Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
ThC20.5
Hybrid Control of a Supermarket Refrigeration Systems Daniel Sarabia, Flavio Capraro, Lars F. S. Larsen, and César de Prada Abstract—This paper presents a non-linear model predictive control (NMPC) of a supermarket refrigeration system. This is a hybrid process involving switching nonlinear dynamics and discrete events, on/off manipulated variables, like valves and compressors, continuous controlled variables like goods temperatures and finally, several operation constraints. The hybrid controller is based on a parameterization of the on/off control signals in terms of time of occurrence of events instead of using directly binary values, on this way, we can reformulate the optimization problem as a NLP problem. A rigorous model of a real supermarket refrigeration system provided by Danfoss is presented as well as results of the hybrid controller operating on it. The paper describes the hybrid process, presents the control problem formulation and provides some results of the proposed approach and comparisons with the traditional control.
A
I. INTRODUCTION
supermarket refrigeration system consists of a central compressor bank that, besides other process units, maintains the required flow of refrigerant to several refrigerated display cases located in the supermarket sales area. Each display case operates with an inlet on/off refrigerant valve to keep the air temperature in the display case within a range, in order to ensure a high quality of the goods. For many years, the control of supermarket refrigeration systems has been based on standard control systems. In particular, each display case is equipped with an independent hysteresis controller that regulates the air temperature in it by manipulating the inlet valve. The major drawback, however, is that the control loops are vulnerable to self-inflicted disturbances caused by the interaction between the distributed control loops.
Manuscript received September 22, 2006. This work was supported in part by the Spanish Ministry of Education and Science (former MYCT) through project DPI2003-0013, as well as the European Commission through VI FP NoE HYCON, Contract no.: 511368. D. S. is with the Department of Systems Engineering and Automatic Control, University of Valladolid, Valladolid, c/ Real de Burgos s/n, 47011, Spain (phone: 34-983-423162; fax: 34-983-423161; e-mail:
[email protected]). F. C. is with Instituto de Automática (INAUT), Universidad Nacional de San Juan, Avda. San Martín 1109 (oeste), 5400, San Juan, Argentina (email:
[email protected]). L. F. S. L. Author is with Central R&D – RA – DP, Danfoss A/S, Nordborg, Denmark (e-mail:
[email protected]). C. P. is with the Department of Systems Engineering and Automatic Control, University of Valladolid, Valladolid, c/ Real de Burgos s/n, 47011, Spain (e-mail:
[email protected]).
1-4244-0989-6/07/$25.00 ©2007 IEEE.
In particular, practice and simulations show that the distributed hysteresis controllers have the tendency to synchronize [1] and [2], meaning that the opening and closing actions of the valves coincide. Consequently, the compressor periodically has to work harder to keep up the required flow of refrigerant, which results in low efficiency, inferior control performance and a higher wear. The use of other control techniques, such as MPC, is significantly complicated by the fact that many of the control inputs are restricted to integer values, such as the opening/closing of the inlet valves and the on/off of the compressors. Furthermore, the system features switched non linear dynamics, including also slow and fast dynamics and several constraints on controlled variables. As a result, classical control does not fit very well with the overall operation of this kind of process and other approaches must be considered. These processes that involves continuous as well as on/off variables and certain logic of operation can be formulated as hybrid systems where continuous variables are represented by real variables and the logic and on/off variables can be converted into binary variables in a set of inequalities, using for example the Mixed Logical Dynamical (MLD) framework [3]. This leads, in the context of Model Predictive Control (MPC), to a mixed integer optimization problem that must be solved every sampling time, see, for instance, [4] and [2]. This kind of formulation in the discrete time domain using linear models allows for a systematic controller design, but in practice, due to the high dimensionality of the problem, except in a limited number of cases, it is not possible to solve the associated MILP or MIQP optimization problem on-line. Additionally, linear models may not provide an adequate representation of the non-linear dynamics that takes place in refrigeration systems which convert the optimization problem in a MINLP one, not suitable for real-time control. This paper presents an alternative approach of the hybrid controller where the on/off actions are formulated in terms of time instants of occurrence of events, instead of as binary values on each sampling time. In contrast with conventional MPC, the non-linear internal model of the controller is described in continuous time and includes the logic of operation. This approach has the advantage that all decision variables are of real type, which means that the associated optimization problem is NLP and can solve this kind of non-linear hybrid problems more efficiently.
