ON THE OPTIMAL DYNAMIC DISPATCH OF REAL POWER

Thomas E. Bechert Harry G. KwatnyDrexel University, Philadelphia

Abstract-Electric utilities presently use static optimization tech-niques to solve the economic load allocation problem. Experience hasshown that a number of difficulties arise when these solutions areincorporated in the feedback control of dynamic electric power net-works. This research attempts to overcome the disadvantages of suchcontrollers by combining economic load allocation and supplementarycontrol action into a single dynamic optimal control problem.

Necessary conditions for an optimal controller are obtained andthese conditions are then used to synthesize the optimal feedbackcontroller for the special case of two-generator load sharing. Digitalcomputer simulations compare the performance of this optimal feed-back controller against that of a conventional controller in operatingone area of a power interconnection.

INTRODUCTION

Optimization techniques have been used by the power industryfor several years to solve the problem of economic dispatch.3 If a givenload is to be shared by a number of generators in an area, thesetechniques determine what fraction of the load should be carried byeach generator, so that overall operating cost is minimized. The pro-cedures now in use are of a static nature. They are based on steadystate operating cost characteristics and do not consider the dynamiccosts involved in passing from one operating state to another. Conse-quently, it is not surprising that poor transient behavior can and doesresult when these solutions are incorporated in the feedback control ofdynamic electric power networks. In this paper, optimal load allocationis formulated as a dynamic optimal control problem, which takes intoaccount not only the steady state cost characteristics, but also thedynamic costs involved in changing the level of megawatt generation.In this way it is intended to achieve a combination of good transientresponse together with economical operation.

It is necessary to briefly review certain salient characteristics ofpower system behavior and operation. Any displacement in the systemvariables from equilibrium results in unbalance between the mechanicalpowers delivered to and the-electrical powers removed from the turbine-generator rotors thus causing the rotors to accelerate. The resultingvariations in speeds and subsequent variations in machine rotor anglescause changes in electric power flows which produce restoring forces onthe rotors. In this way oscillations in speeds, rotor angles and electricalpowers are established with periods ranging up to a few seconds -

depending on rotor inertias as well as inter-machine electrical stiffnesses.In situations where groups of machines are closely tied electrically

each group appears to behave as a single large equivalent machine andthe relevant oscillations occur between weakly linked groups ofmachines. Oscillations of this type have been observed in practice withperiods up to 10 seconds.

Damping of inter-machine (or inter-group) oscillations comes fromtwo sources. The first is the system load itself since small changes insystem freqeuncy produce proportional changes in connected load.The second is the primary speed control (governor) loops which areessential precisely for this reason. While the primary speed controlloops provide damping for the inter-machine oscillations, the slowacting supplementary controls, which act through the governor speedchanger motors, are intended to provide the final control action whichmanipulates the individual generator outputs to produce the desiredgeneration schedule.

Paper 71 TP 552-PWR, recommended and approved by the Power SystemEngineering Committee of the IEEE Power Engineering Society for presentation atthe IEEE Summer Meeting and International Symposium on High Power Testing,Portland, Ore., July 18-23, 1971. Manuscript submitted August 31, 1970; madeavailable for printing April 29, 1971.

The analysis presented below is confined to the situation wherethe prime movers are conventional steam power plants. Throttle valvevariations due to the primary speed control loop are of relatively highfrequency and of small magnitude. Thus, so far as the primary speedcontrol loop is concerned, the boiler appears to be an infinite steamsource. The situation is quite different with the supplementary controlsignals, which are likely to require large and sustained changes in steamflow. It is common practice, therefore, to limit the rate of change ofthrottle valve position due to supplementary control demands in orderto allow the steam generator combustion controls to maintain steamconditions and to avoid a plant upset.1

The steam generator combustion controls act to maintain thestored energy level within the boiler. Thus, the combustion controlsdo not react significantly to governor action and the primary speedcontrol loop has negligible effect on total fuei usage. The boiler itselfoperates at nearly constant efficiency over its load range, however, themultiple valve arrangement used in most modern steam turbines resultsin very significant variations in actual energy delivered to the turbineshaft per unit steam flow. In essence, these valves divide the total rangeof power output for each generator into several "valve regions" andgive rise to piecewise smooth heat rate characteristics. The importanceqf recognizing the piecewise nature of these characteristics has been welldocumented.4-6

In subsequent sections, these dynamic and economic characteris-tics shall be more precisely delineated and incorporated in the formula-tion of the dynamic optimal dispatch problem. Necessary conditionsare then obtained for the solution of the general case in which anarbitrary number of generators share the assigned load. These condi-tions are used to synthesize the feedback control system for the two-generator control area in order to gain insight into the structure of theoptimal controlleqr. Finally, computer simulation results are presentedwhich compare the performances of the optimal controller and a con-ventional control system.

MATHEMATICAL MODEL

For the purpose of the present study it is necessary to adequatelydescribe the medium term (minutes) dynamic and economic character-istics of the system. It will be assumed that the system is electricallystable and that frequency and bus voltage variations are sufficientlysmall that it is permissible to decouple the dynamics of real power andfrequency from the dynamics of reactive power and voltage. In addition,it will be assumed that the system is composed of a number of controlareas which have the distinguishing characteristic that the electricaloutputs of all individual generators within the same area are so closelycoupled together that all generators may be considered to be operatingat the same frequency. It should be noted that the term "control area"is often applied to power networks whose boundaries are defined byother considerations, such as to coincide with corporate boundaries. Insuch cases, the area must be further divided into sub-areas for theanalysis to apply.2

Under these conditions standard dynamic models of intercon-nected power networks have evolved for the analysis of the loadfrequency control problem.2'3 Design of the primary speed controlloop is not of concern here, and it will be assumed that a properlydesigned governor is available and may be treated as an ideal damper.Furthermore, in accordance with the above discussion, it is assumedthat governor action does not affect fuel costs. Also, the supple-mentary control signal rate limits are sufficiently low, in practice, sothat the steam generator combustion controls have no difficulty main-taining pertinent process variables within tight tolerances. Hence, the

889

Fig. 1. Simplified System Model.

dynamics of the steam generator as well as the turbine and governorare negligible with respect to this limitation. Figure I illustrates atypical representation for a two-area interconnection. In the presentstudy, it is necessary to characterize the economic behavior of thesystem and hence to explicitly consider, to some extent, the individualgenerating stations within each area. For simplicity of exposition, areaA is illustrated with only two generating stations.

In order to more closely examine the assumptions made withregard to generating station fuel costs, the simulation studies describedbelow utilize a more accurate computation of the total fuel usage. Forthis purpose steam usage due to both governor action and supple-mentary control are included along with a simple dynamic model of thesteam generator.

