Multiarea Stochastic Unit Commitment for High Wind Penetration …€¦ · methodological contribution of the paper is the presentation of a computationally tractable approach for

OPERATIONS RESEARCHVol. 61, No. 3, May–June 2013, pp. 578–592ISSN 0030-364X (print) � ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.2013.1174

© 2013 INFORMS

Multiarea Stochastic Unit Commitment forHigh Wind Penetration in a Transmission

Constrained Network

Anthony PapavasiliouDepartment of Mathematical Engineering, Center for Operations Research and Econometrics, Catholic University of Louvain,

B-1348 Louvain la Neuve, Belgium, [email protected]

Shmuel S. OrenDepartment of Industrial Engineering and Operations Research, University of California at Berkeley, Berkeley, California 94720,

[email protected]

In this paper we present a unit commitment model for studying the impact of large-scale wind integration in power systemswith transmission constraints and system component failures. The model is formulated as a two-stage stochastic programwith uncertain wind production in various locations of the network as well as generator and transmission line failures.We present a scenario selection algorithm for selecting and weighing wind power production scenarios and compositeelement failures, and we provide a parallel dual decomposition algorithm for solving the resulting mixed-integer program.We validate the proposed scenario selection algorithm by demonstrating that it outperforms alternative reserve commitmentapproaches in a 225 bus model of California with 130 generators and 375 transmission lines. We use our model toquantify day-ahead generator capacity commitment, operating cost impacts, and renewable energy utilization levels forvarious degrees of wind power integration. We then demonstrate that failing to account for transmission constraints andcontingencies can result in significant errors in assessing the economic impacts of renewable energy integration.

Subject classifications : unit commitment; stochastic programming; wind power; transmission constraints.Area of review : Environment, Energy, and Sustainability.History : Received July 2011; revisions received August 2012, October 2012, January 2013; accepted February 2013.

Published online in Articles in Advance May 24, 2013.

1. IntroductionThe large-scale integration of renewable energy resourcessuch as wind and solar power in power systems is limitedby two adverse characteristics of renewable power supply.Certain renewable sources are unpredictable and, in con-trast to conventional generators, the energy source is notcontrollable. Because of the need for maintaining a con-stant balance between supply and demand in the systemand also because of the prohibitively high cost of electric-ity storage, these two adverse characteristics greatly influ-ence our ability to utilize renewable resources at a largescale by (a) imposing costs on power system operations,1

(b) reducing the amount of renewable power that can beabsorbed in the network and, (c) necessitating the deploy-ment of reserves that can ensure the reliable operation ofthe system. The focus of this paper is on quantifying theseimpacts in the presence of transmission network conges-tion and contingencies through the use of a stochastic unitcommitment model.

Because of the large-scale integration of renewableenergy sources, the solution to the stochastic unit com-mitment problem is becoming highly relevant to theelectric power industry. Given the current organization

of electricity markets (e.g., the two-settlement systemsin California, Texas, the Midwest, Pennsylvania-Jersey-Maryland, New York, the United Kingdom, the Nordicregion, and New Zealand to name a few examples), stochas-tic unit commitment simulates an idealized two-settlementsystem and provides a good indication as to how to pro-cure reserves in the system. Moreover, the model is relevantin an operational setting in the residual unit commitmentprocess. Residual unit commitment takes place after theday-ahead market closes, and in this phase the independentsystem operator evaluates if the market has provided suffi-cient resources to meet the needs of the real-time electricitymarket in order to deal with supply uncertainty, demanduncertainty, and network element failures.

The paper contributes to existing literature in terms ofboth modeling and methodology. The modeling contribu-tion of the paper relies on the use of authentic and detaileddata for evaluating the cost of large-scale renewable energyintegration and its impact on day-ahead reserve require-ments while accounting for transmission constrains andcomponent failures. Because of the fact that the modelincludes transmission constraints, it is necessary to modelrenewable power production in each region separatelyand capture the spatial correlations of renewable power

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Papavasiliou and Oren: Multiarea Stochastic Unit CommitmentOperations Research 61(3), pp. 578–592, © 2013 INFORMS 579

production, as well as numerous other features of therenewable power production process (temporal autocorrela-tion, nonlinearity in the conversion of wind speed to windpower, non-Gaussian distribution of wind speed data). Themethodological contribution of the paper is the presentationof a computationally tractable approach for implementingstochastic unit commitment and comparing it to alternativeday-ahead reserve commitment approaches. Computationaltractability is achieved by the development of a scenarioselection procedure inspired by importance sampling forreducing the representation of uncertainty, as well as theimplementation of a parallel decomposition algorithm forsolving the stochastic unit commitment problem. In thefollowing section we review relevant literature regardingthe modeling and computational methods for assessing theimpacts of uncertainty on power system operations, and weelucidate how our paper contributes in each of these areas.

2. Literature Review

2.1. Modeling

The operation of power systems under uncertainty can beperceived as a multistage decision process, where resourcesare committed prior to an operating interval and decisionsare updated as the operating interval approaches and con-ditions in the system are revealed. We follow the approachof Ruiz et al. (2009b) in modeling system operations as atwo-stage decision process. The first-stage decisions rep-resent the day-ahead commitment of generators based ondemand forecasts. Subsequently, uncertainty is revealed andin the second stage the commitment of fast-respondingunits and the dispatch of all committed units is updatedin order to respond to system conditions. In deregulatedpower systems (e.g., the two-settlement markets referencedearlier), the first and second stage of the model can be inter-preted as simulating the day-ahead and real-time markets,respectively.

A stochastic unit commitment model is especially appro-priate for quantifying the impacts of renewable energy inte-gration on renewable energy utilization, operating costs,and day-ahead generating capacity requirements. Renew-able power utilization is an explicit decision variable inthe problem, operating costs are quantified in the objectivefunction of the problem, and reserve capacity requirementsare quantified indirectly by the fact that the commitmentof generators is an endogenous decision variable in theproblem. The estimation of day-ahead generating capac-ity requirements is an especially challenging aspect of ouranalysis. To commit reserves, system operators and analystsoften resort to suboptimal ad hoc rules in a deterministicformulation of the unit commitment problem. These com-mitment policies are not calibrated to an environment withlarge amounts of renewable resources and may result ininefficiencies and miscalculation of the economic impactsof renewable energy integration. These drawbacks havemotivated the work of Piwko et al. (2010) who propose

adaptive reserve commitment approaches in order to over-come the drawbacks of current practice in an environmentof large-scale renewable energy integration.

2.1.1. Transmission Constraints and Contingencies.Transmission constraints result in shedding power supplybecause of the fact that the system, often, cannot sup-port power flows during periods of increased renewableenergy supply. This phenomenon has been observed in pre-vious studies by Sioshansi and Short (2009) and Moraleset al. (2009b) and it may result in waste of renewableenergy supply, thereby undermining the benefits of renew-able energy integration. Moreover, transmission constraintsgreatly complicate the task of determining reserves in vari-ous locations of the system and render the use of stochasticunit commitment especially relevant. Arroyo and Galiana(2005) demonstrate that an ad hoc allocation of reservesin various locations of a transmission-constrained networkresults in suboptimal system performance. The complexinfluence of transmission constraints on locational reserverequirements is also demonstrated by Galiana et al. (2005),as well as Bouffard et al. (2005), on smaller scale systems.

Alternative formulations of the unit commitment prob-lem have been recently proposed in order to address uncer-tainty. This work is motivated by the fact that systemoperators tend to operate the system so as to protect againstworst-case outcomes, and also by the fact that the stochas-tic programming formulation requires detailed knowledgeabout the underlying uncertainty since it requires the gen-eration and weighting of scenarios. Ozturk et al. (2004)formulate a chance-constrained optimization of the unitcommitment problem without transmission constraints andramping constraints. Jiang et al. (2012) use a robust opti-mization formulation of the unit commitment problem,where locational demand is assumed to obey the polyhedraland cardinal uncertainty models defined by Bertsimas andSim (2004). The authors present a Benders’ decompositionalgorithm for solving the problem with transmission andramping constraints. A similar formulation is proposed byBertsimas et al. (2013). However, neither of these modelsaccounts for contingencies explicitly.

