Economic dispatch of chiller plant by gradient method for saving energy

Applied Energy 87 (2010) 1096–1101

Contents lists available at ScienceDirect

Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Economic dispatch of chiller plant by gradient method for saving energy Yung-Chung Chang *, Tien-Shun Chan, Wen-Shing Lee Department of Energy and Refrigerating Air-conditioning Engineering, National Taipei University of Technology, Taipei, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 27 January 2008 Received in revised form 29 April 2009 Accepted 2 May 2009 Available online 27 May 2009 Keywords: Lagrangian multiplier method Gradient method Decoupled system

a b s t r a c t This study employs gradient method (GM) to solve economic dispatch of chiller plant (EDCP) problem. GM overcomes the flaw that with the Lagrangian multiplier (LM) method the system may not converge at low demand. In this study, the load balance constraint and the operating limit constraints of the chillers are fully accounted for. After analysis and comparison of the two cases studies, we are confident to say that this method not only solves the problem of convergence, but also produces results with high accuracy within a rapid timeframe. It can be perfectly applied to the operation of air-conditioning systems. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The UN Intergovernmental Panel on Climate Change (IPCC) estimates that, by the end of this century, the average global air temperature will increase by 1.8–4.0 °C and sea level will rise by 0.6 m due to global warming. Global warming is causing glaciers to diminish or disappear. Many island nations are being submerged by the rising sea. Climate change has altered global rainfall patterns, bringing floods, torrential rains, and alternating drought and flood. Global warming thus poses a severe threat to the survival of human civilization. Carbon dioxide accounts for 80% of the greenhouse effect, and rising carbon dioxide levels are the main cause of global warming. The UN will inevitably impose strict limits on carbon dioxide emissions in the future in order to reduce the rate of warming. When that time comes, national governments worldwide will have to focus increasing attention to ways of reducing carbon dioxide emissions. Taiwan currently depends on thermal power generation for 75% of its power supply, and thermal power plants produce 0.67 kg of carbon dioxide for every kilowatt-hour generated. Air conditioning accounts for approximately 30% of summertime power consumption. Chillers account for the biggest share, roughly 60% of airconditioning system power consumption [1]. As a consequence, increasing chiller operating efficiency can save power, reduce power costs for owners, lessen carbon dioxide emissions, and thereby ease greenhouse warming. The question of how to improve chiller operating efficiency is one of the most important issues in contemporary building energy conservation. To maintain high coefficient of performance, centralized air-conditioning * Corresponding author. Tel.: +886 2 2771 2171x3518; fax: +886 2 2731 4919. E-mail address: [email protected] (Y.-C. Chang). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.05.004

systems often utilize centrifugal chiller units. Each unit has different features. The longer the machine is run, the more apparent the differences are. Inappropriate operation will waste power. When the usage is substantial, like the semiconductor industries, the effects will be profound. Braun et al. [2] is the earliest team that devoted to systematically solving air-conditioning system optimization with quadratic equation. When rated COPs of all chillers are equal and partial load characteristics are consistent, all chillers running in the same partial load ratio is the optimal dispatch. When chiller characteristics differ, LM method is used to solve the EDCP problem. Sun and Reddy [3] used Taylor-Series expansion to linearize air-conditioning system model, and employed linear model to replace quadratic model, so as to improve solving speed and utilized CSB-SQP (Complete Simulation-based Sequential Quadratic Programming) to solve the EDCP problem. Lu et al. [4,5] expressed the chiller power consumption model with quadratic equation of partial load ratio, chilled water supply temperature and cooling water return temperature. While power of chilled water pump and fan were represented by water flow rate and air flow rate, respectively, they used GA (Genetic Algorithm) to solve the optimal setting point. The experiment was carried out in a self-built mini air-conditioning system in laboratory, with much discrepancy from a real system. Ahn and Mitchell [6] used quadratic regressive equation to predict the power consumption of air-conditioning system, and used PID controller to regulate chilled water pump and fan speeds to reach optimal chilled water temperature, cooling water temperature and supply air temperature, in order to minimize overall power consumption. Ma et al. [7] utilized quadratic equation of delta enthalpy in Pressure–Enthalpy curve to predict the chiller power consumption, while considering both chiller and tower fan power consumption, used PMES (Performance Map and Exhaustive

