stochastic scuc solution with variable wind energy using constrained ordinal optimization

Stochastic SCUCSolutionWith VariableWind EnergyUsing Constrained Ordinal Optimization

Hongyu Wu, Member, IEEE, and Mohammad Shahidehpour, Fellow, IEEE

Abstract—This paper proposes a constrained ordinal optimiza-tion (COO) based method for solving the scenario-based stochasticsecurity constrained unit commitment problem. The basic idea is tosample a large number of candidate unit commitment (UC) solu-tions by a crude model and then use an accurate model on a smallselected subset to find good enough UC solutions over all scenarios.To facilitate the proposedmethod, a feasibility model is utilized thatapplies analytical conditions for identifying the feasibility of UCs.A blind picking approach based on the feasibility model is incorpo-rated in the COO-based method for seeking good enough solutions.Numerical tests are performed on a modified IEEE 118-bus systemwith a high penetration of wind energy, in which hourly forecasterrors of wind speed and loads and random outages of systemcomponents are considered.The simulation results show the validityand the effectiveness of the proposed method. Comparative evalua-tions of the proposed COO-basedmethod withmixed-integer linearprogramming solvers are considered in this paper.

Index Terms—Mixed-integer linear programming (MILP),Monte Carlo simulation, NP-hard limitation, ordinal optimization(OO), security constrained unit commitment (SCUC), variable windenergy.

NOMENCLATURE

A. Constants, Sets, and ParametersNT Number of time periods.NI Number of thermal units.NF Number of transmission lines.NW Number of wind generators.NS Number of scenarios.NG Number of cost curve segments.

Index for time period, .Index for thermal unit, .Index for transmission line, .Index for wind generators, .Index for base case and scenarios, repre-sents base case and representsindividual scenario.Index for cost curve segments, .Occurrence probability of scenario .Load demand in scenario at time .No-load cost of thermal unit , in $.Maximum/minimum generation level of thermalunit .

Maximum generation level of thermal unit atsegment .Incremental cost for segment of unit .Forecasted upper bound on the generation ofwind generators at time .Maximum ramping rate of thermal unit .Set of committed thermal units at time .Set of available wind generators at time .

, which is a set of dispatchablegeneration sources including committed thermalunits and available wind generators at time .Maximum/minimum generation of the dispatch-able generators in .

B. VariablesStartup cost of thermal unit at time .Power dispatch of thermal unit at timescenario .Power dispatch of thermal unit at time atsegment in scenario .Corrective dispatch of thermal unit at time atsegment in scenario .Power dispatch of wind generators at time inscenario .Incremental cost for segment of unit .Discrete decision variable (status) of thermal unitat time in base case and scenarios: “1” for ON

and “0” for OFF.C. Matrices and Vectors

Bus-generator/bus-load incidence matrix.Shift factor matrix.Vector of upper limit for power flow.Vector of unit commitment with the dimension of

.Vector of generation dispatch in scenario withthe dimension of .

I. INTRODUCTION

R ENEWABLEenergy sources, especiallywind generation,represent important components of energy portfolio.

In the United States, wind energy is expected to provide 20%of the U.S. energy production portfolio by 2030 [1]–[3]. How-ever, the variability of wind energy has led to various challengesin energy infrastructure operations including those associatedwith the security of electric power systems [4], [5].

Powersystemanalystshavemadeconsiderableefforts toaddressthe impact of uncertainties on the solution of security constrainedunit commitment (SCUC). The solution of stochastic unit

Manuscript received February 20, 2013; revised June 23, 2013 and September06, 2013; accepted October 06, 2013. Date of publication December 23, 2013;date of current version March 18, 2014.

The authors are with the Electrical and Computer Engineering Department,Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: [email protected]; [email protected]).

Color versions of one ormore of the figures in this paper are available online athttp://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSTE.2013.2289853

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commitment (UC) with a high penetration of wind is examinedunder rollingplanningwith scenario trees [6].The rolling planningalgorithmiscarriedout for rescheduling,whichisbasedonthemostup-to-date wind forecasts and daily schedules [7]. The impact ofrandomgeneratoroutagesandloadforecasterrorsonthehourlyUCis analyzed by scenario trees [8], [9]. A multi-scenario long-termSCUCmodel for calculating the cost of power system reliability isproposedin[10], inwhich the lossof loadexpectation isconsideredasaconstraint for calculating thecostof supplyingsystemreserves.A long-term stochastic SCUC model is presented in [11], whichconsidersunit and transmission lineoutagesaswell as loadforecasterrors inmulti-scenario analyses. A chance-constrained stochasticUC problem is solved by a sample average approximation algo-rithm, in which a large portion of the hourly wind energy isguaranteed to be utilized [12]. A chance-constrained method forUC with the joint energy and reserves scheduling underreliability criterion is proposed in [13]. A UC model is presentedfor balancing the required spinning reserves with a high penetra-tion level of wind energy [14].

