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Incorporating fan control into air-conditioning systems to improve energy efficiency and transient response
Applied Thermal Engineering 29 (2009) 1955–1964
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Incorporating fan control into air-conditioning systems to improve energy efficiency and transient response T.-J. Yeh *, Yun-Jih Chen, Wei-Yang Hwang, Jin-Long Lin Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 2 April 2008 Accepted 24 September 2008 Available online 4 October 2008 Keywords: Air-conditioning system Fan control Feedback control System identification Cascading structure
a b s t r a c t Modern air-conditioners frequently incorporate variable-speed compressors and variable-opening expansion valves with feedback control to improve performance and power efficiency. Because making the fan speeds adjustable adds flexibility to the control design and thus can lead to further improvements in performance and efficiency, this paper proposes two control algorithms, respectively, incorporating the outdoor fan and the indoor fan as the additional control inputs for air-conditioning systems. Both of the control algorithms are designed based on a low-order, linear model obtained from system identification. The first algorithm, which modulates the outdoor fan speed, can reduce the steady state power consumption if the temperature difference between the condenser and the outdoor environment is controlled properly. The second algorithm, which adds one more degree of freedom to control by modulating the indoor fan speed, can improve the transient response because actuator saturations become less likely to occur. The two control algorithms are implemented on a split-type residential air-conditioner and their respective performance is validated experimentally. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The air-conditioner is a machine operated on the vapor compression cycle. As shown in the schematic of Fig. 1, in the machine the vapor-phase refrigerant is compressed by a compressor, causing the temperature to increase. The hot, high-pressure refrigerant then enters a condenser, where it is cooled by transferring its heat to the outdoor environment through the air flow induced by an outdoor fan. As a result of the heat transfer, the refrigerant condenses from vapor to liquid. The liquid refrigerant passes through an expansion valve, in which the pressure and temperature both decrease, and enters an evaporator. In the evaporator, the cold refrigerant absorbs heat from the room via the air flow induced by an indoor fan. The heat absorption causes the refrigerant to leave the evaporator as superheated vapor and the vapor is routed back into the compressor. In conventional air-conditioners, the expansion valves are mostly mechanical and the compressors, fans are operated at constant speeds. Although actuations are simple and the associated control methods are intuitive, conventional machines often suffer from poor efficiency and large variations in indoor temperatures. Recently, modern air-conditioners have begun to incorporate electronic controllable expansion valves, variable-speed compressors and fans as the actuation devices. With the new component tech* Corresponding author. Tel.: +886 3 574 2922; fax: +886 3 572 2840. E-mail address: [email protected] (T.-J. Yeh). 1359-4311/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2008.09.017
nology, it becomes possible to implement more advanced control schemes to achieve better performance and furthermore better energy efficiency. Intuitively, simultaneously controlling all the actuation devices in the modern air-conditioners can fully exploit the control authorities so as to maximize efficiency and performance. However, in the literature, to reduce controller complexity as well as the hardware cost, most researches focus on modulating only selected actuators while keeping the other(s) at constant actuation level(s). Particularly, control algorithms that use the expansion valve opening and the compressor speed as the control inputs are most frequently investigated. For instance, in [1], a controller assuming such control inputs was proposed. The control algorithm possesses a cascading structure to deal with the fast and slow dynamics in the air-conditioning system. Experiments show that it can achieve satisfactory transient response in the indoor temperature, and improve energy efficiency at steady state. In [2], the decoupling control of automotive air-conditioner was studied. The controller therein is PI-based and is designed using a transfer function matrix from the expansion valve opening and the compressor speed to the superheat and evaporator temperatures. Using the same input– output pairs, He et al. [3] proposed a model-based LQG control system. The same authors also numerically investigated the possibility of using model-based nonlinear control for the vapor compression systems [4,5]. In the references mentioned above, both the indoor and outdoor fans are kept at constant speeds. It should be noted that the idea of
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
Nomenclature T v
xc av xof xif Tco Tco_set d s a(.),b(..)
temperature (°C) contribution from the indoor thermal load to the indoor dynamics (°C/s) compressor speed (rpm) expansion valve opening (pulse) outdoor fan speed (rpm) indoor fan speed (rpm) temperature difference between condenser and outdoor environment (°C) reference setting for the temperature difference between condenser and outdoor environment (°C) perturbation symbol Laplace variable parameters in the transfer function
modulating the fan speed during the control process is relatively unexplored. Among the limited literature, Shah et al. [6] applied a multivariable adaptive control strategy to automotive air-conditioning systems. In this work, the electronic expansion valve and the indoor fan are modulated by the controller to make the evaporator pressure track a reference signal while regulating the superheat at a set point. The control performance is validated solely by simulations in which the compressor speed and the outdoor fan speed, instead of being viewed as the control inputs, are varied stepwisely to simulate disturbances to the control systems. In [7], the authors employed the steady-state model of a vapor compression cycle to determine the optimal control set-points that minimize the total power consumption. Since the power consumption considered is contributed by the compressor and the outdoor fan, these two actuators are implicitly treated as control inputs. However, the work therein is completely based on numerical analysis of the steady-state model. No control method is specifically discussed and the results are not validated experimentally either. As one can see from the literature survey, there is a lack of systematic and experimental studies on the simultaneous control of the compressor, the expansion valve, and the fan(s) in modern air-conditioners, so the achievable efficiency and performance of the modern air-conditioner are not fully exploited. The purpose of this paper is to address this issue by adding fan controls to the control structure proposed in [1] where the expansion valve and the compressor are the control inputs. Specifically, two control algorithms will be proposed. The objective of the first control algorithms is to properly regulate the outdoor fan speed to enhance the steady-state efficiency, and the objective of the second algorithm is
Condenser
Outdoor fan Expansion valve
Compressor Indoor fan
Evaporator Fig. 1. The schematic of a vapor compression cycle.
p(.) z(.) H n(.) C(.)(s) a, b, c
pole zero humidity new control input controller weighting factors
Subscripts sh superheat c condenser e evaporator i indoor environment o outdoor environment set reference setting
to properly control the indoor fan speed to improve the transient response. 2. Influence of fan speeds on performance and efficiency The control algorithms in this paper basically follow a similar cascading structure as the one proposed in [1] but with the indoor or outdoor fan speed added as an additional control input. To begin with, we implement the original control structure proposed in [1] on an experimental machine and investigate how modulating fan speeds changes the system behavior. As will be shown, such an investigation reveals crucial control characteristics for further improving efficiency and performance. 2.1. Introduction of the basic control structure The control block diagram of Lin and Yeh [1] is depicted in Fig. 2. Controllers C1(s) and C2(s) together regulate the indoor temperature according to the user’s setting temperature Ti_set, and the controller C3(s) allows one to regulate the superheat Tsh, which is defined as the temperature excess of the superheated vapor over the two-phase refrigerant upstream the evaporator, to the desired setting Tsh_set. Specifically, C2(s) closes the outer control loop which regulates the indoor temperature Ti based on Ti_set and produces the desired evaporator temperature setting Te_set, and C1(s) closes the inner control loop which makes the evaporator temperature Te track Te_set. Because the controller design is based on a decoupled plant obtained from performing an input transformation, the outputs of C1(s) and C3(s) have to be multiplied by an input coupling matrix B1 to generate the physical inputs to the air-conditioner: the compressor speed and the expansion valve opening. Notice that C1(s), C2(s), and C3(s) are all PI controllers. To avoid integrator windup due to saturation in the physical inputs, a model-based anti-windup compensator originally developed in [11] is also incorporated. In this diagram, the anti-windup compensator receives the difference between the computed control and the saturated control as the input and produces two compensating signals v1, v2 to account for integrator windup. 2.2. Description of the experimental machine The control system is verified using a split-type residential air-conditioner with R-410A as the refrigerant and cooling capacity rated at 2900 W. The total weight of the refrigerant in the machine is 1050 g. To simulate the actual usage of the air-condi-
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
Tsh_set
+
C3(s)
_ +
Te_set
MIMO Controller B
+
-1
u
C1(s)
ωc
u
αv
+
Vapor Compression Dynamics
Indoor Dynamics
Ti
_ Tsh
+
Te
_
v1
Anti-windup compensator
v2 + +
_ C2(s)
Ti_set
+
Fig. 2. The control block diagram proposed in [1].
In the first experiment, Ti_set and Tsh_set are, respectively, set at 25 and 5 °C the indoor fan is set at high speed mode (mode 3), the outdoor fan is set at 800 rpm and the outdoor temperature To is 27 °C. Once the control responses reach steady state, the speed of the outdoor fan is decreased incrementally to investigate how the power efficiency and performance are affected. It can be seen from the experimental responses in Fig. 3 that when the fan speed decreases from 800 to 560 rpm, both the indoor temperature and the compressor speed are barely affected, but the total power consumption is decreased. However, when the fan speed decreases
rp m
800 600 400 200 0
0
10
20 time[min]
30
(b) Indoor Temperature
(c) Compressor Speed
30
2000 o
Ti set = 26 C
1500
Ti
o
C
rp m
28
1000
26 24
500 0
10
20 time[min]
0
30
0
(d) Superheat
30
Tsh set = 5oC Tsh
500 W
C
o
20 time[min]
600
8 6
400
4 2
10
(e) Power
10
0
10
20 time[min]
300
30
(f) expansion valve opening 50
140
45 C
130
0
10
20 time[min]
30
(g) condenser Temperature
150
o
2.3. Experimental results on changing fan speeds
(a) Outdoor Fan Speed 1000
p u l se
tioner, the experiments are not conducted in controlled chambers and instead the machine is installed in a common office building. The indoor unit of the machine is situated in a 5.3 m 4 m 2.8 m room. In the machine, T type thermocouples with range from 200 to 400 °C and accuracy ±0.1 °C are attached to proper locations to measure the relevant temperatures. Specifically, the measurements from the inlet of the evaporator and from the midpoint of the condenser where the refrigerant is two-phased are, respectively, viewed as the evaporator temperature and the condenser temperature. The temperature difference between the measurements taken from the inlet of the compressor and the evaporator temperature is treated as the superheat. As for the actuators, the opening of the expansion valve is controlled by a stepping motor that 480 pulses correspond to the full opening. Both the compressor and the outdoor fan are driven by DC motors and their speeds can be varied, respectively, from 1000 to 5000 rpm and from 200 to 950 rpm in a continuous manner. Notice that the limitation of the driver electronics and efficiency consideration prevents the speeds of the motors from being varied continuously to 0 rpm, so during the control process, if a speed lower than the lower threshold is desired, one simply shuts down the motor. The indoor fan is driven by an AC motor and its speed can only be set at one of the four modes: stop, low, medium, and high. Since the speed increases linearly with respect to the mode of operation, in the following discrete numbers 0–3 are used to denote the indoor fan speed. The power ratings for the compressor, the outdoor fan and the indoor fan are, respectively, 1400, 200, and 40 W. Finally, the control algorithm is implemented on a personal computer. The A/D and D/A converters both have 16-bit resolution. The sampling period employed is 4 s.
