C (air-conditioning) system of EVs (electric vehicles)

Energy 66 (2014) 342e353

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Energy journal homepage: www.elsevier.com/locate/energy

Near-optimal order-reduced control for A/C (air-conditioning) system of EVs (electric vehicles) Chien-Chin Chiu, Nan-Chyuan Tsai*, Chun-Chi Lin Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 May 2013 Received in revised form 7 January 2014 Accepted 9 January 2014 Available online 6 February 2014

This work is aimed to investigate the regulation problem for thermal comfortableness and propose control strategies for cabin environment of EVs (electric vehicles) by constructing a reduced-scale A/C (air-conditioning) system which mainly consists of two modules: ECB (environmental control box) and AHU (air-handling unit). Temperature and humidity in the ECB can be regulated by AHU via cooling, heating, mixing air streams and adjusting speed of fans. To synthesize the near-optimal controllers, the mathematical model for the system thermodynamics is developed by employing the equivalent lumped heat capacity approach, energy/mass conservation principle and the heat transfer theories. In addition, from the clustering pattern of system eigenvalues, the thermodynamics of the interested system can evidently be characterized by two-time-scale property. That is, the studied system can be decoupled into two subsystems, slow mode and fast mode, by singular perturbation technique. As to the optimal control strategies for EVs, by taking thermal comfortableness, humidity and energy consumption all into account, a series of optimal controllers is synthesized on the base of the order-reduced thermodynamic model. The feedback control loop for the experimental test rig is examined and realized by the aid of the control system development kit dSPACE DS1104 and the commercial software MATLAB/Simulink. To sum up, the intensive computer simulations and experimental results verify that the performance of the nearoptimal order-reduced control law is almost as superior as that of standard LQR (Linear-Quadratic Regulator).
2014 Elsevier Ltd. All rights reserved.

Keywords: Singular perturbation technique Near-optimal control Thermal comfort A/C (air-conditioning system)

1. Introduction Due to global warming and rapid decay of fossil energy reserve, recently HEVs (hybrid electric vehicles) and EVs (electric vehicles) have become new targets for the vehicle market. Among them, EV is specifically emphasized, owing to its property of zero emission. The air cooling mode of A/C (air-conditioning) system for gasoline vehicle is realized by the compressor driven by engine, but in EV it is by motor (or called electric A/C). On the other hand, the heating mode of A/C system for gasoline vehicle is to employ the exhaust heat out of engine, but in EV it is driven by electric heater. For EV, the electric heater definitely consumes significantly partial battery energy. The evolution for EV is somehow deterred due to the problem of short mileage. There are many factors which affect or determine the practically achievable mileage for an EV. The major ones are listed in Table 1 [1]. From Table 1, the A/C system and heating system together constitute the major energy consumption

* Corresponding author. Tel.: þ886 6 2757575x62137; fax: þ886 6 2369567. E-mail address: [email protected] (N.-C. Tsai). 0360-5442/$ e see front matter
2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2014.01.029

up to 65%. Hence, how to reduce energy loss for A/C system and heating system in EV becomes a significant research target nowadays. In addition, how comfortable the cabin environment is to the driver is an influential factor for driving safety. Daanen et al. reported that the cabin temperature is an important factor that could lead to traffic accidents and staying thermoneutral in cabin could make the driver more focused on car driving [2]. Kaynakli et al. proposed that the air temperature and humidity in cabin are the two key factors for the comfortableness perceived by passengers [3]. Fanger proposed the approach of PMV (Predicted Mean Vote) which is the most commonly referred thermal comfort model [4]. It is noted that PMV model is based on two assumptions: (i) the overall effect and symmetry of environment, and (ii) steady-state experimental condition. Jones studied the capabilities and limitation of thermal models for use in thermal comfort standards [5]. However, recently a few advanced studies attempt to unveil the effects of asymmetry of environment in cabin and the localization of air flow. The following citations are some of these types of researches. Jones et al. presented the basis of the new thermal comfort model and its predictions for transient warm-up and cooldown conditions [6]. Instead of considering symmetric and uniform

C.-C. Chiu et al. / Energy 66 (2014) 342e353

Nomenclature

qO,S qO,W

outdoor temperature (summer mode) outdoor temperature (winter mode) It,S solar irradiation intensity (summer mode) It,W solar irradiation intensity (winter mode) cp specific heat of air ra density of air hg specific enthalpy of saturated vapor bq, bp, bc parameters of ASHRAE thermal sensation scale bn h natural convective heat transfer coefficient b h forced convective heat transfer coefficient f CW heat capacity of cabin wall VIN volume of air in cabin AO exterior surface area of cabin wall W AIN interior surface area of cabin wall W ASH exterior irradiated surface area of cabin wall W a absorptance coefficient of solar radiation s transmittance coefficient of solar radiation SHGC Solar-Heat Gain Coefficient

air flows, Zhang et al. reported the development of thermal sensation and comfort models for non-uniform and transient environments [7e9]. In order to analyze and improve the dynamic behavior of the A/C system, at first it is necessary to establish the dynamic models of the ACS (air-conditioned space) and AHU (air-handling unit). Modeling techniques on A/C system can be divided into two categories. One is based on thermodynamic/heat transfer theory [10e 14], while the other is based on experimental measurement data and system identification such as black-box method [15] and greybox modeling technique [16]. For controller design, there are many types of control strategies have been presented such as traditional PID control, Zieglere Nichols tuning rule [18], gain scheduling control [17], fuzzy control [18,19] and ANN (artificial neural network) [20,21]. For the integrated controller design, it is well-known that the self-tuning fuzzy adaptive control was proposed by Soyguder et al. in 2009 [22,23]. However energy consumption is beyond the primary concern by these researches. Instead, the propose of this work is to employ an appropriate control policy particularly for the EV A/C system such that the comfortableness for passengers, least energy consumption and electric power efficiency can be all taken into account together. To maintain the comfortableness of cabin environment, it is necessary to regulate not only interior temperature but also humidity. In order to reduce the energy consumption for general A/C process, a few optimal control strategies was reported [24e28]. Table 1 Impact of accessories on EV range [1].

