xu effects of a gpc-pid control strategy with

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Energy Conversion and Management 47 (2006) 132–145

Effects of a GPC-PID control strategy withhierarchical structure for a cooling coil unit

Min Xu a, Shaoyuan Li a,*, Wen-jian Cai b, Lu Lu b

a Department of Automation, Institute of Automation, Shanghai Jiao Tong University, 1954 Hua-Shan Rd.,

Shanghai 200030, PR Chinab School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore

Received 19 February 2004; received in revised form 29 September 2004; accepted 30 March 2005

Available online 23 May 2005

Abstract

This paper presents a GPC-PID control strategy for a cooling-coil unit in heating, ventilation and air

conditioning systems. By analysis of the cooling towers and chillers, different models in the occupied period

are considered in each operating condition. Because of the complication of components, well tuned PID

controllers are unsatisfied, and the results are poor over a wide range of operation conditions. To solve this

problem, a GPC-PID controller with hierarchical structure is proposed based on minimizing the general-

ized predictive control criterion to tune conventional PID controller parameters. Simulation and experi-

ments show that the proposed controller is able to deal with a wide range of operating conditions andto achieve better performance than conventional methods.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Heating, ventilation and air-conditioning systems; Generalized predictive control; A cooling coil unit; PID

0196-8904/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2005.03.012

* Corresponding author. Tel./fax: +862162932114.

E-mail address: [email protected] (S. Li).

mailto:[email protected]

Nomenclature

Symbols

Tcwi chilled water inlet temperatureTcwo chilled water outlet temperatureTai dry bulb temperature in on-coil stateTao dry bulb temperature in off-coil stateTaiwb wet bulb temperature in on-coil stateTaowb wet bulb temperature in off-coil stateFa air flow of cooling coilFcw chilled water flow rateQ real cooling load provided by all chillersccw chilled water specific heatca air specific heatc1,c2,e parameters in cooling coil modelf non-linear time varying functionKP proportional gainKi integral parameterKd derivative parameterN0 minimum predictive horizonN1 maximum predictive horizonNu control horizony(k + j) system output at time k + j

yðk þ jÞ system predictive output at time k + je(k) error at time kw0,w1,w2 proposed controller parametersJ performance indexr(k) system output at time k

G,E,F,H two Diophantine equation parameter

AbbreviationsAHU air handling unitPID proportional integral derivativeGPC generalized predictive controlFOPDT first-order plus dead timeFIR finite impulse responseCARIMA controlled autoregressive and integrated moving average modelVSD variable speed drivesCCU cooling coils unitHVAC heating, ventilation and air-condition

M. Xu et al. / Energy Conversion and Management 47 (2006) 132–145 133

134 M. Xu et al. / Energy Conversion and Management 47 (2006) 132–145

1. Introduction

In most large air conditioned buildings, air handling units (AHUs) account for a significantportion of the total building energy consumption and have a major effect on comfort conditionsand maintenance costs. Besides, because of the practical characteristics of AHUs and the con-straints imposed by non-ideal actuators, AHUs can be considered as highly non-linear systems[1]. Therefore, the control problem for AHUs is both difficult and important. Generally, AHUsadopt the standard proportional integral derivative (PID) control method to maintain the supplyair temperature at the set point value by changing the chilled water flow. This control method hasa simple structure, a few parameters and robustness to disturbance. This method is favorable onthe assumption that the system model parameters vary in a narrow range [2]. However, a control-ler for an AHU should be designed to cope with a wide range of operating conditions because ofthe multiple time varying processes taking place in the system. The use of fixed gain PID control-lers is not likely to give satisfactory performance, even with a well tuned PID controller, whenapplied to another system with different model parameters and results in poor response. There-fore, an adaptive controller, which can solve the AHU problem and obtain better performanceover a large range of operating condition, is highly desirable.

Over decades, numerous studies related to advanced algorithms for AHU systems appear in theliterature [3–5]. Generalized predictive control (GPC), which intends to solve an optimizationproblem subject to system constraints for a finite future at the current time and to implementthe first optimal control input as the current control input, is another self tuning techniqueattempted by many researchers [6].

