Applied Energy 105 (2013) 418–422

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Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

Stochastic unit commitment with sub-hourly dispatch constraints

0306-2619/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.apenergy.2013.01.008

⇑ Corresponding author. Tel.: +1 630 252 1474.E-mail address: jianhui.wang@anl.gov (J. Wang).

Jiadong Wang a, Jianhui Wang b,⇑, Cong Liu b, Juan P. Ruiz c

a Industrial and Systems Engineering Department, Lehigh University, PA 18015, United Statesb Center for Energy, Environmental, and Economic Systems Analysis (CEEESA), Argonne National Laboratory, IL 60439, United Statesc Department of Chemical Engineering, Carnegie Mellon University, PA 15213, United States

h i g h l i g h t s

" Stochastic unit commit with sub-hourly constraints." Modified benders decomposition." Wind power variability and uncertainty modeling.

a r t i c l e i n f o

Article history:Received 10 August 2012Received in revised form 8 December 2012Accepted 2 January 2013Available online 14 February 2013

Keywords:Wind powerElectricity marketsUnit commitmentDispatch

a b s t r a c t

In this paper, we propose a new unit commitment model that captures the sub-hourly variability of windpower. Scenarios are included in the stochastic unit commitment formulation to represent the uncer-tainty and intermittency of wind power output. A modified Benders decomposition method is used toimprove the convergence of the algorithm. The numerical results show that the proposed model basedon finer granularity outperforms the conventional model of hourly resolution.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Unit commitment (UC) and economic dispatch has been one ofthe most important applications in power system operations [1].Traditionally, the UC problem is based on hourly dispatch con-straints regardless of the fact that security-constrained economicdispatch (SCED) is run multiple times an hour with updated infor-mation such as sub-hourly load in practice. Recently, a multi-timeframe generator control and dispatch model is briefly discussed in[2] as an effective approach to address the uncertainty and sto-chasticity of renewable energy. Also a new UC model is proposedin [3] that includes the consideration of feasible energy deliveryunder large-scale sub-hourly wind integration. While refining thetime interval into sub-hourly granularity is not a new concept, ithas been only considered in economic dispatch models, not UCmodels. In other words, in all the UC models presented in the liter-ature so far, the economic dispatch is of hourly resolution. With thegrowing penetration of renewable generation such as wind power,the sub-hourly variability of renewable generation has increased

unprecedentedly. The renewable energy output can ramp up ordown quickly within an hour [4,5]. Meanwhile, the uncertaintyinherent in wind power forecasting also needs to be addressedwith improved power system operation methods including sto-chastic unit commitment [6] in which wind power uncertaintycan be represented by various realizations of wind power outputin scenarios. Hence, a new stochastic unit commitment model thataddresses both the sub-hourly variability and uncertainty of windpower is proposed in this paper. The main contribution of the pa-per is summarized as follows:

(1) The uncertainty information in wind power forecasting isincorporated through the stochastic formulation while variabilityis explicitly modeled by sub-hourly dispatch constraints in eachscenario. The proposed model differs from the previous ones inthat the traditional hourly constraints are enhanced by sub-hourlyrepresentations of economic dispatch in the scenarios. (2) To re-duce the computational expenses due to the inclusion of a largenumber of scenarios, an improved Benders decomposition algo-rithm is applied to solve the stochastic unit commitment modelwith sub-hourly dispatch constraints. The constraints associatedwith certain scenarios are added to the master UC problem toaccelerate the process in finding the optimal solution.

Nomenclature

Indicesi,j bus index for power generation units i, j = 1 . . . It index for hourly time period, t = 1 . . . Ts index for sub-hourly time period, s = 1 . . . Sk index for scenario, k = 1 . . . K

ConstantsC1, C2 unit production cost function coefficientsPDits load or demand, at bus i, time t and s,Pmaxi , Pmini

maximum and minimum output, unit iSUi, SDi start-up ramp limit and shut-down ramp limit, unit iUOi, POi Initial ON–OFF state indicator and output, unit iPrk probability of wind occurrence for scenario kSRDt, ORRDt requirements of spinning and operating reserve,

time tDTi, UTi minimum down-time and on-time, unit iCens cost of unserved loadWitsk forecasted wind generation at wind farm i, time t, sub-

period s and scenario kPLmaxij maximum capacity of transmission line between bus i

and jMSR maximum spinning reserve for unit iQSC quick start capacity for unit i

VariablesSTCit, SDCit start-up and shut-down cost of a unit i, time tPitsk power output of unit i at time t, s and scenario kZit,Yit start-up, shut-down indicator of unit i at time tBit ON–OFF status of generation unit i at time tWitsk wind generation of unit i at time s, t and scenario kensitsk dispatch curtailment for wind unit i, time t, s and sce-

nario kSRit spinning reserve of unit i at time tORRit operating reserve of unit i at time tditsk bus phase angle at i, time t, s and scenario k