343

Nevertheless, the dynamics of A/C system for EVs is pretty complicated, high-order and highly non-linear. The singular perturbation theory is usually employed to deal with the plant with the presence of parasitic parameters. By singular perturbation technique, the plant can be split into two lower-order subsystems in two time scales. For controller synthesis, by singular perturbation method, the control laws can be developed for slow-mode subsystem and fast-mode subsystem separately. Therefore, the resulted composite control can be obtained by summation of the outputs of these two controllers [29e31]. In this paper, a scale-down experimental test rig of A/C system to imitate the temperature/humidity of cabin in EV is constructed. After thermodynamic model linearization of system by taking Taylor’s Expansion around the operational point, the thermodynamic models of the full system are derived and decoupled into two subsystems, “Slow Mode” and “Fast Mode”, based on the singular perturbation technique. By considering the comfortableness for passengers, least energy consumption and electric power efficiency, a series of optimal controllers, namely near-optimal regulator, near-optimal order-reduced control and corrected LQR (linear-quadratic regulator), is proposed for the EV A/C system. In addition, the standard LQR plays the role of benchmark and is to be compared with the preliminarily proposed controllers. By computer simulation results, the efficacy of various optimal controllers proposed by this paper is all acceptable, with respect to the benchmark by LQR. The near-optimal order-reduced control strategy is the simplest and carries the least computation load so that it is chosen for the efficacy examination by experiments. The experimental results show that under near-optimal order-reduced control strategy, the preset temperature and humidity for cabin can be achieved and then retained not only at summer mode but also at winter mode. 2. Thermodynamic model of HVAC A reduced-scale test rig of HVAC (Heating, Ventilation, and AirConditioning) to imitate the environment of cabin in vehicle is constructed and studied. For the purpose of later-on system analysis and controller design, in this section the thermodynamic model of cabin is developed at first. 2.1. Configuration of HVAC module The HVAC module, shown in Fig. 1, mainly consists of two major portions. One is the cabin, referred to “ECB (Environmental Control Box)” in the experimental setup. The other is “AHU” which can actively adjust the temperature and humidity of ECB. These two subsystems are to be analyzed and discussed respectively in the following sections. 2.2. Thermodynamic model of cabin

Accessory

Range impact

Comment

Air-conditioning

Up to 30%

Heating

Up to 35%

Highly dependent on use, ambient temperature, cabin temperature, and air volume Highly dependent on use and ambient and cabin temperatures

Power steering Power brakes Defroster Others, such as lights, stereo, power-assisted seats, windows, door locks

Up Up Up Up

to to to to

5% 5% 5% 5%

Depending on use Depending on use

The cabin is mainly constituted by two physical elements: cabin wall and the air in cabin. The governing equations of energy conservation for the air in cabin and mass conservation, due to water vaporization, can be described respectively as follows:

ra cp VIN ðdqIN =dtÞ ¼ QAHU þ QW þ QSHG þ QV þ QINF þ Qload

(1)

_ s ðuS  uIN Þ þ m _ V ðuV  uIN Þ ra VIN ðduIN =dtÞ ¼ m _ INF ðuO  uIN Þ þ Mload þm

(2)

where ra is the air density in cabin, cp the specific heat constant, VIN the volume of the air in cabin, qIN the temperature in cabin, and

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b AIN ðq  q Þ QW ¼ h IN W W IN

(6)

b is the heat transfer coefficient, AIN the interior surface where h IN W area of the cabin wall, qW the temperature of the cabin wall. The equation of energy conservation for the cabin wall can be expressed by

CW ðdqW =dtÞ ¼ QO þ QIN

(7)

where CW is the overall thermal capacitance of the cabin wall, QO the heat transfer rate from ambient environment to the cabin wall, and QIN the heat transfer rate to the air inside the cabin from the cabin wall. Finally, based on Eqs. (1) and (2), the equations of energy conservation for air in cabin and mass conservation for water vaporization in cabin can be rewritten respectively as follows [33,34]:

dqIN =dt ¼ ~f s ðqS  qIN Þ=VIN þ hg ~f s ðuS  uIN Þ=cp VIN   b g2=3 ðq  q Þ=r c V þ AIN b h þh W

þ

Fig. 1. Schematic of environmental control for electric vehicles.

(3)

where h is the net enthalpy of the atmosphere, ha the enthalpy of air inside the cabin, hg the enthalpy of saturated vaporization and u the absolute humidity. Hence QAHU can be written in terms of air flow rate and humidity as follows:

_ s hg ðuS  uIN Þ _ s cp ðqS  qIN Þ þ m QAHU ¼ m ¼ ra~f s cp ðqS  qIN Þ þ ra~f s hg ðuS  uIN Þ

(4)

where ~f s is the volume flow rate of air by the air conditioner to the cabin and qS the temperature of AHU. On the other hand, based on Ref. [32], the SHG can be approximated by

QSHG ¼ ASH W  SHGF  ðSHGC=0:87Þ

n

f

IN

W

IN

a p IN

 SHGF  ðSHGC=0:87Þ=ra cp VIN

(8)

