Suppose you are planning a hybrid power plant consisting of a photovoltaic (PV) plant, a battery and backup diesel power to supply a given electric load at a defined remuneration per delivered kWh.
During the planning stage, the exact PV production will not be known (only its distribution). The required amount of diesel to cover the demand at times with no or little sun will thus also be uncertain. The same applies to the profit of the plant which is determined as acquired remuneration minus the fuel cost for the diesel.
Projects with highly uncertain profits, however, can only obtain financing at high interest rates. If we could quantify and bound the uncertainty in more detail, one would be able to acquire cheaper financing and thereby greatly improve the economic viability of such a project. In sum, exact and efficient uncertainty quantification methods could thus help to move forward the energy transition towards a low carbon society.
Mathematically, the plant operation can be described as a linear program (LP), with the solar power availability determining the right hand side of the boundary conditions. Assuming a (multivariate) Gaussian distribution for possible PV timeseries, the task is to determine the distribution of the linear program’s optimal values (or parameters of it).
It seems that such random LPs have not been considered before. Prekopa [1966] investigated random LPs in which the optimal basis does not change.
The following approaches should be investigated:

• It might be easier to investigate the dual program in which the objective coefficients are random, but everything else is fixed.

• For a given basis, one needs to estimate the probability that this basis is optimal.

• Varying the right hand side or objective coefficients yields a decomposition of the space into polyhedra on which a certain basis is optimal.

• In two dimensions, the computation might be possible analytically.

• In general, one needs to estimate this probability, e.g., using sampling.

• Then the (relevant) bases should be enumerated.

• Possibly, one can start with a uniform distribution (cubes) and then generalize to ellipsoids and normal distributions.

This is a cooperation thesis with research group optimization of the department of mathematics.