A REAL TIME OPTIMIZER FOR SECURITY DISPATCH

B. F. Wollenberg W. 0. StadlinLeeds & Northrup CompanyNorth Wales, Pennsylvania

Abstract - As power systems have evolved in size and complexity all voltage magnitudes, phase angles, bus P and Q loads, and P and Qso has the need for sophisticated and comprehensive real time control generations. Methods using a full set of variables reach a solution byand dispatch techniques. Solutions presented in this paper are intended solving the following mathematical program:for on-line applications and are further designed to accommodate "net-work" as well as "nonnetwork" (e.g., environmental) and interacting subje (iu 0operating restrictions. The paper reviews the criteria for the constrained and: h gju) < 0 (1)optimization of real power; discusses practical approaches towards im-plementation; presents numerical results of tests on a large network Where F(x,u) is the system objective function, g(x,y) is the set of acmodel including the effects of convergence, external equivalents, and power flow equations describing the transmission network, and h(x,u)reference bus locations; proposes extensions to include reactive power is the set of inequality constraints specified for the system. The twodispatch and contingency constraints, vectors x and u are the dependent and independent (or controlled)

variables respectively. This technique requires that a complete loadOBJECTIVES OF SECURITY DISPATCH flow be solved simultaneous to solving for the optimum dispatch. The

equations are similar in sparsity to the load flow equations and requireConventional Lambda dispatch programs (ED) provide economic optimally ordered sparsity programmed matrix solution techniques. The

generation base point and participation factors to an automatic gen- inherent advantage of this technique is the simplicity with which volt-eration controller (AGC) which controls system frequency and net in- age and reactive as well as real power constraints can be formulated.terchange. The major thrust of this paper is to present a viable technique Various techniques have been tried512 with the most successful beingfor the inclusion of specified security and environmental constraints. the generalized reduced gradient8 and the Hessian matrix1 1 approaches.An example of a security constrained dispatch includes a regulating A disadvantage to these methods lies in the amount of computation permargin constraint.1 More recently, there has been interest in dispatch iteration as well as the number of iterations to reach a solution. This isconstraints such as transmission line loadings, area interchange, and bus discussed more fully below with respect to the generator cost curves.phase angle spread. There has also been the necessity of imposing en- Alternative approaches utilize a REDUCED SET of variables invironmental constraints which are not a function of the transmission which only the controlled variables appear in the mathematical pro-network, but of plant emissions and atmospheric conditions.2 gram. 13-17 Since the network equations are not solved explicitly, the

Present state-of-the-art allows a system operator to apply the re- effects of the network must be modeled by mathematical functions in-sults of an on-line or off-line contingency analysis and, if necessary, in- volving only the controlled variables. Thus the classical economic dis-corporate corrective action into the dispatch of the system by various patch problem is formulated using the vector of controlled generationapproximate methods.3 Similarly, in the area of environmental con- P and a B matrix equation to represent the system losses as a functionstraints only the minimization of total system pollutant output has been of P:implemented,4 leaving regional environmental constraints to be handled Nby approximate methods. For real time use, a security optimization pro- -gram must provide a solution rapid enough to allow sufficiently fre- minimize: F(P) - F(rj), totaloperatingcostquent repetition to follow changing system conditions. Required output i=lshould consist of generation base points and "constrained" participationfactors allowing the AGC program to control the system so as to track Nthe constraints between base point updates. Rapid execution is also re- subject to: K Pi = PLoss + PLoadquired for the various study programs incorporated in a dispatch com-puter (such as economy A and B transactions evaluation and unit com-mitment) where many dispatch solutions must be executed. The objec- and: Pmini < Pi 5 Pmaxi i =1 **, Ntives of a real-time optimizer are summarized as follows:

-Provide an economic generation schedule for the AGC.-Incorporate network and environmental constraints. wr o = - P + BW P + Bo0 (2)-Provide "constrained" participation factors for the AGC to main-

tain security between optimizations. and: N = Number of controlled generators-Provide a fast solution for both on-line dispatching and back-

ground studies.The Lagrange multiplier solution leads to the coordination equation:

OPTIMIZATION TECHNIQUES

Programs for attaining the desired solution to the constrained dis- 1 (3)patch problem are divided into two general categories differentiated bythe type of variables manipulated. A FULL SET of variables consists of which is incorporated by most present day economic dispatch programs.

The disadvantages of this technique are the need for generating an up-Paper T 74 149-1, recommended and approved by the IEEE Power System dated B matrix as system configuration or load level changes and the in-

Engineering Committee of the IEEE Power Engineering Society for presentation at ability to include functional constraints. The advantage is a fast solutionthe IEEEPESWibniter Meeutingi, New7orkNaYd ,JanaulbeafoY 27-nFenbgruaorer1,1974. time, making this an attractive'technique for both real time dispatching1973. and study programs.

1640

The optimizer technique presented in this paper is based on a re- Nduced set of variables for program speed, but derives transmission loss

iPi + aj hmaxj for each such constraint<penalty factors and constraint sensitivity coefficients from a system load *, (12)flow. Penalty factors are based on Shipley's reference bus dispatch iI(2approach18 with computations carried out as shown by Dommel and Where ajo is a constant determined at the start of the dispatch compu-Tinney.14 If one bus on a network is designated as a reference bus tation so as to make the linear equation match the value of the con-taking up any generation slack created by a change in another generator, strained variable as computed from the current state vector (or tele-a set of derivatives,i can be defined as follows: metered). The coefficients, otjp have been derived by several authors

using different methods. 13,19,20 Basically, they are seen to be injection=

pi=e i for all generators (4) distribution factors since (e.g., letting h = MW flow on a specified line):

note that 1ref = +1 . = [iL iii joj ji (13)

Shipley and others have also shown how to perform economic dispatchwith these "reference bus" penalty factors. Dommel and Tinney solve Thus aji is simply the rate of change of hj (MW flow on the specifiedfor the vector 13 as follows: line) with respect to change in generation at plant i.

The resulting mathematical program has the following form:JT ffi - ~ref' where gef is the vector of partial derivatives

of Pref with respect to the system miNvoltage magnitude and phase ,) Fi (P)angles. i-ljT = transpose Jacobian matrix

=- the vector of reference bus N Nderivatives -3Pref/aPi. (5) subject to: E iPi = Ziipi (constant)

i=l i=lThe solution of the above equation requires the availability of a bus (orstate voltage) vector and a load flow data base representing the existing and: Pminj < Pi < Pmaxpower system status conditions.