A brief discussion of the steam generator follows in which theindicated notation is used:

G = Steam power withdrawn from boiler, MW,F = Thermal power input to boiler, MW,E = Excess energy stored in boiler, MW-sec,v = Rate of change of thermal input power (control signal),

MW/sec,H =Boiler excess power input, MW,T = Fuel consumed, MW-sec.

Boiler operation may be described by the following equations:

H-F- G, ( )

Z H, (2)

F-v, (3)

T P 4.T .F. ~~~~~~~~~~~~~(4)During operation, it is desired to specify v so as to make the thermalpower input F match the steam power output G. In addition, it isrequired to minimize stored energy deviations E in the boiler (signifiedphysically primarily by pressure). Also, stability of the combustionprocesses dictates that rapid fluctuations in firing rate be avoided. Theserequirements may be used in formulating a simplified boiler controldesign as follows:

Assuming constant steam demand G, choose the control u toninimize the cost functional

J,.fE2 + bH2 + &2v2 dt, b >o, a > o.

This is the well known quadratic regulator problem7 with solution

v . -1 E -

1 V,2a+ b H.a a

STRATEGY FOR CONTROL SYSTEM DESIGN

Partition Into SubsystemsThroughout the interconnected power system, frequency devia-

tions and tie-line power flow deviations should be minimized duringthe transient period following a load change. Furthermore, within eachcontrol area, any mismatch between total power generated and totalpower load should be minimized, generation should be distributedamong the several generating stations so as to minimize overall fuelconsumption and equipment wear, during both transient and steadystate conditions, and the new stable equilibrium condition should bereached in minimum reasonable time. A control problem might beformulated for the entire interconnected power system, attempting tominimize some cost functional based on these control objectives. Themagnitude of this control problem becomes prohibitive, even for asmall number of generating stations and control areas.

The present study approaches the control problem by partition-ing the overall interconnected power system into subsystems. Thesystem is partitioned into a subsystem representing the electrical powernetwork, plus separate subsystems representing the mechanical powerdeveloped in each control area. The network subsystem control prob-lem can be solved to find values for the pseudocontrols SA, SB, SC,etc. - representing the mechanical powers delivered to the network bythe respective areas - which minimize the frequency and tie-line devia-tions during the transient period. Then the area A control problem canbe solved to find values for the supplementary control variables whichsteer the actual mechanical power output xA to track its target valueSA, while minimizing a cost functional involving fuel costs and costsrelated to response rate, transient time, and megawatt mismatch. Simi-lar solutions, of course, can also be obtained separately for ControlAreas B, C, etc. This decomposition approach is a practical way to findsolutions; solutions have been obtained for cases of engineering interest.

The detailed nature of these subproblems - the network controlproblem and the area control problem - will be discussed in the follow-ing sections.

The Network Control Problem

The basic objective in the optimal control of the electrical powernetwork subsystem is to find the amount of mechanical power (Si)required from each area, such that the frequency deviations (wi) andthe tie-line power deviations (PAd) are minimized. The control effortnecessary to accomplish this minimization will be the deviation (AS1)of mechanical power (Si) from the power requirements (Pi+Qi) for itsown area.

The optimization problem may be formulated as a standardquadratic regulator problem as follows. First, an n-dimensional columnstate vector y is formed, whose components are the frequency and tie-line power deviations, w1 and P.

Y' -IOAj wB.-tXC, , PAB,- PDCP PCk,*-

The behavior of the network is described by the following equa-tions. Let

-81. S1 - (Pi + (6)

Then,

wi = (-Di w - zJZ + AS)/Hi, Di = Di+ EjRj1 (7)

Pj - T1j ( '} - 0J)In vector-matrix form:

(5)y - Ay + Bu

890

(8)

Fig. 2. Approximate Heat Rate Characteristics.

where, for the system of Figure 2

3. Costs Based on Megawatt Output. Each individual generatingstation has its own characteristic rate of fuel consumption (h(x)), basedon its steady-state megawatt output (x). The functional form of this"heat rate characteristic" is discussed in more detail below. The costfunctional will include the total fuel costs for the n generating stationswithin the control area:

fT hi (xi )dt.

4. Costs Based on the Rate of Change of Megawatt Output. Thevery act of changing a generating station's megawatt output from onelevel to another involves costs which must be considered in the dynamicoptimization of the control. These costs reflect, among other things,reduction of machinery life due to increased mechanical and thermalstresses. A precise formulation of costs based on rate of change (u) ofmegawatt output is not presently available; however, it is clear that themagnitudes of such rates should be penalized. Hence, a term is intro-duced into the cost functional of the following form:

[-iDA/A -1/1

A 0 -D4/HB 1/l

TAB -TAB 0

1 A SA

41 J, u .

a ~ ~~~~_ B

BA]

EIB .

The control vector u is sought which will minimize the quadratic costfunctional J(u)

J(u) = . I{y'Qy + u'Ru } dt

where Q and R are positive definite matrices. The solution to thisproblem is given by 7

u(t) . - [iR-B' K] y(t) (9)

where the n x n symmetric matrix K is the unique positive definite solu-tion of the matrix Riccati equation:

0 - -KA - A'K + KBR-1B'K-Q (10)

The Area Control Problem

Having discussed the network control problem, attention is nowfocused on the second subproblem, i.e., area control. The function ofthe area supplementary control system is to steer the area generatoroutputs from any arbitrary initial state to a desired "target" state whichwill be specified below. Under normal conditions this can always beaccomplished in a finite time interval which will be designated (O,T).Performance evaluation of candidate controllers will be based on costsincurred within the control area during the transition. These are dis-cussed below.

1. Duration of Control Interval (0,T). The power system shouldbe steered to its target state in the minimum reasonable time. Thecorresponding cost term is f|Tdt.

2. Control Area Megawatt Error. The total mechanical powerdelivered by all the n stations in the control area should match the totalarea power demand (L). In designing the area controller L will be con-sidered an arbitrary constant over the time interval (O,T). When imple-menting the control system the target load, L, will be replaced by thecorresponding pseudo control S specified by the network controller. Aquadratic cost term may be associated with this megawatt error:

f[ T x1 _ LI 2 dt.0

|20 M, {I U1 1 ti dt,

where mi and qi are constants associated with the ith generating station.

Heat Rate Characteristics

The steady-state fuel consumed per unit time, or heat rate (h),increases with output power generation (x). In most applications, theheat rate characteristics are approximated by a smooth convex func-tion3 although some applications have explicitly recognized the incre-mental heat rate discontinuities due to valve points.5'6 These discon-tinuities are considered to be of central importance in the operation ofelectric power systems and will be included in the present analysis. Inthis study, the heat rate characteristics will be approximated by piece-wise linear curves as shown in Figure 2.

