## 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 CO_{2} 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.

## 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

In fact,

In the language of stochastic control, the decision

Indeed, the available energy at the end of the

### 2.2 The payoff

We want now to write the payoff to be optimized. During the period

In our set up, the pair

## 3. Dynamic programming

### 3.1 Bellman equation

The value function is defined by

Writing

we get also

We can then write Bellman equation

It is convenient to make the change of variables

with final equation

We have

### 3.2 Main result

We state the following proposition:

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

**Proof.** For

Consider the function

then, for

We have, for

and

and

Consider now the function

For

and

and thus cannot reach a maximum for

Again, this function is concave and

and the property (8) is proven with

The proof is completed.

### 3.3 Optimal feedback

We define by

The recursion (10) writes

It is worth emphasizing that the function

## 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

In the sequel, we have chosen the penalty

Some analysts would prefer 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.

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

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

With these assumptions, the payoff with respect to the size of the storage is given in Figure 2 for an empty storage

## 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

where

The energy produced per unit of time at time

Define

which represents the excess of production of energy over the delivery on the interval

### 5.2 Rewriting the payoff functional

Because of the delay, we cannot consider the pair

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

We have

and we need to compute

The function

Then by stationarity of the Markov process

Therefore

The next transformation concerns

The first integral does not depend on the control and is

We note that

Recalling the definition of

with

The argument

We can also write

so we have

We can give a similar formula for the second penalty term

We introduce the function

and we can write

Combining results, we obtain the formula

or

with

The stochastic control problem becomes

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

in which

### 5.3 Dynamic programming

Let us define the value function

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

A priori

### 5.4 The case h = 0

The case

The optimal feedback is then

so (32) becomes

The solution for

For

### 5.5 Analytical problems for θ and χ

The functions

This allows to compute

## 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.

## Acknowledgments

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