Generalized modelA, b, c, d, E, F, g coefficient vector, matrix of MILPs index of scenariosS, ps number of scenarios and their probabilitiesx, y, t first-stage, second-stage and slack variablesm1, n1, m2, n2 dimensions of first-stage and second-stage prob-

lemsl dual variablesz, h, / variables representing objective function valuesm02; n02; m002; n002 dimensions of second-stage problemsiter number of iterations� tolerance level

J. Wang et al. / Applied Energy 105 (2013) 418–422 419

2. Problem formulation

The proposed stochastic unit commitment problem is formu-lated as a mixed integer linear programming model (MILP) shownbelow:

minX

i;t

ðSTCit þ SDCitÞ þX

i;t

C1iBit þX

i;t;s;k

PrkC2iPitsk þX

i;t;s;k

CensPrkensitsk ð1Þ

s:t:X

i¼1

SRit P SRDt 8 t ð2ÞX

i¼1

ORRit P ORRDt 8 t ð3Þ

Bit � Bi;t�1 ¼ Zit � Yit 8 i; t

Zit 6 Bit ;Yit 6 1� Bit 8 i; t ð4ÞBit � Bi;t�1 ¼ Zit � Yit 8 i; t P 2 ð5ÞBit � UOi ¼ Zit � Yit 8 i; t ¼ 1Zit 6 Bit 8 i; t ð6ÞYit 6 1� Bit 8 i; t ð7ÞSTCit ¼ SUiZit 8 i; t ð8ÞSDCit ¼ SDiYit 8 i; t ð9ÞSRit 6 10MSRi � Bit 8 i; t ð10ÞORRit 6 SRit þ QSCið1� BitÞ 8 i; t ð11Þf ðPOi; Pi;j;subt;k;RDi;Bi;t ;RS Di;Yi;t ;RS Ui; Zi;t ;RUiÞ 6 0 ð12ÞgðPi;j;subt;k; PLi;j;t;ts;k;Wi;t;subt;k; ensi;t;subt;k; PDi;t;subt ; PLi;j;t;ts;k;

PLmaxi;j; xi;j; di;t;subt;kÞ 6 0 ð13ÞWi;t;ts;k 6Wmaxi;t;ts;k 8 i; j; t; ts; k ð14Þ

The decision variables and parameters in the formulation areillustrated at the beginning of this paper in the Nomenclature sec-tion. The objective of the problem is to minimize the total costincluding start-up and shut-down cost, production cost and un-served load cost as shown in the objective function (1). Inequalities(2) model the spinning and operating reserve constraints. Inequal-ities (3)–(7) impose the on and off condition of generators. Equal-ities (8) and (9) are the shut-down and start-up cost constraints.Inequalities (10) and (11) model the quick start cost and availablespinning reserves of the generators. Series of Constraints (12) exert

minimum and maximum generation capacity constraints, modelthe ramp-up and ramp-down constraints on each generator. Seriesof Constraints (13) model the power balance constraints in the net-work, the phase angles of power flow on the transmission lines andthe capacity limits on the transmission line. Inequalities (14) im-pose the available forecasted wind power output constraints.When the sub-hourly index set is empty, the model is simplifiedas the traditional unit commitment model with hourly intervals.When the sub-hourly refined time scale is considered, the unitcommitment model becomes computationally challenging to solveto optimality due to the increased size both in number of variablesand constraints. Instead of using the explicit model describedabove, the condensed model in a more generalized form displayedbelow is employed to illustrate the proposed modified Bendersdecomposition methods such that the potential computationalbottleneck can be resolved.