þ Qload =ra cp VIN

QAHU the heat transfer rate from the supplied air by air conditioner to the cabin. QW is the inward heat transfer rate caused by temperature difference between the cabin wall and the air in cabin, QSHG the SHG (Solar-Heat Gain), QV and QINF the heat loss from the cabin to its ambient environment due to the overall effect of ventilation and infiltration by the air in cabin respectively and Qload the heat load which causes the temperature of the cabin increased. uIN, uS, uV and uO represent the absolute humidities for the air inside the cabin, the supplied air by air conditioner, the air during ventilation process and ambient environment, respectively. _ s; m _ V and m _ INF are the air mass flow rates of the supplied air by air m conditioner, the air during ventilation process and the air due to infiltration, respectively. Mload is the moisture load in the cabin, i.e., the mass of water vaporization inside the cabin. Assume the effects of ventilation and infiltration by air flow can be both neglected. _ INF ðuO  uIN Þ and m _ V ðuV  uIN Þ in Eqs. (1) and Therefore, QV, QINF, m (2) hence vanish. The net enthalpy of the atmosphere inside the cabin can be obtained as

h ¼ ha þ uhg

ASH W

(5)

where ASH W is the effective surface area for the solar-heat radiation applied upon the surface of cabin, SHGF the corresponding SolarHeat Gain Factor, and SHGC the Solar-Heat Gain Coefficient. The heat transfer rate due to the difference of cabin wall temperature and the temperature inside the cabin, QW, can be obtained by Newton’s Equation

_ s ðuS  uIN Þ þ Mload =ra VIN duIN =dt ¼ ½m i. h ¼ ra ~f s ðuS  uIN Þ þ Mload ra VIN

(9)

b n the heat transfer where gIN is the speed of air flowing in cabin, h b coefficient of natural convection, h f the heat transfer coefficient of forced convection. On the other hand, the energy conservation for cabin wall can be described in terms of qW as follows:

 h  i. b b 2=3 bn þ h b g2=3  q qO þ aIt = h CW dqW =dt ¼AO W f O W h n þ h f gO   b b 2=3 ðq  q Þ=C þ AIN IN W W W h n þ h f gIN (10) where AO W is the exterior surface area of the cabin wall, a the absorptance of solar radiation, It the irradiation of the exterior surface of cabin wall and gO the wind speed of ambient environment. 2.3. Thermodynamic model of AHU The AHU module, shown in Fig. 1, is mainly constituted by the air cooling/heating section, the mixing section and the humidifying section. The heating mode of A/C system for gasoline vehicle is to employ the exhaust heat out of engine. However, the amount of heat generated by EVs is much less than that by the gasoline vehicles. Though there are many other heat sources, e.g., the exhaust heat out of AC motor, the induced heat by mechanical braking system and the heat out of transmission units, any of them cannot be provided constantly all the time or be able to promptly heat the cabin up within a short time period. Especially, for the regions suffered from snowing or the countries at high latitude, if merely the exhaust heat is employed, the preset temperature in the EV cabin cannot be effectively retained to make passengers comfortable. The purpose of the air cooling/heating section is to control the temperature of air in the cabin. Once the temperature of air in the cabin was lower than the desired temperature of air in the cabin, the electric heater in heating section would be engaged and generate heat flux to the mixing section. On the contrary, once the temperature of air in the cabin was higher than the desired temperature of air in the cabin, the cold air steam by refrigerator unit in

C.-C. Chiu et al. / Energy 66 (2014) 342e353

cooling section would be transported to the mixing section. After mixing the hot air stream and the cold air stream in mixing section, the mixed air stream then flows through the humidifying section into the cabin. Since the cold air stream in cooling section is imported from the refrigerator unit, the air temperature at outlet of cooling section and the relative humidity of air at outlet of cooling section can be preset as constants as follows:

qC ¼ 5  C fC ¼ 100%

(11)

345

b , the and the heat transfer coefficient of forced convection, h f equation of energy conservation for the heater (i.e., Eq. (11)) and the equation of energy conservation of the air inside the heating section (i.e., Eq. (13)) can be rewritten respectively as follows:

  b þh b g2=3 ½ðq þ q Þ=2  q =C þ P =C dqHT =dt ¼ AHT h n IN H HT HT HT HT f h (15)   bn þ h b g2=3 ½q  ðq þ q Þ=2=r cp V dqH =dt ¼ AHT h HT IN H HT f h a þ ~f h ðqIN  qH Þ=VHT

where qC is the air temperature at outlet of cooling section and fC the relative humidity of air at outlet of cooling section. In this manner, the thermodynamics model of AHU can be simplified and reduced to the one merely consisting of heating section and humidifying section.

These two thermodynamic equations will be employed later to construct the thermodynamic model of the AHU system and synthesize the required controller.

2.3.1. Heating section The heating section of AHU is composed by two physical elements: the electric heater and the air flow inside. The equation of energy conservation by the heater can be described as follows:

2.3.2. Humidifying section Based on the experimental setup shown in Fig. 1, the equations of thermodynamics and absolute humidity inside the humidifying section can be expressed respectively as follows:

  outlet _ h cp qinlet þ PHT CHT ðdqHT =dtÞ ¼ m  qh h

dqS =dt ¼ ~f mix

(12)

where CHT is the heat capacitance of the heater, qHT the temperature _ h the mass flow rate of the of the facial surface of the heater, and m inlet outlet air inside the heating section. qh and qh are the air temperatures of inlet and outlet of the heating section, respectively. PHT is the supplied electric power by electric heater. Assume the heat transfer between the heater and the air in heating section is convective. The corresponding equation of heat transfer can be written as follows:









outlet inlet outlet _ h cp qinlet ¼ ra~f h cp qh  qh  qh m h h . i outlet b A qinlet 2 qHT þ qh ¼h HT HT h