In this paper, we are interested in developing a hierarchical structure control scheme forincorporating GPC into the PID controller. The reasons lie in:

• Once advanced algorithms (such as GPC) are adopted, the existing equipment, especially hard-ware parts, has to be upgraded at a large cost.

• The engineering level and the complicated algorithm will prohibit the implementation of theadvanced control method.

• The advanced tuning methods usually lack explicit specifications and the plant operators areunfamiliar with the parameters tuning.

The proposed hierarchical structure control strategy consists of two levels, a basic level and anoptimization level. The basic level is a conventional PID controller, which is not likely to give sat-isfactory performance as operating conditions change. Through predicting system output andminimizing the GPC criterion in the optimization level, the gain of the PID controller is a vari-able, changing with time, to cater for the changing system and the environmental uncertainties.As a matter of fact, the closed loop performance equals that of the standard GPC, while the prac-tical controller still remains a PID structure to plant personnel. Hence, it is not necessary thatthey grasp the advanced algorithm, and the system performance can be improved with only aminor cost. Simulations and experiments are conducted for a cooling coil unit in an AHU sys-tem, and the results show performance improvement and a significant reduction of the operatingcost.


2. Air handing unit and control

2.1. Main components of the system

A schematic diagram of an AHU is shown in Fig. 1. It consists of coils, air dampers, fans,pumps, filters and valves.

Fresh air from outdoors enters the AHU through the outdoor air damper and may be mixedwith air passing through the recirculation air dampers depending on the mixing box damper set-tings. The temperatures and flow rates of the outdoors and recirculation air streams determine theconditions at the exit of the mixed air plenum. Air, which is exiting the mixed air plenum, passesthrough heating or cooling coils. At most, only one of the two coils will be active at any given timeif implemented properly and there are no valve leaks or other faults in the system. Here, we onlyconsider a cooling coil. After being conditioned in the coils, the air is distributed to the zonesthrough the supply air ductwork. The supply air temperature is measured downstream of the sup-ply fan. Return air is drawn from the conditioned zones by the return fan and is exhausted orrecirculated, depending once again on the position of the mixing box dampers. The return airtemperature is measured downstream of the return fan.

Control parameters of an AHU system are the temperature and flow rate of the conditioned air.To build a closed loop control system, the parameters to be measured are the following: dry bulbtemperatures, wet bulb temperature and the air flow of the cooling coil.

2.2. Plant modeling

Because thermal loads for the zones can vary markedly, it is common for a cooling coil unit tobe controlled to maintain the supply air temperature Tao at a set point value that is sufficiently low

Damper Motor

Outer Air Damper

Exhaust Fan

SupplyFan

Recirculation Air Damper

ReturnFan From Zones

To Zones

Mixed Air Plenum

Return Air Plenum

Filter

Supply Air

Temperature & Flow SensorT& F

T& FFa Fa

Cooling

Coil

FilteraiT aiwbT aoT aowbT

T&F Valve

Chilled water

cwT cwF

Exhausted Air

Fig. 1. Schematic diagram of conventional AHU control system.


to satisfy the zone with the largest cooling load at any given time. The output Tao can be describedas follows and is influenced by Fcw, Fa, Tai and Tcwi:

T ao ¼ f ðF cw; F a; T ai; T cwiÞ ð1Þ
In the steady-state, Eq. (1) can be explicitly expressed by [7]
Q ¼ c1F ea

1 þ c2F a

F cw

� �e ðT ai � T cwiÞ ð2Þ

and

Q ¼ ccwF cwðT cwo � T cwiÞ ¼ caF aðT ai � T aoÞ ð3Þ
Combining Eqs. (2) and (3), Tao can be written as follows
T ao ¼ T ai �ðc1=caÞF e�1

a F ecw

F cw þ c2F ea

ðT ai � T cwiÞ

These system variables change with the air and water flow rate. If the air flow rate or chilled waterflow rate is high, the time constant and time delay will be smaller, and vice versa.