The work presented in this paper extends the modelof Papavasiliou et al. (2011) by introducing transmissionnetwork constraints as well as generator and transmissionline failures. Both features are crucial in accurately assess-ing the economic impact of renewable energy integration.As we demonstrate in §7, ignoring transmission constraintsand contingencies can result in a significant overestimationof the day-ahead generation capacity savings and operatingcost savings of wind power. In line with existing determin-istic and stochastic unit commitment literature, a DC loss-less representation of the transmission network has beenused that ignores reactive power, voltage limits, and linelosses. The results of such optimization are, therefore,regarded as advisory and in practical applications furtherpost-processing of such results is used to assure AC powerflow feasibility and voltage stability.

Papavasiliou and Oren: Multiarea Stochastic Unit Commitment580 Operations Research 61(3), pp. 578–592, © 2013 INFORMS

2.1.2. Multiarea Wind Modeling. Although transmis-sion constraints and contingencies cannot be ignored whenanalyzing the economic impacts of renewable energyintegration, they also introduce certain modeling and com-putational challenges that are addressed in this paper.The modeling of wind power production as an aggregateresource located in a single bus is not adequate. Instead,it is necessary to develop a wind power production modelthat accounts for the nonlinear relationship between windspeed and wind power, the diurnal and seasonal charac-teristics of wind power supply, as well as the spatial andshort-term temporal correlation of wind power supply.

Early work on the time series modeling of wind powerproduction in order to address short-term temporal corre-lation and the nonlinear relationship between wind speedand wind power was performed by Brown et al. (1984) andlater by Torres et al. (2005). In §5 we present a multiareawind production model that captures both the temporal aswell as spatial correlations of wind power production thatare observed in the data set. The modeling and calibrationmethodology described in this section extends the approachof Morales et al. (2010) by also accounting for the diurnaland seasonal patterns of wind power supply.

2.2. Methodology

Recently, various studies have utilized unit commitmentmodels for assessing the impacts of wind integration.A variety of modeling approaches and solution strategiesare employed in these studies. Ruiz et al. (2009a) usethe two-stage stochastic unit commitment model of Ruizet al. (2009b) to analyze the impact of wind integrationin the Colorado power system, without considering trans-mission constraints. A method employed in Ruiz et al.(2009b) that we also adopt in our paper is to classify gener-ators as either “fast” or “slow” resources. Slow generatorsare committed in the first stage, and fast generators canbe committed in the second stage of recourse. Sioshansiand Short (2009) study the impact of wind integrationin the Electricity Reliability Council of Texas (ERCOT)system with a deterministic unit commitment model thatincludes transmission constraints. Wang et al. (2008) usea two-stage formulation that ensures a feasible dispatchin the second stage for all possible wind production out-comes given first-stage commitment decisions. Recent workby Constantinescu et al. (2011) also uses the two-stagestochastic programming formulation of Ruiz et al. (2009b).The authors test their model on a system with 10 genera-tors without transmission constraints. Tuohy et al. (2009)develop a stochastic unit commitment model that accountsfor load and wind uncertainty and perform a simulationof the Irish power system. Morales et al. (2009a) use atwo-stage stochastic programming model for studying windintegration in the IEEE RTS 96 model of Grigg et al. (1999)that includes transmission constraints but not contingencies.The model assumes explicit bids for spinning and nonspin-ning reserves. A similar two-stage stochastic programming

model is used by Bouffard and Galiana (2008) for a casestudy of wind integration in a test system with four gen-erators without transmission constraints and contingencies.With the exception of Wang et al. (2008), none of the afore-mentioned wind integration studies employ decompositiontechniques, which limits the size of the models that arestudied. By comparison, our model uses a scenario selec-tion technique inspired by importance sampling in conjunc-tion with a dual decomposition algorithm implemented in aparallel environment in order to examine a reduced modelof California with 130 generators, 375 lines, and 225 buses.

2.2.1. Scenario Selection. In §6 we address the issueof scenario selection. As in the case of locational windpower production, the introduction of generation contingen-cies requires identifying generator failures depending onwhere they occur in the network. This should be contrastedwith modeling the loss of aggregate generation capacitywhich is a common approach in simpler models withouttransmission constraints such as Takriti et al. (1996) andRuiz et al. (2009a). Because of the fact that wind produc-tion uncertainty is interleaved with contingencies, an enor-mous set of scenarios needs to be considered. To developa systematic approach for selecting and weighing scenar-ios, we introduce a scenario selection algorithm inspiredby importance sampling techniques that selects uncertainscenarios on the basis of their likelihood of occurrence andthe severity of their impact on operating costs. The algo-rithm that we present formalizes the intuition used in thescenario selection algorithm of Papavasiliou et al. (2011)and is shown to outperform alternative deterministic andstochastic day-ahead commitment rules in all the case stud-ies that are presented in §7.

2.2.2. Decomposition of the Stochastic Unit Commit-ment Problem. Even with a limited scenario set, theresulting problem requires the use of decomposition tech-niques in order to achieve computational tractability. Theuse of decomposition algorithms for solving the stochasticunit commitment problem was pioneered by Takriti et al.(1996), who use a multistage stochastic unit commitmentformulation for studying load uncertainty and generatorfailures. The authors use the progressive hedging algorithmof Rockafellar and Wets (1991) to decompose the stochas-tic formulation to single-period subproblems. In Carpentieret al. (1996) the authors use the augmented Lagrangianalgorithm to decompose a multistage stochastic program tosingle-generator subproblems. Nowak and Römisch (2000)develop a Lagrangian decomposition algorithm for optimiz-ing the operations of a hydrothermal system under loaduncertainty. Shiina and Birge (2004) develop a column gen-eration algorithm for decomposing a multistage stochas-tic program into single-generator subproblems. In §4we present an extension of the decomposition algorithm ofPapavasiliou et al. (2011) for decomposing the stochasticunit commitment problem to individual scenarios and solv-ing the more general model presented in this paper.


3. Unit Commitment andEconomic Dispatch

In our study we consider a set of generation resources Gthat is partitioned among a set of slow generators Gs andfast generators Gf . Uncertainty is modeled as a discrete setof realizations S. In the stochastic unit commitment model,the commitment of slow generators is a first-stage deci-sion that cannot be altered in the second stage, whereas thecommitment of fast generators and the production level ofall generators can be adjusted once a realization s ∈ S isobserved. Analogously, in the deterministic model, whichfollows Sioshansi and Short (2009), slow generators canonly provide slow reserves whereas fast generators can pro-vide slow as well as fast reserves.

3.1. Notation

We begin this section by introducing the notation that isused in the subsequent models.

SetsG: set of all generators, Gs: subset of slow generators,

Gf : subset of fast generators;Gn: set of generators that are located in bus n;S: set of scenarios, T : set of time periods, L: set of lines,

N : set of nodes;LIn =8l ∈ L2 l = 4k1n51 k ∈ N9, LOn = 8l ∈ L: l = 4n1 k5,

k ∈N9;IG: set of import groups, IGj : set of lines in import

group j .

Decision variablesugst: commitment, vgst: startup, pgst: production of genera-

tor g in scenario s, period t;�nst: phase angle at bus n in scenario s, period t;wgt: commitment, zgt: start-up of slow generator g in

period t;sgt: slow reserve, fgt: fast reserve provided by generator g

in period t;elst: power flow on line l in scenario s, period t.

Parameters�s: probability of scenario sKg: minimum load cost, Sg: start-up cost, Cg: marginal

cost of generator g;Dnst: demand in bus n, scenario s, period t;P+gs1P

−gs: minimum and maximum capacity of generator gin scenario s;

R+g 1R

−g : minimum and maximum ramping of generator g;

UTg: minimum up time, DTg: minimum down time of gen-erator g;

Treqt : total reserve requirement, Sreq

t : slow reserve require-ment in period t;

Bls: susceptance of line l in scenario s;TCl: maximum capacity of line l;FRg: fast reserve limit of generator g;ICj : maximum capacity of import group j;�jl: polarity of line l in import group j .

