Using cooling load forecast as the optimal operation scheme for a large multi-chiller system

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

available at www.sciencedirect.com

w w w . i i fi i r . o r g

journal homepage: www.elsevier.com/locate/ijrefrig

Using cooling load forecast as the optimal operation scheme for a large multi-chiller system Yan-Yue Lu a, Junghui Chen a,*, Tzu-Chi Liu b, Min-Hsien Chien b a b

Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan 320, Republic of China Green Energy & Environment Research Labs, Industrial Technology Research Institute, Hsinchu, Taiwan 310, Republic of China

article info

abstract

Article history:

Energy saving is one of the most important issues in high-tech manufacturing industries,

Received 25 May 2010

such as semiconductor and electronics, because large chilled water systems are used to

Received in revised form

satisfy big cooling load requirements. In this paper, a new optimal integrity scheme based

5 April 2011

on a two-stage strategy, including a scheduling stage and an operating stage, is proposed to

Accepted 19 May 2011

minimize the system energy consumption within a future time period. Instead of a lag

Available online 30 May 2011

scheme used in the general method, a forecasting scheme consisting of a series of optimal schemes at each sub-time period is also proposed for the two-stage design. The perfor-

Keywords:

mance of the proposed method is examined through an industrial case. The cost of the

System

proposed method is much less than that of the conventional method, so the proposed

Chiller

method is cost-efficient in applications of large air-conditioning systems. ª 2011 Elsevier Ltd and IIR. All rights reserved.

Energy savings Modelling Optimal design

Utilisation de la charge thermique pour pre´voir le fonctionnement optimal d’un grand syste`me a` plusieurs refroidisseurs Mots cle´s : Syste`me ; Refroidisseur ; E´conomies d’e´nergie ; Mode´lisation ; Conception optimale

1.

Introduction

In the high-technology manufacturing industry, such as semiconductor factories and electronics factories, the cooling load of the air-conditioning is heavy as the operation plant is strict with the cleanliness and the temperature of air. In

a typical semiconductor plant, more than ten sets of chilled water units are needed to satisfy heavy-load requirements. The power consumption of the chilled water system accounts for about 60%e70% of the total costs of facility & utility systems in a hi-tech industry. Using the optimization control strategy has tremendous potential to reduce operating costs

* Corresponding author. Tel.: þ86 886 3 265 4107; fax: þ86 886 3 235 4199. E-mail address: [email protected] (J. Chen). 0140-7007/$ e see front matter ª 2011 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2011.05.014

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

Nomenclature Aeva Acon ha,i ha,o hw,i,t hw,o,t hs,w,i

hs,w,o

Keva Kcon COP cs Cw fe fc N ma ma,r mew mcw mcw,i mcw,o m* P

2

heat transfer area of the evaporator (m ) heat transfer area of the condenser (m2) enthalpy of air entering the cooling tower (kJ kg1) enthalpy of air leaving the cooling tower (kJ kg1) enthalpy of water entering the cooling tower (kJ kg1) enthalpy of water leaving the cooling tower (kJ kg1) enthalpy of saturation air corresponding to the temperature of water entering the cooling tower (kJ kg1) enthalpy of saturation air corresponding to the temperature of water leaving the cooling tower (kJ kg1) heat transfer coefficient of the evaporator (Wm2K1) heat transfer coefficient of the condenser (Wm2K1) coefficient of performance of the chiller state variable specific heat capacity of water (kJ kg1K1) evaporator state variable condenser state variable number of transfer units of the cooling tower mass flowrate of air entering the cooling tower (kg s1) rated flowrate of air entering the cooling tower (kg s1) mass flowrate of chilled water (kg s1) mass flowrate of water entering the condenser (kg s1) mass flowrate of water entering the cooling tower (kg s1) mass flowrate of water leaving the cooling tower (kg s1) state variable atmospheric pressure

and increase energy efficiency (Wang and Ma, 2008; Gordon et al., 2000). Many efforts have been undertaken to develop the optimal operation and control strategies for the heating, ventilation and air-conditioning (HVAC) systems in the residential buildings or office buildings (Wang and Ma, 2008; ASHRAE, 2007). Austin (1991) stated that the true optimum loading point of centrifugal chillers could lead to the increase of the chiller plant efficiency by 20% or more. A significant increase in operation efficiency is possible when a chiller optimum loading point is correctly determined. Later Gordon et al. (2000) developed a simple thermodynamic model for the chiller performance using the measured performance data. The model succeeded in predicting the fundamental relation between coefficient of performance (COP) and the cooling rate for the centrifugal chiller. Yao et al. (2004a) investigated a large cooling system of residential buildings. The relationships among the controlled variables, the uncontrolled variables

Ps Pc Pfan Pfan,r Ppump,con Ppump,var PLR Qeva Qcr Qcon Ta,w Te Tc Tw,i,eva Tw,o,eva Tw,i,con Tw,o,con Tw,i,t Tw,o,t Dt Z ua,i ua,o uawo ea gw hp hf

2051

saturated water vapor pressure power consumption of the chiller (kW) power consumption of the cooling tower fan (kW) rated power consumption of the cooling tower fan (kW) power consumption of the constant-speed pump (kW) power consumption of the variable-speed pump (kW) partial load ratio of the chiller heat transfer quantity of the evaporator (equivalent to the chiller cooling capacity) (kW) rated cooling capacity (kW) heat transfer quantity of the condenser wet-bulb temperature of air entering the cooling tower ( C) evaporating temperature ( C) condensing temperature ( C) chilled water returning temperature ( C) chilled water supply temperature ( C) condenser water entering temperature ( C) condenser water leaving temperature ( C) temperature of water entering the cooling tower (equivalent to Tw,o,con) ( C) temperature of water leaving the cooling tower (equivalent to Tw,i,con) ( C) time interval binary variable moisture content of air entering the cooling tower (kg (kg dry air)1) moisture content of air leaving the cooling tower (kg (kg dry air)1) moisture content of saturation air leaving the cooling tower (kg (kg dry air)1) airside heat transfer effectiveness of the cooling tower specific weight of water (kNm3) efficiency of the pump efficiency of the variable-frequency driver

and the chillers’ performance were obtained empirically with the test data. These studies aforementioned are useful to comprehend the process and to improve the chiller performance, but these models were only applied under specified conditions. Based on the studies of chillers’ performance, many optimal operation strategies and solving methods have been developed. Hackner et al. (1984) presented an equal loading rate method to operate chillers. It is a conventional method and is still commonly used in HVAC systems. Chang (2004) used the Lagrangian method to solve the optimal chiller loading problem and to improve the deficiencies of conventional (equal loading rate) methods. Braun and Diderrich (1990) developed optimal and near-optimal control strategies using quadratic relationships for chiller systems. In the system based methodology, an overall empirical cost function of the total power consumption of a chiller plant was developed using a quadratic function. Nassif et al. (2005) used