1097

Y.-C. Chang et al. / Applied Energy 87 (2010) 1096–1101

Search) method to solve the optimal cooling water temperature to minimize power consumption, and compared against GA solution result. Yu and Chan [8] presented a load-based speed control strategy to achieve the optimum operation of a HVAC system, integrated with variable condenser water flow and speed control for tower fans, in an office building. The studied method could reduce the annual system electricity use and operating cost compared to the equivalent system using constant speed fans and pumps. Lee et al. [9] proposed particle swarm algorithm to solve the optimal problem of ice-storage air-conditioning system in an office building. The minimum life cycle and CO2 emission were also taken into consideration. The approaches not only achieve the optimization of this system, but also analyze the increase in power consumption and CO2 emission and its potential influences on the optimization. Most of the above studies solved the optimal operation of an air-conditioning system conducted simulation analysis on a small-to-medium system, and assumed that all chillers have the same characteristic curve. However, there are more chillers in a large system, even as many as tens of thousands of refrigeration tons (i.e., semiconductor factories). The chiller efficiency varies with prolonged running, and there are different characteristic curves. Most conventional methods use quadratic equations to express their characteristic curves, some even approximate with linear equations. Since cubic equation model has higher accuracy, this study used it to establish the power consumption model of chiller. The LM method has been adopted to solve EDCP problem in [10] based on the convex function of the P–Q curve. The LM method that uses lambda-iteration method, however, can cause a problem to not reach convergence at low demand. This study solves the problem by using GM to overcome this shortcoming. After extensive experiment, GM produced high accuracy and rapid execution speed.

Fig. 1. Decoupled system.

by measuring the water flow and the temperature in the main pipe on the load side.

2. System structure

3. EDCP by Lagrangian multiplier method [10]

The designers of air-conditioning systems often develop multiple-chiller systems because they provide operational flexibility, standby capacity and less disruption maintenance. Such a system has a reduced starting in-rush current, reduced power cost under partial-load conditions and a set of chillers that can be operated at the best efficiency [11]. Fig. 1 depicts the structure of a decoupled chilled water system (called a decoupled system), which includes multiple chillers [11]. A decoupled system offers constant flow on the primary side (the chiller side), which prevents excessively low temperatures from impairing the evaporator and prevents the chillers from being frequently shut down. On the secondary side (the load side), the 2-way valves can regulate the chilled water that flows into the cooling coils in accordance with the load variation. Therefore, the decoupled system yields stable control and is applied in air-conditioning systems. The cooling load of an air-conditioned room can be removed to the chiller unit using the chilled water system, it is then emitted to the atmosphere using the cooling water system. Therefore, the cooling load (in refrigeration tons) of a chiller can be calculated from [12],

ASHRAE proposed the concept of EDCP in the Chapter 40 of a 1999 Handbook [13], in which the power consumption model of the chiller is shown with the cooling water return temperature (T cwr;i ), chilled water supply temperature (T chws;i ), and chiller load (Q ;i ) as independent variables. Therefore, the power consumption of the chiller could be expressed as

Q ¼ fmðT CHr  T Chs Þ=3517

ð1Þ 1

where f = the flow rate of chilled water (kg s ), m = the specific heat of chilled water (J kg1 K1), TCHr = the return temperature of chilled water (K), TChs = the supply temperature of chilled water (K). The cooling load of a chiller can be determined by measuring the water flow, the supplied temperature and the return temperature of the chilled water. The input power (kW) can be measured using a power meter and then the P–Q curve can be obtained by regression. The variation in the demand can also be determined

Pi ¼ k0;i þ k1;i X þ k2;i Q i þ k3;i XQ i þ k4;i X 2 þ k5;i Q 2i þ k6;i X 2 Q i þ k7;i XQ 2i þ k8;i X 3 þ k9;i Q 3i

ð2Þ

where k0;i —k9;i are regression coefficients.

X ¼ T cwr;i  T chws;i Q i ¼ output of ith chiller: There are two methods to execute EDCP. One is to control individual chiller load by adjusting the chilled water flow with identical chilled water supply temperature. The other is by using different chilled water supply temperature set point with fixed chilled water flow. The presented paper addressed the first one due to this method saving more power, as shown in Ref. [13]. In a system with all-electric cooling, the best performance occurs when the total power consumption of chillers is minimized while the load demand is satisfied. The chilled water supply temperatures are always specified a value (7 °C, for example) and the cooling water return temperatures are almost remain unchanged during the EDCP process (needs a short period). That is, the Pi of a centrifugal chiller is a convex function of its Qi for a given cooling water temperature and Eq. (2) can be expressed as

Pi ¼ ai þ bi Q i þ ci Q 2i þ di Q 3i

ð3Þ

1098

Y.-C. Chang et al. / Applied Energy 87 (2010) 1096–1101

where ai, bi, ci, di are coefficients of P–Q curve of ith chiller and can be computed immediately from Eq. (2) due to X being a constant. The partial load energy consumption of centrifugal chillers are higher at low loads due to motor losses, but the increased input powers at high load are due to thermal heat exchange inefficiencies [14]. The EDCP problem is to find a set of chiller output which does not violate the operating limits while minimizing the objective function: I X