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The controller uses the framework of non-linear model predictive control, using a nonlinear continuous-time model to make the predictions and calculating the control actions by means of an optimization problem solved on-line every sample time. The paper is organized as follows: In section II the process is described while section III presents the nonlinear model obtained from first principles. Section IV gives an overview of the traditional control systems of supermarket refrigeration systems and section V and VI describe the control objectives and the hybrid model predictive control respectively. Results of some test for evaluation of the proposed approach are given in section VII showing the process response to setpoint changes, disturbances, etc. Finally, the paper ends with some conclusions and references.
evaporates while absorbing heat from the surrounding air circulating through the evaporator. The resulting air flow creates an air-curtain at the front of the display case. The fact that the air-curtain is colder than the goods leads to a heat transfer from the goods Qgoods-air and (as a side effect) from the surrounding Qairload to the air-curtain. Inside the display cases, a temperature sensor is located, which measures the air temperature Tair close to the goods. This temperature measurement is used in the control loop as an indirect measure of the goods’ temperature Tgoods. Furthermore, an on/off inlet valve is located at the refrigerant inlet of the evaporator, which is used to control the temperature in the display case.
II. SUPERMARKET REFRIGERATION SYSTEMS. In a typical supermarket, some of the goods need to be refrigerated to ensure preservation for consumption. These goods are normally placed in open refrigerated display cases that are located in the sales area for self service. A simplified supermarket refrigeration circuit with two display cases is shown in Fig. 1. The heart of the system is the compressor rack, which consist of a number of compressors connected in parallel. They supply the flow of refrigerant to the system by compressing the low pressure refrigerant from the suction manifold, which receives the vapors returning from the display cases. The compressors keep a certain constant pressure in the suction manifold, thus ensuring the desired evaporation temperature. From the compressors, the refrigerant flows to the condenser and further on to the liquid manifold. The evaporators inside the display cases are fed in parallel from the liquid manifold through an expansion valve. The outlets of the evaporators lead to the suction manifold and back to the compressors thus closing the circuit.
Fig. 2. Cross section of a refrigerated display case.
III. NONLINEAR CONTINUOUS-TIME MODEL In this section, the main features of the nonlinear continuous-time model are presented. More details can be found in [1]. The model of the supermarket refrigeration system is composed of individual models for the display cases, the suction manifold, the compressor rack and the condensing unit. The main emphasis in the modeling is laid on the suction manifold and the display cases which contain the most relevant dynamics for controlling the display cases. A. The Display Cases The dynamic in each the display case i can be described by three states: the goods temperature (Tgoods,i), the temperature of the evaporator wall (Twall,i) and the mass of liquefied refrigerant in the evaporator (Mref,i). The model of the display cases encompasses three parts, namely: the goods, the evaporator and the air-curtain in between (see Fig. 2). By setting up the energy balance for the goods and the evaporator, the following two state equations can be derived, assuming a lumped temperature model.
Fig. 1. A simplified layout of a supermarket refrigeration system.