2052

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

a multi-objective genetic algorithm which permitted the optimal operation of the building’s mechanical systems installed in parallel with a building’s central control system. Using this proposed optimization procedure, the supervisory control strategy setpoints were optimized with respect to the energy use and the thermal comfort. The studies mentioned above focused on improving the chiller’s performance and efficiency by means of experiments and model simulation, and developing a number of theoretical models, empirical models, and semi-empirical chiller models. Most of the models could exactly describe the chiller performance and help to optimize the chiller operation. However, from the energy saving viewpoint, the interaction between chillers, cooling towers and pumps should be considered for the whole chilled water system. The efficient operation of the individual unit as well as the whole chilled water systems has been studied in some published work. Lu and Cai (2004) presented a model-based optimization strategy for the condenser water loop of centralized HVAC systems. The optimization problem was formulated to minimize the total operating cost of all energy consuming devices. A modified genetic algorithm for this particular problem was proposed to obtain the optimal setpoints. Yu and Chan (2008) investigated the energy performance of the chiller and the cooling tower systems integrated with the condenser water flow, the speed of tower fans, and the condenser water pumps. Load-based speed control was introduced for the tower fans and condenser water pumps to achieve optimum system performance. Most of the control strategies and operation methods mentioned above were applied to public buildings, not to the hi-tech industry. They are appropriate to the chiller system consisted of 2e4 chillers. The chilled water system scale in public buildings is generally smaller than that in the hi-tech industry. In the past, each chiller in the operation method taking on the same proportional loads by regulated setpoints is commonly employed. With world today experiencing energy revolution and the increased emphasis placed on preserving the environment, we all need to play our roles in saving energy. By saving energy, it is worthy to explore new design methods of the chilled water system processes, but the design work for a large multi-chiller system has rarely been tackled from a system engineering perspective. Furthermore, these control schedules in the past were obtained at a point of time, so they were appropriate for a short period of time. When the cooling load changes with time, the procedure has to be recalculated to obtain a new operation scheme. Therefore, these methods have a hysteretic characteristic; the scheme may not be optimal until a long period of time. The aim of this paper is to present a design methodology for the operation of a large chilled water system in the whole operation period. Two design stages based on forecasted cooling load, including the scheduling stage and the operating stage, are presented. This method can minimize energy consumption while maintaining comfortable conditions in the industrial building. The rest of the paper is organized as follows. Section 2 classifies the types of design problems and the chilled water system model is built in Section 3. Techniques for the design problem, the cooling load forecast, the scheduling stage and the operating stage are respectively discussed in Section 4. Section 5

then details a case study from a semiconductor factory to assess the performance and applicability of the proposed method before some concluding remarks.

2.

Problem description

In high-tech industries, a large set of chilled water systems usually consists of more than 10 chillers with different cooling capacities and several cooling towers (shown in Fig. 1). The interactions between different unit operations are complex. Obtaining a proper operation mode of chiller units is not an easy task because the chiller’s performance is affected by several factors, such as the chilled water supply temperature, the partial load ratio and the temperatures of the water entering condenser, etc. In addition, from the viewpoint of energy saving, for the operation of a large set of chilled water systems, the load requirement should be considered for the whole period of time instead of a single time point only. Fig. 2 shows the cooling load demand changes with the time. In order to obtain an optimal operation scheme of the large set of chilled water systems and to keep the system operation stable without frequent turning on/off of chillers, a design method based on the whole period of the cooling load is presented in this paper. The design objective, represented by Eq. (1), is to minimize the system energy consumption in a whole period when the cooling load changes with time. minJ ¼

" K N h X X  k¼1

 Pc;n ðkÞ þ Pcp;n ðkÞ þ Pfan;n ðkÞ þ Ptp;n ðkÞ

n¼1

M h i X   Zn ðkÞ  Dt  fe þ Zn ðkÞ  fc;n þ Pc;m ðkÞ þ Pcp;m ðkÞ m¼1

i  þ Pfan;m ðkÞ þ Ptp;m ðkÞ  Zm ðkÞ  Dt  fe þ Zm ðkÞ  fc;m

# ð1Þ

where k denotes a sub-period point within Dt time interval in each sub-period. As shown in Fig. 2, the whole period of the operations can be divided into K intervals. For the design of operation stability, two sizes of the chillers are classified,

Cooling Tower 1

Chiller 1

Cooling Tower 2

Cooling Tower m-1

Chiller 2

Cooling Tower M

Chiller N

Air-Conditioning Cooling Load

Fig. 1 e Schematic diagram of a large set of chilled water systems.

2053

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

Pfan

mcw,a Tw,o ,t

Tw,i ,t

Cooling Tower

Ptp

Tw,i ,con

Cooling Water Loop Qcon

mcw

Tw,o ,con

Condenser Tc Fig. 2 e Designed cooling load in the time period.

Expansion

Compressor

Chiller Pc Te

where N and M denote the number of large-sized chillers and small-sized chillers, respectively. Pc,n(k) denotes the power consumption of the n th large chiller at the k th time point. Pcp,n(k), Ptp,n(k) and Pfan,n(k) denote the power consumption of the chilled water pump, the condenser water pump and the cooling tower fan, respectively, and they are equipped with the large chillers. Analogously, Pc,m(k), Pcp,m(k), Pfan,m(k), and Ptp,m(k) denote the corresponding power consumption of the small chillers. Z(k) is a binary variable; when it is 1, the chiller is running during the k th sub-period; on the other hand, when it is 0, the chiller is turned off. fe is the electricity fee (energy charge) . fc,n and fc,m denote the start-up cost of the large and the small chillers from the beginning till reaching a steady operation. Start-up costs are the energy costs incurred at the warm-up period of chillers. If chillers are shut down to save on energy consumption from time to time, the start-up costs would be high as the chillers need to be turned on several times. It may be economical if the chillers are not shut down to reduce the start-up costs. Directly solving Eq. (1) is difficult, because it is an MINLP (mixed integer nonlinear programming) problem in multiple time periods. The binary variable, Z, is used to determine the chiller loading or unloading in Eq. (1). It is hard to compute the convergence of this type of problem. In order to obtain an optimal operation scheme of the large chilled water system and to simplify the calculation effort of mixed integer nonlinear programming, a two-stage mathematic programming for the design problem is proposed in this paper.

3.

