Applying High Performance Computing to

Multi-Area Stochastic Unit Commitment for

Renewable Penetration

Anthony Papavasiliou, Shmuel S. Oren

May 15, 2012

Abstract

We present a parallel implementation of a Lagrangian relaxation al-gorithm for solving stochastic unit commitment subject to uncertaintyin wind power supply and generator and transmission line failures. Wepresent a scenario selection algorithm inspired by importance samplingin order to formulate the stochastic unit commitment problem and vali-date its performance by comparing it to a stochastic formulation with avery large number of scenarios, that we are able to solve through paral-lelization. We examine the impact of narrowing the duality gap on theperformance of stochastic unit commitment and compare it to the impactof increasing the number of scenarios in the model. We examine the re-lation between duality gap in Lagrangian relaxation and the number ofscenarios in the model and relate that to theoretical results provided inthe literature. We finally report results regarding speedup and efficiencyand discuss the scalability of our proposed algorithm.

1 Introduction

Market clearing and dispatch methods can benefit significantly from high per-formance computing capabilities. As power systems planning and operationsmove towards explicitly dealing with uncertainty and distributed resources dueto renewable penetration and demand response integration through approachessuch as stochastic optimization, robust optimization and topology control, highperformance computing is essential. Parallelization and multiprocessor com-puting is the way to deal with the computational challenges posed by theseapproaches.

High performance computing is currently not a broadly accessible resourcefor practitioners such as Independent System Operators, utilities and regulators.The purpose of this research is to perform scoping studies for industrial scaleproblems in anticipation of the technological progress that is expected to takeplace in the coming years in the area of parallel computation. Our objective inthe current publication is to identify the appropriate scale for stochastic unit

1

commitment formulations by analyzing the sensitivity of the resulting unit com-mitment policies and their performance to the number of scenarios consideredin the model.

Distributed computation has a rich history in the area of power systemsplanning and operations. A review of the application of high performance com-puting in power systems is presented by Falcao [8]. Falcao presents variousapplications of parallelization, including security constrained optimal powerflow and composite generation-transmission reliability evaluation. Pereira etal. [20] present three applications of distributed computing in power systemsoperations: reliability evaluation for composite outages, scenario analysis for ahydro dominated system and a Benders decomposition algorithm for securityconstrained dispatch. Monticelli et al. [15] present a distributed Benders al-gorithm for solving the security constrained optimal power flow problem withcorrective rescheduling. The authors present results for a 118 bus IEEE testsystem and demonstrate that accounting for corrective rescheduling can yieldeconomic benefits in the dispatch of the system. Although the authors high-light the possibility of parallelizing their algorithm, they do not present such aparallel implementation.

Kim and Baldick [12] develop a parallel algorithm for solving distributed op-timal power flow across multiple operating regions. The authors argue that thealgorithm they propose is consistent with the communication protocols of sys-tem operators and market operators, requires minimal communication betweensubregions and also reduces the communication time of telemetry. The authorsuse an augmented Lagrangian algorithm to solve the problem and present ef-ficiency and speedup results, although these are estimated as the authors donot implement the algorithm in parallel. Bakirtzis and Biskas [1] propose a de-centralized Lagrangian relaxation algorithm for solving the optimal power flowproblem posed by Kim and Baldick [12]. The authors test three test systems,including a 3-area version of the IEEE RTS 96 system, a scaled-up version ofthe latter with more areas and a full model of the Balkan system. The authorsdemonstrate that the decentralized algorithm converges in about 35 iterationsfor the largest cases studied. A parallel implementation of the algorithm inBakirtzis and Biskas [1] using PVM is presented by Biskas et al. [4].

