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A Computational Framework for UncertaintyQuantification and Stochastic Optimization in Unit

Commitment with Wind Power GenerationEmil M. Constantinescu, Victor M. Zavala,

Matthew Rocklin, Sangmin Lee, and Mihai Anitescu

Abstract—We present a computational framework for integrat-ing a state-of-the-art numerical weather prediction (NWP) modelin stochastic unit commitment/energy dispatch formulations thataccount for wind power uncertainty. We first enhance the NWPmodel with an ensemble-based uncertainty quantification strategyimplemented in a distributed-memory parallel computing architec-ture. We discuss computational issues arising in the implementationof the framework and validate the model using real wind-speeddata obtained from a set of meteorological stations. We build asimulated power system to demonstrate the developments.

Index Terms—weather forecasting, wind, unit commitment, en-ergy dispatch, closed-loop.

NOMENCLATURE

Numerical Weather Predictionγ Ensemble inflation factorM Numerical weather prediction modelMx Number of model statesNS Number of ensemble membersQ Covariance of the model errorU , V , T Horizontal wind components and the tem-

perature atmospheric fieldsV, C Covariance and correlation matrices of the

initial ensemblex Atmospheric fieldx, S2 Ensemble sample average and covariance

matrixxNARR Atmospheric state reconciled with observa-

tions

Unit Commitmentaj ,bj Coefficients of production cost function of

thermal unit jccj ,hcj ,tcold

j Startup cost function coefficients of thermalunit j

cdj,k Shutdown cost of thermal unit j in period k

cpj,k Production cost of thermal unit j in periodk

cuj,k Startup cost of thermal unit j in period k

Cj Shutdown cost of thermal unit j

E.M. Constantinescu, V.M. Zavala and M. Anitescu are with the Mathematicsand Computer Science Division, Argonne National Laboratory, Argonne, IL60439, USA. E-mail: {emconsta, vzavala, anitescu}@mcs.anl.gov.

M. Rocklin is with the Department of Computer Science, University ofChicago, 1100 East 58th Street, Chicago, IL 60637, USA.

S. Lee is with Courant Institute of Mathematical Sciences, New YorkUniversity, New York, NY 10012, USA.

Dk Load demand in period kDTj Minimum down time of thermal unit jKt

j Cost of interval t of the stairwise startupfunction of thermal unit j

N Number of thermal unitsNS Number of wind-power scenariosNwind Number of wind unitsNDj Number of intervals of the stairwise startup

function of thermal unit jνj,k On/off state of thermal unit j in period kps,j,k Power output of thermal unit j in period k

and scenario spwinds,j,k Forecasted power output of wind unit j in

period k and scenario s

pwind,trues,j,k Observed power output of wind unit j in

period k and scenario sP j Maximum power output of thermal unit jP j Minimum power output of thermal unit jRk Reserve in period kRDj Ramp-down limit of thermal unit jRUj Ramp-up limit of thermal unit jSDj Shutdown limit of thermal unit jSUj Startup limit of thermal unit jT Number of periodsUTj Minimum up time of thermal unit j

Inference AnalysisA, b Coefficients of first-stage constraintsd Coefficients of first-stage costf jNS

Suboptimal sample average cost for batch jLNS ,M , s2L,NS ,M Mean and variance of lower boundM Number of data batchesq Coefficients of second-stage costQ, Q Second-stage cost and realizationT,W Coefficients of second-stage constraintsUNS ,M , s2U,NS ,M Mean and variable of upper boundvjNS

Optimal sample average cost for batch jξs Realization s of random variabley Second-stage decision variables for scenario

kz First-stage decision variables

I. INTRODUCTION

Wind power is becoming worldwide a significant componentof the power generation portfolio. In Europe, several countriesalready exhibit adoption levels in the range of 5-20% of the

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total annual demand. In the U.S. an adoption level of 20%is expected by the year 2030 [1]. Such a large-scale adoptionpresents many challenges to the operation of the electrical powergrid because wind power is highly intermittent and difficultto predict. In particular, unit commitment (UC) and energydispatch (ED) operations are of great importance because oftheir strong economic impact (on the order of billions of dollarsper year) and increasing emissions concerns.

Several UC studies analyzing the impact of increasing adop-tion levels of wind power have been performed recently. In [21],a security-constrained stochastic UC formulation that accountsfor wind-power volatility is presented together with an efficientBenders decomposition solution technique. In [19], a detailedclosed-loop stochastic UC formulation is reported. The authorsanalyze the impact of the frequency of recommitment on theproduction, startup, and shutdown costs. They find that increas-ing the recommitment frequency can reduce costs and increasethe reliability of the system. None of these previous stochasticoptimization studies present details on the wind-power forecastmodel and uncertainty information used to support their con-clusions. In [12], [15], artificial neural network (ANN) modelsare used to compute forecasts and confidence intervals for thetotal aggregated power for a set of distributed wind generators.A problem with empirical (data-based) modeling approaches[5], [20], [22], however, is that their predictive capabilities relystrongly on the presence of persistent trends. In addition, theyneglect the presence of spatio-temporal physical phenomenathat can lead to time-varying correlations of the wind speedsat neighboring locations. Such approaches can thus result ininaccurate medium- and long-term forecasts and over- or un-derestimated uncertainty levels [14], [8], [13], which in turnaffect the expected cost and robustness of the UC solution. Acomparison between uncertainty quantification techniques withempirical and physical weather prediction models for ambienttemperature forecasting is presented in [23].

