(522f) Benchmarking Stochastic and Deterministic MPC: A Case Study in Stationary Battery Systems

Kumar, R., University of Wisconsin-Madison
Jalving, J., University of Wisconsin Madison
Zavala, V. M., University of Wisconsin-Madison
Model predictive control (MPC) revolutionized the field of control due to its ability to anticipate and counteract uncertain disturbances in order to maximize performance and satisfy system constraints [1]. In a standard MPC implementation, a deterministic representation of uncertain disturbances (typically the most likely realization of the disturbance) acts as a summarizing statistic of the entire disturbance uncertainty space. The selected statistic is colloquially known as the disturbance forecast and is used to compute the control action [2],[3]. Forecast errors are introduced due to the deviations of the forecast from the actual disturbance realization, which MPC counteracts by adjusting the control action at the next time step. This feedback mechanism provides inherent robustness to the MPC and seeks to maintain the system within constraints [4],[5]. However, this inherent robustness property can be insufficient to avoid constraint violations. For instance, constraint violations might occur during the sampling period when the pre-computed control action is fixed while the system faces the actual disturbance realization. Moreover, a large forecast error might also lead to the loss of recursive feasibility of the MPC.

In this work, we make the observation that certain types of disturbances cannot be adequately represented using summarizing statistics (e.g., most likely realization) and this can induce inconsistencies in the computation of the MPC control action. For instance, for a disturbance signal with zero mean or a discrete (e.g., ON/OFF) disturbance signal, the most likely realization will lead to misleading forecasts. To capture such disturbances, it is necessary to use an explicit characterization of the uncertainty space in the MPC formulation. In this work, we present a stochastic MPC framework that can capture diverse types of uncertainty characterizations [6]-[8]. We consider the scenario-based, two-stage stochastic MPC formulations that provide computational flexibility and achieve significant benefits over deterministic MPC. In a two-stage stochastic MPC, uncertainty of the disturbances is captured when computing the next immediate control action, and uses a receding horizon implementation to include new information and update the uncertainty description [9].

We present a computational framework that integrates disturbance forecasting, uncertainty quantification, and MPC to provide a critical assessment of deterministic and stochastic MPC [10]. Our framework focuses on a case study of the management of stationary battery systems providing frequency regulation (FR) services to the power grid while simultaneously mitigating peak monthly demand charge costs of a university campus [9],[10]. Real disturbance data and a rigorous benchmarking procedure are used to systematically compare performance of deterministic and stochastic MPC policies. We show that that FR dispatch signals from the power grid, which are zero-mean random signals (like white noise), cannot be properly captured by deterministic MPC formulations, and this results in ineffective control actions causing constraint violations and losses in performance. In a deterministic MPC, a back-off term that allocates battery reserve capacity can prevent the constraint violations, but this approach is ad hoc and prevents full utilization of the battery asset and thus decreasing its value. This indicates that feedback alone is insufficient to counteract certain types of disturbances. Stochastic MPC provides a more systematic framework to account for diverse disturbances, satisfy constraints, and maximize asset utilization and value. We show that stochastic MPC provides significant benefits over deterministic MPC in such cases. We also derive a quasi-stochastic MPC formulation using a coarse representation of the uncertainty space, and we show that this simple approach already avoids most of the limitations of deterministic MPC while decreasing the computational complexity of stochastic MPC.


[1] Qin, S.J. and Badgwell, T.A., 2003. A survey of industrial model predictive control technology. Control engineering practice, 11(7), pp.733-764.

[2] Oldewurtel, F., Parisio, A., Jones, C.N., Gyalistras, D., Gwerder, M., Stauch, V., Lehmann, B. and Morari, M., 2012. Use of model predictive control and weather forecasts for energy efficient building climate control. Energy and Buildings, 45, pp.15-27.

[3] Zavala, V.M., Constantinescu, E.M., Krause, T. and Anitescu, M., 2009. On-line economic optimization of energy systems using weather forecast information. Journal of Process Control, 19(10), pp.1725-1736.

[4] Magni, L. and Scattolini, R., 2007. Robustness and robust design of MPC for nonlinear discrete-time systems. In Assessment and future directions of nonlinear model predictive control (pp. 239-254). Springer, Berlin, Heidelberg.

[5] Limon, D., Alvarado, I., Alamo, T. and Camacho, E.F., 2010. Robust tube-based MPC for tracking of constrained linear systems with additive disturbances. Journal of Process Control, 20(3), pp.248-260.

[6] Bernardini, D. and Bemporad, A., 2009, December. Scenario-based model predictive control of stochastic constrained linear systems. In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference (pp. 6333-6338). IEEE.

[7] Lucia, S., Andersson, J.A., Brandt, H., Diehl, M. and Engell, S., 2014. Handling uncertainty in economic nonlinear model predictive control: A comparative case study. Journal of Process Control, 24(8), pp.1247-1259.

[8] de la Penad, D.M., Bemporad, A. and Alamo, T., 2005, December. Stochastic programming applied to model predictive control. In Proceedings of the 44th IEEE Conference on Decision and Control (pp. 1361-1366). IEEE.

[9] Kumar, R., Wenzel, M.J., Ellis, M.J., ElBsat, M.N., Drees, K.H. and Zavala, V.M., 2018. A stochastic model predictive control framework for stationary battery systems. IEEE Transactions on Power Systems, 33(4), pp.4397-4406.

[10] Kumar, R., Jalving, J., Wenzel, M.J., Ellis, M.J., ElBsat, M.N., Drees, K.H. and Zavala, V.M., 2019. Benchmarking stochastic and deterministic MPC: A case study in stationary battery systems. AIChE Journal, p.e16551.