Robust randomized model predictive control for energy balance in smart thermal grids

This paper presents a stochastic model predictive control approach for a thermal grid with uncertainties in the consumer demand profiles. This approach leads to a finite-horizon chance-constrained mixed-integer linear optimization problem at each sampling time, which is in general non-convex and hard to solve. Earlier approaches for such problems are either suboptimal ad-hoc methodologies, or computationally intractable formulations. We provide a unified framework to deal with production planning problems for uncertain systems, while providing a-priori probabilistic certificates for the robustness properties of the resulting solutions. Our methodology is based on solving a random convex optimization problem to compute the uncertainty bounds using the so-called scenario approach and then, solving a robust mixed-integer optimization problem with the computed randomized uncertainty bounds at each sampling time. Using a tractable approximation of uncertainty bounds, the proposed problem formulation retains the complexity of the problem without chance constraints. In the presented thermal grid application this implies that a robust mixed-integer program is solved to provide a day-ahead prediction for the thermal energy production plan in the grid. The performance of the proposed methodology is illustrated using Monte Carlo simulations and employing two different problem formulations: optimization over input sequences (open-loop MPC) and optimization over affine feedback policies (closed-loop MPC).