(199d) Data-Driven Design of Hierarchical MPC Architectures

A well-known challenge arising in model predictive control (MPC) is the computational complexity associated with the length of the planning horizon and with the time resolution of the state and control policies [1]. These issues are often faced in applications in energy systems exhibiting phenomena and disturbances emanating at multiple timescales. For instance, energy systems involve low-frequency (e.g., seasonal) supply/demand variations, peak electricity costs (e.g., demand charges) for which long horizons need to be considered, while high-frequency supply/demand variations (e.g., from wind/solar supply) and participation in real-time electricity markets requires formulations with fine time resolutions [2]. These computational complexity issues are currently handled using ad hoc receding horizon (RH) approximations, which are practical but do not guarantee optimality in the long run [3]. Hierarchical MPC schemes [4, 5, 6] have recently been proposed to handle multiple timescales and achieve stability, but these also do not provide optimality guarantees.

In this work, we present data-driven hierarchical model predictive control (MPC) schemes for systems that exhibit periodicity in state policy over long horizons. Systems driven by exogenous factors with strong periodic components, for instance energy demands and prices in applications involving energy systems, exhibit the periodicity property [7, 8]. The proposed hierarchical schemes are based on a key observation that if the optimal policy of an infinite-horizon problem is periodic (or can be approximated as periodic), a stochastic programming (SP) problem can be used to pose the problem. Under the SP abstraction, the inter-period trajectory of the exogenous factors is interpreted as a realization of a random variable that triggers a periodic trajectory of the states (i.e., the states at the beginning and end of the period are the same). Moreover, the periodic states are interpreted as design variables and operational policies within the periods are interpreted as recourse variables. The SP representation facilitates construction of hierarchical MPC schemes with a long-term (supervisory) MPC controller providing periodic targets to guide a short-term MPC controller operating at finer time resolution [9, 10]. We have shown that under nominal conditions with perfect forecasts, the hierarchical scheme delivers an optimal policy for the infinite horizon problem. For the more relevant case of imperfect forecasts, the hierarchical scheme needs to re-compute periodic targets. We derive retroactive hierarchical MPC schemes under a periodic setting and using statistical approximations, that accumulate real historical data to asymptotically deliver optimal targets [10]. We show that the retroactive design principle offers optimality guarantees and, notably, does not require data forecasts.

Thus the retroactive approach provides key advantages over standard proactive RH schemes which use historical data to compute forecasts, and associated control actions and periodic targets. A fundamental issue with proactive approaches is that no optimality guarantees can be provided unless the forecast is perfect. The targets delivered by the retroactive scheme are used to guide a low-level controller operating at fine time resolutions within the periods. We also derive a specialized retroactive scheme by using incremental cutting-plane (CP) techniques for the case of linear systems [10, 11]. The SP abstraction also provides opportunities to construct retroactive schemes for nonlinear systems, and to obtain the desired stability properties. We demonstrate the concepts using an application of management of stationary battery systems in buildings, where the proposed retroactive hierarchical scheme is used to obtain the optimal charge-discharge policy for the battery system to minimize the peak demand charge of the buildings. We compare the performance of the proposed retroactive hierarchical MPC scheme with a proactive MPC approach for periodic systems.

References:

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[9] Kumar, R., Wenzel, M.J., Ellis, M.J., ElBsat, M.N., Drees, K.H. and Zavala, V.M., 2018, June. Handling Long Horizons in MPC: A Stochastic Programming Approach. In 2018 Annual American Control Conference (ACC) (pp. 715-720). IEEE.

[10] Kumar, R., Wenzel, M.J., Ellis, M.J., ElBsat, M.N., Drees, K.H. and Zavala, V.M., 2019. Hierarchical MPC Schemes for Periodic Systems using Stochastic Programming. Automatica, Accepted.

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