In this work, we seek to exploit recent advances in numericalweather prediction (NWP) models to perform UC/ED studieswith wind-power adoption. The use of physical models is desir-able because consistent and accurate uncertainty information canbe obtained [13]. In a previous study, we have found that NWPmodels allow one to obtain much tighter uncertainty intervals oftemperature forecasts that translate into lower operating costs inbuilding systems [23]. On the other hand, we have also foundthat the practical capabilities of NWP models are limited. Oneof the major limiting factors is their computational complexity.For instance, performing data assimilation every hour at ahigh spatial resolution is currently not practical. In addition,extracting uncertainty information from NWP models quicklybecomes intractable from the point of view of both simulationtime and memory requirements. The question is: From anoperational point of view, how suitable and practical are theforecasting capabilities of state-of-the-art NWP models? Thisis an important question because NWP models are expected tobe used to make real-time operational decisions with importanteconomic implications. To analyze this issue, we present aframework that integrates the Weather Research and Forecast(WRF) model with a closed-loop stochastic UC/ED formulation.In particular, we are interested in analyzing computational issuesand the effects of wind uncertainty on UC/ED operations.

Arguably, more sophisticated hybrid methods that combineboth NWP wind speed forecasts and empirical models areneeded to map the resolution of NWP forecasts down to aspecific domain and to account for system-specific character-istics (e.g., power curves, orography) [13], [16], [6]. We pointout, however, that our approach offers several advantages overprevious work involving wind forecast, such as [16], [6]. Thefact that we have control over both the UC/ED model andthe WRF model allows us to refine the wind forecast withuncertainty as needed. In particular, as presented in Sec. IV-A,we can run the WRF at higher resolution than the data in [16],[6], and we also have control over the number of scenarios used.The latter capability can have a large impact on the UC/EDsolution feasibility and efficiency and can be used in conjunctionwith the confidence estimation described in Sec. III-D eitherto increase the number of scenarios in order to improve theuncertainty precision, if needed, or to use a more calculatedconservative solution. Full quantification of these benefits is animportant medium-term goal of our project. The goal of thiswork, however, is to describe the benefits of the integrationframework and for UC/ED problems.

We model the uncertainty of the wind-speed forecasts using asampling technique that generates an ensemble of the future re-alizations in the targeted geographical region. The ensembles areobtained by using a scalable implementation on a distributed-memory parallel computing and are sent to a stochastic UC/EDproblem. A resampling technique is developed to assess thequality of the stochastic UC/ED solutions. We validate theforecasts and spatial correlations using real wind-speed dataobtained from a set of meteorological stations. We also performan economic analysis of the impact of increasing adoption levelsof wind power. The novelty of our work lies in the integrationof uncertainty quantification and stochastic optimization, topicsnormally analyzed independently. From this integration, wecan analyze the economic effects of forecast accuracy anduncertainty bounds. An addition novelty of our work is acomputational analysis of WRF, which is important in orderto understand its limitations and capabilities in an operationalsetting.

The paper is structured as follows. Section II presents detailson the WRF model and on uncertainty quantification. SectionIII describes the stochastic unit commitment formulation andpresents a resampling technique used to perform inferenceanalysis. Section IV presents numerical validation results forWRF and the closed-loop UC simulation results. We concludewith a summary and directions for future work.

II. WIND FORECAST AND UNCERTAINTY ESTIMATIONUSING WRF

In this section, we describe the procedures used to forecastthe wind speed using WRF. We present in detail the ensembleinitialization and restarting procedures required in an operationalframework. The WRF model [17] is a state-of-the-art numericalweather prediction system designed to serve both operationalforecasting and atmospheric research needs. WRF is the result ofa multi agency and university effort to build a highly paralleliz-able code that can run across scales ranging from large-eddy toglobal simulations. WRF has a comprehensive description of the

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atmospheric physics that includes cloud parameterization, land-surface models, atmosphere-ocean coupling, and broad radiationmodels. The terrain resolution can go up to 30 seconds of adegree (less than 1 km2).

To initialize the NWP simulations, we use reanalyzed fields,that is, simulated atmospheric states reconciled with observa-tions, because the entire atmospheric state space is required bythe model as initial conditions whereas only a small subset ofthe state space is available through measurement at any giventime [9]. In particular, we use the North American RegionalReanalysis (NARR) data set that covers the North Americancontinent (160W-20W; 10N-80N) with a resolution of 10 min-utes of a degree, 29 pressure levels (1000-100 hPa, excludingthe surface), every three hours from 1979 until present. We usean ensemble of realizations to represent uncertainty in the initial(random) wind field and propagate it through the WRF nonlinearmodel. The initial ensemble is obtained by sampling froman empirical distribution, a procedure similar to the NationalCenters for Environmental Prediction (NCEP) method [10]. Inthe following sections we describe in more detail the proceduresneeded for generating the forecast and its uncertainty. A similarapproach is presented in [23].

1) Ensemble Initialization: In a normal operational mode, theNWP system evolves a given state from an initial time t0 to afinal time tF . The initial state is produced from past simulationsand reanalysis fields. Because of observation sparseness in theatmospheric field and the incomplete numerical representationof its dynamics, the initial states are not known exactly and canonly be represented statistically. Therefore, we use a distributionof the initial conditions to describe the confidence in theknowledge of the initial state of the atmosphere. We assumea normal distribution of the uncertainty field of the initial state,a typical assumption in weather forecasting. The distributionis centered on the NARR field at the initial time, the mostaccurate information available. In other words, the expectationis exactly the NARR solution. The second statistical momentof the distribution described by the covariance matrix V isapproximated by the sample variance or pointwise uncertaintyand its correlation, C. The initial NS-member ensemble fieldxt0s := xs(t0), i ∈ {1 . . . NS}, is sampled from N (xNARR,V):

xt0s = xNARR +V

12 ξs , ξs ∼ N (0, I) , s ∈ {1 . . . NS} , (1)

where C = Vij/√ViiVjj and Vii is the variance of variable i.

