A robust model predictive control strategy for improving the control performance of air-conditioning systems

Energy Conversion and Management 50 (2009) 2650–2658

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

A robust model predictive control strategy for improving the control performance of air-conditioning systems Gongsheng Huang a, Shengwei Wang a, Xinhua Xu b,a,* a b

Department of Building Services Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong Department of Building Environment and Services Engineering, School of Environmental Science and Engineering, Huazhong University of Science and Technology, Wuhan, China

a r t i c l e

i n f o

Article history: Received 5 March 2008 Received in revised form 24 November 2008 Accepted 21 June 2009 Available online 16 July 2009 Keywords: Time-delay uncertainty Robust model predictive control Air-conditioning system Robustness

a b s t r a c t This paper presents a robust model predictive control strategy for improving the supply air temperature control of air-handling units by dealing with the associated uncertainties and constraints directly. This strategy uses a first-order plus time-delay model with uncertain time-delay and system gain to describe air-conditioning process of an air-handling unit usually operating at various weather conditions. The uncertainties of the time-delay and system gain, which imply the nonlinearities and the variable dynamic characteristics, are formulated using an uncertainty polytope. Based on this uncertainty formulation, an offline LMI-based robust model predictive control algorithm is employed to design a robust controller for air-handling units which can guarantee a good robustness subject to uncertainties and constraints. The proposed robust strategy is evaluated in a dynamic simulation environment of a variable air volume air-conditioning system in various operation conditions by comparing with a conventional PI control strategy. The robustness analysis of both strategies under different weather conditions is also presented. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Air-conditioning systems are widely used in commercial and office buildings to provide safe and comfortable environment for building occupants. To maintain a safe and comfortable environment, the basic issue among controls is to maintain the system output at its set points although many studies presented different control strategies to reset control set points or change operation modes for improving system energy efficiency [1–4]. The control of air-conditioning processes is difficult to achieve due mainly to the associated uncertainties, nonlinearities and constraints. The sources of uncertainties may be attributed to measurements and the model structure used for control purpose [5,6]. Nonlinear behaviors happen due to hysteresis in actuators/valves and variations in fluid flow rate and the heat transfer coefficients. In addition, the system has time-varying dynamics and persistent timevarying disturbances, i.e., inlet air flow rate, temperature and humidity due to ever-changing cooling loads and weather conditions. Constraints exist because of the use of actuators, which usually operate with a rate limit and a range limit. Due to the complexity of air-conditioning systems, there are a number of approaches proposed for the local control of the supply

* Corresponding author. Address: Department of Building Environment and Services Engineering, School of Environmental Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei Province 430074, China. Tel.: +86 27 8779 2165x403; fax: +86 27 8779 2101. E-mail address: [email protected] (X. Xu). 0196-8904/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2009.06.014

air temperature. They can be roughly categorized into proportional-integral and/or derivative (PI or PID) control [7,8], artificial intelligence control [6,9,10], and robust control [11,12]. PI or PID feedback control algorithm is widely used in HVAC (heating, ventilation and air conditioning) fields due to its simplicity. Usually, the air-conditioning process is assumed to be a firstorder plus time-delay system [11,13], and site tests are conducted to determine the process parameters including process gain, time constant and delay time. These parameters are then used to compute the PID control parameters for practical control using Ziegler and Nichols settings or some other settings [14]. However, the control parameters at the testing point may result in aggressive or sluggish response at other work conditions due to that the control parameters cannot cover all the working range of the airconditioning systems [15]. Artificial intelligence is a newly developing technology, such as neural network control and fuzzy logic control, to deal with nonlinearities or uncertainties in HVAC processes [6,9,10]. Artificial neural network (ANN) has strong modeling capability for nonlinearities; while fuzzy logic control can deal with uncertainties in a straightforward manner. Both methods can find their application in HVAC systems. However, both ANN-based control and fuzzy control require a large amount of system operation data for model training, which should cover the whole process operating range. The collection of such complete data is not easy. Besides, they cannot take account constraints directly in the control design. Therefore, the application of such intelligent methods might be limited in practice.

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Nomenclature AIAE A, B a, b D DCV d e em F f h K N Np Q R S T

average of integrated absolute value of the supply air temperature error (°C) matrix coefficients of state-space model coefficients of the discrete model control input range constraint demand-controlled ventilation integer time delay tracking error (°C) largest possible tracking error (°C) gain of the state feedback rule cost function sampling interval (s) process gain or system gain (°C1) prediction horizon or number of process sampling number of the nested subspaces weight matrix used in the cost function weight factor used in the cost function matrix to denote state subspace time constant (s)

