A Stochastic Optimization Formulation of Unit Commitment With Reliability Constraints

2200 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013

A Stochastic Optimization Formulation of UnitCommitment With Reliability ConstraintsPeng Xiong, Student Member, IEEE, and Panida Jirutitijaroen, Senior Member, IEEE

Abstract—This paper proposes a new formulation for a sto-chastic unit commitment (UC) problem that incorporates twomajor and common sources of uncertainty in short-term gener-ation scheduling, namely, unavailability of generators and loaduncertainty. The objective is to minimize operating cost andexpected loss of load cost subject to reliability constraints in termsof loss of load probability. The problem is first formulated as atwo-stage recourse model in stochastic programming frameworkwhere generator unavailability is expressed by a discrete set ofoutage scenarios and system demand is set to be a nominal valuein power balance equations. Then, load uncertainty is representedas a continuous random variable in loss of load probability con-straint, which is approximated by a mixed-integer piecewise linearfunction and integrated to the second stage problem. As a result,the UC problem has a much smaller dimension as compared tothe original two-stage recourse model. The proposed formulationcan be solved in a timely manner even though the formulationrequires some extra binary decision variables. Although loss ofload probability constraint is approximated in the optimizationproblem, simulation results show that the optimal solutionsyield desired reliability performance. Several case studies areconducted to examine the impact of reliability requirements andsystem uncertainties on UC decisions.

Index Terms—Mixed-integer linear programming, reliability,stochastic programming, uncertainty, unit commitment.

NOMENCLATURE

Indicesand Sets

Indices of linear functions that approximate theconvex generation cost function.

Set of indices of linear functions thatapproximate the convex generation costfunction.

Indices of linear functions that approximate theconvex complementary cumulative distributionfunction.

Set of indices of piecewise upper bounds of theconvex complementary cumulative distributionfunction.

Indices of generators.

Manuscript received September 19, 2012; revised March 01, 2013 and May22, 2013; accepted July 18, 2013. Date of publication November 04, 2013; dateof current version November 25, 2013. This work is supported by Ministryof Education Academic Research Fund, Grant No. WBS R-263-000-691-112.Paper no. TSG-00584-2012.The authors are with the Department of Electrical and Computer Engineering,

National University of Singapore, 119077, Singapore (e-mail: [email protected]; [email protected]).Digital Object Identifier 10.1109/TSG.2013.2278398

The feasible operating region of units in scenario, defined by ramp-rate limitations.

The feasible operating region of units, definedby minimum up/down constraints.

Indices of time steps.

Indices of generator outage scenarios.

Set of all outage scenarios.

Func-tions

The complementary cumulative distributionfunction (CCDF) indicating the probability thatthe actual power demand at time step is higherthan a specified value.

Generation cost functions of unit .

Start-up/shut down cost functions of unitduring time step .

Con-stants

Coefficients of piecewise linear upper boundsof the complementary cumulative distributionfunctions.

Value of load curtailment (in dollars permegawatt hour).

Shut-down cost of unit (in dollars).

Start-up cost of unit (in dollars).

The forecasted electricity demand during timestep (in megawatts).

The intersection point of the linear piecewiseupper bound and the approximation

The maximum capacity of unit , in scenario, it equals zeros if unit is in the failure state(in megawatts).

The minimum capacity of unit , in scenario, it equals zeros if unit is in the failure state(in megawatts).

The number of generators.

A sufficiently large constant.

Coefficients of linear piecewise generation costfunction of unit .

1949-3053 © 2013 IEEE

XIONG AND JIRUTITIJAROEN: A STOCHASTIC OPTIMIZATION FORMULATION OF UNIT COMMITMENT 2201

The maximum reserve provided by generator(in megawatts).

The ramp-down limitation of unit .

The ramp-up limitation of unit .

The shut-down ramping limitation of unit .

The start-up ramping limitation of unit .

The number of time steps of the decision period.

The minimum-down time of unit .

The minimum-up time of unit .

The pre-specified upper limit of the probabilitythat load loss occurs in the decision horizon.

The standard deviation of total electricitydemand at time step .

The probability of outage scenario .

DecisionVariables

The total available capacity scheduled duringtime step , in scenario (in megawatts).

Power output of unit during time step , inscenario (in megawatts).

Load loss during time step , in scenario (inmegawatts).

The upper bound of loss of load probability forall time steps in scenario .