A noteworthy approach in the literature to the solution ofstochastic SCUC is the scenario-based method, in which multiplescenarios are generated by the Monte Carlo sampling (MCS)method to simulate the possible realization of uncertainties[8]–[11], [15]. The required number of scenarios to guarantee areasonablesolution isoftenquite largedue to the slowconvergenceof MCS. The accuracy (confidence interval) of MCS is at the best

, where is the number of simulated scenarios [16]. Anonconvex, Nondeterministic polynomial-time (NP)-hard, deter-ministic SCUCwould have to be solved in each scenario. TheNP-hard limitation suggests that an effective algorithm for the solutionof SCUC can seldom be utilized. Accordingly, a rather blindsearch is the only alternative solution [17]. The total search spaceof UC would be enormous even for a small-scale problem, inwhich the feasible region is confined to a small space especiallywhen transmission flow constraints are considered. Furthermore,the hard coupling constraints which would link scenarios areimposed so that the hourly commitment of nonquick start thermalunits in all scenarios is the same as that in the base case. The

and NP-hard limitation, with multiplicative rather thanadditive impacts, could add inherent difficulties to the solution ofscenario-based stochastic SCUC, resulting in an intractable SCUCproblem with exponentially expanding size.

The uncertainties resulted from the large penetration of rene-wable energy sources would require a real-time or near real-timecontrol with large computation requirements. Accordingly, thesolution of large stochastic SCUC problems with a large numberof scenarios would require computation resources that are com-monly unavailable.

The ordinal optimization (OO) [18]–[20] is considered here tofacilitate the scenario-based solution method with manageablecomputation requirements for stochastic SCUC. The goal of OOis to seek some top -percentile of large solutions, called goodenough solutions, with high probability instead of searching thebest solutionswith certainty. TheOO includes two tenets: ordinalcomparison and goal softening. Ordinal comparison indicatesthat it is much easier to identify a better solution than to find outhow much better the solution is. Goal softening suggests thatwhen it is computationally infeasible to calculate the exact

optimal solution, we can find some good enough solutions witha specified high probability. Both tenets are intuitively reason-able and have been proven mathematically [21], [22].

In recent years, OO has been successfully applied to powersystem problems. An OO-based bidding strategy is proposed in[23], in which an approximate model is used to describe theinfluence of bidding strategies onmarket clearing prices. AnOO-basedalgorithm is suggested in [24] for solving theoptimal powerflow problem. A simplified model is adopted to select a subset ofmost promising solutions, and an exact model is used to evaluatesolutions in the reduced set. Amulti-year transmission expansionplanning problem is investigated based on the OO theory in [25],and crude models are used to derive a small set of plans in whichsimulations are necessary to find good enough solutions.

The OO application to stochastic SCUC has not been utilizedin the literature. The following two important issues must beresolved before applying OO to stochastic SCUC.

1) Given that OO cannot be applied directly to the scenario-based stochastic SCUC, and many infeasible solutionscannot be excluded from the ordinal comparison,wewouldneed to establish fast analytical approaches rather thancostly simulations for identifying feasible solutions.

2) A systematic and efficient method is required to seek goodenough solutions with high probability instead of search-ing the best solution with certainty.

This paper addresses the above issues and a COO-basedmethod is proposed for solving the stochastic SCUC problem.The idea behind the COO-based method is to sample a largenumber of UC solutions by a crude model and then use anaccurate model on a small selected subset to find good enoughsolutions. To facilitate the COO-based solution to the stochasticSCUC, a feasibility model is utilized in which feasibility con-ditions would excludemost of the infeasible UCs with negligiblecomputation efforts. A blind picking (BP) approach based on thefeasibility model is incorporated in the COO-based method forseeking the good enough solutions.

Numerical tests are performed on the modified IEEE 118-bussystem with a high penetration level of wind, in which theforecast errors of wind speed and hourly load and randomoutages of system components are considered. The proposedCOO-based solution for stochastic SCUC is compared with thatof a mixed-integer linear programming (MILP) solver [26], [27].Numerical results in this study suggest that the COO-basedmethod yields good enough schedules in terms of MCS estima-tion accuracy. In addition, COO is computationally more effi-cient for solving the stochastic SCUC problem.

The rest of the paper is organized as follows: the mathematicformulation of stochastic SCUC is given in Section II. TheCOO-based stochastic SCUC is laid out in Section III. Numer-ical testing results are presented and analyzed in Section IV.The concluding remarks are provided in Section V.

II. SCENARIO-BASED STOCHASTIC SCUC FORMULATION

MCS generates a large number of scenarios to simulate windspeed uncertainty, which follows the Weibull distribution func-tion with the correlation factor and diurnal pattern [28]. Otherprobability distributions can be similarly considered in this

380 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 5, NO. 2, APRIL 2014

study. The hourly wind energy is calculated using the hourlywind speed and the power curve of wind turbines. Each scenariois assigned a probability based on the historical and forecasteddata. The scenario reduction technique is used to aggregateclosely related scenarios based on the probability metrics forreducing the computation effort [29]. MCS also considers ran-dom component outages and load forecast errors [11].