Robust H1 control has also been found in the temperature control of air-handling units to deal with dynamic uncertainties [11,12]. Time-delay and process gain uncertainties have been taken into account in the robust control design since they are the major factors affecting the control performance. In robust H1 control, the time-delay and the process gain are assumed to lie in an uncertainty set. Pade approximation is used to describe the timedelay. Difficulty with the robust control design may be the selection of the control parameters, including model uncertainty weights and optimization criteria weights, which is the major part of the controller design. Although robust H1 control can deal with uncertainties directly, it cannot deal with the associated constraints in the HVAC processes as well, which probably influence the robustness and stability of the controlled system. This paper presents a robust model predictive control strategy for HVAC processes which can deal with the associated uncertainties (dynamics variations) and constraints in a straightforward manner. When constraints in the process are likely concerned, model-based predictive control (MPC) is a good choice for control design. The development of MPC is considered as the recent major achievement in control literature, which has been widely accepted as the next generation of a practical control technology [16]. The application of MPC in HVAC systems can be found in the Ref. [17], which dealt with constraints but not uncertainties. Therefore, robust MPC is developed for HVAC applications in order to enhance the closed-loop robustness and stability. In this proposed strategy, the dynamics is represented using a first-order plus time-delay model, of which the time delay and the process gain are assumed to vary in a known range as in the Ref. [11]. An uncertainty polytope is developed to formulate the time-delay and process gain uncertainties, which can bridge the gap between robust model predictive control and its application in air-conditioning systems. An offline model predictive control algorithm developed by Wan and Kothare [18] is employed in this study. It will be shown that the proposed strategy can guarantee a good robustness when uncertainties and constrains are present and can be easily implemented without much user intervention. A typical Variable-Air-Volume (VAV) air-conditioning system for the application of this strategy is presented and the main control problems are identified in Section 2. The robust strategy is described in details in Section 3. In Section 4, the control performance of this robust strategy is demonstrated by comparing

u u_ x y

control input (–) control input varying rate state vector outlet supply air temperature (°C)

Greek symbols k variable c variable indicating the upper bound of the cost function d constant XAB polytope s delay time (s) Subscripts act actual measurement k current time instant r indicate temperature set point L lower bound U upper bound

with that of a conventional PI control strategy. The robustness analysis is also presented in this section. Conclusions are given in Section 5. 2. Description of an air-conditioning system The diagram of a typical AHU VAV air-conditioning system and its control instrumentations is shown in Fig. 1. The fresh air is mixed with the air through the recycled air damper, depending on the fresh air damper position (the fresh air damper, recycled air damper and exhaust air damper are interlocked as usual). The AHU controller generates control signals for manipulating the heating coil valve (or cooling coil valve) and the fresh air damper to control the supply air temperature at its set point if the outdoor air is favorable for free cooling. The AHU controller also generates the control signal to the fresh air damper for controlling the fresh airflow at its set point reset by the demand-controlled ventilation (DCV) strategy maintaining acceptable indoor air quality in the heating mode or the mechanical cooling mode. The conditioned air is distributed to the occupied zones via the supply air ducts and VAV terminals. The pressure controller maintains the supply air static pressure at its set point by modulating the fan pitch angle or the frequency of the supply fan. The return air fan is controlled by a controller to maintain the flow rate difference to maintain a positive pressure in the conditioned area. Usually, an air-handling unit needs to provide heating, free cooling and mechanical cooling at different seasons due to the ever-changing weather conditions in many regions such as Hong Kong. The actual outlet temperature of the AHU is used for the temperature control modulating the heating coil valve, the fresh air damper and/or the cooling coil valve. Fig. 2 presents the conventionally used split-range sequencing control logic [19]. In each of the heating and cooling modes, the air-handling unit usually experiences a wide range of weather conditions and a large change of the internal heat gain of the conditioned area. These external weather disturbances seriously affect the dynamic characteristics (process gain, time constant and time delay) of the air-handling unit together with the ever-varying internal disturbances. In this study, only the temperature control associated with the mechanical cooling processes (i.e., in the partial free cooling mode and in the mechanical cooling mode) is concerned. However, the proposed method can be extended to the control of other processes

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G. Huang et al. / Energy Conversion and Management 50 (2009) 2650–2658

Exhaust Air Damper

Temp. & Humidity Sensor

Flow Station

Return air

Return Fan

Damper Motor

CO2 sensor

Re-circulation Air Damper Cooling Coil Mix Air Plenum Outdoor Filter Air Damper

Pressure controller

M-1

Heating Coil

C

VAV

VAV

VAV

VAV

Supply Fan

H

Pressure Sensor

C

C

VAV

Heating coil control DCV-based fresh Fresh air Cooling coil control airflow set point set-point DCV-based fresh Enthalpy/temperature based air damper control reset strategy fresh air damper control

VAV

VAV

VAV

Conditioned Area

AHU Controller Temperature set-point

Control Signals

Fig. 1. Schematics of an air-handling unit and its control.