40
120 35
110 100
0
10
20 time[min]
30
30
0
10
20 time[min]
30
Fig. 3. Experimental responses when the outdoor fan speed changes decrementally (To = 27 °C).
further to 250 rpm, both the compressor speed and the total power increase slightly.
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(a) Indoor Fan Speed
4 3 2 1 0
0
50 time[min]
100
(b) Indoor Temperature
(c) Compressor Speed
30
5000 Ti set = 25 oC
4000
Ti rpm
o
C
28 26 24 22
3000 2000 1000
0
50 time[min]
0
100
0
(d) Superheat
50 time[min]
100
(e) Power
10 Tsh set = 5 C
1400
Tsh
1300
o
W
o
C
8 6
1200 1100
4
1000 2
0
50 time[min]
100
0
20
180
15 C
160
100
(g) Evaporator Temperature
200
o
pul s e
(f) expansion valve opening
50 time[min]
10
140 5
120 100
0
50 time[min]
100
0
0
50 time[min]
100
pass the corresponding compressor power increase. For the airconditioner considered, the power rating of the indoor fan is much less than that of the compressor, so the power saved by lowering the indoor fan speed is insignificant compared to the compressor power increase. Therefore, the total power consumption increases with the decrease of indoor fan speed. On the other hand, the power rating of the outdoor fan is more comparable to that of the compressor, so there exist possibilities that reducing the outdoor fan speed can lead to total power saving. From the above argument, one can conclude that controlling the speed of the outdoor fan properly can lead to power saving, but the indoor fan should be running at its highest speed to avoid extra power consumption. Motivated by this, a control algorithm which incorporates the outdoor fan speed as the third control input (in addition to the compressor speed and expansion valve opening) is proposed. Besides the indoor temperature and the superheat, the algorithm also regulates the temperature difference between the condenser and the outdoor environment. The reason for doing so is that the temperature difference not only contains the influence from the outdoor temperature, but also dictates the heat transfer rate from the condenser to the outdoor environment. Therefore, regulating it could render the controller the ability to automatically compute ‘‘optimal” speed settings for the outdoor fan at different operating conditions. Notice in this control algorithm, the indoor fan is set at high speed mode. Although it makes no sense to modulate indoor fan speed for improving power efficiency, one cannot rule out the possibility of controlling the indoor fan speed to improve the transient response. To see this, the control parameters for C1(s), C2(s), and C3(s) in the control structure of Fig. 2 are tuned more aggressively so as to speed up the transient response. As shown in Fig. 5, the aggressiveness of the controllers tends to saturate both the compressor and the expansion valve. Regardless of the anti-windup compensation, the control saturation still makes the response oscillatory which in turn leads to a longer settling time. A straightforward way to improve the transient response is to add one more degree of freedom in control by incorporating the indoor fan speed as a control input. By doing so, there is one more actuator to share the control duty that saturation becomes less likely to occur. As a result, it can be expected that fast, less
Fig. 4. Experimental responses when the indoor fan speed changes decrementally (To = 33 °C).
(a) Indoor Temperature
(b) Compressor Speed
30 Ti set = 26 C Ti
4000 rpm
o
C
28 26
3000 2000
24 22
1000 0 0
20
40 60 time[min]
80
0
(c) Superheat
20
40 60 time[min]
80
(d) Expansion Valve Opening
20 o
Tsh set = 3 C Tsh
15
400
C
pul s e
10 o
Next, the influence of modulating the speed of the indoor fan on the system performance and efficiency is investigated. To do so, the same Ti_set, Tsh_set, and initial fan speeds as before are adopted and the control experiment is re-performed. At steady state, the indoor fan is decrementally switched to medium speed (mode 2) and low speed (mode 1). As shown in the experimental responses of Fig. 4, although reducing the indoor fan speed has little influence on the indoor temperature, it does increase both the compressor speed and the power consumption considerably. Specifically, the power consumption is increased by 8% at mode 2 and by 24% at mode 1. Intuitively, reducing outdoor fan speed would lead to an increase in condensing temperature due to less heat rejection; hence more compressor power is needed to raise the discharge temperature due to the increase in condensing temperature. As for the indoor fan, reduction of its speed decreases the heat transfer coefficient from the room to the evaporator. In order to maintain the same heat flux to counteract the thermal load in the room, the evaporator temperature has to even cooler and this demands more compressor power to do so. The total power consumption is not only attributed by the compressor but also the fans. Therefore, whether the power consumption can be reduced depends on whether the power saving due to fan speed reduction can sur-
5000
o
5
300 200
0 100 -5
0
20
40 60 time[min]
80
0
20
40 60 time[min]
80
Fig. 5. Control results with more aggressive controllers and superheat setting.
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
(a) Evaporator temperature 20 Simulation Experiment
C
18 o
oscillatory transient response is easier to be achieved. The second control algorithm proposed in this paper is based on such reasoning except that the algorithm also demands the indoor fan to be operated in the high speed mode at steady state for power efficiency purposes.
16 14
3. System identification
12
2
b11 2 3 6 s þ a1 6 dT e 6 6 7 6 b21 4 dT sh 5 ¼ 6 6 s þ a1 6 dT c 4 b31 s þ a2
b12 s þ a1 b22 s þ a1 b32 s þ a2
3 b14 2 d xc s þ a1 7 7 76 dav b24 76 76 4 dxof s þ a1 7 7 b34 5 dx
b13 s þ a1 b23 s þ a1 b33 s þ a2
In this model structure, the symbol d is added to the relevant variables to address the fact that the model is based on linearization, so both inputs and outputs are perturbed from their values in the nominal condition (For example, dTe is the difference between the evaporator temperature and its value at the nominal condition). Moreover, because both Te and Tsh pertain to the responses occurring in the evaporator, the same dominating pole (a1) is assumed for them. The parameter a2 represents the most dominating pole for Tc, and bij’s are the control gains.
(b) Expansion valve opening 140
pulse
rpm
800 600 400 200 0
120 110
10 20 time[min]
100
0
rpm
(c) Compressor speed
(d) Indoor fan speed
3000
4
2500
3
2000
2
1500
1
1000 0
10 20 time[min]
10 20 time[min]
0
0
10 20 time[min]
Fig. 6. The inputs of the system identification experiment.
25
Simulation Experiment
C o
6
0
5
10
15 time[min]
20
25
(c) Condenser temperature 50 Simulation Experiment
C
45 40
0
5
10
15 time [min]
20
25
Fig. 7. The comparsion of experimental and simulated outputs with the same inputs.
During the identification, the air conditioner is first operated in the nominal condition with Te = 15.2 °C, Tsh = 5.3 °C, Tc = 42.3 °C, xc = 2000 rpm, av = 130 pulse, xof = 500 rpm, xif = 3 (mode number), the outdoor temperature (To) equal to 34 °C the outdoor humidity (Ho) equal to 50%, the indoor temperature (Ti) equal to 27 °C and the indoor humidity (Hi) equal to 57%. The inputs are then perturbed by square waves and the outputs are recorded. The waveforms for the inputs and the outputs are shown, respectively, in Figs. 6 and 7. The input–output data are substituted into the model structure in (1) and least-square method is used to solve for the model parameters [8]. The identified model is given by
Te
3
2 0:1768
sþ0:0279
6 7 6 0:141 3 4 T sh 5 ¼ 10 4 sþ0:0279 0:0565 Tc sþ0:0194
130
20
4
2
1000
15 time[min]
8
ð1Þ
s þ a2
150
10
(b) Superheat
35
if
(a) Outdoor fan speed
5
10
3 7 7 7: 5
0
o
In order to devise the two control algorithms, the dynamics of the vapor compressor cycle in the experimental machine is identified. The identification uses the compressor speed (xc, rpm), the expansion valve opening (av, pulse), the outdoor fan speed (xof, rpm), and the indoor fan speed (xif, mode number) as the inputs and the evaporator temperature (Te), the superheat temperature (Tsh), and the condenser temperature (Tc) as the outputs. Although as shown in [1] the dynamics of vapor compressor cycle is characterized by a set of high order, coupled, nonlinear differential equations, assuming that the operation of the machine is not far from the nominal condition, the associated dynamics can be linearized into linear, time-invariant (LTI) state equations. Furthermore, for the purpose of simplifying the controller design, we also assume that in these LTI state equations, the mapping between any of the perturbed-input-perturbed-output pair is characterized by a first-order transfer function of the form bðÞ =ðs þ aðÞ Þ. By the assumptions, the model is in the following structure:
2:849 sþ0:0279 2:747 sþ0:0279 0:532 sþ0:0194
0:0605 sþ0:0279 0:0926 sþ0:0279 0:199 sþ0:0194
2
4:7147 3 sþ0:0279 6 8:883 76 sþ0:0279 56 4 1:5996 sþ0:0194
xc 3 av 7 7 7; xof 5 xif
ð2Þ
in which the perturbation symbols (d’s) are omitted for simplicity. The same inputs are applied to the identified model. The simulation outputs are plotted together with the experimental ones in Fig. 7. The figure shows that the identified model matches the major dynamics of the actual system. In order to further examine the applicability of the identified model to other operating conditions, the same input wave forms as in Fig. 6 are applied to the air-conditioner under another nominal operating condition with Te = 14.0 °C, Tsh = 6.9 °C, Tc = 38.6 °C, xc = 2000 rpm, av = 130 pulse, xof = 500 rpm, xif = 3, To = 29.5 °C, Hi = 72%. The experimental responses in Te, Tsh, and Tc are shown in Fig. 8. Also shown in the same figure are the outputs simulated by the identified model of (2). Although the operating condition is different, one can see that the identified model can still represent the major dynamics of the actual system.
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
where a3 is a parameter determined by the geometry, the heat transfer coefficient of the evaporator, and the thermal capacitance of the indoor environment, and t represents the contribution from the indoor thermal load. Combining (4) and (5), the dynamics for the air-conditioning system can be depicted by the plant block in Fig. 9. The block consists of three decoupled subsystems: the n 1– Te–Ti subsystem, the n2–Tsh subsystem, and the n3–Tc subsystem. Therefore, if one independently closes the loop of each subsystems in a stable manner, the overall system is guaranteed to be stable. By this reasoning, the proposed control algorithm employs a similar control structure as the one presented in [1]. In Fig. 9, while C1(s), C2(s) control the n1–Te–Ti subsystem and C3(s) controls the n2–Tsh subsystem, there is another controller C4(s) to regulate the temperature difference Tc–To to a preset value Tco_set. Since all Ci(s)’s deal with first-order dynamics, similar to those in Fig. 2, they will be implemented as PI control for simplicity and reducing steady-state error. However, it should be noted that due to the inner-outer loop structure in the n1–Te–Ti subsystem, the design of C1(s), C2(s) may not be as straightforward as the rest of the controllers. Regarding this, one can directly employ the following theorems proposed in [1] to facilitate the design of C1(s), C2(s) for stability.