In Fig. 2, a cross section of an open display case is depicted. The refrigerant is led into the evaporator located at the bottom of the display case, where the refrigerant
dTgoods ,i
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dt
=−
UAgoods − air M goods ,i Cp goods ,i
(T
goods , i
− Tair ,i
)
(1)
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dTwall ,i dt
= −
UAair − wall (Tair ,i − Twall ,i ) M wall Cp wall UAwall − ref ,i ( M ref ,i ) M wall Cp wall
(T
wall , i
− Te,i
)
(2)
where UA is the heat transfer coefficient with the subscript denoting the media between which the heat is transferred, M denotes the mass, and Cp the heat capacity, where the subscript also denotes the media (wall, air and refrigerant “ref”) the parameter refers to. Moreover, Te is the evaporation temperature. Te(Pe) is a refrigerant dependent function of the evaporation pressure Pe Assuming that there are no pressure drop in the suction line, the suction pressure Psuc is equal to the evaporation pressure (Pe,i = Psuc). As indicated in (2), the heat transfer coefficient between the refrigerant and the evaporator wall (UAwall-ref,i) is a function of the mass (Mref,i) of the liquefied refrigerant in the evaporator i. It is assumed here that this relation can be described by the following linear function: UAwall − ref ,i ( M ref ,i ) = UAwall − refrig,max
M ref ,i M ref,max
(3)
where Mref,i = Mref,max when the evaporator is completely filled with liquid refrigerant. The accumulation of refrigerant in the evaporator is described by equation (4)
dM ref ,i dt
⎧ 0 ⎪ ⎪ Q = ⎨− e ⎪ Δhlg ⎪⎩ 0
(4)
if u disp ,i = 1
(6) if u disp ,i = 0
Finally, the air temperature (Tair,i) for each display case i can be found by setting up an energy balance for the air-curtain. Tair ,i =
Qairload + Tgoods ,iUAgoods − air + Twall ,iUAair − wall
(UA
goods − air
+ UAair − wall
)
(7)
where Qairload is a given external heat load on the air- curtain and it is assumed equal for all display cases. B. The Suction Manifold The dynamic in the suction manifold is modelled by one state, namely the suction pressure (Psuc). Setting up the mass balance, and assuming an ideal gas, the corresponding state equation is obtained:
hin − suc f in − suc − hout − suc Fcomp ρ suc dPsuc =R dt CvVsuc
(8)
by the compressors, f in − suc = ∑i =1 f i is the total mass flow nd
from
if u disp ,i = 0 and M ref ,i = 0
the
display
cases
to
the
suction
manifold,
hin − suc = ∑i =1 f i hoe is the inlet enthalpy to the suction nd
where Δhlg is the specific latent heat of the remaining liquefied refrigerant in the evaporator, which is a nonlinear function of the suction pressure. Qe,i is the cooling capacity which can be found by setting up the energy balance for the evaporator: Qe,i = (Twall ,i − Te )UAwall − ref ,i ( M ref ,i )
⎧ Qe,i ⎪ − hie ⎪h f i = ⎨ oe Qe,i ⎪ ⎩⎪ Δhlg
where Vsuc is the total volume of the suction manifold, Fcomp is the volume flow out of the suction manifold determined
if u disp ,i = 1 if u disp ,i = 0 and M ref ,i > 0
enthalpy which is a nonlinear function of the suction pressure and the superheat Tsh.
(5)
The amount of liquid refrigerant in the evaporator Mref,i follows a switched nonlinear dynamic governed by (4). It is assumed that the evaporator i is filled instantaneously when the valve i is opened (udisp,i = 1). Furthermore, when the valve is closed (udisp,i = 0) and all of the enclosed refrigerant has evaporated (Mref,i = 0), then dMref,i/dt = 0. So, the on/off manipulated variables are udisp,i (i=1,2,..., nd), where nd is the total number of display cases. When the refrigerant leaves the evaporator it flows on to the suction manifold. The mass flow fi out of the evaporator is given by equation (6). Where hie is the inlet enthalpy which is a nonlinear refrigerant specific function of the condensing pressure and the subcooling and hoe is the outlet
manifold which is a refrigerant specific non linear function of Psuc and the superheat Tsh, nd is the total number of display cases, and fi is the mass flow of the refrigerant out of the i’th display case. fi is determined by position of the valve and the amount of enclosed refrigerant when the valve is closed, given by (6). ρsuc is the density of the refrigerant in the suction manifold, which is a non-linear refrigerant dependent function of the suction pressure and the superheat in the manifold. hout-suc is assumed equal to hoe, the specific enthalpy at the outlet of the evaporator.