Heating Mode (DCV control)

Total Free Cooling Partial Free Cooling Mode Mode

100%

B uh

udT

Mechanical Cooling Mode (DCV control)

hrtn>hfr

Outdoor air Damper

Cooling Valve

Heating Valve A

uc

ud

hrtn
0% 200% 100% Output of the temperature controller

0%

-100% hrtn : return air enthalpy

hfr : fresh air enthalpy

uh : control signal for the heating valve

uc : control signal for the cooling valve ud : control signal for the damper based DCV control udT : control signal for the damper based on temperature control Fig. 2. Sequence control logic of the air-handling unit.

in the air-conditioning systems without difficulties when the timedelay and process gain uncertainties are of major concern. For the temperature control purpose, the dynamics of the typical air-handling unit concerning temperature control can be described using a first-order plus time-delay model (1) subject to the constraints (2) and (3)

yðsÞ K ¼ ess uðsÞ 1 þ Ts _ 6d juj

ð2Þ

DL 6 u 6 DU

ð3Þ

ð1Þ

where u is the control input, indicating the openness of the valve which controls the chilled water flow rate through the cooling coil; y is the outlet air temperature after the cooling coil. K is the process gain, T is the time constant and s is the time delay. The constraint (2) is the rate limit of the valve actuator; and the constraint (3) denotes the operating range of the valve. Usually, the control input u is in the range of u e [0, 1], where u = 0 indicates the valve is fully closed and u = 1 indicates the valve is fully open. At certain times, DL may be slightly larger than zero while DU may be slightly smaller than unit in practice due to a known mechanic misalignment between the valve and actuator. Note that when the air is cooled down the process gain K is always negative. Taking account of the

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uncertainties or dynamic variations, the process gain and the time delay are assumed to vary in the following uncertainty sets

K 2 ½K L ; K U  and

s 2 ½sL ; sU 

ð4Þ

Control design will then be performed based on the uncertain and constraint model (1)–(4). Note that the time constant T should be also uncertain. Since its influence on the control performance is less significant, its uncertainty is not considered and a nominal value of T is used, as in the Ref. [11]. 3. Robust model predictive control strategy

When the system gain uncertainty is taken into account, i.e., K e [KL, KU], K can be written as a linear combination of KL and KU, and hence the sub-model Ei is a linear combination of Ei,1 and Ei,2, defined by Eqs. (10a) and (10b), respectively. Therefore, the discrete model (7) is a linear combination of Ei,1 and Ei,2 (i = 1, . . . , dU  dL + 1) and can be written as

E¼

dUX dL þ1

ðki;1 Ei;1 þ ki;2 Ei;2 Þ

i¼1

with

dUX dL þ1

ki;1 þ ki;2 ¼ 1; ki;1; k;i2 > 0

i¼1 M

3.1. Formulation of time-delay uncertainty and system gain uncertainty

Ei;1 : ekþ1 ¼ ð1 þ aÞek  aek1 þ bL DukdL iþ1 ; M

with bL ¼ K L ð1  aÞ

Computer control needs to sample the first-order model (1) into a discrete model (5) using a sampling interval h. When the process is required to track a predefined set point yr, Eq. (5) is rewritten as Eq. (6).

ykþ1 ¼ ð1 þ aÞyk  ayk1 þ b1 Dukd þ b2 Dukd1

ð5Þ

ekþ1 ¼ ð1 þ aÞek  aek1 þ b1 Dukd þ b2 Dukd1

ð6Þ

~ where a = eh/T, b1 ¼ Kð1  eðhs~Þ=T Þ; b2 ¼ Kðeðhs~Þ=T  eh=T Þ, d and s ~; 0 6 s ~ < h, yr is the temperature set point, and satisfy s ¼ dh þ s ek = yk  yr is the tracking error. According to the suggestion from the Ref. [16], the sampling interval should satisfy h 6 T/10. Since the process gain and the time delay are uncertain, the parameters of the discrete model (6) are also uncertain. The discrete time delay d lies in the range d e [dL, dU], where dL = fl(sL/h) is the minimum discrete time delay and fl(x) is a function to round x to the nearest integer towards positive, dU = cl(sU/h) is the maximum discrete time delay and cl(x) is a function to round x to the nearest integer towards positive. In this case, a general form for the discrete model (6) is given by Eq. (7).

ð10aÞ

Ei;2 : ekþ1 ¼ ð1 þ aÞek  aek1 þ

M bU DukdL iþ1 ;

M

with bU ¼ K U ð1  aÞ

ð10bÞ

Define a state vector xk as Eq. (11). Then, the discrete model (7) is converted into a state-space model as Eq. (12). The uncertainties associated with (A, B) can now be described using a polytope XAB, defined by Eq. (13). This is because (A, B) is affine with respect to a, bi. The definitions of Ai1, Bi1, Ai2, Bi2, i = 1, . . . , dU  dL + 1 are given in Appendix A.