3.2. Stochastic Unit Commitment

The stochastic unit commitment (SUC) problem can bestated as follows:

(SUC): min∑

g∈G

∑

s∈S

∑

t∈T

�s4Kgugst + Sgvgst +Cgpgst5 (1)

s0t0∑

l∈LIn

elst +∑

g∈Gn

pgst =Dnst +∑

l∈LOn

elst1

n ∈N1 s ∈ S1 t ∈ T 3 (2)

elst = Bls4�nst − �mst51 l = 4m1n5 ∈ L1

s ∈ S1 t ∈ T 3 (3)

−TCl ¶ elst ¶ TCl1 l ∈ L1 s ∈ S1

t ∈ T 3 (4)

P−

gsugst ¶ pgst ¶ P+

gsugst1 g ∈G1 s ∈ S1

t ∈ T 3 (5)

−R−

g ¶ pgst −pgs1 t−1 ¶R+

g 1 g ∈G1 s ∈ S1

t ∈ T 3 (6)t∑

q=t−UTg+1

zgq ¶wgt1 g ∈Gs1 t ¾UTg3 (7)

t+DTg∑

q=t+1

zgq ¶ 1 −wgt1 g ∈Gs1

t ¶ �T � −DTg3 (8)

t∑

q=t−UTg+1

vgsq ¶ ugst1 g ∈Gf 1 s ∈ S1

t ¾UTg3 (9)

t+DTg∑

q=t+1

vgsq ¶ 1 − ugst1 g ∈Gf 1 s ∈ S1

t ¶ �T � −DTg3 (10)

zgt ¶ 11 g ∈Gs1 t ∈ T 3 (11)

vgst ¶ 11 g ∈G1 s ∈ S1 t ∈ T 3 (12)

zgt ¾wgt −wg1 t−11 g ∈Gs1 t ∈ T 3 (13)

vgst ¾ ugst − ugs1 t−11 g ∈Gf 1 s ∈ S1

t ∈ T 3 (14)

�sugst =�swgt1 g ∈Gs1 s ∈ S1

t ∈ T 3 (15)

�svgst =�szgt1 g ∈Gs1 s ∈ S1 t ∈ T 3 (16)

pgst1 vgst ¾ 01 ugst ∈ 801191 g ∈G1

s ∈ S1 t ∈ T 3 (17)

zgt ¾ 01 wgt ∈ 801191g ∈Gs1 t ∈ T 0 (18)

The objective of the problem is to minimize expectedoperating costs, which consist of minimum load costs, start-up costs, and fuel costs. Equation (2) requires balancingthe amount of power that flows into and out of each bus.


Equation (3) represents a linearized, lossless model of thepower flow equations (Kirchhoff’s law) according to whichthe power flow on a line l is proportional to the phase angledifference between the two end buses of the line. The pro-portionality factor Bls is the susceptance of line l underscenario s, where Bls = 0 for scenarios s ∈ S in which line lis out of service, thereby forcing the flow in line l to equalzero. Constraints (4) define thermal capacity limits on thetransmission lines. Constraints (5) impose minimum andmaximum generator capacity limits. Similarly to the caseof transmission line failures, P+

gs = 0 and P−gs = 0 holds for

those scenarios s ∈ S in which generator g is out of ser-vice, thereby forcing power production for generator g toequal zero. The constraints defined by Equation (6) repre-sent ramping constraints on the rate of change of generatoroutput. Equations (7)–(10) represent the minimum up anddown time constraints of both fast and slow generators.Constraints (11) and (12) place upper bounds on the start-up variables. Note that although start-up variables are inprinciple binary, following O’Neill et al. (2010) we are ableto relax them as continuous variables since they are penal-ized in the objective function of the model, thus forcingthem to an extreme value. In this way we are able toreduce the size of the branch-and-bound tree. The transitionrule for generator start-up variables is imposed in Equa-tions (13) and (14). Equations (15) and (16) are the nonan-ticipativity constraints that imply that, for slow generators,second-stage commitment and start-up 4ugst1 vgst5 needs tobe consistent with the first-stage commitment and start-up4wgt1 zgt5 under each scenario s ∈ S. Integrality and nonneg-ativity constraints are imposed in Equations (17) and (18).Note that the model determines the amount of generationcapacity that needs to be committed in the day-ahead timeframe by endogenously accounting for uncertainty withinthe model formulation. This motivates the use of the two-stage stochastic unit commitment model for quantifying theamount of day-ahead reserve requirements that are imposedby system operation uncertainty.

3.3. Deterministic Unit Commitment

A simpler approach to protect the system against uncer-tainty is to introduce exogenous requirements on excessgeneration capacity rather than explicitly modeling the abil-ity of the system to observe and respond to uncertainty.More specifically, in contrast to modeling renewable pro-duction and load uncertainty, the deterministic formula-tion imposes ad hoc requirements on reserve supply, whichis generation capacity that can be called upon in thecase of renewable supply and load fluctuations as well astransmission line and generation outages. Likewise, ratherthan modeling contingencies explicitly, the deterministicmodel imposes exogenous import constraints that ensurethat the system can withstand the failure of major gen-eration resources within load pockets as well as the fail-ure of major transmission interties. Since reserve require-ments and import limits in practice have relied upon user

experience, these procedures may need to be revised inlight of new uncertain generation sources. For the deter-ministic formulation, we follow the model of Sioshansi andShort (2009):

(DUC): min∑

g∈G

∑

t∈T

4Kgwgt + Sgzgt +Cgpgt5 (19)

s0t0∑

l∈LIn

elt +∑

g∈Gn

pgt =Dnt +∑

l∈LOn

elt1

n ∈N1 t ∈ T 3 (20)

−TCl ¶ elt ¶ TCl1 l ∈ L1 t ∈ T 3 (21)

elt = Bl4�nt − �mt51 l = 4m1n5 ∈ L1

t ∈ T 3 (22)

pgt + fgt ¶ P+

g wgt1 g ∈G1 t ∈ T 3 (23)

pgt + sgt + fgt ¶ P+

g 1 g ∈G1 t ∈ T 3 (24)

pgt ¾ P−

g wgt1 g ∈G1 t ∈ T 3 (25)

pgt −pg1 t−1 + sgt ¶R+

g 1

g ∈G1 t ∈ T 3 (26)

pg1 t−1 −pgt ¶R−

g 1 g ∈G1 t ∈ T 3 (27)∑

g∈Gf

fgt +∑

g∈G

sgt ¾ T reqt 1 t ∈ T 3 (28)

fgt ¶ FRg1 g ∈G1 t ∈ T 3 (29)∑

g∈G

sgt ¾ Sreqt 1 t ∈ T 3 (30)

∑

l∈IGj

�jlelt ¶ ICj1 j ∈ IG1 t ∈ T 3 (31)

t∑

q=t−UTg+1

zgq ¶wgt1 g ∈G1

t ¾UTg3 (32)

t+DTg∑

q=t+1

zgq ¶ 1 −wgt1 g ∈G1

t ¶ �T � −DTg3 (33)

zgt ¶ 11 g ∈G1 t ∈ T 3 (34)

zgt ¾wgt −wg1 t−11 g ∈G1 t ∈ T 3 (35)

pgt1 zgt1 sgt1 fgt1¾ 01 wgt ∈ 801191g ∈G1 t ∈ T 0 (36)

Note that decision variables in the resulting model arenot contingent on scenarios, thereby reducing the size of themodel. The maximum capacity constraint in Equation (23)is modified to account for the provision of fast reserves andEquation (24) is added to account for the provision of fastand slow reserves. The total reserve requirement is imposedin Equation (28) and an upper limit on the provision offast reserves is imposed in Equation (29). The online slowreserve requirement is imposed in Equation (30) and theimport constraints are imposed in Equation (31).