C. The Compressor In most refrigeration systems, the compressor capacity is discrete-valued, as compressors can be switched only either on or off. Let nc denote the total number of compressors. The compressor bank is modeled using a constant volumetric efficiency ηvol and the maximal displacement volume Vsl. Thus, the volume flow Fcomp,i out of the suction manifold created by the i’th compressor can be determined by (9) where Ccomp,i is the i’th compressor capacity
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( ∑i =1 C comp ,i = 100 ), and Fcomp = ∑i =1 Fcomp ,i is the total nc
nc
volume flow, finally ucomp,i is the on/off manipulated variable associated with each compressor i (i=1,2,..., nc). Fcomp ,i = u comp ,i C comp ,i
1 η volVsl 100
i = 1, 2,....., nc
(9)
D. Refrigerant Properties In the model described before, a number of refrigerant depended properties are used: Te, Δhlg, hie, hoe and ρsuc. These properties can be computed by using the freeware software package “RefEqns” [5] available for language C, C++, Matlab and others. The corresponding libraries in C, RefrigC.lib and RefrigC.dll, have been incorporated in our simulation of the process. Table I shows the functions used to calculate these properties. For a more detailed description of these thermodynamics properties, and the explanation of the vapor compression cycle in a h-log(p) diagram, see [1].
very similar switching frequencies. Therefore, the valves have a tendency to synchronize leading to periodic high and low flow of evaporated refrigerant into the suction manifold thus creating large variations in the suction pressure and a high wear on the compressors. These effects are displayed in the Fig 3 a) and 3 b). The suction pressure controller is normally a Proportional-Integrator controller (PI-controller) with a dead band (±DB) around the reference of the suction pressure.
TABLE I REFRIGERANT DEPENDED PROPERTIES Property Property Te = TBubP(Psuc) hoe = HTP(Te+Tsh,Psuc) Δhlg = RT(Te+273.15) ρsuc = 1/VTP(Te+ Tsh,Psuc)
IV. TRADITIONAL CONTROL The control systems used in today’s supermarket refrigeration systems are decentralized, as previously mentioned. Each of the display cases is equipped with a temperature controller and a superheat controller that controls the filling level of the evaporator. The compressor rack is equipped with a suction pressure controller. Furthermore, the condenser is equipped with a condensation pressure controller, and on top, various kinds of supervisory controllers may be used to help adjusting the set-points. The Tair in the display cases are controlled by a hysteresis controller that opens and closes the inlet valve. This means that when Tair reaches a certain upper temperature bound max ( Tair ) the valve is opened and Tair decreases until the lower
temperature bound ( Tairmin ) is reached and the valve is closed again. When the valve is closed, the air temperature continues to decrease below the lower bound creating an “undershoot”. This is caused by two things: First, the remaining refrigerant contained in the evaporator evaporates, and secondly, the thermal capacity of the evaporator wall adds an extra order to the system dynamics, see Fig. 3 a), where Tairmax = 5º C and Tairmin = 0º C . This figure depicts a simulation result with the traditional control setup on a system with two display cases and two compressors. In the supermarket, many of the display cases are alike in design and they are working under uniform conditions. As a result, the inlet valves of the display cases are switched with
Fig. 3. a) Effects of synchronization on air temperature using traditional control. b) The effect of this synchronization in the suction pressure and c) The compressor capacity. The simulation of 180 min. has been made for two display cases and two compressors with the same capacity 50%.
V. CONTROL PROBLEM The control challenge of the supermarket refrigeration system is to development an efficient and optimal control strategy that presents better performance than the traditional control presented in section IV. The controlled variables are, respectively, the suction pressure Psuc and the air temperature Tair,i in each display case (i=1,..., nd). They are measured as indicated in Fig. 1. The on/off manipulated variables are the start/stop of compressors (ucomp,i ∀ i=1,..., nc) and the opening/closing of the inlet valves (udisp,i ∀ i=1,..., nd). The non measured disturbances are the heat load on the display cases (Qairload) and the variation of mass of goods (Mgoods,i) inside the display cases. The goods temperature Tgoods,i in each display case are non measured variables that are controlled indirectly using the air temperature .
