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 DOI: 10.1177/003754979105600408

1991 56: 241SIMULATIONFarshid Forouzbakhsh, Robert M. Deiters and Bahman S. Kermanshahi

Optimal dynamic scheduling of a power generation system to satisfy multiple criteria  

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Optimal dynamic scheduling of a powergeneration system to satisfy multiple criteria

Farshid Forouzbakhsh, Robert M. DeitersSophia University

Faculty of Science and TechnologyDepartment of Electrical & Electronics Engineering

7-1 Kioi-cho, Chiyoda-Ku, Tokyo 102 Japan

Bahman S. Kermanshahi

Tokyo Metropolitan UniversityFaculty of Technology

Department of Electrical Engineering2-1-1 Fukazawa, Setagaya-ku, Tokyo 158, Japan

A computer algorithm for the optimalscheduling of generators in a power system ispresented and tested. The algorithm, based ongoal programming, automatically and dynami-cally schedules the output of each generator inthe system for optimal operation. The optimaloperation can take into consideration multipleobjectives such as economy, security, andreduction of pollution as well as practicalconstraints.

To validate and test the algorithm, an examplesystem of 5 generators, 10 busses, and 11 transmission lines is optimized for two objec-tives: minimal generation cost and minimalemission of nitrous oxides (NOx). Hourlychanges in total power demand in the range of90% to 110% are considered together with aconstraint of maximum permissible total NOxemission. Other practical equality and inequalityconstraints are incorporated into the optimiza-tion algorithm.

The simulation results demonstrate that the

outputs of the generators can be changedsmoothly and dynamically. Furthermore, usingthe algorithm, computer control is practicableeither by direct on-line optimization or by usingthe feasible operation region generated as a smalldata-base by off-line computation.

Keywords: electrical power system, optimalscheduling, goal programming, multiple-objective optimization, dynamic scheduling

Introduction

A power system operates under a number of constraints,and it operates most efficiently when optimized with respectto a number of objectives, or criteria. Considering only asingle criterion, researchers (Gent et al., 1971; Delson, 1974;Kothari et al., 1977), have proposed various techniques foroptimizing the performance of a power system. Multiplecriteria, however, must be considered simultaneously toattain a more meaningful, practical optimal schedule ofoperation. Moreover, these multiple objectives are oftenmutually non-commensurable, although there may be trade-off relations among them.Some researchers have proposed a way of optimizing a

power system for several objectives, but the inter-relationsbetween the criteria are not taken into account (Nanda etal.,1987). One of the authors has applied the technique of&dquo;goal programming&dquo; (Ignizio,1975) to the multiple-objectiveoptimization of a power system for a fixed total powerdemand (Kermanshahi, 1986 and 1989).

In the actual operation of a power system the load powerdemand varies greatly over a 24-hour period and rather largehourly changes are common. Consequently, the optimalscheduling of a power system should be dynamic, that isgenerator outputs should be scheduled to adapt continuouslyfor changing load conditions.An actual power system should be optimized simulta-

neously with respect to multiple objectives, such as costcontrol, security control, quality control, and environmentalpollution control. Our algorithm uses goal programming toachieve the dynamically optimal scheduling of generatoroutputs. To demonstrate the effectiveness of the algorithm weconsider two important mutually non-commensurableobjectives, namely the cost of power generation and the rateof emission of nitrous oxides (NOx) pollutant.

Using a model power system consisting of 5 generators, 10

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busses, and 11 transmission lines, we present the numericalresults for the optimal scheduling of the outputs of the fivegenerators for various system loads. Dynamic optimizationof the generator system under the hourly changes commonlyoccurring in a power system can be carried out easily andflexibly under either operator or computer control by usingthe small data base created by the GP algorithm for variousoperational conditions. These same optimal schedulingresults are used here to simulate the optimal performance ofthe system under some anticipated operational conditionsand restraints.

Statement of the Multi-Objective ProblemHere the various objectives and restraints are expressed in

mathematical form and aggregated into a statement of anoptimization problem.