This is equivalent to perturbing the NARR field with N (0,V).That is, xs = xNARR +N (0,V). In what follows, we describethe procedure used to estimate the correlation matrix.

2) Estimation of the Correlation Matrix: In weather modelsthe correlation structure typically is localized in space. There-fore, in creating the initial ensemble one needs to estimate thespatial scales associated with each variable. To obtain thesespatial scales, we build correlation matrices of the forecast errorsusing the WRF model. These forecast errors are estimated byusing the NCEP method [10], which is based on starting severalsimulations staggered in time in such a way that, at any time,two forecasts are available. In particular, we run a month ofday-long simulations started every twelve hours so that everytwelve hours we have two forecasts, one started one day beforeand one started half-a-day before. The differences between twostaggered simulations is denoted as dij ∈ RN×(2×30days), that

is, the difference at the ith point in space between the jth pair offorecasts, where N is the number of points in space multipliedby the number of variables of interest. We can then define ϵsas the ith row, each of which correspond to the deviations fora single point in space. Therefore, the covariance matrix canbe approximated by V ≈ ddT. Calculating and storing theentire covariance matrix are computationally intractable. Con-sequently, we describe the correlation distance at each verticallevel and for each variable by two parameters representing theEast-West and North-South directions. This approach capturesthe Coriolis effect and the Earth rotation, as well as faster andlarger-scale winds in the upper atmosphere. We assume thatcorrelations and winds are roughly similar in nature acrossthe continental U.S. This process is repeated in the verticaldirection. To create the perturbations from these length scales,we take a normally distributed noisy field and apply Gaussianfilters in each direction with appropriate length scales to obtainthe same effect as in (1).

3) Ensemble Propagation through the WRF Model: Theinitial state distribution is evolved through the NWP modeldynamics. The resulting trajectories can then be assembled toobtain an approximation of the forecast covariance matrix:

xtFs =Mt0→tF

(xt0s

)+ ηs(t) , s ∈ {1 . . . NS} , (2)

where xt0s ∼ N (xNARR,V

t0), ηs ∼ N (0,Q), and Mt0→tF (•)represents the evolution of the initial condition through theWRF model from time t0 to time tF . The initial condition isperturbed by the additive noise η that accounts for the variouserror sources during the model evolution. An analysis of thecovariance propagation through the model is given in [23].

In this study, we assume that the numerical model (WRF) isperfect, that is, η ≡ 0, and given the exact real initial conditions,the model produces error-free forecasts. For long predictionwindows, this is a strong assumption. In this study, however, werestrict the forecast windows to no longer than one day ahead,thus making this assumption reasonable.

4) Accounting for Error Underestimation: In an operationalsetting, observations become available periodically and can beassimilated in the atmospheric state. In order to account forthe new information, the ensemble needs to be recentered onthe new reanalyzed field. In our example, we consider 12-hourwindows between restarts. However, the errors given by theensemble variance may be over- or underestimated because ofsimulation and sampling errors. In other words, the ensemblestatistics may diverge from the true statistics. Therefore, theerror levels need to be re-estimated before each initialization.Since correlations between entries in the state vector are morerobustly estimated – their values are accurate under fewerassumptions compared to variance – by our approach [9], [23],variance is the only parameter that needs to be adjusted. One ap-proach is to consider the reanalyzed field xNARR as the true state,for computing corrections purposes only, and require that thissolution be on average within one standard deviation as givenby the ensemble spread. This approach corresponds to finding afactor γ that inflates the ensemble spread about its expectation.Let us consider again the ensemble xs, s = 1, . . . , NS and thereanalyzed solution xNARR. Denote by x = 1

NS

∑NS

s=1 xs thesample expectation and by σj =

√S2

jj , j ∈ {1 . . .Mx}, the

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standard deviation, where S2 = 1NS−1

∑NS

s=1(xs− x)(xs− x)T

is the sample covariance estimation. Then, we have

γ = max (1,min (meanU,V,T (|xNARR − x|/σ) , 4)) ,

where U , V , and T are the the wind-field components and thetemperature, the ensemble variables under consideration. Forthis comparison, we consider only the first five layers, whichinclude grid points located below 300 m. The new ensembleis then obtained by xs ← x + γ (xs − x) , s ∈ {1 . . . NS} .The factor is bounded between one, because the model error isunderestimated in our case, and four to avoid large jumps inthe solution and destabilize the NWP model. Experimentally,however, we noticed that γ ≈ 2, which confirms that thisapproach tends to underestimate the uncertainty. This fact isnot unexpected because the model error is not considered.

III. UNIT COMMITMENT AND ENERGY DISPATCH

In this section, we describe the unit commitment and energydispatch formulation used and discuss extensions to account forwind-power uncertainty. The main idea behind our computa-tional framework is to design a closed-loop UC/ED strategyusing a stochastic programming formulation that incorporatesweather forecast and uncertainty information from the WRFmodel and the ensemble approach described in the previoussection. With this, we explore the accuracy of the forecasts andthe effect of assimilating measurement information at differentfrequencies (WRF model reanalysis).

A. Deterministic Formulation

The UC problem has been studied extensively in literaturereports. The interested reader can refer to [3], [21], [18], [7]. TheUC formulation considered here is based on the mixed-integerlinear programming (MILP) formulation of Carrion and Arroyo[3]. The formulation is shown below. The sets T := {1 . . . T},N := {1 . . . N}, and Nwind := {1 . . . Nwind} represent thetime periods, thermal units, and wind generators, respectively.The demand at each time period k is denoted by Dk, and thereserve requirement is Rk. The power output of unit j at time kis given by the continuous variable pj,k. The expected value ofthe output of the wind unit j at time k, E[pwind

j,k ] is approximatedby 1

NS

∑NS

s=1 pwinds,j,k . The continuous variable pj,k represents the

maximum power output of unit j at time k. This variable isintroduced in order to model the spinning reserves given by thedifferences pj,k − pj,k. The units of all the power outputs areMW. The on/off status of unit j at time k is given by the binaryvariable νj,k.

minpj,k,pj,kνj,k

∑j∈N

∑k∈T

cpj,k + cuj,k + cdj,k (3a)

s.t.∑j∈N

pj,k +∑

j∈Nwind

E[pwindj,k ] = Dk, k ∈ T (3b)∑

j∈Npj,k +

∑j∈Nwind

E[pwindj,k ] ≥ Dk +Rk, k ∈ T (3c)

(4)− (11).

