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Assessment of the controlling envelope of a model-based multivariable controller for vapor compression refrigeration systems
Applied Thermal Engineering 30 (2010) 1538e1546
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Assessment of the controlling envelope of a model-based multivariable controller for vapor compression refrigeration systems Leonardo C. Schurt a, Christian J.L. Hermes a, *, Alexandre Trofino Neto b a b
POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil Department of Automation and Systems Engineering, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 February 2010 Accepted 19 February 2010 Available online 3 March 2010
This paper explores the controlling characteristics of a first-principles model-based controller specially developed for vapor compression refrigeration systems. Mathematical sub-models were put forward for each of the system components: heat exchangers (condenser and evaporator), variable-speed compressor and variable-orifice electric expansion device. The dynamic simulation model was then used to design a multivariable controller based on the linear-quadratic-Gaussian technique using a Kalman filter for the estimator design. A purpose-built testing apparatus comprised of a variable-speed compressor and a pulse-width modulated expansion valve was used to collect data for the system identification, and model and controller validation exercises. It was found that the model reproduces the experimental trends of the working pressures and power consumption in conditions far from the nominal point of operation (30%) with a maximum deviation of 5%. Additional experiments were also performed to verify the ability of the controller of tracking reference changes and rejecting thermal load disturbances. It was found that the controller is able to keep the refrigeration system running properly when the thermal load was changed from 340 to 580 W (460 W nominal), and the evaporator superheating degree was varied from 9.5 C to 22 C (16.6 C nominal). Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Control Refrigeration system Modeling Experimentation Controlling envelope
1. Introduction Household and commercial refrigerators consume by 10% of the electrical energy produced worldwide, a figure that has motivated both customers and governments to push the manufacturers for high efficiency products. However, in spite of the large effort put in refrigeration systems advancements in the past years, just a modest effect has been observed suggesting that the conventional vapor compression refrigeration technology is reaching its limits. Vapor compression refrigeration systems usually comprise a single-speed compressor, a fixed-orifice expansion device, two heat exchangers (i.e., the condenser and the evaporator), and a volatile working fluid (named refrigerant) that undergoes a reversed Rankine thermodynamic cycle. Furthermore, the temperature of the refrigerated compartment is controlled by a thermostat that switches the refrigeration system on and off according to a cycling pattern. It has been advocated in the open literature that the refrigeration systems with electronic-controlled equipment (i.e., compressor speed, and valve opening, among others) may improve significantly the overall energy performance when compared to the conventional * Corresponding author. Present address: Department of Mechanical Engineering, Federal University of Paraná, P.O. Box 19011, 81531-990 Curitiba, PR, Brazil. Tel.: þ55 41 3361 3239; fax: þ55 41 3361 3129. E-mail address: [email protected] (C.J.L. Hermes). 1359-4311/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2010.02.024
refrigeration appliances. For instance, variable-speed compressors have been employed in the past decades to obtain a continuous matching between the cooling capacity and the thermal load with remarkable gains in energy performance [1,2]. Moreover, it has been also reported [3,4] that the system performance improves further in cases where a variable-orifice expansion valve is additionally employed, as the evaporator is kept fully activated (i.e., flooded) during most of the system runtime. It is noteworthy that such refrigeration systems require proper control strategies to account for the simultaneous operation of a variable-speed compressor and a variable-orifice expansion valve. Albeit there have been in the past years some publications with regard to the analysis and development of control strategies for variable-speed compressors and variable-orifice expansion valves operating simultaneously [5e8], their focus was on large capacity refrigeration applications (>6 kW). Pottker and Melo [4] carried out an experimental study to evaluate the effect of variable-opening electric expansion valves on the performance of variable-speed (VSC) and single-speed (SSC) vapor compression refrigeration systems. A purpose-built breadboard refrigeration apparatus consisting of a variable-speed compressor, an electric expansion valve (EEV), and two secondary flow rate and temperature controlled aqueous solution loops (condensing and evaporating media) was designed and constructed to emulate refrigeration systems with capacities ranging from 0.3 to 1.5 kW
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
Nomenclature
Roman A cross-sectional area, m2 A, B, C, E, W state-space matrices valve opening, % Av D coil inner diameter, m h specific enthalpy, J/kg J cost function K controller gain matrix L length, m L observer gain matrix m mass flow rate, kg/s N compressor speed, s1 NTU number of transfer units p pressure, Pa Q heat transfer rate, W q heatflux, W m2 Q, R weighting matrices T temperature, K t time, s u controllable inputs matrix UA overall conductance, W/K V secondary coolant flow rate, L/min v specific volume, m3/kg w disturbance input matrix
(see Fig. 1). It was found that, independently of the secondary fluid temperature entering the heat exchangers, the VSC/EEV system always operates with a maximum coefficient of performance (COP) once it keeps the evaporator flooded and also delivers a cooling capacity that exactly matches the thermal load. Later, Marcinichen et al. [9] put forward an empirical dual-SISO (single-input, single-output) strategy for the simultaneous control of compressor speed and expansion valve opening using the experimental rig constructed by Pottker and Melo [4]. The control strategy was devised to obtain a maximum COP within a range of cooling capacity from 0.3 to 0.8 kW. The refrigeration system was identified using the step-response method, which provided first order linear models for both the evaporator superheating and the brine outlet temperature. The empirical models were used to derive two singleinput, single-output (SISO) proportional-integral (PI) controllers, one for the evaporator superheating as a function of the EEV opening, and another for brine outlet temperature as a function of the compressor speed. Both controllers were implemented into a dual-SISO control strategy that operated satisfactorily in terms of reference tracking and disturbance rejection. Nonetheless, it was found that the empirical identification has constrained the controller to a region close to the point of operation. In order to design controllers that can be applied to a broad range of operation, the use of first-principles simulation models have been suggested in the literature [10] for the dynamic identification of vapor compression refrigeration systems. Albeit there are several publications concerning the modeling for controlling vapor compression refrigeration systems [11e16], all of them conducted the system identification and model validation exercises for regions close to the point of operation (w5%). Therefore, in a prior study, Schurt et al. [17] put forward a multivariable model-driven controller for vapor compression refrigeration systems. Mathematical sub-models were developed for each of the system components: heat exchangers (condenser and evaporator), variable-speed compressor and variable-orifice
x y
1539
dynamic state matrix output matrix
Greek
a 3ll-sl g l r
heat transfer coefficient, W/m2 K internal heat exchanger effectiveness void fraction boundary position, m specific mass, kg/m
Subscripts c condenser e evaporator, static i inlet, integral K Kalman l saturated liquid o outlet r refrigerant ref reference s secondary coolant, isentropic process sur surroundings v saturated vapor Other symbols hxi volume average of x x_ ¼ dx=dt time derivative of x
expansion valve. The dynamic simulation model was then used to design a MIMO (multi-input, multi-output) controller based on the linear-quadratic-Gaussian (LQG) technique using a Kalman filter for the estimator design. The purpose-built testing apparatus constructed by Pottker and Melo [4] was then used to collect data for the system identification and model validation exercises. It was found that the model reproduces the experimental trends of the working pressures and power consumption in conditions far from the operation point (30%) with a maximum deviation of 5%. However, Schurt et al. [17] have not explored the controlling envelope of the MIMO control system, which is, therefore, the main focus of the present study. 2. Dynamic simulation model Modeling of the refrigeration system relies on the development of sub-models for each of the cycle components (see Fig. 1), which have followed both dynamic (refrigerant flow through the heat exchangers) and quasi-steady (compressor, expansion valve, internal heat exchanger, and condensing and evaporating media) modeling approaches. Further details can be found in Schurt et al. [17]. 2.1. Heat exchangers The model for the refrigerant flow through the condenser and evaporator was based on the following key assumptions: (i) 1-D flow; (ii) straight, horizontal and constant cross-sectional coil; (iii) negligible diffusion effects; (iv) negligible pressure drop; and (v) negligible thermal inertia of the walls. The governing equations, derived from the mass and energy conservation principles, are represented by
v 1 v ðrf Þ þ ðmf Þ ¼ S vt A vz
(1)
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L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
Fig. 1. Schematic representation of the test rig (Pottker and Melo [4]).