The O's are incorporated in a linear system model by the following Nperturbation equation: and: 7 ajiPi +P jo± <hmax. for each

JO~~ constraint .jtref = ~Pref =Pi=i (6) (14)

The distinguishing feature of this mathematical program is the form of

Then,-if each generator i is perturbed by APi the total reference bus the cost curves. For linear or quadratic generator plant cost curves, the

perturbation becomes: problem can be classified as a standard linear (LP) or quadratic pro-perturbation becomes:gramming (QP) problem, respectively. The actual cost curves for a plant

N are usually much more complex than a simple linear or quadratic func-

'Pref = ZE 3j«i .t(7) tion. A plant may be made up of several units each represented by a

i/ref third or higher order polynomial or even by piecewise linear or piece-.wise quadratic functions. For the solution proposed the plant cost func-

The beta for the reference bus is set to +1 by definition, therefore the tions must be continuous, piecewise differentiable, and convex. Theabove equation can be written as: plant incremental cost function must be monotonic increasing, piece-

wise differentiable, and not necessarily continuous. Figure 1 shows an

Pref aPref = Pi aiP or Pi APi = 0 PRODUCTION COST FUNCTIONSiiref all i 8 UNIT UNIT 2 PLANT

F ~~~~F2 Fi USIf the perturbations APi are replaced by Pi-P9, where P9 is the operating

2 SPpoint where the ,B's were derived, then: r -P. ~~~~~~P.

Z Pia = i(Pi-P.) = SFiP - =Pi = (9) 1Pil 100 12 200iTherefore, INCREMENTAL COST FUNCTIONS

PiP, = £3iPi0 = constant (10) > |UNIT I UNIT 2 PLANT

This linear equality constraint "models" the system behavior about anoperating point. In practice, the constant is set to the linear sum shown Pwith the Pi's at their telemetered or computed "initial" value P9°. lp 100 i 2 200

The security and environmental constraints are also modeled as Fig. 1linear constraints. Thus if

h(P) < hmax (11) example of a generating plant consisting of two units with different-_ cost functions illustrating an overall plant cost function with a "cusp"~

is an inequality constraint representing either a system limit such as area and the corresponding plant incremental function having a jump.interchange, tie MWt or MVA flow, phase angle difference or an environ- The jump in the plant incremental cost function may present diffi-mental constraint, it can be represented as follows: culties to optimization techniques requiring a gradient of the system

1641

equations since a small perturbation in the generated powers may result constrained p e - pin a jump in some elements of the gradient vector. This effect can be participation Pj -even worse when the cost curves are piecewise linear functions which factor 0 (17)give rise to stepped incremental cost functions. Overcoming any adverse This procedure takes account of all unit limits and cost curve dicon-effects from such cost functions may require special techniques, extra tinuities as well as functional constraints and constraint status changes.iterations, or both. Use of a technique employing a reduced set of vari- If desirable and if the AGC program is designed to handle them, two setsables allows for extra iterations without increasing overall solution time of p's can be generated - for increasing or decreasing load. This ap-beyond a reasonable limit. proach requires two or possibly three complete dispatch calculations

Our objective in choosing a solution technique is to improve itera- each dispatch interval instead of one. The Dantzig-Wolfe algorithmtion speed, primarily to allow for the extra iterations necessary to over- allows quick re-solutions at the incremented load level by addition of acome plant incremental cost function discontinuities and still provide a special slack variable to the mathematical program. In effect, the slackfast overall solution time. Having met this objective, the proposed solu- variable (a slack generator as it were) is given a negative incrementaltion has been found suitable for real time applications as well as back- cost and a high limit of zero for the base solution. Its Beta is set to minusground study programs requiring many repeat solutions. Depending on one for a positive load increment. After the base solution is obtainedthe AGC program design, the real time program runs one or two repeat the slack generator's high limit is increased from zero to APG. Thesolutions to obtain constrained participation factors. Dantzig-Wolfe procedure then starts out with the optimum base solution

A characteristic feature of the power system optimization problem and corresponding simplex tableau, and quickly reconverges with theis that the objective function is usually separable. (Common header or slack generator at its new high limit. Since the slack generator Beta isinteracting generating units may be treated as special cases.) This means opposite in sign to the load increase, the other generators will have tothat the objective function is the sum of individual nonlinear functions increase their output to meet the Beta equality constraint. The advantageeach involving only a single independent variable. Thus, electric power of this procedure is a solution for the incremented load in a fractioh ofdispatch problem can usually be classed as a separable convex problem. the time to reach the base solution. Tests using constrained participationThis can be of great advantage as shown by Dantzig21 and also by factors have shown that constraints can be held during a load ramp. TheLasdon.22 Several straightforward approaches to the solution of this same constraint would drift over limit between dispatch calculations iftype of problem are available by means of generalized linear program- unconstrained participation factors were used.ming. Hadley23 has shown two variations of this technique called N and6 methods, both applicable to separable nonlinear objective function

CONSTRAINT VIOLATION TECHNIQUESproblems which may also include separable nonlinear constraints. Anadditional technique introduced by Dantzig is the Dantzig-Wolfe de-composition method which, as shown by Dantzig and by Lasdon, readily The security dispatch program must handle three classes of con-fits the context of a separable convex problem. This technique has been straints. Two of them, the individual generator unit high and low limittested in a stand-alone version as well as a version coupled to a Newton inequality consttraints and the total generation requirement equalityload flow as described later. A description of the technique is included constraint, are considered hard constraints which must always be satis-inadAppendixIs ofhisdpaper. As wellscriptionthereferencsciuedaboveified. (When the hard constraints cannot be satisfied the problem is con-in Appendix I of this paper as well as the references cited above.

sdrdifail,uls ews ocnie odo eeaiisidered infeasible, unless we wish to consider load or generationshedding.) The third class of constraints consists of the functional con-straints which should be met if possible but may be violated if hecessaryfor shorter or longer periods of time. Thus, the optimization programmust be able to handle cases where some or all of the functional con-

Participation factors, Pi, are passed to the AGC program to de- straints cannot be met. Such cases are referred to as OVERCON-termine eadh unit's generation change in relation to a change in the STRAINED cases.total generation as the system load level shifts. If all units are at their t h o c

If the ability to handle overconstrained cases were not pro'vided,base point generation arid a load shift occurs requiring a change in total t pthe program would simply abort on an error condition. The only alterna-

generation of APG, each unit is moved to a new output according to:' ~~~~~~~~~~~~tivewould be to repeatedly reexecute the program with an altered setrnew base point + p (15) of constraints or limits until a satisfactory solution is reached. Our ex-i3 i i '~PG perience has shown this procedure to be undesirable. An alternative is to

where Xgi = 1 (i for all units on control) set the program up to solve the overconstrained cases in a convenientmanner. The standard method for handling this situation is to add

In practice, the unit's physical response rate, the unit's being off control penalty functions to the constraints which cause the optimization pro-for vatious reasons, etc. will cause varying deviations from the output gram to minimize the desired function of constraint violations. Thedetermined by the above formula and must be accounted for in the AGC penalties are set up inr such a way as to override the power generationprogram itself. costs of those generators that affect the violated constraints. When all