Reserve provided by unit during time step ,in scenario (in megawatts).

Binary variables indicating the status of unitduring time step .

The upper bound of loss of load probabilityduring time step , in scenario .

Binary variables indicating in scenario ,whether the available capacity is larger thanor not.

Others

The random variable indicating the actual totalelectricity demand at time step .

The event that load loss occurs during time step, in scenario .

I. INTRODUCTION

U NIT commitment (UC) problem is a decision makingprocess that determines the optimal schedule (on/off de-

cisions) and the optimal output of each generator to minimizetotal cost over a specified time horizon, typically 24 hours.These decisions have to be committed prior to unforeseencircumstances of future economical and operational data, such

as unexpected changes of electricity demand or a sudden failureof generating units. The performance of obtained decisionswill be greatly improved if information of these uncertaintiesis properly modeled into the UC problem.Stochastic programming is one of the frequently used tech-

niques in optimization with uncertainty. The main advantage ofstochastic unit commitment over its deterministic counterpart isthat the optimal decision ensures the minimum total cost in anexpected value sense [1]–[5]. A common practice is to representsystem uncertainty by a large number of scenarios which resultsin a very large deterministic equivalent problem. The problemmay grow exponentially when multiple sources of uncertaintyare considered. Sample Average Approximation (SAA) is usu-ally adopted to reduce the number of scenarios [6]–[9]. Suchapproximations are sometimes paired with decomposition algo-rithms [10]–[13] to further enhance solution efficiency.Besides economic efficiency, security and reliability are other

concerns in UC problems. The risk of load loss, usually ex-pressed as expected energy not served (EENS) or loss of loadprobability (LOLP) in the decision horizon, should be properlymanaged. A common tool for limiting the risk of load loss ischance-constrained programming (CCP). This method enforcesconstraints in the form of chance of attainment [14] so that someconstraints are satisfied with desired probabilities. CCP methodhas been widely used to address uncertainties in power systemsoptimization, such as transmission network expansion [15], andprobabilistic optimal power flow [16].CCP may not be an appropriate tool for unit commitment

problem when detailed power flow models are incorporated.The power flow models introduce joint chance constraints tothe problem which are extremely difficult to solve as multi-di-mensional integration is required to compute loss of load prob-ability. This is why chance-constrained UC problem is eitherbased on a single-bus assumption [17], or is solved by sampleaverage approximation (SAA) techniques [18], [19]. The lattermethod, however, requires at least one binary decision variablefor each realization to indicate an event of load loss. A numberof samples are usually very large in order to achieve acceptableconfidence levels [20], [21].In this paper, a new formulation of a stochastic UC problem

is proposed to capture stochastic nature of power systems underboth generation unavailability and load uncertainty. In the pro-posed formulation, unit failures are represented by a number ofoutage scenarios so that this type of uncertainty can be incor-porated into a two-stage stochastic programming framework.Although the randomness of load can be modeled as discretescenarios [22], a combination of both sources of uncertaintyleads to a much larger problem dimension. Load uncertainty isthen represented as a continuous distribution function in loss ofload probability constraint. This reliability constraint is approx-imated by a mixed-integer piecewise linear function and is thenintegrated to the second stage problem. Similar to [23]–[25], asingle-bus assumption is adopted here to attain tractable solu-tions.It can be seen that this formulation has a smaller problem di-

mension because the number of scenarios is greatly reduced.Besides, a LOLP function is explicitly defined so the systemoperators are able to directly adjust a level of system reliability


to an appropriate value. Simulations are conducted to demon-strate that the reliability indices are well within a pre-specifiedvalue.

II. STOCHASTIC UNIT COMMITMENT FORMULATION WITHEXPLICIT RELIABILITY CONSTRAINTS

A unit commitment problem is formulated to minimize theexpected total cost while satisfying a predetermined reliabilityrequirement. Two types of uncertainty in short-term unit com-mitment scheduling namely generation unavailability and loaduncertainty are considered. Generation unavailability is mod-eled as a set of outage scenarios while the impact of load uncer-tainty is restricted by reliability constraints.