(13a)

where ~f h is the volume flow rate of air inside the heating section, b the heat transfer coefficient of the facial surface of the heater, h HT and AHT the equivalent surface area constituted by the fins of the heater. The temperatures of inlet and outlet of the heating section, outlet qinlet and qh , are assumed to be identical to the ones of air in h cabin and air at outlet. The so called “air at inlet” and “air at outlet” are referred to the air at Point A and Point B of the heating section, shown in Fig. 1, respectively. Based on continuity principle, they are related to Eqs. (10) and (11) by the following identities:

qinlet ¼ qIN h outlet qh ¼ qH

(13b)

where qH denotes the current highest temperature of air supplied by the heater. The equation of energy conservation of the air inside the heating section and the mass conservation of the water vaporization inside the heating section can be written respectively as follows:

b A ½q  ðq þ q Þ=2 ra cp VHT ðdqH =dtÞ ¼ ra ~f h cp ðqIN  qH Þ þ h HT HT HT IN H uH ¼ uIN (14) where VHT is the total volume of the heating section and uH the absolute humidity of air at outlet of the heater. By replacing the heat b , by the air transfer coefficient of the facial surface of the heater, h HT bn, flow rate, gh, the heat transfer coefficient of natural convection, h

h .  i. ~f q þ ~f q ~f þ ~f  q VHF S c C h H c h h .  i. ~f þ ~f  u cp VHF þhg ~f mix ~f c uC þ ~f h uH S c h

duS =dt ¼ ~f mix

(16)

(17)

h .  i. ~f u þ ~f u ~f þ ~f  u VHF þMHF =ra VHF S c C h H c h (18)

where

~f ~ ~ mix ¼ f c þ f h

(19)

~f mix is the volume flow rate of mixed air composed by the cold air component, ~f c , and the hot air component, ~f h . VHF is the total volume of humidifying section and MHF the mass of moisture load in humidifying section. 2.4. Comfort temperature and humidity Though in some high-latitude continental regions (e.g., USA), the influence of humidity upon comfortableness is not evident, this statement would be invalid for some other tropical or subtropical regions, particularly for most islands, such as Taiwan, Malaysia and Philippines. Even under the same temperature (say, 25  C), people feel comfortable for humidity 30% but uncomfortable for humidity 70%. That is, whether it is dry air flow or wet air flow still partially determines the comfortableness perceived by the passengers in the car. According to the available literature [4], humidity is also one of the factors of comfortableness. In summary, the two key factors on thermal comfortableness perceived by driver and guests in the car are: (1) air temperature (or called dry-bulb temperature) and (2) relative humidity of air [3]. The corresponding linear regression equation of thermal comfortableness can be expressed by [35]

Yðq; pv Þ ¼ bq q þ bp pv þ bc

(20a)

where Y(q, pv) is the so called “thermal sensation scale”, q the drybulb temperature, and pv the partial pressure of water vaporization. bq, bp and bc are the corresponding coefficients of thermal sensation scale defined by ASHRAE (American Society of Heating, Refrigerating and Air-Conditioning Engineers) [36]. The physical value of Y(q, pv) indicates the degree of thermal sensation by human. The typical values are shown in Table 2. Since qIN is defined as the

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C.-C. Chiu et al. / Energy 66 (2014) 342e353

n h .  i. ~f þ ~f  q dYS =dt ¼ bq ~f mix ~f c qC þ ~f h qH VHF S c h h   ~ ~ ~ ~ ~ þ hg Up f mix f c pC = f c þ f h þ f h ðYIN  bq qIN  bc Þ=  i. o   ~f c þ ~f h bp  ðYS  bq qS  bc Þ=bp cp VHF n h   þ bp ~f mix ~f c pC = ~f c þ ~f h þ ~f h ðYIN  bq qIN  bc Þ=bp  i.   ~f c þ ~f h  ðYS  bq qS  bc Þ=bp VHF o þ MHF =ra VHF Up

Table 2 Thermal sensation scale by ASHRAE [36]. Value

Thermal sensation

þ3 þ2 þ1 0 1 2 3

Hot Warm Slightly warm Neutral Slightly cool Cool Cold

temperature of the cabin in this paper, q in Eq. (20) is nothing but referred to qIN

q ¼ qIN

(20b)

Based on classical control theory and state-space representation [37], the thermodynamics of the studied system has to be expressed by “state equations” at first. For this argument, Eqs. (22) and (25), which will be shown later, are expressed by the first derivatives of Eq. (20). Since the optimal controller is to be synthesized according to the thermodynamic model, state-space representation is definitely necessary. To sum up, the thermodynamic model of cabin and AHU can be described by the seven state equations as follows:

    dqIN =dt ¼ ~f c þ ~f h ðqS  qIN Þ=VIN þ hg Up ~f c þ ~f h

þ ASH W  SHGF  ðSHGC=0:87Þ=ra cp VIN þ Qload =ra cp VIN (21) n    ~f þ ~f ðq  q Þ=V þ hg Up ~f þ ~f IN IN S c h c h ½ðYS  bq qS  bc Þ  ðYIN  bq qIN  bc Þ=bp cp VIN n h . i2=3 o b b ~f þ ~f SIN þ AIN c h W hn þ hf  ðqW  qIN Þ=ra cp VIN þ ASH W  SHGF  ðSHGC=0:87Þ=ra cp VIN þ Qload =ra cp VIN n  þ bp ~f c þ ~f h ½ðYS  bq qS  bc Þ

(26)

 ½qHT  ðqIN þ qH Þ=2=ra cp VHT     b þh b ~f =S 2=3 ½ðq þ q Þ=2 q =C dqHT =dt ¼ AHT h n IN H HT HT f h h þPHT =CHT

(27)

Since the A/C system is assumed to operate in a long time frame but be controlled to remain at certain constant temperature and relative humidity, the state equations, Eqs. (21)e(27), can be linearized around an operational point. However, the typical operational point is quite different with respect to different seasons. In order to unveil the discrimination, two operational modes are studied in this paper: “Summer Mode” and “Winter Mode”. The preset physical values of the operational points at summer mode and winter mode are shown in Table 3. At summer mode, due to the electric heater stops working at all, Eq. (27) can be ignored and Eq. (26) can be simplified as follows:

dqH =dt ¼ ~f h ðqIN  qH Þ=VHT

(28)

Table 3 Physical values of operational point.