The references above are focused on simulation studies and the decomposi-tion of the optimal power flow and mid-term hydro scheduling problem. Re-cently, and due to the increasing integration of renewable energy sources, nu-merous researchers have been focusing their attention to the stochastic unitcommitment problem [5], [22], [18]. The objective of this problem is the short-term, day-ahead scheduling of conventional generators in order to manage theunpredictable fluctuations of renewable resources such as wind and solar powersupply. The computational contribution of this paper is to implement a dis-tributed a parallel algorithm for solving the stochastic unit commitment prob-lem in order to assess the sensitivity of the solution on the number of scenariosincluded in the model. The remaining paper is organized as follows. In Sec-tion 2 we formulate the two-stage stochastic unit commitment model. We thenpresent a Lagrangian relaxation algorithm for solving the problem in Section 3,

2

and a scenario selection algorithm for formulating the model in Section 4. Weconclude our analysis with a study of a reduced model of the Western ElectricityCoordinating Council in Section 5.

2 Model Description

The focus of this paper is the day-ahead scheduling of generators subject toreal-time renewable power supply uncertainty and outages of transmission linesand generators. The problem is cast as a two-stage optimization, where thefirst stage represents day-ahead decisions and the second stage represents thereal-time recourse to the revealed system conditions. The stochastic unit com-mitment problem was introduced by Takriti and Birge [25].

In the following model formulation, u represents a binary variable indicatingthe on-off status of a generator, v is a binary startup variable and p is theproduction level of each generator. The minimum load cost of a generator isdenoted as Kg, the startup cost as Sg and the constant marginal cost as Cg.The set of generators is denoted as G. The model that we present in this paperaccounts for transmission constraints, with power flows over transmission linesdenoted as e and bus angles denoted as θ. L represents the set of transmissionlines. The demand for each hour t at each bus of the network n is denotedas Dnt, where N is the set of buses in the network. Operating constrains aredenoted compactly in terms of a feasible set D, and vectors are denoted in bold.Thus, the notation (p, e, θ,u,v) ∈ D encapsulates the minimum/maximum runlimits, minimum up/down times and ramping rate limits of generators, as wellas Kirchhoff’s voltage and current laws and the thermal limits of lines.

The objective is to minimize the cost of serving forecast demand. The prob-lem in the deterministic setting (assuming an accurate forecast of wind powerproduction) can be described as follows:

(UC) : min∑g∈G

∑t∈T

(Kgugt + Sgvgt + Cgpgt) (1)

s.t.∑g∈Gn

pgt = Dnt, n ∈ N, t ∈ T (2)

P−g ugt ≤ pgt ≤ P+g ugt, g ∈ G, t ∈ T (3)

elt = Bl(θnt − θmt), l = (m,n) ∈ L, t ∈ T (4)

(p, e, θ,u,v) ∈ D, (5)

where Bl represents the susceptance of line l and P−g , P+g represent the minimum

and maximum run limits of generator g. The set of generators located in eachbus n is denoted by Gn. The horizon T is 24 hours, with hourly increments.A detailed formulation of the constraints represented by the domain D can befound in Papavasiliou and Oren [17].

3

The stochastic formulation involves a two-stage process, where the set of un-certain outcomes is represented as S. First-stage unit commitment and startupdecisions are represented respectively as w and z and apply for slow-respondingunits Gs for which commitment decisions need to be made in advance, in theday-ahead time frame. The problem to be solved is the following:

(SUC) :

min∑g∈G

∑s∈S

∑t∈T

πs(Kgugst + Sgvgst + Cgpgst) (6)

s.t.∑g∈Gn

pgst = Dnst, n ∈ N, s ∈ S, t ∈ T (7)

P−gsugst ≤ pgst ≤ P+gsugst, g ∈ G, s ∈ S, t ∈ T (8)

elst = Bls(θnst − θmst), l = (m,n) ∈ L, s ∈ S, t ∈ T (9)

ugst = wgt, vgst = zgt, g ∈ Gs, t ∈ T (10)

(p, e, θ,u,v) ∈ Ds, (11)

where decision variables are now contingent on the scenario s ∈ S. Note thatthe domain D = ×s∈SDs is decomposable across scenarios. Each scenario isweighted in the objective function by a probability πs. The sources of uncer-tainty in the model include:

• The hourly supply of renewable energy, which is reflected in the net de-mand Dnst.

