monto carlo computation of power generation production cost under unit commitment constraints

8/3/2019 Monto Carlo Computation of Power Generation Production Cost Under Unit Commitment Constraints

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Monte Carlo Computation of Power Generation Production Costunder Unit Commitment ConstraintsJ. Valenzuela & M. M a z u m d a r

Industr ia l Engineer ing Depar tm entUniversity of PittsburghPittsburgh, PA 15261

Abstract: A highly efficient Monte Carlo procedure forestimating the mean and variance of electric powerproduction cost under unit commitment constraints isproposed. Such estimates are useful in making near-termoperational decisions. We show that for this purpose it isessential to consider the stochastic processes associated withgeneration unit outages. When unit commitment constraintsare taken into account in the production-costing model, acombined combinatorial and continuous optimizationproblem needs to be solved at each hour for each MonteCarlo run. This tends to make the computations quite long.We propose a method to reduce the required number ofMonte Carlo runs by using a control variate technique onbatches. We present the results for a small electric powergenerating system w here we achieved a reduction of samplesize by a factor of 10 for the variance and by a factor of 42for the mean.Keywords: power generation, production costs, M onte Carlo,unit comm itment, stochastic models.

costs because it has the capability of capturing the unitcommitment considerations such as start-up costs, shutdowncosts, minimum up and down times. Typically, in thesesimulations, the time-varying aspect of the load is explicitlymodeled in a deterministic fashion, and th e randomgenerating unit outages are sampled fkom distributionsconsidering their FORS only [ 2 ] . t is the purpose of thispaper to point out that the steady state assumptions inherentin the use of the FOR index may lead to incorrect resultsparticularly when the initial operating status of th e gene ratingunits is know n. Section 11 illustrates this fact by p roviding anum erical exam ple for a small generation system.When o perating constraints are considered in the produ ction-costing model, a combinatorial optimization in the form ofcost minimization is carried out for each Monte Carlo run.This tends to make the computations quite lengthy. SectionIII proposes a variation reduction scheme [3,5],whichconsiderably reduces the required number of Monte Carloruns.r[. INCORPORATING FREQUENCY AND DURATIONOFUNIT OUTAGES1. INTRODUCTION

Production costing models are widely used in the electricpower industry to forecast the cost of producing electricity.These forecasts are used as inputs in financial planning, fuelmanagement, and operational planning. It is the purpose ofthis paper t o describe a Mon te Carlo procedure for estimatingthe mean and variance of production costs that has a muchhigher degree of precision than that of 8 direct Monte Carloprocedure and will require much fewer runs. We consideroperating constraints and start-up costs that typically appearin the unit commitm ent problem. We also demonstrate thatin these simulations it is imperative that the stochasticprocesses associated with generator forced outages andrepairs be modeled. An illustration of such a simulation ofproduc tion costs using stocha stic mod els is given in [ 6 ] .The cost of producing electric power is a random variablebecause it is dependent upon the uncertain mix of availablegenerators and the uncertain demand. The well-knownBaleriaux formula [ l ] accounts only for the uncertaintyresulting from the forced outages of generating units andprovides a form ula for the expected value of production costsin the stead y state. It does n ot consid er operating constraints.Monte Carlo chronological simulation of production costs isfrequently carried out for predicting short-term production

We illustrate here the importance of considering thestochastic processes with the help of an example for a smallgenerating system taken from Wood and Wollenberg [lo].Th e values of the cost parameters and those related to the unitcommitm ent constraints are reproduced in Table 1. The lasttwo columns give the assumed values of the mean up anddown times for each generating unit, which do not appear in[lo]. The hourly load values are given in Table 2, and areconsidered to be deterministic. We assume that this loadpattern is repeated every eight hours.A . The Production Costing Model

It is assumed that the costs are being calculated for a po wergeneration system consisting of N generating units over atime interval [O,T]. (For our example, N = 4.) The followingadditional assumptions are m ade:1. The generating units are committed such that the totalproduction cost is minimized subject to the constraintson minimum up time, minimum down time, s tartup Costs,no load costs, and spinning reserve. The committed unitsare loaded using an economic dispatch algorithm. Th e ithunit has a maximum capacity P, (MW), minimumcapacity P y MW ), no load cost a, ($), variable energy0-7803-5935-6/00/$10.00 (c ) 2000 IEEE 927


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cost b, ($/MWH ), minim um up time tyP (h), minimumdown time t? (h), avera ge full load cost Ki ($/MWH),mean time to failure hi- (h), mean time to repair kv1h) ,and a forced outage rate, pi , i=1,2,...,n. The cost ofunserved energy is F($/MWH).The operating state of each generating unit i follows atwo-state continuous-time Markov chain, Yi(t), withfailure rate hi and repair rate pi.The forced outage rate p,is related to these quantities by the equationpi =ai (a, + p , ) . For i f j, Yi(t> and Yj(s) arestatistically independent for all values of t and s.The chronological load, u(t), is assumed to be completelyknown (deterministic) for each hour t over the timehorizon [O,T].Transm ission losses are considered to be negligible.All units are assumed up at the beginning of the timeinterval.