Unconstrained participation factors are defined by taking the de- constraints are feasible, the constraint penalty function costs becomerivative of Pi with respect to total generation. zero.

unconstrained dPi Three methods of handling constraint violation or overload areparticipation Pi (16) generally proposed:factor: & PiCPibase 1) Minimize the weighted sum of the absolute values of the over-

loads. (ABSSUM method)Unconstrained participation factors are simply a linearization about the 2) Minimize the maximum absolute overload among all the con-base points and have a limited range of validity due to unit limits and straints. (MINMAX method)cost curve discontinuities. These participation factors become even less 3) Minimize the weighted sum of squares of the overloads. (SUM-desirable with constrained dispatching where a constraint may have to SQUARED method)be held during aload change or pass from inactive (nonlimiting) to active The penalty functions associated with each mnethod are created to(limiting) status. A practical solution for constrained participation fac- guarantee that the system will always find a feasible solution whentors is to first solve for the optimum dispatch at the base level and then possible. Each method has been tested on representative power systemat the load level expected at the end of the dispatch interval (2-5 min- data, and each has its own peculiarities. The ABSSUJM method tends toutes). lump all constraint violations on one constraint, whereas the MINMAX

1642

tends to spread all constraint violations evenly over all constraints. amount (and redundancy) of telemetry which is available. A "real time"Neither of these tendencies is desirable. Lumping all violations on a load flow matches bus injections (which have been telemetered or esti-single constraint may cause it to be severely overloaded while an even mated by factors) while a "state estimator" matches telemetered linespread of overload may result in an "umbrella" effect in which many flows as well as injections. The purpose of the load flow or state esti-violations are allowed to be large when one constraint must be violated mator is to build a mathematical model of the power system which candue to the hard constraints. The SUMSQUARED method distributes be used to perform the following functions:error more than ABSSUM although not as much MINMAX, and is 1) Estimation of electrical flows for which there is no telemetryusually preferable for this reason. The weights put on the constraint or the telemetry is invalid.violations remain a matter of judgment. The results of constrained cases 2) Computation of sensitivity coefficients for economic dispatchpresented in the results section used equal weighting for similar con- and security constraints.straints. 3) Contingency analysis for analyzing hypothetical emergencies.

The system operator is not only advised of "off-normal" operatingAPPLICATION conditions detected by the model but is also informed of excessive re-

siduals due to modeling errors, i.e., large differences between telemeteredThe real time optimizer described in this paper can be applied to flows and computed flows.

small as well as large power systems. Justification, however, depends onseveral factors: EXTERNAL NETWORK INTERFACE

1) Flexibility of generation schedules.2) Extent to which the electrical network configuration varies. The usual practice is to build a detailed model of the power system3) Number and type of operating restrictions which may be ex- under dispatch (Area Control Error (ACE) boundary) and couple this

pected. model to an "equivalent" network which is intended to represent theFigure 2 summarizes the basic structure of a security dispatch sys- outside world.

tem, beginning with real time input telemetry and producing corrective Analysis has shown that transmission loss factors are affected bycontrol action.24 the nature of the external equivalent, i.e., the more accurate the equiva-

lent the better the "economic dispatch." A more important considerationis the effect of external equivalents on constraint sensitivity coefficients

TELEMETRY PROCESSORS ALARMS and contingency analysis. Investigation of the effects of one type of ex-DATA 1 ternal equivalent is presented in the results section.

SYSTEM STATUS PROCESSORS ALARMSTEST RESULTS

LOAD FLOW OR ESTIMATOR ALARMS Verification of Optimization MethodologyA large number of tests were run on a representative test system

SENSITIVITY COEFFICIENTS (see acknowledgements) to verify that the optimization method per-SYSTEM CONSTRAINTS forms properly. The tests run without functional constraints were quite

CONSTRAINEDOPTiMIZATION similar in form to those run by Happ25 with correspondingly similar re-

ALARMS sults. As with Happ's tests, it was found that the optimum obtained bycycling through load flow derived penalty factors was virtually identical

GENERATOR BASE POINTS to the optimum reached by a standard economic dispatch algorithm usingPARTICIPATION FACTORS

a B matrix generated for the same test data. The rate of convergence

AUTOMATIC(beginning from a no-loss dispatch) to the optimum dispatch through

GENERATION CONTROL- ALARMS the load flow was about four iterations with a penalty factor gain of0.75.

Fig. 2 Security Dispatch Tests were run to verify that the optimum of the mathematical

Initialzation program obtained by the Dantzig-Wolfe method is correct. For thesetests, a large number of problems were created to give special charac-

The first stage of Security Dispatch is to "scan" the power system teristics such as narrow objective function ridges and various combina-and review and "purify" all the available input telemetry. At this point, tions of generator and constraint function limiting conditions. Bothobviously bad data is rejected, credibility tests are made, and appropri- second and third order polynomial cost curves were used. Each of theate alarm messages are displayed to the system operator. In the event of test problems were solved by the Dantzig-Wolfe algorithm and a standarduntelemetered data or bad data the system operator is allowed to pro- quadratic programming algorithm. Both algorithms reached the- samevide the missing information or if this is unrealistic, programs are de- solution for all problems, with the Dantzig-Wolfe algorithm taking aboutsigned with telemetry "'default" options, i.e., the security dispatch pro- one tenth as much computer run time as the QP algorithm. The decadegram is designed to run with a minimum (enough to execute AGC) of lower execution time can be attributed to the much smaller tableau sizetelemetered data (in the event of massive telemetry failures). used in the Dantzig-Wolfe algorithm than the standard QP algorithm as

The next step is to examine the network configuration as described shown in the following formulas:by the telemetered status of breakers and switches. Every substation is Dantzig-Wolfe Algorithm = (N-M+2)2scanned and analyzed to derive an electrical model composed of nodes Tableau Size(busses) and links (lines, transformers, etc.). At this point the computerdetermines if the system has "islanded," i.e., separated into independent Quadratic Programming = (2N+M+-i2 )x (5N+2N+5)electrical networks.