Chilled water system models

Since this design method is a mathematical programming method based on operation process models, the model units should be firstly set up. The energy consumption of the chilled water system mainly depends on four primary units, including chillers, cooling towers, chilled water pumps and condenser water pumps. The schematic diagram of a single chilled water system is shown in Fig. 3. Appropriate unit models are necessary to exactly describe the heat transfer process as well as to avoid the complicated computation in optimization study. The system model will be built up based on the empirical and the semi-empirical unit models, in which

Evaporator

Tw,o ,eva mew

Qeva

Tw,i ,eva Pcp

Chiller Water Loop Pcp

Cooling Load Fig. 3 e Schematic diagram of a single chilled water system.

the interaction among chillers, cooling towers and pumps will be considered. The four primary units used in this paper are listed in Appendix in detail.

4.

Optimal flexible operation design

Most previous studies concerning the system operation design aimed at the cooling load at a fixed time point. When the cooling load is changed, it is necessary to obtain a new required cooling load before any corrective action is taken. This design may lag behind and it does not fit for the practical operation. It would be preferable to have the required cooling load in the long period and adjust the schedule before the cooling load is changed. In this paper, the design method based on the cooling load forecasting is an operation scheme within the future time period. This design scheme can offer an operating schedule for operators to enhance the performance of the chilled water system and to keep the system operation stable. In addition, for a large chilled water system in the hi-tech industry, the chillers are the main section of energy consumption. The frequent operation of turning on/off chillers may save energy to a certain degree when the cooling load is changed, but it is not conducive to maintaining the system stability and extending the chillers’ life span. Instead of the system design at a point of time, the operation scheme obtained from the optimization design within a time period can prevent the disadvantages mentioned as the interaction

2054

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

Divide a working day into multiple compartments.

Predict the cooling load of each compartment.

According to the peak load, a working day is divided into two periods.

MINLP Sscheduling Stage

Design the scheme of chiller load or unload in the monotonic increase period.

Operating Stage

MIP

Design the scheme of chiller load or unload in the monotonic decrease period.

Design the operation scheme when the chiller number is fixed.

NLP

The system operation scheme in a working day

Fig. 4 e The logics of the solution to the design problem.

of chiller running state between sub-time periods is considered. The integrated expression of this operation optimization problem is described by Eq. (1). The design problem in this paper is solved based on the two-stage procedure: the scheduling stage and the operating stage. The logics of the solution can be described by the block diagram shown in Fig. 4. First, the whole period is divided into several compartments for cooling load prediction. With the estimated trend of the cooling load, may it be increase or decrease, the operation scheme for the loaded or unloaded chillers within each short period is determined at the scheduling stage. Then at the operating stage, the operation conditions of the chilled water system would be designed to minimize the system energy consumption under the needed chillers to be running at the scheduling stage.

4.1.

Cooling load forecast for a time period

Most existing design methods of the chilled water system can just satisfy the cooling load demand at one time point. The cooling load is adjusted without considering the running chillers at the adjacent time points. When the cooling load is changed, the operation scheme has to be redesigned. This may cause the stability problem of the chiller operation system. The dynamic design method based on the cooling load forecasting is needed; it can schedule the optimal operation in a whole period. There are various approaches applied to the load forecasting, including multiple linear regression techniques, time-series approaches, artificial neural networks, etc. (Soliman et al., 1997; Yao et al., 2004b; Hahn et al., 2009) Among these methods, the time-series approach is a very attractive forecasting tool because of its simplicity, recursiveness, and economy of use. Dynamics is an important inherent characteristic of daily operation processes. In some cases, such dynamics exist not only within a particular day but also from day to day. Several different reasons may cause daily-wise dynamics, such as slow property change of refrigerant, drifts of the process characteristics, the effects of slow response variables, and so

Fig. 5 e Two-dimensional representation of the cooling load data.

on. All of these reasons are common in daily operation processes. Thus, the dynamics can be viewed with two different time axes: a within-day time axis and a cross-day time axis, shown graphically in Fig. 5. Based on such an idea, a two-dimensional, time-series approach is proposed with the within-day time direction as one dimension, and the cross-day time direction as the second dimension. As illustrated in Fig. 5, the cooling load data generated by D number of successive days and K sampling intervals of each day are considered. Let Q(d,k) be the cooling load at the sampling interval k in day d (d¼1,2,.,D ; k¼1,2,.,K ), which can be arranged in a 2-D field with two directions d and k standing for day and time, respectively. This means that the current values of the cooling load Q(d,k) will depend on the past values of the current day in the time direction, Q(d,k1),.Q(d,ks),in day direction, Q(d1,k),.,Q(d[,k), and in the cross direction, Q(d1,k1),.Q(dl,ks), where [ and s are the autoregressive orders in the two directions. The region covering the above lagged values is shown in Fig. 5. Mathematically, the forecasting model is expressed as:

Qðd; kÞ ¼ a0 þ

l X

ax Qðd  x; kÞ þ

x¼1



s X

s X

by Qðd; k  yÞ þ

y¼1

cx;y Qðd  x; k  yÞ

l X x¼1

(2)

y¼1

where a, b and c are the regression coefficients.

4.2.

The scheduling stage

For the purpose of maintaining the stability of the chilled water system, the operation mode of alternate running between chillers with the large cooling capacity and chillers with the small cooling capacity is used. In this mode, the large chillers take on most of the cooling load while the small chillers mainly run in the transitional stage of the large chillers within loading and unloading. This means that the small chiller is firstly loaded at the beginning before the large

2055

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

chiller is hooked up whenever the cooling load increases. Likewise, the small chiller is first unloaded to regulate the refrigerating output when the cooling load decreases. In this way, the chiller water system is adjusted in a more stable manner. It can supply enough refrigerating outputs as well as save energy consumption. From the estimated trend of the cooling load during the period, the whole operation period can be classified into two operation modules based on the changing trend of the cooling load. The trend can be either a monotonic increase or decrease. Two operation modules during each time period are sequentially designed. (i) When the trend of the cooling load is a monotonic increase, the scheduling objective function is defined as minJinc ¼

" Kinc X N M  X X ðZn ðkÞ  Zn ðk  1ÞÞ  fc;n þ Zm ðkÞ k¼1

n¼1

m¼1



N X

 Zm ðk  1Þ þ yi;m  fc;m þ

Pcr;n  Dt  fe  Zn ðkÞ ð3Þ

(ii) When the trend of the cooling load is a monotonic decrease, the optimal schedule of the operation can be represented as follows:

(4)

minJdec ¼

n¼1

þ

M X

# Pcr;m  Dt  fe  Zm ðkÞ

th and the (k1) th sub-time period, respectively. ym(k) is also a binary variable. Its value is determined by constraints Eqs. (5) and (6). For example, when the m th small chiller is loaded at the (k1) th time point (i.e. Zm(k1)¼1) and unloaded at the k th time point (i.e. Zm(k)¼0), the value of ym(k) should be equal to 1. In this situation, the fixed cost in the second term of the right side in Eq. (3) is not calculated at the k th time point. The operating logics for the starting up cost of the small chiller could be guaranteed by the constraints of Eqs. (5) and (6), where U is a large enough positive integer. The third and fourth terms of the right side in Eq. (3) are used to calculate the operation cost when the chillers are running in the k th subtime period. Pcr,n and Pcr,m are the rated power of the large chiller and the small chiller, respectively. In this scheduling stage, the rated powers of chillers are employed to determine the scheme of the chiller loading or unloading. The sufficient refrigerating output that satisfies the demand of airconditioning is guaranteed with the constraint of Eq. (7).

m¼1

s:t

Zn ðkÞ  Zn ðk  1Þ

  Zm ðkÞ  Zm ðk  1Þ < U 1  ym ðkÞ

" Kdec X M X  k¼1

N X  Pcr;n Zm ðkÞ  Zm ðk  1Þ þ ym ðkÞ  fc;m þ

m¼1

n¼1

 Dt  fe  Zn ðkÞ þ

(5)

M X

# Pcr;m  Dt  fe  Zm ðkÞ

ð8Þ

m¼1

Zm ðk  1Þ  Zm ðkÞ  Uym ðkÞ N X n¼1

Qeva;r;n  Zn ðkÞ þ

M X

Qeva;r;m  Zm ðkÞ  Qcooling load ðkÞ

(6)

(7)

Zn ðkÞ  Zn ðk  1Þ

s:t

(9)

  Zm ðkÞ  Zm ðk  1Þ < U 1  ym ðkÞ

(10)

Zm ðk  1Þ  Zm ðkÞ  Uym ðkÞ

(11)

m¼1

where the subscript k denotes the sub-time period k. The length of the period is Kinc. The aims of the design are to determine an optimal scheme of the chiller running at each sub-time period and to minimize the operational cost Jinc in the whole time period. In Eq. (3), Zn(k) and Zn(k1) are binary variables and they indicate the on/off state of the n th large chiller in the k th and the (k1) th sub-time period, respectively. If both values of Zn(k) and Zn(k1) are 1, the n th large chiller keeps running at these two sub-period points. Therefore, the first term of the right side in the objective function does not cover the cost of the chiller starting up from the k th sub-time period. If the n th large chiller is loaded from the k th sub-time period, a fixed cost of the chiller starting up would be added to the objective function. The constraint of Eq. (4) is used to ensure that the number of large chillers operated monotonously increases as the cooling load increases continuously. Instead of getting large chillers loaded only, the small chillers could be loaded or unloaded to regulate the refrigerating output of the system and prevent the intensive and drastic change of the chilled water temperature. When the small chillers are loaded, the fixed cost of the chiller startup should be calculated. On the other hand, when the small chillers are unloaded, the fixed cost should not be calculated. The second term of the right side in Eq. (3) is used to determine when the fixed cost of the small chiller loading should be calculated. In Eq. (3), Zm(k) and Zm(k1) are binary variables which indicate the on/off state of the m th small chiller in the k

N X

Qeva;r;n  Zn ðkÞ þ

n¼1

M X

Qeva;r;m  Zm ðkÞ  Qcooling load ðkÞ

(12)

m¼1

where the length of the period is Kdec. Since there is no additional expense requirement for unloading the chillers as the cooling load decreases, the fixed cost term of the large chillers in Eq. (8) is removed. Except for the fixed cost term of the large chillers, the model of the cooling load decreasing is similar to that of the cooling load increasing. Therefore, the two operation modules of the chiller scheduling design can be grouped without dividing the cooling load requirements into increasing or decreasing cooling load. To avoid complex calculation resulting from too many integer variables, the optimization models of the cooling load increasing and decreasing are employed separately in this paper.

4.3.

The operating stage

In the scheduling stage, the rated power of the chillers is employed to determine the scheme of the chiller loading or unloading while the practical power consumption would be calculated in this operating stage. Now, the binary variables Z in Eq. (1) are known because they are already designed at the scheduling stage. Thus, the optimization problem (Eq. (1))

2056

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

becomes a NLP problem that is relatively easy to be solved. The mathematical programming model consists of the primary unit models presented in Appendix. Also, some thermodynamic constraints for the whole chiller system shown in Fig. 1 are needed: mtew ðkÞ ¼

N X

mew;n ðkÞ  Zn ðkÞ þ

n¼1

cw  Taevp ðkÞ  mtew ðkÞ ¼

M X

mew;m ðkÞ  Zm ðkÞ

(16)

m¼1

N X ðTw;o;eva;n ðkÞ  mew;n ðkÞ  cw  Zn ðkÞÞ n¼1

þ

M X

ðTw;o;eva;m ðkÞ  mew;m ðkÞ  cw

m¼1

 Zm ðkÞÞ   mtew ðkÞ  Cw  Tw;i;evp ðkÞ  Taevp ðkÞ  Qcoolingload ðkÞ

ð17Þ (18)

where mtew(k) indicates the total chilled water flowrate. Taevp(k) indicates the average temperature of the chilled water system outputs. T(k)w,i,evp is the temperature of the return chilled water. Eq. (16) denotes that the total chilled

Before executing the optimal operation design of this chilled water system, predicting the cooling load of the whole duration is essential. In this case study, the cooling load data from April 8 to 20 are collected to predict the cooling load on April 21 using the aforementioned twodimensional time-series approach. The forecasted results of the cooling load during 8:00 a.m.e6:00 p.m. on April 21 are plotted in Fig. 6. In Fig. 6, the blue points represent the actual load at each time point; the black line, the forecasted cooling load. For the short-term load forecasting, the twodimensional, time-series approach can make an effective prediction. In order to determine the parameters of these chiller unit models, a great deal of operational data are collected. The related parameters employed in the cooling tower model and the pump model are given in Table 1. The empirical relationship of the chiller’s COP (coefficient of performance) shown in Eq. (A2) is used to evaluate the chiller efficiency. The parameters in Eq. (A2) for the large chillers are listed as follows:

½ p0 p1 p2 p3  ¼ ½ 26:0906 51 33:1815 1:8364  ½ q0 q1 q2  ¼ ½ 51 5:4111 0:1425  ½ R0 R0 R0 R0 R0 R0  ¼ ½ 7:192e  05 0:0001848 0:00021335 0:000431 1:7188e  05

water flowrate of the operation system at the k th time point is equal to the summation of the chilled water flowrate of each loading chiller. Eq. (17) is the enthalpy balance relationship of the chilled water. The constraint of Eq. (18)

5.