!........_._.......-..........Figure I :Tnnsition states.The operating states of the generating units are firstclassified according to their availabilities (Fig. 1). If a unit i savailable (up), it may be in either of two states: committed(on) or decommitted (off). The latter state indicates that the

unit is available but is not being used for the purpose ofgenerating power. If a unit is unavailable (down), it is underrepair, and upon the completion of such repair it proceeds tothe decommitted or off state. The transitions from anavailable sta te to the unavailable state and vice versa occ ur inaccordance with the assum ed failure and repair rates of ea chgenerating unit.In the Monte Carlo simulation, a unit commitment problem(UCP) is solved at each hour, t, during each run of the M onteCarlo simulation. The units with status available at hour tare the only units considered for being committed at thathour. We use forward dynamic programming [lo] todetermine the optimum commitment schedule for a timehorizon of eight hours. Table 3 gives estimated values of themean and variance of production costs for the generationsystem of Tables 1and 2 for a period of 168 hours based on1000 Monte Carlo runs for two different scenarios. The firstscenario (unit commitment constraints or UCC) refers to thesituation in w hich the units are com mitted and dispatched inaccordance with their operating constraints. The secondscenario (merit order loading with average full load cost orAFLC) refers to the situation in which th e units aredispatched in accordance with a merit order based on the

average full load cost and the operating constraints are notconsidered. This table shows the estimated mean andvariance of the production cost for a set of several differentvalues of the mean times to failure and repair for thegenerating units. These values are obtained by multiplyingthe initial values of these parameters as given in Table 1by aconstant k. This multiplication does not change the assumedforced outage rates for the units. Yet, it is found in Table 3that as the value of k is increased, the estimated values of themean and variance change significantly. This table clearlydemonstrates that the mean and variance depend not onlyupon the unit forced outage rates but also on the frequencyand duration of unit outages. To further illustrate this point,we run the Monte Carlo simulation using just the forcedoutage rates [2]. The results are given in Table 3 in the toprow labeled FOR. We observe that when we take in toaccount the initial operating status of the gene rating units, theestimates based on the FOR may be quite different from thoseobtained using th e actual mean up and down times of thegenerating units.The reasons why a consideration of the stochastic processesunderlying the frequency and duration of generation outagesis necessary for the computation of the variance has beendescribed in detail in [4] and 173. For the computation of themean, we have assumed that the initial operating status of thegenerating units is known. Thus, it is important that thetransient conditions at the beginning of the interval beaccounted for. It is incorrect to d o the computations via thelorced outage rate, which is a steady state measure .

Table 3: Estimate o f the m ean and variance of productioncost for 168 hours (1,000 samples).1 k I Mean I Mean IS2[UCClIS2[AFLCl I

111. A VARIANCE REDUCTION SCHEME FOR THEMONTE CARLO PROCEDUREThe particular variance reduction technique adopted is theControl Variate method, which was also used in a similar

context by Breipohl et al . [3]. It consists in generating inaddition to the primary random variable of interest anotherrandom variable that is highly correlated with the former butin whose case the expected value can be analyticallycomputed. In ou r application, we use the same set of randomnumbers to simultaneously produce two estimates ofproduction cost for each Monte Carlo run: (a) one usingoperating constraints, and (b) the other ignoring these

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Table 1 :Unit characteristics.

Table 2:Load profile.Hour 1 1 1 2 1 1 4 1 5 1 1 1L o a d ( M W ) I 4 5 0 I 53 0 I 6 0 0 I540 I 400 1 280 I 2 9 0 I 5 0 0

constraints and using a strict merit order loading based onaverage full load cost. Figure 2 shows that these twoestimates are highly correlated. To obtain estimates of thevariance, we subdivide the entire ensemble of n Monte Carloruns into m batches each consisting of k mutually exclusiveruns (n=mxk) [8]. For each batch, we compute the samplemean and variance of production costs com puted according tothe above two sets of assumptions. Somew hatserendipitously we found that the batch variance of the costsunder scenario (a) is highly correlated (see Figure 3) with thesample batch average under scen ario (b). A similar techniquewas used by Mazumdar and Kapoor [5 ] for reducing thevariance of production costs under scenario (b). They havealso provided reasons for the observed correlation.

IV. NUMERICALRESULTSThe Monte Carlo simulation was done with the followingparameters: n=2,000 replications, m=50 (number of batches),k=40 (batch size), T=24 hours (load profile from Table 3 wasrepeated three times), F=$135 (penalty cost $/MwH), andN=4generating units.From the simulation output, we obtained an approximate95% confidence interval for the mean, variance, and standarddeviatio n of the production costs under scena rio (a):

Mean: [230,117 - 232,0731Variance: [334,420,915 657,387,0481Standard deviation: [18,287 - 25,6391

190 200 210l3uwionCast&AFU:Figure 2: Correlation between ProductionCosts with OperatingConstraints and Merit Order Loading with AFLC.