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Several constraints apply on suction pressure Psuc and on air temperatures Tair,i. VI. HYBRID MPC A natural approach to many decision problems is Model Predictive Control (MPC): An internal model of the process is used to predict the future controlled variables behavior as a function of the present and future control actions, which are selected in order to minimize some performance index or cost function. The optimal control signals corresponding to the present time are applied to the process and the whole procedure is repeated in the next sampling period. In our case, the performance index to be minimized (according to the control objectives describe in section V) is the function nd T ⎛ ⎞ ref 2 J = ∫ ⎜⎜ α 1 ( Psuc − Psuc ) + ∑ α 1+ i (Tair ,i − Tairref,i ) 2 ⎟⎟ dt 0 i =1 ⎝ ⎠ p
(10)
ref where Psuc and Tairref,i are the set points of the controlled
variables (suction pressure and temperatures of each display case i), αi (i=1,...,1+nd) are appropriate weight factors and Tp is the prediction time. Moreover, besides the internal model, several constraints can be taken into account as the ones imposed by the range of the controlled and manipulated variables: min max Psuc ≤ Psuc ≤ Psuc
Tairmin,i ≤ Tair ,i ≤ Tairmax,i ∀ i = 1,..., nd u i ∈ {0,1}
∀ i = 1,..., nd , nd + 1,..., nd + nc
(11) (12)
In the following, for simplicity, we will unify udisp,i and ucomp,i into a single variable ui, with i ranging from 1 to nd+nc, so that the first nd variables are associated with the display cases, and the second nc variables are associated with the compressors.
A. The Internal Model The internal model used in the MPC controller is similar to the continuous one based on first principles presented in section III. However, it is formulated in a different way. Notice that, in (10), the manipulated variables ui are integers, that correspond to the opening/closing of the valves in each display case udisp,i(t) and the start/stop of each compressor ucomp,i(t) at every sampling time in the control horizon. The optimization of (10) implies then solving a MINLP optimization problem every sampling time which is not possible in real time. Nevertheless, we can avoid the use of these integer variables by means of a new parameterization converting the integer decision variables into real decision ones that correspond to the time instants of the switching. This approach allows using conventional non
linear optimization techniques (in terms on real variables) instead of mixed-integer non linear programming, decreasing the complexity of the optimization procedure and saving computation time.
u 1(t)
pulse 1
pulse 2
1 off
···
on
0
T1 off1 T1 on1
pulse k
T 2off1 T2on1
···
Tkoff1 Tkon1
Fig. 4. Parameterization of integer decision variable u1(t). Definition of the pulse and the new real decision variables Tkon1 and Tkoff1 along the time.
The parameterization consists on define two new real variables Tkon,i(t) and Tkoff,i(t) (k∈ℕ ) for each integer manipulated variable ui (i=1, ...,nd, nd+1, ..., nd+nc). Tkon,i and Tkoff,i denote the duration of the integer variable ui when this variable is one and when it is zero respectively in each pulse k, see Fig. 4. The times Tkon,i and Tkoff,i are unknown variables that must be estimated along the prediction horizon.
