Open access peer-reviewed chapter

Optimal Bidding in Wind Farm Management

Written By

Alain Bensoussan and Alexandre Brouste

Submitted: February 14th, 2019 Reviewed: September 19th, 2019 Published: March 25th, 2020

DOI: 10.5772/intechopen.89806

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Abstract

We study the problem of wind farm management, in which the manager commits himself to deliver energy in some future time. He reduces the consequences of uncertainty by using a storage facility (a battery, for instance). We consider a simplified model in discrete time, in which the commitment is for the next period. We solve an optimal control problem to define the optimal bidding decision. Application to a real dataset is done, and the optimal size of the battery (or the overnight costs) for the wind farm is determined. We then describe a continuous time version involving a delay between the time of decision and the implementation.

Keywords

  • optimal control
  • stochastic control
  • wind farm management
  • wind production forecast
  • storage

1. Introduction

A higher penetration level of the wind energy into electric power systems plays a part in the reduction of CO2 emissions. In the meantime, traditional operational management of power systems is transformed by taking into consideration this fluctuating and intermittent resource. Smart grids and storage systems have been developed to overcome these challenges.

For wind power plants, storage is a straightforward solution to reduce renewable variability. It can be used to store electricity in hours of high production and inject electricity in the grid later on. The performance of the operational management can be therefore improved by considering simple charge-discharge plans based on short-term forecasts of the renewable production [1]. For instance, optimal management of wind farms associated with hydropower pumped storage showing economic benefit and increasing the controllability have been studied in [2, 3, 4]. Other examples are the sizing of a distributed battery in order to provide frequency support for a grid-connected wind farm [5] and the optimal operation of a wind farm equipped with a storage unit [6, 7].

For the specific case of isolated systems, which is the aim of our paper, it is necessary to think about distributed energy storage as battery [8], ultra-capacitors [9], or flywheels [10]. In this setting, the question of economic viability in isolated islands without additional reserves arises. Here, the storage unit allows wind farms to respect the scheduled production.

The storage costs will represent a large part of the overnight capital costs and motivate the different researches on storage. Generally the sizing of the storage device is reduced to a minimization problem of the fixed and variable costs of the storage and its application (see [11, 12], for a complete analysis of the cash flow of the storage unit).

In this paper, we present a simplified model in discrete time, in which the commitment is for the next period. We solve an optimal control problem to define the optimal bidding decision. The mathematical setting of the problem is described in Section 2. The main result is detailed in Section 3. Application on a real dataset is described in Section 4. The continuous version of the problem is also described in Section 5. A conclusion ends the paper.

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2. Setting of the problem

2.1 General description

In our problem, the manager has to announce an energy production to be delivered to the next period. Considering the kth period, we may think that the announcement is made at the beginning of the period and the delivery at the end of the period. Of course the real delivery will be split along the k+1th period. This splitting will be omitted in this stylized model. It is convenient to consider the full delivery at the end of the kth period which is the beginning of the k+1th period. So, at the beginning of the kth period (day or hour), the manager commits himself to deliver vk units of energy (kWh or MWh). To simplify, we discard margins of tolerance. To decide, he knows the amount of energy stored in the battery, called yk. The second element concerns the windfarm. The energy produced by the windfarm is a stochastic process Zk. More precisely, Zk is the energy produced during the k1th period, which we consider to be available at the beginning of the kth period. So Zk and all previous values are known. However to fulfill his commitment, the manager will rely on Zk+1, the energy produced during the kth period, which we consider, with our convention, to occur at the end of the kth period, which is the beginning of the k+1th period. So it is not known by the manager, when he takes his decision. We model the process Zk as a Markov chain with transition probability density fkζz. A key issue concerns the choice of this density which is discussed in the application in Section 4. Precisely, although formally

ProbZk+1=ζZk=z=fkζzE1

In fact, Zk is obtained through the power law, operating on another Markov chain, the wind speed (see [13, 14] for examples of such Markov chains). We denote by Fkζz the CDF of the transition probability. We also set F¯kζz=1Fkζz.

In the language of stochastic control, the decision vk (control) is measurable with respect to Fk=σZ1Zk. The storage yk is also adapted to Fk. The evolution of yk is defined by the equation

yk+1=minMyk+Zk+1vk+E2

Indeed, the available energy at the end of the kth period is yk+Zk+1. If this quantity is smaller than vk, then the manager cannot fulfill his commitment; he delivers what he has, namely, yk+Zk+1; and the storage becomes empty. If the available energy yk+Zk+1 is larger than vk, then the manager delivers his commitment vk and tries to store the remainder yk+Zk+1vk. This is possible only when this quantity is smaller than M, which represents the maximum storage of the battery. If yk+Zk+1vk>M, then he charges up to M, and the quantity yk+Zk+1vkM is lost (given free to the grid). This results in formula (2). In this equation, we do not consider the constraint to keep a minimum reserve. We also are considering the battery as a reservoir of kWh, which we can reduce or increase as done in inventory of goods. Finally, we neglect the losses in the battery.

