A Robust Optimization Approach using CVaR for UnitCommitment in a Market with Probabilistic Offers

W. A. BukhshInstitute for Energy and Environment,

Department of Electronic and Electrical Engineering,University of Strathclyde, Glasgow, United Kingdom.

waqquas.bukhsh@strath.ac.uk

A. Papakonstantinou, P. PinsonCentre for Electric Power and Energy,Department of Electrical Engineering,

Technical University of Denmark, Kgs. Lyngby, Denmark{athpapa,ppin}@elektro.dtu.dk

Abstract—The large scale integration of renewable energysources (RES) challenges power system planners and opera-tors alike as it can potentially introduce the need for costlyinvestments in infrastructure. Furthermore, traditional marketclearing mechanisms are no longer optimal due to the stochasticnature of RES. In this paper we propose a risk-aware marketclearing strategy for a network with significant shares of RES.We propose an electricity market that embeds the uncertaintybrought by wind power and other stochastic renewable sourcesby accepting probabilistic offers and use conditional value-at-risk (CVaR) to evaluate the risk of high re-dispatching cost dueto the mis-estimation of renewable energy. The proposed modelis simulated on a 39-bus network, whereby it is shown thatsignificant reductions can be achieved by properly managing therisks of mis-estimation of stochastic generation.

Index Terms—conditional value-at-risk; market clearing; op-timal power flow; risk analysis.

NOMENCLATURESets

B Buses, indexed by b.L Lines (edges), indexed by l.G Generators, indexed by g.W Wind generators, indexed by w.D Demands, indexed by d.Bl Buses connected by line l.Lb Lines connected to bus b.Gb Generators located at bus b.Db Loads located at bus b.Sb Scenarios, indexed by s.

Parameters

bl Susceptance of line l.τl Off-nominal tap ratio of line l.PG−g , PG+

g Min., max. real power outputs of conventionalgenerator g.

PDd Real power demand of load d.fg(pG

g ) Cost function for generator g.γ Prescribed probability level.CPP

w Purchase price at node w.CS

g Cost of committing generator at bus g.R±g Min./max. regulation of generator g.

PW+w Max real power generation from wind farm w.

∆PWw,s Change in wind power generation forecast.

η Threshold level for the loss function.ω Weighting on the risk measured as defined by

CVaR.

Variables

pGg Real power output of generator g.pWw,s Real power output of wind generator w.θb,s Voltage phase angle at bus b.pLl,s Real power injection at bus b into line l (which

connects buses b and b′).∆PG

g,s Regulation of conventional generator g.∆PG±

g,s Upward/downward regulation of conventional gen-erator g.

ug Unit commitment variable for generator g.

I. INTRODUCTION

Modern power systems are in the midst of a comprehensivechange, primarily driven by the liberalization of electricitymarkets and an increased focus on renewable energy sources(RES). Over the last decade, there has been a substantialincrease in the installed capacities of the RES challengingpractices in both transmission system planning and operation[1] as well as in the electricity markets given that theywere designed under the domination of dispatchable and fullypredictable sources of energy [2]. Consequently, the supportmechanisms that were put in electricity markets in orderto safeguard stochastic producers from the price volatilitybrought by the intermittent and uncertain nature of RES,tend to become inefficient as shares of RES increase andimbalance costs are transferred to the consumers [3], [4]. Suchdevelopments highlight the need for electricity markets thatembed the very nature of RES in the market mechanism itselffirst by replacing deterministic offers (e.g. point forecasts)with probabilistic estimates (e.g. quantiles), and second, byincluding risk-aware dispatching mechanisms that are robustenough to accommodate the underlying uncertainties inherentto modern power systems.