(a) Evaporator temperature 20
o
C
Simulation Experiment 15
10
0
5
10
15 20 time[min]
25
30
(b) Superheat 12 Simulation Experiment
o
C
10 8 6 0
5
10
15 20 time[min]
25
30
(c) Condenser temperature 44 Simulation Experiment
Theorem 1. (Stability) Assume that C 1 ðsÞ ¼ fK p1 ðs þ z1 Þg=s and C 2 ðsÞ ¼ fK p2 ðs þ z2 Þg=s. If C1(s) renders the closed-loop poles of the inner-loop at p1, p2, then C1(s) and C2(s) together stabilize the n 1– Te–Ti subsystem regardless the choice of Kp2 if and only if the two conditions are simultaneously satisfied.
o
C
42 40 38 18
20
22
24 26 time [min]
28
30
32
z1 þ z2 < p1 þ p2 þ a3 ;
ð6Þ
2
Fig. 8. Match of the simulated outputs with the experimental outputs under different operating condition.
4. Control algorithm which incorporates the outdoor fan speed control to improve power efficiency A control algorithm which incorporates the outdoor fan speed as the third control input (in addition to the compressor speed and expansion valve opening) to further improve power efficiency is presented in this section. In this algorithm, the indoor fan speed is kept at high speed mode. Therefore, there will be three control inputs: xc, av, and xof. Ignoring the contribution by the indoor fan speed and defining the following input transformation:
2
3 2 3 xc n1 6 7 6 7 4 n2 5 ¼ B1 4 av 5; xof n3
ð3Þ
2
3 b11 b12 b13 where B1 ¼ 4 b21 b22 b23 5 and n1, n2, and n3 are new control inb31 b32 b33 puts, the dynamics for the vapor compression cycle in (1) can be transformed into a decoupling form
2
3 2 a1 T_ e 6_ 7 6 4 T sh 5 ¼ 4 0 T_ c
0
0 a1 0
0
32
Te
3
2
n1
3
76 7 6 7 0 54 T sh 5 þ 4 n2 5: Tc n3 a2
ð4Þ
Next for the indoor dynamics, it follows from the physical modeling in [1] that the influence of the vapor compression cycle dynamics towards the indoor temperature Ti can be approximated by the following first-order equation:
T_ i ¼ a3 T i þ a3 T e þ v;
ð5Þ
ðp1 þ p2 þ a3 Þ 1 1 < þ : ðp1 þ p2 Þðp2 þ a3 Þðp1 þ a3 Þ z1 z2
ð7Þ
Corollary 2. Under the same assumptions as in Theorem 1, C1(s) and C2(s) together stabilize the n 1–Te–Ti subsystem regardless of a3 and the choice of Kp2 if the two conditions are simultaneously satisfied.
z1 þ z2 < p1 þ p2 ; 1 1 1 1 þ < þ : p1 p2 z1 z2
ð8Þ ð9Þ
For the indoor environment considered, the transfer function for the indoor dynamics is estimated using physics [1] as 0:00036=ðs þ 0:00036Þ when the indoor fan is maintained at high speed. Instead of directly tuning of the proportional and integral gains in the controllers Ci(s)’s, the estimated indoor dynamics and the identified vapor-compression-cycle dynamics in (2) are substituted into LQR methodology to obtain the control parameters. The reason why LQR is used is because it is well-known for providing 6 db gain margin and 60° phase margin [9]. These stability margins can provide further robustness to modelling errors caused by possible variations in operating condition (including outdoor humidity, initial indoor humidity, outdoor temperature, initial indoor temperature, indoor thermal disturbance, user’s indoor temperature setting, refrigerant inventory, and so on). The objective of LQR design is to minimize a cost function in the R1 form of 0 xT Qx þ uT Ru dt, where x is the state vector, u is the control, Q is the state weighting matrix and R is the control weighting matrix. LQR utilizes the weighting matrices and the state equation which describe the system dynamics to formulate a Riccati equation to solve for the control parameters [10]. In this research, the tuning of the control parameters are accomplished by iteratively changing weighting matrices (Q, R) so that the resultant controllers can satisfy the stability conditions in Theorem 1 and Corollary 2 and can achieve satisfactory experimental performance
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
_ To
Tco
+
_ Tco _ set
+
ξ3
C4(s)
1 s + a2
Tc Decoupled plant block
_
Tsh _ set
+
ξ2
C3(s)
+ Te _ set
1 s + a1
ξ1
C1(s)
1 s + a1
_
Tsh Te
a3
+
1 s + a3
+
Ti
ν
_ C2(s)
+
Ti _ set
Fig. 9. Outdoor fan control block diagram.
in terms of rise time, overshoot, and so on. The controllers designed eventually are given by
(a) Power
(b) Compressor Speed 3000
800
0:001618ðs þ 0:0771Þ ; s 6ðs þ 0:0035Þ ; C 2 ðsÞ ¼ s 0:001618ðs þ 0:0771Þ ; C 3 ðsÞ ¼ s 0:001518ðs þ 0:0381Þ C 4 ðsÞ ¼ : s
C 1 ðsÞ ¼
2500 [rpm]
Watt
700 600 500
ð10Þ
400
1500 8
10 T
(a) Power
(b) Compressor Speed 3000
12
14
16
o
co set
[ C]
8
10
12
14
16
o
T
co set
[ C]
(c) Fan Speed 800 [rpm]
Notice that to avoid integrator windup due to the saturation in the physical inputs, a model-based anti-windup compensator as the one used in [1] is incorporated in the control algorithm. Moreover, the algorithm also demands appropriate settings for Tsh_set and Tco_set. In the experiments, Tsh_set = 5 °C is adopted because, from prior control experiences in [1], such a setting is high enough to prevent liquid refrigerant from entering the compressor during the control transience and yet it is low enough to ensure acceptable power efficiency. On the other hand,
2000
o
600
T =32 C;H =53%
400
T =27 C;H =52%
o
o
o
o
o
200 8
10 T
12
14
16
o
co set
[ C]
Fig. 11. Effects of changing Tco_set under low outdoor humidity.
800 2500 [rpm]
Watt
700 600 500 400
2000 1500
8
10 T
12
14
16
o
co set
[ C]
8
10
12
14
16
o
T
co set
[ C]
(c) Fan Speed
[rpm]
800 o
600
To=28 C;H o=64%
400
T =26 C;H =62%
o
o
o
200 8
10 T
12
14
16
o
co set
[ C]
Fig. 10. Effects of changing Tco_set under high outdoor humidity.
to determine an appropriate setting for Tco_set, control experiments with different Tco_set are conducted. In these experiments, Ti_set is set at 25 °C and Tco_set are alternatively set as 9, 11, 13, and 15 °C. In Figs. 10 and 11, the total power consumption, the compressor speed, and the fan speed, measured at the steady state of the experiment, are plotted as a function of Tco_set under four outdoor conditions (To = 28 °C, Ho = 64%, and To = 26 °C, Ho = 62%, and To = 32 °C, Ho = 53%, and To = 27 °C, Ho = 52%). It is found that regardless of the change of the outdoor temperature and the outdoor humidity, Tco_set power curves are concave up, and Tco_set = 11 °C seems to minimize the power consumption. The characteristics in the Tco_set–power curves can be explained by observing how the compressor speed and the fan speed vary with Tco_set: at low Tco_set, the condenser temperature, consequently the condenser pressure, is low.In this case, to maintain sufficient heat transfer rate to the outdoor environment, the outdoor fan has to be run at the high speed, so the associate fan power is high.
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(a) Indoor Temperature
Ti (w/o fan control)
Ti set = 26 C
=5 C
10
T sh (w/o fan control)
27
60
(c) Temperature Difference Tco
0
20
40 time[min]
60
0
(d) Evaporator Temperature
Tco set = 11 C
30
Te
Tco
25
Te (with fan control)
20
Te (w/o fan control)
10
C
set
o
C
o
o
C
15
15 10 0
20
40 time[min]
5
60
0
(e) Compressor Speed
40 time[min]
60
20
40 time[min]
60
rpm 0
(g) Power
40 time[min]
100 time[min]
0
800 600
800
Energy Saving: 8.45% ↓ ↑
600
W
1000
Watt-h ou r
1200
400 200
400 0
20
40 time[min]
60
0
0
20
40 time[min]
60
Fig. 12. Experimental results of outdoor fan control with To = 30.5 °C, Ho = 55%, initial indoor humidity = 60%.
Contrarily, at high Tco_set, the condenser temperature as well as condenser pressure are high, so by the increase in the compressor speed observed in Fig. 10(b), the compressor power increases. Due to the high temperature difference between the condenser and the outdoor environment, the outdoor fan does not have to be run at high speed to maintain the heat transfer rate. Therefore, the fan power decreases. In summary, the compressor power tends to increase with Tco_set, but the outdoor fan power decreases with Tco_set. Since the total power is contributed by both the compressor and the fan, there is no surprise that an optimal Tco_set which minimizes the total power consumption is found from the control experiments. Once the optimal Tco_set is found, more experiments are performed to verify the control performance. Fig. 12 shows the experimental responses corresponding to To = 30.5 °C, Ti_set = 26 °C, Ho = 55%, initial indoor humidity = 60% by solid curves. Also shown in the figure by dash–dot curves are the responses associated with the control system which does not incorporate fan speed as the control input. It can be seen that although the outdoor fan speed initially rises to 920 rpm, the proposed control algorithm can automatically regulate the fan speed to 430 rpm at steady state. As a result of this regulation, the steady-state energy is 8.45% less than the case without outdoor fan control. The experimental responses corresponding a different outdoor temperature and different humidity levels are also shown in Fig. 13. This figure corresponds to the operating con-
100 time[min]
150
(f) Outdoor Fan Speed with fan control w/o fan control
1000 500
0
50
100 time[min]
0
150
0
(g) Power
with fan control w/o fan control
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with fan control w/o fan control
(h) Watt hour
with fan control w/o fan control
Te (w/o fan control)
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Te (with fan control)
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W
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(h) Watt hour 5000
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W att-h our
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Te
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Tco
3000
2000
100 time[min]
(d) Evaporator Temperature
Tco set = 11 C
5000
500
50
30
(e) Compressor Speed
rp m
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with fan control w/o fan control
1000
150
o
0
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4000
20 18 16 14 12 10 8
(f) Outdoor Fan Speed
5000
rp m
20
100 time[min]
(c) Temperature Difference Tco
35 o
50
C
40 time[min]
Tsh (w/o fan control)
10
o
20
rpm
0
= 5 oC
5
26 0
20
5
Ti (w/o fan control)
28
set
Tsh (with fan control)
15
5
26 25
Tsh
Ti (with fan control)
30
C
28
set
T sh (with fan control)
o
T sh
15
C
29
Ti (with fan control)
20
o
o
o
Ti set = 26 C
C
30
(b) Superheat
32
20
o
o
C
o
(a) Indoor Temperature
(b) Superheat
31
with fan control w/o fan control
4000
Energy Saving: 7.71%
3000
↓ ↑
2000 1000
0
50
100 time[min]
150
0
0
50
100 time[min]
150
Fig. 13. Experimental results of outdoor fan control with T0 = 29 °C, H0 = 50%, initial indoor humidity = 73%.