2.2 The payoff

We want now to write the payoff to be optimized. During the period k, if the manager delivers his commitment vk, he receives the normal income pvk. If he fails and delivers only yk+Zk+1<vk, there is a penalty. In the sequel, we have chosen the following penalty ϖyk+Zk+1vk, which is common in inventory theory. The parameter ϖ can be adjusted, for instance, to compare with the conditions on the spot market.

In our set up, the pair yk,Zk is a Markov chain. So we have a controlled Markov chain and the state is two-dimensional. We introduce a discount factor denoted by α, which is useful if we consider an infinite horizon. We can have α=1, otherwise. Initial conditions are given at time n and denoted by x,z. We call V=vn,vN the control, where N is the horizon. Finally, we want to maximize the functional

JnxzV=k=nNαknEpminvkyk+Zk+1ϖyk+Zk+1vkE3
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3. Dynamic programming

3.1 Bellman equation

The value function is defined by

Unxz=supVJnxzVE4

Writing

minvkyk+Zk+1=vkyk+Zk+1vk

we get also

JnxzV=k=nNαknEpvkp+ϖyk+Zk+1vkE5

We can then write Bellman equation

Unxz=supv>0{pv+E[p+ϖx+Zn+1v++αUn+1minMx+Zn+1v+Zn+1Zn=z]}E6

It is convenient to make the change of variables vx=y and obtain

Unxz=px+supx+y>0{py+E[p+ϖZn+1y++αUn+1minMZn+1y+Zn+1Zn=z]}E7

with final equation

UN+1x=0

We have x0M and z>0.

3.2 Main result

We state the following proposition:

Proposition 1. The solution of (7) is of the form

Unxz=px+KnzE8

Proof. For n=N, we have

UNxz=px+supx+y>0pyp+ϖEZN+1yZN=z

Consider the function

φN+1y=pyp+ϖEZN+1yZN=z

then, for y<0, we have φN+1y=py. It is monotone increasing, so the maximum cannot be reached at a point y<0. It follows that (8) is satisfied at n=N, with

KNz=supy>0pyp+ϖEZN+1yZN=zE9

We have, for y>0

φN+1y=pyp+ϖ0yyζfζz

and φN+1y is concave, since

φN+1y=pp+ϖFyzφ"N+1y=p+ϖfyz<0

and φN+10=p,φN+1+=ϖ. So the maximum is uniquely defined. Assume now (8) for n+1. We insert it in (7) to obtain

Unxz=px+supx+y>0{py+E[p+ϖZn+1y++αpminMZn+1y+Zn=z]}+αEKn+1Zn+1Zn=z

Consider now the function

φn+1yz=py+E[p+ϖZn+1y++αpminMZn+1y+Zn=z]

For y<0, it reduces to

φn+1yz=py+αpMF¯n+1y+M++0y+M+ζyfn+1ζz

and

φn+1yz=pαpFn+1y+M+pαpFn+1M>0

and thus cannot reach a maximum for y<0. Considering y>0, we have

φn+1yz=pyp+ϖ0yyζfn+1ζz++αpMF¯n+1y+M+yy+Mζyfn+1ζz

Again, this function is concave and

φn+1yz=pp1α+ϖFn+1yzαpFn+1y+Mz
φn+1yz=p1α+ϖfn+1yzαpfn+1y+Mz<0
φn+10z=pαpFn+1Mz>0
φn+1+z=ϖ

and the property (8) is proven with

Knz=α0+Kn+1ζfn+1ζz+maxy>0{pyp+ϖ0yyζfn+1ζz++αpMF¯n+1y+M+yy+Mζyfn+1ζz}E10

The proof is completed.

3.3 Optimal feedback

We define by Snz the point at which φn+1yz attains its maximum. It is positive and uniquely defined. It follows that the optimal feedback in Bellman equation (6) is v̂nxz=x+Snz and Snz is the unique solution of

pp1α+ϖFn+1SnzαpFn+1Sn+Mz=0pp+ϖFN+1SNz=0E11

The recursion (10) writes

Knz=α0+Kn+1ζfn+1ζz+p0SnzF¯n+1ζz+αpSnzSnz+MF¯n+1ζzϖ0SnzFn+1ζzKNz=p0SNzF¯N+1ζzϖ0SNzFN+1ζzE12

It is worth emphasizing that the function Knz increases with M, as can be expected. The feedback has an easy interpretation. The bidding is the level of inventory plus a fixed amount depending on the last value of energy produced by the turbine. It is interesting to note that the quantity Snz decreases with M. This is not so obvious. Clearly, the larger M, the better it is, as captured by the increase of Knz. We can understand why Snz decreases with M, as follows: When M is large, the risk of wasting energy by lack of storage is reduced, so it makes sense to focus on the other risk, to overbid and pay the penalty. Hence it makes sense to reduce the bid.