In this context, there has been significant academic interestin addressing several of the challenges brought by RES. Most

of the day-ahead clearing optimization models are posed astwo-stage stochastic programming problems [5], where oftenthe first stage of the problem is to schedule conventionalgenerators and second stage realizes the uncertainties fromRES. Authors in [6] consider the deployment of reservecapacity in their stochastic programming model and proposean energy-only settlement where the capacity is ‘converted’ toenergy through the market mechanism. One limitation of suchapproach is that the participants can speculate and therefore in-fluence the market clearing mechanism. This issue is addressedby [7] where the authors propose a single auction that clearsthe market and arranges the financial settlement. However, thestochastic clearing requires the flexible generators to acceptlosses for some wind power production realization.

A stochastic optimal power flow (OPF) model is proposedin [8] that captures risks of demand-generation imbalancescaused by wind power using the risk measure conditionalvalue-at-risk (CVaR). The model is based on DC power flow,while unit commitment (UC) is not considered as part of theproblem. This is addressed in [9] presenting a multi-stagestochastic OPF model that gives optimal policies regarding thescheduling of controllable devices in the power system. Bothapproaches in [8], [9] contribute on the overall discussion,however they face some limitations. Specifically, in [8] riskis only associated to a part of operational uncertainty, whilethere is no consideration for the impact of real-time expectedimbalance in the day-ahead schedule. In [9] the probabilisticchance constraints introduce significant computational issuesand challenges in determining the prices corresponding to thedual variable of the constraints. Finally, both papers offer arelatively restricted view of uncertainty as they consider con-ventional market clearing mechanisms using only deterministicoffers.

In this paper we address the aforementioned limitationsby proposing a model that produces a robust optimal policyconsisting of day-ahead generation schedules that minimizethe total cost of generation while. Furthermore, the proposedmodel manages the risk of imbalances due to stochasticgeneration, with stochastic producers’ bids modeled as prob-abilistic estimates following [10]. Specifically, we contributeto the state of the art by proposing a two-stage stochasticprogramming formulation that takes into consideration theeffect of unit commitment decisions and ramp rate restrictionson the optimal policy, while using a CVaR risk measure in amarket where stochastic producers report probabilistic offers.The proposed formulation is tractable as it can account fora large number of scenarios for the future power systemoperation. Finally, we give insights into questions about riskmeasures and modeling details including network constraintsand technical restriction and test the proposed model on a39 bus network with real world data from RES. Throughnumerical simulations we demonstrate the impact of inaccurateforecasts of low predictive value, while we show that bycarefully managing the overall risk of system, a more robustpolicy can be obtained.

The rest of the paper is organized as follows: In Section

II we define uncertainty and risk management in the contextof electricity markets, while Section III introduces the generalformulation, which is described in detail in Section IV. InSection V we numerically evaluate the proposed model and inSection VI we conclude and give future research directions.

II. UNCERTAINTY AND RISK ASSESSMENT IN ELECTRICITYMARKETS

In this section we provide a general framework regardinguncertainty and risk assessment in electricity markets. In Sec-tion II-A we model uncertainty in stochastic production, whilein Section II-B we describe the bidding process in a marketdesigned to utilize probabilistic offers. Finally, in SectionII-C we formally define risk measurement in consistence withliterature [8]. Based on these definitions we can expand [8] ina two-stage stochastic programming formulation that can alsoaccommodate probabilistic offers.

A. Uncertain production

Let a stochastic producer w such as a wind farm face anupper limit PW+

w in its output, defined by the specific technicalspecifications of the deployed turbines and let the real-timegeneration pWw,s,t be equal to ywPW+

w , where yw ∈ [0, 1] is arealization of the random variable Y which models the windfarm’s stochastic output. The variable Y follows a distributionG defined by a set of parameters θ s.t. Y ∼ G(y; θ), where θcan be equal to mean and variance depending on the definitionof the used distribution.