Case study

The chilled water system of a semiconductor factory in Taiwan has been investigated to improve the energy consumption with the method presented in this paper. This chilled water system consists of 12 large chillers (1750 RT) and 3 small chillers (500 RT). Each chiller is equipped with a chilled water pump, a condenser water pump and a cooling tower. Table 1 summarizes the detail of the large and the small chilled water systems.

(19)

For the small chillers, the regressed parameters in Eq. (A2) are listed below:

½ p0 p1 p2 p3  ¼ ½ 0:8028 0:1292 0:0148 0:0006  ½ q0 q1 q2  ¼ ½ 70:7580 1:4468 0:01  ½ R0 R0 R0 R0 R0 R0  ¼ ½ 0:0588 0:1471 0:3052 1:658 2:5652 1:1246 

means that the refrigerated output of the system is supposed to meet the demand of the cooling load. In this mathematic programming problem, the design variables include the flowrate and the temperature of the supply chiller water, the flowrate of cooling water, the output water temperature of the cooling tower, and the air flowrate of the cooling tower. Thus, at each time period (k), an optimal operation can be designed.

0:00016 

(20)

PLR is a key variable in Eq. (A2). In general, for the sake of the energy saving, the chillers are operated within a certain range of PLR which can maximize COP. For this chilled water system of the semiconductor factory, when the evaporator temperature (Te) and the condenser temperature (Tc) are fixed at 9  C and 32  C, the large chillers’ and the small chillers’ relationships between COP and PLR are given in Figs. 7 and 8, respectively. From Figs. 7 and 8, it can be seen that the values of COP are all relatively high within thePLR scope of 0.7e0.8 for these two types of chillers. The field data indicates that PLR corresponding to the maximum COP is around 0.75 with the consideration of the evaporator temperature (Te) and the condenser temperature (Tc) change. Therefore, in the following design, PLR of each chiller should be set to be 0.75 for keeping the best chiller performance. Thus, the required refrigerated output (Qcooling load(k)) of the operating system in the scheduling stage is equal to the predicted cooling load of each hour divided by 0.75. The enlarged forecasting loads are added into the design model equations, Eqs. (7) and (12), as the lower bound values. An optimal design scheme of the operation system based on

2057

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

6

Table 1 e Detail of the chilled water system for the large chillers and the small chillers.

The number of chillers, n,m Rated cooling capacity of chillers (RT), Qcr,n, Qcr,m Rated power of chillers (kW), Pcr,n, Pcr,m Design chilled water supply/return temperatures ( C), Tw,o,eva, Tw,i,eva Design condenser water entering/leaving temperatures ( C), Tw,i,con, Tw,o,con Rated power of the chilled water pump (kW), Pcp,r,n Rated flowrate of the chilled water pump (kg/s), mew,r,n Rated pumping head of the chilled water pump (m), Pc,h,n Rated power of the condenser water pump (kW), Ptp,r,n Rated flowrate of the condenser water pump (kg/s), mcw,r,n Rated pumping head of the condenser water pump (m), Pt,h,n Air mass flowrate of the cooling tower (kg/s), ma,r,n Fan motor power of the cooling tower (kW), Pfan,r,n N’, the parameter in Eq. (15) p, the parameters in Eq. (15) Design wet-bulb and dry-bulb temperatures ( C), T The electricity cost ($/kW.h), fe Fixed cost of chiller loading ($),fc

5.5

Small chiller

12 1750

3 500

957.4 8/16

280.7 8/16

30/35

30/35

30

20

180

50

12

12

5 4.5 4 COP

Large chiller

3.5 3 2.5 2 1.5 1 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PLR

75

47

250

120

23

23

150

75.6

100

22.8

1.26 0.63 22/32

0.5 1.5 22/32

0.08 10

0.08 5

Fig. 7 e Relationship of COP and PLR of the large chillers.

the forecasting cooling load can be solved using the proposed two-stage solution strategy to minimize the operation cost during the design period. GAMS/CPLEX (The General Algebraic Modeling System) is used to solve the mixed integer

programming (MIP) problem at the scheduling stage, and GAMS/MINOS is used to solve the nonlinear programming (NLP) problem at the operating stage. In this case, the whole period shown in Fig. 6, K, can be divided into two periods: K1 (from 8:00 a.m.e1:00 p.m) and K2 (from 1:00 p.m.e6:00 p.m) based on the estimated trends of the cooling load. The optimal design results of the scheduling stage during period K1 and period K2 are shown in Table 2. The design results in Table 2 indicate that the number of running large chillers increases with the increase of the cooling load, while the small chillers would be loaded or unloaded to regulate the refrigerated output of the system during the transitional period. During the transitional period of loading the large chillers from 10 to 11, the small chillers are gradually loaded. When the cooling load further increases, a large chiller is loaded while all the small chillers are unloaded. In the later period (from 12:00 a.m to 1:00 p.m.), a small chiller is loaded again due to the appearance of the

4

1.46

x 10

7

1.44

6.5 6 1.4

5.5 1.38

COP

Predicted Load (RT)

1.42

1.36

4.5

1.34

1.32

5

4

8

9

10

11

12

13

14

15

16

17

18

Time (h)

Fig. 6 e Results of the cooling load prediction: the blue point is actual load at each time point and the black line is the curve of forecasted load.

3.5 3 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PLR

Fig. 8 e Relationship of COP and PLR of the small chillers.

2058

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

Table 2 e The optimal design results of the chilled water system operation. Number of loaded chillers