x0k ¼ ðek ; ek1 ; DukdU ; . . . ; Duk1 Þ

ð11Þ

xkþ1 ¼ Axk þ Buk (

ð12Þ

ðA; BÞ ¼

XAB :¼

dUX dL þ1

ki;1 ðAi;1 ; Bi;1 Þ þ ki;2 ðAi;2 ; Bi;2 Þ;

i¼1 dUX dL þ1

)

ki;1 þ ki;2 ¼ 1; ki;1; k;i2 > 0

ð13Þ

i¼1

3.2. Controller design

E : ekþ1 ¼ ð1 þ aÞek  aek1 þ b1 DukdL þ b2 DukdL 1 þ    þ bdU dL þ1 DukdU

ð7Þ

The values of bi, i = 1, . . . , dU  dL + 1, vary with respect to the change of the time delay as shown in Fig. 3. If the process gain does not change its sign, i.e., positive or negative always, bi lies in the range between zero and K(1  a). Note that at each time only two bis are nonzero and the nonzero bis satisfy bi + bi+1 = K(1  a). Hence, Eq. (8) holds true. dUX dL þ1

bi ¼ Kð1  aÞ

ð8Þ

i¼1

Therefore, the discrete model (7) can be written as PdU dL þ1 PdU dL þ1 ki Ei with ki P 0 and i¼1 ki ¼ 1, i.e., a linear combiE ¼ i¼1 nation of Ei, i = 1, . . . , dU  dL + 1. Ei is defined as follows: M

M

Ei : ekþ1 ¼ ð1 þ aÞek  aek1 þ b DukdL iþ1 ; with b ¼ Kð1  aÞ ð9Þ bi

bdU − d L +1

K (1 − a )

b1

b2

bdU − d L

The application of the uncertainty polytope as Eq. (13) to describe the time-delay and system gain uncertainties makes it possible to use robust model predictive control. The control objective is to produce control signals for controlling the cooling coil valve allowing the supply air temperature to track a preset temperature set point subject to the constraints (2) and (3). In discrete time, the constraints have the following forms

jDuk j 6 dh

ð14aÞ

DL 6 uk 6 DU

ð14bÞ

In this study, an offline model predictive control algorithm using linear matrix inequality (LMI) technique is employed to design the controller [18]. The LMI-based robust MPC designs control laws, which are in the form of a state feedback controller as described by (15). The gain of the control laws is optimized by minimizing the cost function (16a), which is defined using the tracking errors. According to the definition of the state vector (11), this cost function can be rewritten as Eq. (16b).

Duk ¼ F j xk N X fk ¼ ðek Þ2 fk ¼

τ

0 dLh + h

dU h − h

ð16aÞ

k¼1

...

dLh

ð15Þ

dU h

Fig. 3. Variations in the values of bi when time delay changes.

N X ðx0kþ1 Qxkþ1 þ RðDukþ1 Þ2 Þ

ð16bÞ

i¼1

where N is the prediction horizon, Q = diag(1, 0) and R = 0 . The state space is partitioned into a sequence of nested subspaces, denoted as x0 S 1 j x 6 1 (j = 1, . . . , Np), where Np is the number of the nested subspaces. Each subspace is assigned to a state

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feedback rule Fj. If the state lies in the jth subspace, then the control law is Du = Fjx. The subspace can be computed according to the size of the tracking error. When denoting the largest possible tracking error with em and setting (x1 = em, em, 0), a sequence of states for computing the nested subspaces can be defined as

Np  j þ 1 ; Np

xj ¼ x1 

j ¼ 1; . . . ; N p

ð17Þ

Given xj, the matrix Sj is computed using a LMI-based minimization, mincj ;Sj ;Y j cj , subject to the constraints (18a)–(18d), where cj is the upper bound of the cost function (16b) in the state space Sj. The constraint (18a) defines the subspace Sj; the constraint (18b) comes from the constraint (14a); and constraint (18c) ensures that Sj is nested into the subspace Sj1, ensuring that the convert of the control law in Sj from that in Sj1 will not violate the robust stability; the constraint (18d) indicates that the optimizer is obtained at the corners of the uncertainty polytope. j 0

1

ðx Þ

j

x

Sj

d2 I

Y

Y

0 j

j

! P0

ð18aÞ

P0

ð18bÞ

!

Q

S j1  S j P 0 0 Sj B B Al S j þ Bl Y j B B B Q 1=2 S j @ R1=2 Y j

S j A0i;l

þ

Y 0j B0i;l

SjQ

1=2

Y 0j R1=2

Sj

0

0

0

cj I

0

0

0

cj I

i ¼ 1; . . . ; dU  dL þ 1

1

ð18cÞ

C C C C P 0; C A

where Yj are temporary variables. Then, the feedback law Fj is given by

F j ¼ Y j S 1 j

ð19Þ

The computation of Fj and Sj (j = 1, . . . , Np) are performed offline, and they are stored in a lookup table. The online application of the lookup table is to find the appropriate feedback Fk according to the current state xk. This application requires finding the smallest subspace x0 S 1 m x 6 1, m e [1,2, . . . , Np], in which xk lies by checking whether xk satisfies the inequalities (20a), (20b). When m < Np, Fk is a linear combination of Fm and Fm+1 as Eq. (21) [18]. The coefficient ak is positive value and satisfies Eq. (22). When m = Np, Fk is equal to F Np as Eq. (23).