A. A Two-Stage Recourse Model for Stochastic UnitCommitment Considering Unit Outages

Generation unavailability is represented by a set of unitoutage scenarios in the two-stage recourse model. If a unitis in a failure state, its output is forced to be zero and itsmaximum and minimum capacities are set to zero. The objec-tive function is the expected total cost over all scenarios. Itincludes start-up/shut-down cost, power generation cost andcost of load loss. The unit commitment decisions are usuallymade day-ahead or many hours earlier before the generationunavailability is realized. They are referred to as first-stagedecisions. Other decisions such as power dispatch, reserve mar-gins, and some intermediate variables are determined after theobservation of possible unit failures. Variables in this categoryare defined as second-stage decisions. The two-stage stochasticunit commitment formulation considering unit outages is for-mulated as follows.

(1)subject to

(2)

(3)

(4)

(5)

(6)

(7)

where and are power production cost and start-up/shut-down cost function of unit , respectively. The mathemat-ical expressions of all cost functions are given as (8)–(13).

(8)

(9)

(10)

(11)

(12)

(13)

In the expressions above, and are the start-upand shut-down cost coefficient of unit . For simplicity, it isassumed here that and are fixed constants.Constraints (2) are power balance equations. Generation and

reserve capacities of each unit are defined by (3)–(5). The setand in (6)–(7) give feasible regions defined by minimum upand minimum down time constraints and ramp rate limitations[26]. Detailed formulation of and is given in Appendix A.Electricity demand in power balance equations is set to be itsexpected value. It should be noted that load uncertainty is notconsidered in this part of the formulation. The effect of loaduncertainty is modeled in reliability constraints in the followingsection.

B. Reliability Constraints Formulation and Approximation

The objective of reliability constraints is to ensure that suf-ficient reserve is maintained so that the probability of havingpower deficiency in the decision horizon is lower than a pre-se-lected level , i.e., the maximum probability of loss of load.Let be the event that load loss happens during the time

interval in scenario and be the probability of outagescenario . Assuming that uncertainties in generation outagesand load are statistically independent, the reliability requirementthat restricts load loss probability of the entire decision horizonto be below can be expressed as (14).

(14)

Suppose that is a random variable indicating the value oftotal electricity demand at time . Load loss will not happenduring time interval , in scenario , if the total available ca-pacity on line, denoted by in (15), is larger than the randomdemand .

(15)

Let be a complementary cumulative probability dis-tribution function (CCDF) that the random demand is higherthan . By (16), it can be seen that is the same asthe probability that load loss occurs during time interval , inscenario . is a non-increasing function because higheravailable generation capacity leads to lower probability ofload loss, shown as the dash line in Fig. 1.

(16)

Since the function is non-convex and non-linear, it isthen approximated by its upper bound, shown by the solid linein Fig. 1. Such approximation can be applied to all unimodaldistributions. As an example, it is assumed here that the loaduncertainty follows a normal distribution [22], as in (17).

(17)


Fig. 1. Complementary cumulative distribution function and its upperbound.

where is the expected value of . It is considered as the fore-casted demand during time interval . is a standard deviationof load uncertainty at time step . Information on and canbe obtained from historical load data.For normal distributions, if is larger than a threshold de-

mand , the function can be approximated by a linearpiecewise function denoted as an upper bound of the CCDFshown in Fig. 1. In scenarios with any smaller than , thesystem is considered as very unreliable, because load loss maystill occur even if the actual load is much lower than the fore-casted value. The upper bounds of for such cases, there-fore, are set to be one. It can be seen from Fig. 1 that this approx-imation is most conservative when the total available capacityis close to the threshold value . It should also be noted thatis almost always higher than . In fact, it is more likely to

be larger than the expected demand in order to maintain de-sired system reliability. Therefore, this approximation is not asconservative as it appears to be, especially when very tight re-liability requirement is adopted.It is also assumed that forecasting error of electric demand in

each time step is probabilistically independent, which a naturalresult of univariate time series forecasting methods. Accordingto Boole’s inequality and constraint (16), if is higher than, the probability of load loss in scenario satisfies (18).

(18)Since Boole’s inequality is a tighter approximation, this for-

mulation is likely to generate conservative solutions, especiallywhen the electric demand is a significant source of uncertainty.If is less than the threshold demand , as the upper bound

of equals one, the upper bound of load loss probabilityfor all time in such scenario is supposed to be one. A set of bi-nary variables is used to indicate whether the total availablecapacity is larger than , so that the two situations afore-mentioned can be combined. Let be the intermediate variablefor the upper bound calculation of load loss probability for the

TABLE IFUEL COST OF TEST SYSTEM

TABLE IIPROBLEM DIMENSIONS

entire decision horizon in scenario . The reliability constraintsare thus formulated as follows.