• The availability of a generator g. If a generator fails for scenario s, thenP−gs = P+

gs = 0 (assuming the outage lasts for the entire horizon).

• The availability of a line l. If a lines fails for scenario s, then Bls = 0(assuming the outage lasts for the entire horizon).

Upon realization of the uncertain outcome, fast generators Gf adjust theirunit commitment schedule ugst, whereas slow generators Gs are forced to main-tain their day-ahead unit commitment schedule, as indicated by the non-anticipativityconstraints of Eq. 10. All generators can adjust their production levels accord-ing to the realization s, regardless of whether they are slow-responding or fast-responding resources. This two-stage stochastic unit commitment formulationfollows the model of Ruiz et al. [22].

3 Solution Methodology

The Lagrangian relaxation algorithm relies on the observation that the relax-ation of the non-anticipativity constraints in Eq. (10) results in unit commit-ment subproblems (UCs), as in Eqs. (1) - (5), that are independent acrossscenarios. The Lagrangian dual function is obtained as:

4

L =∑s∈S

πs(∑g∈G

∑t∈T

(Kgugst + Sgvgst + Cgpgst)

+∑g∈Gs

∑t∈T

(µgst(ugst − wgt) + νgst(vgst − zgt)) (12)

The problem is solved by maximizing the Lagrangian dual function using thesub-gradient algorithm. The solution of the Lagrangian involves one second-stage unit commitment problem for each scenario (UCs), and one first-stageoptimization (P1). The first-stage optimization is formulated as:

(P1) : max∑g∈Gs

∑s∈S

∑t∈T

πs(µgstwgt + νgstzgt) (13)

s.t.

(w, z) ∈ D1, (14)

where D1 represents the minimum up and down time constraints of slow units.The solution of the Lagrangian dual provides a lower bound for the model.

By introducing redundant second-stage decision variables on startup decisions,we are able to enforce minimum up and down times on slow units, as in Eq. (14).Given these unit commitment schedules, we can solve a second-stage economicdispatch model (EDs), which is the unit commitment model for scenario s,(UCs), with ugst, vgst fixed for g ∈ Gs. This provides feasible solutions at eachiteration as well as an upper bound that can be used for computing the dualitygap and terminating the algorithm. The algorithm is parallelized both in thesolution of (UCs), as well as the solution of (EDs), as indicated in Fig. 1.Further details about the solution methodology are discussed in Papavasiliou etal. [18].

4 Scenario Selection

An early motivation for the formulation of stochastic programming models hadthese models serve as extensions to scenario analysis for policy models [7]. Thepurpose of stochastic programming was the development of robust decisions inpolicy models that hedge against a small number of large-scale outcomes thatcould influence policy decisions. Originally, only a handful of scenarios wereconsidered that represented a few major outcomes in the context of policy anal-ysis. Scenario selection attracted the attention of researchers when stochasticprogramming was extended to an operational tool in various industrial appli-cations, including finance [21], [10] and electricity [13], [16]. In the context ofoperational models, the space of uncertain realizations became too large for anexhaustive enumeration to result in a computationally tractable model. Theseoutcomes had to be reduced to a tractable size by selecting and appropriatelyweighing only a few representative outcomes. With the advent of distributed

5

Figure 1: The parallel implementation of the Lagrangian relaxation algorithm.

computation, the enumeration of a large number of outcomes in the stochas-tic programming formulation presents no significant computational difficulty,and the natural question arises how such policies compare to existing scenarioselection methods. This task is undertaken in the present publication.