Fuel Cost

Minimizing the cost of fuel is, of course, a fundamentalcriterion in the operation of a thermal power plant. Usuallyfuel consumption is given as a time-rate of heat production,for example, as mega Btu’s per hour (MBtu/h). Since the costof fuel is measured in cost per Btu, fuel cost is usually givenas the time-rate of cost ($/h).The widely accepted second-order polynomial function for

the cost-rate of power generation is:

in which Pgl is the power output of the i th generator in asystem of N generators. a, f3 , and y are the generation costcoefficients for each generator. I is, therefore the time-rate of generation cost for the entire system. a, /3, and yarepositive coefficients determined to fit the observed character-istics of each generator.

E~nission of Noxious Gases

Local environmental agencies set the maximum allowablerates for the emission of noxious gases produced by fuelcombustion in a thermal power plant. Therefore, the rate ofemission of gases such as the oxides of nitrogen, carbon, andsulphur must be minimized, or at least kept under permis-sible limits. Here we consider only the emission of the oxidesof nitrogen (NOx). The rate of NOx emission is representedby the widely accepted function consisting of a second-orderpolynomial with an exponential term.

Here a , b, c, d, and e are the NOx emission coefficients foreach of the N generators; I D1D170mnent is the time-rate of emissionof the pollutant NOx. a, c, d, and e are positive and b anegative coefficient. They are determined to fit the observedcharacteristics of each generator.

Other Constraints

The optimization of a power system under static, orconstant operating conditions is subject to the followingconstraints:At a load bus:

Here If and Q; are the respective specified values foractive and reactive power; P. and QA are the actual values ofactive and reactive power.

I I

At a generator bus:

where Vis is the specified effective value of the output voltageat the bus, and Viz its actual value.To ensure a balance of active power in the system:

in which PD is the total load consumption of active power andP is the total loss of active power in the transmission linesfrom generators to load. Here, the transmission loss P~ will betaken to be a nonlinear function of the bus voltages of thegenerators.

It should also be noted here that the active and reactive

power outputs of each generator are nonlinear functions ofthe voltages at the generator busses.

Since the active and reactive power outputs of eachgenerator as well as the generator bus voltages have opera-tional upper and lower limits, expressed by

The Multi-Objective Index Function

Equations (1) to (10) express in mathematical form acombination of objectives and constraints from which a multi-objective optimization problem can be formulated. I ---&dquo;and I

-u..1IIfIÐÛ can be minimized independently of one another

- and that is actually done as a preliminary step in ouralgorithm. But finally, one desires to find an acceptablemulti-objective minimum. For this purpose, we consider acriterion function F which has I and I ~_~~ as argu-ments.

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subject to some equality constraints

and some inequality constraints

where h is a vector representing the equality constraints, andg a vector representing the inequality constraints.

Setting up the Goal Programming I

Basically, the &dquo;goal programming&dquo; is a way of optimizing aset of objective functions or functionals with respect to someargument so that they approach as closely as possible to a setof goal values. Expressed in a generalized way, a set (vector)of objective functionals F ( x )

is considered together with a corresponding set (vector) ofgoal values A

In these general terms the GP problem is to optimize a&dquo;distance function&dquo;

with respect to the argument x. Here w. are weightingcoefficients used to give more or less importance to each ofthe various objective functionals F¡ ( x ). This optimization iscarried out under the constraints

where the product

and

Here oa are called &dquo;over-attainment values&dquo; and ua, &dquo;under-attainment values.&dquo; If one minimizes F ( x ) while approach-ing as closely as possible to the goal values, the values ( oa -ua ) are positive and

If one maximizes F ( x ), the values ( oa - ua ) are negative and

Here in our specific application, since both generation costand NOx emission are to be minimized, Equation (17) takesthe form of Equation (20). In terms of our power generationproblem, Equation (20) becomes

in which the &dquo;trial&dquo; set of values are those obtained at each

step of the optimization algorithm while searching for thedesired GP optimum. The &dquo;goal values&dquo; are those optimum(minimum) values obtained when each objective is optimizedindependently, that is without considering its effects on theother objectives. The distance function of Equation (16)becomes

It is, then, this Equation (24) which is to be minimized.