min z ¼ c � xþX

s

ps � ds � ys ð15Þ

s:t: A � x 6 b ð16ÞEs � xþ Fs � ys

6 gs 8 s ¼ 1;2; . . . ; S ð17Þx 2 f0;1gn1 ð18Þy 2 Rn2

þ ð19Þ

The objective of the generalized model is to minimize the totalcost including start-up and shut-down cost, production cost andunserved load cost as shown in the objective function (15). Sce-nario-independent inequalities A � x 6 b (16) corresponding to theexplicit model constraints (2)–(11) model the spinning and operat-ing reserve constraints, unit start-up and shut-down indicators,minimum on and off time constraints, start-up and shut-down costconstraints. And scenario-dependent inequalities Es � x + Fs � ys

6 gs

(17) with respect to the explicit model constraints (12)–(14) are forthe economic dispatch in each scenario including the generationcapacity, ramp down/up, power balance, transmission and windgeneration capacity constraints accordingly. The binary decisionvariables x indicate whether the unit is on or off and the continu-

420 J. Wang et al. / Applied Energy 105 (2013) 418–422

ous variables y represent the output level of different energysources. The constraints (16) are the first-stage constraints and(17) are the second-stage constraints. We denote the second stageobjective function value ps � ds � ys as hs and the objective functionvalue of the overall optimization problem as z. The lower boundof z is z ¼ c � xþ

Psh

s. Conventionally, the second-stage constraintsconsist of hourly interval based constraints. However, the model inour paper is refined with sub-hourly periods to capture the windpower output variability within the hour, which leads to more con-straints for economic dispatch in each scenario. This enhanced for-mulation makes this already NP-hard MILP problem [7] moredifficult to solve to optimality. Hence, we present a modified Bend-ers decomposition below to reduce the computational require-ments of the proposed model.

3. Methodology

Benders decomposition can exploit the problem structure andreduce the size of the problem by decomposing the original prob-lem into subproblems based on scenarios [8]. As the standardBender decomposition could result in slow convergence, a modi-fied Benders decomposition method is implemented in order toaccelerate the convergence of lower and upper bounds as shownin Fig. 1 below. In comparison with the standard Benders decom-position algorithm, we include a subset of the second-stage eco-nomic dispatch constraints from one scenario into the masterproblem that originally only contains the first-stage constraints.The purpose of modifying the traditional Benders decompositionthis way is to obtain a better initial feasible solution for the sub-problems and reduce the number of generated feasibility cuts. Ifwe decompose the second-stage constraint matrix into two matri-

Fig. 1. A modified Benders decomposition framework.

ces with dimensions m01 � n01 and m002 � n002 (assuming n02 6 n1), themaster problem at the initial iteration including part of the sec-ond-stage constraints is presented as follows:

min c � x ð20Þs:t: A � x 6 b ð21Þ

Esm02�n02

� xþ Fsm02�n02

� y 6 gsm028 s ¼ 1; . . . ; S ð22Þ

x 2 f0;1gn1 ; y 2 Rn02þ ð23Þ

After the first iteration, the algorithm proceeds as the standardBenders decomposition,

min c � xþX

s

/s ð24Þ

s:t: A � x 6 b ð25Þ

x 2 f0;1gn1 ; /s 2 Rn02þ ð26Þ

The upper bound of z is �z ¼minf�zp; c � xþP

s/sg where �zp is the

�z at the previous iteration and the formulation of the subproblemfor a scenario is shown below:

min /s ¼ ps � ds � ys ð27Þs:t: Fs � ys þ ts ¼ gs � Es � x 8 s ¼ 1;2; � � � ; S ð28Þ

y; t 2 Rn2þ ð29Þ

Instead of generating aggregated optimality cuts, one subprob-lem is solved at a time to determine whether a cut should be gen-erated for that scenario. If the subproblem is not feasible, afeasibility cut will be generated. If the subproblem is feasible butthe difference between the upper and lower bounds is larger thanthe tolerance level, an optimality cut will be generated. The masterproblem is solved again with these cuts added. The algorithm ter-minates when no Benders cuts are generated.

Consider ws as the extreme rays when the subproblem is un-bounded, the feasibility cuts are expressed by the duality theorembelow:

ws � ðgs � Es � xÞ 6 0 ð30Þ

Consider ls is the dual variable vector corresponding to con-straint (25), The optimality cuts are generated if the difference ofupper bound and lower bound satisfies ð�z� zÞ=�z > � as below:

ls � ðgs � Es � xÞ 6 hs ð31Þ

Proposition 1. If the unit commitment problem has a finite optimum,the modified Benders decomposition gives an optimal solution in afinite number of iterations.

