1

Optimal Dispatch of Wind FarmsFacing Market Prices

Gilles Bertrand, Anthony Papavasiliou

Abstract—This paper discusses different bidding strategies forwind farm producers. Up to now, wind power producers (WPP)are paid following feed-in tariffs in Belgium. This system willcome to an end soon and the producers will have to bid inthe day-ahead market. As wind owners can not forecast theirproduction perfectly, they will face imbalance costs or revenues.Our contribution is twofold, first we implement a model forimbalance price forecasting based on machine learning methods,then we propose a new model to obtain optimal bids if weconsider that our producer influences the imbalance market.These models are applied on a hypothetical wind farm site inBelgium.

Index Terms—Wind power, electricity market, imbalance mar-ket, optimization, machine learning.

I. INTRODUCTION

THE share of renewable energy supply has increasedsubstantially in recent years and will continue in the

future due to the ambitious goal plan [1] of the EuropeanCommission to cover 20% of European energy consumptionby renewable energy supply by 2020 and 27% by 2030.To this end, Belgium has used feed-in tariffs, but this willchange in the future. This change of remuneration mechanisminduces the need for new bidding strategies for Belgian windproducers.

A. Literature review

One of the first papers in which the problem of biddingstrategies for wind farms is considered is [2]. A Markov chainis used to model the transition between the production outputsat two consecutive periods, afterwards a policy is implementedand compared with several classical strategies for wind fore-casting (e.g. persistence, persistence with scaling factor) andthe revenue is computed with respect to different values ofimbalance prices. In subsequent literature, bidding strategiesare proposed for this problem in different case studies, in-cluding the Nordic power market [3] and the Dutch market[4]. These models consist of three parts: initially a model togenerate production scenarios, then a model which producesimbalance price scenarios, and finally an optimization modelthat uses the previous scenarios to obtain an optimal bid. In[3], the authors use a wind forecast from a meteorologicalmodel and employ an autoregressive moving average (ARMA)model to generate error forecast scenarios that they add tothe meteorological forecast. In [4], the authors model directly

G. Bertrand and A. Papavasiliou are with the Center for Operations Re-search and Econometrics (CORE), Université catholique de Louvain, Belgium.e-mail: gilles.bertrand@uclouvain.be, anthony.papavasiliou@uclouvain.be

power output by using a seasonal autoregressive integratedmoving average (SARIMA) model. In terms of imbalanceprice scenarios modelling, the authors in [3] use the pricesfrom the days before and after the tested day as scenarios.The authors also propose an alternative model using ARMAin [5]. In [4], the authors use a SARIMA model. Finally, in[3] and [4] a mixed integer program (MIP), with the powerand price scenarios as input, is solved in order to obtain theoptimal bid.

B. Paper organization

In order to analyse optimal bidding strategies, we use theapproach shown in Fig. 1. The methodology includes gen-erating production scenarios, price scenarios and using thesescenarios to obtain the optimal bids by solving a stochasticprogram. In section 2, we describe briefly the Belgian market.Section 3 explains the model used to generate power scenarios.In section 4, we present different methods for forecastingimbalance prices. Section 5 focuses on a price-taking modelfor obtaining the bids and on a simplification of this for theBelgian market. Section 6 presents a model that takes intoaccount the influence of the producer on the imbalance market.In section 7 we present our results on a test case.

Fig. 1. Model decomposition.

II. OVERVIEW OF THE BELGIAN MARKET

In Belgium, there are two markets in which a wind farmproducer can participate:• In the day-ahead market, producers need to submit their

bids before 12AM the day before.• In the intraday market, which is not modelled in this

paper due to its lack of liquidity.

2

After these markets, if the producer does not produce itscleared quantity, it will have to pay the difference at the im-balance price. The imbalance settlement scheme is explainedin table I, inspired from [6].

The price that will be charged to the producer depends ontwo properties, the state position of the system either long(too much production) or short (not enough production) andthe state of the producer, either over or underproduction. Letus first define the notation used in table I, afterwards we willpresent an example to illustrate how to read this table:• MIP : Highest price of activated bid to increase produc-

tion.• MDP : Lowest price of activated bid to decrease produc-

tion.• α : Additional incentive applied on top of the regulation

costs in case of major system imbalances. It is equal tozero if the system imbalance is lower than 140MW.

