Optimal Energy Consumption in Refrigeration Systems - Modelling and Non-Convex Optimisation Tobias Gybel Hovgaard, 1, 2,3 * Lars F. S. Larsen, 1,2 Morten J. Skovrup 4 and John Bagterp Jørgensen 3 1. Danfoss A/S, Nordborgvej 81, DK-6430 Nordborg, Denmark 2. Vestas Technology R&D, Hedeager 42, DK-8200 Aarhus N, Denmark 3. DTU Informatics, Technical University of Denmark, Richard Petersens Plads, Building 321, DK-2800 Kgs, Lyngby, Denmark 4. IPU Technology Development, Building 403, DK-2800 Kgs, Lyngby, Denmark Supermarket refrigeration consumes substantial amounts of energy. However, due to the thermal capacity of the refrigerated goods, parts of the cooling capacity delivered can be shifted in time without deteriorating the food quality. In this study, we develop a realistic model for the energy consumption in super market refrigeration systems. This model is used in a Nonlinear Model Predictive Controller (NMPC) to minimise the energy used by operation of a supermarket refrigeration system. The model is non-convex and we develop a computational efficient algorithm tailored to this problem that is somewhat more efficient than general purpose optimisation algorithms for NMPC and still near to optimal. Since the non-convex cost function has multiple extrema, standard methods for optimisation cannot be directly applied. A qualitative analysis of the system’s constraints is presented and a unique minimum within the feasible region is identified. Following that finding we propose a tailored minimisation procedure that utilises the nature of the feasible region such that the minimisation can be separated into two linear programs; one for each of the control variables. These subproblems are simple to solve but some iterations might have to be performed in order to comply with the maximum capacity constraint. Finally, a nonlinear solver is used for a small example without separating the optimisation problem, and the results are compared to the outcome of our proposed minimisation procedure for the same conceptual example. The tailored approach is somewhat faster than the general optimisation method and the solutions obtained are almost identical. Keywords: modelling and simulation, energy efficiency, optimisation, model predictive control, thermodynamics INTRODUCTION S upermarket refrigeration and refrigeration systems in gen- eral have been modelled for both analysis and control in several previous publications. These are both concerned with the overall system (Larsen, 2005; Larsen et al., 2007a,b; Hovgaard et al., 2010a) and the complex thermodynamics of the individual parts such as evaporators and condensers (Willatzen et al., 1998; Rasmussen and Larsen, 2009). However, the focus in this study is on describing the power consumption of super- market refrigeration systems in a form that enables us to use optimisation methods like Model Predictive Control (MPC) (see e.g. Maciejowski, 2002; Rawlings and Mayne, 2009) to minimise the total cost of the system. This is not a completely new idea. In Larsen (2005) and Sarabia et al. (2008) MPC is applied to refriger- ation systems and in Larsen et al. (2007b) optimisation is applied in order to utilise the daily variations to minimise power con- sumption. However, the models used in such papers tend to be rather simple in their description of, for example, the work done in the compressor in order to make the models fit into standard forms suitable for optimisation and MPC. In the latter only one decision variable is used in the objective function and the power ∗ Author to whom correspondence may be addressed. E-mail address: togho@vestas.com Can. J. Chem. Eng. 9999:1–8, 2012 © 2012 Canadian Society for Chemical Engineering DOI 10.1002/cjce.21672 Published online in Wiley Online Library (wileyonlinelibrary.com). | VOLUME 9999, 2012 | | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | 1 |