(19)

(20)

(21)

(22)

(23)

(24)

where is a sufficiently large constant, which can be set aslarge as the maximum total capacity of all units. If the binaryvariable equals to one, must be higher than in sce-nario , suggested by constraint (19). In this case, the constraint(20) is equivalent to constraint (15) and load loss probabilityis defined by constraints (21) and (22) where the Boole’s in-equality applies. Otherwise, constraint (19) becomes inactive,and constraints (20)–(22) always hold. Load loss probabilityequals one as defined by (23). Considering both conditions, ifconstraint (24) is satisfied, the probability of load loss in the en-tire decision horizon is lower than the pre-selected value .It is observed that in (24), is unlikely to be one if the prob-ability is larger than , therefore we can force undersuch circumstance to be one, as implied by (23), so that the solu-tion time caused by branch-and-bound process may be reduced.Note that unlike the deterministic reserve criterion, the

reserve margin here is decided completely in a probabilisticfashion according to the probability distribution of demandand a predefined level of reliability requirement. If the outagecontingencies are properly expressed by selected scenarios, theprobability of no load loss occurring in the decision intervalshould be limited under the required level . Since thisformulation introduces some approximation into reliabilityrequirements, the true values of load loss probability are stillnot known. In the following sections, load loss probability,denoted by , is simulated to assess the actual reliabilityperformance of obtained decisions.


TABLE IIIPERFORMANCE UNDER VARIOUS RELIABILITY REQUIREMENTS

III. CASE STUDIES

A. Base Case

The proposed approach is implemented on the IEEE Relia-bility Test System—1996 [27] for a spring weekday, where peakload of this day is 2565 MW. The fuel cost data is shown inTable I. The cost of load shedding is assumed to be 3000$/MWh.The outage contingencies are ranked based on the amount of

capacity loss due to unit failures. Eight single-outage and sixdouble-outage contingencies with the largest capacity loss areselected as the scenario set. Probability of each scenario is cal-culated according to a two-state Markov process model whileassuming that each unit outage is mutually independent. De-tailed procedure can be found in [1]. It is observed from nu-merical studies that scenarios with smaller capacity loss haverelatively insignificant impact on system operation, so they arenot included in the outage scenario set in order to reduce com-putational complexity. Load uncertainty is modeled as the errorterm deviated from the forecasted demand. It is assumed thatthis error term in each hour follows normal distribution withstandard deviation of 0.03 of the total load. The CCDFof load is approximated by a four-segment piecewise linear ex-pression. Numerical values of the parameters and aregiven in the Appendix B.Hydro and nuclear units as well as generators with 350 MW

capacity are defined as must run units. Since the commitmentof fast-start units is not decided in the first-stage, the start-up/shut-down decisions for fast-start units are defined as contin-uous variables ranging from 0 to 1, as suggested in [28]. Theproblem dimension is presented in Table II. In the base case,the probability that load loss occurs in these 24 hours is set tobe lower than . The relative dual gap is set to be0.1%. It takes 295.9 s to solve the base case problem and theoptimal objective value is $443 919.03.The actual performance indices, such as expected cost, proba-

bility of load loss, and EENS, are computed using Monte Carlosimulations considering the original combined impact of unitoutages and load uncertainty. An economic dispatch problem issolved for each randomly sampled realization to calculate thecost and power deficiency measures. In this section and subse-quent case studies, all performance metrics are calculated from

TABLE IVVALUES OF PARAMETERS AND

repeating simulation 5 times, and each run has a sample size of20 000.It is found from the simulation that the expected operating

cost is $443 290.66. This simulated cost is lower than the ob-jective value of the optimization problem by 0.14% as a resultof the inaccuracy caused by reduced unit outage scenarios. Theprobability of load loss is 0.00318, which is lower than the re-quired upper bound , and the EENS throughout the deci-sion horizon is 1.53 MWh.