Let us present an example in order to illustrate how to usethe table. We consider the case in which the market is shortin production and the producer produces less than its dayahead cleared quantity. As the market requires more power, itactivates upward regulation and since the producer producesless than its promised quantity, it needs to buy the differencein megawatts at MIP + α. It is worth noting that the produceralways pays more for being in underproduction than what itis paid for being in overproduction.

TABLE IIMBALANCE SETTLEMENT RULES.

Net downward regulation Net upward regulationOverproduction MDP − α MIP

Underproduction MDP MIP + α

III. WIND SPEED MODEL

We have at our disposal a day-ahead forecast y and therealization of wind speed y. Moreover, we have the powercurve (curve that maps wind speed to power) of the windfarm. As the wind forecast is not perfect, we will generateerror scenarios to model the different possible outcomes. Letus explain the procedure:• First we obtain the error time series: e = y − y.• Then we suppress the seasonality: e−µs

σswith s a given

period (a month and an hour).• After that, we fit an ARMA model. The general formu-

lation of an ARMA(p,q) is

Xt =

p∑i=1

αi xt−i +q∑j=1

θ jεt− j + εt

It makes sense to use an autoregressive process to gen-erate the error scenarios because we expect a certaincontinuity in the error process. Indeed if we overestimatethe wind speed at a quarter, it is likely the case that wewill also overestimate it for the next quarters.The problem is that, as shown in Fig. 2, we have discon-tinuity in the series from the last quarter of a given day tothe first quarter of the following day. This figure shows

the wind speed forecast for two consecutive days and themoment at which these forecasts are made. The forecastof the first day is made at hour 11 and the forecast ofthe second day is made at hour 35. In a given day, weknow that the forecast error of a quarter will likely bepropagated to the following quarters. On the contrary, itis not the case between the last quarter of a day and thefirst quarter of the following day because for the secondday, the producer has received more information whichcan be used for forecasting. For this reason, in order tomodel the first quarter of the day, we will draw randompoints from past realizations (at the first quarter, due tothe discontinuity, we do not have any starting point foran autoregressive model). To forecast the second quarterwe only have access to one lag of the process (Xt−1)and zero lag from the error. Indeed, we do not know thevalue of two quarters ago. Therefore, we can only usethe best possible AR(1). For the third period, we can usethe best ARMA(1,2). This is represented in Fig. 3 for thecase in which the best possible model is an ARMA(3,3).After determining the order of the ARMA model, we canlearn the parameters of the model by using maximumlikelihood estimation.

• Finally, we convert wind speed to production by usingthe power curve of the wind farm.

0 10 20 30 40 50 60 70

0

5

10

15

20

25

30

Time [hour]

Win

d s

peed [m

/s]

First day forecast

Second day forecast

First day forecast time

Second day forecast time

Fig. 2. Wind forecast of two days and the time at which there are made.

Fig. 3. Available data for the producer for each quarter and the best associatedmodel.

3

IV. IMBALANCE PRICE FORECASTING

In this section, we will describe different methods we testfor imbalance price forecasting. The aim of these methodsis to generate a point forecast1. We will denote this forecastas ¯pimb. Initially, we introduce three base cases using onlypast information of the series, afterwards we propose somerelevant features for forecasting imbalance prices, next wepropose a procedure for selecting the useful features. Finally,we propose two models for forecasting imbalance prices withthese exogenous variables.