3.4. Economic Dispatch

The deterministic and stochastic unit commitment modelsrepresent two different methods for committing slow gener-ation resources. These resources need to have their sched-ules determined in the day-ahead scheduling time framebecause of their limited operating flexibility. Once slowresources are committed, the performance of the systemis tested by performing Monte Carlo simulations of itsresponse to net demand and contingency outcomes, giventhe unit commitment schedule of slow generators. The netdemand outcomes are generated using the multiarea windpower production time series model presented in §5 andthe generation of contingencies is described in §7.3. Theeconomic dispatch of units for each outcome c requiressolving the following problem:

4EDc52 min∑

g∈G

∑

t∈T

4Kgwgt + Sgzgt +Cgpgt5 (37)

s0t0∑

l∈LIn

elt +∑

g∈Gn

pgt =Dnct +∑

l∈LOn

elt1

n ∈N1 t ∈ T 3 (38)

elt = Blc4�nt − �mt51

l = 4m1n5 ∈ L1 t ∈ T 3 (39)

P−

gcugt ¶ pgt ¶ P+

gcugt1 g ∈G1 t ∈ T 3 (40)

pgt −pg1 t−1 ¶R+

g 1 g ∈G1 t ∈ T 3 (41)

wgt =w?gt1 g ∈Gs1 t ∈ T 3 (42)

zgt = z?gt1 g ∈Gs1 t ∈ T 3 (43)

42151 42751 43251 43351 43451 4355

pgt1 zgt1¾ 01 wgt ∈ 801191g ∈G1 t ∈ T 0 (44)

Equations (42) and (43) set the commitment of slowgenerators to the optimal solution of the unit commitmentproblem.

4. Decomposition AlgorithmBy dualizing the constraints of Equations (15) and (16) weobtain the following Lagrangian:

L=∑

g∈G

∑

s∈S

∑

t∈T

�s4Kgugst +Sgvgst +Cgpgst5

+∑

g∈Gs

∑

s∈S

∑

t∈T

�s4�gst4ugst −wgt5+�gst4vgst −zgt550 (45)

We can then decompose the Lagrangian to one subprob-lem for each scenario, that determines optimal second-stagedecisions:

4P2s52 min∑

g∈G

∑

t∈T

�s4Kgugst +Sgvgst +Cgpgst5

+∑

g∈Gs

∑

t∈T

�s4�gstugst +�gstvgst5 (46)

s0t0 4251 4351 4451 4551 4651 4951 41051 41251 4145

pgst ¾01 vgst ¾01 ugst ∈801191

g∈G1 t∈T 0 (47)

Note that the constraint vgst ¶ 11 g ∈Gs of Equation (12),although redundant for the model (SUC), is necessary forbounding P2s when applying the decomposition algorithm.We also obtain a single subproblem that determines optimalfirst-stage decisions:

4P152 min −∑

g∈Gs

∑

s∈S

∑

t∈T

�s4�gstwgt+�gstzgt5 (48)

s0t0 4751 4851 41151 4135

wgt ∈801191 zgt¾01 g∈Gs1 t∈T 0 (49)

The dual variables are updated as follows:

�k+1gst =�k

gst +�k�s4wkgt − uk

gst51

g ∈Gs1 s ∈ S1 t ∈ T 3 (50)

�k+1gst = �k

gst +�k�s4zkgt − vkgst51

g ∈Gs1 s ∈ S1 t ∈ T 1 (51)

where wkgt and zkgt are the optimal solutions of (P1) at iter-

ation k; ukgst and vkgst; are the optimal solutions of (P2s)

at iteration k; and �k is the step size at iteration k. Wecould have relaxed only the nonanticipativity constraint onthe commitment variables. The advantage of also relax-ing the nonanticipativity constraint on the start-up variablesis that (P2s), s ∈ S, is a smaller problem, since the con-straints on the unit commitment of the slow generators area part of (P1). An additional advantage of this choice ofdecomposition is that, at each step, the slow generator unitcommitment solutions of the first subproblem can be usedfor generating a feasible solution to the original problemby solving an economic dispatch problem (EDs), Equa-tions (37)–(44), for each scenario s ∈ S. As a result, at eachstep of the algorithm we get an upper bound on the opti-mal solution that can be used for terminating the algorithm,as well as a feasible schedule. This should be contrastedwith the case where we would have chosen to relax onlythe nonanticipativity constraints on the unit commitmentvariables, and not the start-up variables.

An important feature of the proposed algorithm is thefact that the second-stage subproblems (P2s) and the eco-nomic dispatch problems (EDs) can be solved in parallel.As we discuss in §7.5, we have implemented a parallelalgorithm for the problem in a cluster of 1,152 nodes, witheight CPUs per node at 2.4 GHz and 10 GB per node. Thestep size rule follows Fisher (1985) and Held et al. (1974)and is presented in Papavasiliou et al. (2011).


5. Multiarea Wind Power ModelBecause of the highly nonlinear but static relationshipbetween wind speed and wind power production it is com-mon in the wind power modeling literature to model windspeed rather than wind power with time series models andthen convert speed to power using a static conversion curve.The task of modeling wind speed consists of removing sea-sonal and daily patterns, fitting the data to a parametric ornonparametric distribution and fitting an appropriate timeseries model to the underlying “noise” in order to capturethe strong temporal dependency of wind speed.

Given a multiarea data set ykt , where k indexes locationand t indexes time period, the first step is to remove hourlyand seasonal patterns by subtracting the hourly mean anddividing by the hourly standard deviation in order to obtaina stationary data set ySkt for each location.2

We then filter the data set in order to obtain an approx-imately stationary Gaussian data set yGS

kt . Brown et al.(1984), Torres et al. (2005), and Morales et al. (2010)use this approach for transforming Weibull-distributed windspeed data to Gaussian data, and Callaway (2010) uses anonparametric transformation. Since no single parametricdistribution provides a close fit for the observed data in alllocations of our data set, we fit an empirical distributionFk4 · 5 to the data for each location k. The resulting dataset yGS

kt can be modeled by an autoregressive process:

yGSk1 t+1 =

p∑

j=0

�kjyGSt−j + �kt1 (52)

where �kt is the estimated noise and �kj1 j ∈ 811 0 0 0 1 p9 arethe estimated coefficients of the autoregressive model. Thecalibration process is summarized in the following steps:

Step (a). Remove systematic seasonal and diurnal effects

ySkt =ykt − �kmt

�kmt

1

where ykt is the data, ySkt is the transformed data that isstationary, and �kmt and �kmt are the sample mean and stan-dard deviation, respectively, for location k, season m, andhour t.

Step (b). Transform the data in order to obtain aGaussian distribution on the data set

yGSkt =N−14Fk4y

Skt551

where yGSkt is the transformed data that is stationary

Gaussian distributed, N−14 · 5 is the inverse of the cumu-lative distribution function of the normal distribution, andFk is the cumulative function of the fit for the data inlocation k.

Step (c). Use the Yule-Walker equations (Box andJenkins 1976) to estimate the autoregressive parameters �kj

and covariance matrix è of the estimated noise in theautoregressive model of Equation (52). We have observed

that the inclusion of spatial correlations in the covariancematrix è improves the fit of the model, compared to usinga diagonal covariance matrix that only captures the vari-ance of the noise in each location but ignores its correlationwith the noise in other locations.

Once the parameters of the statistical model are esti-mated, the inverse process can be performed for simulatingwind speed. The relationship between wind speed and windpower typically depends on wind turbine size, manufac-turer design, and various other factors, and an aggregatedpower curve can be used to represent a group of turbines.To simulate wind power production, we use a piecewiselinear approximation of such an aggregate power curve foreach location. We provide details about the data used in ourmodel and the goodness of fit in §EC.2 of the electroniccompanion (available as supplemental material at http://dx.doi.org/10.1287/opre.2013.1174).