The production cost for each thermal unit is approximated byusing the linear model [2]

cpj,k = ajνj,k + bjpj,k, j ∈ N , k ∈ T , (4)

where aj and bj are cost coefficients. To model the startup costcuj,k we use a staircase cost Kt

j , t = 1, ..., NDj , where NDj

is the number of intervals. This leads to the following set ofinequality constraints:

cuj,k ≥ Ktj

(νj,k −

t∑n=1

νj,k−n

),

j ∈ N , k ∈ T , t = 1, ..., NDj , (5a)cuj,k ≥ 0, j ∈ N , k ∈ T . (5b)

The formulation of the shutdown cost is given by

cdj,k ≥ Cj

(νj,k−1 − νj,k

), j ∈ N , k ∈ T , (6a)

cdj,k ≥ 0, j ∈ N , k ∈ T , (6b)

where Cj is the shutdown cost of unit j. The power output ofeach unit at each period must satisfy the bounds

P jνj,k ≤ pj,k ≤ pj,k, j ∈ N , k ∈ T , (7a)

0 ≤ pj,k ≤ P jνj,k, j ∈ N , k ∈ T , (7b)

where P j and P j are the maximum and minimum capacities ofunit j, respectively. The thermal power outputs must also satisfythe ramp-up limits

pj,k ≤ pj,k−1 +RUjνj,k−1 + SUj

(νj,k − νj,k−1

)+ P j(1− νj,k), j ∈ N , k ∈ T , (8)

and the shutdown and ramp-down limits are

pj,k−1 ≤ pj,k +RDjνj,k + SDj

(νj,k−1 − νj,k

)+P j(1− νj,k−1), j ∈ N , k ∈ T . (9)

Here, RDj , RUj , SDj , and SUj are the ramp-down, ramp-up, shutdown, and startup limits of unit j, respectively. Theminimum uptime constraints are

Gj∑k=1

(1− νj,k) = 0, j ∈ N (10a)

k+UTj−1∑n=k

νj,n ≥ UTj

(νj,k − νj,k−1

), j ∈ N

k = Gj + 1, . . . , T − UTj + 1 (10b)T∑

n=k

(νj,n − (νj,k − νj,k−1)

)≥ 0, j ∈ N ,

k = T − UTj + 2, . . . , T , (10c)

where UTj are the minimum uptime limits and Gj =min(T, (UTj − U0

j )νj,0) is the number of periods that unit jmust be initially ON. The initial state of unit j is denoted by νj,0and is a fixed parameter. The minimum downtime constraints

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are formulated asLj∑k=1

νj,k = 0, j ∈ N (11a)

k+DTj−1∑n=k

(1− νj,n) ≥ DTj

(νj,k−1 − νj,k

), j ∈ N ,

k = Lj + 1, . . . , T −DTj + 1 (11b)T∑

n=k

(1− νj,n − (νj,k−1 − νj,k)

)≥ 0, j ∈ N ,

k = T −DTj + 2, . . . , T , (11c)

where DTj denote the minimum downtime limits and Lj =min(T, (DTj−S0

j )(1−νj,0)) is the number of periods that unitj must be initially OFF. Note that it is possible to use this modelto simulate the performance of the energy dispatch problem byfixing the commitment variables νj,k.

B. Stochastic Programming Formulation

We extend the previous deterministic formulation by consid-ering corrective actions on the power outputs of the thermalgenerators to account for the uncertainty in the wind poweroutputs. The problem can be cast as a two-stage stochasticprogramming problem similar to the ones proposed in [2],[21], [19]. The first-stage decision variables are the currentthermal power outputs pj,1, pj,1 and the commitment profilesover the entire planning horizon νj,k. The power outputs arenonanticipatory (here and now) because it is assumed that thecurrent wind-power outputs pwind

j,1 are known and given bypwind,truej,1 . To formulate the second stage, we consider multiple

realizations of the wind power outputs pwinds,j,k , and we define

scenario-dependent thermal power outputs ps,j,k and ps,j,k withk > 1 (wait and see). Note that we do not define second-stagescenario-dependent commitment variables because we wish tokeep the problem computationally tractable. The formulation ofthe stochastic optimization problem is

minps,j,k,ps,j,k,νj,k

1

NS

∑s∈S

∑j∈N

∑k∈T

cps,j,k + cuj,k + cdj,k

(12a)

s.t.∑j∈N

ps,j,k +∑

j∈Nwind

pwinds,j,k = Dk, s ∈ S, k ∈ T (12b)∑

j∈Nps,j,k +

∑j∈Nwind

pwinds,j,k ≥ Dk +Rk, s ∈ S, k ∈ T

(12c)ps,j,1 = p1,j,1 s ∈ S, j ∈ N (12d)ps,j,1 = p1,j,1 s ∈ S, j ∈ N (12e)

(4)− (9), s ∈ S(10)− (11),

where S := {1 . . . NS}. The ramp and power-limit constraintsare defined over each scenario, s ∈ S where pj,k ← ps,j,kand pj,k ← ps,j,k. The nonanticipativity constraints for thepower outputs in the first time step are given by equations(12d) and (12e). For the known wind power outputs we setpwinds,j,1 ← pwind,true

j,1 . Note that if the stochastic formulation isable to capture the uncertainty of the wind power accurately, the

reserve requirements can be reduced to less conservative levelsor even be removed. Note also that one can solve a closed-loopstochastic dispatch problem by fixing the commitment actions.