Operating strategy 1

14000 RT (10:00 a.m.)

No. 1e10

No. 1e10

No. 1e10

No. 1

No. 1

No. 1e3

Tw,i,con

mcw

PLR

Tw,o,eva

Tw,i,con

mcw

PLR

Tw,o,eva

Tw,i,con

mcw

PLR

10.0 10.1 716.3

29.0 29.3

215.2 89.6

0.74 0.76

9.9 10 725.8

29.0 29.3

218.0 92.3

0.75 0.79

10.0 10.1 755.6

29.0 29.3

215.9 90.3

0.74 0.77

mcw

PLR

mcw

PLR

mcw

PLR

215.9 92.5

0.73 0.78

29.0 29.3

218.7 95.1

0.75 0.80

216.3 92.9

0.74 0.79

Tw,i,con

mew

All large chillers All small chillers Operation cost, $

134.3 37.3 714.5

Operating strategy 3 All large chillers All small chillers

13400 RT (9:00 a.m.)

Tw,o,eva

All large chillers All small chillers Operation cost, $ Operating strategy 2

13200 RT (8:00 a.m.)

No. 1 No. 2 No. 3

Operation cost, $

29.0 29.3

mew

Tw,i,con

136.0 38.3 724.5

mew

Tw,i,con

134.5 37.4 753.8

29.0 29.3

Tw,o,eva

mew

Tw,i,con

mcw

PLR Tw,o,eva

mew

Tw,i,con

mcw

PLR Tw,o,eva

mew

Tw,i,con

mcw

PLR

8.9 5.0

150.9 27.1

29.0 29.3

215.2 92.7

0.74 0.78

152.9 27.8

29.0 29.3

218.0 95.2

0.75 0.80

150.2 37.9 26.8 26.8

29.0 29.6 29.3 29.3

214.1 117.6 91.7 91.7

0.73 1 0.77 0.77

712.3

8.9 5.0

722.3

Number of loaded large chillers

8.9 6.0 5.0 5.0 752.1

14200 RT (11:00 a.m.)

14400 RT (12:00 a.m.)

14500 RT (1:00 p.m.)

No. 1e11

No. 1e11

No. 1e11 No. 2

Operating strategy 1 All large chillers All small chillers Operation cost, $

Tw,i,con

mcw

PLR

10.0

29.0

216.1

0.74

770.8

Operating strategy 2

134.7

29.0

mcw

PLR

216.5

0.74

767.5

Operating strategy 3

Tw,o,eva 9.9

Tw,i,con

mcw

PLR

Tw,o,eva

Tw,i,con

mcw

PLR

29.0

218.0

0.75

10.0 10.1 784.3

29.0 29.3

214.8 89.1

0.74 0.76

mcw

PLR

mcw

PLR

219.3

0.75

215.4 92.1

0.73 0.78

778.9 Tw,i,con

mew

All large chillers All small chillers Operation cost, $

All large chillers All small chillers

Tw,o,eva

mew

Tw,i,con

136.3

29.0

Tw,i,con

134.0 37.1 782.2

777.6

29.0 29.3

Tw,o,eva

mew

Tw,i,con

mcw

PLR Tw,o,eva

mew

Tw,i,con

mcw

PLR Tw,o,eva

mew

Tw,i,con

mcw

PLR

8.9

150.9

29.0

215.2

0.74

152.9

29.0

218.0

0.75

150.2 27.0

29.0 29.3

214.1 92.3

0.73 0.78

8.9

8.9 5.0

No. 1 No. 2 No. 3

Operation cost, $

mew

765.1

775.2

Number of loaded large chillers

779.8

14300 RT (2:00 p.m.)

14250 RT (3:00 p.m.)

14050 RT (4:00 p.m.)

No. 1e11

No. 1e11

No. 1e10 No. 1e3

Operating strategy 1 All large chillers All small chillers Operation cost, $

mcw

PLR

Tw,o,eva

Tw,i,con

10.0

29.0

217.5

0.74

10.0

135.8

29.0

mcw

PLR

218.4

0.74

774.2

Operating strategy 3

mcw

PLR

Tw,o,eva

Tw,i,con

mcw

PLR

29.0

216

0.74

10.0 6.2 758.1

29.0 29.3

216.6 91.7

0.74 0.78

Tw,i,con

mcw

PLR

mcw

PLR

216.9

0.74

216.4 93.2

0.74 0.79

770.9 Tw,i,con

mew

All large chillers All small chillers Operation cost, $

Operation cost, $

Tw,i,con

775.6

Operating strategy 2

All large chillers All small chillers

Tw,o,eva

mew 134.9

29.1

mew

Tw,i,con

150.0 41.8 754.6

768.8

29.1 29.3

Tw,o,eva

mew

Tw,i,con

mcw

PLR Tw,o,eva

mew

Tw,i,con

mcw

PLR Tw,o,eva

mew

Tw,i,con

mcw

PLR

8.9

152.6

29.0

217.6

0.74

151.6

29.0

216.2

0.74

151.8 27.4

29.0 29.3

216.3 93.8

0.74 0.79

8.9

No. 1 No. 2 No. 3 771.8

766.4

8.9 5.0

754.4

2059

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

Table 2 (continued) Number of loaded chillers

Operating strategy 1

No. 1e10

No. 1e3

No. 1e3 mcw

PLR

Tw,o,eva

Tw,i,con

mcw

PLR

10.0 6.4 749.2

29.0 29.3

214.3 89.5

0.73 0.76

10.1 6.7 740.3

29.0 29.3

211.8 87

0.72 0.73

mcw

PLR

mcw

PLR

214.0 91.0

0.73 0.77

211.6 88.7

0.72 0.75

Tw,i,con

mew 148.4 40.8 745.3

Operating strategy 3

Operation cost, $

No. 1e10

Tw,i,con

All large chillers All small chillers Operation cost, $

All large chillers All small chillers

13700 RT (6:00 p.m.)

Tw,o,eva

All large chillers All small chillers Operation cost, $ Operating strategy 2

13800 RT (5:00 p.m.)

No. 1 No. 2 No. 3

29.0 29.4

mew

Tw,i,con

146.7 39.8 736.0

29.0 29.3

Tw,o,eva

mew

Tw,i,con

mcw

PLR

Tw,o,eva

mew

Tw,i,con

mcw

PLR

8.9 5.0

150.1 26.8

29.0 29.3

213.9 91.6

0.73 0.77

8.9 5.0

148.4 26.1

29.0 29.3

211.5 89.3

0.72

745.1

peak load. Thus, with determination of chillers loaded or unloaded during each time interval, the fixed cost of the P PM P chiller loading ð Kk¼1 ½ N n¼1 Zn ðkÞ  fc;n þ m¼1 Zm ðkÞ  fc;m Þ is $145 for the whole period. By this operation mode of the alternative running between large chillers and small chillers, the chilled water system is adjusted in a more stable manner. The chiller’s operating scheme given by the above design results is in agreement with the trend of the predicted cooling load shown in Fig. 6. When the design period changes from period K1 to period K2, the chiller running state at the last sub-time period in period K1, i.e. at 1:00 p.m., would become the initial value of the first sub-time period in period K2. This way, the consistent operation scheme would be obtained. Table 2 shows that the number of running large chillers decreases during this period with the decrease of the cooling load. Similarly, the small chillers are loaded during the transitional period of unloading the large chillers from 11 to 10. The proposed methodology is so flexible that it can tolerate inaccurate peak points of the cooling load. During this transition, the main adjustment is made for loading or unloading small chillers, because the operation rules for the small chillers are the same in both monotonic increase period and monotonic decrease period. Thus, even if we have inaccurate peak points of the cooling load, the design results are not affected. The operation conditions of the multi-chiller system would be designed after the scheduling of loading or unloading chillers is determined. Now, the binary variables Z in the objective function (Eq. (1)) are given. Thus, the optimization problem becomes a NLP problem at each sub-time period. Three feasible operation strategies are tested here: Strategy 1regulating the supplying temperatures of the chilled water Strategy 2regulating the chilled water flowrates Strategy 3regulating both the supplying temperatures of the chilled water and the chilled water flowrates.