In the event of system separation, security dispatch may be applied whr N ubro eeainpat' M~~~~~~~~~1 = numzber of functional constraints

to one or more of the network islands.The last step in the initialization process is to execute a "real time" The large tableau size for the QP algorithm is mainly attributed to the

load flow or "state estimator." The choice of program depends on the need to explicitly represent the unit high and low limits by an extra N

1643

inequality constraints rather than implicitly by column generation as inthe Dantzig-Wolfe algorithm. r----

Application Testsm

Equivalent Circuit: One area of application of on-line optimization I Iwhich remains open for better solution is the use of an equivalent circuit I NEWTON LOAD FLOWin the system load flow model. I

Tie line flow to neighboring systems will vary as internal generation I PENALTY FACTORS Ishifts occur and may cause significant errors in the optimum dispatch OPT!MIZATION LOOP lunder some conditions. Four separate tests were run to show the differ- NOences which might arise when using various tie flow equivalents. The IV Gequivalent used fixed generator and load busses with one slack generator. L _ - YES OPT MAL P.. p __ _The tests were all run with the load flow and mathematical programconverging to an optimum. The four tests were:

1) Optimization with the equivalent attached to the load flow ES rIMATED TOTAL GENmodel with no area net interchange control. r -

2) Optimization with the equivalent attached as in (1) but with - p NEW = p OLD + p A, CLAMP p NEWthe slack bus in the equivalent always adjusted to hold area net I TO WITHIN HARD OR RATE LIMITS |interchange across the ACE boundary. lT

3) Optimization without an equivalent and with all border ties TOTAL GEN = 7 P. NEWtreated as fixed generators set to the flows resulting from (2). 1

4) Optimization without an equivalent and with all border ties AGC Itreated as fixed generators or loads as determined by the busses j|CONVERGE ADJUST Athey are attached to in the original equivalent. SIMULATOR

Test (1) was run mainly for comparison and as might be expected, is in NEWTON LOAD FLOW WITH p jNEW Ierror in total generation by about the same error as the net interchange.lTest (2) was used mainly as a basis of comparison for nonequivalent I SW! NG BUS NO Imethods such as tests (3) and (4). The results from tests (3) and (4) are CONVERGE Imuch closer to test (2) than the results of (1). Both tests (3) and (4) L__ _ YES NEW TOTAL GENconverged to a higher optimum than (2) with (4) having double the -

Fdeviation of (3). All further tests of the optimizer were run with anequivalent circuit with area net interchange control (test 2). If no equiva- NO YE S

lent is available, it is recommended that all the border ties be represented T A

as fixed generators (test 3). When running under constrained optimiza- Fig. 3 Load Ramp Simulatortion it is very important that the constraint sensitivity coefficients bederived from a load flow model with an equivalent since constraineddispatch may have a very great effect on border tie flows. Optimization Loop: This loop solves the optimum dispatch as pre-

Reference Bus Location: Tests were run on the optimizer with the viously described with a mathematical programming algorithm followedreference at two different locations in the load flow model. The loss by a net interchange constrained load flow and penalty factor calculation.factors (,B') calculated for the two cases were quite different since one Load Increment Logic: All loads are incremented a specific per-test located the reference bus at one extreme of the system near a large centage of peak system load while holding all power factors constant.load center and the other test located it near the opposite extreme of The losses at the new load level are estimated as the square of the loadthe system where load is more spread out. No differences were noted in increase times the previous losses.either the rate of convergence of the optimum. AGCSimulator: The first pass through the AGC simulator uses

Load Ramp Tests: Normal use of an optimizer program in a real- the estimated new total generation and ramps all units in proportion totime environment will entail sampling the power system at specified time their participation factors while observing both the unit hard limits andor load change intervals and computing the optimum unit base point rate limits. When this first section has converged, a load flow is run (withgenerations and participation factors. The proposed method of operation net interchange control) so as to calculate the correct total losses. Theof the optimizer described in the application section of this paper runs unit generators are then reconverged on the correct total generationsone load flow on the current state of the system (state estimator or real and another load flow run, etc. The result of this process is a solved loadtime load flow) from which loss factors and constraint sensitivity coef- flow at the new load level with the generations set according to the op-ficients are obtained and passed to the Dantzig-Wolfe optimization timal trajectory determined by the previous dispatch with all hard limitsalgorithm. The optimization algorithm's solution is then used directly and unit rate limits being accounted for. The final load flow from thisas the base points and participation factors for the next interval. Not simulation represents the state estimate which would be calculated forrepeating load flow-optimization steps would thus ignore second order the next dispatch.changes in the loss factors, constraint sensitivity coefficients, and border Tests using the load ramp simulator were made to establish thetie flows. The assumption of convergence over a few dispatches and then rate of convergence of the optimization algorithm when only a singlereaching a tracking status must be challenged for conditions of continu- load flow is run at each load level. Basic tests consisted of several loadous load ramping. The AGC program will carry generation units along increments each allowing the AGC simulator to converge (three iterationsthe trajectory determined by the base points and participation factors being sufficient), and then running three iterations of the optimizationuntil a new dispatch is called for at which time the system is sampled and loop. The object of this procedure is to show the deviation from thea state estimate or real time load flow is run. Whatever error is present true optimum remaining if no repeat load flow optimizations are mademust be corrected by one execution of the optimization algorithm. To beyond the initial state estimate load flow and optimization. Load in-verify that this approach is acceptable, a simulation was run, as shown crement rates used were 35, 70, 105, 140, and 175 megawatts/minute.in Figure 3. The simulation consists of an optimization loop, load incre- The faster load increase rates are only to be expected when interchangement logic, and an AGC simulation loop, contracts are being adjusted or when a unit comes on line and is ramped

1644

to its base load point. Under these conditions, the system will have many\FIG . 4 TEST SYSTEM 2 velocity limited units so that economic base points will not be attainable

anyway. Of major importance therefore is the fact that for steady loador slow rates, the one step load flow-optimization procedure is able to

f°STEAM UN TS attain at least; 1% accuracy, except when a unit comes off limit (where4 HYDRO UNI TS velocity limiting would also occur). This is deemed sufficiently accurate

C° 4 to warrant the use of a single load flow-optimization step approach.

O I 5 a^£ r T E CONSTRAI NT ) Conflicting Constraints

The map of the test system shown in Figure 4 depicts the location0a 8 | of the major generating plants and the constraints tested. Each of the

constraints shown (one angle constraint, 3 line MVA constraints, and)O2 5 / one area export constraint) was imposed separately, and then combina-

tions of two or three were imposed. Table I summarizes the results ofL - -A R E A 1 the constraint tests. The base case column indicates the unconstrained

optimum cost and the constraint values at this optimun.ThefollowingSI 0 five columns show the resu?ts of imposing each constraint separately.