B. Modeling probabilistic offers from stochastic producers

We consider an electricity market that accepts probabilisticoffers from the stochastic producers, instead of deterministicoffers e.g. point forecasts. Based on the stochastic frameworkintroduced in [10], producers are asked to submit their pre-dictive distributions; in essence estimates of distribution G.Stochastic producers can report a CDF, a set of quantiles, orin case of a parametric distributions, they can report the param-eters of a known distribution. Given the introduced notation,let θ̂w be the set of reported parameters during the biddingstage of a day-ahead market. The parameter set θ̂w definesdistribution F (y; θ̂), like θ does for G. It should be notedthat G may not necessarily be equal to F , as F representsthe predictive distribution and G the actual distribution of thestochastic output. In terms of the parameter sets, let θ̂w = εθw,with ε representing the imperfect nature of an estimate.

Using predictive distributions as offers in the day-aheadmarket has significant implications in the context of a two-stage stochastic optimization model. Now, the finite set ofscenarios that models stochastic production is sampled fromthe predictive distribution F as it is the only distributionavailable to the market and system operators prior to theactual production. Given this intrinsic link between the day-ahead schedule and expected balancing stage it becomes clearthat the reported offers in the day-ahead market can heavilyinfluence the optimal policy.

C. Risk measures

Due to the use of probabilistic offers, it is important fora market operator to derive a summary statistic from thepredictive distribution and clear the day-ahead market basedon it. In this paper we assume that this statistic is the meanwhich the market operator can either extract from the reportedCDF, or the empirical CDF derived from the set of quantiles,given that µ =

∫∞0

(1−F (t))dt, or just use the reported valuein case of a parametric predictive distributions. The selectionof the mean is a necessary link with the existing electricitymarket setups where it is a common practice for stochasticproducers to use point forecasts that correspond to conditionalexpectation estimates [11].

Let PWw = µwP

W+w be the summary statistic used to clear

the day-ahead market and let ∆PWw,s be the change in the

generation availability corresponding to the scenario s for thegenerator w in the time period t, respectively. The wind poweroutput for the generator w is modeled as follows:

0 ≤ pWw,s ≤ PW

w + ∆PWw,s (1)

Now, integration of renewable energy sources, especiallywind power, requires careful assessment of risk when itcomes to commitment of conventional power plants. Normallytransmission system operators (TSOs) err towards risk aversepolicy. They tend to commit more conventional generationthan required in order to account for potential power imbal-ances [12]. However such a risk averse policy is not optimalas it may result in large amounts of renewable energy beingcurtailed when it comes to clearing the market in the real-time.Therefore there is a need for a risk-aware policy that takes intoaccount the uncertainties from the stochastic producers andoptimally manages the risk and rewards in a power system.Risk measures like value-at-risk (Var) and CVaR are two riskassessment measures that are used largely in the financialindustry to manage assets. It is commonly known that CVaRis a superior measure for risk when compared to VaR [13].Given a cost function L(x, ξ) : X ×Ξ→ R, the γ −CVaR is

φγ(x) =1

1− γ

∫L(x,ξ)≥ηγ(x)

L(x, ξ)p(ξ)dξ (2)

where ηγ(x) is γ − VaR.Calculation of γ − CVaR is equivalent to minimization of

the following function

Fγ(x, η) = η +1

1− γ

∫ξ∈Ξ

[L(x, ξ)− η]+p(ξ)dξ (3)

Let the cost function be a linear function describing the costof real power purchased at a spot market. The above equationcan then be approximated such that

F̂γ(pWw , η) = η +

1

1− γE

[∑w∈W

CPPw [PW

w − pWw,s]− η

]+

(4)

where CPPw is the purchase price of power at at bus w.

III. CONCEPTUAL FRAMEWORK

Our conceptual framework is as follows: a TSO has to de-termine the commitment of conventional power plants. Thereis a great level of uncertainty from generation from the windpower producers and this uncertainty will obviously effect thecommitment of conventional power plants. In this situationa transmission system needs to find a decision that optimalbalances the risks of over-committing expensive conventionalgeneration versus under-utilization of low cost and cleangeneration from wind power and other stochastic producers.