C. Closed-Loop Implementation

To simulate the closed-loop performance of the power system,we consider a rolling-shrinking horizon approach. The startingrolling time is reset to one each time new fossil fuel andelectricity price information is obtained from the commodityand day-ahead markets. The latter information is availablefrom the independent system operators. This period is assumedto be T = 24 hours. At the start of the rolling time, weassume that the wind-power forecast becomes available fromWRF for the next 24 hours. At this point, the stochastic unitcommitment problem is solved by using the current wind-power outputs pwind

s,j,ℓ ← pwind,truej,ℓ and the future forecasts

pwinds,j,k , k = ℓ + 1, ..., T , where ℓ is the current time step.

The solution of this problem gives the commitment profiles νj,kover the 24-hour rolling horizon. At each step inside the rollinghorizon ℓ = 2, ..., T , the horizon is shrunk by one time stepT ← T − ℓ, and the stochastic energy dispatch is solved overthe remaining horizon with the new, true wind power but thesame WRF forecasts of the current day. Each of these shrinkinghorizon problems gives the current power outputs pj,ℓ and pj,ℓat current time ℓ.

D. Inference Analysis

In the above stochastic formulation, the wind-power outputsare assumed to have a probability distribution P. In moststochastic optimization studies this distribution is assumed tobe known. As seen in Section II, obtaining this distribution ispart of the modeling task. Since many different forecast models(autoregressive, ANN, physics based) can be used to constructthe error distribution, there is not a unique distribution. Froma practical point of view, we expect that such a distributionis able to encapsulate the actual realizations of wind powerand has tight confidence intervals. We model the wind speeddistribution by propagating an assumed Gaussian distributionof the initial state conditions through the WRF model. Becauseof the complexity of the model, we are limited to a singlebatch of a few (less than a hundred) samples. From a stochasticoptimization point of view, this is an issue because we are notsolving the problem with the full distribution. Consequently, wemust perform an inference analysis to assess the quality of thesolution.

1) Computation of Confidence Intervals: The two-stagestochastic UC problem with fixed binary variables can beexpressed in the following abstract form [11]:

minz≥0

dT z +Q(z), s.t. Az = b. (13)

Here, z are the first-stage decision variables, and Q(z) =E [Q(z, ξ)] is the second-stage cost. We assume that the proba-bility distribution P of ξ has finite support; that is, ξ has a finitenumber of scenarios {ξ1, ..., ξK} with probabilities πs ∈ (0, 1),so we have Q(z) = 1

K

∑Ks=1 Q(z, ξs), where

Q(z, ξs) = minys≥0

qT ys, s.t.Tz+Wys = ξs, s = 1, ...,K. (14)

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Here, ys are the second-stage decision variables, and ξs are therealizations of the wind-power outputs. Since K is a very largenumber, it is impractical to solve the stochastic problem exactly.Therefore, given a fixed number of realizations NS ≪ K fromWRF, we solve the approximate problem,

minz≥0

dT z +1

NS

NS∑s=1

Q(z, ξs), s.t. Az = b. (15)

This smaller problem is known as a sample-average-approximation (SAA) of the original problem (13), which isusually computationally intractable. We seek to estimate lowerand upper bounds of the true optimal solution v∗ (using theentire set of K realizations) and their corresponding con-fidence intervals. Here we use the methodology put forthin [11]. A lower bound can be estimated generating j =1, ...,M batches, each of size NS , and we can then solve(15) for each batch. If we denote as vjNS

the optimal costof each SAA problem, we can estimate the lower bound asLNS ,M = 1

M

∑Mj=1 v

jNS

. The sample variance estimator is

given by s2L,NS ,M = 1M−1

∑Mj=1

(vjNS− LNS ,M

)2. The mean

and variance can be used to construct confidence intervalsof the lower bound. To estimate the upper bound, we picka given value for the first-stage variables z and generate anew set of j = 1, ...,M batches of data. We then evaluate(13), leading to f j

N (z). Note that each evaluation involves thesolution of the second-stage problem (14). As before, we havethe mean UNS ,M = 1

M

∑Mj=1 f

jNS

and variance s2U,NS ,M =

1M−1

∑Mj=1

(f jNS− UNS ,M

)2.

2) Weighted Average Sampling: The inference analysis taskrequires multiple batches of realizations. As expected, obtainingthese from WRF is not practical. Here, we present a heuristicresampling technique to avoid this limitation. To create newtime series from the existing batch of WRF realizations, weexpress a new realization as a weighted average of the availableones. Suppose the WRF model is x(t) = M(t, x(0)), wherex(t) is the state vector at time t. If we are given NS samplesxj and we can write x(0) =

∑j∈S wjxj(0), the propagation of

x(0) is x(t) =M(t, x(0)) =M(t,∑

j∈S wjxj(0)). Assumingthe variance of the samples is small, we can write xj(0) =x(0) + ϵj(0). We justify the computation of weighted averagesof the time series by observing that

x(t) = M

t, x(0) +∑j

wjϵj(0)

≈ M(t, x(0)) +

∑j

wj∂M∂x

(t, x(0))ϵj(0)

≈∑j

wjM(t, x(0) + ϵj(0)

)=∑j

wjxj(t).

In other words, the weighted average approximates, to firstorder, the nonlinear propagation of weighted samples of theinitial conditions. The weights are chosen to be Gaussiannear the unit vectors in the standard basis on a hyperplane∑

j∈S wj = 1 in the w space.