Target State

The target state is defined to be the optimal steady state genera-tion schedule. For simplicity, it will be assumed that transmissionlosses within a given area are negligible. This is a reasonable firstapproximation in view of the relatively close electrical proximity ofmachines within the area. Note that this assumption does not precludeaccounting for significant losses which might occur on inter-area ties.Such losses could be considered in an overall steady state optimizationwhich would schedule the tie-line power flows.

Let H denote the overall area heat rate, equal to the sum of theindividual station heat rates. Let g denote the area megawatt deficiency.It is required to find the values of x1, x2, . . ., xn which minimize H, sub-ject to the constraint g = 0. With a slight abuse of notation, and if it isagreed to interpret dh/dx at a valve point as assuming any value betweenthe left and right hand derivatives, the necessary conditions for a mini-mum are:

xl + X2 + *--- + xn = L,

dhl/dxl = dh2/dx2 - .. = dhn/dXn ' A

(11)

(12)

The target state is selected so that the total generation will match theload, and so that all generating stations will operate at the same incre-mental heat rate, X. To each fixed value of area load L there correspondsa fixed target state XT.

In general, for the staircase incremental heat rate curves assumedin this study, the target state determining equations will be satisfiedsimultaneously when one generating station operates at a point withinsome valve region with dh/dx = X, while all the remaining stationsoperate at the valve point generation which corresponds to X.

A special case arises when the incremental heat rate curves fortwo or more stations have some off valve point values equal to one

891

OA

ZAB

1/BA 0

B - O 1/4. O

Mi- .: 0, qj > 1,

another. In these cases, an increment of load may be absorbed by anyof these generating stations, with the same effect on steady-state cost.An arbitrary decision can be made for selecting target states for valuesof L corresponding to these special cases.

Statement of the Area Control Problem

Let xi = mechanical power delivered by the ith generating stationUi = rate of change of mechanical power output of ith generating

stationUi= maximum allowable rate of change of megawatt output of

ith generating stationn = number of generating stations in the control area

The dynamic system may be represented by the following set,of firstorder linear differential equations:

S UiIi 1, 2 *.. n

The control vector u(t) lies in the restraint set, defined by the following:

Q i i Ui i s 1, 2, *., n

Admissible controls are the set of bounded piecewise continuouscontrol vectors u(t) in which steer the state vector x(t) from somearbitrary initial state x(0) = x0 to the fixed target state x(T) = xTcorresponding to a fixed load L. The control problem is to find theoptimal control u*(t): the vector u(t) which steers the system from xto xT while minimizing the cost functional J(u).

T

J(u) +J^f i{xi - L,} + 3{m ui1o~~~~~~~~Y ,Ju Iq)

where N dhii- r2 dxi

and again, dhi/dxi may assume any value between the left and righthand derivatives at a valve point. The Hamiltonian function is definedas

H( n, i, u) = n ofo(x, u) + E n juj (16)

The maximum value of the Hamiltonian over all values of u in therestraint set is denoted by M, i.e.,

m ( 6 b£ ) = R( n,x,u).

For the autonomous optimal control problem formulated above,Pontryagin's Maximum Principle may be stated as follows:

If u*(t) is an optimal controller in Q2, steering the optimal responsex*(t) from xo to xT, while minimizing the cost C(u), then it is necessarythat:

(Hl) There exists a nontrivial adjoint response *(t),(H2) H (4*, Ax, u*) =M (*, x*) almost everywhere on

0< t< T,(H3) M ( *, x*) = 0 everywhere, and(H4) 7n0<0

By the maximum principle, i, < 0. It may be shown that % cannot bezero, hence, n0 < 0. The conditions of the maximum principle are stillsatisfied if the adjoint vector A* is replaced by

A

70. Without loss ofgenerality, it is henceforth assumed that n0, = -1.

Consider next the maximization of H with respect to u. It qi > 1then it is not difficult to show that

Y4 {Zbhi(xi)}. dt (14)

SOLUTION OF THE AREA SUPPLEMENTARY CONTROLPROBLEM

Optimal Controller Necessary Conditions

Necessary conditions for the optimal area control problem can bedeveloped from the Pontryagin Maximum Principle.7 For conveniencea new cost functional C(u) is defined by dividing J(u) by the positivenumber 2. Then

C(u) = fTfT (x(t), u(t)) dt0

where

fo (x,u) l + {2xj - L} 2 + a { 2:m,u, qj} +

N { 2 hj(xj)}and

1 = rl/ Y2 3 = 3/ 2 al; = 4/ 2The state vector x(t) is extended to be an (n + 1 ) vector

x(t) = (x0(t), x1(t), *',p xn(t))'

by defining an additional state variable

Xo = fo(xsu), x0(O) = 0.

The (n + 1) -dimensional adjoint vector n (t) is defined as a continuoussolution of the system of equations

ri o(t) constant

l(t)- 2n0r{ Z Xi - L 2fno i (15)

i=5 1, ***, n.

Ui

Ui* = A

-Ui

where A ; 1f

iqi ] zlIf qi = 1, then

ui

Ui* = 0

-Us,

A >U1

1A1 < Ui

A< -Ui

(17)

sgn ni, f.=-3 mim

Dl > Si

ni < Si

ni '- ij

(18)

The special points ni = ± Pi are of interest in the event that the adjointresponse ni(t) might dwell at one of these values for a finite timeinterval. These points are singular points; here maximizing the Hamil-tonian does not define the optimal control.7 Additional informationcan be obtained from the state and adjoint equations.

Suppose, for the first k of the n generating stations in the area,the adjoint response is ?i = ±fli on some finite time interval (a,b). Then,the optimal control ui* is well defined for i = k + 1, . . ., n, but is singu-lar for i = 1, . . ., k. Then, on (a,b), according to the adjoint canonicalequations, for all i = 1, . . ., k,

i O 2 [ xJ - L] + N4d i) (k< n)

Thus, these k stations all operate with the same value of incrementalheat rate, which is proportional to the area MW deficiency. Also, on(a,b), for i = 1, . ., k, 7ni 0 which leads to

uj + d2hi(xi) 0. (19)

892

I

Since a staircase function has been assumed for the incremental heatrate curve, the second derivative vanishes (except at the valve point,where it is undefined). Hence, the following relation holds

PLiu = E1k+u (20)

Thus the summation of the singular controllers must equal the negativesummation of the non-singular controllers. It may happen that theboundedness of the controllers prevents fulfillment of this condition;this means that the first k adjoint equations cannot be satisfied on thefinite time interval (a,b) by the solutions ni7 ± i, i = 1,..., k andhence such a singular condition cannot exist.