Previous research on scenario selection inspired by the interpretation of sce-nario selection as a transportation problem [6], [9] is inadequate for the purposesof our model for two reasons: firstly, it is only directly applicable to continuoussources of uncertainty, rather than discrete element failures, and secondly, theresulting algorithms that the authors propose ignore the objective function ofthe problem that is being optimized, which in our case leads to excessively poorperformance due to the great sensitivity of the objective function on load losses.Moment matching methods [10], [13] suffer from the same drawbacks.

4.1 Importance Sampling

The scenario selection algorithms that we propose in this section are inspired byimportance sampling. Importance sampling is a statistical technique for reduc-ing the number of Monte Carlo simulations that are required for estimating theexpected value of a random variable within a certain accuracy. For an exposi-tion see Mazumdar [14] and Infanger [11]. As Pereira and Balu [19] report, thistechnique has been used in reliability analysis in power systems with composite

6

generation and transmission line failures, where the estimated random variableis a reliability metric (e.g. loss of load probability or expected load not served).

Given a sample space Ω and a measure p on this space, importance samplingdefines a measure q on the space that reduces the variance of the observedsamples of the random variable C, and weighs each simulated outcome ω byp(ω)/q(ω) in order to un-bias the simulation results. The measure q is ideallychosen such that it represents the contribution of a certain outcome to theexpected value that is being computed, i.e.

q?(ω) =p(ω)C(ω)

EpC(15)

Of course, it is not possible to determine this measure since EpC is thequantity we wish to compute. Nevertheless, the intuition of selecting samplesaccording to their contribution to the expected value can be carried over toscenario selection.

4.2 Proposed Algorithm

The extension of the intuition of importance sampling to the case of scenarioselection is straightforward: if the ideal measure q? of Equation (15) were closelyapproximated by a measure q, then selecting a small number of outcomes accord-ing to this measure and weighing them according to p(ω)/q(ω) would provide anaccurate estimate of the expected cost. Therefore, samples selected accordingto q can be interpreted as representative scenarios that need to be weightedaccording to p(ω)/q(ω) relative to each other in order not to bias the result.

We proceed by generating an adequately large subset of the sample spaceΩS = ω1, . . . , ωM and we calculate the cost of each sample against a determin-

istic unit commitment policy CD(·). Since C =

M∑i=1

CD(ωi)

Mprovides an accurate

estimate of expected cost, we interpret the sample space of the system as ΩS

and the measure as the uniform distribution over ΩS , hence p(ω) = M−1 for allω ∈ ΩS . We then obtain q(ωi) = CD(ωi)/C, i = 1, . . . ,M , and each selected sce-nario is weighed according to πs = p(ω)/q(ω), hence πs/πs′ = CD(ωs′)/CD(ωs)for each pair of selected scenarios ωs, ωs′ ∈ Ω. Hence, the proposed algorithmselects scenarios with a likelihood that is proportional to their cost impact, anddiscounts these scenarios in the stochastic unit commitment in proportion totheir cost impact in order not to bias the stochastic unit commitment policy.We therefore propose the following algorithm:

Step (a). Define the size N of the reduced scenario set Ω = ω1, . . . , ωN.Step (b). Generate a sample set ΩS ⊂ Ω, where M = |ΩS | is adequately

large. Calculate the cost CD(ω) of each sample ω ∈ ΩS against the best deter-

minstic unit commitment policy and the average cost C =

M∑i=1

CD(ωi)

M.

7

Table 1: Generation mix for the test caseType No. of units Capacity (MW)Nuclear 2 4,499Gas 88 18,745.6Coal 6 285.9Oil 11 2,952Dual fuel 23 4,599Import 22 12,691Hydro 6 10,842Biomass 3 558Geothermal 2 1,193Wind (moderate) 5 6,688Wind (deep) 10 14,143Fast thermal 82 11,856.1Slow thermal 42 19,225.4

Step (c). Choose N scenarios from ΩS , where the probability of picking ascenario ω is CD(ω)/(MC).

Step (d). Set πs = CD(ω)−1 for all ωs ∈ Ω.