B. Influence of Reliability Requirements

The main contribution of the proposed formulation is that de-sired reliability performance can be achieved by adopting ap-propriate reliability requirement . The effects of differentreliability requirements are investigated by several numericalcase studies with changing from 0.001 to 0.010. Otherparameters are set to be the same as the base case. Results aregiven in Table III. The first two columns present the values of

and the optimal objective values obtained from the pro-posed UC optimization problem. The other columns show theperformance of the optimal decision in terms of reliability mea-sures and expected cost from Monte Carlo simulation.Results suggest that the system is properly protected from the

risk of load loss because reliability indices are alwaysbelow a pre-selected upper bound of . It is also observedthat the level of conservatism in UC scheduling can be adjustedby adopting different reliability requirements. This trade-off be-tween reliability and economic efficiency is demonstrated byFig. 2. In the case of , the probability of load lossduring the entire 24 hours is around 0.0005 and the simulatedEENS is 0.1959 MWh. Although this case has the best relia-bility level, the expected total cost is much higher than other


Fig. 2. Simulated cost under different reliability requirements.

Fig. 3. Decisions in terms of total spinning reserve for cases under differentreliability requirements.

cases. As increases, implying that less conservative reli-ability criteria are applied, the total cost decreases because theoperating cost drops dramatically and the system becomes lessreliable.The level of conservatism is also reflected in the optimal UC

decisions. As shown in Fig. 3, when is assigned to be0.001, 0.005, and 0.010, respectively, the reserve margin sched-uled online may vary greatly especially during the peak hours.It can be seen that the most reliable case, i.e., the one with

requires online reserve as high as 420MWwhilein the other two cases the maximum reserve is no higher than330 MW. Fig. 4 illustrates the influence of in terms ofthe number of online units. For stringent reliability constraints,more units are required to be turned on to increase the total avail-able capacities and to hedge the risk of generator outages.

C. Influence of Generator Outages on Decisions

Several numerical case studies are conducted to examine theimpact of generation unavailability on UC decisions and theireconomic and reliability performance. The same test systemas the base case is employed with the reliability requirement

. The studies are done by varying the failure rateof every generator by a factor of ranging from 0.25 to 2.00.Fig. 5 shows how UC decisions change according to different

failure rates. For small values of , the chance of outage is lowso the operating cost is much lower as a smaller number of units

Fig. 4. Decisions in terms of the numbers of on line units for cases under dif-ferent reliability requirements.

Fig. 5. Decisions in terms of on line unit numbers under various failure rates.

are required to be scheduled on. When the generators becomeless reliable, indicated by larger , more expensive units areturned on to maintain the desired reliability performance. There-fore, the operating cost increases dramatically for cases withbetween 0.75 and 1.25. As the failure rate factor keeps in-creasing, it may become uneconomical in terms of the expectedcost to turn on the remaining expensive units. More load curtail-ment is allowed under contingency, as long as the probability ofload loss is bounded below a selected level. Consequently, theoperating cost does not increase much when is above 1.25while the EENS increases. Fig. 7 shows the reliability perfor-mance of the optimal unit commitment decisions under differentfailure rates. The probability of load loss in the entire day is con-stantly below the upper bound and the simulated EENSis restricted below 2.5 MWh for all cases.

D. Influence of Load Uncertainty on Decisions

The following case studies investigate how UC decisions areaffected by load uncertainty. Parameters of test system are thesame as the base case, except that the standard deviation of loadforecasting error is varied from 0.01 to 0.07 of the total demand.The decisions corresponding to three representative load un-

certainty levels are shown in Fig. 8. As the electricity demandbecomes more volatile, more units are scheduled online to pro-tect the system from possible load loss. The operating cost rises


Fig. 6. Simulated total cost and operating cost under various failure rates.

Fig. 7. Reliability performance of unit commitment decision under variousfailure rates.

Fig. 8. Decisions in terms of on line unit numbers under different levels of loaduncertainty.

because some expensive units are turned on to provide extra ca-pacity and this is why the expected total cost increases as shownin Fig. 9.As the level of load uncertainty increases, it is observed that

the loss of load probability is always restricted under the upperlimit , and the total amount of EENS is constantlybelow 2 MWh, shown by Fig. 10. It is also observed that theprobability of load loss and the EENS are almost non-increasingas the level of load uncertainty increases. This is because theupper bound approximation and the Boole’s inequality used inthe proposed reliability constraints are quite tight. Thereby, the

Fig. 9. Simulated total cost and operating cost under different levels of loaduncertainty.