Goal Programming for Dynamic Scheduling of aPower System

For the optimal scheduling of a power system underhourly changes in the total load, GP can be used. Referring toFigure 1, one can see how the goal programming algorithm isused repeatedly over some range of values of the weightingfactors wl and w2. This produces a locus of optimal points inthe generation cost vs. NOx emission region (Figure 3). Thena point on this locus is selected to satisfy other constraints,for example, maximum permissible NOx emission. Thegenerator outputs corresponding to these points are theoptimum values for scheduling.When the load changes, this process is repeated for the

new load value. Thus the system automatically, on-line (ifthe computer response is fast enough), and dynamically seeksthe new optimal generator outputs. Alternatively, a goalregion (as in Figure 3) can be previously constructed off-line -to cover the range of total load variations which can be

expected. Figure 3 shows the goal region computed for ourmodel power system. As can be surmised from Figure 3, thevariations are smooth enough that optimal points in theregion which were not actually calculated can be easilyinterpolated from those that were calculated. Thus, one canprepare a data base off-line that can be used to schedule the

power system dynamically, on-line, and with very littlecomputational delay.A constraint such as maximum permissible NOx emission

rate is shown as a line which further limits the goal region toa &dquo;feasible goal region.&dquo; A practical operating point should, ofcourse, lie in this feasible goal region if possible.

Numerical and Simulation Results for a Test SystemWe tested our GP optimization technique on the model

system shown in Figure 2. (Yokoyama, 1974) The values ofthe coefficients used for each of the five generators are givenin Table 1. Table 2 gives the real and reactive power on eachbus at 100% (base) power demand. Table 3 gives the physical

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Figure 1. Flow diagram for dynamic scheduling

parameters of each transmission line.Table 4 shows the results of the first stage of the computa-

tion in which each objective is independently optimized. The’goal values’ of Equations (22) and (23) are those values of&dquo;Total Generation Cost&dquo; and ’Total NOx Emission’ enclosedin boxes.

Table 5 presents the coordinated optima obtained by GP forspecific values of load demand and specific weighting factors.Since only the relative values of wl and w2 are of concern, onlythe wi weighting factor for generation cost, has been varied.When wI=1()6, the generation cost is given far more impor-tance than NOx emission, and the optimum value obtained

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Figure 2 Single Line Diagram of Test System

for generation cost is very close to its goal value. Thus, inFigure 3 the points for w, = 1()6lie along the lower edge of thegoal region and those for w1=1 1 lie along the upper edge. Forthis model system one can see, therefore, that the coordinated

optimum for 105% load demand cannot, because of theconstraint on NOx emission, be that for wl =106. It must be ata lower value of wl for which the optimal point is just to theleft of the NOx limit line.

In Figure 4 we show how the outputs of each generatoractually change as the total demand varies from 90% to 110%of base load. These simulation results were obtained with the

power system model described and the NOx limit of about0.253 (ton/h) shown in Figure 3. The scheduling strategyused is to give much greater emphasis to minimizinggeneration cost wl =106 as long as NOx is below the limit.After the NOx limit is reached (at about 100% of base load),generation cost is progressively de-emphasized, but only tothe degree necessary to keep NOx at the limit. As the totalload further increases (above about 107%), the NOx limit canno longer be observed. From here on, minimizing NOxemission is given equal weight, that is w, = 1.

This strategy can be easily visualized in the goal region ofFigure 3. The locus of optimal operating points moves upalong the lower edge of the goal region until it reaches theNOx limit. Then it climbs vertically up this limit line until itreaches the upper edge of the goal region. Thereafter it movesupward along the upper edge. It now exceeds the limit onNOx which is, nevertheless, being minimized as much aspossible.

Conclusions

The GP technique has been used to schedule the generatoroutputs in a power system so as to minimize both generationcost and NOx emission. The technique easily incorporatesvarious practical equality and inequality constraints. The GP

Figure 3. Trade-off relations for dynamic scheduling

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Table 1. Generation Cost & NOx Emission Coefficients

Table 2. Specified Bus Data

algorithm is embedded in a total algorithm which dynami-cally optimizes the power system under load demandchanges. This could be executed even with a small personalcomputer once the feasible goal region has been computedand stored as a data base. Such a data base would also allowthe operator to change the outputs of generators smoothly asthe power demand changes.

Furthermore, this technique can easily be used to simulatethe optimal operation of any power system. One need knowonly the configuration corresponding to Figure 2 and theparameters corresponding to Tables 1, 2, and 3.Although we considered only two objectives, other

objectives can be added in a straightforward manner. Ofcourse, then the goal region would become an n-dimensionalspace and the constraints on the objectives n - dimensionalsurfaces. With more than two objectives, the technique ofcomputing the goal region off-line and then interpolating itdynamically on-line would probably be more practical.