5 Results

In this section we analyze a test system of the California Independent SystemOperator interconnected with the Western Electricity Coordinating Council.The system is composed of 225 buses, 375 lines and 130 generators. The fuel mixof the generators and their classification among fast and slow units is presentedin Table 1. The schematic of the system under consideration is presented in Fig.2. The wind penetration level that we analyze corresponds to the 2030 windintegration targets of California. The wind data is calibrated against one yearof data from the National Renewable Energy Laboratory database. We studyeight day types corresponding to weekdays and weekends of each season.

5.1 Sensitivity in number of scenarios

We first assess the sensitivity of the stochastic unit commitment policy on thenumber of scenarios. We solve one stochastic unit commitment model with1000 scenarios (referred to as S1000) and equal weighting for all scenarios andcompare its performance to the performance of our scenario selection algorithmapplied on 10, 50 and 100 scenarios (referred to, respectively, as S10, S50 andS100). The S1000 policy is derived by running the Lagrangian relaxation algo-rithm for 100 iterations. The S10, S50 and S100 policies are derived by runningthe Lagrangian relaxation algorithm until the gap of the S1000 is attained, inorder to isolate the effect of the number of scenarios on the performance of the

8

Figure 2: A schematic of the WECC model studied in the Results section.

9

Table 2: Daily operating cost ($ M)S10 S50 S100 S1000 PF

WinterWD 5.857 5.914 5.874 5.867 5.722SpringWD 5.635 5.595 5.610 5.596 5.436

SummerWD 11.773 11.769 11.764 11.773 11.594FallWD 8.151 8.175 8.159 8.170 7.983

WinterWE 4.259 4.273 4.289 4.281 4.130SpringWE 3.801 3.789 3.798 3.792 3.637

SummerWE 9.495 9.469 9.461 9.482 9.315FallWE 6.149 6.142 6.124 6.148 6.010Average 7.303 7.307 7.299 7.301 7.138

Table 3: Day-ahead reserve capacity (MW)S10 S50 S100 S1000

WinterWD 8846 8575 8885 8600SpringWD 9173 8639 9077 8572

SummerWD 12185 12327 12261 12497FallWD 10039 10182 9771 9989

WinterWE 7700 8074 6978 7170SpringWE 7588 7001 7105 7032

SummerWE 11041 10545 10795 10810FallWE 9476 8669 8665 8637Average 9744 9542 9538 9485

policies. We also run a perfect foresight policy that commits units by anticipat-ing outcomes in advance.

5.1.1 Sensitivity of Cost on Number of Scenarios

The cost results are shown in table 2. The performance of the policies is as-sessed by running a Monte Carlo simulation on 1000 outcomes of wind powerproduction and contingencies. We note from the results that the proposed sce-nario selection algorithm performs very close to the S1000 policy even for asfew as 10 scenarios. Increasing the number of scenarios has a negligible impacton performance improvement.

5.1.2 Sensitivity of Optimal Policy on Number of Scenarios

The amount of day-ahead reserve capacity that is committed is shown in Table3. There is a tendency for more scenarios to result in less committed capacity.Nevertheless, this does not have a notable impact on cost performance, as weobserve in Table 2. The hourly committed capacity of slow reserves for eachday type and each policy is shown in Fig. 3.

10

Figure 3: The daily unit commitment schedule for each day type for S10, S50,S100 and S1000.

11

Table 4: Lower and upper bound distance from average cost ($ 1000s)S10 S50 S100 S1000

WinterWD (254, 325) (100, 169) (-165, -93) (-180, -105)SpringWD (1073, 1123) (135, 28) (97, 154) (115, 164)

SummerWD (-367, -234) (48, 87) (187, 304) (-76, 62)FallWD (-146, -45) (-292, 397) (-191, -77) (-108, 7)

WinterWE (185, 295) (-323, 413) (-504, -411) (-84, 17)SpringWE (668, 783) (-121, 202) (-228, -153) (52, 128)