Proof 1. The added constraints in the initial iteration are valid forthe convex hull of the feasible region. By generating optimalitycuts by the linear programming duality theorem, which supportsthe recourse function

Psp

s � ds � ys, the non-optimal solution is sep-arated until no cuts are generated. The difference between themodified Benders decomposition and the standard one is on howto initialize the Benders method. After the initial iterations, theconclusion from the standard Benders method applies. From theseminal paper by Benders [8], it is known that the Benders methodgives an optimal solution to the original problem in a finite numberof steps. Since at the first iteration, the added constraints are a sub-set of the second-stage constraints, the optimal solution is guaran-teed within the feasible region of the initial iteration. So thegenerated feasibility cuts and optimality cuts are both valid forthe original problem such that the Benders method gives an opti-mal solution in a finite number of iterations as indicated inFig. 1. h

J. Wang et al. / Applied Energy 105 (2013) 418–422 421

The proof can also be obtained as a special case of the two-stagestochastic programs from the book by Birge and Louveaux [11],where the optimal solution computed in a finite number of itera-tions is proved for a generic two-stage stochastic program wherethe first-stage problem is an integer program.

0.6

0.8

1

1.2

1.4

1.6

1.8

2 x 104

Cos

t

UCED

4. Numerical example

An IEEE 118 bus system is used to illustrate the effectiveness ofthe proposed formulation and solution approach. In the system,there are 92 generators and we assume one wind farm is locatedat bus 59. We assume 10 min as the unit sub-hourly dispatch res-olution. In other words, the economic dispatch is assumed to bedone sub-hourly within each hour. It should be noted that thesesub-hourly dispatch constraints are embedded in the scenariosand associated with the master UC problem. The simulation hori-zon is 6 h. The number of wind power scenarios is 10. A time-adap-tive quantile-copula estimator based kernel density forecast isused to generate high-quality hourly and sub-hourly wind powerscenarios [9]. The cost for the unserved load is 3000$/MW h. Toillustrate the effectiveness of the proposed model, we first obtainthe UC binary solutions, then we run economic dispatch of thesame sub-hourly resolution with the unit commitment solutionsobtained from the traditional hourly model and the proposedsub-hourly model for comparison. The 10-min wind power pointforecast used in this economic dispatch model and the point fore-cast used in the traditional hourly deterministic UC are obtainedfrom National Renewable Energy Laboratory [10].

Table 1 below illustrates the differences between the traditionaldeterministic unit commitment model with hourly dispatch con-straints and the proposed model in terms of model sizes. If the sim-ulation period is 1 h, the number of decision variables for theproposed model is 4066, which doubles the size of the traditionalmodel. The number of constraints for the proposed model is79062, which is more than four times that for the traditional mod-el. It can be seen that the size of the proposed sub-hourly modelincreases a lot compared with the traditional model and the diffi-culty of solving the resultant MILP problem to optimality increasesnonlinearly. Hence the modified Benders decomposition methoddescribed above is needed to exploit the L-shaped structure ofthe MILP problem. Without using the proposed modified Bendersdecomposition approach, the problem is solved longer than usingthe standard approach.

The key procedure of the modified Benders decomposition is toconstruct the initial master problem. The selection of the second-stage constraints included into the initial master problem is case-dependant. The general guideline is to select a small subset ofthe second-stage constraints to produce a good initial feasible solu-tion while not increasing the computational requirement of the ex-tended master problem. In our case we select a subset of thetransmission line, power balance, ramp and capacity constraintsassociated with a sub-hourly period in a scenario such that the ex-tended constraint set of the master problem better approximatesthe feasible region of the original problem. Among the differentchoices for this test instance we have tried, the constraint subsetwith the first sub-hourly period in the first scenario gives the bestresult. To illustrate the impact of sub-hourly constraints, weconducted two case studies.

Table 1Model comparison between traditional and proposed models.