dition that To = 29 °C, Ho = 50%, initial indoor humidity = 73%. Because the increase in the initial indoor humidity increases the thermal capacitance of the indoor room, which in turn slows down the indoor dynamics [1], the settling time for the indoor temperature is slower than that in Fig. 12. The steady-state fan speed in this case becomes 510 rpm and the energy saving is 7.71%. 5. Control algorithm which incorporates the indoor fan speed control to improve transient response A control algorithm which can further improve the transient response is presented in this section. In this algorithm, the indoor fan speed is incorporated as an additional control input, but the outdoor fan speed is kept constant. Therefore, there will be three control inputs: xc, av, and xif. Ignoring the contribution by the outdoor fan speed and defining the following input transformation:
n1 n2
2
xc
3
6 7 ¼ B2 4 av 5;
ð11Þ
xif
b11 b12 b14 and n1, n2 are new control inputs, the b21 b22 b24 dynamics for Te and Tsh in (1) can be transformed into a decoupling form where B2
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¼
a1 0
0 a1
Te n1 þ : T sh n2
(a) Indoor Temperature
ð12Þ
5000
o
Ti set = 26 C Ti with fan control Ti w/o fan control
30
o
C
With such decoupled dynamics, the plant for the air-conditioning system is again in a decoupling form as in Fig. 9 except that the dynamics associated with the condenser temperature (Tc) does not appear. Consequently, a similar control structure as the control algorithm in Fig. 9 can be employed. Specifically, n1, which controls the evaporator temperature and indoor temperature, is determined simultaneously by the inner controller C1(s) and the outer controller C2(s); n2, which controls the superheat temperature, is determined by the controller C3(s). It should be noted that n1 and n2 only act as intermediate variables for control system design. In actual implementation, one has to resolve the physical inputs xc, av, and xif from n 1, n2 using (11). Because there is one control redundancy in T (11), infinite solutions of ½ xc av xif exist. One way to determine a viable solution for the physical inputs is to formulate a conT strained linear optimization problem. In detail, once ½ n1 n2 is given, it is desired to find a solution that satisfies (11) and minimizes the quadratic cost J ¼ ax2c þ ba2v þ cx2if , where a, b, c > 0 are the weighting factors. The problem can be easily solved with the method of Lagrangian multipliers and the optimal solution is given by
28
with fan control w/o fan control
4000 3000 2000
26 24
1000 0 0
20
15 10
0
with fan control w/o fan control
400 300
100 0
20
40 60 time[min]
80
0
(e) Evaporator Temperature Error
20
C1(s)
3 10 o
2 0 1 -10 0
20
40 60 time[min]
80
0
0
Tsh_set
C3(s)
-
40 60 time[min]
ξ2
80
Fig. 15. Indoor fan control experiment.
0:001518ðs þ 0:0771Þ ; s 8ðs þ 0:0028Þ ; C 2 ðsÞ ¼ s 0:001015ðs þ 0:0768Þ C 3 ðsÞ ¼ : s C 1 ðsÞ ¼
ð14Þ
It should be noted that C1(s) and C2(s) are also designed using the theorems presented in the section. The weighting matrix h previous 2 1 2 0:852 i 1 . The control reW is selected to be diag 2000 120 1 sponses of the proposed control algorithm are shown in Fig. 15. In the figure, with similar experimental conditions as those for Fig. 5
-
u
u
Vapor Compression Dynamics
+ + + -
+
20
+
Ti_set
ξ1 T -1 W -1B-1 2 ( B 2 WB 2 )
+
80
(f) Indoor Fan Speed
C2(s)
-
40 60 time[min]
with fan control w/o fan control
20
ð13Þ
where W ¼ diag ½ a b c is the weighting matrix. The block diagram of this control algorithm is shown in Fig. 14. In this diagram, an anti-windup compensator is also incorporated to avoid integrator windup due to the saturation in the physical inputs. Notice that, before the input associated with the indoor fan speed is actually injected to the air conditioner, it is quantized to an integer so that it can correspond to the mode of operation of the indoor fan. Moreover, in order for the indoor fan to operate at high speed mode (mode 3), one can increase c, the weighting factor for xif, so that the indoor fan speed is more penalized. By doing so, the control authority in the indoor fan speed is reduced and xif tends to stay around zero at steady state. Because xif represents a control perturbation from the nominal indoor fan speed (mode 3), increasing the associated weighting factor can make the indoor fan return to its nominal mode of operation at steady state. In the control experiment, the same C1(s), C2(s), C3(s) used for conducting the experiment in Fig. 5 are adopted. They are, respectively, given by
+
80
4
xc 1 n 1 6 7 1 T 1 T ; 4 av 5 ¼ W B2 B2 W B2 n2 xif
Te_set
40 60 time[min]
200
5
3
20
(d) Expansion Valve Opening
Tsh set = 3 C Tsh with fan control Tsh w/o fan control
20
-5
0
o
25
C
80
(c) Superheat
30
o
40 60 time[min]
C
2
(b) Compressor Speed
32
rp m
T_ e _T sh
#
p u lse
"
Anti-windup Compensator
+
Fig. 14. Indoor fan control algorithm.
Indoor Dynamics
Ti
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(To = 33 °C, Ho = 55%), the indoor temperature (Ti) response for the proposed control algorithm is less oscillatory and is settled within 26 ± 0.3°C in 15 min. The mode of operation for the indoor fan follows the sequence 3–2–1–2–3 that the fan speed is reduced during the control transience. Particularly, the initial reduction of the fan speed decreases the heat transfer from the room to the evaporator. This makes the dynamics of the vapor compression cycle less coupled with the indoor dynamics which in turn makes the compressor speed (xc) and the expansion valve opening (av) easier to regulate the superheat (Tsh) and the evaporator temperature (Te). From Figs. 15(b) and 15(d) where the histories for xc and av are shown, it can be seen that incorporating the indoor fan speed as an additional control input makes the saturation in xc and av occurred less frequently, so the responses for Tsh and Te, respectively, in Figs. 15(c) and 15(e) become less oscillatory. The improvements in the responses of Tsh and Te can offset the decrease of the initial heat transfer from the room to the evaporator caused by fan speed reduction, so a shorter settling time is observed in the indoor temperature response (Fig. 15(a)). Notice that at steady state, the indoor fan speed automatically settles at high speed mode, so the air-conditioning system is operated efficiently. 6. Conclusions Two control algorithms, respectively, incorporating the outdoor fan and the indoor fan as the additional control inputs for air-conditioning systems are presented in this paper. While the first algorithm can enhance the steady-state power efficiency, the second one can improve the transient response. In practical applications in which the design is usually constrained by hardware cost, system complexity, or the computation power available, one can implement only one of the algorithms to improve either the efficiency or the transient behavior. By doing so, only one fan speed needs to be modulated in real-time, the other is maintained at constant speed. If cost, complexity, and computation power are not an issue, then both algorithms can be simultaneously implemented in the air-conditioning machine in the way that the second algorithm is responsible for the control action during the startup/transient phase of operation, and the first algorithm takes over the control when steady state is reached. A bumpless transfer scheme [11,12] can be incorporated in this case to ensure the control action is transferred smoothly between the
two algorithms. Besides, the air-conditioner considered in this paper has only one evaporator. As the demand for multi-room airconditioning service for common households is growing, the market attention is shifting from the single-evaporator machine to the multi-evaporator air-conditioner (MEAC). Because making the fan speeds continuously adjustable for control purposes apparently adds flexibility to the controller design and thus can lead to performance and/or efficiency improvements, on-going researches are devoted to incorporating indoor/outdoor fan speeds as the control inputs for MEAC’s. Acknowledgement The authors gratefully acknowledge the support provided by the Energy and Environment Research Laboratories of Industrial Technology Research Institute in Taiwan. References [1] J.-L. Lin, T.-J. Yeh, Modeling, identification and control of air-conditioning systems, International Journal of Refrigeration 30 (2) (2007) 209–220. [2] M. Hattori et. al., Automotive refrigeration system controller with a simple precompensator, in: Proceedings of the 29th Conference on Decision and Control, Hawaii, 1990, pp. 1590–1591. [3] X.-D. He, S. Liu, H.H. Asada, Hiroyuki Itoh, Multivariable control of vapor compression systems, HVAC&R Research 4 (3) (1998) 205–230. [4] T. Cheng, X.-D. He, H.H. Asada, Nonlinear observer design for two-phase flow heat exchangers of air conditioning systems, in: Proceeding of the American Control Conference, USA, 2004, pp. 1534–1539. [5] T. Cheng, X.-D. He, H.H. Asada, Shinichi Kasahara, Heat exchanger dynamic observer design, ASHRAE Transactions 111 (2005) 328–335. [6] R. Shah, B.P. Rasmuussen, A.G. Alleyne, Application of a multivariable adaptive control strategy to automotive air conditioning systems, International Journal of Adaptive Control and Signal Processing 18 (2004) 199–221. [7] L.S. Larsen, C. Thybo, Potential energy savings in refrigeration systems using optimal set-points, in: Proceedings of the 2004 IEEE International Conference on Control Applications, Taipei, 2004, pp. 701–704. [8] R. Johansson, System Modeling and Identification, Prentice-Hall, Englewood Cliffs, NJ, 1993. [9] M.G. Safonov, M. Athans, Gain and phase margins for multiloop LQG regulators, IEEE Transactions on Automatic Control AC-22 (2) (1977) 1735–1744. [10] J.M. Maciejowski, Multivariable Feedback Design, Addision-Wesley, Reading, MA, 1989. [11] L. Zaccarian, A.R. Teel, A common framework for anti-windup, bumpless transfer and reliable designs, Automatica 38 (2002) 1735–1744. [12] S.F. Graebe, A.L.B. Ahlén, Dynamic transfer among alternative controllers and its relation to anti-windup controller design, IEEE Transactions on Control System and Technology 4 (1) (1996) 92–99.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Incorporating fan control into air-conditioning systems to improve energy efficiency and transient response T.-J. Yeh *, Yun-Jih Chen, Wei-Yang Hwang, Jin-Long Lin Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 2 April 2008 Accepted 24 September 2008 Available online 4 October 2008 Keywords: Air-conditioning system Fan control Feedback control System identification Cascading structure