2.3. Basic level control strategy

In a cooling coil unit, plant engineers usually adopt the conventional velocity PID controller inthe basic level:

DuðkÞ ¼ ½Kp þ K i þ Kd�eðkÞ þ ½�Kp � 2Kd�eðk � 1Þ þ Kdeðk � 2Þ¼ w0eðkÞ þ w1eðk � 1Þ þ w2eðk � 2Þ

ð4Þ

where w0 = Kp + Ki + Kd, w1 = �Kp � 2Kd and w3 = Kd.As system exhibits nonlinear behavior, the resulting system performance with conventional PID

controller is reduced, especially when the controllers operates over a wide range of operating con-ditions. One solution to alleviate the problem is continually to retune the basic level controllerparameters over a wide range operating conditions. The other is to design advanced controllerto enable a wider operating conditions. Because of constraints on time availability of plant per-sonnel and running cost of the plant, the former solution is commonly adopted. Therefore, wedesign the advanced predictive control algorithm incorporating basic level PID controller, toretune the basic level controller parameters through receding horizon window on the minimiza-tion of GPC criterion.

Over a large range of operating conditions, the controller parameters vary with time, so thecontroller can be expressed in the following format:

DuðkÞ ¼ w0ðkÞeðkÞ þ w1ðkÞeðk � 1Þ þ w2ðkÞeðk � 2Þ ¼X2

i¼0

wiðkÞeðk � iÞ ¼ WTðkÞeðkÞ ð5Þ

where

WTðkÞ ¼ w0ðkÞ w1ðkÞ w2ðkÞ½ � eðkÞ ¼ eðkÞ eðk � 1Þ eðk � 2Þ½ �T


3. A GPC-PID strategy with hierarchical structure

3.1. Optimization level control strategy

The drawback of the conventional PID controller is that it has three degrees of freedom in tun-ing, which is difficult for plant engineers to tune the parameters to meet different specifications.Hence, in this paper, a GPC-PID control strategy with hierarchical structure is proposed. A basiclevel and an optimization level are consisted of the hierarchical control structure. The basic levelcontroller parameters are retuned via the optimization level based on the identification modelthrough minimizing the performance index. Since the identification model and criterion is thesame with that of the standard GPC control algorithm, better performance than conventionalPID controller is obtained in the hierarchical structure.

The cooling coil unit to be controlled can be described as a CARIMA model, and j-step systemoutput can be derived by using two Diophantine equations [8]:

yðk þ j jkÞ ¼ GjDuðk þ j� 1Þ þ HjD�uðk � 1Þ þ F j�yðkÞ

where Gj and Hj are polynomials with respect to q�1 of degree j � 1 and maxðnEB � jþ 1; n� 1Þ,respectively. D�uðk � 1Þ ¼ Duðk � 1Þ=C and �yðkÞ ¼ yðkÞ=C, where C is assumed as the minimumphase and is supposed to be C(z�1) = 1.Substituting Eq. (5) into the above equation yields

yðk þ j jkÞ ¼ Gj

X2

i¼0

wiðkÞeðk � iÞ !

þ HjDuðk � 1Þ þ F jyðkÞ

Therefore, the optimal control variable can be obtained through the GPC criterion.When model and environment uncertainty is presented, the prior well tuned parameters are no

longer suitable, the optimization level starts to work and finds the optimal controller parameters{w0(k),w1(k),w2(k)} via a receding horizon optimization method based on a multistage costfunction,

J ¼ EXN1

j¼N0

½yðk þ j jkÞ � rðk þ jÞ�2 þXNu

j¼1

kðjÞ½Duðk þ j� 1Þ�2( )

ð6Þ

Substituting Eq. (5) into Eq. (6), the cost function becomes:

J ¼ EXN1

j¼N0

½eðk;W Þ�2 þXNu

j¼1

kðjÞ½W ðkÞeðkÞ�2( )

Suppose W*(k � 1) is the optimal parameter vector for minimization of J(k � 1,W) at the timek � 1.