The first of the constraint combinations (Test 6) imposes both tie con-57 / straint I and the area 2 export constraint. Since these constraints are

PHASE relatively independent, both can be met exactly. Test 7 pairs tie con-Jo9 /ANGLE straint 2 with the area 2 export which move more or less together and

CONSTRA NT thus, the optimizer puts the most restrictive constraint on limit (tie con-straint 2) which then leaves the area 2 export constraint at a value con-

T I.E-p/*I }_sX t siderably below its limit. The resulting dispatch in this case virtually/ CONSTRA NT IXS(Tl E I t \ \ matches the dispatch when tie constraint 2 (the most restrictive of the

2 ONSTRAINT two constraints) is imposed alone. As stated previously, the constraintsI#3 o/ o a 4 were set up so as to allow the program to find a solution in the case ofi\

11 °°t3 6 constraint infeasibility, the method choosen for the tests described hereG X12 / being equally weighted least squares. Tests 8, 9 and 10 are cases repre-

'\A R E A 2 V senting such infeasibility. Test 8 imposed tie constraints 1 and 2 and re-

1 00 KM - < s)sults in both being overloaded (one by 15% of its limit, the other by 1%of its limit). Note that in these tests the overload condition is set tominimize the sum of squares of constraint overload. By properly scaling

TABLE I 85% LOADLIMITS BASE CASE TEST TEST TEST TEST TEST TEST 'TEST TEST TEST TEST

STATION MW MW 2 3 4 5 6 7 8 9 10

LCATH 160 111 133 I116 140 131 133 147 140 160 160 1602 COUCH 110 54 73 68 94 77 75 I10 94 110 110 1103 LYNCH 125 65 82 65 101 86 86 106 100 125 125 1254 MOSES 134 30 L 30 L 30 L 30L 30 L 30 L 47 30 L 134 134 1345 RITCH 830 620 672 528 758 620 650 4841 758 508 505 5586 STERL 242 242 242 242 242 242 242 2142 242 242 242 2427 BAXTER 510 507 510 510 510 510 510 510 510 510 510 5108 DELTA 195 138 172 191 180 166 164 195 180 195 195 1959 NATCH 65 15 L 1 5L 15 L 15L 61 15 L 62 15 L 65 65 6510 REXBN 240 240 240 240 240 240 240 240 240 240 240 240II GYPSY 630 630 601 630 550 621 626 626 551 389 379 56312 NINEM 220 220 167 220 III 163 174 174 III 180 188 10013 MARKET Fixed 28 28 28 28 28 28 28 28 28 28 28114 MI CH(Bus I) 335 32 5 293 335 258 279 283 283 257 335 335 21315 MI.CH(Bus 2) 500 500 500 500 500 500 500 500 500 500 500 50016 PATER 87 66 30 L 78 30L 30 L 30 L 30 L 30 L 56 60 30 LTOTAL (AREA I & 2)44111 3794 3788 3796 3787 3784 3786 3784 3786 3777 3776 3773COST $/HR. 8633 8695 8686 8796 8704 8683 8803 8796 8984 8989 8936

PHASE ANGLE*CONSTRAINT Q13.1110 16.80 I3.5 17.2 I .20 13.70 I14.10 11.10 10.20 10.20*| 10.20 9

TIE CONSTRA! NT #1 1112 MVA 178 195 1112 225 180 1 90 1112 225 1611* 1611* 1 80 *T, E CONSTRAINT#2 11456 MVA | 593 1522 |6011 1156 |531 |5311 5111 11456 11460* 11461 * 11115*|TIE CONSTRAIlNT #3 28 MVA 35 30 36 25 28 31 29 25 23 211 22AREA 2 EXPORT 1502 MW 1 626 11480 61167 3111 11483 1502 150 11 3411 350 1353 1300 1

LOW Li MiT ON LIMITr UNDER LIMIT * OVER LIMiT__1645

each constraint equation, the criterion could have been set to minimize Since the number of combinations of line outage effects on remainingthe sum of squares of the percent of constraint overloads. Tests 9 and lines is very large it is advisable to perform a discrimination analysis10 impose three constraints simultaneously. In Test 9 all three tie con- prior to optimization in order to choose only the worst possible linestraints are imposed resulting in tie constraint 3 being under limit. In, outages.Test 10 the area 2 export limit is lowered to 300 MW and imposed The LP structure of the optimizer allows the system operator toalong with tie constraints I and 2. This results in a least squares over- answer security related "what if" questions when the program is in theload solution with the two tie constraints overloaded and the area ek- study mode. The following is a brief list of questions that can be readilyport on limit. adapted to the methods of this paper.

* What is the maximum secure load level that can be supplied by aAPPLICATION EXTENSIONS given network configuration and given generator availability?

* What is the maximum available spinning reserve that does notVoltage and Reactive Optimization violate tie flow constraints?

* What is the maximum secure power import, export, or wheeling?Voltage and reactive optimization can be executed more or less in- ' W

dependently of real power optimization and can be applied at different * What is the shortest time to meet security constraints, starting.. ~~from a given isecure state?time intervals (usually less frequently than real power dispatch). As in fromat given insecuretyate?

the case of real power the reattive power problem is formulated in termsof a linear network model. The independent variables are the reactive

Constrained Interchange Transaction Evaluationsources (such as shunt capacitors and generators) and the reference busvoltage. These independent variables (Qi and Eref) are considered to be Interchange Transaction Evaluation (ITE) is a program which de-bounded by hard constraints. Security constraints take the form of upper termines the incremental cost of a given amount of interchange with aand lower bounds on voltage at selected busses in the network. An neighboring utility. A series of economic dispatch calculations are exe-equality constraint can also be defined which maintains a constant total cuted with the interchange stepped through some range specified by thereactive input. The objective function is usually to minimize total real system operator. The interchange itself is reflected in the ED as borderpower transmission losses. Mathematically the voltage and reactive power tie flow increments obtained by multiplying the total interchange MWoptimization is summarized as follows: by a tie distribution factor for each tie. If a system dispatch is to be per-

formed in real time by an optimization program capable of handlingminimize:ref real powfunctional constraints, there may be a need to replace the ED's in the

7Qi relpoerlsse ITE program by separate constrained optimizations at each step of pro-

posed interchange. This may be necessary if a known system constraint0 needs to be held while evaluating interchange levels, or perhaps more

subject to: Z Qi = S Qi. (constant) importantly, if the interchange itself causes a constraint to become active.

and: Qmini S5 Qi S5 Qmaxi hard limit A method has been tested which accomplishes the incrementaloptimizations by adding a single slack variable to the mathematicalprogram as was done to obtain constrained participation factors. The

and: EminRef S ERef 5 EmaxRef hard limit corresponding tenns in the system equality and inequality (functional)N constraints are set up to reflect the total system effect of the interchange

and: Emiflj < (nRef +~ Qij+cjoe) < Emax; appearing over the tie lines. The generation on the slack variable theni=i- :1