Figure 3: Correlation betw een and S: .

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Evaluating the analytical expression given in [9], we obtainIhe expected value of production cost under scenario (b) to be$214,918.Using the same simulation output, we obtained anapproximate 95%confidence interval for the mean, variance,and standard deviation of the production costs under scenario(a) using the control variate technique:Mean: [230,821 - 231,1211Variance: [464,682,585- 564,594,9321Standard deviation: [21,556- 23,7611

These intervals are much shorter than the ones obtainedwithout using variance reduction. In order to obtain the samewidth for the confidence interval as the one above usingdirect Monte Carlo, we would need 84772 replicationsinstead of 2000 for estimating the mean, and 522 batchesinstead of 50 for estimating the variance. For this example,use of this particular control variate reduces th e sam ple sizeby a factor of 42 when we are estimating the mean and by afactor of 10when we a re estimating the variance.V . CONCLUSIONS

The results of this paper show that an accurate evaluation ofthe mean and variance of electric power production costsrequires the explicit modeling of the stochastic processesassociated with the generator forced outages and repairs.Analytical computation of these quantities is difficult andMonte Carlo appears to be the only feasible alternative.

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Monte Carlo simulation might also become quite timeconsuming when operating constraints are considered andeach run requires solving a combinatorial optimizationproblem. To reduce the computational burden of Monte Carlothis paper has proposed a variation reduction procedure usingthe control variate technique. For this procedure we havechosen the con trol variate to be th e production cost based onpriority loading. A prerequisite for the application of thistechnique is that an analytical expression for the expectedvalue of the control variate is available. We have obtainedthis expression for a small system us ing direct enumeration ofoutage states. For a larger system it will be necessary to use amore refined algorithm for carrying out this analytical

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computation. Ou; example shows- that the chosen controlvariate results in a significant sample siz e reduction. [ I O ] Wood A. and B. Wollenberg. Power Generation-Operation and Control,Wiley & Sons, Inc., Second Ed., 1996, pp. 145.

Lee, F.N., M. in, and A.M. Breipohl, Evaluation of the Variance ofProduction Cost Using a Stochastic Outage Capacity Model, IEEETrans. Power Syst;, 5, 1990,pp . 1061-1067.Mazumdar M. and A. Kapoor, Variance Reduction in Monte CarloSimulation of Electric Power Production Costs, American Jour nal ofMathematical and Management Sciences, 17,3 & 4,1997 , pp-239-262.Mazumdar, M. and A. Kapoor, Stochastic Models for PowerGeneration System Production Costs, Elec. Pow er Syst. Res., 35.Ryan, S.M. nd M. Mazumdar, Effect of Frequency and Duration ofGenerating Unit Outages o n Distribution of System Production Costs,IEEE Trans. ower Syst.. 5, 1990, pp. 191-197.Schmeiser B., Batch Size Effects in the Analysis of SimulationOutput, Operations Research, 30.3, May-June 1982, pp. 556-567.Valenzuela, J . and M. Mazumdar, Stochastic Monte CarloComputation of Power Generation Production Cost under OperatingConstraints, submitted to IEEE Tmns. Power Syst. (under review).

1995, pp. 93-100.

VI. REFERENCESBaleriaux, H., E. Jamoulle, and Fr. L. de Guertechin, Simulation de1Exploitation dun Parc de Machines Thermiques de ProductiondE1ectricite Couple a des Stations de Pompage, Revue E (editionBreipohl A., F. Lee, and J. Chiang, Stochastic Production CostSimulation, Reliability Engineering and System Safety, 46, 1994, pp.Breipohl A., F.N. ee, J . Huang, and Q. Feng, Sample Size Reductionin Stochastic Production Simulation, IEEE Trans. Power Syst-, 5, 3,

SRBE), 5 , 1967, pp. 22.5-245.IO1 107.

August 1990, pp. 984-992.

VII. BIOGRAPHIESMainak Mazumdar received his Ph.D. in Industrial Engineering fromCornell University in 1966. He worked as a Research Scientist at theWestinghouse R&D C enter during the period 1966-1981. Since 1981, he hasbeen a faculty member in the Department of Industrial Engineering at theUniversity of Pittsburgh.Jorge Valenzuela received his BS in Electronic Engineering from NorthernCatholic University in Chile in 1984. He obtained an MS degree in lndustrialEngineering at Northern Illinois University in 1996. Currently he is a Ph.D.candida te at the University of Pittsburgh.

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monto carlo computation of power generation production cost under unit commitment constraints

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