where f ¼ 1 and S ¼ 0 retrieve the conservation of mass, whereas f ¼ h and S ¼ dp/dt þ 4q/D retrieve the conservation of energy, where q is the heat flux [W/K] and D the coil inner diameter [m]. To be solved Eq. (1) must be integrated in both time and space domains. The latter was performed following the so-called movingboundaries approach [11], according to which the spatial domain is divided into two zones namely single and two-phase flow regions, as illustrated in Fig. 2. Since the interface position varies with time, the Leibnitz's rule has to be invoked for the integration of Eq. (1), l2 ðtÞ Z
l1 ðtÞ
supplying a heat flux with the same magnitude of that calculated assuming two-phase refrigerant [17]. The averaged terms of Eq. (2) were approximated by a 2nd order scheme (single-phase zone) and by the mean void fraction (twophase zone), respectively:
v d ðrf Þdz ¼ ðl2 l1 Þ ðhrf iÞ vt dt dl dl þ hrf i rl2 fl2 2 hrf i rl1 fl1 1 dt dt
(3)
hrf i ¼ rv fv hgi þ rl fl ð1 hgiÞ
(4)
The condenser and evaporator mean void fraction hgi were fitted to experimental data as a part of the system identification exercise [17]. The spatial integration of the mass and energy conservation equations (1) in each flow region represented in Fig. 2 provided three ordinary differential equations that were re-organized in the form of a linear set of differential equations:
ð2Þ
where l1 ¼ 0 and l2 ¼ l(t) for the two-phase region, whereas l1 ¼ l(t) and l2 ¼ L for the single-phase region (i.e., evaporator superheating and condenser subcooling). It is worth noting that the condenser gascooling region was modeled as if it was saturated since the vapor provides a lower heat transfer coefficient, but at higher logmean temperature difference which compensate each other, thus
Fðx; u; wÞ ¼ A1 ðmi mo Þ
1 ðr fo þ rl fl Þ 2 o
hrf i ¼
CðxÞ$x_ ¼ Fðx; u; wÞ
(5)
where x ¼ [p l ho]T, u ¼ [mi mo]T, w ¼ [Qtp Qsp]T, and F(x, u, w) is obtained from
mi ðhi hl Þ þ Qtp
mo ðhl ho Þ þ Qsp
T
(6)
Qsp
Qtp mi, hi
mλ,, hλ
λ (t) L
Fig. 2. Schematic representation of the spatial discretization of the heat exchangers.
mo, ho
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The elements of matrix C(x) are shown in Table 1. It is noteworthy that, l ¼ v for the evaporator, l ¼ l for the condenser, and the superscripts ()0 and ()00 represent the pressure and enthalpy derivatives, respectively. Moreover, it is worth noting that mi ¼ meev and mo ¼ mvsc for the evaporator, whereas mi ¼ mvsc and mo ¼ meev for the condenser. The heat transfer rates were calculated according to the 3-NTU approach presented in [17].
the evaporator inlet, improving the cooling capacity and avoiding compressor slugging. The internal heat exchanger sub-model determines the states of the refrigerant at the compressor (1) and valve (4) inlet sections (see Fig. 1) based on the information available at the condenser (3) and evaporator (6) exits (see Fig. 1), respectively. On one hand, the compressor inlet temperature T1 was obtained from the temperature effectiveness:
2.2. Compressor
T1 ¼ 3llsl T3 þ ð1 3llsl ÞT6
(12)
h1 ¼ hv ðTe Þ þ cp;v ðTe ÞðT1 Te Þ
(13)
The compressor sub-model provides the refrigerant mass flow rate suctioned from the evaporator and discharged to the condenser, and the power consumed by the compression process [18], being respectively calculated from
mvsc ¼
St cSt ðpc =pe Þcv;v =cp;v 1 N=v1
(7)
Wvsc ¼ a þ b$mvsc ðh2s h1 Þ
h4 ¼ h3 þ ðh6 h1 Þ
(14)
T4 ¼ Tc þ ðh4 hl ðpc ÞÞ=cp;l ðpc Þ
(15)
(8)
where cp,v and cv,v were calculated for the saturated vapor at the evaporating pressure, a, b, c and St were fitted to experimental data as a part of the model identification exercise [17], and T2s was obtained from
T2s ¼ Tc þ ðh2s hv ðpc ÞÞ=cp;v ðpc Þ
(9)
where h2s ¼ h(pc,s1) and s1 ¼ s(pe,h1). The refrigerant enthalpy at the compressor discharge was obtained from the following energy balance:
h2 ¼ h1 þ ðWvsc UAvsc ðT2s Tsur ÞÞ=mvsc
(10) C),
where Tsur is the surrounding air temperature (¼25 and UAvsc is the conductance of the compressor obtained empirically from the system identification exercise [17]. 2.3. Expansion valve The refrigerant mass flow rate through the expansion valve, meev, was calculated from the orifice equation as follows
meev
On the other hand, the refrigerant enthalpy at the evaporator inlet h4 was obtained from the following energy balance:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ceev Ae 2$r4 $ðpc pe Þ
(11)
where meev is given in [kg/s], Ae ¼ AoAv/100 is the effective passage flow area, Av is the valve opening [%], Ao is the nominal orifice crosssectional area (0.1238 cm2), r4 is the specific mass at the valve inlet, and Ceev is the discharge coefficient calculated as a power-law function of the subcooling degree, Ceev ¼ d(T3 Tc)e, where the constants d and e were obtained from the system identification exercise [17]. 2.4. Liquid line suction line heat exchanger The liquid line/suction line heat exchanger (also named internal heat exchanger) aims to increase the amount of liquid refrigerant at Table 1 Elements of matrix C(x). c11
lðhgir0v þ ð1 hgiÞr0l Þ þ ðð1=2ÞðL lÞðr0l þ r0o ÞÞ
c12
hgirv þ ð1 hgiÞrl ð1=2Þðrl ro Þ
c13
ð1=2ÞðL lÞr00o
c21
lððhgirv h0v þ r0v ðhv hl ÞÞ þ ð1 hgiÞðrl h0l þ r0l ðhl hl ÞÞ 1Þ
c22
hgirv ðhv hl Þ þ ð1 hgiÞrl ðhl hl Þ
c23 c31
0 ð1=2ÞðL lÞðrl h0l hl r0o 2Þ
c32
ð1=2Þro ðhl ho Þ
c33
ð1=2ÞðL lÞðro þ r00o ðho hl ÞÞ
2.5. Solution algorithm The overall system simulation relies on the time integration of an equation set comprised of six ordinary differential equations. The initial conditions are the evaporating and condensing pressures, the refrigerant enthalpies at the evaporator and condenser outlet sections, and the positions of the evaporation and condensation boundaries, all obtained from the prior steady-state solution of the equation set enforcing the time-derivatives to be nil. The boundary conditions are the compressor speed, the valve opening, and the temperatures and flow rates of the secondary coolants, all of them obtained from experimental data. The numerical integration of the differential equations was carried out through the ODE23S adaptive method for a stiff set of equations [19]. The thermodynamic properties were obtained from REFPROP7 [20]. 3. Controller design In order to simplify the controller design, a linear state-space representation was adopted, which required the linearization of the dynamic simulation model using a Taylor expansion series, yielding
dx_ ¼ A$dx þ B$du þ W$dw dy ¼ C$dx þ E$dw
(16)
The vectors dx, dy, du and dw indicate respectively the difference between the static (x0, y0, u0, w0) and the dynamic values of the state variables, x ¼ [pe le ho,e pc lc ho,c]T, the controlled variables, y ¼ [DTsup Ts,e,o]T, the input variables, u ¼ [N Av]T, and the perturbations, w ¼ [Vc Ve]T. The matrices A, B, W, C and E were obtained from the model linearization exercise and their values are summarized in [17]. The MIMO controller was designed based on a LQG scheme with an integrator, as illustrated in Fig. 3. The controller design was focused not only on matching the cooling capacity to the thermal loads but also on keeping the evaporator superheating at a predefined level by driving the compressor speed and the electric valve opening simultaneously. In Fig. 3, yref ¼ [DTsup,ref Ts,e,o,ref]T is the reference input signal, x_ ¼ yref y is the tracking error, x is the output of the integrator, xK is the estimated state, u ¼ Ke$xK þ Ki$x is the control signal, and Ke and Ki are the state-feedback and integral gain matrices, respectively. The LQG controller consists of a combination of a linearquadratic-regulator (LQR) and an optimal state estimator, which have been developed independently according to the separation principle [21]. The former consists of finding the matrix K ¼ [KejKi]
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
Ki -
++
x = Ax + Bu y = Cx
u
-
Ke
xK
x y
Kalman Filter
Fig. 3. Schematic representation of the controller.