(18) represents the net interchange. The first complete optimization is per-for each constraint j formed with the slack variable's high limit at zero (representing zero

increment of interchange) and then its high limit is "opened" in stepsThe reactive power sensitivity coefficients aPRef/aQi are computed at which directly correspond to the interchange steps desired. Each solutionthe same time the aPRef/aPi are computed. The voltage versus reactive at a new incremental step is again a fraction of the original time neededsensitivity coefficients aEj/aQi are computed outside the optimization for the base solution. A test on the fifteen plant system used in the testloop since they are affected mainly by changes in network configuration. series required 22 CPU seconds on an XDS Sigma 5 for ten interchangeFor practical purposes we can assume that aEj/aQi = aO0/aPi. In other steps. This test was performed with only one unit per bus and wouidwords the sensitivity coefficients for phase angle constraints might also take somewhat longer with multiple units.be used for bus voltage constraints. Depending on the specific applicationwe may also wish to incorporate transformer tap positions, cost func- CONCLUSIONStions for the reactive sources, or the option of switching lightly loadedtransmission lines. An optimization technique for real power has been presented and

shown to be suitable for large power system optimization problems.Contingency Constraints Optimization is performed on a reduced set of variables and does not

In addition to continuous operating constraints, the system operator require convergence of a load flow as the optimum is approached. The

may wish to dispatch generation in such a manner that the loss of any validity of this approach has been demonstrated in tests with actual.selected line will notcasethreaiinlnpower system incremental cost curves while optimizing under static as

selecte*linewllnotausethremainngline to become overloaded, well as dynamic load conditions. The ability to handle single and multipleThis type of security constraint can be incorporated by means of line network constraints has also been demonstrated on a large system model.outage factors 6jk ahj/ahk where "k" represents the tripped line and

Unique features include network constraints, constrained participation"j" represents a remaining line. Thus, factor output, and the ability to handle infeasible constraints by any ofhj + bjkhk < hmaxj several overload criteria.