A. Base cases

The aim of these base cases is to have a reference in order tocheck if we get improvements by using exogenous variables.• The value of the same quarter the day before (two days

before if the quarter is after 12 AM).

pimb,t ,q =

pimb,t−1,q ∀q ≤ 44pimb,t−2,q ∀q > 44

• The mean value of the day before.

pimb,t ,q =1

96

44∑l=1

pimb,t−1,l +52∑l=1

pimb,t−2,97−l

• The mean value of the same quarter in the last 50 days.

pimb,t ,q =

150

∑50k=1 pimb,t−k ,q ∀q ≤ 44

150

∑50k=1 pimb,t−k−1,q ∀q > 44

B. Exogenous data

We now proceed with describing two alternatives to the basecases that rely on explanatory factors: (i) a linear model, (ii)radial basis function networks. To build our model we requireexogenous variables. We consider the following explanatoryfactors for these models:• r1, the index of the quarter in the year of the price we

want to forecast (a number between 1 and 35040).• r2, the index of the quarter in the day of the price we

want to forecast (a number between 1 and 96).• r3, the available generation forecast for the considered

quarter.• r4, the load forecast for the considered quarter.• r5, the solar forecast expectation for the considered

quarter.• r6, the wind forecast expectation for the considered

quarter.• r7, the wind forecast 10th quantile for the considered

quarter.• r8, the wind forecast 90th quantile for the considered

quarter.• r9, the day-ahead price for the hour in which the quarter

is.• r10, the last available gas price at Zeebrugge hub.• r11, the imbalance price mean of the the last day.

1The reason for which we do not generate scenarios will be explained inthe optimization model part.

C. Feature selection

From this initial set of features, we will try to find theminimal subset of features that maximizes mutual information.The mutual information is defined as:

I ( ¯pimb , xi ) = H( ¯pimb) − H( ¯pimb | xi )

with• xi ∈ P({r1 , r2 , r3 , r4 , r5 , r6 , r7 , r8 , r9 , r10 , r11}).2

The mutual information could be understood as the differencebetween the entropy of ¯pimb and the entropy of ¯pimb when xiis known. Consequently, mutual information is always positiveand is equal to 0 if the two variables are independent.In our case, for every subset of features, we compute themutual information and we take the subset of the smallestsize that gives the maximum mutual information.

Regression model

Having selected relevant features, our goal now is to fita parametric function fα that links the input x (which isthe optimal set of features obtained by the feature selectionprocess) and the output ¯pimb by the relation ¯pimb = fα(x).These parameters are estimated from historical data using thefollowing two alternative approaches.

1) Linear model: The regression function is defined by:

fw = wT x + w0

To compute the weights w and w0, we minimize the sum ofthe quadratic error for the N historical data.

minw ,w0

N∑k=1

‖ pimb,k − wT xk − w0‖22

2) Radial basis function network: This method is nonlinear,the regression function is defined by:

fc ,σ,w (x) =n∑i=1

wiφ(x , ci , σi )

The idea of this method is to take a weighted sum of basisfunction φ(x , ci , σi ). In our case, we use Gaussian function asbasis function:

φ(x , ci , σi ) = exp

− || x − ci | |22

2σ2i

Each Gaussian function is centred at a centroid ci and hasa standard deviation σi to represent its influence range. Formore details about RBFN, the reader may refer to [7].

V. PRICE TAKER MODEL

As we consider a wind farm, we do not have any cou-pling constraints between different quarters. Therefore we cansolve the problem independently for each quarter. The aimis to obtain a function f (pDA) that will indicate for a givenday-ahead price which quantity should be bid. We use thefollowing notation.• qDA: the bid quantity in the day-ahead market.

2P() represents the power set.

4

• qimb ,p ,i : the excess production in real time.• qimb ,n ,i : the missing production in real time.• pDA: the day-ahead price.• pimb ,p , j : the price paid for 1MW of excess production

in scenario j.• pimb ,n , j : the price paid by the wind producer for a

missing megawatt in scenario j.• Prodi : the wind farm production in scenario i.• ri : the probability of wind power production scenario i.• s j : the probability of imbalance price scenario j.• I : the set of wind scenarios.• J : the set of imbalance price scenarios.• ai : a binary variable that ensures that we cannot be in

under and in overproduction at the same time.• M: a sufficiently large constant.• PWF : the wind farm maximal production.The optimization problem that wind power producers are

called to solve in the day-ahead frame can be formulated asfollows:

maxqDA ,qimb,p ,i ,qimb,n ,i

[pDAqDA +∑i∈I

riqimb,p ,i∑j∈J

s j pimb,p , j

−∑i∈I

riqimb,n ,i∑j∈J

s j pimb,n , j ] (1)

st qDA + qimb,p ,i − qimb,n ,i = Prodi ∀i ∈ I

qDA ≥ 0qDA ≤ PWF

qimb,p ,i ≥ 0 ∀i ∈ I

qimb,n ,i ≥ 0 ∀i ∈ I

qimb,p ,i ≤ Mai ∀i ∈ I (2)qimb,n ,i ≤ M(1 − ai ) ∀i ∈ I (3)

ai ∈ {0, 1} ∀i ∈ I (4)

This formulation is inspired by [3] but there is a possiblesimplification for the Belgian market because qimb,p ≤ qimb,n .Indeed, as explained in table I, qimb,n = qimb,p +α with α ≥ 0.This property ensures that it is not possible to benefit by beingat the same time in over and in underproduction. It means thatby dropping constraints (2), (3) and (4), we will not influencethe optimal solution.

In fact there exists an analytical solution to this problem. Toprove this, we will use the fact that the problem is concave.By analyzing the problem, we see that when qDA is fixed, theother variables are also fixed. We can study 3 cases:

• The maximum is at qDA = 0. It is the case if the derivativeat 0 is negative. This derivative is pDA − pimb,p . It meansthat if the day-ahead price is lower than the positiveimbalance prices, the producer prefers to bid 0 and bepaid at the imbalance price.

• The maximum is at qDA = PWF. It is the case if thederivative at PWF is positive. The derivative at PWF ispDA − pimb,n . It means that if the day ahead price ishigher than the negative imbalance prices, the producerprefers to bid its maximum capacity. By doing so, for theunserved quantity, he will be paid at the day-ahead priceand will have to pay it back at the imbalance price whichis lower.

• There is a maximum in the interval 0 < qDA < PWF. Weobtain (the proof is proposed in Appendix A.)

qDA = H−1 (κ) (5)

with κ =pimb,p − pDA

pimb,p − pimb,n

with H() the cumulative distribution of Prod.The results justify that we do not generate imbalance pricescenarios. Indeed, in a risk neutral case, the results of themodel only depend of the mean of the imbalance pricescenarios.It is also interesting to notice that we are in the first two casesif pDA is not in the interval [pimb,p , pimb,n], which is verylikely (99.92% of the quarter from 2013 to 2015). Moreover,for these two first cases, the wind producer bid is independentwith respect to the production scenarios.

VI. NON PRICE TAKER IN IMBALANCE

In this section, we develop a model which accounts for theproducer influence on the imbalance price.

A. Procedure

To model the dependence between the output of wind powerproducers and the imbalance price, we consider that the pro-ducer influences the imbalance price linearly as a function ofhis imbalance quantity. Mathematically, the optimum biddingproblem can be described by the following model:

maxqDA ,qimb,p ,i ,qimb,n ,i

[pDAqDA

+∑i∈I

riqimb,p ,i

∑j∈J

s j [pimb,p , j + βqimb,p ,i ]

−∑i∈I

riqimb,n ,i

∑j∈J

s j [pimb,n , j − βqimb ,n ,i ]] (6)

st qDA + qimb,p ,i − qimb,n ,i = Prodi ∀i ∈ I

qDA ≥ 0qDA ≤ PWF

qimb,p ,i ≥ 0 ∀i ∈ I

qimb,n ,i ≥ 0 ∀i ∈ I

This model requires the estimation of the β coefficient.For this purpose, we represent the imbalance price againstthe system imbalance. The result is shown in figure 4. Next,we estimate the coefficient of the linear regression and obtaina value of −0.2429. This can be interpreted as follows: ifwe increase voluntarily our output by 1MW , the imbalanceprice will decrease by 0.2429 euro

MWh. Notice that the producer’s

influence is always unfavourable. Indeed, if it is in overpro-duction, the price decreases and if it is in underproduction weincrease it. Consequently, this model could only reduce thegain of the WPP with respect to the case in which the WPPis assumed not to influence the imbalance price. However, themodel represents better the revenue the producer will get fromthe market.

