7. Distributed Stochastic Thermal Energy Management in Smart Thermal Grids

This work presents a distributed stochastic energy management framework for a thermal grid with uncertainties in the consumer demand profiles. Using the model predictive control (MPC) paradigm, we formulate a finite-horizon chance-constrained mixed-integer linear optimization problem at each sampling time, which is in general non-convex and hard to solve. We then 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 program 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 formulation retains the complexity of the problem without chance constraints. We also present two distributed approaches that are based on the alternating direction method of multipliers (ADMM) to solve the robust mixed-integer problem. 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).