x0k S 1 m xk 6 1

ð20aÞ

x0k S 1 mþ1 xk

ð20bÞ

>1

F k ¼ ak F m þ ð1  ak ÞF mþ1 x0k ð k S 1 m

a

F k ¼ F NP

þ ð1  a

1 k ÞS mþ1 Þxk

ð21Þ ¼1

3.2.1.2. Online application algorithm. Step 2_1: Search the smallest subspace for xk as Eqs. (20a), (20b); Step 2_2: If m < Np, compute ak as Eq. (22) and compute the control law Fk as Eq. (21); else compute the control law as Eq. (23); Step 2_3: Compute Duk by (15) and uk by uk = uk1 + Duk; Step 2_4: Apply the control signal uk to the system; Step 2_5: At the next sampling time, go back to step 2_1. The application of the proposed control strategy is simple. User only needs to specify the uncertainty sets for the process gain and the time delay and the number of the nested spaces Np. Other computational work can be done without user intervention. The only parameters for control performance tuning are Q and R in the cost function [20]. However, if the uncertainty sets are appropriately specified, the tuning of Q and R is unnecessary because model mismatches (uncertainties) are considered in control design.

4. Tests and result analysis

ð18dÞ

l ¼ 1; 2

3.2.1. Control algorithm 3.2.1.1. Offline optimization. Step 1_1: Choose Np and em, and compute xj (j = 1, . . . , Np) as Eq. (17); Step 1_2: Compute Sj based on min cj and Fj by Eq. (19), respeccj ;Sj ;Y j tively, and stored them in a lookup table for online use.

ð22Þ ð23Þ

The offline LMI-based model predictive control algorithm can guarantee the robust stability of the closed-loop system if the minimization subject to the constraints (18a)–(18d) is feasible for x1. It should be noted that infinite prediction horizon and control horizon are used in the above approach, i.e. N = 1. The constraint (14b) cannot be directly taken into account in the minimization as the constraint (14a). However, its violation may be avoided by reducing the value of d. The online computational burden is not heavy since much of the design work is completed offline. The control algorithm is summarized as follows.

The robust model predictive control strategy (i.e., robust strategy thereafter) was tested for the cooling valve control allowing the outlet temperature tracking the preset temperature set point in a dynamic simulation environment as described in Section 4.1. The performance of this strategy was verified by comparing its performance data with that using a conventional PID control algorithm (i.e., conventional strategy thereafter) as presented in Section 4.2. The robustness of this robust strategy was also analyzed by comparing with that of the conventional strategy as presented in Section 4.3. 4.1. Test facility and test conditions The air-conditioning system and the building system as illustrated in Fig. 1 were simulated. The conditioned area is about 1200 m2. The design air flow rate is 7.2 kg/s, and the design supply air temperature is 13 °C. The design supply chilled water flow rate and temperature are 7.0 kg/s and 7 °C, respectively. The building envelopes and internal mass were represented using a first-order thermal network model to calculate the heat transfer due to the external disturbances and internal disturbances resulting in cooling load. For the air-handling unit, the cooling/heating coils were simulated using a dynamic model representing the nonlinear nature of the coils. The resistance characteristic of the damper is nonlinear and was simulated using the Legg’s exponential correlation [21]. The flow characteristics of the heating and cooling coil valves are equal percentage with dead bands [13]. Dynamic sensor model was used to simulate the temperature, pressure sensors etc. using time constant method. An actuator model was used to simulate the realistic characteristics of the actuators. The detailed description on the models of air-conditioning system and digital controllers can be found in the article [22]. In the simulation tests, occupancy, lighting and equipment loads in each zone were provided as input data files. The solar gains of each zone transmitted through the windows, sol–air temperature of each exterior wall, the outdoor air temperature, humidity and CO2 concentration at different weather conditions, were also

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G. Huang et al. / Energy Conversion and Management 50 (2009) 2650–2658

provided as input data files. A dynamic occupancy detection method [23] is used in DCV control to determine the total occupancy for calculating the fresh air set point for the fresh air damper control in the heating mode and the mechanical cooling mode, respectively. The air-conditioning system worked from 7:50 am to 20:00 pm. The control parameters of the conventional strategy (only P term and I term were used) were computed using Ziegler and Nichols settings [14] based on the process parameters of the airconditioning system. The process parameters at a typical summer operation condition in Hong Kong were used to determine the control parameters for the cooling process control all the year. The proportional gain is 5% °C1, and the integral time is 150 s. These parameters were finely tuned for the summer operation conditions. For control algorithm of the robust strategy, the range of the process gain is from 10 to 1, the time constant is 150 s, and the range of the delay time is between 40 s and 70 s. The largest possible tracking error em (i.e., the largest difference of the measured temperature and its set point) is 18 °C, and the number of the nested subspace, Np, is 4. The sampling interval is 15 s for both strategies which is equivalent to T/10. Because the actuator of the valve takes 150 s moving from fully close to fully open, the rate limit for the control signal was set to be |Duk| 6 dh = 15/150 = 0.1. The operating range of the control signal was 0 6 u 6 1. Therefore, DL = 0 and DU = 1. 4.2. Control performance evaluation Two typical inlet conditions of the air-handling unit were used for the performance test of the robust strategy together with the step change profile of the supply air temperature set point as shown in Fig. 4a. These conditions were also used to test the control performance of the conventional strategy (i.e., PI algorithm) for comparison. The temperature set points had a step change from 13 °C to 16 °C, and then returned to 13 °C after 2 h, and this process