735.7

The design results shown in Table 2 (row Strategy 1) are obtained when the chilled water flowrate, mew, is set to be 180 kg s1 for the large chillers and 50 kg s1 for the small chillers. The measured returning chilled water temperature, Tw,i,eva, is 16  C. This operation strategy keeps the chilled water flowrate constant by using a constant-speed pump and adjusts the refrigerated output of each chiller by regulating the respective chilled water supply temperature. From row Strategy 1, it can be seen that the operation condition of each large chiller is the same at each time point. The chillers’ operation is all regulated around PLR of 0.75 since the chillers have an excellent performance under this condition as mentioned earlier. When the cooling load experiences a big change, the chiller loading or unloading is regulated to maintain the system operation state within the scope of high performance. When a little change to the cooling occurs, the chilled water supply temperatures are adjusted. In this way, a stable and consistent operation scheme is obtained. Row Strategy 2 in Table 2 is the design results of the regulated chilled water flowrates. The chilled water supplying and returning temperatures, Tw,o,eva and Tw,i,eva, are fixed at 8  C and 16  C, respectively. The refrigerated output of each chiller is regulated by altering the respective chilled water flowrate. Therefore, the variable-speed pumps are needed in the chilled water supply side. The design results shown in row Strategy 2 are similar to the results in row Strategy 1, but the chilled water supply temperatures and the chilled water flowrates are lower than those of strategy 1 with the same cooling load. The relatively low chilled water flowrate would cause the low power consumption of chilled water pump, but the relatively low chilled water supply temperature would cause higher chiller power consumption. However, the system power consumption of strategy 1 and strategy 2 depends on the trade-off between the chiller power and the chilled water pump power. In this case, the design result indicates that strategy 2 has less system power consumption compared with strategy 1.

2060

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

Table 3 e Comparisons of the total costs of the three operation strategies. Chiller operation methods Strategy 1(regulating Tw,o,eva) Strategy 2(regulating mew) Strategy 3 (regulating Tw,o,eva and mew)

Total operation cost, $ 8325.8 8299 8280.2

When the returning temperature of measured chilled water is 16  C, the design result of the regulated chilled water supply temperatures and chilled water flowrates are simultaneously designed and listed in row Strategy 3 in Table 2. From row Strategy 3, it can be seen that the system operation is stable during the whole design time period. When the cooling load is changed, the chillers are regulated for loading or unloading to maintain the system stability. After that, the operation variables, such as the chilled water flowrates and the cooling water flowrates, are further adjusted slightly to keep the chillers effectively running. The design results listed in row Strategy 3 indicates that the chillers’ performance in this condition is the best when the chillers’ PLRs are mostly kept around 0.75. Thus, for the time point of 10:00 a.m., PLRs of the three small chillers are maintained to get close to the optimal value. By using the variable load of the distribution method and regulating operation variables, the chillers would take on different cooling loads, and the chilled water system can still maintain a relatively high efficiency. Because the loads of each chiller are not the same, the operation strategy of the variable load, not the average load, yields better system performance. For these three operation strategies, their total operation costs during the whole design time period are compared in Table 3. The result showed that the total cost of strategy 3 is the smallest among the three operation strategies since this method regulates more control variables, making the system operation more flexible. However, for practical consideration, this operation strategy to regulate two variables in the chilled water supply side is not easy to implement. Strategy 1 and strategy 2 are commonly employed in practice. Which one is chosen depends on the trade-off of the power consumption between the chiller and the chilled water pumps. From the viewpoint of energy saving and stability in operation, the operation strategy of regulating chilled water flowrates can be a good choice.

6.

Conclusions

This paper investigates how the optimal operation scheme helps improve the operating cost of a large multi-chiller system subject to time-of-day operating schedule. In the large multi-chiller system, the large-sized chillers are used for the heavy-load requirements. On the other hand, the smallsized chillers are used for the minor adjustment based on the small differences in the required load. In order to maintain the system’s stability and to save energy to a certain degree whenever the cooling load is changed, this application rests on the optimal scheduling of chiller loadings and its design operation condition to avoid frequent loading or unloading of

the chillers; the need of the cooling load at each sub-time period is considered rather than a single time point only. The merits of the proposed model can be drawn:  Unlike the conventional reported design methods which are applied at one time point only, this forecasting scheme can offer an operating reference for operators to enhance the energy performance of the chilled water system and to keep the system operation stable.  The two-stage optimization design method that minimizes the energy consumption is presented. It consists of two steps. The first step is the scheduling stage for determining which chillers to be loaded or unloaded at each sub-time period; the other is the operating stage for calculating the operation condition of the optimal design at each sub-time period. The procedures can simplify the calculation effort, so they can be easily used in industry. The study case demonstrates energy saving potential in the large chilled water system of the hi-tech industry. The results highlight that the supplying temperatures of the chilled water and the chilled water flowrates should be allowed to vary throughout the day to enhance their sustainability. The collocation running mode of the large chillers and the small chillers can efficiently reduce the chiller system energy consumption when the cooling load varies. In this modelbased approach, these models have high prediction accuracy, require less calibration efforts and maintain certain physical significances of parameters. However, if the model parameters are changed during the operation, the fault diagnosis and calibration of the model are necessary. They will be studied in the future research.

Acknowledgments This work is supported by the National Science Council (NSC) and Bureau of Energy (Grant No. 100-D0309), Taiwan, R.O.C.

