Supermarket refrigeration system
|Supermarket refrigeration system|
|Discrete control functions:||3|
|Interior point equalities:||9|
For almost everywhere the mixed-integer optimal control problem is given by
Here the differential state describes the suction pressure in the suction manifold (in bar). The next three states model temperatures in the first display case (in °C). is the goods' temperature, the one of the evaporator wall and the air temperature surrounding the goods. then models the mass of the liquefied refrigerant in the evaporator (in kg).
describe the corresponding states in the second display case.
describes the inlet valve of the first display case, respectively the valve of the second display case. denote the activity of a single compressor.
The following polynomial functions are used in the model description originating from interpolations:
These fixed values are used within the model for the day scenario. A night scenario is also available, see Variants.
|3000.00||Disturbance, heat transfer from outside the display case|
|0.20|| Disturbance, constant mass flow of refrigerant
from unmodeled entities
|200.00||Mass of goods|
|1000.00||Heat capacity of goods|
|300.00|| Heat transfer coefficient between goods
|260.00||Mass of evaporator wall|
|385.00||Heat capacity of evaporator wall|
|500.00|| Heat transfer coefficient between air and
|50.00||Mass of air in display case|
|1000.00||Heat capacity of air|
|4000.00|| Maximum heat transfer coefficient between
refrigerant and evaporator wall
|40.00|| Parameter describing the filling time of the
evaporator under opened valve
|10.00||Superheat in the suction manifold|
|1.00||Maximum mass of refrigerant in evaporator|
|5.00||Total volume of suction manifold|
|0.08||Total displacement volume|
For the relaxed problem (we only demand instead of the optimal solution is 12072.45. The illustrated solution with integer controls has a (suboptimal) objective function value of 12252.81.
Model descriptions are available in
- C code at Supermarket refrigeration system (C)
- optimica at Supermarket refrigeration system (optimica)
Since the compressors are parallel connected one can introduce a single control instead of two equivalent controls. The same holds for scenarions with parallel connected compressors.
In the paper <bibref>Larsen2007</bibref> mentioned above, the problem was stated slightly different:
- The temperature constraints weren't hard bounds but there was a penalization term added to the objective function to minimize the violation of these constraints.
- The differential equation for the mass of the refrigerant had another switch, if the valve (e.g. ) is closed. It was formulated this way:
This additional switch is redundant because the mass itself is a factor on the right hand side and so the complete right hand side is 0 if .
- A night scenario with two different parameters was given. At night the following parameters change their value:
Additionally the constraint on the suction pressure is softened to .
- No periodicity was required but the solution on a fixed time horizon 4 hours - 2 in day scenario and 2 in night scenario - with was asked.
- The number of compressors and display cases is not fixed. Larsen also proposed the problem with 3 compressors and 3 display cases. This leads to a change in the compressor rack's preformance to . Unfortunately this constant is only given for these two cases although Larsen proposed scenarios with more compressors and display cases.