5

−1500 −1000 −500 0 500 1000−400

−200

0

200

400

600

800

1000

System Imbalance (MW)

Imb

ala

nce

price

(E

uro

/MW

h)

Data

Linear Fit

Fig. 4. Positive imbalance price versus System Imbalance.

B. Analytical solution

We make the following approximations in order to obtainan analytical solution to this problem:• pimb,p , j = pimb,n , j• The WPP always delivers entire available quantity.

With these assumptions we obtain the following optimal day-ahead bid:

qDA = ¯Prod +pimb − pDA

2β(7)

The proof is provided in Appendix B. The result can beinterpreted as follows: The first term is the expected productionof the wind farm. The second term could be decomposed asfollows:• The numerator is the difference between the imbalance

price mean and the day-ahead price. It is clear that thisdifference is the expected reward when the WPP decidesto move away from the expected value.

• The denominator represents the influence of the WPP onthe market. The greater the influence on the market, thecloser to its expected production the WPP has to bid.This is logic because as we explained earlier, the WPPcan only influence the market in an unfavourable way.

It is possible that the optimal solution is not in the interval[0, PWF]. In that case, we project the solution on this interval.

VII. CASE STUDY

The aim of this section is to:1) Assess the quality of the different price forecasting

methods.2) Compare the revenue between the two optimization

models.For this purpose, we use three years of data from 2013 to

2015 of the imbalance prices and the relative features. Thesedata are available at the site of the Belgian transmission systemoperator Elia3. Table II presents results for each model. Webegin by describing the different columns:

3Available at http://www.elia.be/en/grid-data/data-download

• The second column contains the root mean square error(RMSE) of the imbalance model. It seems that it doesnot explain completely the results for the revenue shownin the last column (some methods have better RMSE butworse revenue).

• The third column contains the percentage of right clas-sification of the imbalance price model. We count aright classification if either both the forecast and therealization of imbalance are lower or higher than theday-ahead price. Consequently, the percentage of rightclassifications is clearly related to the revenue. Howeverit does not explain completely the results. For instance,the method using the value of the same quarter the daybefore (line 1) has one of the best percentages but oneof the worst final revenues.

• The fourth column contains the average absolute differ-ence between the day-ahead price and the imbalance pricewhen the classification is correct. In fact it represents theaverage surplus obtained by deviating from the expecta-tion of 1MW if the classification is correct.

• The fifth column presents the average absolute differencebetween the day-ahead price and the imbalance pricewhen the classification is incorrect. It represents theaverage surplus lost by deviating from the expectationof 1MW if classification is incorrect.

• The sixth column is the expected revenue for a deviationof 1 MW from the expected value. It is the expected sumof the two last columns with the probability of column2. This criterion predicts almost perfectly the ranking ofthe method. Therefore it provides a quality criterion ofthe imbalance model without solving any optimizationproblem.

• The last column is the revenue on the three year period.From the last column of this table, we can conclude that us-

ing the optimization model with RBFN improves the revenueby:• 41% with respect to the case in which we bid expected

wind power production. It proves that developing theoptimization model is useful.

• 5.4% with respect to the best base case (mean value ofthe last 50 days).

• 5.7% with respect to the linear model.

TABLE IIRESULTS FOR DIFFERENT MODELS.

Method/param RMSE True % Mean win Mean loss Crit RevenuePrevious quarter 60.82 0.589 33.34 28.07 8.1 4.0765e+06

Mean previous day 48.03 0.567 33.98 27.49 7.36 4.0446e+06Mean 50 quarter 45.61 0.595 34.19 26.74 9.51 4.2931e+06

Linear 44.98 0.588 33.85 27.36 8.63 4.2805e+06RBFN 51.78 0.602 34.39 26.31 10.23 4.5263e+06

Expectation 3.2089e+06

We now compare the revenue for models of sections V andVI. For this purpose, we use the data available at the site ofthe Belgian transmission system operator Elia4. The data givethe imbalance prices for different value of system imbalance.Notice that between two points of this table, we consider a