IV. INTEGRATIVE STUDY

In this section we integrate the wind-speed forecasts producedby WRF by following the procedure described in Section II withthe stochastic unit commitment/energy dispatch formulationsdescribed in Section III. The entire computational frameworkis sketched in Figure 1.

WRF

Model

Energy

Dispatch

Thermal Power

Generators

Ensemble

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Meteo StationsWind Power

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Analysis

Uncertainty Quantification

Stochastic Optimization

Fig. 1. Schematic representation of computational framework.

A. Wind Forecast and Uncertainty Quantification

We use the WRF model to forecast the wind speed in aspecific region that covers the state of Illinois. We set upa computational nested domain structure including a high-resolution sector that covers the target area and two additionaldomains of larger coverage but lower resolution. The parentdomains supply the boundary conditions for the nested ones,and the largest domain has prescribed boundary conditions fromcoarser ones. This setup is illustrated in Fig. 2. A similar setupwith one coarse domain is described in [23]. We generate sixensemble data sets, each containing the predicted wind speed forIllinois corresponding to domain # 3 in Fig. 2. Each ensemblehas NS = 30 members. The data is sampled every 10 minutes,and each ensemble is evolved one day ahead. The starting timeof the experiment t0 corresponds to June 1, 2006, 6:00 PM CT(local time), with each data set restarted from the reanalyzedsolution at time t0 + (k − 1) × (12 hours) with k = 1, . . . , 6.In other words, each data set is started at the revalidation timewith 12-hour increments.

1) Validation Using Wind and Temperature Data Measure-ments: The weather station observations were obtained fromthe National Climatic Data Center (NCDC), and their locationsare illustrated in Fig. 3.b. In Fig. 4 we show the wind-speed(±2σ) predictions and measurements for Peru and Chicago, IL(denoted by ∇ in Fig. 3.b). Each ensemble evolves for 24 hours,and new ones are started every 12 hours. We remark that the

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a) Wind farm location b) Weather stationsFig. 3. Wind farms (circles) and meteorological stations (triangles) locationsin Illinois.

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Fig. 4. Wind-speed (±2σ) predictions and measurements (o) for Peru (left,RMS error=1.56, R2=0.32) and Chicago (right, RMS error=1.51, R2=0.35), IL.The vertical dashed lines denote the beginning of a new 24-hour predictionwindow; different colors are used to indicate ensembles started at differenttimes.

wind-speed measurements obtained from NCDC are given inmiles per hour rounded to the nearest integer. Doing so hasthe unfortunate effect of diminishing the wind variability andyielding more pessimistic than real validation results. Despitethis, the wind-speed uncertainty intervals generated by WRFcapture the trends well, with few exceptions. It is also clear fromFig. 4 that the forecasts do not improve much when updatedevery 12 hours instead of every 24 hours. Note that the forecastsare not improved significantly at the middle of the day, perhapsbecause measurements assimilated during the day are not asinformative as those assimilated during the night, where thewind currents tend to be stronger. We have also observed thatthe wind-speed trends are much more difficult to predict thantemperature trends. This point is enforced by the correlogramfor the temperature and wind speed at Peru, IL shown in Fig. 6,where it is clear that the time correlations of wind speed decaymore quickly than those of the temperature.

We present validation results at six active wind-farms inthe state of Illinois to analyze their magnitude and correlation

structure. The order of the windows goes from left to rightand coincides with the wind-farm location numbering shownin Fig. 3. Currently, the power produced by wind turbinesdepends on the wind speed at elevations of about 40-120 meters.The wind-speed fields at these heights can be extracted fromWRF. Unfortunately, the NCDC data available for validationis reported only at 10 meters. Obtaining real wind-speed dataat higher altitudes requires access to proprietary databases ofoperational wind farms. The wind-speed fields at 10 metersabove the ground for three consecutive days of June 2006 arepresented in Fig. 5. The WRF realizations are able to capturethe general trends of the actual observations at all locations. Inaddition, they are able to encapsulate the observations. Note thatthe wind speed is relatively low at this height. The maximumaverage is around 6-7 meters per second. We have found thatthe wind speeds reach a maximum average of around 10 metersper second at 100 meters in the studied region. In addition, wehave observed that the uncertainty levels increase significantlyat this height as a result of the larger range and variability.This increase is also expected because most of the wind speeddata assimilated in WRF is near ground level. The 100 metersprofiles are not presented here because of space restrictions. Formore details, please refer to Section 4.1 in the technical report[4]. In Fig. 7 we show the spatial correlations of the wind speedfor a particular wind farm on June 5, 1:50 AM, as inferred fromthe 30-member WRF ensemble simulation. The wind speed ishighly correlated over the studied region, and it has a nontrivialspatial structure.

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Fig. 5. Wind-speed realizations for 6 wind farm locations in Illinois at 10m andobservations (dots) at nearest meteorological stations. Vertical lines representbeginning of day (12:00 AM).

2) Implementation Considerations: In this study we usedversion 3.1 of WRF [17]. The ensemble approach taken for esti-mating the uncertainty in the weather system is highly paralleliz-able because each scenario evolves independently through WRF.The most expensive computational element is the evolution ofeach sample through the WRF system. We therefore considera two-level parallel implementation scheme. The first level is acoarse-grained task decomposition represented by each sample.A secondary finer-grained level consists in the parallelization ofeach sample. This approach yields a highly scalable solution.The simulations were performed on the Jazz Linux cluster atArgonne National Laboratory. Jazz (now decommissioned) had350 compute nodes, each with a 2.4 GHz Pentium Xeon with1.5 GB of RAM and used Myrinet 2000 and Ethernet forinterconnect. Our running times given in Fig. 8 indicate thataround 32 CPUs were sufficient to generate forecasts with WRF

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Fig. 6. Correlogram for the wind and temperature measurements and simula-tions at Peru, IL.