o

(22)

qeW qeH qeHT

nh

i o.  b g2=3 q qO þaIt = bh n þ h CW dqW =dt=¼AO W f O W n h . i2=3 o b b ~f þ~f SIN ðqIN qW Þ=CW þAIN c h W hn þ hf (23) h .  i. ~f þ ~f  q VHF dqS =dt ¼ ~f mix ~f c qC þ ~f h qH S c h h   þhg Up ~f mix ~f c pC = ~f c þ ~f h þ ~f h ðYIN  bq qIN  bc Þ=  i.   ~f c þ ~f h bp ðYS  bq qS  bc Þ=bp cp VHF

Physical value

YSe

where YIN is the thermal sensation scale for the air in cabin. 2=3 b þb h n h f gO

    b ~f =S 2=3 hn þ h dqH =dt ¼ ~f h ðqIN  qH Þ=VHT þ AHT b f h h

o

 ðYIN  bq qIN  bc Þ=VIN bp þ Mload =ra VIN Up



where YS is the thermal sensation scale for the supplied air by AHU, and pC the partial pressure of water vaporization of the cold air stream inside AHU.

2.5. Linearization of system model

 ½ðYS  bq qS  bc Þ  ðYIN  bq qIN  bc Þ=bp cp VIN    2=3 2=3 b b ~ ~ þ AIN =SIN ðqW  qIN Þ=ra cp VIN W hn þ hf f c þ f h

dYIN =dt ¼ bq

(25)

~f e c ~f e h e MHF e PHT e YIN e Qload e Mload

qeO geO

(24)

Ite SHGFe

qeIN qeS CMH: m3/h.

Summer

Winter

29.02  C 2.94 25.0  C N/A 9.24 CMH 12.71 CMH 0.038 kg/h N/A 0 2.065 W 0.18 g/h 32  C 0.1 m/s 450 W/m2 360 W/m2 25  C 13  C

15.02  C 2.45 45  C 38.29  C 19.07 CMH 14.64CMH 0.00 kg/h 115.1 W 0 2.065 W 0.18 g/h 7 C 0.1 m/s 225 W/m2 180 W/m2 22  C 32  C

C.-C. Chiu et al. / Energy 66 (2014) 342e353

The system thermodynamics at summer mode can be expressed by Eqs. (21)e(25) and (28). The linearized state-space representation of the system thermodynamics can be obtained by taking Taylor’s Expansion around the operational point. Hence the linearized summer-mode thermodynamics can be rewritten as follows: s

(29)

where

~ s ¼ us  ues ; h ~ s ¼ ys  hðxes Þ ~ s ¼ xs  xes ; u ~ s ¼ hs  hes ; y x

us ðtÞ ¼

h

~f c

YIN ~f h

qS YS qH  T

qW

3

(30a)

(30c)

MHF

qC ¼ qIN

On the other hand, the linearized winter-mode thermodynamics can be rewritten by taking Taylor’s Expansion around the associated operational point as follows:

(32)

where

(33a)

~f c

~f h

qS YS qH qHT  T iT

MHF

Cluster Cluster #2 #2 (Slow (SlowModes) Modes)

-10

-8

-6

-4

-2

0

Real Axis Fig. 2. Groups of eigenvalues of open-loop system.

3. Model reduction by singular perturbation The singular perturbation theory is often employed to deal with the plant with parasitic parameters. By singular perturbation approach, the original system model can be split into two reducedorder subsystems in two time scales, i.e., slow-mode subsystem and fast-mode subsystem. In this paper, the ratio of volume of AHU with respect to that of cabin is defined as the singular perturbation parameter for the studied A/C system. That is, the singular perturbation parameter, ε, is defined as

PHT

hw ðtÞ ¼ ½ Qload Mload qO gO It SHGF  T

(33b) (33c) (33d)

where F˛<77 is the system matrix at winter mode, G˛<74 the corresponding input matrix, J˛<76 the disturbance matrix and H˛<37 the output matrix. Both of the system outputs at summer mode and winter mode are the temperature of cabin, qIN, and the associated thermal sensation scale by ASHRAE, YIN. For simplicity, the linearized system model around the operational point will be named as “nominal plant” hereafter. The eigenvalues of the nominal open-loop system are shown in Fig. 2. It is observed from Fig. 2 that the eigenvalues of the nominal open-loop system can be clustered into two groups: fast mode and slow mode. In other words, the HVAC system possesses two-time-scale property. Therefore, the HVAC is a type of singular perturbation system.

(34)

Assume the linearized plant model and the measurement can be expressed as follows [38]:



~ w ¼ uw  uew ; h ~ w ¼ yw  hðxew Þ ~ w ¼ xw  xew ; u ~ w ¼ hw  hew ; y x

h

-1

ε ¼ ðVHT þ VHF Þ=VIN

,

~ w þ Jh ~ w þ Gu ~ w ¼ Fx ~w x ~w ~ w ¼ Hx y

qW

0

-4

(30d)

(31)

pC ¼ ðYIN  bq qIN  bc Þ=bp

YIN

(Fast (FastModes) Modes)

1

-2

(30b)

where the superscript e is referred to the operational point. A˛<66 is the system matrix at summer mode, B˛<63 the corresponding input matrix, D˛<66 the disturbance matrix and C˛<36 the output matrix. At winter mode, due to the air conditioner stops working, it only provides bypass air flow. Therefore, the air temperature at outlet of cooling section, qC, and the partial pressure of water vaporization at outlet of cooling section, pC, can be obtained respectively as follows:

xw ðtÞ ¼ ½ qIN

Cluster#1#1 Cluster

-3

iT

hs ðtÞ ¼ ½ Qload Mload qO gO It SHGF  T

uw ðtÞ ¼

4

2

~ s þ Dh ~ s þ Bu ~s ¼ Ax ~s ~ s ¼ Cx y

xs ðtÞ ¼ ½ qIN

Groups of eigenvalues of open-loop system

Imaginary Axis

,

~ x

347

x_ εz_



 ¼

y ¼ ½ M1

A12 A22

A11 A21 M2 

    x B1 þ u; B2 z



  x z

xð0Þ zð0Þ



 ¼

x0 z0

 (35a)