6. Scenario SelectionThe challenge of selecting scenarios for the stochastic unitcommitment problem is to discover a small number ofrepresentative outcomes that properly guide the stochas-tic program to produce a unit commitment schedule thatimproves average costs, as compared to a schedule deter-mined by solving a deterministic unit commitment model.The basic trade-off that needs to be balanced in dispatch-ing fast reserves is the flexibility that fast units offer inutilizing renewable generation versus their higher operat-ing costs. Fast generators typically incur higher marginalfuel costs. In addition, the start-up and minimum load costsof these units are similar to those of slow units, howevertheir capacity is smaller; hence, their start-up and minimumload cost per unit of capacity is greater than that of slowgenerators. The advantage of largely relying on fast unitsis that the system is capable of discarding less renewablepower, which results in significant savings in fuel costs.Unlike fast generators that can shut down on short noticein the case of increased renewable power generation, slowgenerators can only adjust their output level but cannotdeviate from their day-ahead commitment schedule, as indi-cated by the nonanticipativity constraints of Equations (15),(16), because of limitations on their operational flexibil-ity. As a result, slow units cannot back down from theirminimum generation levels and therefore, in the case ofoversupply, require the waste of excess renewable energyin order to stay online. We note that the dispatch of thesystem is performed based on the objective of minimizingoperating costs, as indicated in the models of §3. There areno must-take or priority dispatch requirements for renew-able resources, or a penalty that must be paid for curtailingrenewable power injections.3

The introduction of transmission constraints complicatesscenario selection considerably because of the fact thatthe fast reserves are not readily accessible when certaintransmission lines are congested and the availability of


renewable resources may be limited because of transmis-sion constraints. The further introduction of composite out-ages introduces the complication of protecting the systemagainst very low probability outcomes that can severelyimpact system reliability.

The stochastic unit commitment literature has reliedextensively on the scenario selection and scenario reduc-tion algorithms proposed by Dupacova et al. (2003) andtheir faster variants that were proposed by Heitsch andRömisch (2003). The effectiveness of these algorithms inthe stochastic unit commitment problem was first demon-strated by Gröwe-Kuska et al. (2002) who apply the algo-rithms for scheduling hydro and thermal units in a Germanutility. Morales et al. (2009a) propose a variant of the sce-nario reduction algorithm of Heitsch and Römisch (2003)that removes the scenarios that cause the least change inthe second-stage costs of the optimization problem. Thesescenario selection and scenario reduction techniques relyon clustering scenarios in such a way that the transformedmeasure of the stochastic program is perturbed minimally.General scenario clustering techniques have been used pre-viously in the stochastic optimization literature by Pflug(2001) and Latorre et al. (2007).

Despite the theoretical justification of the scenario reduc-tion algorithm of Dupacova et al. (2003) and Heitsch andRömisch (2003), as dicussed in Papavasiliou et al. (2011)these algorithms perform poorly in a unit commitmentmodel without transmission constraints and contingencies,where wind power production is the only stochastic input.This is attributed to two reasons. Firstly, the algorithms ofDupacova et al. (2003) and Heitsch and Römisch (2003)are not guaranteed to preserve the moments of hourly windgeneration. Because of the predominant role of fuel costsin the operation of the system, the accurate representationof average wind supply in the case of large-scale windintegration is crucial for properly guiding the weightingof scenarios. Moreover, the modeler cannot specify certainscenarios that are deemed crucial. For example, the real-ization of minimum possible wind output throughout theentire day needs to be considered explicitly as a scenario.Otherwise, there is the possibility of under-committingresources and accruing overwhelming costs from load shed-ding in economic dispatch. To overcome these drawbacks,Papavasiliou et al. (2011) generate a large number of sam-ples from the statistical model of the underlying processand select a subset of samples based on a set of prescribedcriteria that are deemed important. The authors then assignweights to each scenario such that the first moments ofhourly wind output are matched as closely as possible.Scenario selection that strives to match certain statisticalproperties of the underlying process has been proposed inprevious literature, e.g., by Hoyland and Wallace (2001)and Hoyland et al. (2003).

The introduction of transmission constraints and contin-gencies complicates the task of scenario selection consider-ably, as it is not clear how to extend the scenario selectionalgorithms of Dupacova et al. (2003) and Papavasiliou et al.(2011) to networks with multiarea wind production andhow to account for network component failures. Assum-ing independence among net load outcomes and contingen-cies, which is a very reasonable assumption, one naturalapproach for generating scenarios would be to decouplethe selection of contingencies from the selection of netload scenarios. The previously discussed algorithms couldthen be adapted for selecting multiarea net load scenarios(resulting from random scenarios of wind generation), andwe can take the cross product of these net load scenarioswith a set of significant contingencies. Such an approachis described in §EC.3 of the electronic companion. Onenatural question that this approach raises is which contin-gencies to select and how to weigh them relative to eachother. For example, if the failure of any given generatorover an entire day has a likelihood of 1% (Pereira and Balu1992), in a network with 130 generators the chances ofa single-generator failure are approximately 35.6%.4 Thequestion arises, then, if a scenario includes the failure ofa single generator, how should that scenario be weighedagainst other scenarios? Which generator failures shouldwe include in the scenario set? Should we consider com-posite failures, for example, the failure of multiple gener-ators, multiple lines, or generators and lines in the samescenario? It becomes clear that a methodical approach forscenario selection is needed.

6.1. Importance Sampling

The scenario selection algorithm that we propose in thissection is inspired by importance sampling. In contrastto scenario generation techniques that aim to match themoments of the underlying stochastic process such as thoseproposed by Hoyland and Wallace (2001) and Hoylandet al. (2003), the proposed scenario selection algorithmaims at selecting scenarios that best represent the averagecost impact of uncertainty on the problem. Importance sam-pling is a statistical technique for reducing the number ofMonte Carlo simulations that are required for estimating theexpected value of a random variable within a certain accu-racy. For an exposition see Mazumdar (1975) and Infanger(1992). As Pereira and Balu (1992) report, this techniquehas been used in reliability analysis in power systems withcomposite generation and transmission line failures, wherethe estimated random variable is a reliability metric (e.g.,loss of load probability or expected load not served).

Given a sample space ì and a measure p on this space,importance sampling defines a measure q on the space thatreduces the variance of the observed samples of the ran-dom variable C, and weighs each simulated outcome �by p4�5/q4�5 in order to unbias the simulation results.The measure q is ideally chosen such that it represents thecontribution of a certain outcome to the expected value that


is being computed, i.e.,

q?4�5=p4�5C4�5

ƐpC0 (53)

Of course, it is not possible to determine this mea-sure since ƐpC is the quantity we wish to compute. Nev-ertheless, the intuition of selecting samples according totheir contribution to the expected value can be carriedover to scenario selection. For example, in Papavasiliouet al. (2011) the authors include the wind power produc-tion outcome with the lowest aggregate production overthe entire day. Although the likelihood of this outcomeis very low, its impact on system costs can be extremelyhigh, making its contribution to expected cost p4�5C4�5significant.

6.2. Proposed Algorithm

The extension of the intuition of importance sampling tothe case of scenario selection is straightforward: if the idealmeasure q? of Equation (53) were closely approximated bya measure q, then selecting a small number of outcomesaccording to this measure and weighing them accordingto p4�5/q4�5 would provide an accurate estimate of theexpected cost. Therefore, samples selected according to qcan be interpreted as representative scenarios that need tobe weighted according to p4�5/q4�5 relative to each otherin order not to bias the result.

We proceed by generating an adequately large subsetof the sample space ìS = 8�11 0 0 0 1�M9 and we calculatethe cost of each sample against a deterministic unit com-mitment policy CD4 · 5. Since C =

∑Mi=14CD4�i5/M5 pro-

vides an accurate estimate of expected cost, we interpretthe sample space of the system as ìS and the measure asthe uniform distribution over ìS , hence p4�5 = M−1 forall � ∈ ìS . We then obtain q4�i5 = CD4�i5/4MC51 i =

11 0 0 0 1M , and each selected scenario is weighed accordingto �s = p4�5/q4�5, hence �s/�s′ = CD4�

s′5/CD4�s5 for

each pair of selected scenarios �s1�s′ ∈ ì. Hence, the pro-posed algorithm selects scenarios with a likelihood that isproportional to their cost impact, and discounts these sce-narios in the stochastic unit commitment in proportion totheir cost impact in order not to bias the stochastic unitcommitment policy. We therefore propose the followingalgorithm:

Step (a). Define the size N of the reduced scenario setì= 8�11 0 0 0 1�N 9.