a b s t r a c t Modern air-conditioners frequently incorporate variable-speed compressors and variable-opening expansion valves with feedback control to improve performance and power efficiency. Because making the fan speeds adjustable adds flexibility to the control design and thus can lead to further improvements in performance and efficiency, this paper proposes two control algorithms, respectively, incorporating the outdoor fan and the indoor fan as the additional control inputs for air-conditioning systems. Both of the control algorithms are designed based on a low-order, linear model obtained from system identification. The first algorithm, which modulates the outdoor fan speed, can reduce the steady state power consumption if the temperature difference between the condenser and the outdoor environment is controlled properly. The second algorithm, which adds one more degree of freedom to control by modulating the indoor fan speed, can improve the transient response because actuator saturations become less likely to occur. The two control algorithms are implemented on a split-type residential air-conditioner and their respective performance is validated experimentally. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The air-conditioner is a machine operated on the vapor compression cycle. As shown in the schematic of Fig. 1, in the machine the vapor-phase refrigerant is compressed by a compressor, causing the temperature to increase. The hot, high-pressure refrigerant then enters a condenser, where it is cooled by transferring its heat to the outdoor environment through the air flow induced by an outdoor fan. As a result of the heat transfer, the refrigerant condenses from vapor to liquid. The liquid refrigerant passes through an expansion valve, in which the pressure and temperature both decrease, and enters an evaporator. In the evaporator, the cold refrigerant absorbs heat from the room via the air flow induced by an indoor fan. The heat absorption causes the refrigerant to leave the evaporator as superheated vapor and the vapor is routed back into the compressor. In conventional air-conditioners, the expansion valves are mostly mechanical and the compressors, fans are operated at constant speeds. Although actuations are simple and the associated control methods are intuitive, conventional machines often suffer from poor efficiency and large variations in indoor temperatures. Recently, modern air-conditioners have begun to incorporate electronic controllable expansion valves, variable-speed compressors and fans as the actuation devices. With the new component tech* Corresponding author. Tel.: +886 3 574 2922; fax: +886 3 572 2840. E-mail address: [email protected] (T.-J. Yeh). 1359-4311/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2008.09.017
nology, it becomes possible to implement more advanced control schemes to achieve better performance and furthermore better energy efficiency. Intuitively, simultaneously controlling all the actuation devices in the modern air-conditioners can fully exploit the control authorities so as to maximize efficiency and performance. However, in the literature, to reduce controller complexity as well as the hardware cost, most researches focus on modulating only selected actuators while keeping the other(s) at constant actuation level(s). Particularly, control algorithms that use the expansion valve opening and the compressor speed as the control inputs are most frequently investigated. For instance, in [1], a controller assuming such control inputs was proposed. The control algorithm possesses a cascading structure to deal with the fast and slow dynamics in the air-conditioning system. Experiments show that it can achieve satisfactory transient response in the indoor temperature, and improve energy efficiency at steady state. In [2], the decoupling control of automotive air-conditioner was studied. The controller therein is PI-based and is designed using a transfer function matrix from the expansion valve opening and the compressor speed to the superheat and evaporator temperatures. Using the same input– output pairs, He et al. [3] proposed a model-based LQG control system. The same authors also numerically investigated the possibility of using model-based nonlinear control for the vapor compression systems [4,5]. In the references mentioned above, both the indoor and outdoor fans are kept at constant speeds. It should be noted that the idea of
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Nomenclature T v
xc av xof xif Tco Tco_set d s a(.),b(..)
temperature (°C) contribution from the indoor thermal load to the indoor dynamics (°C/s) compressor speed (rpm) expansion valve opening (pulse) outdoor fan speed (rpm) indoor fan speed (rpm) temperature difference between condenser and outdoor environment (°C) reference setting for the temperature difference between condenser and outdoor environment (°C) perturbation symbol Laplace variable parameters in the transfer function
modulating the fan speed during the control process is relatively unexplored. Among the limited literature, Shah et al. [6] applied a multivariable adaptive control strategy to automotive air-conditioning systems. In this work, the electronic expansion valve and the indoor fan are modulated by the controller to make the evaporator pressure track a reference signal while regulating the superheat at a set point. The control performance is validated solely by simulations in which the compressor speed and the outdoor fan speed, instead of being viewed as the control inputs, are varied stepwisely to simulate disturbances to the control systems. In [7], the authors employed the steady-state model of a vapor compression cycle to determine the optimal control set-points that minimize the total power consumption. Since the power consumption considered is contributed by the compressor and the outdoor fan, these two actuators are implicitly treated as control inputs. However, the work therein is completely based on numerical analysis of the steady-state model. No control method is specifically discussed and the results are not validated experimentally either. As one can see from the literature survey, there is a lack of systematic and experimental studies on the simultaneous control of the compressor, the expansion valve, and the fan(s) in modern air-conditioners, so the achievable efficiency and performance of the modern air-conditioner are not fully exploited. The purpose of this paper is to address this issue by adding fan controls to the control structure proposed in [1] where the expansion valve and the compressor are the control inputs. Specifically, two control algorithms will be proposed. The objective of the first control algorithms is to properly regulate the outdoor fan speed to enhance the steady-state efficiency, and the objective of the second algorithm is
Condenser
Outdoor fan Expansion valve
Compressor Indoor fan
Evaporator Fig. 1. The schematic of a vapor compression cycle.
p(.) z(.) H n(.) C(.)(s) a, b, c
pole zero humidity new control input controller weighting factors
Subscripts sh superheat c condenser e evaporator i indoor environment o outdoor environment set reference setting
to properly control the indoor fan speed to improve the transient response. 2. Influence of fan speeds on performance and efficiency The control algorithms in this paper basically follow a similar cascading structure as the one proposed in [1] but with the indoor or outdoor fan speed added as an additional control input. To begin with, we implement the original control structure proposed in [1] on an experimental machine and investigate how modulating fan speeds changes the system behavior. As will be shown, such an investigation reveals crucial control characteristics for further improving efficiency and performance. 2.1. Introduction of the basic control structure The control block diagram of Lin and Yeh [1] is depicted in Fig. 2. Controllers C1(s) and C2(s) together regulate the indoor temperature according to the user’s setting temperature Ti_set, and the controller C3(s) allows one to regulate the superheat Tsh, which is defined as the temperature excess of the superheated vapor over the two-phase refrigerant upstream the evaporator, to the desired setting Tsh_set. Specifically, C2(s) closes the outer control loop which regulates the indoor temperature Ti based on Ti_set and produces the desired evaporator temperature setting Te_set, and C1(s) closes the inner control loop which makes the evaporator temperature Te track Te_set. Because the controller design is based on a decoupled plant obtained from performing an input transformation, the outputs of C1(s) and C3(s) have to be multiplied by an input coupling matrix B1 to generate the physical inputs to the air-conditioner: the compressor speed and the expansion valve opening. Notice that C1(s), C2(s), and C3(s) are all PI controllers. To avoid integrator windup due to saturation in the physical inputs, a model-based anti-windup compensator originally developed in [11] is also incorporated. In this diagram, the anti-windup compensator receives the difference between the computed control and the saturated control as the input and produces two compensating signals v1, v2 to account for integrator windup. 2.2. Description of the experimental machine The control system is verified using a split-type residential air-conditioner with R-410A as the refrigerant and cooling capacity rated at 2900 W. The total weight of the refrigerant in the machine is 1050 g. To simulate the actual usage of the air-condi-
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Tsh_set
+
C3(s)
_ +
Te_set
MIMO Controller B
+
-1
u
C1(s)
ωc
u
αv
+
Vapor Compression Dynamics
Indoor Dynamics
Ti
_ Tsh
+
Te
_
v1
Anti-windup compensator
v2 + +
_ C2(s)
Ti_set
+
Fig. 2. The control block diagram proposed in [1].
In the first experiment, Ti_set and Tsh_set are, respectively, set at 25 and 5 °C the indoor fan is set at high speed mode (mode 3), the outdoor fan is set at 800 rpm and the outdoor temperature To is 27 °C. Once the control responses reach steady state, the speed of the outdoor fan is decreased incrementally to investigate how the power efficiency and performance are affected. It can be seen from the experimental responses in Fig. 3 that when the fan speed decreases from 800 to 560 rpm, both the indoor temperature and the compressor speed are barely affected, but the total power consumption is decreased. However, when the fan speed decreases
rp m
800 600 400 200 0
0
10
20 time[min]
30
(b) Indoor Temperature
(c) Compressor Speed
30
2000 o
Ti set = 26 C
1500
Ti
o
C
rp m
28
1000
26 24
500 0
10
20 time[min]
0
30
0
(d) Superheat
30
Tsh set = 5oC Tsh
500 W
C
o
20 time[min]
600
8 6
400
4 2
10
(e) Power
10
0
10
20 time[min]
300
30
(f) expansion valve opening 50
140
45 C
130
0
10
20 time[min]
30
(g) condenser Temperature
150
o
2.3. Experimental results on changing fan speeds
(a) Outdoor Fan Speed 1000
p u l se
tioner, the experiments are not conducted in controlled chambers and instead the machine is installed in a common office building. The indoor unit of the machine is situated in a 5.3 m 4 m 2.8 m room. In the machine, T type thermocouples with range from 200 to 400 °C and accuracy ±0.1 °C are attached to proper locations to measure the relevant temperatures. Specifically, the measurements from the inlet of the evaporator and from the midpoint of the condenser where the refrigerant is two-phased are, respectively, viewed as the evaporator temperature and the condenser temperature. The temperature difference between the measurements taken from the inlet of the compressor and the evaporator temperature is treated as the superheat. As for the actuators, the opening of the expansion valve is controlled by a stepping motor that 480 pulses correspond to the full opening. Both the compressor and the outdoor fan are driven by DC motors and their speeds can be varied, respectively, from 1000 to 5000 rpm and from 200 to 950 rpm in a continuous manner. Notice that the limitation of the driver electronics and efficiency consideration prevents the speeds of the motors from being varied continuously to 0 rpm, so during the control process, if a speed lower than the lower threshold is desired, one simply shuts down the motor. The indoor fan is driven by an AC motor and its speed can only be set at one of the four modes: stop, low, medium, and high. Since the speed increases linearly with respect to the mode of operation, in the following discrete numbers 0–3 are used to denote the indoor fan speed. The power ratings for the compressor, the outdoor fan and the indoor fan are, respectively, 1400, 200, and 40 W. Finally, the control algorithm is implemented on a personal computer. The A/D and D/A converters both have 16-bit resolution. The sampling period employed is 4 s.