B. Predictions and Control Horizons The prediction horizon (N2) in MPC corresponds to the future time interval used to compute predictions with the internal model. In standard MPC, it is chosen longer than the process settling time. Notice that, in our framework of on/off manipulated variables, the process will never settle in a stationary point, but will oscillate eventually reaching a stable operational pattern. So, instead, the prediction horizon will be translated into Np, number of pulses of the slower integer manipulated variable ui, assumed to be required for obtaining a stable operational pattern. This means that, in order to compute the cost function (10), the internal model will be integrated until the slowest train of pulses completes Np pulses. Notice that each integer manipulated variable can perform a different number of pulses in the same period of time Tp. Fig. 5 displays an example with two integer variables u1 and u2 and Np=4 where the prediction horizon is the time needed by u2 to implement four full pulses. We can observe, that u1 makes more pulses (5 pulses). The concept of control horizon (Nu) in continuous MPC corresponds to the time interval where present and future control actions are computed. After Nu sampling times, the controller maintains the last control signal computed at Nu until the end of the prediction horizon N2 in order to compute the output predictions. In the hybrid environment, the control horizon Nb,i corresponds to the number of pulses of each manipulated variable ui for which the controller compute the decision variables Tkon,i and Tkoff,i, (on and off periods of every pulse (k=1,2,3,…)). From the pulse number
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Nb,i until the end of the prediction horizon Tp, the values of the decision variables Tkon,i and Tkoff,i will be taken equal to the ones of the pulse Nb,i, so, the schedule of ui will be forced to follow the pattern of the Nb,i pulse, assuming that the system should reach a stationary schedule, in a similar way to how the future control signals are treated in continuous MPC. In the example considered in Fig. 5, The variable u1 has Nb,1=1, so, only the times T1on,1 and T1off,1, that correspond to the first pulse will be computed and the same pattern will be applied in the following ones, that is to say, Tkon,1 = T1on,1 and Tkoff,1 = T1off,1 ∀ k > Nb,1 = 1. For u2 has been assigned Nb,2=2, so, the controller compute T1on,2, T1off,2, T2on,2 and T2off,2 and from the second pulse on, follow the pattern defined by the last T2on,2 and T2off,2 times computed, so Tkon,2 = T2on,2 and Tkoff,2 = T2off,2 ∀ k > Nb,2 = 2. u1(t)
pulse 1
pulse 2
pulse 3
pulse 4
pulse 5
1
(implemented in a commercial library NAG for C) using a schematic like the one of Fig. 6, that corresponds to a sequential approach where the cost function J (10) is computed by integration of the dynamical internal model (1)-(9). The simulation package integrates the internal model equations along the prediction horizon Np taking as initial conditions the current process state and evaluating the formulated cost function J at the end of the integration. The internal model was simulated using the EcosimPro simulation language [7] which allows combining DAE (Differential Algebraically Equations) with events, while performing a correct integration in spite of the model discontinuities. It belongs to the category of the so called “modeling languages” that are object oriented, allows an easy re-use of the components and generates automatically C++ code corresponding to the simulation. This code of the internal model is embedded in the controller, which has been programmed in C++.
0
T2off1 T2on1
···
T5off1 T5on1
CONTROLLER
Nb1=1 u2(t)
pulse 1
pulse 2
pulse 3
Internal model: DAE equations
pulse 4
1
Tkon,i Tkoff,i
0
T1on2 T1off2
T2on2
T2off2 Nb2=2
Start prediction
···
T4on2
(15) (17) (11) (13)
Np=4
∑i 1
nd + nc =
T
≤ Ton ,i ≤ T
T
≤ Toff ,i ≤ T
Tkon,i Tkoff,i
ui(t)
Process
y(t)
Fig. 6. Non-linear MPC controller implementation.
From a technical point of view, it is necessary to add a real/discrete converter between the controller and the process. Hybrid MPC calculates the exact time instants Tkon,i and Tkoff,i of changes for each on/off manipulated variable, and the converter transform these times in a discrete manipulated signal ui, when the events will occur. That is, the application of the on/off signals is decoupled from the sampling time.