Step (b). Generate a sample set ìS ⊂ ì, whereM = �ìS � is adequately large. Calculate the cost CD4�5of each sample � ∈ ìS against the best determin-stic unit commitment policy and the average cost C =∑M

i=14CD4�i5/M5.Step (c). Choose N scenarios from ìS , where the prob-

ability of picking a scenario � is CD4�5/4MC5.Step (d). Set �s =CD4�5

−1 for all �s ∈ ì.

6.3. Discussion

In contrast to the scenario selection algorithms that buildthe scenario set sequentially by evaluating the impact ofcandidate scenarios on the existing scenario set (see, forexample, Dupacova et al. 2003, Gröwe-Kuska et al. 2002,and Latorre et al. 2007), the proposed algorithm selects andweighs scenarios in one shot (as in, for example, Hoylandand Wallace 2001). The proposed algorithm ensures that,as long as the cost impacts of all selected scenarios areof the same order of magnitude, which is commonly thecase, then so are the weights in the stochastic unit com-mitment formulation. As a result, each scenario influencesthe first-stage decisions. This should be contrasted withthe case where the probabilities of certain scenarios arevery small compared to the probabilities of other scenarios.In that case, as we can see in Equation (48), the influenceof scenarios with very small probability is minimal in theobjective function of (P1), and consequently these scenarioswill not tend to influence the optimal solution of (P1) andtherefore the first-stage decision, i.e., the commitment ofslow generators. In addition, from Equations (50) and (51)we see that scenarios with small probability exhibit verysmall changes in the values of their dual multipliers, whichfurther supports the argument that these scenarios do notinfluence (P1). In fact, we observed that the stochastic unitcommitment policy remained completely unaffected by sce-narios that were 100 times less likely to occur than theircompeting scenarios in the stochastic unit commitment for-mulation. Therefore, the inclusion of these scenarios intro-duced superfluous computational load. This is preventedwith our proposed scenario selection algorithm.

Commonly occurring scenarios are most likely to popu-late the original set of candidate scenarios ìS in step (b),and are therefore also most likely to populate the reducedscenario set ì in step (c). Thus, highly likely outcomesare represented. At the same time, adverse scenarios areaccounted for (by including them in the scenario set) with-out being overemphasized (by discounting their probabilityweighting).

An additional appealing feature of the proposed algo-rithm is that it selects a rich set of multiarea net load out-comes. This should be contrasted to the case where wewould multiplex net load scenarios selected according tothe scenario selection algorithms of Dupacova et al. (2003),Gröwe-Kuska et al. (2002), and Papavasiliou et al. (2011)with contingency scenarios. Moreover, in contrast to thescenario selection method proposed in Papavasiliou et al.(2011), the scenario selection algorithm proposed in thispaper does not depend on the judgement of the modeler forspecifying criteria that are deemed important. The proposedprocedure can be applied to a broad setting of problemsunder uncertainty in a straightforward fashion. In §7 it isshown that the resulting stochastic unit commitment pol-icy outperforms the approach of Dupacova et al. (2003)as well as common deterministic rules for four case stud-ies of uncertainty, one of which is the case study that wasperformed in Papavasiliou et al. (2011).


7. ResultsIn this section we present simulation results for a reducedmodel of the California power system with 225 buses,130 generators, and 375 transmission lines. The modelincludes a sparse representation of the entire Western Elec-tricity Coordinating Council Western interconnect outsideof California, which is necessary in order to capture loopflow effects as well as imports and exports into and out ofCalifornia. A schematic description of the model showingthe wind sites, zones, and import constraints of the net-work is presented in §EC.1 of the electronic companion.The model is also used in Yu et al. (2010) and Papavasiliouet al. (2011). We study three levels of wind integration, thezero wind integration case as well as a case of moderate anddeep wind integration corresponding to the 2012 and 2020renewable energy integration targets of California, respec-tively. Ex post we have observed that the moderate inte-gration case corresponds to approximately 7% wind energypenetration, and the deep integration case corresponds toapproximately 14% wind energy penetration. In the rest ofthe paper we refer to these cases as moderate and deep inte-gration, respectively. We also perform the deep integrationcase study for the case where transmission constraints andcontingencies are not accounted for, in order to quantifythe impact of these effects on the analysis. The latter caseis denoted as deep simple.

7.1. Day Types

We focus our analysis on eight day types that representweekdays and weekends for each season. We assume thatthe day-ahead forecast of wind power production for eachday type is equal to the hourly average wind profile ofthe day type, as estimated by the available data. Since thesystem operator is facing the same source of uncertaintyfrom day to day within days of the same type, we assumethat unit commitment decisions are identical for days of thesame type. This determines initial conditions of the systemat the start of each day for the unit commitment models.Following the approach of Bertsimas et al. (2013), in theeconomic dispatch model of §3.4 the initial conditions ofthe system are ignored.

By assuming an identical unit commitment policy foreach day type, and by ignoring boundary conditions in theeconomic dispatch model, we are able to parallelize compu-tation in the evaluation phase of the Monte Carlo simulationsince each day is treated independently and can be simu-lated by a different processor. As a result, we are capableof simulating a total of 8,000 days (1,000 days for each daytype), by contrast to Sioshansi and Short (2009), Ruiz et al.(2009a), and Tuohy et al. (2009), who only simulate oneyear of operations on a rolling horizon basis. As a result,we are able to obtain more accurate results regarding costperformance. The ability to parallelize computation throughour assumptions comes at the price of ignoring bound-ary conditions in the economic dispatch model and tran-sitions between different day types, e.g., from a weekend

to a weekday or from a weekday to a weekend. In futureresearch our intention is to quantify the impact of theseassumptions by comparing our batch processing model witha serial model based on rolling planning.6

7.2. Data

We do not use the wind production data from Yu et al.(2010) since our wind production model is more detailed.As we describe in §EC.2, we are using 2006 wind produc-tion data from the National Renewable Energy Laboratorydatabase. We also use load data from the same year, whichis publicly available at the California ISO Open AccessSame-Time Information System, CAISO (CAISO 2011).The average load in the system is 27,298 MW, with a min-imum of 18,412 MW and a peak of 45,562 MW.

We use a more general model for thermal generatorswithin CAISO, with 124 generators, compared to the modelin Yu et al. (2010), which uses 23 aggregated thermal gen-erators. The value of lost load is set to 5,000 $/MW-h. Thenumber of generators and the capacity for each fuel typeare shown in Table 1. The last two rows of Table 1 describehow the fossil fuel generation mix is partitioned into fastand slow generators. The entire thermal generation capacityof the system is 30,231.5 MW. The total capacity of the sys-tem, not including wind power resources, is 53,665.5 MW.

7.3. Relative Performance of Policies

In this section we discuss the relative performance ofthe stochastic and deterministic unit commitment policies.The results are obtained by running the economic dispatchmodel against 1,000 Monte Carlo outcomes of wind powerproduction and contingencies for each day type, with aprobability of generator failure of 1% (Pereira and Balu1992) and a probability of transmission line failure of0.1% (Grigg et al. 1999). We assume that network elementfailures occur over the entire day. As we clarify in §7.1,by parallelizing the Monte Carlo simulations we are able tosimulate 8,000 days of operations and obtain more accurateresults regarding average cost performance. In §EC.4 we

Table 1. Generation mix for the test case.

