Stochastic MPC for Energy Management in Smart Grids with Conditional Value at Risk as Penalty Function

This paper considers imbalance problems arising in Energy Management in Smart Grids (SG) as discrete-time stochastic linear systems subject to chance constraints, and proposes a Model Predictive Control (MPC) approach to solve them. It is well-known that handling the closed-loop constraint feasibility of such systems is in general difficult due to the presence of a potentially unbounded uncertainty source. To overcome such a difficulty, we propose two new ideas. We first reformulate the chance constraint using the so-called Conditional Value at Risk (CVaR), which is known to be the tightest convex approximation for chance constraints. We then relax the CVaR constraint using a penalty function depending on a coefficient parameter. An optimal solution is therefore obtained by solving a single unconstrained problem which, intuitively, takes into consideration a risk of the system trajectories in an undesirable state. A case study using an academic example is presented to estimate the a-posteriori probability of the coefficient parameter in order to show when such a penalty function is exact by means of probabilistic constraint fulfillment.