The objective of stochastic SCUC, expressed in (1), is todetermine the day/week-ahead hourly schedule of thermal andwind units such that the sum of the operation cost in base case( ) and the expected dispatch cost in individual scenarios( ) is minimized

The first line in the objective function (1) is the base-casegeneration cost of thermal units, which includes constant no-loadcost, time-dependent startup cost, and piecewise linear energyproduction cost. Corrective actions in each scenario representthe deviations of thermal generation dispatch from the base case.The second line in (1) represents the expected corrective cost inscenarios for accommodating the uncertainties. To facilitate thecomparison of COO- and MILP-based methods, only nonquickstart thermal units are considered in this paper, with the hourlyscenario commitments the same as those in the base case.However, the quick start units can further be considered inthis study [30]. By substituting and

into (1), objective function (1) is equivalently trans-

formed into

subject to the following constraints

The constraints in base-case and individual scenarios includethe system load balance, capacity limits of thermal generating

units, limits of thermal generation on segments, limits of windenergy, and transmission network constraints, which are listed in(3)–(7), respectively. Corrective dispatch capabilities of thermalunits are shown in (8), which are restricted by the ramping up/down rate limits. Constraints (8) can be implicitly included inconstraint (4) as follows:

Note that constraint (8) ensures the secure and feasible transferof the system operation status from the base case to scenarios.Therefore, the system reserve requirements are implicitly repre-sented by deviations in the dispatch of thermal units between thebase case and scenarios and would be optimally determinedthrough corrective dispatch [15].

Objective (2) is also subjected to other individual thermal unitconstraints such as minimum ON/OFF time and ramping up/downrate [31], [32]. Other resources such as energy storage, demandresponse, and distributed generation can also be considered here.The scenario-based stochastic SCUCmodel in (2)–(9) is a large-scale, nonconvex, NP-hard optimization problem. The SCUCsolution is difficult to find due to the computation burden causedby and NP-hard limitations. The application of theproposedCOO-basedmethod to the solution of stochastic SCUCis discussed in Section III.

III. APPLICATION OF COO TO STOCHASTIC SCUC

A. Constrained Ordinal Optimization

Mathematically, (2)–(9) is formulated as a mixed-integerprogramming (MIP) problem

where for any candidate solution in the entire discretesolution space , the true value ; can be obtained byaveraging the infinite number of MCS. In (10), represents therandomness in scenario and is the observed systemperformance. Let represents theset of feasible candidate solutions. TheMIP problem (10) is thentransformed into a nonconstrained (normal) OO as follows:

As computation resources could be limited for large-scalesystems, the sample average over a finite number of MCSscenarios is used to estimate

WU AND SHAHIDEHPOUR: STOCHASTIC SCUC SOLUTION WITH VARIABLE WIND ENERGY 381

where is the estimated value of . The true valueof the objective is given as

where is the observation noise stated as an independentlyand identically distributed random variable.

To apply COO, one typically selects the candidate solutionsfrom randomly. The feasibility of the candidate solutions isroughly estimated by a feasibility model and the observed(estimated) feasible solutions form the set . Let representa good enough subset, where each solution belongs to the top-percentile of candidate solutions in a given set. The good

enough subset is easy to specify but difficult to calculate. Byapplying a crude model randomly, COO selects a subset ofobserved feasible candidate solutions, which is called selectedsubset . A degree of alignment between the selected subsetand the good enough subset is quantified as follows:

where represents the alignment level, which is measured bycounting how many good enough solutions exist in the selectedsubset ; is the alignment probability (AP); and is theprobability criteria, which is normally set to 0.95. The relation-ship among the observed feasible set , good enough subset

, and selected subset is shown in Fig. 1, in which thefield between and is labeled as alignment.

According to the OO theory, the optimization problems arecategorized into several classes by identifying the optimizationproblem’s ordered performance curves (OPCs) [33]. The deter-mination of the selection size is dependent on the shape of itsOPC. If the problem category is identified, then the selection size

will be calculated based on a table given in [33].Once the subset is chosen from , the performance of each

candidate solution in would be evaluated by an accurate modelfor the optimization problem. It is guaranteed that the selectedsubset would contain good enough solutions with a prespecifiedhigh probability [18]–[20]. COO saves the computation efforts byat least one order ofmagnitude when comparing the size of withthat of . In addition, the convergence rate of OO would beexponential [21] which is much faster than that of O( ).

Two factors that influence the efficiency of COO for solvingstochastic SCUC are 1) a feasibility model for the UC solutionsand 2) a systematic method within the COO framework for

finding good enough solutions with a high probability. Theabove two issues are discussed as follows.

