governor action ca.n equally well be t.aken into acc.ount. [39]. Anderson proves t,hat

V(X) = XTPX + 2 j(f)'Qd< LCX ( 1 3

*,,s-s, is a Lyapunov funct,ion for system (6) if Q and N are * positive semidefinite diagonal nmtrices, t,he sum being

posit,ive definite, such that ( N + Qs)C(ls - A)-lB = H(s ) is a poshive real matrix, arid if P is a positive definite solution (n-hich always exists) of the equations

PA + ATP = -LLT PB = CTN - LW + ATCTQ (16)

W'W = QCB + BTCTQ

Fig. 2. St.abi1it.y regions for a numerical example. where W and L are auxiliary matrices. The solution P of the (16) can be found in various n-al-s, by solving the nonlinear algebraic equations, or by spect.ra1 fact.orizat,ion,

nlac.hies is then considered as an infinite bus \ i t h respect, 01 bg obiailling the steady-state solution of a ma,t,rix to the machine under consideration [35], [36]. Moreover a Riccati equation, ~,SaDurlov funct,ion (15) proves t,he t,wo-machine system is exactly equivalent t,o a single- synlptotic in the large of (6) if the nonlinearit,y nmchine syst.em provided al/Ml = a2/M7rIp, t,hat is in satisfies particular when damping is neglected (al = a? = 0).

fi(0) = 0, V. LYAPUNOV METHODS FOR MULTIMACHINE SYSTEMS f J & ) > 0, YE, # 0, i = 1,2, . . . , m. (17) The early approaches t o the multimachlne problem by

means of Lya.punov theory consist. of using the energy function as a Lyapunov fundon [23], [37]. In these studies stability improvement, by means of damping torques cannot be a.tlalyzed, and governor a.ction is not included. Act.ually two types of energy functions have been used; t.he difference is t,he expression of kinetic energy. The first expression of kinetic energy is [23], [37 1

n E k i n = .dfiW$'/2 (13)

i = l

whereas the second expression that has been used is [211, ~ 5 1 , 1381

E k i n = x. l1JiJfj(Wi - Wj)2/(2cL1fj). (14) 1 ,I i

It will be shown in the sequel that (14) is only a valid part of a Lyapunov fundon if damping torques are uniform, i.e., if

aI/Ml = @ / M z = . . . = an /Mn.

A different approach to the a.pplication of Lyapunov methods t.o the analysis of power system stability is to t,ake advant.age of t,he fact that the mathemat.ica1 model (8) is of the multivariable Popov t.ype. The procedure developed by Anderson for constructing Lya.punov functions for the multivariable Popov criterion for asymptotic stability in the large can hence be used here to compute finite stability regions [39], [40]. This approach is similar to t.he procedure discussed in the preceeding section for single-machine systems. It is discussed in some det,ail in

However for the transient st,ability problem, inequalit,ies (17) are only satisfied for a limited range of the arguments a.bout, the zero values. Severt.heless 1,yapunov function (15) is useful to obtain an estimate of t.he stabi1it.p region. For t,he mathemat,ical model of the pou-er system discussed in Section 11, we have

CB = BTCT = O,,,,

and hence

w = omm. The set (16) becomes much simpler in th1, case:

PA + AZ'P = negative semidefinite PB = C'N + ATCTQ.

If P is partitioned

\i-e have

PIM-lR + M-lRPI + Pz + PZT = negative semidefinite (18)

Pa + PJW'R = O n , (19)

P1MP1DT D'Q (20)

P2M-'DT = DTN. (2 1)

Xoreover it. should be borne in mind that the multimachine power system (6) is not of order (212), but, of order (2n - l ) , with st.at.e variables wl, w?, . . . , w n , 61 -

338 IEEE TRANSACTIOKS ON ~~UTOMATIC CONTROL, AUGUST €971

6,, 82 - 6,, 6,-1 - 6,. The Lyapunov function should hence be a function of the load angle differences only. This requires

eP2 = eP3 = 0, (22)

where e is a row vector u-ith n components :

e = [l 1 . - - 1 ] .