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Fig. 7. Spatial correlation for the wind field for wind farm #8 on June 5,1:50 AM, denoted by “X.” The circle markers denote the other wind farms inIllinois.

in a closed-loop UC/ED setting. The times also suggest that, inorder to generate forecasts every hour, one would need about500 CPUs.

CPUs Time[hr]

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Fig. 8. Scalability of WRF on the computer cluster Jazz for 24 hours.

B. Economic Study Unit Commitment/Energy Dispatch

Because of the lack of detailed design data of thermal andwind-power units in the open literature, we have constructedan artificial simulation study. We first describe the thermal andwind-power assumptions used and then discuss our results fromthe simulation.

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Fig. 9. Wind power realizations for 6 wind farm locations in Illinois at 10 m andobservations (dots) at nearest meteorological stations. Vertical lines representbeginning of day (12:00 am).

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Fig. 10. Closed-loop profiles for thermal units. Solid thin line is optimalprofile (with perfect information), solid thick line is stochastic UC solution,and thick gray lines are planned scenarios at the beginning of each day.

1) Power System Description: The thermal power systemspecifications used in this work are based on those reported in[3]. The system contains a total of 10 thermal generators witha total installed capacity of 1662 MW. The peak demand is1326 MW. The ramp limits of the units are not reported, so wehave assumed them to be 50% of the corresponding maximumcapacity. The reserve requirements are assumed to be 10% of thedemand. To simulate increasing level of wind power adoption,we increase the number of wind turbines at 12 existing windfarm locations in Illinois.

2) Results: To generate wind-power forecasts, we propagatethe wind-speed observations and the WRF realizations at aheight of 10 meters through a typical wind-power curve witha maximum capacity of 1.5 MW. The nominal curve has acut-in speed of 3 meters per second and reaches the ratedcapacity at 12 meters per second. The wind-speed observations,forecast, and ensembles used are summarized in Fig. 5. Aspreviously mentioned, we used the height of 10 meters becausethe NCDC data used for validation are reported only at thislevel. As expected, the wind speeds are relatively low at thislevel, thus leading to small power outputs. Instead of usingthe wind speed WRF forecasts at 100 meters, we have keptthe 10 meters WRF forecasts and observations and mappedthese using a shifted power curve obtained by displacing thenominal cut-in speed from 3 to 2 meters per second. With this,the rated capacity is reached at around 11 meters per second.This strategy allowed us to obtain more consistent validationresults for wind power compared to linear interpolation of thewind-speed observations. The resulting wind-power realizations

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al P

ower

[MW

]

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Fig. 11. Closed-loop total power profiles obtained with stochastic UC formu-lation. Top thick line is demand profile, medium thick line is the implementedthermal profile, gray lines are planned realizations at beginning of each day,bottom thick line is actual total wind power, and adjacent gray lines are forecastprofiles.

and observations are presented in Fig. 9. The wind-powerdistribution is clearly affected by the nonlinear structure of thepower curve, increasing the spread of the distribution. The WRFrealizations are able to encapsulate the actual power observa-tions. The largest differences are observed at the beginning ofthe third day.

We have run the closed-loop UC/ED system assuming arolling horizon and a forecast frequency of 24 hours. TheED problem runs every hour. A total of 30 WRF realizationsare used to solve the stochastic problem. The resulting MILPproblems are implemented in AMPL and are solved with theCBC solver from the COIN-OR repository. The MILP contains38,651 variables from which 240 are binary, 783 equalityconstraints, and 40,747 inequality constraints. The averagesolution time for the stochastic UC problem in a quad-core Intelprocessor running Linux is about 9 minutes in cold-start mode.The solution time of the energy dispatch problem is less than 10seconds. The results for the 20% penetration study are presentedin Figs. 10 and 11. In Fig. 10, we present the policies forthe power levels of six thermal units. The solid lines representthe predicted and the realized power profiles, while the graylines represent the forecasted realizations at the beginning ofthe day. We notice that the sensitivity of the power levels ofsome units to the uncertainty of the wind power is very small.Generators #2 and #5 are the most sensitive, while generators #3and #4 exhibit no sensitivity. We have found that the sensitivitylevels depend strongly on the design characteristics and pricesof the generators. We have also found that the optimal costof the stochastic strategy over three days of operation is onlyabout 1% larger than that of the perfect information strategy.We also performed an inference analysis using the resamplingstrategy of Section III-D for the first day of operation usingM = 30 different batches. The upper bound mean was foundto be UNS ,M = $474, 064 with variance s2U,NS ,M = 1, 082$2.The lower bound mean was found to be LNS ,M = $474, 317with variance s2L,NS ,M = 1, 656$2. Both variances are lessthan 0.25% of the mean cost. The smaller variances indicatethat 30 WRF realizations might be sufficient to estimate theoptimal cost accurately. Note, however, that the mean of theupper bound is lower than that of the lower bound. This canoccur because the estimates are produced on different scenarios(the estimates would be exactly equal to each other if usedwith the same scenarios). It indicates the presence of a biasin the estimates which has been noted in other applications

of stochastic programming [11]. This bias can be reducedusing variance reduction techniques such as Latin Hypercubesampling instead of Monte Carlo sampling and by increasingthe number of scenarios [11]. We have also found that updatingthe WRF forecast every 12 hours instead of every 24 hours doesnot bring important economic benefits. The reason is twofold:minor improvements in forecast accuracy, as pointed out inSection IV-A, and the properties of the power system underconsideration. A different outcome could be obtained with adifferent generator mix.