repeated. One inlet condition (i.e., hot summer condition thereafter) of the air-handling unit is that the inlet air temperature is 26 °C, the air humidity is 0.021 kg/kg, and the supply air flow rate is 7.0 kg/s. This inlet condition is representative for the operation conditions of the air-handling unit in hot summer season in Hong Kong. The other inlet condition (i.e., cool winter condition thereafter) is that the inlet air temperature is 17 °C, the air humidity is 0.009 kg/kg, and the supply air flow rate is 3.0 kg/s. This inlet condition is representative for operation conditions in cool winter season in Hong Kong. In the tests of the conventional strategy, the control parameters kept unchanged. Fig. 4a shows the outlet air temperature of the air-handling unit using the robust strategy (i.e., robust model predictive control strategy) at the hot summer operation condition. The outlet temperature followed the supply air temperature set point very quickly when there was a sudden step change of the supply air temperature set point and reached the steady state whether the step change was upward or downward. The robust strategy took about 550 s to reach the set point 16 °C when the step change of the set point occurred at 11:00 am. Fig. 4b shows the control signals and increments. It can be observed that the control signal satisfied all the constraints under consideration. The control performance of the conventional strategy is presented for comparison. Fig. 5a shows the control performance of the conventional strategy (using PI algorithm) in the same hot summer operation condition. The outlet air temperature can also track the set point well. Fig. 5b illustrates the control signal and its increments, which shows that at the beginning of the transients the PI control strategy usually produced significant changes in the control signal which were larger than the rate limit of the actuator. Due to this rate limit, the output cannot respond as fast as the control signal. This can explain why this strategy took about 500 s to reach the temperature set point of 16 °C when the step (a) 18.00

(a) 18.00

Outlet temperature

17.00

Temperature ( C)

17.00

16.00

o

o

Temperature ( C)

Outlet temperature

16.00 15.00 14.00

15.00 14.00 13.00

13.00 Step change profile of temperature set point

12.00 Step change profile of temperature set point

12.00

11.00 10

11.00 10

11

12

13

14

15

16

17

11

12

13

14

18

15

16

17

18

Time (h)

Time (h)

(b) 1.20 (b) 1.20

0.60

0.10 Increment of control signal

0.40

0.00

Control signal (-)

0.20

Increment (-)

0.80

0.30

Control signal

0.30

0.80

0.20

0.60

0.10

0.40

0.00

Increment (-)

1.00

Control signal

1.00

Control signal (-)

0.40

0.40

Increment of control signal

0.20 0.20

-0.10

0.00

-0.20

-0.10

0.00 10

11

12

13

14

15

16

17

18

-0.20 10

11

12

13

14

15

16

17

18

Time (h)

Time (h) Fig. 4. Control performance of the robust strategy in the hot summer condition.

Fig. 5. Control performance of the conventional strategy in the hot summer condition.

G. Huang et al. / Energy Conversion and Management 50 (2009) 2650–2658

Further robustness analysis was conducted to evaluate the capability of these strategies for the disturbance rejection since the air-conditioning system always experiences the internal disturbances due to occupations and the use of lighting and equipments etc., and the external disturbances due to ever-varying weather conditions. Once again, the conventional strategy was also implemented for comparison. In the summer test day, the air temperature is between 27.1 °C and 29.7 °C, and the humidity is around 0.02 kg/kg as shown in Fig. 8. It’s very representative for the summer weather in Hong Kong. In the test day, the air-conditioning system operated at the mechanical cooling mode, and the DCV control was activated to introduce the enough and just enough fresh air (outdoor air) maintaining acceptable indoor air quality for saving cooling energy according to the indoor occupants. In the representative winter test day, the air temperature is between 13.3 °C and 15.8 °C, and the weather is also dry. The air-conditioning system operated at the partial free cooling mode as shown in Fig. 2. The maximum fresh air was introduced to reduce the mechanical cooling energy. Mechanical cooling still was required to cool down the mixing air to the temperature set point of 13 °C. For the robustness analysis, the average of integrated absolute value of the supply air temperature error with respect to the temperature set point (i.e., AIAE) is used as Eq. (24).

17.00 Outlet temperature o

16.00 15.00 14.00 13.00 Step change profile of temperature set point

12.00 11.00 10

11

12

13

14

15

16

17

18

Time (h) Fig. 7. Control performance of the conventional strategy in the cool winter condition.

