(136g) Hierarchical MPC Schemes for Periodic Systems Using Stochastic Programming Techniques

Kumar, R., University of Wisconsin-Madison
Zavala, V. M., University of Wisconsin-Madison
Model Predictive Control (MPC) is a powerful planning framework that can handle inequality constraints, multivariable interactions, and accommodate different kinds of cost functions and disturbances [1, 2]. A well-known challenge arising in MPC is the computational tractability associated with the length of the planning horizon and with the time resolution of the state and control policies. These tractability issues are often encountered in energy system applications that exhibit phenomena and disturbances emanating at multiple timescales. For instance, long horizons are often required to respond to low-frequency (e.g., seasonal) supply/demand variations, peak electricity costs (e.g., demand charges) while fine time resolutions are needed to tackle high-frequency supply/demand variations (e.g., from wind/solar supply) and to participate in real-time electricity markets [3-5]. Tractability issues are currently handled by using ad-hoc receding horizon (RH) approximations, which are practical but do not provide optimality guarantees [6,7].

Hierarchical MPC schemes [8, 9] have recently been proposed to handle multiple scales and achieve stability. These schemes, however, do not provide optimality guarantees in the sense that the computed policies match those of the long-horizon optimal control problem of interest. The
hierarchical scheme proposed in [10] uses adjoint information obtained from a long-term and coarse MPC controller to guide a short-term MPC controller operating at fine time resolutions. Computational experiments are provided to demonstrate that this approach can achieve optimality but no guarantees are given. Moreover, such an approach requires smoothness and continuity of the adjoint profiles, which is not guaranteed in general applications. The hierarchical scheme proposed in this work relies on the observation that, if the optimal policy of an infinite horizon optimal control problem is periodic (or can be approximated with a periodic policy), the problem can be cast as a stochastic programming (SP) problem. Periodicity is a property that is commonly observed in systems driven by exogenous factors with strong periodic components (e.g., energy demands and prices) [11-13]. Under the SP abstraction, the periodic states are interpreted as design variables and operational policies over the periods are interpreted as recourse variables.

We have recently observed that the SP representation provides a mechanism to construct hierarchical MPC schemes in which a long-term (supervisory) MPC controller provides periodic targets to guide a short-term MPC controller [14]. Under nominal conditions with perfect forecasts, we have shown that the hierarchical scheme delivers an optimal policy for the infinite horizon problem. For the more practical case of imperfect forecasts, the hierarchical scheme needs to re-compute periodic targets. While this can certainly be done using an RH scheme (e.g., computes targets by anticipating multiple future periods), such an approach would not provide optimality guarantees. In fact, to the best of our knowledge, no RH scheme currently exists that can provide optimal policies in the presence of imperfect forecast information.

The key contribution of this work is the observation that, under a periodic setting, one can derive retroactive optimization schemes that accumulate real historical disturbance information to asymptotically deliver optimal targets. We argue that this retroactive design principle offers a key advantage over proactive RH schemes (which rely only on forecast information). The targets obtained with a retroactive scheme are used to guide a low-level MPC controller operating at fine time resolutions within the periods. In the case of linear systems, one can derive a specialized retroactive scheme by using incremental cutting-plane (CP) techniques [15,16]. The SP setting also reveals strategies to construct retroactive schemes for general nonlinear systems and to derive metrics to monitor optimality. We demonstrate the concepts using an application in buildings with electrochemical storage.
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