40
120 35
110 100
0
10
20 time[min]
30
30
0
10
20 time[min]
30
Fig. 3. Experimental responses when the outdoor fan speed changes decrementally (To = 27 °C).
further to 250 rpm, both the compressor speed and the total power increase slightly.
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(a) Indoor Fan Speed
4 3 2 1 0
0
50 time[min]
100
(b) Indoor Temperature
(c) Compressor Speed
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5000 Ti set = 25 oC
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Ti rpm
o
C
28 26 24 22
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0
50 time[min]
0
100
0
(d) Superheat
50 time[min]
100
(e) Power
10 Tsh set = 5 C
1400
Tsh
1300
o
W
o
C
8 6
1200 1100
4
1000 2
0
50 time[min]
100
0
20
180
15 C
160
100
(g) Evaporator Temperature
200
o
pul s e
(f) expansion valve opening
50 time[min]
10
140 5
120 100
0
50 time[min]
100
0
0
50 time[min]
100
pass the corresponding compressor power increase. For the airconditioner considered, the power rating of the indoor fan is much less than that of the compressor, so the power saved by lowering the indoor fan speed is insignificant compared to the compressor power increase. Therefore, the total power consumption increases with the decrease of indoor fan speed. On the other hand, the power rating of the outdoor fan is more comparable to that of the compressor, so there exist possibilities that reducing the outdoor fan speed can lead to total power saving. From the above argument, one can conclude that controlling the speed of the outdoor fan properly can lead to power saving, but the indoor fan should be running at its highest speed to avoid extra power consumption. Motivated by this, a control algorithm which incorporates the outdoor fan speed as the third control input (in addition to the compressor speed and expansion valve opening) is proposed. Besides the indoor temperature and the superheat, the algorithm also regulates the temperature difference between the condenser and the outdoor environment. The reason for doing so is that the temperature difference not only contains the influence from the outdoor temperature, but also dictates the heat transfer rate from the condenser to the outdoor environment. Therefore, regulating it could render the controller the ability to automatically compute ‘‘optimal” speed settings for the outdoor fan at different operating conditions. Notice in this control algorithm, the indoor fan is set at high speed mode. Although it makes no sense to modulate indoor fan speed for improving power efficiency, one cannot rule out the possibility of controlling the indoor fan speed to improve the transient response. To see this, the control parameters for C1(s), C2(s), and C3(s) in the control structure of Fig. 2 are tuned more aggressively so as to speed up the transient response. As shown in Fig. 5, the aggressiveness of the controllers tends to saturate both the compressor and the expansion valve. Regardless of the anti-windup compensation, the control saturation still makes the response oscillatory which in turn leads to a longer settling time. A straightforward way to improve the transient response is to add one more degree of freedom in control by incorporating the indoor fan speed as a control input. By doing so, there is one more actuator to share the control duty that saturation becomes less likely to occur. As a result, it can be expected that fast, less
Fig. 4. Experimental responses when the indoor fan speed changes decrementally (To = 33 °C).
(a) Indoor Temperature
(b) Compressor Speed
30 Ti set = 26 C Ti
4000 rpm
o
C
28 26
3000 2000
24 22
1000 0 0
20
40 60 time[min]
80
0
(c) Superheat
20
40 60 time[min]
80
(d) Expansion Valve Opening
20 o
Tsh set = 3 C Tsh
15
400
C
pul s e
10 o
Next, the influence of modulating the speed of the indoor fan on the system performance and efficiency is investigated. To do so, the same Ti_set, Tsh_set, and initial fan speeds as before are adopted and the control experiment is re-performed. At steady state, the indoor fan is decrementally switched to medium speed (mode 2) and low speed (mode 1). As shown in the experimental responses of Fig. 4, although reducing the indoor fan speed has little influence on the indoor temperature, it does increase both the compressor speed and the power consumption considerably. Specifically, the power consumption is increased by 8% at mode 2 and by 24% at mode 1. Intuitively, reducing outdoor fan speed would lead to an increase in condensing temperature due to less heat rejection; hence more compressor power is needed to raise the discharge temperature due to the increase in condensing temperature. As for the indoor fan, reduction of its speed decreases the heat transfer coefficient from the room to the evaporator. In order to maintain the same heat flux to counteract the thermal load in the room, the evaporator temperature has to even cooler and this demands more compressor power to do so. The total power consumption is not only attributed by the compressor but also the fans. Therefore, whether the power consumption can be reduced depends on whether the power saving due to fan speed reduction can sur-
5000
o
5
300 200
0 100 -5
0
20
40 60 time[min]
80
0
20
40 60 time[min]
80
Fig. 5. Control results with more aggressive controllers and superheat setting.
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
(a) Evaporator temperature 20 Simulation Experiment
C
18 o
oscillatory transient response is easier to be achieved. The second control algorithm proposed in this paper is based on such reasoning except that the algorithm also demands the indoor fan to be operated in the high speed mode at steady state for power efficiency purposes.
16 14
3. System identification
12
2
b11 2 3 6 s þ a1 6 dT e 6 6 7 6 b21 4 dT sh 5 ¼ 6 6 s þ a1 6 dT c 4 b31 s þ a2
b12 s þ a1 b22 s þ a1 b32 s þ a2
3 b14 2 d xc s þ a1 7 7 76 dav b24 76 76 4 dxof s þ a1 7 7 b34 5 dx
b13 s þ a1 b23 s þ a1 b33 s þ a2
In this model structure, the symbol d is added to the relevant variables to address the fact that the model is based on linearization, so both inputs and outputs are perturbed from their values in the nominal condition (For example, dTe is the difference between the evaporator temperature and its value at the nominal condition). Moreover, because both Te and Tsh pertain to the responses occurring in the evaporator, the same dominating pole (a1) is assumed for them. The parameter a2 represents the most dominating pole for Tc, and bij’s are the control gains.
(b) Expansion valve opening 140
pulse
rpm
800 600 400 200 0
120 110
10 20 time[min]
100
0
rpm
(c) Compressor speed
(d) Indoor fan speed
3000
4
2500
3
2000
2
1500
1
1000 0
10 20 time[min]
10 20 time[min]
0
0
10 20 time[min]
Fig. 6. The inputs of the system identification experiment.
25
Simulation Experiment
C o
6
0
5
10
15 time[min]
20
25
(c) Condenser temperature 50 Simulation Experiment
C
45 40
0
5
10
15 time [min]
20
25
Fig. 7. The comparsion of experimental and simulated outputs with the same inputs.
During the identification, the air conditioner is first operated in the nominal condition with Te = 15.2 °C, Tsh = 5.3 °C, Tc = 42.3 °C, xc = 2000 rpm, av = 130 pulse, xof = 500 rpm, xif = 3 (mode number), the outdoor temperature (To) equal to 34 °C the outdoor humidity (Ho) equal to 50%, the indoor temperature (Ti) equal to 27 °C and the indoor humidity (Hi) equal to 57%. The inputs are then perturbed by square waves and the outputs are recorded. The waveforms for the inputs and the outputs are shown, respectively, in Figs. 6 and 7. The input–output data are substituted into the model structure in (1) and least-square method is used to solve for the model parameters [8]. The identified model is given by
Te
3
2 0:1768
sþ0:0279
6 7 6 0:141 3 4 T sh 5 ¼ 10 4 sþ0:0279 0:0565 Tc sþ0:0194
130
20
4
2
1000
15 time[min]
8
ð1Þ
s þ a2
150
10
(b) Superheat
35
if
(a) Outdoor fan speed
5
10
3 7 7 7: 5
0
o
In order to devise the two control algorithms, the dynamics of the vapor compressor cycle in the experimental machine is identified. The identification uses the compressor speed (xc, rpm), the expansion valve opening (av, pulse), the outdoor fan speed (xof, rpm), and the indoor fan speed (xif, mode number) as the inputs and the evaporator temperature (Te), the superheat temperature (Tsh), and the condenser temperature (Tc) as the outputs. Although as shown in [1] the dynamics of vapor compressor cycle is characterized by a set of high order, coupled, nonlinear differential equations, assuming that the operation of the machine is not far from the nominal condition, the associated dynamics can be linearized into linear, time-invariant (LTI) state equations. Furthermore, for the purpose of simplifying the controller design, we also assume that in these LTI state equations, the mapping between any of the perturbed-input-perturbed-output pair is characterized by a first-order transfer function of the form bðÞ =ðs þ aðÞ Þ. By the assumptions, the model is in the following structure:
2:849 sþ0:0279 2:747 sþ0:0279 0:532 sþ0:0194
0:0605 sþ0:0279 0:0926 sþ0:0279 0:199 sþ0:0194
2
4:7147 3 sþ0:0279 6 8:883 76 sþ0:0279 56 4 1:5996 sþ0:0194
xc 3 av 7 7 7; xof 5 xif
ð2Þ
in which the perturbation symbols (d’s) are omitted for simplicity. The same inputs are applied to the identified model. The simulation outputs are plotted together with the experimental ones in Fig. 7. The figure shows that the identified model matches the major dynamics of the actual system. In order to further examine the applicability of the identified model to other operating conditions, the same input wave forms as in Fig. 6 are applied to the air-conditioner under another nominal operating condition with Te = 14.0 °C, Tsh = 6.9 °C, Tc = 38.6 °C, xc = 2000 rpm, av = 130 pulse, xof = 500 rpm, xif = 3, To = 29.5 °C, Hi = 72%. The experimental responses in Te, Tsh, and Tc are shown in Fig. 8. Also shown in the same figure are the outputs simulated by the identified model of (2). Although the operating condition is different, one can see that the identified model can still represent the major dynamics of the actual system.
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
where a3 is a parameter determined by the geometry, the heat transfer coefficient of the evaporator, and the thermal capacitance of the indoor environment, and t represents the contribution from the indoor thermal load. Combining (4) and (5), the dynamics for the air-conditioning system can be depicted by the plant block in Fig. 9. The block consists of three decoupled subsystems: the n 1– Te–Ti subsystem, the n2–Tsh subsystem, and the n3–Tc subsystem. Therefore, if one independently closes the loop of each subsystems in a stable manner, the overall system is guaranteed to be stable. By this reasoning, the proposed control algorithm employs a similar control structure as the one presented in [1]. In Fig. 9, while C1(s), C2(s) control the n1–Te–Ti subsystem and C3(s) controls the n2–Tsh subsystem, there is another controller C4(s) to regulate the temperature difference Tc–To to a preset value Tco_set. Since all Ci(s)’s deal with first-order dynamics, similar to those in Fig. 2, they will be implemented as PI control for simplicity and reducing steady-state error. However, it should be noted that due to the inner-outer loop structure in the n1–Te–Ti subsystem, the design of C1(s), C2(s) may not be as straightforward as the rest of the controllers. Regarding this, one can directly employ the following theorems proposed in [1] to facilitate the design of C1(s), C2(s) for stability.