2 × N b ,i number of real decision
variables that are subjected to lower and upper limits: max on , i max off , i
(10) J (14)
End prediction
C. NMPC Controller The aim of the hybrid MPC is to minimize (10) under the dynamics (1)-(9), static constraints (11) with respect to the decision variables Tkon,i and Tkoff,i (k=1,..., Nb,i and i=1,.., nd, nd+1,..., nd+nc) corresponding to the periods when every integer manipulated variable ui have to be on and off respectively along the control horizons. The optimization problem involves
Objective Function to minimize
Constraints Non-linear Optimizer min J
T4off2
Fig. 5. Prediction (Np) and control horizons (Nb1, Nb2) for integer variables u1 and u2 in hybrid MPC.
min on , i min off , i
Setpoints
Simulator Package
Converter
T1off1 T1on1
∀ i = 1,..., nd , nd + 1,..., nd + nc
(13)
In our approach, the path constraints (11) are not included explicitly in the optimization problem, but, they are considered in the cost function (10), when we penalize the error between references and controlled variables. All terms in the cost function are normalized. The resulting optimization problem is solved periodically, with a given sampling period using an SQP algorithm [6]
VII. SIMULATIONS AND CONTROL RESULTS The process model described in section III has been simulated and tested using also the simulation language EcosimPro. To illustrate the behavior and performance of the hybrid MPC, we have made an experiments of 24 hours (86400 sec., from 0:00 to 24:00) of simulation of a supermarket refrigeration system. This supermarket is composed by two compressor (with the same capacity Ccomp,1=Ccomp,2=50%) and two equally sized display cases. We have strong non-measured disturbances on external heat load (Qairload), being great when the supermarket is open (from 9:00 to 21:00) and more people are present in the sales area, see Fig. 7. When the supermarket opens (at 9:00)
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and at the beginning of the evening (at 13:00), the display cases are loaded with 400 Kg of goods at 3ºC of temperature, see Fig. 7 and Fig. 8. The mass of goods in both display cases follow different patterns along the day, see Fig. 7. The nominal values for these disturbances are Qairload=900 J/s and Mgoods,1=Mgoods,2=400 Kg. Notice that, the internal model uses always these constant values to make the predictions because they are nor measured. On the other hand, at the beginning of each prediction, the internal model fix in 3ºC the initial value for the non measured states, the temperature of goods (Tgoods,1 and Tgoods,2). The heat capacity of the goods are equal in both display cases (Cpgoods,1=Cpgoods,2=1000 J/(Kg K)). Other parameters values and constants used in the non linear model are shown in Table II. Zoom (from 6:00 to 9:00)
the prediction horizon was fixed in Np=3, which corresponds to predictions of around half an hour (1800 sec.), and the control horizons for every manipulated integer variable were selected as Nb,1=Nb,2=Nb,3,=Nb,4=2. So the optimization problem has 16 decision variables. Finally, the control parameters (setpoints and maximum, minimum values permitted for controlled and decision variables) and weights αi in cost function (10) are given in Table. III.
maximum minimum setpoint weight
TABLE III CONTROL PARAMETERS OF HYBRID MPC Ton,i (s) Toff,I (s) T Psuc Tair,1 (ºC) air,2 (bar) (ºC) valves 4.4 4 4 100 600 4.0 0 0 10 100 4.2 2 2 1 1 1 -
Ton,i (s) Toff,i (s) compressors 600 600 60 60 -
The evolution of the controlled variables, suction pressure and air temperatures and its limits are shown in Fig. 9. We can observe the good behavior and small range of these variables in presence of disturbances, being always within the imposed limits. The temperature of goods (the non measured states), which are actually the real variables to be controlled, are around 2ºC (the set point for both air temperatures), see Fig. 8.
Open supermarket (from 9:00 to 21:00)
Fig. 7. Non-measured disturbances: the external heat load, and the amount of goods in each display case.
Zoom (from 6:00 to 9:00) Open supermarket (from 9:00 to 21:00) Fig. 9. Controlled variables, suction pressure and air temperature in both display cases. The upper and lower limits of these variables are also shown.