35.0

0.034 0.030

Temperature in the summer day

30.0

0.026 25.0

0.022 Humidity in the summer day

20.0

0.018

Humidity in the winter day Temperature in the winter day

15.0

0.014

Humidity (kg/kg)

4.3. Robustness analysis

18.00

Temperature ( C)

change occurred at 11:00 am, which was only slightly less than the robust strategy. The control performance using the robust strategy at the cool winter operation condition is presented in Fig. 6. The outlet air temperature could track the set point well. The time taken by the robust strategy to reach the steady state is about 850 s when the step change occurred at 11:00 am. The responding speed was much slower than that of the summer because the system operated in part load condition in winter and equal percentage valves were used. The control performance is much better than that using the conventional strategy as shown in Fig. 7. This figure presents that this conventional strategy needed more than 1 h for the outlet temperature to track the step-changed temperature set point at the cool winter condition. This shows that the control parameters (i.e., proportional gain and integral time) of the conventional strategy tuned in the summer operation condition might not be appropriate for the winter operation condition. The inherent reason is that the characteristics of the air-conditioning system are always changed with the operation conditions. The robust strategy took account of the nonlinearity and the ever-varying dynamic characteristics of the air-conditioning process directly, and hence achieved a better robustness without tuning any parameters.

Outdoor air dry-bulb temperature ( )

2656

0.010 10.0 0

2

4

6

8

0.006 10 12 14 16 18 20 22 24

Time (h ) Fig. 8. Temperature profiles of test days.

P AIAE ¼

 yact j N

N jyr

ð24Þ

where N is the number of process sampling, yr and yact are the set point and actual outlet temperature of the air-handling unit, respectively. Fig. 9 shows the inlet air conditions of the cooling coil of the airconditioning system in the summer test day. The mixing air temperature was around 26 °C. The supply air flow rate almost reached the maximum air flow rate in the morning due to that the building was warmed up at night and much heat needed to be removed. In most of the office hours, the supply air flow rate was from 5.0 to 6.0 kg/s. After 5:00 pm, people left the office gradually and supply

18.00

40.0

o

o

Temperature ( C)

Outlet temperature

16.00 15.00 14.00 13.00 Step change profile of temperature set point

12.00

8.0

37.0

7.0

34.0

Flow rate

6.0

31.0

5.0

28.0 25.0

4.0

Temperature

22.0

3.0

19.0

2.0

16.0

1.0

13.0 10.0

11.00 10

11

12

13

14

15

16

17

18

Time (h) Fig. 6. Control performance of the robust strategy in the cool winter condition.

8

9

10

11

12

13

14

15

16

17

18

19

Supply air flow rate (kg/s )

Inlet air temperature ( C )

17.00

0.0 20

Time (h ) Fig. 9. Air flow rate and inlet air temperature of the coil in the summer test day.

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G. Huang et al. / Energy Conversion and Management 50 (2009) 2650–2658

14.0

16.0 15.5 15.0

12.5

14.5

12.0

14.0

11.5

13.5

Using the conventional strategy

11.0

13.0

10.5

12.5

10.0

o

Using the robust strategy

13.0

Outlet temperature ( C )

o

Outlet temperature ( C )

13.5

9

10

11

12

13

14

15

16

17

18

19

16.0

13.0

15.0

12.5

14.5

12.0

14.0

11.5

13.0

10.5

12.5

10.0 8

9

10

11

12

13

14

15

16

20

Fig. 10. Comparison of the robustness of the robust strategy and the conventional strategy in the summer test day.

30.0

5.0 Flow rate

25.0

4.0 3.0

20.0

Temperature

2.0 15.0

1.0

10.0

0.0 8

9

10

11

12

13

14

15

16

17

18

19

12.0 20

5. Conclusion A robust model predictive control strategy is presented in this study for improving the robustness of the temperature control of air-conditioning systems by taking directly account of the process gain and time-delay uncertainties as well as the constraints on the control input. This strategy was evaluated in a dynamic simulation environment of a typical AHU VAV system at different operation conditions by comparing with a conventional PI control strategy. The test results show that the proposed strategy has much better robustness than the conventional PI control strategy when the system experiences a wide range of operating conditions. This is because the model uncertainties can be specified to cover a possible large range of the operation conditions of air-conditioning systems. In practical application, the implementation of this robust strategy is not complex since it only requires the possible range of the system gain and the time delay instead of the detailed values at a certain operation point. Besides, no much user intervention is required whether in the offline control law design or in the online control law application. Acknowledgement

Appendix A. Definition of coefficients

2

Supply air flow rate (kg/s )

o

Inlet air temperature ( C )

6.0

19

the robust strategy may perform better than the conventional strategy in the representative test winter day.

8.0 7.0

18

Fig. 12. Comparison of the robustness of the robust strategy and the conventional strategy in the winter test day.

When dL = 0

35.0

17

Time (h )

Time (h )

40.0

13.5

Using the conventional strategy

11.0

o

o

15.5 Using the robust strategy

Outlet temperature ( C )

13.5

The work presented is financially supported by a grant (PolyU 5293/07E) from the Research Grants Council of the Hong Kong SAR and Sun Hung Kai Properties Ltd.