We model this problem as a two-stage stochastic program-ming problem. The TSO has full access to producers’ prob-abilistic offers, publicly available from the market operatorand can now measure the risk of extreme a-priori using aregularizer defined in terms of the CVaR (F̂γ(pW, η) in SectionII-C). Through this model we investigate if the mean of thescenarios is a good estimate to use in the day ahead marketand determine the appropriate risk level.

The general form of a two-stage stochastic program [5] is

min cTx+ EξQ(x, ξ) (5a)subject to

Ax = b, (5b)x ≥ 0, (5c)

where x is a vector of decision variables and Q(x, ξ) =min{qTy : Wy = h − Tx, y ≥ 0} is the optimal value ofthe second stage problem.

First stage decision variables determine which conventionalgenerator will come on-line for the day-ahead operation ofthe power system along with their operating points and thesedecisions are represented by a vector x. In the second stage ofthe problem, full information is received on realization of therandom process determined by the vector ξ, and corrective ac-tions are taken. Corrective actions for the day-ahead decisionsare rescheduling of conventional generators and curtailmentfrom the wind power producers. These corrective actions aremodeled as a recourse action: affecting the values of first stagevariables and hence influence the objective function of theproblem. In next section we give the mathematical formulationof the problem.

IV. PROBLEM FORMULATION

Consider a power network with the set of buses B. LetG be the set of conventional power plants and S be the setof scenarios from the wind power producers. The constraintsand objective function of our optimization problem are givenbelow.

A. Power flow

Let pGg be the real power generation from the conventional

generator g. The power balance equations are given as, ∀b ∈B, s ∈ S:∑g∈Gb

(pGg + ∆pG

g,s

)+∑w∈Wb

pWw,s =

∑d∈Db

PDd +

∑l∈Lb

pLl,s (6)

where pWw,s denotes the real power bid from the renewable

generator w, PDd denotes the real power demand d and pL

l,s isthe flow of real power in the line l in scenario s, respectively.The power flow equations are given as, ∀l ∈ L, ∀s ∈ S:

pLl,s = −bl

τl(θb,s − θb′,s) (7)

where b and b′ are the two ends of the line l. Voltage anglesat the two ends of the line l = (b, b′) are denoted by θb,sand θb′,s, respectively. We consider the DC model of powerflow [14]. In order to keep the formulation linear the modelignores the line losses and the reactive power in a network.The second stage recourse variables ∆pG

g,s in (6) are modeledin terms of the upward and the downward regulation variabless.t.

∆pGg,s = ∆pG+

g,s −∆pG-g,s (8a)

0 ≤ ∆pG+g,s ≤ R+

g (8b)

0 ≤ ∆pG-g,s ≤ R−g (8c)

where R+g,t, R

−g,t are the permissible upward and downward

regulation of the generator g in the time period t, respectively.

B. Unit commitment

Given the single-dimensional time framework we employin this research, the unit commitment constraints are straight-forward. Specifically, let ug , the unit commitment status ofgenerator g, be equal to 1 if the generator is on-line and is0 otherwise. Furthermore, generation from the conventionalgenerators is bounded by the inequality constraints

ugPG-g ≤ pG

g ≤ PG+g ug (9)

where PG-g , P

G+g are the lower and the upper bounds on the

generation output of the generator g, respectively.

C. Operating constraints

The line flow limits are given by the following set ofconstraints: ∀l ∈ L

− Pmaxl ≤ pL

l,s ≤ Pmaxl (10)

where Pmaxl is the real power capacity limit of the line l.