(35b)

where x˛
xs ¼ ½ qIN

zs ¼ ½ qS

YIN

YS

qW  T qH  T

(36a)

(36b)

348

C.-C. Chiu et al. / Energy 66 (2014) 342e353 T T us ¼ R1 0 ðN0 M0 þ B0 Ks Þxs ¼ G0 x s

(2) Winter Mode:

xw ¼ ½ qIN zw ¼ ½ qS

YIN YS

qW  T

(37a)

qH qHT  T

(37b)

(44)

For fast-mode subsystem, the associated optimal control, uf, is synthesized to make the performance index, Jf, minimized

1 Jf ¼ 2

ZN

 yTf yf þ uTf Ruf dt;

R>0

(45)

0

4. Composite controller design By singular perturbation technique, two controllers can be synthesized for slow-mode model and fast-mode model individually. In other words, one control input, us, is designed for slowmode subsystem while the other control input, uf, is designed for fast-mode subsystem. Therefore, the composite control can be obtained by adding these two control components

u ¼ us þ uf ¼ G1 x þ G2 z

(38)

4.1. Near-optimal regulator In this section, the near-optimal regulator [38] for linear timeinvariant singular perturbation system is proposed. For slow-mode subsystem, the optimal control, us, is synthesized to make the performance index, Js, minimized

1 2

ZN ðyTs ys þ uTs Rus Þdt;

R>0

(39)

0

where

(40)

By substituting Eq. (40) into Eq. (39), the performance index can be rewritten as follows:

1 2

(46)

That is, the optimal control for the fast-mode subsystem can be obtained as

uf ¼ R1 BT2 Kf zf ¼ G2 zf

(47)

To sum up, the composite optimal control, uc, can be constructed as follows:

 h 1 T T uc ¼  Ir  R1 BT2 Kf A1 22 B2 R 0 ðN0 M0 þ B0 Ks Þ i 1 T B2 Kf z þ R 1 BT2 Kf A1 22 A21 z  R

(48)

4.2. Near-optimal order-reduced control Since the system submatrix, A22, for either summer mode or winter mode is Hurwitz, the order-reduced controller can be synthesized by solely taking the slow-mode subsystem model into account. In other words, we can directly ignore the fast-mode ð0Þ stable subsystem to obtain the order-reduced controller (i.e., G2 ¼ 0 in Eq. (47)). Hence, the near-optimal order-reduced controller [38] can be constructed by simplifying Eq. (48) as follows: T T ur ¼ R 1 0 ðN0 M0 þ B0 Ks Þx ¼ F

  x z

(49)

where F is the feedback control gain.

ys ¼ M0 xs þ N0 us

Js ¼

0 ¼ Kf A22  AT22 Kf þ Kf B2 R 1 BT2 Kf  MT2 M2

ð0Þ

where the feedback gains G1 and G2 can be separately synthesized to meet the performance specifications based on the slow-mode and fast-mode models respectively. In this paper, three types of optimal control strategies are synthesized. Namely, they are nearoptimal regulator, near-optimal order-reduced control and corrected LQR [38]. By comparing the performance of these three controllers and stander LQR in the later-on computer simulations, the most superior/simplest controller will be chosen for the further verification in realistic experiments at last.

Js ¼

For the fast-mode subsystem triplet, (M2, A22, B2), similarly a unique semi-positive definite solution, Kf, satisfies the corresponding algebraic Riccati equation

ZN ðxTs MT0 M0 xs þ 2uTs NT0 xs þ uTs R0 us Þdt

(41)

0

4.3. Corrected linear-quadratic regulator The corrected linear-quadratic regulator [38] is to utilize the optimal feedback gain, G(ε), as follows:

G1 ðεÞyG1 ð0Þ þ ε½dG1 ðεÞ=dεε¼0

(50)

G2 ðεÞyG2 ð0Þ þ ε½dG2 ðεÞ=dεε¼0

(51)

where where

R0 ¼ R þ NT0 N0

(42)

Assume the slow-mode subsystem triplet, (M0, A0, B0), is stabilizable and observable, there exists a steady-state unique semipositive definite stabilizing solution, Ks, for the algebraic Riccati equation T T 1 T 0 ¼ Ks ðA0  B0 R1 0 N0 M0 Þ  ðA0  B0 R 0 N0 M0 Þ Ks

þ

T Ks B0 R1 0 B0 Ks



MT0 ðI



T N0 R1 0 N0 ÞM0

(43)

Therefore, the optimal control for the slow-mode subsystem can be obtained as

ð0Þ

ð0Þ

1 G1 ð0Þ ¼ ðIr þ G2 A1 22 B2 ÞG0 þ G2 A22 A21

(52)

G2 ð0Þ ¼

(53)

ð0Þ G2

ð1Þ

ð1ÞT

ð0Þ

ð1Þ

½dG1 ðεÞ=dεε¼0 ¼ R 1 ðBT1 K1 þ BT2 K2

½dG2 ðεÞ=dεε¼0 ¼ R 1 ðBT1 K2 þ BT2 K3 Þ ð0Þ

Þ

(54) (55)

G0 and G2 are the uncorrected slow-mode and fast-mode feedback gain matrices defined by Eqs. (44) and (47) respectively. The ð0Þ ð0Þ ð0Þ matrices K1 ; K2 and K3 satisfy the following identities:

C.-C. Chiu et al. / Energy 66 (2014) 342e353 Table 4 Initial values of system output.

Table 6 System parameters.