B. Feasibility Model for UC Solutions

The following notations are introduced to facilitate the pre-sentation of the feasibility model:

�

� �

where is the minimum element of th row of matrix .Accordingly, transmission network constraint (7) is converted to

� �

where � is a non-negative matrix. The non-negativity of�would greatly simplify the presentation of feasibility model

for the UC solutions, which will be discussed as follows.Feasibility model for UC solutions:Inequalities (17)–(19) are satisfied if a given UC satisfies

constraints (3)–(9)

In (19), is the -component of non-negative matrix �,is the -component of matrix �, is the total number of

dispatchable generators in , and is the order ofdispatchable generating units in such that

In (19), and are given as

Here, is an integer number such that

Fig. 1. Illustration of COO.


Inequality (17) indicates that the sum of committed thermalunit capacities plus the maximum available wind energy mustbe no less than the system demand. Likewise, inequality (18)suggests that the systemminimum generation must be no greaterthan the system demand. Inequality (19) shows that the estimatedminimum transmission line flow must be no greater than the linecapacity, which is a necessary but insufficient condition for afeasible UC solution. The detailed proof of inequality (19) isprovided in the Appendix.

The proposed feasibility model could accommodate addition-al system-wide constraints or/and components such as demandresponse and battery. For example, if system emission con-straints are added, an inequality constraint similar to (19) willbe added based on the estimation of lower bound on systememissions. Accounting for additional components would intro-duce extra terms in (17)–(19) for estimating the upper/lowerbounds of power generation.

Fig. 2 exhibits a flowchart for implementing the feasibilitymodel. The flowchart shows that checking the inequality con-ditions analytically could lead to tremendous savings in compu-tation costs as compared to those for the traditional method,which solve the corresponding security constrained economicdispatches (SCEDs) repeatedly for the base case and all scenariosuntil the UC feasibility is known. The feasibility model identifiesan estimated feasible candidate space which is derived fromthe solution space . Hence, it is easier for OO to reach the samealignment level in than in .

Here, certain solutions in may be truly infeasible as shownby the dots in Fig. 1, since a crude and imperfect feasibilitymodelis applied. The imperfectness is in tune with the goal softeningtenet of OO, as OO requires feasibility with certain probabilityinstead of feasible solutions. Some feasible and good enoughsolutions can be found with a high probability if such imperfect-ness is accounted for in determining the size of the selectedsubset.

C. Size Determination of the Selected Subset

The size of selected subset is dependent on the adoptedselection method. According to OO theory, BP is the simplestselectionmethod that involves the random selection of the subsetwithout replacement or comparison in . BP would offer the

same tendency for every solution to be evaluated in the solutionspace [22].

Suppose is the size of , is the size of good enoughsubset, represents the size of selected subset , and is theprobability that an observed feasible decision in is trulyfeasible. Based on the above notations, the AP for BP withfeasibility model (BPFM) [22], [34] is expressed as

where represents the number of different choices when

we choose solutions out of distinguished ones. In COO, (23)would determine the value for the prescribed alignmentprobability. is usually a small number; for example,

when , , , and .Note that OPC cannot be directly applied to COO as thefeasibility model adopted is normally imperfect [22]. To accom-modate such imperfectness, COO may require a larger thanwhen using the OPC.

D. COO-Based Method for the Solution of Stochastic SCUC

The COO-based method, which integrates BPFM to solve thestochastic SCUC, is shown in Fig. 3 and described as follows:

Step 1) Initial sampling: As load andwind energy in scenarioswould fluctuate according to their respective base-case forecasts, it is reasonable to assume that goodenough UC solutions to the stochastic SCUC wouldbe located near the base-case optimal UC solution.Starting with the base-case optimal UC solution,commitment states of thermal units with minimumON/OFF times are adjusted by a branch and bound

Fig. 2. Flowchart of the feasibility model.


(B&B) method [35] without violating the minimumON/OFF time constraints. Here, the B&B method ran-domly selects the commitment states of such thermalunits as branching variables, where every potentiallyfeasible UC solution has the equal chance. Theadjustment will lead to an increase in the systemoperation cost, called opportunity cost [35]. Theopportunity cost, which is an indicator on how gooda UC solution would be, is utilized to sample goodenough UC solutions. Note that the B&B method

is a systematic and implicit enumeration of all can-didate solutions in which the entire search space isexamined.

Step 2) UC feasibility check: Use the flowchart in Fig. 2 toidentify the feasible UC solutions sampled in Step 1)which constitute . If the size of reaches , thengo to Step 3); otherwise, go back to Step 1).

Step 3) Subset size determination: Given , , , , andvalues, determine based on (23).

Step 4) Blind pick: Randomly pick sampled solutionsin , which would constitute

set . The COO theory ensures that there are no lessthan good enough feasible solutions in the selectedsubset with the probability .