For the case k = I, that is, when only one of the n adjoint vari-ables dwells on i7 = + fi, on the finite time interval (a,b), then theoptimal singular control is uniquely specified on (a,b) by Equation (20).

Adjoint Vector Boundary Conditions

Pontryagin's Maximum Principle can be applied to find necessaryconditions on the initial and final values of the adjoint variables. Con-sider the condition H(i , Z, u) = 0, almost everywhere along optimaltrajectories. In particular, for t = 0+ and t = T-, the optimal stateresponse x(t) is equal to xo and xT respectively, and the correspondingboundary values of the adjoint vector are denoted 7o and 7T Theseboundary values of the state vector and the adjoint vector are relatedby the equation:

E [n iui- i I Ui I i] = a, + { Ex,-L} 2 +

4 { E hJ(xi)}(21)

For the initial and the terminal times, the right side of Eq. 21 dependsonly on the known vectors xo and xT. The left side of Eq. 21 dependsonly on the optimal values of 7ni and up. But in the previous sectionsrelations were derived which express the optimal controllers ui* interms of the optimal adjoint variables 7ri*. Therefore, the left side ofEq. 21 depends only on the optimal adjoint vector 7ni*. Solutions ofEq. 21 are hypersurfaces in the n-dimensional adjoint space representingthe locus of allowable initial and target vectors 7nO and 77T, respectively.

For one special case of interest, when q = 1, n = 2, the resultinghypersurface is the polyggn shown in Figure 3.

.6T. K,

A A0 I. ®-.,

.0,

0 0-. .

Fig. 3. Definition of Control Zones in Adjoint Space.

Optimal Solution for Special Case

The optimal feedback controller will be synthesized for the case inwhich the control area contains just two generating stations (n = 2),and the cost functional penalizes the control input to the first power(q = I). For convenience, the station with the larger maximum controlbound is designated station No. 1 (Ul > U2). The solution to thisproblem, then, will be the unique determination of the optimal control(u I *, u2*) for each "present state" (xl, x2) in the state space.

The Hamiltonian maximization conditions provide the resultssummarized in Figure 3. Singular solutions exist on segments A and B.

Note that optimal solutions cannot dwell on the values 72 = ± 02 be-.cause the necessarily resulting singular solutions u2 = ±U1 would violatethe boundedness limitation 1u21 6 U2, since U2 < Ul. Also note thatoptimal solutions cannot dwell on the segment ( 71 = -, ,72 <+ 02),nor on the segment (7 1 = + 01, 772 > -12), both of which violate the con-dition that the Hamiltonian must vanish almost everywhere on (O,T).

The state and adjoint simultaneous differential equations may besolved by dividing the overall control interval (O,T) into subintervals,according to the jumps in u1, u2, a,, and 02; the final values for eachsubinterval then become the initial values for the next. Solution ofeach subinterval in turn continues until the initial point and targetpoint are joined by simultaneous optimal trajectories in both the statespace and the adjoint space.

There is one point in the state space, namely, the target point,through which all optimal trajectories must pass. Thus, in the adjointspace there is a single unique octagon through which all optimal adjointtrajectories must pass, at time t = T. By starting at t = T, with the statevector (xl, x2) at the known target point and with the adjoint vector atsome arbitrary point on the target octagon, an optimal trajectory maybe traced out in the state and adjoint spaces for all subintervals witht < T. This "backing out of the target" process may be repeated for allpoints on the target octagon, thereby tracing out all possible optimaltrajectories.

In particular, the points at which optimal trajectories cross thecontroller switching lines (i?l = ± .1, 712 = ± ,2) may be mapped intothe state space. When all such switching lines have been mapped intothe state space, the state space will have been divided into regions ofconstant optimal control action (ul*, u2*). This will complete thesynthesis of the optimal feedback controller; for any given point in thestate space the corresponding optimal controller will have been deter-mined. All of the state space switching lines have been obtained in thisfashion and are shown in Figure 4. Thus, the optimal feedback control-ler has been obtained for the system with two generating stations and aconstant load L.

I, s-i. 'A. A 4."AAA AAA.AKAi, xl

Fig. 4. Overall State Portrait.Valve Point Singularities

It is noted that the switching lines illustrated in Figure 4 containsegments of trajectories on which one or the other of the generatorsdwells at a valve point. Such a situation arises as a consequence of asingular condition set up due to the discontinuities in the incrementalheat rates.

Time-Optimality of Solutions

Time optimal solutions have the property that at least one stationvaries monotonically at its maximum rate of change U, on the entirecontrol interval (O,T). With some computation, which will be omittedhere, it is possible to show that a sufficiently large, but finite, value ofthe cost functional coefficient a1 may always be selected so that alltrajectories are time optimal. This is the case illustrated in Figure 4.

893

I, b%o,I

I

ov

9

DIGITAL COMPUTER SIMULATION STUDIES

Computer Simulation Model

The general system model used in the simulation is diagrammed inFigure 1. It includes Control Area A, which contains GeneratingStations 1 and 2, Control Area B, which represents the remaininggenerating stations in the interconnection, and a tie-line connecting thetwo areas. The megawatt capacity of the tie-line is assumed to be one-tenth the total capacity of Area A, which, in turn, is assumed to be one-tenth the capacity of Area B.

The simulation program provides two options for the Area AController - either the optimal feedback controller described above, ora conventional megawatt-frequency controller, with automatic eco-nomic dispatch, Figure 5.3 The same response rate limitations areimposed on both controllers. In Area B, a simple, reset, megawatt-frequency controller is assumed to be in operation.

The incremental cost characteristics in this research are assumedto be "staircase" functions. This form would lead to ambiguities indetermining values for target megawatts (x IT) in the conventionalcontroller; therefore the horizontal portions of the curves are assumedto have a shallow positive slope of 0.05 MW/MW, a technique used inpractice, for the digital simulation.

The question remains as to the construction of the optimal con-troller area megawatt target signal (SA), based on available measure-ments. Optimization of the power network subsystem leads to thefollowing expression for the area megawatt target signal, SA:

SA = (PA + QAB) + (CA5) wA + (CA6) tB + (CA7) PAB

The coefficients CA5, CA6, CA7 are determined from the solution tothe Riccati equation for that network, and are referred to in the systemsimulation discussion as the Riccati coefficients.

A numerical value of the first bracket is not readily available bydirect measurement, but examination of the Area A power summationpoint in Figure 1 leads to a means for measuring (PA + QAB) indirectly:

(PA + QAB) = (x1+x2-Rf1 wA.R2 wA-HAWA)-PABDA wA

The term on the right side in brackets is equal to the instantaneousArea A megawatt generation, which is measurable, and the remainingterms on the right side are individually measurable. Thus, in practice,the quantity (PA + QAB) may be measured by this indirect technique.