Preset initial values of system output

Value Summer

qIN(0) YIN(0)

Temperature of air in cabin Thermal sensation scale of air in cabin

ð0Þ K1

¼ Ks ;

ð0Þ K2

¼ Km ;

ð0Þ K3



30 C 1.5062

Parameter Winter 

16 C 2.4197

¼ Kf

(56)

where Ks and Kf are defined by Eqs. (43) and (46) respectively. The ð0Þ new item is the cross-mode gain, K2 .





Km ¼ Km Ks ;Kf h    i ¼ Ks B1 R1 BT2 Kf  A22  AT21 Kf þ MT1 M2 1   A22  B2 R1 BT2 Kf

(57)

ð1Þ

The derivatives, Ki h½dKi ðεÞ=dεε¼0 ; i ¼ 1;2;3, satisfy the following equations: ð1Þ

ð0Þ

ð0ÞT

0 ¼ K1 ðA11  S1 K1  S12 K2 þ þ þ

ð0Þ

Outdoor temperature (summer mode) Outdoor temperature (winter mode) Solar irradiation intensity (summer mode) Solar irradiation intensity (winter mode) Specific heat of air Density of air Specific enthalpy of saturated vapor Parameters of ASHRAE thermal sensation scale

Natural convective heat transfer coefficient Forced convective heat transfer coefficient Heat capacity of cabin wall Volume of air in cabin Exterior surface area of cabin wall Interior surface area of cabin wall Exterior irradiated surface area of cabin wall Absorptance coefficient of solar radiation Transmittance coefficient of solar radiation Solar-Heat Gain Coefficient

Value

qO,S qO,W It,S It,W cp

ra

hg

bq bp bc

b hn b h f

CW VIN AO W AIN W ASH W

a s

SHGC

32  C 7 C 450 W/m2 225 W/m2 1.005 kJ/kg  C 1.225 kg/m3 2546.8 kJ/kg 0.245  C1 0.248  103 Pa1 6.475 15 W/m2 K 160 W/m2 K 4.908 kJ/ C 0.0027 m3 0.451 m2 0.430 m2 0.1 m2 6% 80% 0.87

Þ

ð0Þ ð0ÞT ð1Þ ðA11  S1 K1  S12 K2 ÞT K1 ð1Þ ð0Þ ð0ÞT K2 ðA21  ST12 K1  S2 K2 Þ ð0Þ ð0ÞT ð1ÞT ðA21  ST12 K1  S2 K2 ÞT K2

ð1Þ

349

ð1Þ

(58a)

ð0Þ

0 ¼ K1 ðA12  S12 K3 Þ þ K2 ðA22  S2 K3 Þ ð0Þ

ð0ÞT T

þ ðA21  ST12 K1  S2 K2 þ

ðAT11



ð0Þ K1 S1

ð1Þ



ð1Þ

Þ K3

(58b)

ð0Þ ð0Þ K2 ST12 ÞK2

ð0Þ

ð0Þ

ð1Þ

0 ¼ K3 ðA22  S2 K2 Þ þ ðA22  S2 K3 ÞT K3 þ

ð0ÞT K2 ðA12



ð0Þ S12 K3 Þ

þ ðA12 

ð0Þ ð0Þ S12 K3 ÞT K2

(58c)

and

S1 ¼ B1 R 1 BT1

(59)

S2 ¼ B2 R 1 BT2

(60)

S12 ¼ B1 R1 BT2

(61)

To sum up, the corrected linear-quadratic regulator, ucorrected, can be expressed as follows:

ucorrected ¼ G1 ðεÞx þ G2 ðεÞz

(62)

5. Computer simulations At the first stage, the proposed control policies: Near-optimal regulator, near-optimal order-reduced control and corrected LQR are examined by intensive computer simulations. In addition, Table 5 Desired values of system output. Set values of system output to achieve

Nominal value Summer

Temperature of air in cabin Thermal sensation scale of air in cabin

qeIN e YIN



25 C 0

Winter 22  C 0

Fig. 3. (a) Temperature of air in cabin, qIN, at summer mode. (b) Zoom-up of lowest of temperature of air in cabin, qIN, at summer mode.

350

C.-C. Chiu et al. / Energy 66 (2014) 342e353

Fig. 6. Thermal sensation scale of air in cabin, YIN, at winter mode.

Fig. 4. Thermal sensation scale of air in cabin, YIN, at summer mode.

the standard LQR plays the role of benchmark and is to be compared with the preliminarily proposed controllers by this work. The initial values and the desired values of the system outputs are summarized in Tables 4 and 5 respectively. In addition, all the physical quantities of system parameters are listed in Table 6. Under various control strategies, the corresponding response of the system outputs, i.e., qIN and YIN, are shown in Figs. 3e6. It can be observed that the discrimination of the temperature response under various control policies is fairly limited. In other words, by splitting the original system model into two reduced-order subsystem models: slow mode and fast mode, the efficacy of various optimal controllers proposed by this paper is all acceptable, with respect to the benchmark by LQR. Furthermore, even if merely the slow-mode model, obtained by singular perturbation technique, is employed to replace the original system model, the degree of downgraded performance in terms of time response of the temperature and humidity regulation by the proposed control strategies is still satisfactory, to some extent. Since the near-optimal order-reduced controller is the simplest and carries least computation load, it is chosen for the efficacy verification of later-on experiments in the next section.