OrN(a~8ks.P+, ACKNOWLEDGMENTS

~~~2JShm8j(iPi+C3°<Aaxj (19)

i4= The authors wish to acknowledge the expert assistance and guidancewhere h represents the transmission line of Henry R. Koen of Middle South Services, Inc., who supplied the test

megawatt flow, case described below. Additional gratitude is expressed to Mr. Wayne

1646

Barcelo of MSS and Dr. V. J. Law of Tulane University for their many 25. H. H. Happ, "Optimum Power Dispatch," 1973 IEEE PES Summerideas and reviews of the work while in progress. Power Meeting, Paper T-73-460-3.

The test case supplied by Middle South Services represents a 1966Base Case containing: APPENDIX I

247 Busses431 Lines Draw the composite curve shown in Figure 1 with a line from the15 Generation Plants low cost point to the high cost point:29 Generating Units each with a separate cost function made up

of 4 quadratic segments. Fi

WY x~~~~~i- piREFERENCES

1. W. 0. Stadlin, "Economic Allocation of Regulating Margin," IEEE PT-PAS-71, Jul./Aug. 1971, p. 1776. Low High

2. P. G. Friedmann, "Power Dispatch Strategies for Emission and En- Limit Limitvironmental Control," presented at ISA Power InstrumentationSymposium, Chicago, Ill., May 1973, Paper ISA IPI 73461 (59-64).

3. H. D. Limmer, "Techniques and Applicatioins of Security Calcula- The shaded area will always be a convex set since the cost function istions Applied to Dispatching Computers," Third Power Systems convex. Form the vector xi as follows: where Fi is the plant productionComputation Conference, 1969.ComputationConference, 1969. ~~~cost when producing power Pi. Define the set Xi as:

4. M. R. Gent, John Wm. Lamont, "Minimum-Emission Dispatch,"Proceedings 1971 Power Industry Computer Application Confer-ence.

5. T. E. Dyliacco, R. K. Babickas, T. J. Kraynak, S. K. Mitter, "Multi- Xi = i = !i > (20)Level Approach to Power System Decision-Problems - the Opti-mizing Level," Proceedings 1967 Power Industry Computer Appli-cation Conference. That is, the shaded area in the figure above will be represented as all

6. A. M. Sasson, "Combined Use of the Powell and Fletcher-Powell points which satisfy a set of linear constraints. Thus Bixi (where there isNonlinear Programming Methods for Optimal Load Flow," IEEE really an implicit slack variable for each row) really represents an infiniteTransactions PAS Vol. 88, No. 10, July 1969, p. 1530.

7. F. J. Jaimes, A. H. El Abiad, "Optimization by a Sequence of number of linear constraints which comprise the border of the shadedEquality Constrained Problems-Its Application to Optimal Power area.Flows," Proceedings 1971 PICA Conference, p. 219. Taking the original form (Eq. 14) of the optimization problem, let:

8. J. Peschon, D. W. Bree, L. P. Hajdu, "Optimal Solutions InvolvingSystem Security," Proceedings 1971 PICA Conference, p. 210. FFi1 0 Pi

9. J. Peschon, W. F. Tinney, D. S. Piercy, 0. J. Tveit, M. Cuenod, Pi= ci[1 0], Ai L0 cq"Optimum Control of Reactive Power Flow," IEEE T-PAS-87, iOctober 1968, p. 40. constant 1

10. J. Carpentier, "Contribution a' L'Etude des Dispatching Econo- bo = (21)mique," Bulletin de la Society trancasie des Electriciens, July 1962. hhlhm - ao0

l . A. M. Sasson, F. Viloria, F. Aboytes, "Optimal Load Flow Solu-tion Using the Hessian Matrix," IEEE Transactions PAS Vol. 92, The optimization problem becomes:No. 1, Jan./Feb. 1973, p. 31.

12. G. F. Ried, L. Hasdorff, "Economic Dispatch Using Quadratic minimize: C0xl + C2x2 + + CrProgramming," Paper T-73-217-7, presented at IEEE 1973 WinterPower Meeting, N.Y. Alxl + A2X2 + + AI]-n + s bo

13. A. Thanikachalam, J. R. Tuder, "Optimal Rescheduling of Power A + x +for System Reliability," IEEE Transactions PAS Vol. 90, Sept./Oct. = 1,1971, p.2186. -

14. H. W. Dommel, W. F. Tinney, "Optimal Power Flow Solutions," B2x2 b2T-PAS-87, No. 10, October 1968, p. 1866.

15. A. Chang, P. E. Mantey, "Optimization and Computation Applied bto Power System Scheduling and Control," Automatica Vol. 7, rnxn _n1971, p.417.

16. H. Nicholson, M. J. H. Sterling, "Optimum Dispatch of Active and Xi1 >0 X2 >0x..O (22)Reactive Generation by Quadratic Programming," IEEE Transac-tions PAS Vol. 92, No. 2, Mar./Apr. 1973, p. 644.

17. J. Carpentier, "Results and Extensions of the Methods of Differ- T s o t 22ential and Total Injections," 1972 PSCC Proceedings, Vol. 1 The structure of this problem and lends itself to thePaper 2.1/8. Dantzig-Wolfe decomposition technique where a master program gen-

18. R. B. Shipley, M. Hochdorf, "Exact Economic Dispatch-Digital erates a shadow price vector ir used by N subprograms to generateComputer Solution," AIEE Transactions PAS Vol. 75, December potential new columns for the master algorithm. The subprograms solve1956.

19. H. E. Brown, "Interchange Capability and Contingency Evaluation the following problem for each generating plant:by a Z-Matrix Method," IEEE Transactions, Vol. 91, Sept./Oct.1972,p. 1827. minimize: [Fi WAi] xi (23)

20. M. L. Baughman, F. C. Schweppe, "Contingency Evaluation: RealPower Flows from a Linear Model," IEEE 1970 Summer PowerMeeting, Conference Paper 70CP689-PWR. subject to: xis an eleent of the set Xi

21. G. B. Dantzig, Linear Programming and Extensions, Princeton defined previously.University Press, 1963, Princeton, N. J.

22. L. S. Lasdon, Optimization Theory for Large Systems, MacMillan Ti upormi ovdb ipyatigtepaticeetlcsCo., 1970, New York. Tl uporm1 ovdb ipystm h ln ceetlcs

23. G. Hadley, Nonlinealr and Dynamic Programming, Addison Wesley, according to the coefficient in the resulting linear function shown above1964, New York. and applying a normal "Lambda look up" type procedure to determine

24. R. E. Godfrey, T. 'W Hissey, W. 0. Stadlin, "Modern Application each unit's output and thus the plant's output. No composite plant costPrograms and Techniques for Digital Computer Control of Inter- ..-connected Power Systems," Proceedings 1972 IEEE International function need be generated. The units are all represented separately andConvention. may use any cost function representation.

1647

Discussion 1. In the event of overloads due to a contingency, is it possible to actuatecontrols over the system to alleviate the overload? If the answer is

A. J. WoodHeandr.tJ.c Ringilee (Power Taeucthhnrolofgies, Inlc., Schenectady, yes, is it better to wait for the contingency to happen, if it will, andN.Y.): Hearty congratulations to the authors for a clear and compre- then include overload constrants into the dispatch. If the answer is

Whnire pleasedsto find agreement between the findings reported in negative, it implies that the system is in a highly vunerable operatingthis paper and results previously reported on direct methods for de- state, that dependency on continuous controls is risky and that thetermining penalty factors based upon Shipley's model.26 system is suffering from design considerations.termiing pnaltfactrs bsed uon Shpleys modl.262. The effect of external systems is of greater importance and much

The authors' findings that choice of reference bus is not critical to more sfict than sretio o disatchinternalgerationconvergence is good news; the non-sensitivity of the optimum solution mores.gnificant than its reaction to dispatch intemal generationis an expected outcome with proper application of non-linear pro- The authors comments to these considerations would be appreci-graniming algorithms to convex problems. The authors view the effectiof eorsi cs cure

We concur with the authors' arguments that the discrete model for ateda Finally, how do the authors view the effect of errors in cost curvei data. ~~~~Should they be estimated in real time as part of the overall state

developing economic participation factors is superior to the differential of the system?model.

REFERENCE REFERENCESDi- 1. H. Glavitsch, "Economic Load Dispatching and Corrective Reschedul-

26. 0. J. Denison, N. D. Reppen, R. J. Ringlee, "Direct Economic Ds ing using Online Information of the System State," 1973 PICA Con-patch", IEEE Paptr C 73-098-1, Presented at the 1973 Winter ference Proceedings, Paper XIV-B3 pp. 412E420 June, 1973.

Power Meeting,New York, Feb. 1, 1974. 2. G. Douphin, D. Feingold and G. Spohn, "Methods of Optimizing theProduction of Generating Stations of a Power Network," 1967 PICA

Manuscript received January 28, 1974. Conference Record, pp. 133-140, 1967.3. A. Merlin, "On the Optimal Generation Planning in a large Trans-

mission System (The MAYA Model)," IVth. P.S.C.C. Proceedings,Paper 2.1.6, 1972.

4. 0. J. Denison, N. D. Reppen and R. J. Ringlee, "Direct EconomicDispatch," IEEE Paper No. C 73 098-1, Winter Power Meeting, 1973.

H. H. Happ (General Electric Company, Schenectady, N.Y.): This is an 5. R. B. Gungor, N. F. Tsang and B. Webb, "A Technique for Optimiz-excellent paper. It was good to read that the authors' results, without ing Real and Reactive Power Schedules," IEEE PAS-90, pp. 1781-functional constraints, were similar to those of this discussor's and as 1790, 1971.reported in reference 25. The method of the authors' differs from thatin reference 25 in the use of the constrained optimization approach asshown in Figure 2. The Danzig-Wolfe algorithm appears to be a practicaloptimization procedure for the application as described here, and itwould be helpful if the authors would include it more fully in the clo-sure. Their comparison with Quadratic Programming was particularlynoteworthy. The results of other approaches they may have attemptedwould also be of interest. They indicated that no differences were noted cs F theftcmetand questins and omethe oortunityin either the rate of convergence nor in the optimum, when different to commento an riesthe pnt cone the siilr-buses were chosen as the reference. This has been this discussor's experi- to oureqtion (5) and theeoivtioncof the cici-ence also, and agrees with the theory. The authors may nevertheless lyo u qain()adteegnetrdrvto fteBceEcen asho,quandfyagreeswircommenth ,sincthe