Fig. 8. Non-measured state, the temperature of goods in each display case. TABLE II PARAMETERS VALUES FOR THE NONLINEAR MODEL Component Parameter Value Parameter Value Display UAgoods-air 60 [W/K] Mwall 180 [Kg] 360 [W/K] Mref,max 0.6 [Kg] UAair-wall case τfil 40 [sec] UAwall-ref,max 900 [W/K] Cp,wall 385 [J/(Kg K)] Tsh 10 [K] Vsuc Suction 5 [m3] R 85 [J/(KgK)] 649 [J/(Kg K)] hie 209661.5 [] Cv manifold Compressor Vsl 6 [m3/h] η 0.8 Refrigerant Thermo dynamical properties of R404a
Regarding the controller, sampling time was set at 60 sec.,
A zoom of 180 min. (from 6:00 to 9:00) of the simulation in the nominal case for disturbances is display in Fig. 10. We can compare these results with the traditional control ones in the same conditions (see Fig. 3), and we can view the drastic decrease of amplitude of the oscillations of all variables and what is more important, that the synchronization effect on air temperature never occurs (see Fig. 10 lower graph). Fig. 11 shows the manipulated on/off variables: opening/closing the valves and star/stop the compressors from 6:00 to 9:00 (for the nominal conditions). We can see
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ThC20.5
the perfect periodicity in all signals. This is a direct effect produced by the policy used to force repeated pulses at the end of the predictions (see section VI B). In this approach, we need to make more starts/stops of the compressors than the traditional case, but also, we can observe that compressors never work at the same time, so, for this supermarket and for the control objectives presented before, it is possible use only one compressor.
see [2]), making possible the implementation of this controller to larger refrigeration systems. Another advantage is the decoupled of the manipulated variables from the sampling time, this leads to the possibility to increase the sampling time, but the control signals are always applied in it corresponding exact instant calculated by the hybrid controller. However, for future works it is interesting try to decrease the time consumed by the optimization algorithm to find the minimum. In our approach this time is around 40 seconds every sampling time (we have a sampling time of 60 seconds) but a MPC implementation with MLD models, see [2], the optimization procedure needs only 9.7 seconds to find the minimum. The differences between both procedures are very significantly. On the other hand, in our approach, the start/stop of the compressors is very frequently, because, we are assuming and imposing that all manipulated variables follow a periodical behavior. For the compressors this can not be a desired behavior, and it is important to investigate and try to improve this result, in order to minimize these switching times. REFERENCES [1]
Fig. 10. A zoom of the controlled variables, suction pressure and air temperature in both display cases corresponding with a 180 min. of simulation in nominal conditions (from 6:00 to 9:00).
[2] [3] [4]
[5] [6] [7]
Fig. 11. A zoom of the manipulated variables, valves and compressors corresponding with a 180 min. of simulation in nominal conditions.
VIII. CONCLUSION The control approach followed in this paper presents good results on a realistic simulation under disturbances of a supermarket refrigeration systems consisting in two display cases and two compressors. In this approach, to add more devices (display cases or compressors) implies an increase on the decision variables but not in an exponential form (like in MPC with MLD models and integer decision variables, 4185
L. F. S. Larsen, “Model Based Control of Refrigeration Systems,” PhD thesis, Department of Control Engineering, Aalborg University, Aalborg, Denmark, 2007. L. F. S. Larsen, T. Geyer and M. Morari, “Hybrid MPC in supermarket refrigeration systems,” in 16th IFAC World Congress, Prague, Czech Republic, 2005. A. Bemporad and M. Morari, “Control of systems integrating logic, dynamic, and constraints,” Automatica, vol. 35, 1999, pp 407-427. C. de Prada, D. Sarabia and S. Cristea, “Modelling and Control of Four Tanks Mixed Logic Dynamical (MLD) System,” Dynamics of Continuous, Discrete, and Impulsive Systems B: Applications and Algorithms, vol. 12 (2), 2005, pp 243-264. Skovrup, M. J. “Thermodynamic and Thermophysical Properties of Refrigerants”. Software package in Borland Delphi, Version 3.00. C. T. Lawrence and A. L. Tits, “A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm,” SIAM J. Optimization, vol. 11 no. 4, 2001, pp 1092-1118. EA Int. (1999). EcosimPro User Manual. Available: http://www.ecosimpro.com.
