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.
- optimal control
- stochastic control
- wind farm management
- wind production forecast
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 . 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  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 , ultra-capacitors , or flywheels . 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 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 period. This splitting will be omitted in this stylized model. It is convenient to consider the full delivery at the end of the period which is the beginning of the period. So, at the beginning of the period (day or hour), the manager commits himself to deliver 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 . The second element concerns the windfarm. The energy produced by the windfarm is a stochastic process . More precisely, is the energy produced during the period, which we consider to be available at the beginning of the period. So and all previous values are known. However to fulfill his commitment, the manager will rely on , the energy produced during the period, which we consider, with our convention, to occur at the end of the period, which is the beginning of the period. So it is not known by the manager, when he takes his decision. We model the process as a Markov chain with transition probability density A key issue concerns the choice of this density which is discussed in the application in Section 4. Precisely, although formally
In fact, 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 the CDF of the transition probability. We also set
In the language of stochastic control, the decision (control) is measurable with respect to The storage is also adapted to The evolution of is defined by the equation
Indeed, the available energy at the end of the period is If this quantity is smaller than , then the manager cannot fulfill his commitment; he delivers what he has, namely, ; and the storage becomes empty. If the available energy is larger than then the manager delivers his commitment and tries to store the remainder This is possible only when this quantity is smaller than which represents the maximum storage of the battery. If then he charges up to and the quantity 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 , if the manager delivers his commitment , he receives the normal income If he fails and delivers only there is a penalty. In the sequel, we have chosen the following penalty which is common in inventory theory. The parameter can be adjusted, for instance, to compare with the conditions on the spot market.
In our set up, the pair is a Markov chain. So we have a controlled Markov chain and the state is two-dimensional. We introduce a discount factor denoted by , which is useful if we consider an infinite horizon. We can have , otherwise. Initial conditions are given at time and denoted by . We call the control, where is the horizon. Finally, we want to maximize the functional
3. Dynamic programming
3.1 Bellman equation
The value function is defined by
we get also
We can then write Bellman equation
It is convenient to make the change of variables and obtain
with final equation
We have and
3.2 Main result
We state the following proposition:
Proposition 1. The solution of (7) is of the form
Proof. For we have
Consider the function
then, for we have It is monotone increasing, so the maximum cannot be reached at a point It follows that (8) is satisfied at with
We have, for
and is concave, since
Consider now the function
For it reduces to
and thus cannot reach a maximum for Considering we have
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 point at which attains its maximum. It is positive and uniquely defined. It follows that the optimal feedback in Bellman equation (6) is and is the unique solution of
The recursion (10) writes
It is worth emphasizing that the function increases with , 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 decreases with This is not so obvious. Clearly, the larger the better it is, as captured by the increase of We can understand why decreases with as follows: When 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.
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 EUR/MWh (Official Journal from March 8, 2013) and the discount factor . In this first application, which is the number of periods of 30 min in a day.
In the sequel, we have chosen the penalty which is common in inventory theory. The parameter can be adjusted.
Some analysts would prefer the penalty . 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.
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 . 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 .
With these assumptions, the payoff with respect to the size of the storage is given in Figure 2 for an empty storage , and 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.
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 is a standard one-dimensional Wiener process, built on a probability space , and we denote by 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 is where the function is the power law. So the energy produced on an interval of time is We assume that the manager bids for a delivery program with a fixed delay In other words, if he decides a level per unit of time at time the delivery will be at On the interval of time the delivery is provided , otherwise it is
which represents the excess of production of energy over the delivery on the interval The initial value represents the initial amount of energy in the storage unit. We have We should have similarly , Indeed one cannot deliver more than one produces, and the storage of the excess production is limited by 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 which is a purely mathematical extension. The control is the process , which is adapted to the filtration 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 We note that 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
5.2 Rewriting the payoff functional
Because of the delay, we cannot consider the pair 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 with
and the standard reasoning of dynamic programming will become applicable. The first transformation concerns the term
and we need to compute . We remember that is measurable and that is a stationary Markov process. Let us introduce the transition probability density representing
The function is the solution of Fokker-Planck equation
Then by stationarity of the Markov process , we can write
The next transformation concerns
The first integral does not depend on the control and is when as it will be the case in practice. The second integral is written as
We note that
Recalling the definition of see (16). We then can write
The argument is a real number and . So
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
The stochastic control problem becomes
which is a standard stochastic control problem. To avoid singularities, we impose a bound on the control So we impose
in which is a fixed constant.
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 and (we may assume that We can add growth conditions to get a problem which is well posed. The optimal feedback is obtained by taking the sup in the bracket, with respect to the argument .
5.4 The case
The case has a trivial solution. We note that
The optimal feedback is then
so (32) becomes
The solution for does not depend on and has an easy probabilistic interpretation
For or , we have to solve parabolic problems, considering as a time, backward and forward. We define the values and by defined by (38).
5.5 Analytical problems for and
The functions and are solutions of the parabolic PDE
This allows to compute and .
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.
The research is supported by EREN-GROUPE, the grant NSF-DMS 161 2880 and the grant ANR-15-CE05-0024.