SummerWE (-57, 99) (438, -283) (-150, 93) (-108, 50)FallWE (810, 913) (-530, 624) (-304, -207) (-92, 7)

5.1.3 Sensitivity of Bounds on Number of Scenarios

In Table 4 we examine the distance of the Lagrangian relaxation bounds from theresults of the Monte Carlo simulation. The left and right figure is the distanceof the lower and upper bound respectively from the average performance of thepolicy. Comparing S10 with S1000, there appears to be a notable improvementbetween the two policies in the ability of the bounds to estimate the averagecost performance. Note that the net of the right minus the left distance providesthe duality gap for each solution.

5.2 Improving the Gap Versus Adding More Scenarios

In this section we assess the impact of closing the gap on the quality of thesolution and compare it to the benefit of adding more scenarios to the model.The results in Table 5 are obtained by running the Lagrangian relaxation al-gorithm for 100 iterations without terminating it prematurely when the gap ofS1000 is attained, as was the case in Section 5.1. The resulting policies areindicated by an uppercase asterisk in Table 5. We note that, for the scenarioselection algorithm that we employ in this paper, the benefits of closing thegap outweigh the benefits of adding more scenarios to the model. The practicalimplication of this result is that the S10 model, which is a moderately sizedstochastic unit commitment model that can be solved on a single processor, canoutperform the S1000 model that can currently only be solved in a distributedcomputation environment. This result validates the practical applicability ofour proposed scenario selection algorithm, nevertheless this result can only bevalidated through the use of high performance computing.

5.3 Sensitivity of Duality Gap Bounds on the Number ofScenarios

In Table 5 we notice that the duality gap has a strong influence on the per-formance of the resulting stochastic unit commitment policy. It is thereforepractically important to be able to discern when an instance of a stochasticunit commitment model can admit further improvements in its duality gap.

12

Table 5: Performance Improvement as a Function of Gap ImprovementPolicy Gap ($) Cost ($M)S10 97827 7.303S10? 70559 7.300S50 92413 7.308S50? 62190 7.286S100 93711 7.299S100? 67069 7.289S1000 98485 7.301

An interesting question that arises is the relationship of the duality gap to thenumber of scenarios in the problem.

Previous research has focused on addressing this question. Bertsekas et al.[2] estimate a bound on the duality gap of the deterministic unit commitmentproblem. The authors first demonstrate that the dual relaxation of the unitcommitment problem is equivalent to a linear program whereby one decides onthe probability of a sequence of on-off schedules for units and correspondingoperating points. The authors argue that at most 2|T | units have a schedulethat is not primal feasible. By bounding the cost of changing the schedule ofthese at most 2|T | units to a feasible one the authors conclude that the dualitygap converges to zero as the number of units in the system increases.

Takriti and Birge [24] use a similar approach to Bertsekas et al. [2] providea bound on the stochastic unit commitment problem assuming that there is notemporal interdependence among continuous decisions, i.e. there are no rampingconstraints. The authors demonstrate that the duality gap does not depend onthe number of scenarios, but only on the number of branching points of thescenario tree. For a two-stage formulation, there is only one branching point.Although the authors bound the duality gap, they provide no indication as towhether the gap should decrease with the number of scenarios.

Sen et al. [23] present an example that demonstrates that the duality gapfor two-stage mixed integer programming problems with binary decision vari-ables does not converge to zero, even as the number of equiprobable scenariosincreases. This result contradicts a result by Birge and Dempster [3] wherethe authors claim that under certain mild assumptions, as the probability ofscenarios converges uniformly to zero the duality gap converges to zero.

Therefore, although the literature suggests that the duality gap does notincrease if the number of branching points does not increase (as is the case in ourstudy), there is no indication that the gap actually decreases. This is a questionthat we wish to explore in future research due to its practical implications fordeveloping termination criteria in a distributed computing environment.

13

Figure 4: The running time of the Lagrangian relaxation algorithm for S100.