Case Number of decision variables Number of constraints

Traditional model 2006 14,162Proposed model 4066 79,062

CASE 1 The first case study compares four types of instances:deterministic UC of hourly resolution, deterministic UCof sub-hourly resolution, stochastic UC of hourly resolu-tion and stochastic UC of sub-hourly resolution in termsof UC and dispatch costs. In Fig. 2, ‘D’ indicates deter-ministic UC and ’S’ represents stochastic UC. The differ-ence between deterministic and stochastic instanceshere is the wind power representation. There is onlyone scenario for deterministic UC and ten scenarios forstochastic UC. We use 10 min as the sub-hourly resolu-tion. It shows that for both deterministic and stochasticinstances, the solution with sub-hourly intervals cansignificantly reduce the dispatch cost. For hourly inter-val instances, there is little difference of dispatch costbetween deterministic and stochastic instances. But forsub-hourly interval instances, there is relatively largerdifference, which suggests that increasing interval reso-lution has a larger impact than the increasing number ofscenarios with respect to reducing the dispatch cost.

CASE 2 The second case compares different sub-hourly resolu-tion as 30, 15 and 10 min with the proposed model. InFig. 3, it is clear that the finer the sub-hourly intervalresolution, the more dispatch cost can be reduced. Forthe 30 min case, the dispatch cost is more than eighttimes that for the other two cases. However, as thesub-hourly interval resolution gets finer, the reductionof dispatch cost gets smaller. For this case, the dispatchcost is the same for 15 and 10 min intervals. For the totalUC cost, the finer sub-hourly interval contributes toincreased UC cost, which can be explained by the factthat the increased number of constraints need to besatisfied.

Following the two case studies above, we focus on the sub-hourly model with 10-min resolution and make a detailed investi-gation of the number of committed units and curtailment amountwith respect to UC and dispatch costs. Table 2 below illustrates thedifferences in dispatch cost between the traditional hourly modeland the proposed sub-hourly model with 10-min resolution.

The second column in Table 2 compares the number of commit-ted units in the system. The traditional model requires 13 units tobe on-line while the proposed model commits 19 units. This is be-cause the sub-hourly model imposes additional sub-hourly secu-rity and dispatch requirements of the system at each sub-periodin the scenarios. Also as shown in the third column of Table 2,the sub-hourly model results in more total cost at the UC stage

D + hourly D +Sub−hourly S + houly S+ Sub−houlry0

0.2

0.4

Fig. 2. Comparison of four groups of instances.

30 min 15 min 10 min0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 x 104

Subhourly resolution

Cos

t

UCED

Fig. 3. Comparison of different sub-hourly resolutions.

Table 2Comparisons among different models and sub-hourly resolutions.

Case Number ofcommittedunits

UC cost ($) Dispatchcost ($)

Loadcurtailment(MW)

D + hourly 8 2835.6 18,428 4.9D + sub-hourly 11 3192.3 13,290 3.28S + hourly 13 3307.9 18,428 4.97S + sub-hourly (10 min) 19 5364.6 2228 0S + sub-hourly (15 min) 19 4672.9 2228 0S + sub-hourly (30 min) 15 3450.7 18428 4.97

422 J. Wang et al. / Applied Energy 105 (2013) 418–422

since there are more constraints in the sub-hourly model than thehourly model. The difference of total cost corresponds to the in-creased number of on-line generators between the proposed modeland the traditional model. As mentioned above, the real windpower output is used here in contrast to the forecasts used inUC. As the traditional model does not capture the sub-hourly var-iability of wind power, the total cost and load curtailment are sig-nificantly higher than the proposed model as shown in the last twocolumns in Table 2. This comparison shows the evidence of theeconomic benefits from the adopted sub-hourly constraints inthe proposed model. The numerical results also indicate that ifthe sub-hourly time resolution is not small enough, the impact of

uncertainty may still be difficult to reduce as shown by the sameamount of load curtailment by the 30-min and hourly models.

5. Conclusion

In this paper, we explore the impacts of finer granularity of theunit commitment model by incorporating sub-hourly dispatchconstraints. These additional constraints can reflect the sub-hourlyvariability of renewable energy sources. A modified Bendersdecomposition method is used to reduce the computational costsfrom these additional constraints. The computational experimentsshow that the dispatch cost can be significantly reduced if the unitcommitment decision from the proposed sub-hourly model is usedin comparison with the traditional model of hourly resolution. Theresults illustrate the potential practical use of the proposed meth-od in integrating large amounts of renewables in power systemoperations with moderate modifications to the standard Bendersdecomposition method.

Acknowledgment

Work supported by the U. S. Department of Energy [Office ofElectricity Delivery and Energy Reliability] under Contract No.DE-AC02-06CH11357.

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