Step 5) Accurate evaluation: Let , calculate for eachusing SCED such that the system

operation cost is minimized. If is infea-sible according to SCED, let . Let

; if or there is any computationbudget left, repeat Step 5); go to Step 6) otherwise.

Step 6) Final solution: The final solution for thescenario-based stochastic SCUC is given as

where (24) suggests that the optimal UC solutionhas the minimum expected operation cost over

all scenarios, which is consistent with objectivefunction (2). Equation (25) indicates that the optimaldispatch is the base-case dispatch correspondingto .

An acceleration strategy for eliminating inactive line flowconstraints [36] is implemented prior to Step 1). A reduced set ofline flow constraints are then taken into account in the proposedCOO-basedmethod. The computation cost of the proposed COOmethod is mostly on SCEDs, i.e., solving a series of linearprogramming problems in base case and all scenarios, for theselected subset of UC solutions. The computation burden incr-eases only linearly rather than exponentially when the numberof scenarios increases, which would lead to a significant savingsin computational efforts, or an improvement in the MCS esti-mation accuracy under a given computation budget. As the sizeof the selected subset is small, the computational burden for thesolution of the stochastic SCUC even with a large number ofscenarios is still tractable.

The proposed COO method for solving stochastic SCUC is asystematic method within the COO theory framework, and thefeasibility model is based on the analytical feasibility conditionsrather than the heuristics and empirical results. In general, thefinal solution may not be the optimal solution for the stochasticSCUC problem. The purpose of the proposed COO method is tofind a good enough solution with a high probability instead of anoptimal solution. The COO theory ensures at least of the top-percentile of solutions would be obtained with a prescribed

high probability.

Fig. 3. Flowchart of COO for the solution of stochastic SCUC.


IV. NUMERICAL RESULTS

A. Test System

Numerical tests are performed for a modified IEEE 118-bussystem, which has 54 thermal generators, 186 branches, and 91demand sides. The detailed data of generators, transmissionnetwork, and load profiles are given in [31]. There are threegeographically dispersed wind farms located at buses 14, 54, and95, respectively. The wind energy is calculated based on windspeed forecasts and the wind turbine power curve, which arefrom http://www.nrel.gov/. The daily wind energy forecast is24 631 MWh, i.e., 21.7% of the total daily energy demand inthe system.

The wind speed follows the Weibull distribution and thecontinuous distribution is approximated by the discrete distribu-tion. The wind speed forecast error is characterized by theautoregressive moving average (ARMA) [37]. The load forecasterrors at each time period are modeled by a normally distributedfunction with a standard deviation of 5%. The random outages ofgenerators and transmission lines are simulated based on forcedoutage rates and repair rates by a sampling method for calculat-ing the 0/1 values of system component availability in eachscenario [11]. About 1800 scenarios are originally generatedwhich are reduced to 185 by the scenario reduction technique[29]. Load shedding is considered as the last resort to maintainthe system reliability in the case of system contingency and thevalue of lost load is set at 1000$/MW. The observation noise isdisregarded for simplicity in this system. The numerical tests areimplemented in Microsoft Visual C++ on an Intel Xeon Severwith 64 GB RAM. An optimizer in CPLEX 11.0 [38] is used tosolve SCED. The following cases are studied:

B. The COO-Based Solution for Stochastic SCUC

Table I shows the index of 30 ( ) UC solutionsselected by BPFM with , , , and

, in which the top 5% good enough set is listed forcomparison. The shadowed index numbers in Table I are the UCsolutions that are within both and . As expressed in (23),there is a 95% probability that at least one ( ) UC solutionwould be in . As a result, four UC solutions, i.e., 334,108, 746, and 9, are in the top 5% good enough set and the actualalignment level is higher than that required ( ).

For each UC solution in , SCED is employed to calculatethe optimal dispatch in the base case and scenarios. The UCsolution 746 listed in Table II is the final UC solution for thestochastic SCUC problem with the lowest expected systemoperation cost identified by (24). Here, base-load thermal unitsare always committed. Some expensive units are kept OFF duringthe scheduling horizon to minimize the expected system opera-tion cost. Units 1, 2, 6–9, 15–18, 31, 32, 34, 47, and 51–53are turned ON at certain hours to satisfy the peak load of 6000MWat hour 21 as highlighted in Table II. The expected operationcost is $1 431 840 that includes the base-case operation cost of$1 411 060 and the expected corrective dispatch cost of $20 780in all scenarios.

The system operation cost with the UC solution for all 185scenarios is shown in Fig. 4, in which the dotted line representsthe expectation. All scenario costs are within the interval

[$1 348 660, $1 571 260] and the expensive units are dispatchedmore at certain scenarios where the available wind generation islower. Load shedding take s place at scenarios 20, 143, and 182where the load balance constraint cannot be met in the case of asystem contingency. In such scenarios, the corrective dispatchcosts are much higher than those of other scenarios.