Simulation Computer Runs

The incremental cost curves assumed for Stations Nos. 1 and 2 inthe computer simulations are:

0l 2.5 5.7 9.0 13.0 15.0 17.5 20.0xl 0-10 10-20 20-30 30-40 40-50 50-60 60-70

02 3.5 8.0 1U.0 15.0X2 0-12 12-24 24-36 36-48

increased from 45 MW to 50 MW, with no change in scheduled tie-linepower of 5 MW, and the time responses were computed and recorded.

The new steady state megawatt target for Area A then becomes55 MW. The steady state economic allocation of generation correspond-ing to this load is found to be xlT = 30 MW and x2T = 25 MW.

Many runs were conducted under these load conditions, varyingthe several system free parameters, and comparing the resulting transientresponses and fuel costs. In the case of the Conventional Controller,the parameters varied included the gains Gl, G2, GPA, GRA, and GB.In the case of the Optimal Controller, the parameters varied includednot only GB, but also the coefficients of the cost functional terms, Y2,73, 74* The maximum allowable rate of change of megawatts, UMAX,was set at a realistic 5 per cent of capacity per minute for most runs,but the first series were operated at 18 per cent per minute, in order toaccelerate the time responses and conserve computer time. Most runswere repeated for three settings of steady state speed regulation: 5 percent, 10 per cent, and 15 per cent. Selected from these many runs werethose which gave the best transient response and fuel cost for the Con-ventional Controller, for a given speed regulation and response rate.These "Best Conventional Controller Runs" provide results againstwhich the Optimal Controller results can be compared.

Computer Simulation Results

The results of the computer simulations are presented below, inseveral series of transient response curves. These include comparisons ofvarious settings of the cost functional coefficients (72, 73, 74). In allcases, the "best conventional controller" results are also plotted, forcomparison.

Series I - Transient Response Characteristics. Results of thisseries of runs are shown in Figure 6. System parameters were set at theirnominal values, which are listed here for reference.

Nominal Parameter Settings:

HAHBDADB

TABBABBGIG2GRAGPAGB

7274

U2UMAX

1.60 MW-sec2/rad16.00 MW-sec2/rad0.20 MW-sec/rad2.00 MW-sec/rad

10.69 MW/rad75% of Area Frequency Requirement Characteristic75% of Area Frequency Requirement Characteristic0.50 MW/MW-sec0.50 MW/MW-sec0.05 MW/MW-sec0.10 MW/MW0.20 MW/MW-sec1.00 -

1.00 MW610 MW-sec460 MW-sec18%/min

Riccati Coefficients:

The computer simulations were run under the following initialload conditions:

Area A load PA = 45 MWArea B load PB = 505 MWScheduled Tie-line Flow QAB = 5 MW

The system was initially quiescent, with the above loads being suppliedby the area generators as follows:

Area A Generation XA = PA + QAB = 50 MWArea B Generation XB = PB + QAB = 500 MW

The Area A initial generation was economically allocated xl = 26 MWand x2 = 24 MW. Then, at time t = 0, the Area A load PA was abruptly

a1 a2 a3

*5 *5 .5bl b2

.5 .5CA5 CA6 CA7

-3.5619 3.2588 -0.3864

Figure 6 shows the megawatt output of each generating station,relative to its steady state target value. The conventional controllerrapidly steers station 1 close to its target value; then small oscillationsoccur, due to the coupling with station 2. The output of station 2,meanwhile, rapidly overshoots its target value by a wide margin, thenrapidly returns to its initial value, and finally approaches its targetalong a damped oscillatory path. The strange behavior of station 2 maybe explained by reference to the automatic economic dispatch systemof the conventional controller, in Figure 5. As the Area A lambda signalchanges value, the station 1 target x1T remains constant, at a valve

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Fig. 5. Area A Conventional Controller.

point, while the station 2 target x2T rapidly jumps between the valvepoints at 24 MW and at 36 MW. Finally, as the lambda signal settles outat its final value, the x2T signal oscillates along the linear slope betweenvalve points, until its steady state value is reached, at the 25 MW level.This type of behavior is typical of all the simulations, and indicates thedifficulty encountered when trying to incorporate the staircase incre-mental cost curve into the conventional controller.

The optimal controller, however, steered both stations rapidly totheir desired output levels and then held them at those points.

With respect to fuel consumption, the optimal controllers showlittle improvement. The fuel consumption is essentially the time integralof the megawatt output. Large positive megawatt deviations tend toincrease fuel consumption, while negative megawatt deviations con-serve fuel.

Regulation was set at 5%, 10%, and 15% for both controllers andhad no effect on any of the megawatt results, after the initial transientperiod. But the magnitude of the frequency deviations varied directlywith the regulation setting. The curves of Figure 9 were recorded withthe regulation set at 15 per cent.

Series II - Variations in Gamma 2. Results of this series of runsare presented in Figure 7. In this and all later series of runs, UMAX wasset at the realistic value of ±5 per cent of capacity per minute. Theconventional controller gain GRA was reset to the value 0.02 MW/MW-sec, and gave improved results. All other system parameters remain attheir nominal settings, except 72 which was set at several values. Itwould be expected that increasing the value of 2 would tend to reducethe megawatt deviations, and the curves in Figure 7 support thisconclusion.

The frequency curves shown correspond to a regulation setting of5 per cent. The other curves in this series are essentially identical for allthree settings of regulation.

It is noted here that the relevant switching lines were not suffi-ciently altered by the variations in the parameter 73 to produce anychanges in the transient responses. For some initial point locations inthe state space, the transient responses would be greatly influenced byvariations in 73.

Series III - Variations in Gamma 4. It is to be expected that in-creasing the value of 74 would reduce the fuel consumption. It was ob-served that increasing 74 from 1/4 to I does reduce the fuel cost. Butfurther increases do not affect the results. The reasons may be seen byreferring to the state space switching lines. Increasing the value of 74beyond I simply moves the switching lines outside the range of theobserved trajectory and the resulting transient response is the same asfor 74 = I. Although the fuel consumption increases as 74 is reduced, itwas noted that improvements in transient response also result similar tothose shown for increasing 72*

Fig. 6. Computer Simulation Results - Series 1.CONCLUSIONS

This paper has reported a new formulation of the electric powergeneration control problem, combining the economic load allocationand supplementary control functions. The power interconnection Ispartitioned into the electrical network subsystem plus separate controlarea subsystems, each of which then constitutes a separate controlproblem. The electrical network subsystem is solved to find the shaftpower required from each control area, such that network frequencyand tie-line power errors may be minimized. This demand signal thenserves as a target toward which each control area's power output isdriven.

