Optimal Control of a Dispatchable Energy Source for Wind Energy Management
-
Guglielmo D’Amico
and
Robert Adam Sobolewski
Abstract
The major drawback of wind energy relies in its variability in time, which necessitates specific strategies to be settled. One such strategy can be the coordination of wind power production with a co-located power generation of dispatchable energy source (DES), e.g., thermal power station, combined heat and power plant, gas turbine or compressed air energy storage. In this paper, we consider an energy producer that generates power by means of a wind park and of a DES and sells the produced energy to an isolated grid. We determine the optimal quantity of energy produced by a DES, given the unit cost of this energy, that a power producer should buy and use to hedge against the risk inherent in the production of energy through wind turbines. We determine the optimal quantity by solving a static optimization problem taking into account the possible dependence between the amount of energy produced by wind turbines and electricity prices by using a copula function. Several particular cases are studied that allow the determination of the optimal solution in an analytical closed form. Finally, a numerical example concerning a real 48 MW wind farm located in Poland and Polish Power Exchange shows the possibility of implementing the model in real-life problems.
A Proofs
Proof of Lemma 2.1.
We have
where
(A.1)
Now we calculate the first-order derivative of
The use of the Leibniz formula for differentiation under an integral sign gives
The derivatives of the second addendum is
Interchange the limit and the derivatives and apply Leibniz’s formula for differentiation under an integral sign to obtain
The derivatives of the third addendum is
Apply Leibniz’s formula for differentiation under an integral sign to obtain
(A.4)
Now we have to calculate the derivatives of the fourth addendum:
(A.5)
The summation of formulas (A.2), (A.3), (A.4) and (A.5) and some algebraic manipulations gives
To prove the concavity of
where the latter inequality is due to Assumption 3. ∎
Proof of Lemma 3.1.
The objective is the computation of the integral
where the random variable
A change of variable (standardization)
where
This latter integral can be calculated by introducing again a standardization, i.e.
where Φ denotes the cdf of a standard Normal distribution. Therefore we have
where ϕ denotes the probability density function of a standard Normal distribution. The former integrals have been evaluated by [14] who established that
and
Then a simple substitution gives
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© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Notes on Cumulative Entropy as a Risk Measure
- Comparing Short and Long-Memory Charts to Monitor the Traffic Intensity of Single Server Queues
- Optimal Control of a Dispatchable Energy Source for Wind Energy Management
- Bounded M-O Extended Exponential Distribution with Applications
- On Some Characterizations of the Levy Distribution
Articles in the same Issue
- Frontmatter
- Notes on Cumulative Entropy as a Risk Measure
- Comparing Short and Long-Memory Charts to Monitor the Traffic Intensity of Single Server Queues
- Optimal Control of a Dispatchable Energy Source for Wind Energy Management
- Bounded M-O Extended Exponential Distribution with Applications
- On Some Characterizations of the Levy Distribution