Considering the definition of A it is readily established that sC(sZ - A ) -lB is a. positive real matxix for any nonnegative values of the damping constants ul, . . . , a,. Hence N = Om, and Q = Imm satisfy t,he conditions of Anderson's theorem for t,he construction of Lyapunov functions. This yields [40]

Pz = P3 = on, (23)

P i = M + pMEM ( 2 4

2R + p ( M E R + REM) nega.tive semidefinite (25)

where p is a constant, and where E is an ( n X 7 . ) mat.rix whose elements are all equal to 1. PI (and hence P ) is positive definit.e if

p > po = - 1/(ZXi) (2.6) i

and (25) holds if

P l 6 P 6 0 (27) where p 1 is the negative solut.ion of the quadrat.ic equa.tion

- p1 M i - 1 = 0. (2s) ( i r 1 )

It is readily checked that p a and p l are equal if and only if damping is uniform, i.e.,

U J l I f l = U?/1142 = . . . = U I L / M , . (29)

In all other cases po is smaller than p ] . The derivative of the Lyapunov funct,ion is nega.tive

semidefinite in the whole state space, since Q is a matrix with only zero elements. Using theorem 4 of [39], u-e conclude t,hat, the region conta.ining t,he origin and bounded by the closed surface

V ( x ) = V ( P ) (30)

is a region of asymptotic stabilit.y, if xu is the unstable equilibrium state of (6), which is closest to the origin in the follon-ing sense. Consider the surfaces V ( x ) = k. For small k , these surfaces are closed and surround t,he origin. If k increases, t,hen the first equilibrium state (necessarily unstable) hit by such a surface is called the closest. equilibrium stat.e x".

Note t,hat the value of V(xu) for the Lyapunov funct,ion const,ructed above is independent of the value of the constant. p , since at. any equilibrium state, w1, . . . , w n vanish. The largest stabilit,y region is hence obtained by taking p as sn~nll as possible. From this remark the follon-- ing conclueions are immediate.

1) For p = 0, the Lyapunov function

is obtained, -\vhic,h is the energy funct.ion considered in [23], [37]. It is valid for any values of the damping constants, and yields the transient stability region bounded by

2 llriw*2/2 + Jc;f(t)'dt = 0. (32) i = 1

2) The largest stability region that can be obt.ained by means of t.he above procedure follows from t,he Lyapunov function

It is obt.ained for p = pa, a.nd it is only valid if t.he damping is uniform, that, is, if all U . ~ ; L W ~ are equal. It yields the t.ransient. st ability region bounded by

(34) Note t,hat cases 1) and 2) are equivalent if one of the busbars is infinite.

3) For a given set of u i / M i , the best. obtainable Lya- punov func.tion is

P c x

\There 0 2. p l 2 po, and p 1 depends on the const.ants ut and as explained above. The transient stabilit,y region is bounded by

(36)

The rather surprising result, is that. the obtained transient stability region can get smaller even if the damping constants increase; the important, factow are not only the values of the damping constants, but, their relative values. There are t.wo possible explanations for this phenomenon.

1) Lyapunov methods only yield sufficient, condit.ions; this means that from the comparison of the stabi1it)y regions obtained for tn-o different systems from a class of Ign,punov functions one cannot derive a valid comparison of the actual domains of attraction in both systems.

3 ) The result might give an indication that. the increase of some dumping constants might lead to a, smaller domain of attraction. I t could indeed be argued that nonuniform damping in some \my deteriomtes the equilibrium between the synchronizing torques of the machines. An introductory simulat.ion study has confirmed this conjecture, and has shown that the latker explanation

WILLEMS: mANSIENT STABILITY STUDIES 339

is correct in some cases. Consider the two-machine porn-er system

dwl/dt = -alWl + 0.5 - sin (X + ~ / 6 ) drn2/clt = -u2w2 - 0.5 + sin (z + ~ / 6 )

dx/c l t = w1 - w2

for initia.1 stat,es x(0) = 0, wl(0 ) = w2(0) = 00. For a uniformly damped system (a1 = @ = a) , the solution is

x(t) = 0,

wl(t) = wg(t) = woe-at.

This shows tha.ts the initia.1 state under consideration belongs to the domain of attractmion of t.he equilibrium sta,te (x = w1 = w2 = 0) for all positive a. The case of nonunifornz damping a1 = 0.1, az = 2.0 was simula,ted on a. digital comput.er; convergent solutions were obt,ained

.for wo up to 1.5, but divergent solut,ions for wo = 1.75 a.nd higher. This proves that the generally accept,ed intuitive statement, t,hat da.mping always aids stability is not true in all cases. Results in the same direction are report.ed by McLain [49].