To achieve an optimal control variable at time interval k, a second-order Taylor expansion ofJ(k,W) over W*(k � 1) is given by


Jðk;W Þ Jðk;W ðk � 1ÞÞ þ dJðk;W ÞdW

��W ðk�1Þ

� ðW � W ðk � 1ÞÞ þ 1

2ðW �W ðk � 1ÞÞT

� d2Jðk;W ÞdW 2

��W ðk�1Þ

� ðW �W ðk � 1ÞÞ þ Oð=W � W ðK � 1Þ=3Þ ð7Þ

where O(W) is the higher order expansion terms. By minimizing Eq. (7) with respect to W, theresult is shown as follows:

W ðkÞ ¼ W ðk � 1Þ þ P ðkÞ eðk;W ðk � 1ÞÞ � Sðk;W ðk � 1ÞÞ � kW ðk � 1ÞeðkÞeTðkÞ½ �

PðkÞ ¼ QðkÞ � kQðkÞeðkÞeTðkÞQðkÞ½eTðkÞPðkÞeðkÞ þ 1�

QðkÞ ¼ Pðk � 1Þ � kP ðk � 1ÞSðkÞSTðkÞP ðk � 1Þ½STðkÞP ðk � 1ÞSðkÞ þ 1�

Sðk;W ðk � 1ÞÞ ¼ � deðk;W ÞdW

��W ðk�1Þ

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð8Þ

In practice, the signal Du(k)given by W(k) at time k is computed and executed to the system, andat time k + 1, a new control action Du(k + 1) is recalculated with the estimated model and the cri-terion on a moving horizon window. Since the estimated model is a CARIMA form, the control-ler, through minimizing the criterion via receding horizon optimization, is equal to that ofgeneralized predictive control.

Under the condition k > 0, as the predictive horizon is larger than the amount of the controlhorizon, we can guarantee the closed loop asymptotical stability through adjusting the predictivehorizon and control horizon. Simultaneously, it is guaranteed that the maximum predictive hori-zon is larger than the upper of the time delay and the deterioration of performance will be small.

The GPC-PID control strategy with hierarchical structure is given as follows:

Step 1: Estimate the nominal CARIMA model to yieldG, H and F.Step 2: Set maximum and minimum predictive horizons and control horizon.Step 3: Initialize starting values as W*(0) = 0, P(0) = I.Step 4: Compute the W*(k) based on Eq. (8) through the predictive horizon.Step 4: Set the control law at the sample time k, as Du(k) = W*(k)e(k).Step 5: At the next sample timek + 1, go back to step 3 and obtain the next optimal control

value.

Thus, the optimal parameters are retuned at any sample time to obtain better performance thanthe conventional PID controller.

4. Simulation and experimental results

The experiment and simulation are also conducted on a typical HVAC system [9] (shown inFig. 2). There are five heat transfer loops: indoor air loop that includes fans, cooling coils, terminal

rooms & dampers

coil fans & ducts

cooling coils & valves

pumps &

pipes

evap

orat

ors

cond

ense

rs

compressorwater pumps

cooling towers

indoorair

chilledwater

condenser water

outdoor air

refrigerant

chillers

tower fans

outd

oor

envi

ronm

ent

outd

oor

envi

ronm

ent

Fig. 2. The plant of a typical HVAC system.


units, dampers and ducts; chilled water loop that includes pipes, pumps, cooling coils, chiller evap-orators and valves; refrigerant loop that includes evaporators, compressors, condensers andexpansion valves; condenser water loop that includes cooling towers, chiller condensers andpumps; outdoor air loop includes fans and cooling towers. All motors (fans, pumps and compres-sors) are equipped with variable speed drives (VSD) to control the motor speed.

We consider cooling coils units (CCU) with the dimensions of 25 cm · 25 cm · 8 cm for theAHU system whose schematic diagram is shown in Fig. 3. The system requires a differential pres-sure sensor across each cooling coil to monitor the chilled water flow rate as well as to estimate thecooling load of each coil. The sensors are also to control the opening positions of control valves toconstrain the chilled water flow. Another differential pressure sensor is mounted to monitor thepump head, which is used to control the VSD pump speed. Each coil also requires a tempera-ture sensor to measure the chilled water return temperature together with a common chilled watersupply temperature sensor to estimate the cooling load of each coil.