(a) Evaporator temperature 20
o
C
Simulation Experiment 15
10
0
5
10
15 20 time[min]
25
30
(b) Superheat 12 Simulation Experiment
o
C
10 8 6 0
5
10
15 20 time[min]
25
30
(c) Condenser temperature 44 Simulation Experiment
Theorem 1. (Stability) Assume that C 1 ðsÞ ¼ fK p1 ðs þ z1 Þg=s and C 2 ðsÞ ¼ fK p2 ðs þ z2 Þg=s. If C1(s) renders the closed-loop poles of the inner-loop at p1, p2, then C1(s) and C2(s) together stabilize the n 1– Te–Ti subsystem regardless the choice of Kp2 if and only if the two conditions are simultaneously satisfied.
o
C
42 40 38 18
20
22
24 26 time [min]
28
30
32
z1 þ z2 < p1 þ p2 þ a3 ;
ð6Þ
2
Fig. 8. Match of the simulated outputs with the experimental outputs under different operating condition.
4. Control algorithm which incorporates the outdoor fan speed control to improve power efficiency A control algorithm which incorporates the outdoor fan speed as the third control input (in addition to the compressor speed and expansion valve opening) to further improve power efficiency is presented in this section. In this algorithm, the indoor fan speed is kept at high speed mode. Therefore, there will be three control inputs: xc, av, and xof. Ignoring the contribution by the indoor fan speed and defining the following input transformation:
2
3 2 3 xc n1 6 7 6 7 4 n2 5 ¼ B1 4 av 5; xof n3
ð3Þ
2
3 b11 b12 b13 where B1 ¼ 4 b21 b22 b23 5 and n1, n2, and n3 are new control inb31 b32 b33 puts, the dynamics for the vapor compression cycle in (1) can be transformed into a decoupling form
2
3 2 a1 T_ e 6_ 7 6 4 T sh 5 ¼ 4 0 T_ c
0
0 a1 0
0
32
Te
3
2
n1
3
76 7 6 7 0 54 T sh 5 þ 4 n2 5: Tc n3 a2
ð4Þ
Next for the indoor dynamics, it follows from the physical modeling in [1] that the influence of the vapor compression cycle dynamics towards the indoor temperature Ti can be approximated by the following first-order equation:
T_ i ¼ a3 T i þ a3 T e þ v;
ð5Þ
ðp1 þ p2 þ a3 Þ 1 1 < þ : ðp1 þ p2 Þðp2 þ a3 Þðp1 þ a3 Þ z1 z2
ð7Þ
Corollary 2. Under the same assumptions as in Theorem 1, C1(s) and C2(s) together stabilize the n 1–Te–Ti subsystem regardless of a3 and the choice of Kp2 if the two conditions are simultaneously satisfied.
z1 þ z2 < p1 þ p2 ; 1 1 1 1 þ < þ : p1 p2 z1 z2
ð8Þ ð9Þ
For the indoor environment considered, the transfer function for the indoor dynamics is estimated using physics [1] as 0:00036=ðs þ 0:00036Þ when the indoor fan is maintained at high speed. Instead of directly tuning of the proportional and integral gains in the controllers Ci(s)’s, the estimated indoor dynamics and the identified vapor-compression-cycle dynamics in (2) are substituted into LQR methodology to obtain the control parameters. The reason why LQR is used is because it is well-known for providing 6 db gain margin and 60° phase margin [9]. These stability margins can provide further robustness to modelling errors caused by possible variations in operating condition (including outdoor humidity, initial indoor humidity, outdoor temperature, initial indoor temperature, indoor thermal disturbance, user’s indoor temperature setting, refrigerant inventory, and so on). The objective of LQR design is to minimize a cost function in the R1 form of 0 xT Qx þ uT Ru dt, where x is the state vector, u is the control, Q is the state weighting matrix and R is the control weighting matrix. LQR utilizes the weighting matrices and the state equation which describe the system dynamics to formulate a Riccati equation to solve for the control parameters [10]. In this research, the tuning of the control parameters are accomplished by iteratively changing weighting matrices (Q, R) so that the resultant controllers can satisfy the stability conditions in Theorem 1 and Corollary 2 and can achieve satisfactory experimental performance
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
_ To
Tco
+
_ Tco _ set
+
ξ3
C4(s)
1 s + a2
Tc Decoupled plant block
_
Tsh _ set
+
ξ2
C3(s)
+ Te _ set
1 s + a1
ξ1
C1(s)
1 s + a1
_
Tsh Te
a3
+
1 s + a3
+
Ti
ν
_ C2(s)
+
Ti _ set
Fig. 9. Outdoor fan control block diagram.
in terms of rise time, overshoot, and so on. The controllers designed eventually are given by
(a) Power
(b) Compressor Speed 3000
800
0:001618ðs þ 0:0771Þ ; s 6ðs þ 0:0035Þ ; C 2 ðsÞ ¼ s 0:001618ðs þ 0:0771Þ ; C 3 ðsÞ ¼ s 0:001518ðs þ 0:0381Þ C 4 ðsÞ ¼ : s
C 1 ðsÞ ¼
2500 [rpm]
Watt
700 600 500
ð10Þ
400
1500 8
10 T
(a) Power
(b) Compressor Speed 3000
12
14
16
o
co set
[ C]
8
10
12
14
16
o
T
co set
[ C]
(c) Fan Speed 800 [rpm]
Notice that to avoid integrator windup due to the saturation in the physical inputs, a model-based anti-windup compensator as the one used in [1] is incorporated in the control algorithm. Moreover, the algorithm also demands appropriate settings for Tsh_set and Tco_set. In the experiments, Tsh_set = 5 °C is adopted because, from prior control experiences in [1], such a setting is high enough to prevent liquid refrigerant from entering the compressor during the control transience and yet it is low enough to ensure acceptable power efficiency. On the other hand,
2000
o
600
T =32 C;H =53%
400
T =27 C;H =52%
o
o
o
o
o
200 8
10 T
12
14
16
o
co set
[ C]
Fig. 11. Effects of changing Tco_set under low outdoor humidity.
800 2500 [rpm]
Watt
700 600 500 400
2000 1500
8
10 T
12
14
16
o
co set
[ C]
8
10
12
14
16
o
T
co set
[ C]
(c) Fan Speed
[rpm]
800 o
600
To=28 C;H o=64%
400
T =26 C;H =62%
o
o
o
200 8
10 T
12
14
16
o
co set
[ C]
Fig. 10. Effects of changing Tco_set under high outdoor humidity.
to determine an appropriate setting for Tco_set, control experiments with different Tco_set are conducted. In these experiments, Ti_set is set at 25 °C and Tco_set are alternatively set as 9, 11, 13, and 15 °C. In Figs. 10 and 11, the total power consumption, the compressor speed, and the fan speed, measured at the steady state of the experiment, are plotted as a function of Tco_set under four outdoor conditions (To = 28 °C, Ho = 64%, and To = 26 °C, Ho = 62%, and To = 32 °C, Ho = 53%, and To = 27 °C, Ho = 52%). It is found that regardless of the change of the outdoor temperature and the outdoor humidity, Tco_set power curves are concave up, and Tco_set = 11 °C seems to minimize the power consumption. The characteristics in the Tco_set–power curves can be explained by observing how the compressor speed and the fan speed vary with Tco_set: at low Tco_set, the condenser temperature, consequently the condenser pressure, is low.In this case, to maintain sufficient heat transfer rate to the outdoor environment, the outdoor fan has to be run at the high speed, so the associate fan power is high.
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T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
(a) Indoor Temperature
Ti (w/o fan control)
Ti set = 26 C
=5 C
10
T sh (w/o fan control)
27
60
(c) Temperature Difference Tco
0
20
40 time[min]
60
0
(d) Evaporator Temperature
Tco set = 11 C
30
Te
Tco
25
Te (with fan control)
20
Te (w/o fan control)
10
C
set
o
C
o
o
C
15
15 10 0
20
40 time[min]
5
60
0
(e) Compressor Speed
40 time[min]
60
20
40 time[min]
60
rpm 0
(g) Power
40 time[min]
100 time[min]
0
800 600
800
Energy Saving: 8.45% ↓ ↑
600
W
1000
Watt-h ou r
1200
400 200
400 0
20
40 time[min]
60
0
0
20
40 time[min]
60
Fig. 12. Experimental results of outdoor fan control with To = 30.5 °C, Ho = 55%, initial indoor humidity = 60%.
Contrarily, at high Tco_set, the condenser temperature as well as condenser pressure are high, so by the increase in the compressor speed observed in Fig. 10(b), the compressor power increases. Due to the high temperature difference between the condenser and the outdoor environment, the outdoor fan does not have to be run at high speed to maintain the heat transfer rate. Therefore, the fan power decreases. In summary, the compressor power tends to increase with Tco_set, but the outdoor fan power decreases with Tco_set. Since the total power is contributed by both the compressor and the fan, there is no surprise that an optimal Tco_set which minimizes the total power consumption is found from the control experiments. Once the optimal Tco_set is found, more experiments are performed to verify the control performance. Fig. 12 shows the experimental responses corresponding to To = 30.5 °C, Ti_set = 26 °C, Ho = 55%, initial indoor humidity = 60% by solid curves. Also shown in the figure by dash–dot curves are the responses associated with the control system which does not incorporate fan speed as the control input. It can be seen that although the outdoor fan speed initially rises to 920 rpm, the proposed control algorithm can automatically regulate the fan speed to 430 rpm at steady state. As a result of this regulation, the steady-state energy is 8.45% less than the case without outdoor fan control. The experimental responses corresponding a different outdoor temperature and different humidity levels are also shown in Fig. 13. This figure corresponds to the operating con-
100 time[min]
150
(f) Outdoor Fan Speed with fan control w/o fan control
1000 500
0
50
100 time[min]
0
150
0
(g) Power
with fan control w/o fan control
1000
50
1500
with fan control w/o fan control
(h) Watt hour
with fan control w/o fan control
Te (w/o fan control)
20
150
4000
2000
60
set
Te (with fan control)
10 50
1200
1400
W
20
150
15
1800 1600 1400 1200 1000 800 600
50
100 time[min]
150
(h) Watt hour 5000
with fan control w/o fan control
W att-h our
0
0
Te
25
Tco
3000
2000
100 time[min]
(d) Evaporator Temperature
Tco set = 11 C
5000
500
50
30
(e) Compressor Speed
rp m
3000
0
6000
with fan control w/o fan control
1000
150
o
0
1500 with fan control w/o fan control
4000
20 18 16 14 12 10 8
(f) Outdoor Fan Speed
5000
rp m
20
100 time[min]
(c) Temperature Difference Tco
35 o
50
C
40 time[min]
Tsh (w/o fan control)
10
o
20
rpm
0
= 5 oC
5
26 0
20
5
Ti (w/o fan control)
28
set
Tsh (with fan control)
15
5
26 25
Tsh
Ti (with fan control)
30
C
28
set
T sh (with fan control)
o
T sh
15
C
29
Ti (with fan control)
20
o
o
o
Ti set = 26 C
C
30
(b) Superheat
32
20
o
o
C
o
(a) Indoor Temperature
(b) Superheat
31
with fan control w/o fan control
4000
Energy Saving: 7.71%
3000
↓ ↑
2000 1000
0
50
100 time[min]
150
0
0
50
100 time[min]
150
Fig. 13. Experimental results of outdoor fan control with T0 = 29 °C, H0 = 50%, initial indoor humidity = 73%.