The hourly dispatch ofwind farm2 at bus 54 is shown in Fig. 5.The available wind generation is dispatched without any curtail-ments at hours 1–13 and 18 when the hourly available windgeneration is below 234 MW. Wind curtailments, highlighted inFig. 5, would arise at hours 14–17 and 19–24 when the availablewind generation is beyond 256 MW. The flows on key lines 77and 78 reach their respective limits during curtailment hours at allscenarios. The above results indicate that the proposed COO-basedmethod could produce a good enough schedule with a highprobability.

C. Impact of q on the COO-Based Solution

COOsolutionswith different are compared in Table III.Withlowering from 0.95 to 0.50, which indicates that the feasibility

TABLE ISELECTED SUBSET AND GOOD ENOUGH SET

TABLE IIGOOD ENOUGH SOLUTION OBTAINED BY COO


model becomes less accurate in determining the feasibilities ofUC solutions, increases from 30 to 57 to ensure feasible andgood enough solutions can be found with the prescribed highprobability. The larger size of the selected subset would in turnlead to a higher CPU time. However, as shown in Table III, theexpected operation cost and wind penetration would changeslightly as the is lowered. The results demonstrate the robust-ness of COO-based solution for its applications to large powersystems. Normally, in such cases, the proposed feasibility modelis less accurate, as (19) is a necessary but insufficient conditionfor a feasible UC solution. Such inaccuracy is accounted for inBPFM. Therefore, COO could yield good enough solutions withhigh probabilities in large-scale systems, but the computationtime will be higher.

D. Comparison of COO and MILP Methods for the Solution ofStochastic SCUC

The COO-based solution is compared with that of MILPin Table IV, in which the expected energy not served (EENS)is the average energy not suppliedwhen the load shedding occurs[10]; the relative errors of MCS are given in brackets and “ ”

represents 95% of confidence interval. For example,indicate that 95% of the wind penetration would fall in

. The smaller the confidenceinterval, the more accurate the MCS will be. The MILP-basedsolution is procured by engaging CPLEX 11.0 in which thedefault settings are selected and the maximum threshold ofoptimality gap is set to 0.1%. About 185 scenarios for theMILP-based method are computationally infeasible. Thus, they

are further reduced to 16 by the scenario reduction techniquesuch that it is tractable for the MILP-based method.

The slight differences in expected operation costs and EENSsuggest that the COO-based method procures a good enoughschedule as compared with the MILP-based method. However,the COO-based method obtains a much better stochastic SCUCsolution in terms of higher MCS estimation accuracy andcomputation efficiency. The relative errors of the MILP-basedmethod are more than twice of those of COO. The relative errorwill be smaller if more MCS scenarios are introduced. However,additional scenarios may result in a computationally infeasibleMILP-based solution. In contrast, the COO-based solution has acomputation efficiencywith a CPU time ofof that of the MILP-based solution. The computation burden ofthe COO-based solutionwould increase linearly with the numberof scenarios. So, the MCS estimation accuracy for stochasticSCUC could be enhanced further with a minute addition to thecomputation requirements.

V. CONCLUSION

The critically growing impact of uncertainties on powersystem requires stochastic market operations. Since the MCSis capable of simulating the various uncertainty factors, thescenario-based solution to the stochastic SCUC becomes amajorapproach. However, the intractable computational complexityfor the scenario-based approach calls for a computationallyefficient solution. This paper has proposed aCOO-based solutionto the stochastic SCUC, in which a BP approach based on thefeasibility model is incorporated for obtaining the good enoughsolutions. The numerical results demonstrate the validity andeffectiveness of the proposed COO-basedmethod. The proposedCOO method for the solution of stochastic SCUC yields robustand good enough schedules which are computationally moreefficient than the MILP-based method. This feature makes theCOO-based method a suitable alternative for solving the

Fig. 4. System operation cost versus scenarios.

Fig. 5. Expected dispatch of wind farm 2.

TABLE IIICOO SOLUTIONS WITH DIFFERENT

TABLE IVCOMPARISON RESULTS BETWEEN COO- AND MILP-BASED METHODS

Stochastic SCUC solved by MILP with reduced 16 scenarios.


stochastic SCUC problem applied to large-scale power systems,or in real-time cases with rolling plan where a high computationefficiency is required.

APPENDIX

THEOREM A1

Consider the following linear programming (LP) problem:

The necessary condition for a given UC to be feasible is thatthe following inequalities must be satisfied:

The structure of (A1) can be fully utilized such that the optimalsolution and objective value of (A1) are obtained analyticallyrather than by solving a series of LP problems. The analyticalsteps are described as follows.

1) Let denote the order of dispatchable gener-ating units in such that

where is the total number of units in .2) If a UC is feasible, by substituting (21) into (A1), we have

Rearranging the terms in (A4), we obtain

If inequalities (17) and (18) are satisfied, there must existsome certain integer such that

3) Let

Then, we have

Inequality (19) is obtained by substituting (A8) into (A2).

