## 2 Supermarket Refrigeration Systems
### Hybrid Model Predictive Control, Advanced Process Control  
In response to the aforementioned problems that are inherent to decentralized control, a centralized Hybrid Model Predictive Control, HMPC, has been proposed in [7] but maintaining in the low level only the PI controller for suction pressure (2). HMPC uses a rigorous non-linear continuous-time model to describe the dynamic of the refrigeration system. The main emphasis in the modeling is on the suction manifold and the display cases, which contain the most relevant dynamics for controlling them. The model considers the features of the refrigerant and the compressor. The compressor's capacity in most refrigeration systems is a discrete value, as compressors can either be switched on or off. In addition, the system dynamics based on first principles are considered as well as the manipulated variables that correspond to the opening/closing of the valves in each display case. A new parameterization converts these integer decision variables into real decisions \(T_{on,i}^{k}\) and \(T_{off,i}^{k}\), that is, the time instants when the integer variable \(I\) changes its value. This approach allows us to use conventional non-linear optimization techniques (in terms of real variables) instead of mixed-integer non-linear programming, decreasing the complexity of the optimization procedure and saving computation time. The following variables are considered from the standpoint of system control, where \(i\) refers to a specific display case, and \(n\) is the total number of display cases:  
* The _controlled variables_ are, respectively, the suction pressure (\(P_{suc}\)) and the air temperature (\(T_{air,j}\)) in each display case (\(i=1,\,\ldots,\,n\)). They are measured, as indicated in Figure 1, for each display case. * The on/off _manipulated variables_ are the opening/closing of the inlet valves of the display cases. * The _non-measured disturbances_ are the heat load on the display cases (\(Q_{airload}\)) and the variation in mass of goods (\(M_{goads,i}\)) inside them. * The temperature of the goods (\(T_{goads,i}\)) in each display case are _non-measured variables_ that are controlled indirectly using air temperature. * Several _constraints_ apply to suction pressure (\(P_{suc}\)) and to air temperatures (\(T_{air,i}\)). In the structure of Figure 1, composed of 3 display cases, a non-linear model has to be solved that has 888 variables and 67 equations in the case of two compressors (16 of which are differential equations). In the case of three compressors, there are 889 variables and 69 equations of which 16 are also differential equations. The model describes the behaviour of the system at each future point, based on the present situation, \(t=0\). The future action of the valves (open/close or no change) is based on the mean expected behaviour of the controlled variables up until a temporal horizon \(T_{p}\). The objective is to maintain the controlled variables \"close\" to the reference value, minimizing a cost function \(J\), defined by:  
\[J=\int_{0}^{T_{p}}\Bigg{(}\alpha_{1}\Big{(}P_{suc}-P_{suc}^{ref}\Big{)}^{2}+ \sum_{i=1}^{n}\alpha_{1+i}\bigg{(}\Big{(}\hat{T}_{air,i}^{max}-T_{air,i}^{ref max}\Big{)}^{2}+\Big{(}\hat{T}_{air,i}^{min}-T_{air,i}^{refmin}\Big{)}^{2} \bigg{)}\Bigg{)}dt, \tag{3}\]  
where, \(P_{suc}^{ref}\) is the reference value of the suction pressure in the controller PI, \(T_{air,i}^{ref,\,\,min}\) and \(T_{air,i}^{ref,\,\,max}\) are the minimum and maximum acceptable values for the temperature in each display case, and \(\alpha_{\rm i}\) (\(i=1,\,\ldots,\,1+n\)) is the weight of each addend in Equation (3). Although HMPC does not manipulate the compressors, the first term of Equation (3) penalizes the difference between the suction pressure and its reference. Each of the terms that constitute the second part seeks to maintain a stable periodical pattern of air temperature in the display cases.