Appendix 1. Chiller unit models: The chiller COP is commonly used to evaluate the chiller working efficiency (Chang, 2004) and it is defined as follows: COP ¼

Qeva Pc

(A1)

where Pc is the power consumption of the chiller. Qeva is the quantity of heat transferred at the evaporator. It is equivalent to the chiller’s actual cooling capacity. Most of them indicate that COP is related to the partial load ratio (the ratio of the chiller’s actual cooling capacity to the rated cooling capacity), PLR, the evaporator temperature (Te) and the condenser temperature (Tc). COP is used here (Yao et al., 2004a), COP ¼

3 X  s¼1

2 5  X   X ðRs $PLRs Þ ps $Tse  qs $Tsc  s¼0

s¼o

(A2)

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

PLR ¼

Qeva Qcr

(A3)

The coefficients, p, q and R, employed in Eq. (A2) can be estimated by the regression of the operation data. Qcr denotes the rated cooling capacity. The evaporator heat transfer (i.e. cooling capacity), Qeva, and the condenser heat transfer, Qcon, can be calculated based on the energy balances, 

Qeva ¼ mew  Cw  Tw;i;eva  Tw;o;eva



(A4)

  Qcon ¼ mcw  Cw  Tw;o;con  Tw;i;con

(A5)

Qcon ¼ Qeva þ Pc

(A6)

where mew and mcw are the mass flowrates of the chilled water and the cooling water, respectively. Tw,i,eva and Tw,o,eva are the temperatures of the returning chilled water and the chilled water supply, respectively. Tw,i,con and Tw,o,con are the temperatures of the water entering and leaving the condenser respectively. 2. Cooling tower unit model: The water leaving condenser can be cooled in the cooling tower through processing the heat transfer and the mass transfer between water and air. The states of water and air entering and leaving the cooling tower are important because they simultaneously affect the efficiency of the chiller and the cooling tower. The effective model given by Braun (1988) that considers the influence is employed here:

2061

where cs is the fictitious specific heat defined as the ratio of the enthalpy difference to the temperature difference of the saturation air corresponding to the water entering and leaving the cooling tower. N which denotes the number of transfer units for measuring the cooling tower performance is given: N ¼ Amp m¼

(A14)

ma mcw;i

(A15)

The coefficients A and p in Eq. (A14) could be determined based on the measured data under different flowrates of the air and the water. The property parameters of the moist air employed in the above cooling tower model could be gotten from the papers of Yu and Chan (2008) and Graves (2003). The temperature of the water leaving cooling tower, Tw,o,t, given by Yu and Chan (2008) is Tw;o;t ¼

mcw;i cw Tw;i;t  Qct mcw;o cw

(A16)

The required Tw,o,t can be satisfied through varying the air flowrate; therefore, a variable-speed fan is required to regulate the air flowrate in the cooling tower. According to the fan law, there is a cubic relationship between the fan power and the flowrate factor,  Pfan ¼ Pfan;r

ma ma;r

3 (A17)

3. Pump unit models:

  ma ha;o  ha;i ¼ mcw;i cw Tw;i;t  mcw;o cw Tw;o;t

(A7)

The power of the constant-speed pump and the variablespeed pump can be calculated as follows (Yao et al., 2004a):

  ma ua;o  ua;i ¼ mcw;i  mcw;o

(A8)

Ppump;con ¼

gw Ph mew hp

(A18)

Ppump;var ¼

gw Ph mew hpv hfv

(A19)

where ma is the mass flowrate of the air entering the cooling tower. ha,i and ha,o denote the enthalpies of the air entering and leaving the cooling tower, respectively. Tw,i,t and Tw,o,t denote the temperatures of the water entering and leaving cooling tower; ua,i and ua,o denote moisture contents of the air entering and leaving the cooling tower, respectively. The heatrejection capacity of the cooling tower can be calculated by the following equations:   Q ¼ ea ma hs;w;i  ha;i ea ¼

ha;o  ha;i hs;w;i  ha;i

(A9)

(A10)

where ea is the heat transfer effectiveness of the cooling tower, and it represents the ratio of the actual enthalpy difference in the air side to the maximum enthalpy difference. ea of the cross-flow cooling tower is calculated by Eqs. (A11)e(A13) ea ¼

1  expð  Nð1  m ÞÞ 1  m expð  Nð1  m ÞÞ

m ¼ cs ¼

ma cs mcw;i cw

hs;w;i  hs;w;o Tw;i;t  Tw;o;t

(A11)

(A12)

(A13)

For the constant-speed pump, the efficiency of the pump, hp, remains unchanged, whereas for the variable-speed pump, the efficiencies of the pump and the variable-frequency driver, hpv and hfv, change with the variation of the rotate speed.

references

ASHRAE, 2007. ASHRAE HandbookdHVAC Applications (SI). Chapter 41: supervisory control strategies and optimization. American Society of Heating, Refrigerating and AirConditioning Engineers, Inc. Austin, S.B., 1991. Optimum chiller loading. ASHRAE Journal 7, 40e43. Braun, J.E. 1988. Methodologies for the design and control of chilled water systems. Ph.D. thesis. Madison, WI: University of Wisconsin-Madison. Braun, J.E., Diderrich, G.T., 1990. Near-optimal control of cooling towers for chilled-water systems. ASHRAE Transactions 96, 806e813. Chang, Y.C., 2004. A novel energy conservation methodeoptimal chiller loading. Electr. Power Syst. Res. 69, 221e226. Gordon, J.M., Ng, K.C., Chua, H.T., Lim, C.K., 2000. How varying condenser coolant flow rate affects chiller performance:

2062

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 0 5 0 e2 0 6 2

thermodynamic modeling and experimental confirmation. Appl. Therm. Eng. 20, 1149e1159. Graves, R.D. 2003. Thermodynamic modeling and optimization of a screw compressor chiller and cooling tower system. Msc thesis, The Office of Graduate Studies of Texas A&M University. Hackner, R.J., Mitchell, J.W., Beckman, W.A., 1984. HVAC system dynamics and energy use in buildings. Part I. ASHRAE Trans 90, 523e535. Hahn, H., Meyer-Nieberg, S., Pickl, S., 2009. Electric load forecasting methods: tools for decision making. Eur. J. Oper. Res. 199, 902e907. Lu, L., Cai, W.J., 2004. HVAC system optimizationecondenser water loop. Energ. Convers. Manage. 45, 613e630. Nassif, N., Kajl, S., Sabourin, R., 2005. Optimization of HVAC control system strategy using two-objective genetic algorithm. HVAC&R Research 11, 459e486.

Soliman, S.A., Persaud, S., El-Nager, K., 1997. Application of least absolute value parameter estimation based on linear programming to short-term load forecasting. Int. J. Electr. Power Energy. Syst. 19, 209e216. Wang, S.W., Ma, Z.J., 2008. Supervisory and optimal control of building HVAC systems: a review. HVAC&R Research 14, 3e32. Yao, Y., Lian, Z.W., Hou, Z.J., 2004a. Optimal operation of a large cooling system based on an empirical model. Appl. Therm. Eng. 24, 2320e2321. Yao, Y., Lian, Z.W., Liu, S.Q., Hou, Z.J., 2004b. Hourly cooling load prediction by a combined forecasting model. Int. J. Therm. Sci. 43, 1107e1118. Yu, F.W., Chan, K.T., 2008. Optimization of water-cooled chiller system with load-based speed control. Appl. Energy 85, 931e950.