of the control action, u(t) ¼ K$xa(t), where xa ¼ [xK x]T, that minimizes the following quadratic criterion:
J ¼
ZNh
i xTa ðtÞ$Q $xTa ðtÞ þ uT ðtÞ$R$uðtÞ dt
5. Results and discussion
(17)
0
where Q and R are positive-definite Hermitian matrices determined through the Bryson method [22]. It is noteworthy that Q and R lead to the elements of matrix K straightforwardly through the positive-definite-solution P of the Riccati equation [17]. A discrete-time version of the LQG controller was designed and the controller was then implemented on a digital computer with a sample time of 2 s. It is worth of note that the controller was designed to reject perturbations due to the integrator, and also to keep the time-response of the closed-loop system approximately the same that of the open-loop system. The values of matrices Q, R and K of the discrete-time LQ regulator are summarized in [17]. Since some states cannot be measured directly (e.g., position of the evaporation and condensation boundaries), a state estimator was also required. The estimator was developed using a Kalman filter as indicated below,
x_ K ðtÞ ¼ ðA L$CÞ$xK ðtÞ þ B$uðtÞ þ L$yðtÞ
Moreover, the compression power was measured with an uncertainty of 0.5% (full scale). For steady-state conditions, the difference between the refrigerant-side and the secondary fluid-side heat transfer rates was less than 2%. A control and data acquisition system was used for monitoring the experimental variables and for setting the compressor speed and valve opening. Further details of the experimental apparatus can be found in [4]. The facility was used to collect data for the system identification and model validation exercises. It was found that the model reproduces the experimental trends of the working pressures and power consumption in conditions far from the operation point (30%) with a maximum deviation of 5%, as illustrated in Fig. 4.
The experimental analyzes were carried out to assess the controller performance in terms of reference tracking, controlling envelope, and disturbance rejection. The reference tracking test was conducted imposing step changes to the controlled variables (i.e., references), for instance the evaporator superheating and the brine temperature at the evaporator outlet. The controlling envelope was evaluated by changing the references until the maximum and minimum controlled states were reached. The disturbance rejection was investigated by varying the flow rate of the evaporating and condensing media. 5.1. Reference tracking and disturbance analysis The test was initiated when the steady-state condition was achieved for the following operating conditions: compressor speed of 3000 rpm, valve opening of 45%, evaporating medium flow rate and temperature of 1.24 L/min and 10 C, respectively, and condensing medium flow rate and temperature of 1.17 L/min and 35 C,
(18)
where L and S are calculated according to the procedure described in [17]. 4. Experimental facility Fig. 1 shows a schematic diagram of the HFC-134a refrigeration system used in this study. The compressor is of hermetic variablespeed reciprocating type, with minimum and maximum speeds of 1800 and 4500 rpm, respectively. The condenser and evaporator are both tube-in-tube heat exchangers, connected to two distinct temperature and flow controlled secondary heat transfer loops. Brine of 27% ethylene-glycol aqueous solution is used for the evaporator, whereas pure water is used for the condenser. The expansion device is a pulse-width-modulation (PWM) electric expansion device with variable opening from 0 to 100%. The condenser and evaporator heat transfer rates are obtained from energy balances on the refrigerant and secondary fluid sides. The refrigerant heat transfer rate was calculated by multiplying the refrigerant mass flow rate, measured by a Coriolis mass flow meter (measurement uncertainty of 0.1%) by the enthalpy difference between the heat exchanger inlet and outlet, which provided a maximum measurement uncertainty of 2% [4]. Immersion T-type thermocouples (measurement uncertainty of 0.2 C) and pressure transducers (measurement uncertainty of 0.15%) were installed at selected points of the circuit to obtain the refrigerant enthalpies and the evaporator superheating. The secondary fluid heat transfer rate was calculated using the volumetric flow rate, measured by a turbine flow meter (measurement uncertainty of 1%), the fluid density, the specific heat and the temperature difference between the heat exchanger inlet and outlet.
a 3.0 Evaporating pressure [bar]
yref +
Experimental Simulation
2.5
2.0
1.5
1.0
0
b 16 Condensing pressure [bar]
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50
100
150
200 250 Time [min]
300
350
400
150
200 250 Time [min]
300
350
400
Experimental Simulation
15
14
13
12
11
0
50
100
Fig. 4. Model validation exercise: (a) evaporating pressure; and (b) condensing pressure.
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Table 2 Changes imposed to the evaporator superheating and brine outlet temperature during the reference tracking exercise. Time [min]
t < 20
20 < t < 40
40 < t < 60
60 < t < 80
80 < t < 100
100 < t < 120
t > 120
DTsup [ C]
16.6 4
20.2 4
12.5 4
16.6 4
16.6 5
16.6 3
16.6 4
Ts,e,o [ C]
respectively. At this condition, an evaporator superheating of 16.6 C and a brine temperature at the evaporator outlet of 4 C were achieved and adopted as references. Independent changes in the references were imposed following the sequence shown in Table 2. The dynamic behavior of the controlled variables is depicted in Fig. 5a, where it can be noted that the controller drove the refrigeration system towards the prescribed references. It is noteworthy that a reference change in one of the controlled variables produced a disturbance in its counterpart, which was identified and properly rejected by the controller. The highest disturbance level was observed during the second change in the evaporator superheating degree, at t ¼ 40 min, reaching a minimum value of 10.6 C, which represents an undershooting of 2 C. Such an effect was induced by the magnitude of the imposed change (7.7 C), negligible in the first and third changes of references (at t ¼ 20 min and t ¼ 60 min, respectively). The disturbances caused by changing the reference of the brine outlet temperature were also properly rejected by the controller, which affected the evaporator superheating in 2 C only (at t ¼ 120 min). For the brine outlet temperature, the controller showed a better performance, with a maximum undershooting of 0.5 C at the instant when the highest change of reference was observed, i.e., 2 C at t ¼ 100 min. Furthermore, the controller kept the brine outlet temperature within a 0.5 C band during the first 80 min. Fig. 5b shows the variations experienced by the control actions (i.e., compressor speed and valve opening) in order to drive the controlled variables towards the desired references. It is worthy noting that, during the change of reference of the evaporator superheating (t < 40 min), the control actions presented opposite behaviors: the compressor speed increased when the valve opening decreased. This was so as the cooling capacity tends to decrease inasmuch as the superheating degree increases, forcing the controller to increase the compressor speed in an attempt to keep the cooling capacity (see Fig. 5d), thereby reducing the evaporating pressure (see Fig. 5c) and increasing the evaporator log-mean temperature difference. The opposite behavior was observed within the time interval 40 < t < 60 min. During the change of reference of the brine outlet temperature (t > 60 min), it was noted that both compressor speed and valve opening showed the same behavior (see Fig. 5b). On one hand, to raise the brine outlet temperature (i.e., lower cooling capacity) without changing the evaporator superheating (t ¼ 80 min), the controller increased the evaporating pressure (see Fig. 5c) by reducing both the compressor speed and the expansion valve opening so as to diminish the refrigerant mass flow rate. On the other hand, the cooling capacity increased at t ¼ 100 min as the compressor speed and the valve opening were simultaneously brought up (see Fig. 5b). The disturbance rejection test aims to check whether the controller is able to keep the controlled variables at the reference previously set, even if changes are imposed on the thermal load or the surrounding conditions. Such changes have been respectively emulated in the experimental apparatus by modifying the evaporator and condenser secondary coolant flow rates, as shown in Table 3. Such variations have induced disturbances on the controlled variables that have been recognized by the controller and then rejected, as can be seen in Fig. 6a. The largest deviation observed in the evaporator superheating, in comparison to its
a
b
c
d
Fig. 5. Reference tracking analysis: (a) controlled variables; (b) control actions; (c) working pressures; and (d) cooling capacity.
