Description

 We have studied a stand-alone hybrid system composed of wind turbines, solar photovoltaic (or PV)
panels and batteries. To compensate for a lack of energy from these sources, an auxiliary fuel generator
guarantees to meet the demand in every case but its use induces important costs. Such systems develop
on islands or small isolated towns or regions. The aim is to determine the optimal number of photovoltaic
panels, wind turbines and elements of battery to install in order to serve a given demand while minimizing
the total cost of investment and use.
Moreover, the stochastic behavior of the solar and wind energy production on the one hand, and the
demand on the other hand, needs to search for a robust solution, i.e. a solution which is good enough
whatever the scenario that occurs. We assume that there is no known distributions of the data and
following the approach proposed in [3], we consider that the uncertain data can vary between given
bounds and that there are limits to the total variation of each kind of data.
We can state the robust problem as a two stages mixed-integer mathematical program.

Results
COMPLEXITY RESULTS
We proved that the problem without uncertainty is NP-difficult when the number of integer variables is not
fixed.
Nevertheless, we showed that, for our energy park design problem, the recourse problem, i.e. the second
stage problem, can be solvable in polynomial time by using dynamic programming for any number of
integer variables. We recall that in the general case, the problem is NP-difficult [2].
SOLVING APPROACH
We have proposed an exact approach based on constraints generation to solve the robust problem. The
method requires a linearization of quadratic terms and a dualization of the minimization part of the
recourse problem.
COMPUTATIONAL RESULTS
The test have been performed on a Bi-pro Intel Nehalem XEON 5570 at 2.93 GHz with 24 Go of RAM.
We consider two cases: either there is uncertainty only on the demand, or the uncertainty also concerns
wind and solar energy. We have considered real data. We have tested several
uncertainty levels for up to 8760 time periods in less than two minutes.
GENERALIZATION OF OUR APPROACH
We have also proved that our approach can be applied to general mixed integer problem with continuous
recourse variables, even if the usual recourse property is not verified.