4Available at http://www.elia.be/en/grid-data/data-download

6

linear relation. We compute the revenue for three differentsizes of wind farms. We do this evaluation for every quarterbetween October 2013 (beginning of the data) and December2015 for three sizes of wind farms (the 60 and 120 are justobtained by multiplying both forecast and realization of the 12MW). The results are presented in table III. For a small windfarm, the hypothesis that the producer is a price-taker is valid,therefore taking into account that we influence the marketis not necessary. Nevertheless, for the wind farm of highercapacity, if the producer does not account for his influenceon the market, he will face huge losses. Indeed, for the 120MW case, the producer earns less than 20% of its income ifit takes into account that it influences the market. Finally, it isclear that the case in which the WPP accounts for its influenceperforms at least as well as the other methods and in somecases far better.

TABLE IIIREVENUE FOR DIFFERENT METHODS AND INSTALLED CAPACITY (IN

MILLIONS OF EUROS).

Model/Installed capacity 12 60 120Initial 3.06 10.26 3.90

Influence on the price 3.07 11.99 22.57Expected value 2.25 11.15 22.11

VIII. CONCLUSION

In this paper, we develop a model for optimally biddingwind power under imbalance price and wind power outputuncertainty. For this purpose, we develop a way to generatewind speed error scenarios by dealing with the discontinuitybetween days in the error series. We also test the interest of dif-ferent features for imbalance price forecasting and implementdifferent models to use these features. A model that accountsfor the influence of the wind power producer on the imbalanceprice has also been developed.

The results of our case study demonstrate that the profitof the producer can be increased by introducing exogenousvariables in the imbalance price forecasting process. We alsodemonstrate that the influence of large wind farms on theimbalance price may need to be accounted for in day-aheadbidding.

APPENDIX A

In this section, we prove equation 5. We start by computingthe derivative of the objective function of model (1). Thisreasoning is inspired from [8]:

F′

(qDA) = pDA − pimb,p( ∑qDA<Prodi

ri)− pimb,n

( ∑qDA≥Prodi

ri)

= pDA − pimb,p + (pimb,p − pimb,n)( ∑qDA≥Prodi

ri)

= pDA − pimb,p + (pimb,p − pimb,n)Pr(qDA ≥ Prod)In what follow, we denote as Prod: the randomvariable representing the production distribution.= pDA − pimb,p + (pimb,p − pimb,n)H(qDA)

where H() denotes the cumulative distribution of Prod. Weequate this derivative to 0:

0 = pDA − pimb,p + (pimb,p − pimb,n)H(qDA)

⇔ qDA = H−1 (κ)

with κ =pimb,p − pDA

pimb,p − pimb,n

APPENDIX BIn this section, we prove equation (7). We start by comput-

ing the derivative of the objective function of model (6).

f (qDA) = pDAqDA +∑i∈ I

ri (Prodi − qDA)∑j∈J

s j [pimb, j + β(Prodi − qDA)]

= pDAqDA +∑i∈ I

ri (Prodi − qDA)[pimb + β(Prodi − qDA)]

= pDAqDA + pimb( ¯Prod − qDA) +∑i∈ I

ri (Prodi − qDA)β(Prodi − qDA)

= pDAqDA + pimb( ¯Prod − qDA) + β∑i∈ I

ri (Prod2i − 2qDAProdi + q2

DA)

= pDAqDA + pimb( ¯Prod − qDA) + β(E[Prod2i ] − 2qDA ¯Prod + q2

DA)

Let us take the derivative of f with respect to qDA:

f ′(qDA) = pDA − pimb − 2β ¯Prod + 2βqDA

We equate the derivative to 0:

0 = pDA − pimb − 2β ¯Prod + 2βqDA

⇔ qDA = ¯Prod +pimb − pDA

2β

ACKNOWLEDGMENT

The authors would like to thank Dr. Sebastien Chaumontand Dr. Olivier Daxhelet for their comments and thoughtfuldiscussions during the development of this work. This researchwas funded by the ENGIE Chair on Energy Economics andEnergy Risk Management and by the FSR-FNRS through aFRIA grant.

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