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Table 3 Changes imposed to the evaporator and condenser secondary coolant mass flow rates during the disturbance rejection exercise. Time [min]
t < 20
20 < t < 40
40 < t < 60
60 < t < 80
80 < t < 100
100 < t < 120
t > 120
Vs,c [L/min] Vs,e [L/min]
1.16 1.23
1.16 1.43
1.16 1.00
1.16 1.23
1.38 1.23
0.93 1.23
1.16 1.23
a
b
reference, was 3 C at t ¼ 40 min. Such deviation is related to the sudden reduction of the brine flow rate from 1.43 to 1.00 L/min, which represented a 30% reduction in the thermal load imposed (from 550 to 385 W, see Fig. 6d). It can be also noted in Fig. 6 that the variations observed for t < 70 min are related to the thermal load disturbances, as the changes in the condenser side had no practical influence on the controlled variables and control action, such as the compressor speed and the valve opening (Fig. 6b), the mass flow rate (Fig. 6d), and evaporating temperature (Fig. 6c). The condensing pressure, on the other hand, has experienced some disturbances, although the largest variation was due to the changes on the evaporator brine flow rate (Fig. 6c). Albeit the cooling capacity (see Fig. 6d) was practically not affected by the disturbance imposed in the condenser secondary coolant flow rate (t > 70 min), it changed substantially with the evaporator brine flow rate (t < 70 min). 5.2. Assessment of the controlling envelope The operational envelope beyond which it is not possible to maintain the controlled variables at their references was evaluated by artificially driven the controller through step changes promoted in the references. In addition, changes in the evaporator brine flow rate were also imposed to the system in order to identify the maximum and minimum controllable thermal loads. First, increasing step changes (from 4.0 to 6.0 C, step of 0.5 C) on the brine outlet temperature were imposed, as shown in Fig. 7a,
c
d
Fig. 6. Disturbance rejection analysis: (a) controlled variables; (b) control actions; (c) working pressures; and (d) cooling capacity.
a
b
Fig. 7. Controlling envelope analysis with increasing brine temperature: (a) controlled variables; (b) control actions.
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
a
b
1545
as shown in Fig. 7b. When the minimum compressor speed was reached, the evaporator superheating increased as the valve opening ruled the system while the compressor speed became saturated. Similarly, decreasing step changes (from 4.0 to 2.5 C, step of 0.5 C) on brine outlet temperature were imposed, as shown in Fig. 8a, which is equivalent to a thermal load augmentation. At each new reference change, the controller increased the cooling capacity by opening the valve and raising the compressor speed up to the maximum allowed value (4500 rpm), as shown in Fig. 8b. As the evaporator superheating was fixed at 16.6 C, the brine outlet temperature was limited within the range from 3 to 5.5 C. Alternatively, decreasing step changes (from 16.6 to 8.5 C) on the evaporator superheating were imposed, as shown in Fig. 9a, until the controller is no longer maintaining the controlled variables at their references. It is noteworthy that at superheating degrees lower than 8.5 C the system became unstable due to two factors. First, as the proposed controller is linear, it cannot identify the non-linear dynamics of the variables (see Fig. 9b). In addition, the evaporator dry-out location oscillates at lower superheating degrees due to the natural two-phase flow instabilities [11], which affected the quality of the control signal and therefore the control action.
a Fig. 8. Controlling envelope analysis with decreasing brine temperature: (a) controlled variables; (b) control actions.
which is equivalent to a thermal load reduction. In order to follow the references and also to adapt the system to lower cooling demands, the controller decreased the valve opening while reduced the compressor speed until 1800 rpm, the minimum allowed speed,
a
b
b
c
Fig. 9. Controlling envelope analysis with decreasing evaporator superheating: (a) controlled variables; (b) control actions.
Fig. 10. Controlling envelope analysis with increasing and decreasing thermal load: (a) controlled variables; (b) control actions; and (c) disturbances.
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Additional tests were carried out varying the evaporator brine flow rate, but keeping the evaporator superheating degree and the brine outlet temperature references fixed (see Fig. 10a). During the first 120 min, the flow rate was steeply increased to the maximum value that can be achieved without loosing the system controllability, as showed in Fig.10b. Therefore, the refrigeration system experienced a thermal load variation from 460 to 580 W (see Fig. 10c). In other words, a thermal load 26% higher than the nominal was achieved but keeping the preset references (see Fig. 10b). Between 120 and 140 min, the system was then returned to its initial condition. After 140 min, the brine flow rate was steeply decreased in order to reduce the thermal load to 340 W, i.e., a figure 26% lower than the nominal value (see Fig. 10c). 6. Summary and conclusions A dynamic simulation model for the identification and control of vapor compression refrigeration systems was developed and validated for a broad range of operation. A breadboard refrigeration system comprised of a variable-speed compressor and an electric expansion valve was used to gather the required experimental data for both system identification and model validation exercises. It was found that the model reproduces well the experimental trends of the working pressures even in conditions far from the operation point used in the system identification, with maximum discrepancies between calculated and measured counterparts within 5% error bands. A model-driven multivariate linear strategy for controlling the evaporator feeding and for matching the cooling capacity to the thermal loads was devised. The dynamic simulation model was linearized according to a Taylor expansion series and then used to design a proportional plus integral type controller based on the linear-quadratic-Gaussian (LQG) method, which is based on a state estimator of the Kalman filter type. Additional experimental tests were also carried out using the breadboard facility to verify the controller ability for tracking reference changes and rejecting thermal load disturbances due to the integral action of the controller. It was found that the controller was able to drive the facility for thermal loads spanning from 340 to 580 W, i.e., 26% in comparison to the nominal value. Similarly, the controller kept the refrigeration system running properly for evaporator superheating degrees ranging from 9.5 C to 22 C, albeit it was designed for 16.6 C. However, it is worthy of note that the controller cannot be applied for superheating degrees lower than 9.5 C, when both the controlled system and the control signal became unstable. Acknowledgements The authors would like to thank the CNPq (Grant No. 573581/ 2008-8 e National Institute of Science and Technology in Refrigeration and Thermophysics) for the financial support. Thanks are also addressed to Embraco S.A. for sponsoring this research project. The authors duly acknowledge Prof. C. Melo and Prof. N. Roqueiro
(Federal University of Santa Catarina), and Dr. L. W. da Silva (Embraco S.A.) for their valuable technical contributions. References [1] S.A. Tassou, T.Q. Qureshi, Comparative performance evaluation of positive displacement compressors in variable-speed refrigeration applications. Int. J. Refrigeration 21 (1998) 29e41. [2] C. Aprea, R. Mastrullo, Experimental evaluation of electronic and thermostatic expansion valves performances using R22 and R407C. Appl. Therm. Eng. 22 (2002) 206e218. [3] S.A. Tassou, H. Al-Nizari, Investigation of the steady state and transient performance of reciprocating chiller equipped with an electronic expansion valve. Heat Recov Syst CHP 11 (1991) 541e550. [4] G. Pottker, C. Melo, A study on the relationship between compressor speed and expansion valve opening in refrigeration systems, in: Conf. on compressors and their systems, London, UK, Paper C658, 2007. [5] A. Outtagarts, P. Haperschill, M. Lallemand, The transient response of an evaporator fed through an electronic expansion valve. Int. J. Energy Res. 21 (1997) 793e807. [6] J.M. Choi, Y.C. Kim, Capacity modulation of an inverter-driven multi-air conditioner using electronic expansion valve. Energy 28 (2003) 141e155. [7] D.S. Yang, G. Lee, M.S. Kim, Y.J. Hwang, B.Y. Chung, A study on the capacity control of a variable speed vapor compression system using superheat information at compressor discharge, in: 10th Int. Refrig. Conf. at Purdue, West Lafayette-IN, USA, Paper R-164, 2004. [8] B. Lamanna, Development of an advanced control system for chillers with inverter driven scroll compressor and comparison with traditional on/off systems, in: IIR conference on commercial refrigeration, Vicenza, Italy, 2005, pp. 295e302. [9] J.B. Marcinichen, T. N. Holanda, C. Melo, A dual SISO controller for a vapor compression refrigeration system, in: 12th Int. Refrig. Conf. at Purdue, West Lafayette-IN, USA, Paper 2444, 2008. [10] B. Rasmussen, A. Alleyne, Control-oriented modeling of transcritical vapor compression systems. J. Dyn. Syst. Meas. Contr. 126 (2004) 54e64. [11] G.L. Wedekind, W.F. Stoecker, Theoretical model for predicting the transient response of the mixture-vapour transition point in horizontal evaporating flow. ASME J. Heat Trans. (February 1968) 165e174. [12] X. He, S. Liu, H. Asada, Modeling of vapor compression cycles for multivariable feedback control of HVAC systems. J. Dyn. Syst. Meas. Contr. 119 (1997) 183e191. [13] M. Willatzen, N.B.O.L. Pettit, L. Ploug-Sorensen, A general dynamic simulation model for evaporators and condensers in refrigeration. Part I: movingboundary formulation. Int. J. Refrigeration 21 (1998) 398e403. [14] D. Leducq, J. Guilpart, G. Trystram, Non-linear predictive control of a vapour compression cycle. Int. J. Refrigeration 29 (2006) 761e772. [15] T.L. Mckinley, A.G. Alleyne, An advanced nonlinear switched heat cycles using the moving-boundary method. Int. J. Refrigeration 31 (2008) 1253e1264. [16] Q. Qi, S. Deng, Multivariable control-oriented modeling of a direct expansion (DX) air conditioning (A/C) system. Int. J. Refrigeration 31 (2008) 841e849. [17] L.C. Schurt, C.J.L. Hermes, A. Trofino Neto, A model-driven multivariable controller for vapor compression refrigeration systems. Int. J. Refrigeration 32 (2009) 1672e1682. [18] W.C. Gosney, Principles of Refrigeration. Cambridge University Press, Cambridge, UK, 1982. [19] L.F. Shampine, M.W. Reichelt, The Matlab ODE Suite. SIAM J. Sci. Comput. 18 (1997) 1e22. [20] E.W. Lemmon, M.O. McLinden, M.L. Huber, REFPROP 7.0, Standard Reference Database 23. National Institute of Standards and Technology, Gaithersburg, MD, USA, 2002. [21] U. Mackenroth, Robust Control Systems. Springer Verlag, Berlin, Germany, 2004. [22] M.J. Johnson, M.A. Grimble, Recent trends in linear optimal quadratic multivariable control system design. IEE Proceedings Part D Control Theory and Applications 134 (1987) 53e71.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Assessment of the controlling envelope of a model-based multivariable controller for vapor compression refrigeration systems Leonardo C. Schurt a, Christian J.L. Hermes a, *, Alexandre Trofino Neto b a b
POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil Department of Automation and Systems Engineering, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 February 2010 Accepted 19 February 2010 Available online 3 March 2010
This paper explores the controlling characteristics of a first-principles model-based controller specially developed for vapor compression refrigeration systems. Mathematical sub-models were put forward for each of the system components: heat exchangers (condenser and evaporator), variable-speed compressor and variable-orifice electric expansion device. The dynamic simulation model was then used to design a multivariable controller based on the linear-quadratic-Gaussian technique using a Kalman filter for the estimator design. A purpose-built testing apparatus comprised of a variable-speed compressor and a pulse-width modulated expansion valve was used to collect data for the system identification, and model and controller validation exercises. It was found that the model reproduces the experimental trends of the working pressures and power consumption in conditions far from the nominal point of operation (30%) with a maximum deviation of 5%. Additional experiments were also performed to verify the ability of the controller of tracking reference changes and rejecting thermal load disturbances. It was found that the controller is able to keep the refrigeration system running properly when the thermal load was changed from 340 to 580 W (460 W nominal), and the evaporator superheating degree was varied from 9.5 C to 22 C (16.6 C nominal). Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Control Refrigeration system Modeling Experimentation Controlling envelope
1. Introduction Household and commercial refrigerators consume by 10% of the electrical energy produced worldwide, a figure that has motivated both customers and governments to push the manufacturers for high efficiency products. However, in spite of the large effort put in refrigeration systems advancements in the past years, just a modest effect has been observed suggesting that the conventional vapor compression refrigeration technology is reaching its limits. Vapor compression refrigeration systems usually comprise a single-speed compressor, a fixed-orifice expansion device, two heat exchangers (i.e., the condenser and the evaporator), and a volatile working fluid (named refrigerant) that undergoes a reversed Rankine thermodynamic cycle. Furthermore, the temperature of the refrigerated compartment is controlled by a thermostat that switches the refrigeration system on and off according to a cycling pattern. It has been advocated in the open literature that the refrigeration systems with electronic-controlled equipment (i.e., compressor speed, and valve opening, among others) may improve significantly the overall energy performance when compared to the conventional * Corresponding author. Present address: Department of Mechanical Engineering, Federal University of Paraná, P.O. Box 19011, 81531-990 Curitiba, PR, Brazil. Tel.: þ55 41 3361 3239; fax: þ55 41 3361 3129. E-mail address: [email protected] (C.J.L. Hermes). 1359-4311/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2010.02.024
refrigeration appliances. For instance, variable-speed compressors have been employed in the past decades to obtain a continuous matching between the cooling capacity and the thermal load with remarkable gains in energy performance [1,2]. Moreover, it has been also reported [3,4] that the system performance improves further in cases where a variable-orifice expansion valve is additionally employed, as the evaporator is kept fully activated (i.e., flooded) during most of the system runtime. It is noteworthy that such refrigeration systems require proper control strategies to account for the simultaneous operation of a variable-speed compressor and a variable-orifice expansion valve. Albeit there have been in the past years some publications with regard to the analysis and development of control strategies for variable-speed compressors and variable-orifice expansion valves operating simultaneously [5e8], their focus was on large capacity refrigeration applications (>6 kW). Pottker and Melo [4] carried out an experimental study to evaluate the effect of variable-opening electric expansion valves on the performance of variable-speed (VSC) and single-speed (SSC) vapor compression refrigeration systems. A purpose-built breadboard refrigeration apparatus consisting of a variable-speed compressor, an electric expansion valve (EEV), and two secondary flow rate and temperature controlled aqueous solution loops (condensing and evaporating media) was designed and constructed to emulate refrigeration systems with capacities ranging from 0.3 to 1.5 kW
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
Nomenclature
Roman A cross-sectional area, m2 A, B, C, E, W state-space matrices valve opening, % Av D coil inner diameter, m h specific enthalpy, J/kg J cost function K controller gain matrix L length, m L observer gain matrix m mass flow rate, kg/s N compressor speed, s1 NTU number of transfer units p pressure, Pa Q heat transfer rate, W q heatflux, W m2 Q, R weighting matrices T temperature, K t time, s u controllable inputs matrix UA overall conductance, W/K V secondary coolant flow rate, L/min v specific volume, m3/kg w disturbance input matrix
(see Fig. 1). It was found that, independently of the secondary fluid temperature entering the heat exchangers, the VSC/EEV system always operates with a maximum coefficient of performance (COP) once it keeps the evaporator flooded and also delivers a cooling capacity that exactly matches the thermal load. Later, Marcinichen et al. [9] put forward an empirical dual-SISO (single-input, single-output) strategy for the simultaneous control of compressor speed and expansion valve opening using the experimental rig constructed by Pottker and Melo [4]. The control strategy was devised to obtain a maximum COP within a range of cooling capacity from 0.3 to 0.8 kW. The refrigeration system was identified using the step-response method, which provided first order linear models for both the evaporator superheating and the brine outlet temperature. The empirical models were used to derive two singleinput, single-output (SISO) proportional-integral (PI) controllers, one for the evaporator superheating as a function of the EEV opening, and another for brine outlet temperature as a function of the compressor speed. Both controllers were implemented into a dual-SISO control strategy that operated satisfactorily in terms of reference tracking and disturbance rejection. Nonetheless, it was found that the empirical identification has constrained the controller to a region close to the point of operation. In order to design controllers that can be applied to a broad range of operation, the use of first-principles simulation models have been suggested in the literature [10] for the dynamic identification of vapor compression refrigeration systems. Albeit there are several publications concerning the modeling for controlling vapor compression refrigeration systems [11e16], all of them conducted the system identification and model validation exercises for regions close to the point of operation (w5%). Therefore, in a prior study, Schurt et al. [17] put forward a multivariable model-driven controller for vapor compression refrigeration systems. Mathematical sub-models were developed for each of the system components: heat exchangers (condenser and evaporator), variable-speed compressor and variable-orifice
x y
1539
dynamic state matrix output matrix
Greek
a 3ll-sl g l r
heat transfer coefficient, W/m2 K internal heat exchanger effectiveness void fraction boundary position, m specific mass, kg/m
Subscripts c condenser e evaporator, static i inlet, integral K Kalman l saturated liquid o outlet r refrigerant ref reference s secondary coolant, isentropic process sur surroundings v saturated vapor Other symbols hxi volume average of x x_ ¼ dx=dt time derivative of x
expansion valve. The dynamic simulation model was then used to design a MIMO (multi-input, multi-output) controller based on the linear-quadratic-Gaussian (LQG) technique using a Kalman filter for the estimator design. The purpose-built testing apparatus constructed by Pottker and Melo [4] was then used to collect data for the system identification and model validation exercises. It was found that the model reproduces the experimental trends of the working pressures and power consumption in conditions far from the operation point (30%) with a maximum deviation of 5%. However, Schurt et al. [17] have not explored the controlling envelope of the MIMO control system, which is, therefore, the main focus of the present study. 2. Dynamic simulation model Modeling of the refrigeration system relies on the development of sub-models for each of the cycle components (see Fig. 1), which have followed both dynamic (refrigerant flow through the heat exchangers) and quasi-steady (compressor, expansion valve, internal heat exchanger, and condensing and evaporating media) modeling approaches. Further details can be found in Schurt et al. [17]. 2.1. Heat exchangers The model for the refrigerant flow through the condenser and evaporator was based on the following key assumptions: (i) 1-D flow; (ii) straight, horizontal and constant cross-sectional coil; (iii) negligible diffusion effects; (iv) negligible pressure drop; and (v) negligible thermal inertia of the walls. The governing equations, derived from the mass and energy conservation principles, are represented by
v 1 v ðrf Þ þ ðmf Þ ¼ S vt A vz
(1)
1540
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
Fig. 1. Schematic representation of the test rig (Pottker and Melo [4]).