12.0 8

14.0

Outlet temperature ( C )

air flow rate reduced. This is the representative operation of the air-conditioning system in summer seasons. Fig. 10 presents the outlet temperature profiles of the air-handling unit at this summer test day using the robust model predictive control strategy (i.e., the robust strategy) and the PI control algorithm (i.e., the conventional strategy). It is noted that the time axis starts from 8:00 am while the air conditioning operated from 7:50 am. This robust strategy cooled down the supply air to the set point (13 °C) in a short time, about 28 min, and kept the outlet air temperature around the set point very closely. The error, AIAE, is about 0.10 °C. When the conventional strategy (i.e., PI algorithm) was implemented, it almost took 39 min to bring the supply air temperature to the set point as shown in Fig. 10. Moreover, obvious delays were observed due to the slow response to ever-changing disturbances. The error is 0.12 °C. The comparison results demonstrate that the robust strategy has much better capability to reject the disturbance than the conventional strategy in summer seasons. The inlet air conditions of the cooling coil of the air-conditioning system in the winter test day are presented in Fig. 11. The supply air flow rate was from 3.0 to 4.0 kg/s. The inlet air was about 17 °C, which is the mixture of the fresh air (about 15 °C) and the recycled air (about 25 °C). The fresh air damper was at the fully open position allowing as much as fresh air to be introduced in for reducing mechanical cooling energy. The outlet temperature profiles using the robust strategy and the conventional strategy are presented in Fig. 12. This robust strategy manipulated successfully the outlet temperature to track the set point with the error (i.e., AIAE) of 0.13 °C. The conventional strategy cannot respond as quickly as the robust strategy to the ever-changing disturbances. The temperature took a slightly longer time to reach the set point. The error was also high, about 0.16 °C. The comparison shows that

20

Time (h ) Fig. 11. Air flow rate and inlet air temperature of the coil in the winter test day.

A1;1

1þa 6 1 6 ¼6 4 0dU 1;3 2

a

3 01;dU 01;dU þ1 7 7 7; IdU 1;dU 1 5

01;dU þ2 1þa

a

6 6 1 A2;1 ¼ 6 60 4 dU 1;3 01;dU þ2 2 1þa 6 6 1 AdU þ1;1 ¼ 6 60 4 dU 1;3 01;dU þ2

01;dU 1

b

01;dU þ1 0dU 1;dU 1 a b

ML

ML

3 7 7 7; . . . ; 7 5

3 01;dU 1 7 01;dU þ1 7 7 IdU 1;dU 1 7 5

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G. Huang et al. / Energy Conversion and Management 50 (2009) 2650–2658

2

A1;2

1þa 6 1 6 ¼6 4 0dU 1;3 2

a 01;dU þ1

01;dU

IdU 1;dU 1

3

In these definitions, 0d1 ;d2 is a d1  d2 matrix with all items being zero, Id,d is a d  d identity matrix.

7 7 7; 5

01;dU þ2 MU

References

3

1 þ a a 01;dU 1 b 7 6 7 6 1 0 1;d þ1 U 7; . . . ; A2;2 ¼ 6 7 60 0dU 1;dU 1 5 4 dU 1;3 01;dU þ2 3 2 MU 1 þ a a b 01;dU 1 7 6 6 1 01;dU þ1 7 7 AdU þ1;2 ¼ 6 60 IdU 1;dU 1 7 5 4 dU 1;3 01;dU þ2 2

B1;1

3 ML b 6 7 ¼ 4 0dU ;1 5; 2

B1;2

1 MU

 B2;1 ¼    ¼ BdU þ1;1 ¼

3

b 6 7 ¼ 4 0dU ;1 5;

; B2;2 ¼    ¼ BdU þ1;2 ¼

1

0dU þ1;1 1 

 ;

0dU þ1;1



1

When dL > 0

3 ML 1 þ a a 01;dU dL b 01;dL 1 7 6 7 6 1 01;dU þ1 7; . . . ; A1;1 ¼ 6 7 60 IdU 1;dU 1 5 4 dU 1;3 01;dU þ2 3 2 ML 1 þ a a b 01;dU 1 7 6 6 1 01;dU þ1 7 7 AdU dL þ1;1 ¼ 6 60 IdU 1;dU 1 7 5 4 dU 1;3 01;dU þ2 2

3 MU 1 þ a a 01;dU dL b 01;dL 1 7 6 7 6 1 01;dU þ1 7; . . . ; A1;2 ¼ 6 7 60 I dU 1;dU 1 5 4 dU 1;3 01;dU þ2 3 2 MU 1 þ a a b 01;dU 1 7 6 6 1 01;dU þ1 7 7 AdU dL þ1;2 ¼ 6 60 I dU 1;dU 1 7 5 4 dU 1;3 01;dU þ2 2

B1;1 ¼    ¼ BdU dL þ1;1 ¼ B1;2 ¼    ¼ BdU dL þ1;2 ¼



 0dU þ1;1 ; 1   0dU þ1;1 1

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