Type No. of units Capacity (MW)

Nuclear 2 41499Gas 94 20159506Coal 6 28509Oil 5 252Dual fuel 23 41599Import 22 121691Hydro 6 101842Biomass 3 558Geothermal 2 11193Wind (moderate) 5 61688Wind (deep) 10 141143Fast thermal 88 11100601Slow thermal 42 19122504


provide information about the variance of our results. Windproduction outcomes, generator failures, and transmissionline failures are assumed to be independent. We considertwo deterministic policies. The first policy places a totalreserve requirement T req

t in Equation (28) equal to a frac-tion of peak forecast net load for the day and an onlineslow reserve requirement S

reqt in Equation (30) equal to

half of the total reserve requirement. Forecast net load,here, refers to forecast load minus forecast wind, sched-uled imports and scheduled non-wind renewable resources.To determine the best policy of this type, we find the frac-tion of peak load that yields the best performance. Theother deterministic policy that we consider is a variant ofa reserve commitment policy that was recently proposedin Piwko et al. (2010). The authors propose a heuristicapproach for committing spinning reserves, “the 3+5 rule,”which requires the system to carry hourly spinning reserveno less than 3% of hourly forecast load plus 5% of hourlyforecast wind power. This rule is adapted in our model bysetting this as the online slow reserve requirement and set-ting the total reserve requirement at twice the level of theonline slow reserve requirement. The results are shown inTable 2, where the best peak-load policy is highlighted initalic font and the best deterministic policy is highlightedin bold font. We note that the best peak-load policy outper-forms the 3 + 5 rule for all case studies.

To validate our proposed scenario selection algorithmthat is inspired by importance sampling, denoted as SUC-IS, we compare it to an alternative scenario selection algo-rithm. The alternative stochastic unit commitment policy,denoted as SUC2, is based on crossing the most severesystem contingencies with wind production outcomes pro-duced by the scenario reduction algorithm of Dupacovaet al. (2003). The details of the alternative scenario selec-tion algorithm are described in §EC.3 of the electroniccompanion. We also consider a perfect foresight policy thatcommits and dispatches resources under perfect forecast ofuncertain outcomes. The perfect foresight policy boundsthe attainable cost of any unit commitment rule.

The relative performance of the proposed scenario selec-tion algorithm based on importance sampling with respectto the deterministic policies, the alternative stochastic unitcommitment policy SUC2, and the perfect foresight pol-icy for the four case studies is presented in Figure 1. Theresults are presented in terms of the relative cost of eachpolicy compared to the proposed stochastic unit commit-ment policy for each of the eight day types. In the lastthree rows of Table 3 we present the absolute cost of the

Table 2. Deterministic policy cost comparison.

Case 10% 20% 30% 40% 50% 3 + 5

Deep simple 512511279 512671340 513051363 513341489 513811318 512701797No wind 1115071776 1115051126 1115251722 1115111003 1115951341 1115281686Moderate 915751616 915731424 915751465 915921592 915391965 916051016Deep 716381328 716451104 716121689 716611245 716591490 716371464

proposed scenario selection algorithm SUC-IS as well asits gains over the best deterministic policy and the alterna-tive stochastic unit commitment policy SUC2. We note thatthe benefits of the proposed stochastic unit commitmentpolicy SUC-IS range between $145,261 to $244,226 perday relative to the best deterministic policy, and between$5,022 to $52,622 per day for the alternative stochasticunit commitment policy SUC2. We note that the proposedscenario selection algorithm SUC-IS outperforms SUC2,although the majority of the potential benefits is capturedby either stochastic formulation. Moreover, the introduc-tion of transmission constraints affects the gains of stochas-tic unit commitment, which supports the argument thatstochastic unit commitment is especially valuable for thedetermination of locational capacity requirements.

7.4. Operating Costs, Committed ConventionalCapacity, and Renewables Utilization

In Table 3 we present summary results for renewableenergy waste, operating costs, and committed conventionalcapacity for the four case studies under consideration.Renewable energy losses range between 0.2% of totalrenewable energy production in the case of the deep inte-gration study without transmission constraints and contin-gencies to 2.5% of total renewable energy production whentransmission constraints and contingencies are accountedfor. Operating costs decline steeply as the level of renew-able power penetration increases, due to the decrease infuel costs, which are the predominant cost in the system.Further details regarding operating costs are provided in§EC.4 of the electronic companion. By comparing the costof column 2 (deep simple) to that of column 5 (deep),we note that failing to account for transmission constraintsand contingencies results in an underestimation of operat-ing costs by 31.0%. The significant cost increase resultingfrom transmission constraints can be attributed to the oper-ating cost impacts of contingencies but also to the reducedflexibility of dispatching units in the system.

The commitment of conventional generation capacity,which is the most important factor in analyzing the eco-nomic of renewable energy integration, presents the mostinteresting results. Committed capacity is computed as thesum of slow generation capacity committed in the dayahead, plus the total amount of available fast capacity(11,006.1 MW, as indicated in Table 1) that is readily avail-able in real time. The result reported in Table 3 thereforerepresents the optimal choice of conventional generationcapacity that is required for reliably operating the system


Figure 1. In reading order: Cost comparison for the deep integration case without transmission or contingencies, thezero wind integration case, the moderate integration case, and the deep integration case.

10.00 5.00

4.00

3.00

2.00

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0.00

–1.00

–2.00

Perfect forecast

10% peak load

3 + 5 rule

SUC2

Perfect forecast

20% peak load

3 + 5 rule

SUC2

Perfect forecast

30% peak load

3 + 5 rule

SUC2

Perfect forecast

50% peak load

3 + 5 rule

SUC2

6.006.00

5.00

4.00

3.00

2.00

1.00

0.00

–1.00

–2.00

–3.00

4.00

2.00

0.00

–2.00

–4.00

–6.00

Rel

ativ

e co

st (

%)

Rel

ativ

e co

st (

%)

Rel

ativ

e co

st (

%)

Rel

ativ

e co

st (

%)

8.00

6.00

4.00

2.00

Fal

lWE

Fal

lWE

Sum

mer

WE

Sum

mer

WE

Spr

ingW

E

Spr

ingW

E

Win

terW

E

Win

terW

E

Fal

lWD

Fal

lWD

Sum

mer

WD

Sum

mer

WD

Spr

ingW

D

Spr

ingW

D

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terW

D

Fal

lWE

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mer

WE

Spr

ingW

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terW

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Fal

lWD

Sum

mer

WD

Spr

ingW

D

Win

terW

D

Fal

lWE

Sum

mer

WE

Spr

ingW

E

Win

terW

E

Fal

lWD

Sum

mer

WD

Spr

ingW

D

Win

terW

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inte

rWD

Moderate Deep

No wind

0.00

–2.00

–6.00

–4.00

Deep simple

under uncertainty, given the set of available candidateresources and without accounting for capital investments innew capacity, which are assumed to be a sunk cost. Notethat, since committed slow unit capacity is the only first-stage decision in the stochastic unit commitment model, thethird line of Table 3 (capacity (MW)) indicates the changein first-stage decisions across different day types.

We note that the moderate integration case reduces aver-age conventional committed capacity by a mere 840 MW,which represents 12.6% of the 6,688 MW of installedwind capacity. Average conventional committed capacityfor the deep integration case is reduced by 1,670 MW,which represents 11.8% of the 14,143 MW of installedwind capacity. Most importantly, we note that failing toaccount for transmission constraints in the deep integra-tion study results in an underestimation of the commit-ted capacity by 25.9% of installed wind capacity, relative

Table 3. Summary results for each case study.

Deep simple No wind Moderate Deep

RE daily waste (MWh/%) 163 (0.2%) 0 (0%) 877 (1.9%) 2,346 (2.5%)Capacity (MW) 19,958 23,619 22,779 21,949Cost ($M) 5.106 11.283 9.329 7.405Daily savings relative to best det. ($) 145,261 221,854 244,226 207,698Daily savings relative to SUC2 ($) 39,681 5,022 52,622 15,574

to the estimated 11.8% reduction when these features areaccounted for. This strongly supports the argument that theinclusion of transmission constraints and contingencies iscrucial for accurately assessing the impact of large-scalerenewable energy integration.