theory.horsmaryneverthe1 ents as given by Glavitsch (Sasson's reference 1). First, it must be notedwish to qualify their comments, since even in the ordinary lo'ad flow that the Jacobian matrix used by Glavitsch exten'ds over all busses ofthere are at times detectable differences in convergence depending upon the system, while our Jacobian is that used by Tinney and DommeI I4the choice in the swing bus selected. ln the authors' procedure, iterations and does not include the reference bus. The derivation shown in thethrough the load flow are not made, as seen from Figure 2, and as andjustified by a simulation procedure. Perhaps the authors can indicate e ix of the paper by Glavitsch is the same basic computation givenwhether the option for recycling through the load flow is retained in quation (12) of Tinney and Dommel.There are several reasons why we manipulated the penalty factorseither the off-line or the pl-anned real time version of the program. into the linear equality constraint (or j constraint) shown in our equa-

Manuscript received February 11, 1974. tion (10). We had experimented with various gradient search optimiza-Manuscript received February 11, 1974. tion procedures and had always found them too slow with respect toprogram execution time. We were convinced that one of the mathe-matical programming approaches was required to attain the fast exe-cution time we desired. Incremental linear programming was attemptedand was also abandoned because of poor convergence characteristics.Quadratic programming proved adequate as far as convergence is con-cerned but still took too long to solve. The Dantzig-Wolfe technique has

A. M. Sasson (American Electric Power Service Corp., New York, N.Y.): proved a natural for electric power dispatching because it is not re-The authors have formulated the economic dispatch problem to include stricted by the shape of the cost curves found in practice and, morecontrol and functional constraints with emphasis on a real time applica- importantly, its solution time is very fast.tion. The formulation of. the problem has similarities with that of There are several reasons for requiring a fast optimization algorithmGlavitchl who derives the , coefficients as elements of an eigenvector for a real time dispatch system. For dispatching, the program may beof the load flow Jacobian matrix. There are, however important differ- called upon to reexecute every two to five minutes and is further limtitedences between the two approaches relating to a) stating the problem in by sharing computer time with other teal time functions such as dataterms of the changes in control variables and b) not using penalty func- gathering, display refreshing, etc. Thus, we chose a REDUCED SETtions for functional constraints. It would be of interest to have the solution instead of techniques which of necessity converge an entireauthors' views on the relative merits of the two approaches. By the load flow while calculating the optimum dispatch, that is, the so-calledway, the eigenvector derivation of the j3 coefficients can also be found "optimal load flow" or FULL SET approaches. Dr. Sasson points outin Refs. 2 and 3. In Ref. 4, the 3 coefficients are also used and calcu- that GRG and Hessian approaches could be used -in real time dispatch-lated from a Jacobian submatrix. It seems that this is what the authors' ing since only one iteration would be required to update the optimumEq. (5) should mean and not how it is defined in the paper. In Ref. 5, at each execution. This amounts to a type of tracking optimizationthe ,B coefficients are implicitly used but calculated iteratively from the procedure where execution time is held down by assuming that onlytwo system real and reactive power balance equations. small changes occur between each dispatch. We have not pursued the

We share with the authors the idea of including the power system tracking approach for the following reasons. The chief advantages of ain the dispatch iteration. In this respect, we believe that other ap- constrained optimization approach to system dispatching is to holdproaches, such as the GRG and Hessian, will not only solve just as well constraints when the power system is in trouble. These conditions arein a single iteration but that the time of the iteration is greatly reduced most apt to occur after a disturbance on the system and could causeby obtaining the state from a state estimator. network power flows to become limiting -thus necessitating several

In relation to contingency constraints, theft inclusion in the dis- iterations of an optimization procedure. This argument may also bepatch results in a continuous economic penalty which is difficult to applied to batch versus trackling type state estimation algorithms. Wejustify. There are two considerations to be made with respect to con- definitely prefer batch type algorithms because of their immunity to re-tingencies for the dispatch: initialization and convergence problems when performing an optimiza-

Manuscript received February 13, 1974. Manuscript received April 25, 1974.

1648

tion following a disturbance. Beyond this, a fast batch algorithm has the and most importantly the load increment itself can be reflected in theability to produce participation factors for Automatic Generation Con- resulting participation factors. The possibility also exists of developingtrol (AGC), as well as the advantage of using the same algorithm for real small and large load increment participation factors should such a needtime studies such as interchange transaction evaluation, unit- commit- arise, of course the AGC system would need to be designed to use them.ment, fuel scheduling, and security evaluation. Dr. Happ's request for more information on the Dantzig-Wolfe

Dr. Sasson's comments on contingency constraints raise a very algorithm has also been expressed by other researchers. This algorithmimportant question as to the advisability of incurring an economic was adopted directly from the outline of the technique given in Las-penalty to avoid a possible overload condition. We agree that if suffici- don,22 Chapter 3. The technique itself is not new, having appeared inent time is available to correct an overload by control action then there Dantzig2 1 over ten years ago, however, the algorithm's adaptation tois little justification in constraining the dispatch to guard against it. electric power dispatching is new. We have also extended the algorithmOn the other hand, when a potential contingency would result in an to allow for inclusion of nonlinear separable tonstraints, thus enablingoverload which cannot be corrected by control action and would thus dispatch with regulating margin, heat balance, or nonlinear pollutionresult in the overloaded line being switched out by protective relaying constraints. We plan to publish these results and will try to include ait is definitely a candidate for a contingency constraint. Yes, under more complete description of the basic Dantzig-Wolfe algorithm at thatthese circumstances, the system is in a highly vulnerable state and de- time.pendency on controls is risky, which really gets to the heart of the As mentioned previously, various gradient search techniques asmatter. How much economic penalty is justifiable in the face of such a well as incremental linear programming were attempted before therisk? We agree with Dr. Sasson on the importance and significance of Dantzig-Wolfe algorithm was developed. Though quadratic programmingthe influence of external systems. A steady state equivalent circuit is did not run as fast as the Dantzig-Wolfe algorithm it was an invaluablenecessary to give assurance that the constraint sensitivity coefficients aid in debugging, i.e.1 QP confirmed the then new Dantzig-Wolfe algo-will produce the effect they are supposed to. Beyond this, of course, lies rithm results. We also wish to thank Dr. Kenneth A. Fegley of thethe problem of constraint violations caused by shifts in generation University of Pennsylvania for providing us with the QP algorithm.and/or line outages in external systems. There are well known techniques We would indeed like to qualify our remarks as Dr. Happ suggests,for coordinating systems which are in economic dispatch - how they with respect to the independence of convergence rate and optimumshould be coordinated when attempting constrained economic dispatch when different busses were chosen as the reference. By convergence wehas yet to be fully analyzed. With respect to cost function errors, the meant the number of recycles between the Dantzig-Wolfe optimizationgreatest influencing factor is usually the cost of fuel rather than the ex- algorithm and the load flow supplying the penalty factors. The conver-act shape of the curve. Heat rate data taken at a single operating level gence of the load flow itself was affected by the reference bus location,(preferably at or near full load) should provide a suitable multiplying requiring five base load flow iterations when located at station 15. (seefactor for on-line correction of the cost curve. Estimation of the heat Figure 4) as opposed to four iterations when located at station 6. Afterinput curve over a wide operating range could necessitate uneconomic the base load flow, however, the number of load flow iterations wassystem operation for extended periods of time. The gain in curvature identical for all further optimization-load flow recycles for both ref-accuracy could thus be offset by the cost of obtaining the necessary erence bus locations.data. Finally, Dr. Happ inquires as to our thoughts on the necessity of

We thank Drs. Wood and Ringlee for their comments and would recycling through the load flow. First, we should emphasize that re-like to further elaborate on the use of the discrete model versus differ- cycling through the load flow only updates the penalty factors and theential model for development of participation factors. Many systems flow for constrained lines, areas, etc. Constraint sensitivity factors areare presently experiencing control problems due to severe rate limiting only updated when the network changes configuration. Although theof units to meet pollution control objectives. The result is a controlla- option is available to allow recycling we normally would not use it inbility problem which may benefit from the discrete method of develop- the real time version and would only recommend it for the off-lineing participation factors since rate limits, cost curve discontinuities, version when initializing a base case.

1649