D. Objective function

Let λw,s be the probability of the scenario s for therenewable generator w. The objective is to minimize the costof generation from the conventional generators, and optimallyutilize the generation from the RES while initiating the de-mand response from the distribution system operators. Notethat we do not consider ramping cost of the generators betweenthe time intervals. Day-ahead cost is

CDAg (pG

g ) = f(pGg ) + CS

gug (11)

where f(pGg ) is cost of generation and CS

g is the cost of com-mitting generator g. Close to real-time operation the system

Fig. 1. Modified 39 bus system with 6 conventional generators and 3 windpower producers.

operator has an improved estimate of the actual generationfrom stochastic sources and the generator may need to getregulated in order to meet the demand. The cost of suchregulation is

CREGg (∆pG

g,s) =(CR+g ∆pG+

g,s + CR-g ∆pG-

g,s

)(12)

where CR+g is the up-regulation cost and CR-

g is the down-regulation cost of generator g respectively. The overall objec-tive function of the proposed optimization model is

z =∑g∈G

CDAg (pG

g ) + E(CREGg (∆pG

g,s))

+ ω∑w∈W

F̂β(pWw , η)

(13)

where ω is the weighting on the risk measure, equal toF̂β(pW, η), as defined by the CVaR in equation (4).

E. Overall formulation

The overall formulation of the problem is

min z(pGg , p

Ww,s,∆p

Gg,s

)(14a)

subject to(1, 4, 6− 10) (14b)

where(pGg , p

Ww,s,∆p

Gg,s

)are the decision variables. Depending

on the objective function f(pGg,t), the overall problem is linear

or quadratic (LP or QP). We use CPLEX 12.06 [15] calledfrom a PYOMO [16] model to solve the problem.

V. NUMERICAL RESULTS

In this section, we simulate the proposed model on a39-bus test network derived from the New England 39 bustest case in [17], and modified as shown in Fig. 2. In this

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Stochastic Parameter y

Pro

ba

bil

ity

40% under−estimating

60% over−estimating

Perfect estimate

Fig. 2. CDFs for imperfect and perfect estimation

context, three fossil fuel generators at buses 33, 34 and 36 arereplaced by wind generation with twice the capacity of originalgeneration and more realistic costs where derived from [18].With these modifications the conventional generation accountsfor the 63% of total capacity of the generation in the network.Furthermore, the cost of up-regulation is assumed to be 10%higher than the day-ahead cost of generation and cost of down-regulation is 9% less than the day-ahead cost of generation.The risk quantile, γ, is equal to 0.95 and 50 scenarios areconsidered unless otherwise stated.

In our simulations we assume zero cost of generation fromwind power [19]. Moreover wind power from the producer wcan be spilled continuously to zero at the price of CW

w . Windpower production uncertainty is modeled by a Beta distributionin consistence with the related literature (cf. [19], [20]), with-out this restricting our assumptions regarding uncertainty andrisk measurement in Section II and the theoretical frameworkin Section IV. The used Beta distribution is defined by meanand variance s.t. (µw, σ

2w) = (0.55, 0.05) with parameters αw

and βw defined as

α =(1− µ)µ2

σ2− µ, β =

(1− µ)a

µ(15)

In the proposed day-ahead market wind power producersreport parameters (µ̂w, σ̂

2w) and it is entirely possible that

both mean and variance are mis-estimated i.e. (µ̂w, σ̂2w) 6=

(µw, σ2w), with (µ̂w, σ̂

2w). We simplify our analysis by as-

suming that only the wind power producer located in bus 34may mis-estimate its parameters and by only considering themis-estimation of the mean. This translates to (µ̂w, σ̂

2w) =

(εµw, σ2w) with ε ∈ [0.6, 0.7, ...., 1.3, 1.4] being a parameter

which denotes the imperfect nature of the estimate. Valuesless than 1 represent under-estimation of the mean, and valuesabove 1 represent over-estimation of the mean. This simpli-fication allows us to evaluate the impact of mis-estimationon the overall day-ahead schedule and real-time dispatch, acritical point of assessment for any application of stochasticprogramming. In this context, in Figure 2 we demonstrate the

0 1 2 3 4

0.6

0.8

1u30 = 0

u30 = 1

ω: regularisation weight

Nor

mal

ised

expe

cted

optim

alba

lanc

ing

cost

s

Fig. 3. Optimal exp. balancing costs for varying the regularization weight ω.

differences between the perfect and imperfect estimates byplotting the cumulative distribution function of the Beta dis-tribution with the parameters used in the simulation, alongsidewith the distributions which correspond to over and underestimating of the mean of the actual Beta distribution. Asexpected, the whole shape of the distribution is affected by themis-estimation of the mean, given that as shown in equations(15) the mean influences both α and β.