where f ¼ 1 and S ¼ 0 retrieve the conservation of mass, whereas f ¼ h and S ¼ dp/dt þ 4q/D retrieve the conservation of energy, where q is the heat flux [W/K] and D the coil inner diameter [m]. To be solved Eq. (1) must be integrated in both time and space domains. The latter was performed following the so-called movingboundaries approach [11], according to which the spatial domain is divided into two zones namely single and two-phase flow regions, as illustrated in Fig. 2. Since the interface position varies with time, the Leibnitz's rule has to be invoked for the integration of Eq. (1), l2 ðtÞ Z
l1 ðtÞ
supplying a heat flux with the same magnitude of that calculated assuming two-phase refrigerant [17]. The averaged terms of Eq. (2) were approximated by a 2nd order scheme (single-phase zone) and by the mean void fraction (twophase zone), respectively:
v d ðrf Þdz ¼ ðl2 l1 Þ ðhrf iÞ vt dt dl dl þ hrf i rl2 fl2 2 hrf i rl1 fl1 1 dt dt
(3)
hrf i ¼ rv fv hgi þ rl fl ð1 hgiÞ
(4)
The condenser and evaporator mean void fraction hgi were fitted to experimental data as a part of the system identification exercise [17]. The spatial integration of the mass and energy conservation equations (1) in each flow region represented in Fig. 2 provided three ordinary differential equations that were re-organized in the form of a linear set of differential equations:
ð2Þ
where l1 ¼ 0 and l2 ¼ l(t) for the two-phase region, whereas l1 ¼ l(t) and l2 ¼ L for the single-phase region (i.e., evaporator superheating and condenser subcooling). It is worth noting that the condenser gascooling region was modeled as if it was saturated since the vapor provides a lower heat transfer coefficient, but at higher logmean temperature difference which compensate each other, thus
Fðx; u; wÞ ¼ A1 ðmi mo Þ
1 ðr fo þ rl fl Þ 2 o
hrf i ¼
CðxÞ$x_ ¼ Fðx; u; wÞ
(5)
where x ¼ [p l ho]T, u ¼ [mi mo]T, w ¼ [Qtp Qsp]T, and F(x, u, w) is obtained from
mi ðhi hl Þ þ Qtp
mo ðhl ho Þ þ Qsp
T
(6)
Qsp
Qtp mi, hi
mλ,, hλ
λ (t) L
Fig. 2. Schematic representation of the spatial discretization of the heat exchangers.
mo, ho
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1541
The elements of matrix C(x) are shown in Table 1. It is noteworthy that, l ¼ v for the evaporator, l ¼ l for the condenser, and the superscripts ()0 and ()00 represent the pressure and enthalpy derivatives, respectively. Moreover, it is worth noting that mi ¼ meev and mo ¼ mvsc for the evaporator, whereas mi ¼ mvsc and mo ¼ meev for the condenser. The heat transfer rates were calculated according to the 3-NTU approach presented in [17].
the evaporator inlet, improving the cooling capacity and avoiding compressor slugging. The internal heat exchanger sub-model determines the states of the refrigerant at the compressor (1) and valve (4) inlet sections (see Fig. 1) based on the information available at the condenser (3) and evaporator (6) exits (see Fig. 1), respectively. On one hand, the compressor inlet temperature T1 was obtained from the temperature effectiveness:
2.2. Compressor
T1 ¼ 3llsl T3 þ ð1 3llsl ÞT6
(12)
h1 ¼ hv ðTe Þ þ cp;v ðTe ÞðT1 Te Þ
(13)
The compressor sub-model provides the refrigerant mass flow rate suctioned from the evaporator and discharged to the condenser, and the power consumed by the compression process [18], being respectively calculated from
mvsc ¼
St cSt ðpc =pe Þcv;v =cp;v 1 N=v1
(7)
Wvsc ¼ a þ b$mvsc ðh2s h1 Þ
h4 ¼ h3 þ ðh6 h1 Þ
(14)
T4 ¼ Tc þ ðh4 hl ðpc ÞÞ=cp;l ðpc Þ
(15)
(8)
where cp,v and cv,v were calculated for the saturated vapor at the evaporating pressure, a, b, c and St were fitted to experimental data as a part of the model identification exercise [17], and T2s was obtained from
T2s ¼ Tc þ ðh2s hv ðpc ÞÞ=cp;v ðpc Þ
(9)
where h2s ¼ h(pc,s1) and s1 ¼ s(pe,h1). The refrigerant enthalpy at the compressor discharge was obtained from the following energy balance:
h2 ¼ h1 þ ðWvsc UAvsc ðT2s Tsur ÞÞ=mvsc
(10) C),
where Tsur is the surrounding air temperature (¼25 and UAvsc is the conductance of the compressor obtained empirically from the system identification exercise [17]. 2.3. Expansion valve The refrigerant mass flow rate through the expansion valve, meev, was calculated from the orifice equation as follows
meev
On the other hand, the refrigerant enthalpy at the evaporator inlet h4 was obtained from the following energy balance:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ceev Ae 2$r4 $ðpc pe Þ
(11)
where meev is given in [kg/s], Ae ¼ AoAv/100 is the effective passage flow area, Av is the valve opening [%], Ao is the nominal orifice crosssectional area (0.1238 cm2), r4 is the specific mass at the valve inlet, and Ceev is the discharge coefficient calculated as a power-law function of the subcooling degree, Ceev ¼ d(T3 Tc)e, where the constants d and e were obtained from the system identification exercise [17]. 2.4. Liquid line suction line heat exchanger The liquid line/suction line heat exchanger (also named internal heat exchanger) aims to increase the amount of liquid refrigerant at Table 1 Elements of matrix C(x). c11
lðhgir0v þ ð1 hgiÞr0l Þ þ ðð1=2ÞðL lÞðr0l þ r0o ÞÞ
c12
hgirv þ ð1 hgiÞrl ð1=2Þðrl ro Þ
c13
ð1=2ÞðL lÞr00o
c21
lððhgirv h0v þ r0v ðhv hl ÞÞ þ ð1 hgiÞðrl h0l þ r0l ðhl hl ÞÞ 1Þ
c22
hgirv ðhv hl Þ þ ð1 hgiÞrl ðhl hl Þ
c23 c31
0 ð1=2ÞðL lÞðrl h0l hl r0o 2Þ
c32
ð1=2Þro ðhl ho Þ
c33
ð1=2ÞðL lÞðro þ r00o ðho hl ÞÞ
2.5. Solution algorithm The overall system simulation relies on the time integration of an equation set comprised of six ordinary differential equations. The initial conditions are the evaporating and condensing pressures, the refrigerant enthalpies at the evaporator and condenser outlet sections, and the positions of the evaporation and condensation boundaries, all obtained from the prior steady-state solution of the equation set enforcing the time-derivatives to be nil. The boundary conditions are the compressor speed, the valve opening, and the temperatures and flow rates of the secondary coolants, all of them obtained from experimental data. The numerical integration of the differential equations was carried out through the ODE23S adaptive method for a stiff set of equations [19]. The thermodynamic properties were obtained from REFPROP7 [20]. 3. Controller design In order to simplify the controller design, a linear state-space representation was adopted, which required the linearization of the dynamic simulation model using a Taylor expansion series, yielding