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4. Application

We describe in this section an application on a wind farm project financed by EREN on a French island with national tender process. First we set the energy price p=230 EUR/MWh (Official Journal from March 8, 2013) and the discount factor α=1. In this first application, N=48 which is the number of periods of 30 min in a day.

In the sequel, we have chosen the penalty ϖyk+Zk+1vk which is common in inventory theory. The parameter ϖ can be adjusted.

Some analysts would prefer the penalty p2yk+Zk+111vk>yk+Zk+1. This is rather strange, because it is fixed, whatever be the level of failure. If the failure is very small, the damage is not big, and still the penalty is high. Conversely, if the failure is big, the damage is big, and still the penalty does not change. Even more surprising, for a given level of commitment, the bigger the failure, the lower the penalty.

The production over a period of 30 min is presented on Figure 1. It is worth mentioning that we used directly the data proposed from July 26, 2005 to March 9, 2008 captured by a measurement mast.

Figure 1.

Histogram of the production over a period of 30 min.

For this first application, stationary law is considered as Gaussian. Mean and variance of the model are similar to those of the empirical distribution in Figure 1. This model allows to construct closed-form cumulative distribution function Fk. One-step forecasting error is 24% of the mean and 11% of the nominal power.

But this process does not take into account the stylized facts of the production on a period of 30 min (positive values below nominal power limit, atom for zero production, intraday seasonality, etc.). Consequently, in the optimal control problem, we use the corresponding truncated Gaussian distribution (between 0 and 7 MWh).

Finally, the penalty is fixed (geometrically) to ϖ=34p.

With these assumptions, the payoff with respect to the size of the storage is given in Figure 2 for an empty storage x=0, and z is the average production as initial conditions. Some simple economic models penalizing the size of the battery with its costs would reveal an optimal size of the storage unit between 4 and 6 MWh.

Figure 2.

On the left, daily payoff Unxz in terms of the size of the storage M for α=0.9, p=230, and ϖ=34p. Here z=3 MWh and the initial storage is empty with x=0 MWh. On the right, part of the decision Snz (from the direct wind production) in terms of the size of the storage M.

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5. Continuous time version

In the last section, we present a continuous version of the aforementioned problem. This new problem exhibits interesting questions in control theory when there is a delay between the decision and the application of the decision.

5.1 A continuous time model

We model the wind speed by a diffusion

dz=gzdt+σzdwz0=zE13

where wt is a standard one-dimensional Wiener process, built on a probability space Ω,A,P, and we denote by Ft the filtration generated by the Wiener process. This is the unique source of uncertainty in the model. We suppose that the model has a positive solution.

The energy produced per unit of time at time t is φzt where the function φ is the power law. So the energy produced on an interval of time 0t is 0tφzsds. We assume that the manager bids for a delivery program with a fixed delay h. In other words, if he decides a level vt per unit of time at time t, the delivery will be at t+h. On the interval of time 0t the delivery is htvshds, provided t>h, otherwise it is 0.

Define

ηt=x+0tφzsdshtvshds,t>hηt=x+0tφzsds,thE14

which represents the excess of production of energy over the delivery on the interval 0t. The initial value x represents the initial amount of energy in the storage unit. We have 0xM. We should have similarly 0ηtM, t. Indeed one cannot deliver more than one produces, and the storage of the excess production is limited by M. This constraint is more complex to handle than in the discrete time case. To simplify we shall treat the constraints with penalties and not impose them. In particular, for coherence, we remove the condition 0xM, which is a purely mathematical extension. The control is the process v., which is adapted to the filtration Ft and just positive. We then define the payoff. The payoff will include the penalty terms and the profit from selling the energy. We assume that the manager can sell his production up to his commitment at a fixed price per unit of time and unit of energy p. We note that ηt<0 captures the situation in which the manager delivers less than his commitment, and there is a penalty for it. The payoff is now written as

Jxzv.=pEh+expαsminφzsvshdsϖE0+expαsηsdsπE0+expαsηsM+dsE15

5.2 Rewriting the payoff functional

Because of the delay, we cannot consider the pair zt,ηt as the state of a dynamic system and apply dynamic programming. In fact, we shall see that it is possible to rewrite the payoff (15) in terms of the pair zt,xt with

xt=x+0tφzsds0tvsdsE16

and the standard reasoning of dynamic programming will become applicable. The first transformation concerns the term