In Fig. 11 we present the profiles of total aggregated (sumover the total units) demand, thermal power, and wind power.The solid lines represent the predicted and the realized powerprofiles while the gray lines represent the forecasted realizationsat the beginning of the day. The top solid line is the dailydemand profile, which is assumed to be constant. Note that theaggregated wind-power profile (bottom) does not follow a strongperiodic trend. Nevertheless, the WRF realizations are able toencapsulate the actual profiles (solid lines) during the first twodays. As a result, the optimizer is always able to satisfy the load,even for an adoption level of 20%. On the third day, however,we see a significant mismatch between the forecasted windpower and the realized one in the first 12 hours of operation.In this case, the reserves are sufficient to satisfy the load. Thiseffect could potentially be ameliorated by inflating the initialconditions of the WRF ensembles, but it cannot be predicted apriori. Thus, a high frequency and adaptive inflation/resamplingprocedure should be added to the system. We also found thata deterministic strategy (using only the WRF forecast mean) isnot able to sustain adoption levels of more than 10% even withthe allocated reserves. We observed that increasing the adoptionlevels increases the startup and shutdown costs, but these arenegligible (on the order of $10,000) with respect to the totalproduction costs.

V. CONCLUSIONS AND FUTURE WORK

We presented a computational framework for the integrationof the state-of-the-art Weather Research and Forecasting (WRF)model in stochastic unit commitment/energy dispatch formu-lations that account for wind-power uncertainty. We extendedthe WRF model with a sampling technique implemented in adistributed-memory parallel computing architecture to generateuncertainty information. In addition, we developed a resamplingstrategy that avoids expensive WRF simulations to performinference analysis. Our simulated commitment study indicatesthat using WRF forecasts and uncertainty information is criticalto achieve high adoption levels with minimum reserves. Ourstudy illustrates an operational setting with real data, pointingout several issues and limitations that are not found in idealizedexperiments using artificial forecasts and uncertainty informa-tion. For instance, we have not found significant benefits ofupdating the WRF forecasts in intra-day operations. In addition,the numerical experiments indicate that a relatively large numberof CPUs are required to generate forecasts and uncertainty infor-mation at a higher frequency than 12 hours. We emphasize thatthe integrative framework presented here is preliminary and doesnot consider more detailed issues such as intra-day reschedulingof unit commitment, effects of updating wind power forecasts athigher temporal resolutions (e.g., hourly), as used in the Danish

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power system. These two factors affect the value of wind powerforecasts during intra-day operation. Therefore, as part of futurework, we are interested in developing techniques to generateforecasts at higher spatial and temporal resolution. In addition,we are interesting in generating wind-power forecast modelsby fusing WRF wind-speed forecasts and operational wind-power data. Thanks to our open access to WRF, our frameworkis highly flexible and allows us to consider these extensions.Additionally, we are interested in dealing with networks ofreal size with hundreds of generators, transmission constraints,and intra-day scheduling. To solve these challenging problems,we are developing algorithms for the solution of stochasticoptimization problems in parallel computing architectures. This,together, with variance reduction techniques can enable thesolution of large-scale problems with small statistical errors.

ACKNOWLEDGMENTS

This work was supported by the Department of Energy,through Contract No. DE-AC02-06CH11357. We acknowledgethe insightful comments of the referees, which helped usimprove the quality of the paper and pointed out interestingfuture extensions to our work. We thank Professor MohammadShahidehpour for comments on an earlier version of this docu-ment. We also thank Argonne National Laboratory’s LCRC forthe use of Jazz cluster.

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Emil M. Constantinescu received his B.Sc. and M.S. degrees from the Collegeof Automatic Controls and Computers, Bucharest Polytechnic University, Ro-mania (2001,2002) and the Ph.D. degree from Virginia Tech (2008) in computerscience. He is currently the Wilkinson Postdoctoral Fellow in the Mathematicsand Computer Science Division at Argonne National Laboratory. His researchinterests include uncertainty quantification in weather and climate models andtheir applications to energy systems.

Victor M. Zavala is currently a Director’s Postdoctoral Fellow in the Mathemat-ics and Computer Science Division at Argonne National Laboratory. He receivedthe B.Sc. degree from Universidad Iberoamericana (2003) and the Ph.D. degreefrom Carnegie Mellon University (2008), both in chemical engineering. He hasserved as a research intern at ExxonMobil Chemical Company (2006, 2007) andas a consultant for General Electric Company (2008). His research interests arein the areas of mathematical modeling and optimization of energy systems.

Matthew Rocklin is a Ph.D. computational mathematics student in the Com-puter Science Department at the University of Chicago. He completed hisundergraduate degree in physics and mathematics at the University of California-Berkeley in 2007 and is now focusing on scientific computing.

Sangmin Lee is currently a Ph.D candidate in the Mathematics Departmentof the Courant Institute at New York University. He earned B.Sc. and M.Sc.degrees in physics from Seoul National University, Korea, in 1999 and 2001,respectively. His research interests lie in the area of numerical methods forstochastic processes.

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Mihai Anitescu obtained his Engineer (M.Sc.) Diploma in electrical engineeringfrom the Polytechnic University of Bucharest in 1992 and his Ph.D. degree inapplied mathematical and computational sciences from the University of Iowain 1997. He is currently a computational mathematician in the Mathematics andComputer Science Division at Argonne National Laboratory and a professor inthe Department of Statistics at the University of Chicago. He is the authorof more than 60 papers in scholarly journals and conference proceedings,on numerical optimization, numerical analysis, computational mathematics andtheir applications.

The submitted manuscript has been created by the University of Chicago asOperator of Argonne National Laboratory (“Argonne”) under Contract No. DE-AC02-06CH11357 with the U.S. Department of Energy. The U.S. Governmentretains for itself, and others acting on its behalf, a paid-up, nonexclusive,irrevocable worldwide license in said article to reproduce, prepare derivativeworks, distribute copies to the public, and perform publicly and display publicly,by or on behalf of the Government.