# Stochastic Robust Optimization for Unit Commitment with Wind Curtailment

This week my first journal paper with German Morales was published. It is titled Robust Unit Commitment with Dispatchable Wind Power and published open-access in the journal on Electric Power Systems Research. German joined our group as a post-doc on the Gaming beyond the Copper Plate (GCP) project, where he has already made a number of relevant contributions, but this is the first one where I was involved in. For me doing this research was very interesting, because it got me acquainted with combining stochastic and robust optimization.

Stochastic optimization aims (usually) at finding a decision for which the (weighted) sum of the objective values over a set of scenarios is optimal. Robust optimization aims at finding an optimal decision under the worst-case uncertainty realization over a continuous uncertainty set (i.e., a two-stage stochastic and/or robust optimization problem). In a unified stochastic-robust optimization, the aim is to combine the advantages of both (good expected and robust solution) and overcome the disadvantages of each (high computational burden and expensive over-conservativeness).

The formulation we used is also adaptive (i.e., two stage); this means that we take into account that some decisions can be made after we have observed the actual value of the uncertain parameters (i.e., after the realization of the uncertainty). Generally, taking this adaptive part into account makes the problem harder.

An adaptive robust optimization model is a very good fit to the problem of determining which power generators to put online, given predictions for electricity demand and wind generation: we aim to minimize the costs for generators in expectation, including cost of scenarios where the uncertain realization of wind creates an imbalance and we need to recover (adaptive decisions) from this at some extra costs, for example by redispatching generators or putting expensive reserve generators online if needed, or by wind curtailment (i.e., shutting down wind generators).

Our main contribution is an improved formulation for the problem where the uncertainty for wind 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.

In our paper we also considered the unified stochastic robust model and 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 at an acceptable cost in additional computation time.

A final important contribution, admittedly all work done by German, 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).

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