The control problem is formulated for each control area, placingan upper bound on the allowable rate of change of power output foreach generator. This dynamic optimal control problem is solved to findthe optimal rate at which each generator should be driven toward itsmegawatt target, such that the cost functional is minimized. Thetime-to-target may be minimized for any initial state, by requiring thatthe cost functional coefficient yl exceed a computable constant. Thenone generator, the time-critical generator, will approach its megawatttarget at its maximum allowable rate of change, fixing the duration ofthe control interval. The remaining generators need not approach theirtnegawatt targets at maximum allowable rate of change, but instead aremanipulated during the fixed time interval so as to minimize the costfunctional.

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Scheduled export power from Area A to Area j, MWSteady state speed regulation

Fig. 7. Computer Simulation Results - Series II.

Throttling losses, present in each valve region of multiple-valveturbines, are considered throughout this research. A staitcase functionis used to represent the incremental cost cutve, reflecting this character-istic. It should be noted that the resulting target state has the propertythat all generating stations operate at a valve point except one, whoseoutput trims the total generation to match demand. In the dynamicsituation small changes in demand affect only this generator; the othersremain at constant output. It is interesting to note that this would notbe the case if fuel costs were eliminated from the area transient costfunctional.

Necessary conditions for optimal controllers were derived for an

arbitrary number of participating generators. The optimal feedbackcontroller was synthesized for the special case of two-generator loadsharing. Controller switching lines were stored in the controller;measurement of present values of system variables then dictated theoptimal control action. This procedure would not be suitable for sys-

tems involving a greater number of generatihg stations because of thedifficulty in storing the complex switching surfaces even if they couldbe obtained. A numerical procedure for computer solution of the stateand adjoint differential equations is a reasonable alternative. Standardprocedures do not apply, however, because of the singular nature of thesolutions. Although the controller singularities are easily remedied thevalve point singularities are not. This problem is currently under study.

In the digital simulations, the optimal controller proved farsuperior to conventional controllers in steering the system rapidly tothe new economic operating point following a load change. The desiredimprovement in performance was a direct consequence of including thetransient cost terms in the cost functional. The optimal controller com-pletely eliminated the persistent oscillations in generator outputs andtie-line power flows, bringing these variables to their proper values inone-third the time required by the conventional controller. In addition,reductions in fuel consumption were obtained. In general, varying thecost functional coefficients in the simulations produced the expectedmodifications in system performance.

GLOSSARY OF FREQUENTLY USED SYMBOLS

DA Slope of Area A load (including losses) vs. frequency character-istic, MW-sec

HA Equivalent rotary inertia of Area A, MW-sec2hi(xi) Heat rate associated with ith turbine, Btu/hr

PA Area A load (including losses) at nominal frequency, MWPAj Actual power exported from Area A to Area j, in excess of

QAj MWQA Scheduled total export power from Area A, MW

TAj Synchronizing torque coefficient of tie-line between Area Aand Area j, MW/rad

Xi Mechanical power driving ith turbine, MW (exclusive of

governor)xA Total mechanical power driving Area A rotary inertia, MW

(exclusive of governor)

oij Constant associated with ith turbine incremental cost in jthvalve region

<4A Area A frequency deviation, rad/sec

ACKNOWLEDGMENT

The authors gratefully acknowledge computer services and partialsupport of H.G.K. provided by the Philadelphia Electric Companyunder Drexel University Research Contract No. 416. T.E.B. also ex-

presses appreciation for support provided in the form of predoctoralfellowships by the National Defense Education Act and the NationalScience Foundation.

REFERENCES

[1] C. Concordia, F. P. deMello, L. K. Kirchmayer, R. P. Schulz;"Effect of Prime-Mover Response and Governing Characteristicson System Dynamic Performance," Proc. Amer. Power Conf., 28(1966), 1074-1085.

[21 0. 1. Elgerd, C. E. Fosha, Jr.; "Optimum Megawatt-FrequencyControl of Multiarea Electric Energy Systems," IEEE Trans.,PAS-89 (1970), 556-563.

[3] L. K. Kirchmayer; Economic Control of Interconnected Systems,New York: Wiley (1959).

[4] G. L. Decker, A. D. Brooks; "Valve Point Loading of Turbines,"AIEE Trans., 77 (1958), 481i486.

[5] R. J. Ringlee, D. D Williams; "Economic System Operation Con-sidering Valve Throttling Losses: It-Distribution of System Loadsby the Method of Dynamic Programming," AIEE Trans., 81(1962), 615-622.

[6] L. H. Fink, H. G. Kwatny, J. P. McDonald; "Economic Dispatchof Generation via Valve-Point Loading," IEEE Trans., PAS-88(1969), 805-811.

[7] M. Athans, P. L. Falb; Optimal Control, New York: McGraw-HillBook Co., Inc. (1966).

Discussion

C. W. Ross (Leeds and Northrup Company, North Wales, Pa. 19454):There appears to be an ever increasing number of papers in the technicalliterature implying application of optimal control theories. The questionis raised - "Should some guidelines be established for evaluating thecontents of such a paper?" Presently - each reviewer and reader con-cerned with application must laboriously wade through many theoriesand mathematical gymnastics, often unfamiliar to him, in order toevaluate the practicality of a proposed scheme for his application. Itseems to this discusser that the authors of such papers should carry outthis function since they should be most familiar with theory and theapplication so that the job need only be done once.

It has been the experience of this discusser that any given controlscheme designed to be optimum for certain criteria and conditions maybe far from opti}num for other criteria and operating conditions. There-fore, authors should carefully evaluate all conditions and contingencieswhich may affect control and in particular those which may not havebeen considered in the original design of the control scheme. This is thearea which is most often overlooked and where most pitfalls occur.

Perhaps the best way to approach the establishment of guidelinesfor evaluating optimum control schemes is to consider the commonelements involved, e.g., performance criteria; process modei with itsconstraints; disturbance characteristics; practicality of control design;choice of existing controllers for reference evaluation. Some of theseelements are discussed below with regard to the subject paper to illus-trate guidelines for evaluation.

For more details on these and other points reference is made to theDISCUSSIONS given on the authors' Reference 7, "Optimum Mega-watt-Frequency Control of Multi-area Electric Energy Systems" by0. 1. Elgard and C. E. Fosha. Apparently the authors overlooked these

Manuscript received August 6, 1971.

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QAJRA

DISCUSSIONS and their REFERENCES. Further reference is made toDiscussions on the IEEE Transactions Paper No. 71TP52-PWR, "AnOptimal Linear Systems Approach to Load-Frequency Control," byR. K. Cavin, III, M. C. Budge, Jr. and P. Rusmussen.