dx_ ¼ A$dx þ B$du þ W$dw dy ¼ C$dx þ E$dw
(16)
The vectors dx, dy, du and dw indicate respectively the difference between the static (x0, y0, u0, w0) and the dynamic values of the state variables, x ¼ [pe le ho,e pc lc ho,c]T, the controlled variables, y ¼ [DTsup Ts,e,o]T, the input variables, u ¼ [N Av]T, and the perturbations, w ¼ [Vc Ve]T. The matrices A, B, W, C and E were obtained from the model linearization exercise and their values are summarized in [17]. The MIMO controller was designed based on a LQG scheme with an integrator, as illustrated in Fig. 3. The controller design was focused not only on matching the cooling capacity to the thermal loads but also on keeping the evaporator superheating at a predefined level by driving the compressor speed and the electric valve opening simultaneously. In Fig. 3, yref ¼ [DTsup,ref Ts,e,o,ref]T is the reference input signal, x_ ¼ yref y is the tracking error, x is the output of the integrator, xK is the estimated state, u ¼ Ke$xK þ Ki$x is the control signal, and Ke and Ki are the state-feedback and integral gain matrices, respectively. The LQG controller consists of a combination of a linearquadratic-regulator (LQR) and an optimal state estimator, which have been developed independently according to the separation principle [21]. The former consists of finding the matrix K ¼ [KejKi]
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
Ki -
++
x = Ax + Bu y = Cx
u
-
Ke
xK
x y
Kalman Filter
Fig. 3. Schematic representation of the controller.
of the control action, u(t) ¼ K$xa(t), where xa ¼ [xK x]T, that minimizes the following quadratic criterion:
J ¼
ZNh
i xTa ðtÞ$Q $xTa ðtÞ þ uT ðtÞ$R$uðtÞ dt
5. Results and discussion
(17)
0
where Q and R are positive-definite Hermitian matrices determined through the Bryson method [22]. It is noteworthy that Q and R lead to the elements of matrix K straightforwardly through the positive-definite-solution P of the Riccati equation [17]. A discrete-time version of the LQG controller was designed and the controller was then implemented on a digital computer with a sample time of 2 s. It is worth of note that the controller was designed to reject perturbations due to the integrator, and also to keep the time-response of the closed-loop system approximately the same that of the open-loop system. The values of matrices Q, R and K of the discrete-time LQ regulator are summarized in [17]. Since some states cannot be measured directly (e.g., position of the evaporation and condensation boundaries), a state estimator was also required. The estimator was developed using a Kalman filter as indicated below,
x_ K ðtÞ ¼ ðA L$CÞ$xK ðtÞ þ B$uðtÞ þ L$yðtÞ
Moreover, the compression power was measured with an uncertainty of 0.5% (full scale). For steady-state conditions, the difference between the refrigerant-side and the secondary fluid-side heat transfer rates was less than 2%. A control and data acquisition system was used for monitoring the experimental variables and for setting the compressor speed and valve opening. Further details of the experimental apparatus can be found in [4]. The facility was used to collect data for the system identification and model validation exercises. It was found that the model reproduces the experimental trends of the working pressures and power consumption in conditions far from the operation point (30%) with a maximum deviation of 5%, as illustrated in Fig. 4.
The experimental analyzes were carried out to assess the controller performance in terms of reference tracking, controlling envelope, and disturbance rejection. The reference tracking test was conducted imposing step changes to the controlled variables (i.e., references), for instance the evaporator superheating and the brine temperature at the evaporator outlet. The controlling envelope was evaluated by changing the references until the maximum and minimum controlled states were reached. The disturbance rejection was investigated by varying the flow rate of the evaporating and condensing media. 5.1. Reference tracking and disturbance analysis The test was initiated when the steady-state condition was achieved for the following operating conditions: compressor speed of 3000 rpm, valve opening of 45%, evaporating medium flow rate and temperature of 1.24 L/min and 10 C, respectively, and condensing medium flow rate and temperature of 1.17 L/min and 35 C,
(18)
where L and S are calculated according to the procedure described in [17]. 4. Experimental facility Fig. 1 shows a schematic diagram of the HFC-134a refrigeration system used in this study. The compressor is of hermetic variablespeed reciprocating type, with minimum and maximum speeds of 1800 and 4500 rpm, respectively. The condenser and evaporator are both tube-in-tube heat exchangers, connected to two distinct temperature and flow controlled secondary heat transfer loops. Brine of 27% ethylene-glycol aqueous solution is used for the evaporator, whereas pure water is used for the condenser. The expansion device is a pulse-width-modulation (PWM) electric expansion device with variable opening from 0 to 100%. The condenser and evaporator heat transfer rates are obtained from energy balances on the refrigerant and secondary fluid sides. The refrigerant heat transfer rate was calculated by multiplying the refrigerant mass flow rate, measured by a Coriolis mass flow meter (measurement uncertainty of 0.1%) by the enthalpy difference between the heat exchanger inlet and outlet, which provided a maximum measurement uncertainty of 2% [4]. Immersion T-type thermocouples (measurement uncertainty of 0.2 C) and pressure transducers (measurement uncertainty of 0.15%) were installed at selected points of the circuit to obtain the refrigerant enthalpies and the evaporator superheating. The secondary fluid heat transfer rate was calculated using the volumetric flow rate, measured by a turbine flow meter (measurement uncertainty of 1%), the fluid density, the specific heat and the temperature difference between the heat exchanger inlet and outlet.
a 3.0 Evaporating pressure [bar]
yref +
Experimental Simulation
2.5
2.0
1.5
1.0
0
b 16 Condensing pressure [bar]
1542
50
100
150
200 250 Time [min]
300
350
400
150
200 250 Time [min]
300
350
400
Experimental Simulation
15
14
13
12
11
0
50
100
Fig. 4. Model validation exercise: (a) evaporating pressure; and (b) condensing pressure.