7.5. Computational Performance

The stochastic unit commitment problem of §3.2 for42 scenarios has 909,216 continuous variables, 173,376binary variables, and 2,371,824 constraints. The stochasticunit commitment algorithm was implemented in the Javacallable library of CPLEX 12.4, and parallelized using themessage passing interface (MPI). The algorithm was imple-mented on a high performance computing cluster in theLawrence Livermore National Laboratory on a network of1,152 nodes, 2.4 GHz, with eight CPUs per node and 10 GB


per node. The parallel implementation of the Lagrangianrelaxation algorithm and Monte Carlo simulations is shownin Figure 2. Problems 4P15 and 4P2s5, s ∈ S were run for120 iterations. For the last 40 iterations, 4EDs5 was runfor each s ∈ S in order to obtain a feasible solution andan upper bound for the stochastic unit commitment prob-lem. The average elapsed time for the 42-scenario prob-lem on 20 machines was six hours, 47 minutes. The mixedinteger programming (MIP) gap for 4P15 and 4P2s5, s ∈ Swas set to �1 = 1%, and the MIP gap for obtaining a fea-sible schedule from 4EDs5 was set to �2 = 001%. Notefrom Table 3 that the daily savings of SUC-IS relative toSUC2 fall within the MIP gap of the economic dispatchproblem for the zero wind integration study. For all othercases (moderate, deep, and deep simple), the benefits ofthe SUC stochastic unit commitment policy are guaranteedto reflect a superior scenario selection approach. The sumof the optimal solutions of the first and second subprob-lem yield a lower bound LB on the optimal cost, whereasthe optimal solution of the feasibility run results in anupper bound UB. The average gap, 4UB − LB5/LB, that weobtained is 0.77%. However, to estimate an upper boundon the optimality gap it is also necessary to account for theMIP gap �1 that is introduced in the solution of 4P15 and4P2s51 s ∈ S. The average upper bound on the optimalitygap, 4UB − 41 − �15LB5/441 − �15LB5, is 1.79%.

8. Conclusions and PerspectivesIn this paper we present a two-stage stochastic unit com-mitment model that can be used for assessing the impactof wind power integration on operating costs, committedday-ahead generation capacity, and renewable energy uti-lization. We present a scenario selection algorithm inspiredby importance sampling that is shown to outperformalternative stochastic and deterministic unit commitment

Figure 2. Parallel implementation of the Lagrangian relaxation algorithm and Monte Carlo simulation.

zgt*

vgst*

wgt*

�gst �gst

*ugst

Dual multiplierupdate

Second-stagesubproblems P2s

1 2 Ns. . . .

First-stagesubproblem P1

Second-stagefeasibility runs EDs

1 2 Ns. . . .

Nc1 2 . . . .

Monte Carloeconomic dispatch EDc

approaches for four case studies, and a subgradient algo-rithm for solving the resulting stochastic program. Thesubgradient algorithm has been parallelized in a high per-formance computing cluster in order to solve a moder-ately sized system within an acceptable time frame. Thedaily operating cost benefits of stochastic unit commit-ment are shown to range between $145,261 to $244,226relative to deterministic commitment rules and between$5,022 to $52,622 relative to an alternative stochastic unitcommitment model. Renewable energy waste is negligibleacross all case studies. Failing to account for contingen-cies and transmission results in an underestimation of oper-ating costs by 31.0%. Conventional committed capacitydecreases by 11.8%–12.6%s of installed wind capacity andfailing to account for transmission constraints and contin-gencies results in an underestimation of committed capacityby 25.9% of installed wind capacity relative to 11.8% whenthese features are accounted for.

There are various extensions of the present model thatare intended in future research. In practice, system forecastscan and will be updated, leading to the opportunity for indi-vidual units to be committed or shut down as required. Thefact that forecasts and dispatch decisions are revised duringthe day can be represented through a multistage formula-tion of the stochastic unit commitment problem. Moreover,the seasonal correlation of load with renewable productioncan affect capacity requirement calculations and requiresan extension of the time series modeling framework pre-sented in §5. The model can also be extended to an opti-mal investment model, where the first stage is interpretedas investment in new generation capacity. The inclusion ofinvestment decisions on transmission lines in order to inte-grate increased amounts of renewable resources also rep-resents an exciting area of future research. As we describein §7.1 our analysis focuses on eight representative daytypes and ignores the boundary conditions of slow units


in order to obtain a representative set of unit commitmentschedules and parallelize the Monte Carlo simulations. Thisis justified on computational grounds and also by the factthat the California ISO day-ahead market currently oper-ates with a 24-hour look-ahead and ignores boundary con-ditions, unless resources are self-dispatched or endogenizetheir on/off status by modifying their start-up bids. In futureresearch we look forward to quantifying the implicationsof this simplifying assumption and the trade-offs in termsof computational effort.

Supplemental Material

Supplemental material to this paper is available at http://dx.doi.org/10.1287/opre.2013.1174.

Endnotes

1. Increased operating costs can arise from the wear and tear ofconventional generators that adjust their output in order to bal-ance the supply of renewable power. Balancing generators are alsooften required to operate at inefficient operating levels. In addi-tion, the minimum load levels of balancing resources result inincreased emissions and fuel costs. In addition to the cost of cor-rective actions, the unpredictability of renewable resources resultsin suboptimal commitment of units in the day-ahead time frame.Quantifying the additional operating costs resulting from the large-scale integration of renewable energy sources has recently emergedas an active area of research (see Sioshansi and Short 2009, Ack-ermann 2005, van Hulle 2005).2. The seasonal and diurnal systematic patterns are removed dur-ing calibration in order to obtain an approximately stationary dataset, which is a necessary condition for the Yule-Walker equationsthat are used for estimating the autoregressive parameters to bevalid. The systematic effects are added back to the data set in thesimulation phase. By removing systematic effects we are able toexploit the full year of available data for estimating a time seriesmodel of underlying noise.3. Such restrictions are imposed in many systems in order tofacilitate the penetration of renewable energy and the reduction ofcarbon emissions.4. What we are trying to capture is the probability of having aunit down in an operating day, conditioned on the unit being avail-able during the closure of the day-ahead market. We approximatethis quantity with the forced outage rate (FOR) of a unit. Aca-demic as well as real-world data sets (e.g., Pereira and Balu 1992,Canadian Energy Association (CEA) 2006, Grigg et al. 1999)indicate that the range of FOR varies widely. We use a uniformFOR for all units, which is contained within the range of FORsthat occur in practice and in academic research, although the pro-posed methodology can be applied for varying FORs without anyexcess computational requirements.5. This is the probability of obtaining one success in 130 inde-pendent trials for which the probability of success is 0.01.6. Ignoring boundary conditions between days and treating theunit commitment schedule of each day independently is alsojustified by the current ISO market structure where the day-ahead market is cleared each day separately, whereas the stateof resources offered into the market can only be accounted forimplicitly through the start-up and no-load bids of the unitsor through self-dispatch of units offered as price takers. There

are strong incentives, however, against such behavior since self-dispatched units are not assured cost recovery like centrally dis-patched resources. In addition, the inter-temporal optimization(medium-term scheduling) of hydro resources is performed at theindividual resource level and not as part of the central unit com-mitment, and these resources are subsequently bid into the day-ahead market as single-day price-taking resources.

Acknowledgments

This research was funded by the National Science Foundation[Grant IIP 0969016], by the U.S. Department of Energy throughthe Future Grid initiative administered by the Power Systems Engi-neering Research Center, by the Lawrence Livermore NationalLaboratory, and by the Federal Regulatory Energy Commission.

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Anthony Papavasiliou is an assistant professor in the Depart-ment of Mathematical Engineering at the Catholic University ofLouvain, and a member of the Center for Operations Research andEconometrics. His research interests are focused on energy sys-tems operations, planning and economics, and optimization underuncertainty. He has served as a researcher for Pacific Gas andElectric and the Federal Energy Regulatory Commission and thePalo Alto Research Center.

Shmuel S. Oren is a professor of industrial engineering andoperations research at the University of California at Berkeley.He is also the Berkeley site director of the Power System Engi-neering Research Center. He was an adviser to the Market Over-sight Division of the Public Utilities Commission of Texas and tothe Energy Division of the California Public Utilities Commissionand is a member of the CAISO Market Surveillance Committee.He is a Fellow of INFORMS and IEEE.

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Multiarea Stochastic Unit Commitment for High Wind Penetration …€¦ · methodological contribution of the paper is the presentation of a computationally tractable approach for