Authors often choose, and call conventional, a control schemewhich will dramatize the "improvement" with their control scheme.The control scheme referred to by the authors would appear to be the"'" dispatch system. However, it is not clear from Figures 1 and 2exactly how it was simulated. The authors do not refer to other, andmore serious, difficulties encountered with this type of control. Theseproblems include:

1. Reset Windup Cycling - due to velocity limits on the dispatchunits.

2. Excessive Control Action - due to proportional control ac-tion, on noisy control errors, added to reduce reset windup. Costwisethis is a more important problem than the "target designation" prob-lem discussed in the paper.

3. Reset Dead Zone Cycles - due to dead zones in the governors.4. Steady-State Errors - due to controller deadbands used to

overcome problem in 3 above.The failure to recognize the problems listed above are due in part

to the inadequacy of the authors' "conventional" model to includenonlinear and time varying parameters incurred in electric powersystems.

In this discusser's opinion the authors should have included the"permissive control scheme" as their reference conventional controller.This control scheme has had the widest acceptance in the electric powerindustry for decades. Not only does the permissive control overcomethe above listed control problems (plus others including initialization),it also deals with the area of target state (or destination)l which is themain point of the authors' paper.

With regard to target designation the permissive control schemehas as its primary objective (criteria) - satisfying the area controlerror consistent with minimizing control action; economic allocation isthe secondary objective. Under this definition for control objectives theauthors' control scheme may not be optimum since the paper statesthat all dispatch units will not be driven at their maximum rate tosatisfy the area load requirements. Modern direct digital executions ofthe permissive control principle have a hierarchical structure whichincludes an error adaptive control computation (EACC).2,3 The EACCchanges the control strategy (including the relative weighting of variouscontrol criteria such as economics and control error) depending on thenature of the disturbance and the required control action.

The construction of the authors' optimal controller area megawatttarget signal (SA) raises a number of questions:

1. Do the coefficients CA5, CA6, and CA7 need to be updated online as the system changes? If so, what are computer requirements intime and memory and how are the change data determined?

2. System parameter R and D used to calculate PA+QAB arerarely known. These parameters are nonlinear and time varying. Itshould also be noted that the actual valve points determined fromgeneration only are uncertain.

3. How many neighboring frequencies need to be measured andhow many coefficients similar to CA6 are required?

4. The control action does not appear to have reset. What willforce the area control error to zero under contingencies such as: non-responding units, dead zones, uncorrelated limits and constraints, andinaccuracy in measurements? If reset is added how is windup avoided?

5. How would the use of this control equation affect the per-formance of interconnected system operation? The added complica-tions may actually degrade the overall performance. At present theinterconnection uses common signals to form the area control error,i.e., system frequency, common tie line measurements and schedules,and a frequency bias setting which represents an area's contribution tothe security of the system.

REFERENCES

[1 ] N. Cohn, "Methods of Controlling Generation on InterconnectedPower Systems," AIEE, 80, Pt. III, pg. 270-282 (1962).

[21 C. W. Ross, "A Comprehensive Direct-Digital Load-FrequencyController," presented at the IEEE-PICA Conference, May 1967.

[3] C. W. Ross and T. A. Green, "Dynamic Performance Evaluation ofa Computer Controlled Electric Power System," IEEE Transac-tions 71-TP593-PWR, presented July 20, 1971, Portland, Oregon.

Thomas E. Bechert and Harry G. Kwatny: We are very pleased that Dr.Ross has found sufficient interest in our paper to invest the time re-quired to produce a formal discussion. Dr. Ross has raised severalobjections and questions and we would like to respond to these in-

Manuscript received September 14, 1971.

dividually. Before proceeding in that direction, perhaps it would bebest to briefly review the intent of the paper and to describe thecircumstances under which our interest in the problem developed.

To begin with, our interest in pursuing a radically new approachto the economic dispatch problem grew during the course of develop-ment of economic dispatch and load-frequency control algorithms inconjunction with a major utility and a major interconnection. At thistime we had the opportunity to observe and analyze day-to-day opera-tions of a large system utilizing a standard economic dispatch and load-frequency control system. It was concluded that significant improve-ment of overall performance (on that particular system) could beachieved by modifications which may be considered more-or-lessstandard but that totally satisfactory automatic operation over the fullspectrum of daily disturbances would require a unified analysis com-bining dynamic and static considerations.

It should be noted that various techniques have been employed inthe design of supplementary control systems to obviate problems dueto the coupling of the static (economic dispatch) and dynamic (load-frequency control) aspects of the problem. The most successful tech-nique from the point of view of stability has been to decouple the twofunctions in a temporal sense - i.e., design the economic dispatch sys-tem to be a very slow acting trim on the load-frequency control system.This, of course, was the original philosophy in the design of automaticeconomic dispatch systems. On many systems this philosophy is nolonger adequate and faster response of the economic loop is required ifit is to function at all. The concept of "permissive" control is anotherapproach to the problem of coupling between the static and dynamicaspects of the problem. Permissive control has serious disadvantagesand has not been universally accepted as a suitable control strategy.

The primary purpose of the paper is to provide a framework forthe design of alternative control strategies which take account of boththe static and dynamic aspects of the supplementary control problem ina direct and consistent manner. The vehicle chosen for design is optimalcontrol theory, as that provides the most direct approach. An exampleis provided to illustrate and contrast functional aspects of the unifiedsupplementary controller with a conventional structure.

The authors regret the sparcity and apparent deficiency of theexplanations and interpretations which accompany some of the mathe-matics in the paper. The authors feel that the paper presents the supple-mentary control problem in a new perspective and have tried to providea general outline of the problem formulation and the nature of its solu-tion in order to elicit discussion on all of these aspects. Unfortunately,this objective is not entirely compatible with the new PES emphasis onbrevity.

Dr. Rose is quite right that "... any given control scheme designedto be optimum for certain criteria and conditions may be far fromoptimum for other- criteria and operating conditions." Nevertheless,formal optimization has been a useful guide in engineering design formany years. Economic dispatch represents one of the oldest engineeringapplications of optimization (1). The principles established in the solu-tion of this static optimization problem have served the industry wellover the decades. Our impetus to formulate an expanded optimizationproblem stems from the fact that operating conditions have changed tothe point where the old criterion is inadequate.

No attempt has been made in the present paper to present anexhaustive examination of operation under all operating conditions.This is clearly beyond the scope of a single paper.