AcknowledgmentsWe are especially grateful to our colleagues M. Yabuki, K.

Hieda, and Y. Tanaka, all of Sophia University, for theirfrequent help, discussion, and encouragement-especially inhelping us use the computer effectively.

Table 3. Specified Line Data

References

[Dels 74] Delson, J.K., 1974. "Controlled Emission Dispatch,"IEEE Transactions on PAS, Vol. PAS-93, No.5., pp. 1359-1366.

[Gent 71] Gent, M.R. and Lamont. J. W.,1971. "MinimumEmission Dispatch," IEEE Transactions on PAS, Vol. PAS-90,No.6., pp. 2650-2660.

[Igni 75] Ignizio, J.P.,1975. Goal Programming and Extensions,Lexington Books, London.

[Kerm 86] Kermanshahi, B.S. and Yokoyama, R., 1986. "DynamicGeneration Rescheduling Based on Environment-Economy VectorCriteria, "Proceedings of IEEE MONTECH’86 Conference on ACPower Systems, No. TH0152, pp. 126-133.

[Kerm 89] Kermanshahi, B.S., August 1989. "Coordination ofEconomy and Emission Criteria for Optimal GenerationDispatching of Large-Scale Power Systems,"Proceedings of IFACInternational Symposium on Power Systems and Power Plant Control,Seoul, Korea, No. D-11.

[Koth 77] Kothari, D.P., Maheswari, S.K. and Sharma, K.G., 1977."Minimization of Air Pollution Due to Thermal Plants," Journal ofthe Institute of Engineering of India, Vol. 57, pp. 65-68.

[Nand 87] Nanda, J., D.P. Kothari, and Lingamurthy, K.S., 1987."Economic-Emission Load Dispatch Through Goal ProgrammingTechniques," Proceedings of IEEE PES Winter Meeting, NewOrleans, Louisiana, Paper No. 87 WM 236-3, Feb 1-6,1987.

[Yoko 74] Yokoyama, R., Jan.1974. "A Unified Approach toDetermine Sensitivity Constants for Power System Analysis andControl," Journal of the Institute of Electrical Engineering in Japan,Vol. 94, No.1, pp. 17-24.

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Table 4. Independent Optimization of Objectives

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Table 5. Coordinated Optimal Points & Generator Outputs

FARSHID FOROUZBAKHSH was bom in

Tehran, Iran in 1956. He graduated with a B.S.in electrical engineering from TehranUniversity in 1983. For two years, whileuniversities in Iran were dosed, he worked atthe design and istallation of large-scaletransformer substations. In 1986 he receivedthe M.S. degree in electrical engineering fromTehran University. In the same year, awardeda scholarship by the Japanese goverment, hecame to Japan. After language study and a

year of research at Shibaura Institute of Technology, he entered thedoctoral program at Sophia University in Tokyo. He is now workingin the planning, control, optimization, and operation of large-scalepower systems. He is a member of the IEE and of the IEIE of Japan.

ROBERT M. DEITERS is a U.S. citizen whocame to Japan in 1952 as a Catholic mission-ary, a career which he pursues now as a

professor in the Electrical-ElectronicsEngineering Dept. of Sophia University, aCatholic-sponsored university in the heart ofTokyo. Besides theological studies, hereceived the degree of M.S. in electricalengineering from Marquette University in1963 and the degree of Doctor of Engineeringfrom Tokyo University in 1968. A long-time

member of SCS and of the Japan Society for Simulation Technology,he started out in control systems and analog computation and laterdid work on error compensation in hybrid computer systems. Nowhe works in the field of digital simulation, as well as directingstudents working in the area of distributed computer networks.

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Figure 4. Scheduled Optimal Generator Outputs

BAHMAN SHOUSHTARI KERMANSHAHIwas bom in Kermanshah, Iran in 1956. In 1978he graduated from Iran Air College in Tehran.Coming to Japan, he entered Asia Universityin Tokyo where he received a diploma inJapanese literature in 1981. In 1985 he receivedthe B.S. degree in electrical engineering fromNikon University. He received his M.S. andPh.D. degrees in electrical engineering fromTokyo Metropolitan University in 1987 and1990 respectively. His current research

interests are in the optimization of power systems, and in the theoryand applications of fuzzy sets and neural networks. He is a memberof the IEE and of the IEIE of Japan.

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