Robust Economic MPC for a Power Management Scenario with Uncertainties Tobias Gybel Hovgaard, Lars F. S. Larsen and John Bagterp Jørgensen Abstract— This paper presents a novel incorporation of probabilistic constraints and Second Order Cone Program- ming (SOCP) with economic Model Predictive Control (MPC). Hereby the performance of the controller is robustyfied in the presence of both model and forecast uncertainties. Economic MPC is a receding horizon controller that minimizes an eco- nomic objective function and we have previously demonstrated its usage to include a refrigeration system as a controllable power consumer with a portfolio of power generators such that total cost is minimized. The main focus for our work is power management of the refrigeration system. Whereas our previous study was entirely deterministic, models of e.g. supermarket refrigeration systems are uncertain, as are forecasts of outdoor temperatures and electricity demand. The linear program we have formulated does not cope with uncertainties and thus it is, liable to drive an optimal solution to an infeasible or very expensive solution. The main contribution of this paper is the Finite Impulse Response (FIR) formulation of the system models, allowing us to describe and handle model uncertainties in the framework of probabilistic constraints. Our new solution using this setup for robustifying the economic MPC is demonstrated by simulation of a small conceptual example. The scenario is primarily chosen to illustrate the effect of our proposed method in that it can be compared with our previous deterministic simulations. I. I NTRODUCTION In [1] we introduced economic MPC to control a number of independent dynamic systems that must collaborate to minimize the overall cost of satisfying the cooling demand for some goods while meeting market demands for power at all times. Our control strategy is an economic optimizing model predictive controller, economic MPC. MPC for constrained systems has emerged during the last 30 years as the most successful methodology for control of industrial processes [2]. MPC is increasingly being considered for refrigeration systems [3], [4] and for power production plants [5]. MPC based on optimizing economic objectives has only recently emerged as a general methodology with efficient numerical implementations and provable stability properties [6]–[8]. Our proposed economic MPC controller has previously been formulated in a deterministic setting and the contribution of this paper is to put our strategy into a more realistic scenario where different uncertainties always affect the system. Thus, this paper provides a novel extension to the economic MPC to provide robust performance in the presence of both forecast and model T. G. Hovgaard and L. F. S. Larsen are with Danfoss Refrigeration and A/C Controls, DK-6430 Nordborg, Denmark. {tgh,lars.larsen}@danfoss.com J. B. Jørgensen is with DTU Informatics, Technical University of Den- mark, DK-2800 Lyngby, Denmark. jbj@imm.dtu.dk uncertainties. This is done in a way similar to [9] where energy consumption for climate control is minimized under influence of uncertain weather predictions but also the ability to handle model uncertainties in the closed-loop MPC is an important issue in this paper. The Smart Grid is the future intelligent electricity grid and is intended to be the smart electrical infrastructure required to increase the amount of green energy significantly. The Danish transmission system operator (TSO) has the following definition of Smart Grids which we adopt in this work: ”Intelligent electrical systems that can integrate the behavior and actions of all connected users - those who produce, those who consume and those who do both - in order to provide a sustainable, economical and reliable electricity supply efficiently” [10]. A larger share of intermittent stochastic power-generating sources such as wind turbines makes it difficult to balance demand and supply of electricity in a flexible and cost-efficient manner. To account for this we previously introduced large power consumers, such as cold rooms, or an aggregation of a number of like consumers such as supermarket systems, with the ability to adjust the power consumption profile to the power supply. Due to the large thermal capacity of cold rooms, their consumption of electricity can, to some degree, be shifted in time to benefit the overall system. The thermal capacity in the refrigerated goods is then utilized to store ”coldness” such that the refrigeration system can increase cooling when there is an over production of energy and then lower its consumption at other times. The temperature is allowed to vary within certain bounds, which have no impact on food quality. We exploit that the dynamics of the temperature in the cold room are rather slow while the power consumption can be changed rapidly. [9], [11], [12] also utilized load shifting capabilities to reduce total energy consumption. Several works exist that consider constrained model predictive control (MPC) in the presence of uncertainty [13]. In many applications distributions can be quantified for uncertainty and if this information is ignored (e.g. by defining worst-case costs and invoking constraints over all uncertainty realizations) it can lead to conservative results, and the need for a stochastic extension to constrained MPC is clear [14]. Taking expected values of the cost provides an obvious way to utilize probabilistic information [15]. However constraints often admit a probabilistic formulation too, e.g. a variable should not exceed a certain bound with a given probability. 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 1515