A mean-variance criterion for economic model predictive control of stochastic linear systems

Abstract

Stochastic linear systems arise in a large number of control applications. This paper presents a mean-variance criterion for economic model predictive control (EMPC) of such systems. The system operating cost and its variance is approximated based on a Monte-Carlo approach. Using convex relaxation, the tractability of the resulting optimal control problem is addressed. We use a power management case study to compare different variations of the mean-variance strategy with EMPC based on the certainty equivalence principle. The certainty equivalence strategy is much more computationally efficient than the mean-variance strategies, but it does not account for the variance of the uncertain parameters. Openloop simulations suggest that a single-stage mean-variance approach yields a significantly lower operating cost than the certainty equivalence strategy. In closed-loop, the single-stage formulation is overly conservative, which results in a high operating cost. For this case, a two-stage extension of the mean-variance approach provides the best trade-off between the expected cost and its variance. It is demonstrated that by using a constraint back-off technique in the specific case study, certainty equivalence EMPC can be modified to perform almost as well as the two-stage mean-variance formulation. Nevertheless, we argue that the mean-variance approach can be used both as a strategy for evaluating less computational demanding methods such as the certainty equivalence method, and as an individual control strategy when heuristics such as constraint back-off do not perform well.Simulations demonstrate that while MV-EMPC(M) outperforms CE-EMPC in open-loop, the back-off based modification of CE-EMPC works almost as well in closed-loop. Nevertheless, the performance of MV-EMPC(M) provides important information that can be utilized to evaluate the performance of less computational demanding control strategies. We also emphasize that the preliminary results presented in this paper are based on a single test system. Other examples may show that the performance gap between MV-EMPC(M) and CE-EMPC can be much larger. 53rd IEEE Conference on Decision and Control

Economic Nonlinear Model Predictive Control

“…To this end, it was noted in (Bayer, Müller, et al, 2014) that just transferring robust mpc approaches from a stabilizing to an economic context might not result in an optimal closed-loop performance, necessitating the development of novel schemes. Some robust and stochastic economic mpc approaches can, e.g., be found in (Bayer, Lorenzen, et al, 2016;Bayer, Müller, et al, 2014;Broomhead et al, 2015;Huang et al, 2012;Lucia et al, 2014;Marquez et al, 2014;Sokoler et al, 2014). Overall, the picture is still much less complete compared to the nominal case, and various interesting open problems remain.…”

Application of economic MPC to frequency control in a single-area power system

“…Although the case α = 1 results in even lower generation costs, it is disregarded due to its high risk. A systematic method for trading-off cost variance and cost expectation may be achieved for α = 1, by combining the proposed strategy with mean-variance EMPC [29], [30].…”