dition that To = 29 °C, Ho = 50%, initial indoor humidity = 73%. Because the increase in the initial indoor humidity increases the thermal capacitance of the indoor room, which in turn slows down the indoor dynamics [1], the settling time for the indoor temperature is slower than that in Fig. 12. The steady-state fan speed in this case becomes 510 rpm and the energy saving is 7.71%. 5. Control algorithm which incorporates the indoor fan speed control to improve transient response A control algorithm which can further improve the transient response is presented in this section. In this algorithm, the indoor fan speed is incorporated as an additional control input, but the outdoor fan speed is kept constant. Therefore, there will be three control inputs: xc, av, and xif. Ignoring the contribution by the outdoor fan speed and defining the following input transformation:
n1 n2
2
xc
3
6 7 ¼ B2 4 av 5;
ð11Þ
xif
b11 b12 b14 and n1, n2 are new control inputs, the b21 b22 b24 dynamics for Te and Tsh in (1) can be transformed into a decoupling form where B2
1963
T.-J. Yeh et al. / Applied Thermal Engineering 29 (2009) 1955–1964
¼
a1 0
0 a1
Te n1 þ : T sh n2
(a) Indoor Temperature
ð12Þ
5000
o
Ti set = 26 C Ti with fan control Ti w/o fan control
30
o
C
With such decoupled dynamics, the plant for the air-conditioning system is again in a decoupling form as in Fig. 9 except that the dynamics associated with the condenser temperature (Tc) does not appear. Consequently, a similar control structure as the control algorithm in Fig. 9 can be employed. Specifically, n1, which controls the evaporator temperature and indoor temperature, is determined simultaneously by the inner controller C1(s) and the outer controller C2(s); n2, which controls the superheat temperature, is determined by the controller C3(s). It should be noted that n1 and n2 only act as intermediate variables for control system design. In actual implementation, one has to resolve the physical inputs xc, av, and xif from n 1, n2 using (11). Because there is one control redundancy in T (11), infinite solutions of ½ xc av xif exist. One way to determine a viable solution for the physical inputs is to formulate a conT strained linear optimization problem. In detail, once ½ n1 n2 is given, it is desired to find a solution that satisfies (11) and minimizes the quadratic cost J ¼ ax2c þ ba2v þ cx2if , where a, b, c > 0 are the weighting factors. The problem can be easily solved with the method of Lagrangian multipliers and the optimal solution is given by
28
with fan control w/o fan control
4000 3000 2000
26 24
1000 0 0
20
15 10
0
with fan control w/o fan control
400 300
100 0
20
40 60 time[min]
80
0
(e) Evaporator Temperature Error
20
C1(s)
3 10 o
2 0 1 -10 0
20
40 60 time[min]
80
0
0
Tsh_set
C3(s)
-
40 60 time[min]
ξ2
80
Fig. 15. Indoor fan control experiment.
0:001518ðs þ 0:0771Þ ; s 8ðs þ 0:0028Þ ; C 2 ðsÞ ¼ s 0:001015ðs þ 0:0768Þ C 3 ðsÞ ¼ : s C 1 ðsÞ ¼
ð14Þ
It should be noted that C1(s) and C2(s) are also designed using the theorems presented in the section. The weighting matrix h previous 2 1 2 0:852 i 1 . The control reW is selected to be diag 2000 120 1 sponses of the proposed control algorithm are shown in Fig. 15. In the figure, with similar experimental conditions as those for Fig. 5
-
u
u
Vapor Compression Dynamics
+ + + -
+
20
+
Ti_set
ξ1 T -1 W -1B-1 2 ( B 2 WB 2 )
+
80
(f) Indoor Fan Speed
C2(s)
-
40 60 time[min]
with fan control w/o fan control
20
ð13Þ
where W ¼ diag ½ a b c is the weighting matrix. The block diagram of this control algorithm is shown in Fig. 14. In this diagram, an anti-windup compensator is also incorporated to avoid integrator windup due to the saturation in the physical inputs. Notice that, before the input associated with the indoor fan speed is actually injected to the air conditioner, it is quantized to an integer so that it can correspond to the mode of operation of the indoor fan. Moreover, in order for the indoor fan to operate at high speed mode (mode 3), one can increase c, the weighting factor for xif, so that the indoor fan speed is more penalized. By doing so, the control authority in the indoor fan speed is reduced and xif tends to stay around zero at steady state. Because xif represents a control perturbation from the nominal indoor fan speed (mode 3), increasing the associated weighting factor can make the indoor fan return to its nominal mode of operation at steady state. In the control experiment, the same C1(s), C2(s), C3(s) used for conducting the experiment in Fig. 5 are adopted. They are, respectively, given by
+
80
4
xc 1 n 1 6 7 1 T 1 T ; 4 av 5 ¼ W B2 B2 W B2 n2 xif
Te_set
40 60 time[min]
200
5
3
20
(d) Expansion Valve Opening
Tsh set = 3 C Tsh with fan control Tsh w/o fan control
20
-5
0
o
25
C
80
(c) Superheat
30
o
40 60 time[min]
C
2
(b) Compressor Speed
32
rp m
T_ e _T sh
#
p u lse
"
Anti-windup Compensator
+
Fig. 14. Indoor fan control algorithm.
Indoor Dynamics
Ti
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(To = 33 °C, Ho = 55%), the indoor temperature (Ti) response for the proposed control algorithm is less oscillatory and is settled within 26 ± 0.3°C in 15 min. The mode of operation for the indoor fan follows the sequence 3–2–1–2–3 that the fan speed is reduced during the control transience. Particularly, the initial reduction of the fan speed decreases the heat transfer from the room to the evaporator. This makes the dynamics of the vapor compression cycle less coupled with the indoor dynamics which in turn makes the compressor speed (xc) and the expansion valve opening (av) easier to regulate the superheat (Tsh) and the evaporator temperature (Te). From Figs. 15(b) and 15(d) where the histories for xc and av are shown, it can be seen that incorporating the indoor fan speed as an additional control input makes the saturation in xc and av occurred less frequently, so the responses for Tsh and Te, respectively, in Figs. 15(c) and 15(e) become less oscillatory. The improvements in the responses of Tsh and Te can offset the decrease of the initial heat transfer from the room to the evaporator caused by fan speed reduction, so a shorter settling time is observed in the indoor temperature response (Fig. 15(a)). Notice that at steady state, the indoor fan speed automatically settles at high speed mode, so the air-conditioning system is operated efficiently. 6. Conclusions Two control algorithms, respectively, incorporating the outdoor fan and the indoor fan as the additional control inputs for air-conditioning systems are presented in this paper. While the first algorithm can enhance the steady-state power efficiency, the second one can improve the transient response. In practical applications in which the design is usually constrained by hardware cost, system complexity, or the computation power available, one can implement only one of the algorithms to improve either the efficiency or the transient behavior. By doing so, only one fan speed needs to be modulated in real-time, the other is maintained at constant speed. If cost, complexity, and computation power are not an issue, then both algorithms can be simultaneously implemented in the air-conditioning machine in the way that the second algorithm is responsible for the control action during the startup/transient phase of operation, and the first algorithm takes over the control when steady state is reached. A bumpless transfer scheme [11,12] can be incorporated in this case to ensure the control action is transferred smoothly between the
two algorithms. Besides, the air-conditioner considered in this paper has only one evaporator. As the demand for multi-room airconditioning service for common households is growing, the market attention is shifting from the single-evaporator machine to the multi-evaporator air-conditioner (MEAC). Because making the fan speeds continuously adjustable for control purposes apparently adds flexibility to the controller design and thus can lead to performance and/or efficiency improvements, on-going researches are devoted to incorporating indoor/outdoor fan speeds as the control inputs for MEAC’s. Acknowledgement The authors gratefully acknowledge the support provided by the Energy and Environment Research Laboratories of Industrial Technology Research Institute in Taiwan. References [1] J.-L. Lin, T.-J. Yeh, Modeling, identification and control of air-conditioning systems, International Journal of Refrigeration 30 (2) (2007) 209–220. [2] M. Hattori et. al., Automotive refrigeration system controller with a simple precompensator, in: Proceedings of the 29th Conference on Decision and Control, Hawaii, 1990, pp. 1590–1591. [3] X.-D. He, S. Liu, H.H. Asada, Hiroyuki Itoh, Multivariable control of vapor compression systems, HVAC&R Research 4 (3) (1998) 205–230. [4] T. Cheng, X.-D. He, H.H. Asada, Nonlinear observer design for two-phase flow heat exchangers of air conditioning systems, in: Proceeding of the American Control Conference, USA, 2004, pp. 1534–1539. [5] T. Cheng, X.-D. He, H.H. Asada, Shinichi Kasahara, Heat exchanger dynamic observer design, ASHRAE Transactions 111 (2005) 328–335. [6] R. Shah, B.P. Rasmuussen, A.G. Alleyne, Application of a multivariable adaptive control strategy to automotive air conditioning systems, International Journal of Adaptive Control and Signal Processing 18 (2004) 199–221. [7] L.S. Larsen, C. Thybo, Potential energy savings in refrigeration systems using optimal set-points, in: Proceedings of the 2004 IEEE International Conference on Control Applications, Taipei, 2004, pp. 701–704. [8] R. Johansson, System Modeling and Identification, Prentice-Hall, Englewood Cliffs, NJ, 1993. [9] M.G. Safonov, M. Athans, Gain and phase margins for multiloop LQG regulators, IEEE Transactions on Automatic Control AC-22 (2) (1977) 1735–1744. [10] J.M. Maciejowski, Multivariable Feedback Design, Addision-Wesley, Reading, MA, 1989. [11] L. Zaccarian, A.R. Teel, A common framework for anti-windup, bumpless transfer and reliable designs, Automatica 38 (2002) 1735–1744. [12] S.F. Graebe, A.L.B. Ahlén, Dynamic transfer among alternative controllers and its relation to anti-windup controller design, IEEE Transactions on Control System and Technology 4 (1) (1996) 92–99.