5.4 Running Time

In this section we report results on running time. These results are obtainedfrom running the S100 model for Winter weekdays with 10 to 50 processors, inincrements of 10 processors. The Lagrangian relaxation algorithm was run for40 iterations.

The results are shown in Fig. 4. We note that doubling the number ofprocessors from 10 to 20 reduces computation time by 22%, however the incre-mental benefits rapidly decrease beyond that point. In fact, using 50 processorsresults in a 47% improvement of computation time. The bottlenecks in com-putation time occur on the one hand from the first-stage subproblem (P1),and from those unit commitment subproblems (UCs) and economic dispatchsubproblems (EDs) that require excessive solution times.

6 Conclusions

We present a distributed algorithm for solving the stochastic unit commitmentproblem of deploying day-ahead reserves in order to mitigate the uncertain andvariable supply of renewable energy resources in short-term scheduling. Ourtwo-stage stochastic model accounts for uncertain renewable supply, as wellas generator and transmission line outages. We present a scenario selectionalgorithm for selecting and weighing scenarios that is inspired by importancesampling and we test our scenario selection algorithm on a reduced model of theWestern Electricity Coordinating Council with 225 buses, 375 transmission lines

14

and 130 generators. We use our distributed algorithm to solve a naive scenarioselection model with 1000 scenarios and demonstrate that the scenario selectionalgorithm that we propose is capable of outperforming the naive scenario selec-tion algorithm by selecting as few as 10 scenarios. This indicates that, givenan appropriate scenario selection method, stochastic unit commitment modelsthat can be solved on a single processor are capable of delivering near-optimalresults. We examine the sensitivity of the unit commitment policy on the num-ber of scenarios and find that models with larger numbers of scenarios tend tocommit less capacity. We confirm that a stochastic unit commitment formula-tion with 1000 scenarios is able to closely approximate the average performanceof the policy as evaluated in Monte Carlo simulation, indicating that 1000 sce-narios represent a very close approximation to the full uncertainty in the model.We find that the benefits of reducing the duality gap of a particular instanceof a stochastic unit commitment model can yield superior benefits to increasingthe number of scenarios that are used in the model. Motivated by this resultwe investigate the sensitivity of duality gap bounds on the number of scenariosthat are input into the model and revisit theoretical results provided in the lit-erature. We finally present running time results and find that the incrementalbenefits of using additional processors for an instance of 100 scenarios rapidlydecrease beyond 20 processors.

Acknowledgment

This research was funded by NSF Grant IIP 0969016, the US Department ofEnergy through a grant administered by the Consortium for Electric Reliabil-ity Technology Solutions (CERTS), the Power Systems Engineering ResearchCenter, the Lawrence Livermore National Laboratory and the Federal EnergyRegulatory Commission.

References

[1] Anastasios G. Bakirtzis and Pandelis N. Biskas. A decentralized solutionto the DC-OPF of interconnected power systems. IEEE Transactions onPower Systems, 18(3):1007–1013, August 2003.

[2] Dimitri P. Bertsekas, Gregory S. Lauer, Nils R. Sandell, and Thomas A.Posbergh. Optimal short-term scheduling of large-scale power systems.IEEE Transactions on Automatic Control, AC-28(1):1–11, January 1983.

[3] John R. Birge and M. A. H. Dempster. Stochastic programming approachesto stochastic scheduling. Journal of Global Optimization, 9(3-4):417–451,1996.

[4] Pandelis N. Biskas, Anastasios G. Bakirtzis, Nikos I. Macheras, and Niko-laos K. Pasialis. A decentralized implementation of DC optimal power

15

flow on a network of computers. IEEE Transactions on Power Systems,20(1):25–33, February 2005.

[5] Emil M. Constantinescu, Victor M. Zavala, Matthew Rocklin, Sangmin Lee,and Mihai Anitescu. A computational framework for uncertainty quantifi-cation and stochastic optimization in unit commitment with wind powergeneration. IEEE Transactions on Power Systems, 26(1):431–441, Febru-ary 2011.