Each piping branch consists of a cooling coil and a two-way control valve. The two-way controlvalve is modulated to keep the off coil air temperature constant for the varying supply air volumein response to the varying cooling demand. The pump speed is modulated by a differential pres-sure sensor to keep the pressure differential of the farthest away branch (coil + valve) constant.The set point of a constant pressure differential is often chosen on the basis of the designedfull load condition. Sometimes, a safety factor is added to make sure that each coil has enoughpressure drop.

Chilled water pump

Chilled water in

Cooling coilcwiTcwF

Chilled water out

Air in Air out

cwoT

aF

aiT aiwbT

aF

aoT aowbT

Fig. 3. Schematic diagram of a cooling coil.


The measured signals for the experiment are the air and water flow rates, on coil air dry bulb/wet bulb temperatures and cooling coil inlet and outlet water temperatures. The experiment isconducted under the following conditions: the chilled water supply temperature is fixed; the cool-ing load variation is achieved through the air and water flow rates. The curve fitting results of themodel in Eq. (2) are c1 = 0.45, c2 = 0.70, e = 0.61. This is a non-linear model, and the model islinearized at three different operating condition.

Different controller parameters and system parameters at different operating condition is shownin Table 1. A result is shown in Fig. 4 based on the Table 1 controller parameters. At the stage 1,optimal PID controller parameters are selected [10] and a short rise time and no overshoot areobtained. At the stage 2, a large oscillation is obtained, since controller parameters cannot be se-lected again. Next stage, system is divergence for different operating condition. As can be seen inFig. 3, because conventional PID controller is chosen based on the stage 1 operating condition,system performance is unsatisfied for different operating condition. The same, we switch to theproposed controller in stage 3, better performance than conventional PID controller can beobtained (shown in Fig. 5). Fig. 6 shows the performance of the GPC-PID under a square waveset point change. It is clear that a better performance is achieved through the GPC-PID controlstrategy. Whereas it is difficult to apply one set of PID controller parameters to obtain good per-formance for the whole range of operating condition. Although tuning the controller at the pointof highest process gain is possibly to overcome system oscillation, it may result unsatisfied controlperformance under other stages of operating condition (seen in Fig. 7). To verify the robustness to

Table 1

Three level plant models with wide range (Figs. 3 and 4)

System parameters GPC-PID scheme Optimal PID parameter

a3 a2 a1 a0 b0 d N0 N1 Nu k KP Ki Kd

Stage 1 0 0 1 0.2 0.2 0 0 0 0 0 4 2.8 0.1

Stage 2 0 1 0.6 0.02 0.02 0 0 0 0 0

Stage 3 1 0.52 0.18 0.02 0.04 5 1 5 3 0.7

0 100 200 300 400 500 600 700 800 900 1000(s)

-1

0

1

2

3

syst

em o

utpu

t

Stage 1 Stage 2

Stage 3

Fig. 4. PID control scheme with optimal controller parameters.

0 100 200 300 400 500 600 700 800 900 1000(s)0

0.5

1

1.5

syst

em o

utpu

t

Stage 1 Stage 2 Stage 3

Fig. 5. GPC-PID control scheme for stage 3.

0 100 200 300 400 500 600(s)-0.5

0

0.5

1

1.5

syst

em o

utpu

t

stage 1 stage 2 stage 3

Fig. 6. System output with the proposed strategy for the whole operating condition.

0 100 200 300 400 500 600(s)

-1

0

1

2

syst

em o

utpu

t


Fig. 7. System output with optimal PID controller.



the disturbance, a white noise with 0.1 dithers are introduced in the controlled loop. As can beseen in Fig. 8, a satisfied control performance with small oscillation is obtained.

Figs. 9–11 show the different responses of fixed well tuned parameters PID controller and GPC-PID control strategy in the case of a temperature set point change and chilled water flow ratefluctuation.