For damped systems better Lyapunov functions and hence improved est.imat,es of the t.ransient st.a.bilit.y regions are obta.ined by considering a matrix N in t.he Anderson procedure, which is a nonzero matrix. Then, however, the derivat.ive is only nega,tive semidefinite for a. limited region in the state space. The computation of the stability region then gets much more complex 1391, and the idea of open Lyapunov surfaces can be a.pplied as ex; plained in the preceeding section [33]. If stability condition (36) is used, then the computational difficulties are rather mild. The most difficult aspect. is the determination of the stable and unstable equilibrium stat,e by solving t-he set. of nonlinear a.lgebra.ic equations (5). A common technique is to use the steepest descent method t,o minimize C r ( P e i - PmJe. For the unst.a.ble equilibrium a careful examination of t,he equations is necessary to obtain a good initial guess in t.he it.eration procedure [3T]. The computation of t,he stability boundaly a.nd the critical syitching time is straight,forward [38]. The crit.ical swit.ch1ng time is determined as explained in Section 111, by numerically integrating t,he system equat,ions under fault conditions until the trajectory crosses the surface defined by (36).

To compare t*he Lyapunov a.pproach svith simulation methods, some of the more successful results obta.ined for numerical examples are discussed below. These examples relate to systems for which the mathemat.ica1 model of Section I1 is su%cient.ly accura.t.e. El-Abiad and Nagap- pan [37] consider a four-machine system and c0mput.e the critica.1 switching t.in1e for clearing a symmetrical t*hree- phase fault. The stability condit.ion (35), obtained by t,he Lyapunov approach, yields an estin1at.e of the critical swhching time equal to 160 radians. Direct, simula,t.ion of t.hk system shows tha.t the actual critical swit.ching time lies bet,n-een 160 and 165 radians. A more elaborate comparison of Lyapunov results and simulation is described in 1381, where a porn-er syst.em containing nine machines is

considered. The numerical data are given in the reference. The critical fa,ult clea,ring time following a three-phase fault is computed. Using Lyapunov function (33), an est.imat,e of the critical fault clea.ring time is obtained equal to 0.43 s. Simulation of t,he system on a digital computer reveals that the act,ual critical switching t.ime is 0.44 s.

These examples show that. for systems for which the mathematical model of Section I1 is sufficientJy accurate the results obt,ained by means of t,he Lyapunov method are very sa.t.isfactory and in good .agreement with simulation results. However t.0 make t,he Lyapunov approach more competit.ive ni th simulation techniques further results are needed to deal with improvements in the mat,hematical model. This is necessary in some cases, as is discussed in the next section.

VI. AREAS FOR FURTHER RESEARCH Further research on the t,ransient. power system sta,bility

problem is needed at, least in the following two directions. First, the ma,thematical model should be improved to approach t,he rea.1 physical system, and second, better methods for st,abilit,y ana,lysis are needed to go along vith these improvements of the model.

A fist topic of furt.her research is how to include the effect, of transfer conductances. Only in the single-machine system and the two-machine system with uniform damping (u1/Nl = a 2 / N 2 ) transfer conduct,ances are easy to deal d h . Indeed both cases are described by a differential equation of the second order of the form

d*6(t) dS(t) - f LY ~ f B - A COS (6 + e) = o (37) dt2 cIt

where 6 is the difference of t.he load angles [21], [25], 1381. This singIe-nonlinearit): problem can easily be analyzed by means of the methods discussed in Sectmion IV. How- ever, consider the two-machine system with nonuniform damping and constant, power input

M 1 & + alii1 = Pml - EISG1 - EIE?Yn COS (lil - + e12) 1M2& + a,& = P,? - ES2G2 - E1E21r12 COS ( 6 2

- 61 + e1s. Although this system is also of the mukivariable Popov type, the application of the technique of Section V has not. led to a satisfactory Lyapunov function. This is a fortiori t,rue for higher order syst,ems. El-Abiad and Nagap- pan [37] looked into this case, but obtained a Lyapunov function whose derivative is indefinite. In [21] and [38] a Lyapunov funct.ion is derived for uniformly damped systems, which has a zero derivative if in some terms (not in all) of the system equations ei, is assumed to be 90". This procedure has the following apparent. drawbacks.

1) The result is only applicable t.o uniforndy damped system xithout. governors.

2 ) The technique lacks consistency since in some terms eij is assumed go", but in other terms, involving the same pall. of indices, it. is not.

340 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, AUGUST 1971

3) In this way some t e r n of indefinite sign a.re neglected in V ( x ) , although the finally obtained result is that V ( x ) vanishes identically, so that one wonders to what extent the small t.erms can be neglected. 4) From the set of n second-order differential equat,ions

of the multimachine system, a set of n(n - 1)/2 different,ial equations of the second order is derived involving all load angle differences. This set is to some extent in- compat.ible with the original system equations because of the approximations. However, experimental results [38] show- that the obtained st,a.bilit.y regions are fair approximations of the true stability regions.