[6] Jitka Dupacova, Nicole Growe-Kuska, and Werner Romisch. Scenario re-duction in stochastic programming: An approach using probability metrics.Math Programming, 95(3):493–511, 2003.

[7] Yu Ermoliev. Numerical Techniques for Stochastic Optimization. SpringerSeries in Computational Mathematics, August 1988.

[8] Djalma Falcao. High performance computing in power system applications.Lecture Notes in Computer Science, 1215:1–23, 1997.

[9] Holger Heitsch and Werner Romisch. Scenario reduction algorithms instochastic programming. Computational Optimization and Applications,24(2-3):187–206, 2003.

[10] Kjetil Hoyland, Michal Kaut, and Stein W. Wallace. A heuristic formoment-matching scenario generation. Computational Optimization andApplications, 24:169–185, 2003.

[11] Gerd Infanger. Monte carlo (importance) sampling within a Benders de-composition algorithm for stochastic linear programs. Annals of OperationsResearch, 39(1):69–95, 1992.

[12] Balho H. Kim and Ross Baldick. Coarse-grained distributed optimal powerflow. IEEE Transactions on Power Systems, 12(2):932–939, May 1997.

[13] Jesus M. Latorre, Santiago Cerisola, and Andres Ramos. Clustering algo-rithms for scenario tree generation: Application to natural hydro inflows.European Journal of Operational Research, 181:1339–1353, 2005.

[14] M. Mazumdar. Importance Sampling in Reliability Estimation, Reliabilityand Fault Tree Analysis, pages 153–163. SIAM, Philadelphia, PA, 1975.

[15] A. Monticelli, M. V. F. Pereira, and S. Granville. Security-constrainedoptimal power flow with post-contingency corrective rescheduling. IEEETransactions on Power Systems, 2(1):175–180, February 1987.

[16] Matthias P. Nowak and Werner Romisch. Stochastic lagrangian relaxationapplied to power scheduling in a hydro-thermal system under uncertainty.Annals of Operations Research, 100(1-4):251–272, 2000.

[17] Anthony Papavasiliou. Coupling Renewable Energy Supply with DeferrableDemand. PhD thesis, U.C. Berkeley, October 2011.

16

[18] Anthony Papavasiliou, Shmuel S. Oren, and Richard P. O’Neill. Reserverequirements for wind power integration: A scenario-based stochastic pro-gramming framework. IEEE Transactions on Power Systems, 26(4):2197–2206, November 2011.

[19] M. V. F. Pereira and N. J. Balu. Composite generation/transmission reli-ability evaluation. Proceedings of the IEEE, 80(4):470–491, April 1992.

[20] Mario V. F. Pereira, Mario J. Teixeira, Michael F. McCoy, and HerminioJ. C. P. Pinto. Developing concurrent processing applications to powersystem planning and operations. IEEE Transactions on Power Systems,5(2):659–664, May 1990.

[21] G. Ch. Pflug. Scenario tree generation for multiperiod financial optimiza-tion by optimal discretization. Math Programming B, 89:251–271, 2001.

[22] Pablo A. Ruiz, Russ C. Philbrick, and Peter W. Sauer. Wind power day-ahead uncertainty management through stochastic uc policies. In PowerSystems Conference and Exposition, pages 1–9, March 2009.

[23] Suvrajeet Sen, Julia L. Higle, and John R. Birge. Duality gaps in stochasticinteger programming. Journal of Global Optimization, 18:189–194, 2000.

[24] Samer Takriti and John R. Birge. Lagrangian solution techniques andbounds for loosely coupled mixed-integer stochastic programs. OperationsResearch, 48(1):91–98, January - February 2000.

[25] Samer Takriti, John R. Birge, and Erik Long. A stochastic model for theunit commitment problem. IEEE Transactions on Power Systems, 11(3),August 1996.

17