I=Eh+expαsminφzsvshds

We have

I=expαhE0+expαsminφzs+hvsds=expαhE0+expαsEminφzs+hvsFsds

and we need to compute Eminφzs+hvsFs. We remember that vs is Fs measurable and that zs is a stationary Markov process. Let us introduce the transition probability density mηsz representing

mηsz=Probzs=ηz0=z

The function mηsz is the solution of Fokker-Planck equation

ms122η2σ2ηm+ηgηm=0mη0z=δηzE17

Then by stationarity of the Markov process zs, we can write

Eminφzs+hvsFs=RminφξvsmξszsE18

Therefore

I=expαhE0+expαsRminφξvsmξszsE19

The next transformation concerns

II=E0+expαsηsdsII=E0hexpαsηsds+Eh+expαsηsds=II1+II2.E20

The first integral does not depend on the control and is 0, when x0, as it will be the case in practice. The second integral is written as

II2.=expαhE0+expαsηs+hds

We note that

ηs+h=xs+ss+hφzτ

Recalling the definition of xs, see (16). We then can write

Eηs+h=EEηs+hFs==xszsh

with

θxzs=Ex+0sφzτE21

The argument x is a real number and z0=z. So

II2=expαhE0+expαsxszshds

We can also write

II1=0hexpαsθxzsds

so we have

II=0+expαsθxzsds+expαhE0+expαsxszshdsE22

We can give a similar formula for the second penalty term

III=E0+expαsηsM+dsE23

We introduce the function

χxzs=Ex+0sφzτM+E24

and we can write

III=0hexpαsχxzsds+expαhE0+expαsxszshdsE25

Combining results, we obtain the formula

Jxzv.=ϖ0hexpαsθxzsdsπ0hexpαsθxzsds++expαhE0+expαspRmin(φξvs)m(ξszs)(ϖθ(xszsh)+πχ(xszsh))E26

or

Jxzv.=ρhxz+expαhE0+expαslhxszsvsdsE27

with

lhxzv=pRminφξvmξszϖθxzh+πχxzhE28

The stochastic control problem becomes

dxdt=φztvtdz=gzdt+σzdw,x0=x,z0=zsupv.E0+expαslhxszsvsdsE29

which is a standard stochastic control problem. To avoid singularities, we impose a bound on the control vt. So we impose

0vtφzt+v¯a.s.E30

in which v¯ is a fixed constant.

5.3 Dynamic programming

Let us define the value function

Φxz=supv.0vtφzt+v¯E0+expαslhxszsvsdsE31

Then it is easy to write the Bellman equation for the value function, namely,

αΦ=φz∂Φx+gz∂Φz+12σ2z2Φz2++sup0vφz+v¯lhxzvv∂ΦxE32

A priori xR and z0 (we may assume that σ0=0). We can add growth conditions to get a problem which is well posed. The optimal feedback v̂hxz is obtained by taking the sup in the bracket, with respect to the argument v..

5.4 The case h=0

The case h=0 has a trivial solution. We note that

l0xzv=pminφzvϖxπxM+E33

The optimal feedback is then

v̂0xz=0ifx<0φzif0xMφz+v¯ifx>ME34

so (32) becomes

αΦ=ϖx+φz∂Φx+gz∂Φz+12σ2z2Φz2,ifx<0E35
αΦ=z+gz∂Φz+12σ2z2Φz2,if0<x<ME36
αΦ=zπxMv¯∂Φx+gz∂Φz+12σ2z2Φz2,ifx>ME37

The solution for 0<x<M does not depend on x and has an easy probabilistic interpretation

Φxz=Φz=pE0+expαsφzsds,E38

For x<0 or x>M, we have to solve parabolic problems, considering x as a time, backward and forward. We define the values Φ0z and ΦMz by Φz defined by (38).

5.5 Analytical problems for θ and χ

The functions θxzs and χxzs are solutions of the parabolic PDE

θs+φzθx+gzθz+12σ2z2θz2=0θxz0=xE39
χs+φzχx+gzχz+12σ2z2χz2=0χxz0=xM+E40

This allows to compute θxzh and χxzh.

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6. Conclusions

The problem of the optimal delivery for wind energy in some future time with a storage facility (a battery for instance) is considered. We solve an optimal control problem to define the optimal bidding decision in a simple discrete stochastic problem and apply it to real data. Optimal size of the battery and the overnight costs are discussed.

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Acknowledgments

The research is supported by EREN-GROUPE, the grant NSF-DMS 161 2880 and the grant ANR-15-CE05-0024.

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Written By

Alain Bensoussan and Alexandre Brouste

Submitted: February 14th, 2019 Reviewed: September 19th, 2019 Published: March 25th, 2020