REFERENCES

[1] World Wind Energy Association, Energy World Wind Energy Report, 2010[Online]. Available: http://www.wwindea.org/home/images/stories/pdfs/worldwindenergyreport2010_s.pdf.

[2] U.S. National Renewable Energy Laboratory, 20% Wind Energy by 2030:IncreasingWind Energy’s Contribution to U.S. Electricity Supply [Online].Available: http://www.nrel.gov/docs/fy08osti/41869.pdf.

[3] W. Tian, M. Shahidehpour, and Z. Li, “Analysis of 2030 large-scale windenergy integration in the eastern interconnection using WINS,” Electr J.,vol. 2, no. 3, pp. 71–87, Oct. 2011.

[4] J. Wang, M. Shahidehpour, and Z. Li, “Security-constrained unit commit-ment with volatile wind power generation,” IEEE Trans. Power Syst.,vol. 23, no. 3, pp. 1319–1327, Aug. 2008.

[5] B. C. Ummels, M. Gibescu, E. Pelgrum, W. L. Kling, and A. J. Brand,“Impacts of wind power on thermal generation unit commitment anddispatch,” IEEE Trans. Energy Convers., vol. 22, no. 1, pp. 44–51,Mar. 2007.

[6] A. Tuohy, P. Meibom, E. Denny, and M. O’Malley, “Unit commitment forsystems with significant wind penetration,” IEEE Trans. Power Syst.,vol. 24, no. 2, pp. 592–601, May 2009.

[7] P. Meibom, R. Barth, B. Hasche, et al., “Stochastic optimization model tostudy the operational impacts of high wind penetrations in Ireland,” IEEETrans. Power Syst, vol. 26, no. 3, pp. 1367–1379, Aug. 2011.

[8] J. Valenzuela and M. Mazumdar, “Monte carlo computation of powergeneration production costs under operating constraints,” IEEE Trans.Power Syst., vol. 16, pp. 671–677, Nov. 2001.

[9] S. Takriti, J. R. Birge, and E. Long, “A stochastic model for the unitcommitment problem,” IEEE Trans. Power Syst., vol. 11, no. 3,pp. 1497–1508, Aug. 1996.

[10] L. Wu, M. Shahidehpour, and T. Li, “Cost of reliability analysis based onstochastic unit commitment,” IEEE Trans. Power Syst., vol. 23, no. 3,pp. 1364–1374, 2008.

[11] L. Wu, M. Shahidehpour, and T. Li, “Stochastic security-constrained unitcommitment,” IEEE Trans. Power Syst., vol. 22, no. 2, pp. 800–811,May 2007.

[12] Q. Wang, Y. Guan, and J. Wang, “A chance-constrained two-stagestochastic program for unit commitmentwith uncertainwind power output,”IEEE Trans. Power Syst., vol. 27, no. 1, pp. 206–215, Feb. 2012.

[13] D. Pozo and J. Contreras, “A chance-constrained unit commitment with ansecurity criterion and significant wind generation,” IEEE Trans.

Power Syst., vol. 28, no. 3, pp. 2842–2851, Aug. 2013.[14] J. Morales, A. Conejo, and J. Pérez-Ruiz, “Economic valuation of reserves

in power systemswith high penetration ofwind power,” IEEETrans. PowerSyst., vol. 24, no. 2, pp. 900–910, May 2009.

[15] L. Wu, M. Shahidehpour, and T. Li, “Comparison of scenario-based andinterval optimization approaches to stochastic SCUC,” IEEE Trans. PowerSyst., vol. 27, no. 2, pp. 913–921, May 2012.

[16] H. J. Kushner and D. S. Clark, Stochastic Approximation for Constrainedand Unconstrained Systems, New York, NY, U.S.A.: Springer, 1978.

[17] N. P. Padhy, “Unit commitment—A bibliographical survey,” IEEE Trans.Power Syst., vol. 19, no. 2, pp. 1196–1205, May 2004.

[18] Y. C. Ho, R. S. Sreenivas, and P. Vakili, “Ordinal optimization in DEDS,”J. Discrete Event Dyn. Syst., vol. 2, no. 2, pp. 61–68, 1992.

[19] Y. C. Ho and M. Larson, “Ordinal optimization and rare event probabilitysimulation,” J. Discrete EventDyn. Syst., vol. 5, no. 2–3, pp. 281–301, 1995.

[20] M. Deng and Y. C. Ho, “Sampling selection method for stochastic optimi-zation problems,” Automation, vol. 35, no. 2, pp. 331–338, 1999.

[21] L. Dai, “Convergence properties of ordinal comparison in the simulationof discrete event dynamic systems,” J. Optim. Theory App., vol. 91, no. 2,pp. 363–388, 1996.