Fig. 10. Reliability performance of unit commitment decisions under differentlevels of load uncertainty.

resultant decisions will be more conservative when load fluctu-ation becomes the main contributing factor of uncertainty.It should be noted that in practical power systems, the fore-

casting error of demand alone is typically small, yet the usageof uncertain renewable energy sources may greatly increase thelevel of system uncertainties. The combined impact of uncer-tain load and renewable energy is usually modeled by the termnet load [29], which has similar stochastic nature as electricitydemand. As a result, the proposed approach can be easily ex-tended to address the uncertainty of renewable energy sourceswithout major modification.

E. Computational Experience

The case studies above are all conducted on a one-area testsystem, containing 32 units. All problems are solved by CPLEX12.4 on a Dell Latitude E6420, with the relative dual gap set tobe 0.1%. The solution time of these cases with various valuesof is illustrated by Fig. 11. It is shown that the solutionefficiency greatly relies on the value of . The longest solu-tion time is approximately 450 s, when , and thefastest cases are solved within 50 s.In order to reveal the efficiency of this method on larger scale

systems, extra experiments are then conducted on a three-areatest system. This system is constructed by replicating the one-area test system above three times. The new system contains 96units, and outage contingencies are represented by 23 scenarios.


Fig. 11. Solution time for the one-area test system under different reliabilityrequirements.

Fig. 12. Solution time for the three-area test system under different reliabilityrequirements.

The solution time under different values of is provided inFig. 12 as the optimality gap is relaxed to be 0.5%. It is observedthat all cases are solved within 20 minutes, which shows a goodpotential to apply this formulation to solve practical power sys-tems.

IV. CONCLUSION

This paper proposes a new formulation to solve unit commit-ment problems under uncertainties. Power flow equations arenot considered because it requires joint chance constraints tomodel the loss of load probability. The feasibility set defined byjoint constraints may not be convex, and multi-dimensional in-tegration is needed to calculate the joint probability of load loss,leading to an intractable problem [20].The main contribution of this paper is to incorporate system

uncertainties in terms of generation unavailability and load fluc-tuation in the two-stage recourse model efficiently. Instead ofrepresenting all uncertainties—both generation unavailabilityand demand fluctuation—as a discrete set of scenarios, the gen-eration unavailability is modeled as a set of outage scenarioswhere the load uncertainty is modeled as a loss of load proba-bility constraint. This reliability constraint is then approximatedusing mixed-integer piecewise linear functions.The advantage of this formulation is that it allows the operator

to adjust the level of conservatism in UC scheduling by adopting

proper reliability requirement, which is expressed by the proba-bility that load loss occurs in the decision period. This providesa decision making tool for the system operators to trade-off re-liability and cost in the UC problem. The proposed formula-tion has a much smaller problem dimension as compared to thetwo-stage recourse model and can be solved in a timely mannereven though the formulation requires some extra binary decisionvariables. Case studies on both a 32-unit and a 96-unit systemsuggest that this method may have a good potential to solve UCproblems with practical system sizes.

APPENDIX A

MINIMUM-UP/DOWN TIME AND RAMP-RATE CONSTRAINTS

The feasible set in (6) is a set defined by minimum-up andminimum-down time constraints. Detailed formulation of theseconstraints is expressed by (25)–(27) and (28)–(30).

(25)

(26)

(27)

(28)

(29)

(30)

In each scenario , the decision variables are also subject toramp-rate limitations (31)–(32). This feasible set is denoted by.

(31)

(32)

APPENDIX B

PIECEWISE LINEAR UPPER BOUNDS OF CCDF

In all case studies, the upper bound of CCDF is expressed by afour-segment piecewise linear function. The values of functioncoefficients and are provided in the table below.


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Peng Xiong (S’08) received the B.Eng. degree in electrical engineering fromZhejiang University, Hangzhou, China, in June 2008, and the Ph.D. degree inengineering at the National University of Singapore in April 2013. He is cur-rently a Postdoctoral Researcher at the National University of Singapore. Hisresearch interests are in the area of power system reliability and optimization.

Panida Jirutitijaroen (S’05–M’08–SM’13) received the B.Eng. degree (Hon.)from Chulalongkorn University, Bangkok, Thailand, in 2002, and the Ph.D. de-gree in electrical engineering at Texas A&M University, College Station, TX,USA, in August 2007.She is currently an Assistant Professor at the Department of Electrical and

Computer Engineering, National University of Singapore, since January 2008.She was a Postdoctoral Researcher at Texas A&M University from Septemberto December 2007. Her research interests are power system reliability and op-timization.

A Stochastic Optimization Formulation of Unit Commitment With Reliability Constraints

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