6. Experimental results In order to verify the efficacy of the proposed near-optimal order-reduced control strategy, intensive realistic experiments for HVAC are undertaken. The schematic diagram and its photography of the test rig are shown in Figs. 7 and 8 respectively. The dimension of test rig is: 260 mm (W)  290 mm (H)  410 mm (D). The test rig is to experimentally verify the theoretical analysis described in Section 5. Though the heat radiation transfer mode is non-linear for the actual size of vehicle, the linearized radiation heat transfer coefficients can be used with acceptable accuracy for the scaleddown test rig as long as the surface temperature is not significantly high. In the experimental test rig, by using the ECB to imitate the vehicle cabin, the interface module DS1104 by dSPACE and the commercial software MATLAB/Simulink are employed to realize the proposed control loops. The interface module DS1104 acquires the voltage signals in terms of the temperature in ECB, qIN, by the temperature sensor and the relative humidity in the ECB, fIN, by the humidity sensor respectively. In fact, the ASHRAE thermal sensation scale, YIN, which is nothing but one of the system outputs, is the linear combination of qIN and the partial pressure of water vaporization, pv, according to Eq. (20). The partial pressure of water vaporization, pv, can be obtained as follows:

pv ¼ pg fIN

(63)

where pg is the saturated vapor pressure.

  ln pg ¼ 31:9602  6270:3605  0:46057lnðqIN Þ q IN

255:38 K  qIN  273:16 K

(64a)

and

ln

  pg Rc

¼

Ac þBc qIN þCc qIN þDc qIN þEc qIN 2 Fc qIN Gc qIN 2

3

4

(64b)

273:16 K  qIN  533:16 K

Fig. 5. Temperature of air in cabin, qIN, at winter mode.

The coefficients of Eq. (64b) are listed in Table 9 [39]. By added-on power amplifier circuit modules to drive AHU, humidifier and electric heater, the performance of the proposed control law: near-optimal order-reduced control strategy, is

C.-C. Chiu et al. / Energy 66 (2014) 342e353

351

Fig. 7. Schematic diagram of experimental test rig.

examined. The desired system outputs under summer mode and winter mode of the ECB are preset in Tables 7 and 8 respectively. At summer mode, the local temperature of the atmospheric temperature is about 28e30  C. At winter mode, due to the subtropical environment of Taiwan, it is not cold at all. More importantly, in order to accomplish the experiments both for Summer Mode and Winter Mode during the same week, we always preset the ambient temperature as 28e30  C. However, to imitate Winter Mode, the desired temperature of the air in ECB is set as 42  C so that it makes sense for the heating subsystem, instead of cooling subsystem, to the engaged. In the same manner, the corresponding ASHRAE thermal sensation scale for Winter Mode is referred to the relative humidity 50% and the temperature 42  C. The experimental results are shown in Figs. 9e12. At summer mode, under near-optimal order-reduced control strategy, the AHU is able to regulate the temperature in the ECB to 25  C, and limit the temperature fluctuation in the range of 0.5  C, as shown in Fig. 9. In comparison, once the control loop is removed, the temperature

Table 7 Desired values of output under summer mode. The variables to be preset

Summer mode

Desired temperature of air in ECB ðqIN Þ Desired thermal sensation scale of air in e Þ ECB environment ðYIN e

25  C 0 (Neutral)

Table 8 Desired values of output under winter mode. The variables to be preset

Winter mode e ðqIN Þ

Desired temperature of air in ECB Desired thermal sensation scale of air in e Þ (42  C, 50% RH) ECB ðYIN

42  C 4.8316

in ECB, shown in Fig. 9, will be continuously increased. After a certain period of time, the temperature of air in ECB, qIN, is expected to be far higher than the atmosphere temperature since the solar energy is continuously absorbed by the ECB. The ASHRAE thermal sensation scale inside the ECB can be controlled to merely vary below 0.5, i.e., the thermal comfortableness zone, and maintain at thermoneutral level (Y ¼ 0), shown in Fig. 10. On the contrary, at

Table 9 Coefficients for Eq. (63b).

Fig. 8. Photography of partial experimental test rig (outdoors).

Coefficient

273.16 K  qIN  533.16 K

Rc Ac Bc Cc Dc Ec Fc Gc

22105649.25 27405.526 97.5413 0.146244 0.12558  103 0.48502  107 4.34903 0.39381  102

352

C.-C. Chiu et al. / Energy 66 (2014) 342e353

Fig. 9. Temperature of air in ECB, qIN, at summer mode.

Fig. 12. Thermal sensation scale of air in ECB, YIN, at winter mode.

winter mode, the AHU can heat the air in the ECB to the desired temperature 42  C, shown in Fig. 11. The real-time actual ASHRAE thermal sensation scale, shown in Fig. 12, gradually merges to the desired value specified in Table 8.

7. Conclusions

Fig. 10. Thermal sensation scale of air in ECB, YIN, at summer mode.

In this study, a scale-down test rig of HVAC, which mainly consists of the ECB and the AHU to imitate the environment of cabin in vehicle, is constructed. The AHU module is constituted by the air cooling/heating sections, the mixing section and the humidifying section. The thermodynamic models of ECB and AHU are successfully derived and can be described by seven state equations. By system model linearization and singular perturbation technique, the studied system can be decoupled into two order-reduced subsystems: slow-mode subsystem and fast-mode subsystem. A series of optimal controllers, namely near-optimal regulator, near-optimal order-reduced control and corrected LQR, is proposed for the studied HVAC system. According to the computer simulation results, the efficacy of various optimal controllers developed by our work is all acceptable, with respect to the benchmark by LQR. The near-optimal order-reduced control strategy is the simplest and carries the least computation load so that it is chosen for the followup efficacy examination by experiments. The experimental results show that under near-optimal order-reduced control law, AHU is able to regulate the temperature in the ECB to 25  C, and continue to stay thermoneutral at summer mode. On the other hand, at winter mode, the AHU is able to heat the air in the ECB up to the preset temperature. In addition, the time response of actual ASHRAE thermal sensation scale gradually tends to achieve the desired value. By experiments, it is preliminarily verified that the nearoptimal order-reduced control strategy is almost as superior as that by LQR.

Acknowledgments

Fig. 11. Temperature of air in ECB, qIN, at winter mode.

The authors thank for the financial support and the equipments access which National Science Council (NSC) has provided. (Project #: NSC 102-2221-E-006-229-MY3).

C.-C. Chiu et al. / Energy 66 (2014) 342e353

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