L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546
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Table 2 Changes imposed to the evaporator superheating and brine outlet temperature during the reference tracking exercise. Time [min]
t < 20
20 < t < 40
40 < t < 60
60 < t < 80
80 < t < 100
100 < t < 120
t > 120
DTsup [ C]
16.6 4
20.2 4
12.5 4
16.6 4
16.6 5
16.6 3
16.6 4
Ts,e,o [ C]
respectively. At this condition, an evaporator superheating of 16.6 C and a brine temperature at the evaporator outlet of 4 C were achieved and adopted as references. Independent changes in the references were imposed following the sequence shown in Table 2. The dynamic behavior of the controlled variables is depicted in Fig. 5a, where it can be noted that the controller drove the refrigeration system towards the prescribed references. It is noteworthy that a reference change in one of the controlled variables produced a disturbance in its counterpart, which was identified and properly rejected by the controller. The highest disturbance level was observed during the second change in the evaporator superheating degree, at t ¼ 40 min, reaching a minimum value of 10.6 C, which represents an undershooting of 2 C. Such an effect was induced by the magnitude of the imposed change (7.7 C), negligible in the first and third changes of references (at t ¼ 20 min and t ¼ 60 min, respectively). The disturbances caused by changing the reference of the brine outlet temperature were also properly rejected by the controller, which affected the evaporator superheating in 2 C only (at t ¼ 120 min). For the brine outlet temperature, the controller showed a better performance, with a maximum undershooting of 0.5 C at the instant when the highest change of reference was observed, i.e., 2 C at t ¼ 100 min. Furthermore, the controller kept the brine outlet temperature within a 0.5 C band during the first 80 min. Fig. 5b shows the variations experienced by the control actions (i.e., compressor speed and valve opening) in order to drive the controlled variables towards the desired references. It is worthy noting that, during the change of reference of the evaporator superheating (t < 40 min), the control actions presented opposite behaviors: the compressor speed increased when the valve opening decreased. This was so as the cooling capacity tends to decrease inasmuch as the superheating degree increases, forcing the controller to increase the compressor speed in an attempt to keep the cooling capacity (see Fig. 5d), thereby reducing the evaporating pressure (see Fig. 5c) and increasing the evaporator log-mean temperature difference. The opposite behavior was observed within the time interval 40 < t < 60 min. During the change of reference of the brine outlet temperature (t > 60 min), it was noted that both compressor speed and valve opening showed the same behavior (see Fig. 5b). On one hand, to raise the brine outlet temperature (i.e., lower cooling capacity) without changing the evaporator superheating (t ¼ 80 min), the controller increased the evaporating pressure (see Fig. 5c) by reducing both the compressor speed and the expansion valve opening so as to diminish the refrigerant mass flow rate. On the other hand, the cooling capacity increased at t ¼ 100 min as the compressor speed and the valve opening were simultaneously brought up (see Fig. 5b). The disturbance rejection test aims to check whether the controller is able to keep the controlled variables at the reference previously set, even if changes are imposed on the thermal load or the surrounding conditions. Such changes have been respectively emulated in the experimental apparatus by modifying the evaporator and condenser secondary coolant flow rates, as shown in Table 3. Such variations have induced disturbances on the controlled variables that have been recognized by the controller and then rejected, as can be seen in Fig. 6a. The largest deviation observed in the evaporator superheating, in comparison to its
a
b
c
d
Fig. 5. Reference tracking analysis: (a) controlled variables; (b) control actions; (c) working pressures; and (d) cooling capacity.
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Table 3 Changes imposed to the evaporator and condenser secondary coolant mass flow rates during the disturbance rejection exercise. Time [min]
t < 20
20 < t < 40
40 < t < 60
60 < t < 80
80 < t < 100
100 < t < 120
t > 120
Vs,c [L/min] Vs,e [L/min]
1.16 1.23
1.16 1.43
1.16 1.00
1.16 1.23
1.38 1.23
0.93 1.23
1.16 1.23
a
b
reference, was 3 C at t ¼ 40 min. Such deviation is related to the sudden reduction of the brine flow rate from 1.43 to 1.00 L/min, which represented a 30% reduction in the thermal load imposed (from 550 to 385 W, see Fig. 6d). It can be also noted in Fig. 6 that the variations observed for t < 70 min are related to the thermal load disturbances, as the changes in the condenser side had no practical influence on the controlled variables and control action, such as the compressor speed and the valve opening (Fig. 6b), the mass flow rate (Fig. 6d), and evaporating temperature (Fig. 6c). The condensing pressure, on the other hand, has experienced some disturbances, although the largest variation was due to the changes on the evaporator brine flow rate (Fig. 6c). Albeit the cooling capacity (see Fig. 6d) was practically not affected by the disturbance imposed in the condenser secondary coolant flow rate (t > 70 min), it changed substantially with the evaporator brine flow rate (t < 70 min). 5.2. Assessment of the controlling envelope The operational envelope beyond which it is not possible to maintain the controlled variables at their references was evaluated by artificially driven the controller through step changes promoted in the references. In addition, changes in the evaporator brine flow rate were also imposed to the system in order to identify the maximum and minimum controllable thermal loads. First, increasing step changes (from 4.0 to 6.0 C, step of 0.5 C) on the brine outlet temperature were imposed, as shown in Fig. 7a,
c
d
Fig. 6. Disturbance rejection analysis: (a) controlled variables; (b) control actions; (c) working pressures; and (d) cooling capacity.
a
b
Fig. 7. Controlling envelope analysis with increasing brine temperature: (a) controlled variables; (b) control actions.
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a
b
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as shown in Fig. 7b. When the minimum compressor speed was reached, the evaporator superheating increased as the valve opening ruled the system while the compressor speed became saturated. Similarly, decreasing step changes (from 4.0 to 2.5 C, step of 0.5 C) on brine outlet temperature were imposed, as shown in Fig. 8a, which is equivalent to a thermal load augmentation. At each new reference change, the controller increased the cooling capacity by opening the valve and raising the compressor speed up to the maximum allowed value (4500 rpm), as shown in Fig. 8b. As the evaporator superheating was fixed at 16.6 C, the brine outlet temperature was limited within the range from 3 to 5.5 C. Alternatively, decreasing step changes (from 16.6 to 8.5 C) on the evaporator superheating were imposed, as shown in Fig. 9a, until the controller is no longer maintaining the controlled variables at their references. It is noteworthy that at superheating degrees lower than 8.5 C the system became unstable due to two factors. First, as the proposed controller is linear, it cannot identify the non-linear dynamics of the variables (see Fig. 9b). In addition, the evaporator dry-out location oscillates at lower superheating degrees due to the natural two-phase flow instabilities [11], which affected the quality of the control signal and therefore the control action.
a Fig. 8. Controlling envelope analysis with decreasing brine temperature: (a) controlled variables; (b) control actions.
which is equivalent to a thermal load reduction. In order to follow the references and also to adapt the system to lower cooling demands, the controller decreased the valve opening while reduced the compressor speed until 1800 rpm, the minimum allowed speed,
a
b
b
c
Fig. 9. Controlling envelope analysis with decreasing evaporator superheating: (a) controlled variables; (b) control actions.
Fig. 10. Controlling envelope analysis with increasing and decreasing thermal load: (a) controlled variables; (b) control actions; and (c) disturbances.
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Additional tests were carried out varying the evaporator brine flow rate, but keeping the evaporator superheating degree and the brine outlet temperature references fixed (see Fig. 10a). During the first 120 min, the flow rate was steeply increased to the maximum value that can be achieved without loosing the system controllability, as showed in Fig.10b. Therefore, the refrigeration system experienced a thermal load variation from 460 to 580 W (see Fig. 10c). In other words, a thermal load 26% higher than the nominal was achieved but keeping the preset references (see Fig. 10b). Between 120 and 140 min, the system was then returned to its initial condition. After 140 min, the brine flow rate was steeply decreased in order to reduce the thermal load to 340 W, i.e., a figure 26% lower than the nominal value (see Fig. 10c). 6. Summary and conclusions A dynamic simulation model for the identification and control of vapor compression refrigeration systems was developed and validated for a broad range of operation. A breadboard refrigeration system comprised of a variable-speed compressor and an electric expansion valve was used to gather the required experimental data for both system identification and model validation exercises. It was found that the model reproduces well the experimental trends of the working pressures even in conditions far from the operation point used in the system identification, with maximum discrepancies between calculated and measured counterparts within 5% error bands. A model-driven multivariate linear strategy for controlling the evaporator feeding and for matching the cooling capacity to the thermal loads was devised. The dynamic simulation model was linearized according to a Taylor expansion series and then used to design a proportional plus integral type controller based on the linear-quadratic-Gaussian (LQG) method, which is based on a state estimator of the Kalman filter type. Additional experimental tests were also carried out using the breadboard facility to verify the controller ability for tracking reference changes and rejecting thermal load disturbances due to the integral action of the controller. It was found that the controller was able to drive the facility for thermal loads spanning from 340 to 580 W, i.e., 26% in comparison to the nominal value. Similarly, the controller kept the refrigeration system running properly for evaporator superheating degrees ranging from 9.5 C to 22 C, albeit it was designed for 16.6 C. However, it is worthy of note that the controller cannot be applied for superheating degrees lower than 9.5 C, when both the controlled system and the control signal became unstable. Acknowledgements The authors would like to thank the CNPq (Grant No. 573581/ 2008-8 e National Institute of Science and Technology in Refrigeration and Thermophysics) for the financial support. Thanks are also addressed to Embraco S.A. for sponsoring this research project. The authors duly acknowledge Prof. C. Melo and Prof. N. Roqueiro
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