Against this background, Fig. 3 shows the normalizedoptimal balancing costs while varying the weighting ω on theCVaR risk measure. When the weighting is zero, the modeldoes not take into account the costs about the mis-estimationof wind. We observe that as the weighting is increased thebalancing cost decreases. More conventional generation isscheduled in day ahead market to minimize the mis-estimationin real-time. Note that for 0 ≤ ω ≤ 0.4 generator at bus 30(most expensive generator) is not committed. But for valuesω > 0.4 generator at bus 30 is committed.

Following the preceding analysis, we study the compu-tational tractability of the proposed model by plotting theoptimal cost with respect to the number of scenarios. InFig. 4 the solid red line represents the conventional generationcost with the dashed blue line showing the expected optimalbalancing cost. We observe that estimated balancing cost iszero in the deterministic case i.e. when only one scenario isconsidered. The deterministic case corresponds to the modelwith perfect information about the future realizations. Wefurther observe that the expected balancing costs increase withthe increase in number of scenarios. However the day-aheadgeneration cost has a very steady behaviour after 20 scenarios.This indicates that adding extra scenarios after 20, does nothave a severe influence on the generation cost.

Finally, we investigate the impact of imperfect informationby analyzing the wind power producer at bus 34. Fig.5 showsthat both over and under estimation result in a significantincrease of the overall cost (i.e. balancing costs plus generationcosts). We observe that under-estimation costs are as highas 25% while over-estimation costs are as high 15%. Forunder-estimation, the commitment of expensive generators inmaintaining the demand with generation balance leading to in-creased generation costs, while for over-estimation, generators

0 20 40 60 80 1000

0.5

1

Number of scenarios

Nor

mal

ised

optim

alco

sts

Generation costTransaction cost

Fig. 4. Optimal cost corresponding to number of scenarios.

0.6 0.8 1 1.2 1.4

0.8

0.9

1

OverestimationUnderestimation

ε

Nor

mal

ised

cost

s

Fig. 5. Affect of misestimation of wind power generation on overall costs.

face additional costs in order to buy surplus from the real-timemarket, hence high balancing costs.

VI. CONCLUSION AND FUTURE WORK

This paper solves a two-stage stochastic unit commitmentproblem for a day-ahead market that can accept probabilis-tic offers based on a risk measure defined by the CVaR.We simulate the proposed model on a 39-bus system andshow that as the weighting on the CVaR increases, so doesthe scheduling of conventional generation in the day-aheadmarket. We demonstrate the proposed model’s favourablecomputational properties and show that re-dispatch cost canbe further reduced by carefully managing the risk of mis-estimation.

For future research we intend to extend the proposedmodel so it captures more of the challenging aspects inpower systems. As a starting point, the single-dimensionaltime framework will be extended so that the model can takeinto consideration inter-temporal constraints. In doing so, thisresearch can be extended to a multi-stage unit commitmentmodel that considers the minimum up and down times of thegenerators. Furthermore, we intend to consider several mis-estimating wind power producers and take into account thecorrelation among different wind power producers. Finally,we intend to challenge the use of the mean as the optimalstatistic for the clearing of the day-ahead market and examinethe effect of different statistics on the optimal solution.

ACKNOWLEDGEMENTS

W. A. Bukhsh is supported by EU FP7 project ‘GARPUR’under Grant Agreement No 608540. The other authors arepartly supported by the Danish Council for Strategic Research(DSF) through the project ‘5s-Future Electricity Markets’, No.12-132636/DSF.

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