Dr. Ross has correctly identified the "X" dispatch system. Severaldifficulties with this class of control schemes are mentioned by Dr.Ross. Among these, Reset Windup Cycling takes on a special characterwhen discontinuous incremental cost curves are employed. This is dis-cussed in the paper. Excessive Control Action due to noisy controlerrors has been recognized as a problem, and a control action penaltyhas been included as a factor in the optimization criterion. No simula-tions of a stochastic nature are illustrated in the paper (although somehave been performed); and consequently, this aspect of the problemhas not been emphasized.

As Dr. Ross points out, governor dead zones are not included inthe model and Reset Dead Zone Cycles and Steady-State Errors are notexamined in the paper. A limited number of simulations have been runwith dead zone included, and on the basis of these, no change in thegeneral conclusions would be made. However, there are many otherrefinements which could be included in the model and any proposedcontroller design should certainly be evaluated, in the final analysis,using a more complete representation of the system. The idealizedmodel used is an appropriate vehicle for introducing the main points ofthe paper.

The reference control, scheme used in the paper is one of thestructures recommended in the standard references (2,3) and is in useon numerous systems. We find the arguments against permissive con-trol more convincing than those for it and consequently have elected toemploy a mandatory scheme.

Dr. Ross suggests that the appropriate criteria are " . primaryobjective (criteria) - satisfying the area control error consistent with

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minimizing control action; economic allocation is a secondary objec-tive." In formulating the optimal control problem we have transposedprecisely these words directly into the controller design process. Exam-ination of the optimal controller will reveal that the number of genera-tors in motion depends directly on the deviation of area generationfrom area load. The more serious this error, the more generators areemployed to correct it. When load and generation are matched within"reasonable tolerances", fuel economics and control action becomemore predominant factors. What signify reasonable tolerances are es-tablished by the relative weights in the cost functional (the y's). Sincethe optimal controller is parameterized in terms of these weightingcoefficients, the controller is directly applicable in an adaptive structure.

We regret that it was not possible to elaborate more fully in thepaper on the construction of the area megawatt target signal (SA). AsDr. Ross has pointed out, introduction of the coefficients CA5, CA6,and CA7 has important implications in terms of inter-area communica-tion and coordination. We hope to discuss these points at suitablelength in a future publication. Note, however, that when these co-efficients are zero, SA reduces to the area MW requirement, which is,perhaps, the case of most immediate interest as it corresponds directlyto current operating practice.

Dr. Ross is quite correct in his observation that the optimal con-troller does not have reset in the usual sense. However, it does producea rate command as the control signal to each station. Consequently, thearea MW error will be driven to zero even in the presence of dead zonesand uncorrelated rate limits. On the other hand, area MW error will notbe driven to zero in the case of nonresponding units (or uncorrelatedconstraints which result in nonresponding units). Since unit outputs aremonitored as they are required inputs to the controller, failure of a unitto respond can be easily recognized and appropriate action taken.

[1] M. J. Steinberg and T. H. Smith, Economy Loading of PowerPlants and Electrical Systems, New York: John Wiley and Sons,Inc. (1943).

[2] L. K. Kirchmayer, Economic Control of Interconnected Systems,New York: John Wiley and Sons, Inc. (1959).

[3] N. Cohn, Control of Generation and Power Flow on Intercon-nected Systems, New York: John Wiley and Sons, Inc. (1966).

OPTIMUM POWER FLOW FOR SYSTEMS WITH AREA INTERCHANGE CONTROLS

John Peschon, Hermann W. Dommel, W. Powell, and Donald W. Bree, Jr.*

Abstract-In this paper, the hourly operating cost of an intercon-nected power system constrained by prescribed area interchanges isminimized. The scheduled area interchanges act as constraining equa-tions in the economic optimization procedure. The hourly cost minimi-zation is performed with respect to active and reactive generation and issubjected to constraints on voltage and power flows. This minimizationis carried out under several assumptions, notably: optimum operation ofsome area k of the interconnected system, optimum economic operationof the whole system with the interchange constraints satisfied, andoptimum operation of the whole system with the interchange con-straints disregarded for increased overall economy. These three assump-tions are encountered in the operation of power pools. A numericalexample involving a 30-node system, real and reactive power withadjustable transformers and three areas is solved by two optimizationprocedures: the Penalty Function method and the Generalized ReducedGradient. A system equivalent that includes the effect of the area inter-change controllers upon the flow is developed.

INTRODUCTION

Minimization of the hourly cost of generation for an isolated (non-interconnected) power system has been the subject of several recent

1,2,311,12papers, , 1 ' in which nonlinear programming (NLP) techniqueswere used to formulate and solve this problem. The advantages of thisapproach over the previously known economic dispatch methods9'13were shown in these references.

*J. Peschon and D. Bree are with Systems Control, Inc., Palo Alto, California.H. Dommel and W. Powell are with the Bonneville Power Administration,

Portland, Oregon.

Paper 71 TP 584-PWR, recommended and approved by the Power SystemEngineering Committee of the IEEE Power Engineering Society for presentationat the IEEE Summer Meeting and International Symposium on High Power Testing,Portland, Ore., July 18-23, 1971. Manuscript submitted September 17, 1970; madeavailable for printing May 4, 197 1.

Minimization of hourly cost of operation by NLP is often termedan optimum power-flow solution because the independent variables(controllable parameters such as transformer taps, real power of apower plant, etc.) in a Newton's power flow are determined such thata specified scalar cost function, most frequently the hourly cost ofoperation, is minimum.

The feasible power-flow solution methods in wide use today15were originally designed for non-interconnected systems. Since mostsystems are interconnected, the specified area interchanges were ob-tained by iterative adjustment of the area generations. It was shownrecently4 that these specified area interchanges can be viewed as a setof constraining equations in addition to the standard ac power-flowequations. These area interchanges are maintained at their desiredvalue by the load frequency control system (LFC). Solution byNewton's method of this augmented set of equations constitutes apower-flow solution that also satisfies the specified area interchanges.The computational approaches to solve this augmented set are dis-cussed in Refs. 4 and 8.

With this established knowledge of optimum power-flow (OPF)solutions for non-interconnected systems and feasible power-flow(FPF) solutions for interconnected systems with area interchangecontrols, the obvious next step is to develop an OPF solution forinterconnected systems. Such a solution was suggested in Ref. 5 for atwo-area system, but due to the particular choice of the intercon-nected variables, the proposed procedure cannot be extended to multi-area systems.

One of the aims of this paper is to extend the single-area OPFsolution to the general multi-area case. A second aim is to discuss andclarify the several OPF problems that naturally arise, depending on theobjective pursued; viz. system optimization or area optimization. Athird aim is to discuss possible equivalencing procedures to simplify theoptimization in both contexts of system analysis and operations.

Reference 14 was found to be of great value in combining theconcepts of economic dispatching and of area interchange controls.

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