[22] Y. C. Ho, Q. Zhao, and Q. Jia, Ordinal Optimization: Soft Optimization forHard Problems, New York, NY, U.S.A.: Springer, 2007.

[23] X. Guan, Y. C. Ho, and F. Lai, “An ordinal optimization based biddingstrategy for electric power suppliers in the daily energy market,” IEEETrans. Power Syst., vol. 16, no. 4, pp. 788–797, Nov. 2001.


[24] S. Y. Lin, Y. C. Ho, and C. H. Lin, “An ordinal optimization theory-basedalgorithm for solving the optimal power flow problem with discretecontrol variables,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 276–286,Feb. 2004.

[25] M. Xie and F. F. Wu, “Multiyear transmission expansion planning usingordinal optimization,” IEEE Trans. Power Syst., vol. 22, no. 4,pp. 1420–1428, Nov. 2007.

[26] J. M. Arroyo and A. J. Conejo, “Optimal response of a thermal unit toan electricity spot market,” IEEE Trans. Power Syst., vol. 15, no. 3,pp. 1098–1104, Aug. 2000.

[27] J. M. Arroyo and A. J. Conejo, “Modeling of start-up and shut-down powertrajectories of thermal units,” IEEE Trans. Power Syst., vol. 19, no. 3,pp. 1562–1568, Aug. 2004.

[28] J. F. Manwell, J. G. McGowan, and A. L. Rogers,Wind Energy Explained,New York, NY, U.S.A.: Wiley, 2002.

[29] J. Dupačová, N. Gröwe-Kuska, and W. Römisch, “Scenario reduction instochastic programming: An approach using probability metrics,” Math.Program., vol. 95, no. 3, pp. 493–511, 2003.

[30] L. Wu, M. Shahidehpour, and C. Liu, “MIP-based post-contingencycorrective action with quick-start units,” IEEE Trans. Power Syst., vol. 24,no. 4, pp. 1898–1899, Nov. 2009.

[31] M. Shahidehpour, H. Yamin, and Z. Y. Li, Market Operations in ElectricPower Systems, New York, NY, U.S.A.: Wiley, 2002.

[32] Y. Fu and M. Shahidehpour, “Fast SCUC for large-scale power systems,”IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2144–2151, Nov. 2007.

[33] T. W. E. Lau and Y. C. Ho, “Universal alignment probabilities and subsetselection for ordinal optimization,” J. Optim. Theory App., vol. 93, no. 3,pp. 455–489, 1997.

[34] X.Guan,C.Song,Y.C.Ho,andQ.Zhao,“Constrainedordinaloptimization—Afeasibilitymodelbasedapproach,”J.DiscreteEventDyn.Syst.,vol.16,no.2,pp. 279–299, 2006.

[35] H. Wu, X. Guan, Q. Zhai, et al., “A systematic method for constructingfeasible solution to SCUC problem with analytical feasibility conditions,”IEEE Trans. Power Syst., vol. 27, no. 1, pp. 526–534, Feb. 2012.

[36] Q. Zhai, X. Guan, J. Cheng, and H. Wu, “Fast identification of inactivesecurity constraints in SCUC problems,” IEEE Trans. Power Syst., vol. 25,no. 4, pp. 1946–1954, Nov. 2010.

[37] A. Boone, “Simulation of short-term wind speed forecast errors using amulti-variate ARMA (1,1) time-series model,” Master thesis, KTH,Stockholm, Sweden, 2005.

[38] ILOG CPLEX, ILOG CPLEX Homepage, 2011[Online]. Available: http://www.ilog.com.

HongyuWu (M’09) received the B.S. degree in energy and power engineering,and the Ph.D. degree in system engineering from Xi’an Jiaotong University,Xi’an, China, in 2003 and 2011, respectively.He is a Visiting Scientist at the Robert W. Galvin Center for Electricity

Initiative at the Illinois Institute of Technology, Chicago, IL, USA. His researchinterests include renewable energies, electricity markets, and power systemoptimization.

Mohammad Shahidehpour (F’01) is the recipient of the Honorary Doctorate in2009 from the Polytechnic University of Bucharest in Romania.He is the Bodine Chair Professor and Director of Robert W. Galvin Center for

Electricity Innovation (www.galvincenter.org) at Illinois Institute of Technology,Chicago, IL, USA. He is also a Research Professor at King Abdulaziz Universityin Jeddah, Saudi Arabia, North China Electric Power University in Beijing, andSharif University in Tehran.Dr. Shahidehpour is an IEEE Distinguished Lecturer, Chair of the 2012 IEEE

Innovative Smart Grid Technologies Conference, Chair of the 2013 Great LakesSymposium on Smart Grid and the New Energy Economy, and the Editor-in-Chief of the IEEE TRANSACTIONS ON SMART GRID. He is the recipient of the 2012IEEE PES Outstanding Power Engineering Educator Award.


stochastic scuc solution with variable wind energy using constrained ordinal optimization

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