An improved mathema,tical model and better Lyapunov functions are needed to include flux decay, sa.t.uration, voltage regulation, an excitat,ion dynamics. Flux decay was considered before [41], [42], but t.he results are not entirely satisfact,ory. The difficulties stem mainly from the fact, t,hat the mat.hematica1 model is not of the Popov type any more, but multiplicative nonlinea.rities a.re involved; for such systems much less st,ability results are known in control theory.

It should also be noted that the direct methods discussed in this paper require the integration of tahe system equations during the fault period where a series of SUC-

cessive events occurs, such as circuit reclosing, genera,t,or and load dropping, and capacitor swit.ching [43]-[46]. If one would try to include the fault period in the direct stabilit,y analysis, one obt,ains a nonautononlous nonlinear system for which stability theory is not very n-ell developed. Other aspects t,hat require further research are the effect of t.he kind of fault that occurs, t.he dynanlic modeling of loads [47], and the effect, of transient pole saliency, which int.roduces a nonlinear damping term.

VII. CONCLUSION In this paper an exposition has been given of the power

system transient stability problem. Some interesting recent, contribut,ions and some new results have been treated. The comparison of simula.tion and direct methods is discussed. The core of the paper concerns t.he application of Lyapunov methods to single-machine and multimachine power systems. Although some int,erest,ing and valuable results are already available, much research remains to be done; some of the areas for furt.her research have been indic.ated at. the end of the paper.

AC~YOWLEDGMENT The aut,hor gratefully acknowledges numerous useful

comments by Prof. J. C. Willems of N.I.T., Cambridge, and by Prof. A. Va,n den Meersche of t.he University of Gent, Gent,, Belgium. He also thanks G. Hoffman for his assistance for the simulation described in Sect.ion V.

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Jacques L. Willems was born in Bruges, Belgium, on September 18, 1939. He received a degree in electromechanical engineering from theuniversity of Ghent,, Ghent, Belgium, in 1963, the M.S. degree in electrical engineering from M.I.T., Cambridge, in 1964, and t.he degree of Doct,or of Applied Science from the University of Ghent, in 1967.

Since 1964 he has been on the st.aff of the Engineering School of t.he University of Ghent., except for a leave of absence from

February to August. 1970 at Harvard Universit.y, Cambridge, Mass. He teaches courses on system theory, circuit. theory, and power systems; his main research interests are in the areas of control theory and power syst,em analysis.

A Nonlinear Regulator Problem for a Model of Biological Waste Treatment

GERARD D’ANS, PETAR V . HOKOTOVI~, AND DAVID GOTTLIEB

Abstract-Using the Monod model of bacterial growth the pollutant removal problem is formulated as an optimum state regulator problem. This nonlinear problem is solved by a direct method based on an application of Green’s theorem.

T IXTRODUCTIOS

HE continuous flow cultivation of microorganisms [1]-[3] has not only been used in bact.eriologica1

research [4] and ferment,a.t.ion industry [5], [6], but, has also become one of the most promising methods for biological treat.ment of urban wast,es [7]-[15]. Control

Paper recommended by R. E. Larson, Vice Chairman of the IEEE Manuscript received July 3, 1970; revised February 26, 1971.

S C S Information Disseminat.ion Commit.tee. G. D’Ans and P. Kokotovik are with the Coordinat,ed Science

Laboratory and Department, of Electrical Engineering, University of Illinois, Urbana, Ill.

of Illinois, Urbana., Ill. D. Gottlieb is with t,he Department of Plant Pathology, University

engineers and t,heorists should realize t.ha.t the under- standing and mat.hemat,ical descriptions of microbial growth kinetics [14], [ G I are developing in a direction that can ma,ke realist,ic applications of optimum control theory feasible in t,he near future. Although a generally accept,ed dynamic model of microbial growth is not a,vailable yet,, most mat,hematical models currently in use [15] originate from a model introduced by Monod [16].

In this paper we use t,he Monod model to formulate and solve an optimum control problem for a bact.eria1 growth process. The solution of our problem has two interpreta- t,ions. First, the feedback control obt,ained maximizes, the amount, of bacteria produced during a tmnsient from any admissible initial st.a,te t.o tmhe opt,imum st.eady st,at.e. Second, the sa.me feedback cont,rol also ma.ximizes t,he amount, of the nutrient substrate removed from the cont,inuous flow during this transient. The first, interpreta- tion is of interest in laborat,ory experiments and industria.1