# Stochastic Robust Optimization for Unit Commitment with Wind Curtailment

With the growing share of renewable energy, properly reasoning about the uncertainty in the balance between electricity generation and load becomes increasingly important. In the previous century, demand was predictable, generation fully controlled, and the electricity system often centrally prepared for the worst case. Such an optimization for the worst-case scenario using mathematical programming is called robust optimization, which is typically expressed by a two-stage formulation: minimizing the costs of the maximum-cost scenario. With more uncertainty, the worst case simultaneously becomes more extreme (requiring significantly more costly preparations) and more unlikely. If we always prepare for a situation where there is no wind, installing all these renewables may not have the expected benefit of a reduction in non-renewables, because they need to be stand-by anyway.

An alternative to robust optimization is to minimize the expected costs. Stochastic optimization aims (usually) at finding a decision for which the weighted sum of the objective values over a set of scenarios is optimal. Against some additional computational burden this usually gives outcomes with low expected costs. However, an unrepairable shortage in our electricity system, even if only for a brief moment, leads to a black-out, and extremely high costs to society. This leads to a delicate problem: how are we going to find the right balance between the risk of a black-out versus the costs of robust solutions (both in terms of money and in terms of reduction in fossil fuel-based power plants)?

In a unified stochastic-robust optimization, the aim is to combine the advantages of both (low expected costs and a robust solution) and overcome the disadvantages of both (high computational burden and expensive over-conservativeness). This is typically done by weighing both the costs of the worst-case as well as the expected costs in the objective function in the optimization problem.

In recent work (Robust Unit Commitment with Dispatchable Wind Power), we studied this problem of determining which power generators to put online, given predictions for electricity demand and wind generation. This problem is called the unit commitment problem. In this problem some decisions can be made after we have observed the actual value of the uncertain parameters (i.e., after the realization of the uncertainty). Here these decisions typically are about redispatching generators or putting expensive reserve generators online if needed, or by wind curtailment (i.e., shutting down some of the wind generators). A similar problem occurs in maintaining balance in a micro-grid with PV panels.

We found an improved formulation for the problem where the uncertainty for renewable generation is modeled by a so-called box uncertainty set. Such a box uncertainty set assumes a range for wind power generation for every time step and different locations in the network. We show that the adaptive robust optimization model can then be represented by a single-level optimization program. Intuitively, this is possible by taking into account the lowest amount of wind power generation possible and optimize for that scenario. In general, if the uncertainty is represented by a vector that describes upper bounds for a set of constraints (in our case, the maximum wind power that can be dispatched), the worst case must be a minimal element in the set of all such vectors possible, since all other vectors lead to strictly larger feasible regions and thus better solutions.

For the unified stochastic-robust model we reasoned that it makes sense to allow the actual wind dispatch in a scenario never to be lower than that in the worst-case scenario (with minimal wind). These extra constraints (per scenario per time step and per location) appear to improve the robustness of the unified model significantly at an acceptable cost in additional computation time.

A final important contribution is an extensive set of experiments, where we show the difference in both quality (in terms of cost and wind curtailment) and run time for the stochastic, robust and stochastic-robust formulations. The most significant result is that the run time for the stochastic-robust is lower than a pure robust or a pure stochastic formulation, while outperforming them significantly in all other aspects (costs, robustness and wind curtailment).

This work is titled Robust Unit Commitment with Dispatchable Wind Power, by German Morales, Alvaro Lorca, and Mathijs de Weerdt, and is published open-access in the Elsevier journal on Electric Power Systems Research.