First, we let the off coil temperature stabilize at 18.7 �C, and then the set point goes up to19.5 �C with a stable flow rate for the chilled water at the time t = 250 s.

Second, we increase the air flow rate, which leads to an increase of the water flow rate. The tem-perature set point remains unchanged to the time t = 1500 s.

Third, we decrease the air flow rate, which leads to a decrease of the water flow rate. The tem-perature set point remains unchanged to the time t = 2200 s.

Based on the same operating condition, the fixed well tuned PID controller is designed in spiteof the disturbance of the air flow rate. From Figs. 10 and 11, the overshoot of the output is largerthan 0.7�, while the PID controller tuned via the receding horizon optimization is only up to 0.2�.

0 100 200 300 400 500 600(s)-0.5

0

0.5

1

1.5

syst

em o

utpu

t


Fig. 8. System output with the proposed strategy in the presence of output disturbances.

Fig. 9. The air flow rate change.

Fig. 10. Conventional PID control method.

Fig. 11. The GPC-PID control strategy.


5. Conclusions

A novel receding horizon optimization algorithm based on the GPC criterion and CARIMAmodel, has been developed for an CCU system. Simulation and experiments show that the con-troller is able to reject the effects of both static and dynamic disturbances rapidly. The perfor-mance of the CCU system is improved compared with that of the conventional well tuned PIDcontroller. Without major change of hardware and software, the controller can self tune theparameters of the PID controller with one degree of freedom.

Acknowledgements

The authors would like to acknowledge the financial support of the National Science Founda-tion of China (Grant No. 60474051) and Development Program of Shanghai Science and Tech-nology Department under Grant 04DZ11008, and partly by the Specialized Research Fund forthe Doctoral Program of Higher Education of China under Grant 20020248028. The authorsare grateful to the anonymous reviewers for their valuable recommendations.


Appendix A

The criterion given by Eq. (6) can be rewritten as

Jðk;W Þ ¼ ½rðk þ jÞ � yðk þ j jkÞ�2 þ k½W TðkÞeðkÞ�2 ð9Þ
Differentiating this expression twice with respect toW
dJðk;W ÞdW

��W ðk�1Þ

¼ eðk;W ðk � 1ÞÞdeðk;W ÞdW

��W ðk�1Þ

þ kðW ðk � 1ÞÞTeðkÞeðkÞ ð10Þ

When k is large enough, the optimal W*(k) is very close to W*(k � 1), and then the followingapproximation is rational.

OðW ðkÞ � W ðK � 1ÞÞ ! 0

d2Jðk;W ÞdW 2

��W ðk�1Þ

d2Jðk;W ÞdW 2

��W ðkÞ

ð11Þ

Using Eqs. (10) and (11), the optimal vector W*(k) is written as

W ðkÞ ¼ W ðk � 1Þ þ RðkÞ�1½eðk;W ðk � 1ÞÞ Sðk;W ðk � 1ÞÞ � kW ðk � 1ÞeðkÞeTðkÞ�
where
RðkÞ ¼ d2Jðk;W ÞdW 2

��W ðkÞ

ð12Þ

Sðk;W ðk � 1ÞÞ ¼ � deðk;W ÞdW

��W ðk�1Þ

Applied the known lemma, we can conclude Eq. (9)

W ðkÞ ¼ W ðk � 1Þ þ P ðkÞ½eðk;W ðk � 1Þ Sðk;W ðk � 1ÞÞ � kW ðk � 1ÞeðkÞeT ðkÞ�

PðkÞ ¼ QðkÞ � kQðkÞeðkÞeT ðkÞQðkÞ½eT ðkÞPðkÞeðkÞ þ 1�

QðkÞ ¼ Pðk � 1Þ � kP ðk � 1ÞSðkÞST ðkÞP ðk � 1Þ½ST ðkÞP ðk � 1ÞSðkÞ þ 1�

Sðk;W ðk � 1ÞÞ ¼ � deðk;wÞdW

��W ðk�1Þ

8>>>>>>>>>><>>>>>>>>>>:

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xu effects of a gpc-pid control strategy with

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