J¼

Pi

ð4Þ

The power consumption of each chiller is finally calculated by Eq. (3). But, the lambda-iteration method will not converge at low demand. This shortcoming can be overcome by GM method. 4. Gradient method [16] Eq. (7) is the Taylor-Series expansion of the objective function of Eq. (4), expended about the initial feasible operating point:

J þ DJ ¼ P1 ðQ 1 Þ þ P2 ðQ 2 Þ þ    þ PðQ I Þ dP1 dP2 dPI DQ 1 þ DQ 2 þ    þ DQ I dQ 1 dQ 2 dQ I ! 2 2 1 d P1 d P2 2 2 þ ½ D Q  þ ½ D Q  þ    þ  1 2 2 2 dQ 21 dQ 2

i¼1

þ

Simultaneously, the balance equation must be satisfied: I X

Q i ¼ CL

ð5Þ

ð7Þ

i¼1

where CL = system cooling load. Since EDCP is in search of a manipulation point that minimizes the power consumption of the entire group of chillers while meeting the load requirement, the objective function, Eq. (4), is the total power consumption of chillers and is irrelevant to the power consumptions of cooling towers and auxiliary equipments, adding them into Eq. (4) has no effect. The power consumptions of pumps are very small compared to chillers’ power (about 5%), thus, Eq. (4) does not take into consideration of their power consumptions. The LM method [15] is adopted to find the optimal solution of the convex function. For a system’s cooling load CL, the Lagrangian multiplier k can be evaluated by

k ¼ bi þ 2ci Q i þ 3di Q 2i

ð6Þ

For a system cooling load CL, the Lagrangian multiplier k and Qi can be evaluated from Eqs. (5) and (6) by the lambda-iteration method following the steps below [16]: 1. Assume two Lagrangian multipliers and find the chillers0 outputs for these two values in order to establish two feasible solutions. 2. Extrapolate (or interpolate) the two former solutions to get closer to desired value which has less tolerance, as shown in Fig. 2. 3. The desired Lagrangian multipliers can rapidly be found by keeping track of the cooling load versus the Lagrangian multiplier. 4. If e is less than a specified tolerance, then end; or else, return to Step 2.

If change DQi is small, the second order and higher items could be overlooked. The change of objective function is as follows:

dP1 dP2 dPI DQ 1 þ DQ 2 þ    þ DQ I dQ 1 dQ 2 dQ I

DJ ¼

ð8Þ

If Qi has slight change, load balance Eq. (5) derives that the total output change should be 0, given that the load demand remains the same I X

DQ i ¼ 0

ð9Þ

i¼1

To fit this relationship, at least one chiller is chosen to be dependant variable (as x), thus Eq. (9) derives that the output change of this chiller should equal to the negative value of the total change of others, or:

DQ X ¼ 

X

DQ i

ð10Þ

i–x

When Eq. (10) is substituted into Eq. (8), the change of objective function is the function of output change of N  1 independent units as follows:

X dPi dPx DQ i þ DQ x dQ dQ i x i–x X dP i dP x X ¼ DQ i  DQ i dQ i dQ x i–x i–x X  dP i dPx  ¼  DQ i dQ i dQ x i–x X @J x ¼ DQ i @Q i i–x

DJ ¼

ð11Þ

@Jx The coefficient in the above equation (@Q ) represents the effect i of chiller output on objective function; larger the number, higher the effect. Thus, the chiller with the highest coefficient should be adjusted its output first, and use the dependent unit to absorb the adjusted amount. The output change of ith chiller is

 K 2 for k ¼ 1  10 3  10 2 ¼ 0:1Q i  for k  10 3

DQ i ¼ 0:1Q i 

0

ð12Þ

where k is an iteration step and Q i is the capacity of ith chiller. Use of GM to solve EDCP follows the steps below:

Fig. 2. Lagrangian multiplier projections.

Step 1. Randomly select a feasible solution that satisfies load balance Eq. (5). @Jx Þ of Step 2. Select dependent unit (x), and compute coefficient ð@Q i Eq. (11) with the feasible solution.

1099

Y.-C. Chang et al. / Applied Energy 87 (2010) 1096–1101 Table 1 Chillers data. System

Chiller

ai

bi

ci

di 4

Capacity (RT) 4

Case1

CH-1 CH-2 CH-3 CH-4

104.09 67.15 384.71 541.63

0.3702 2.6173 0.77913 0.41348

21.24  10 1.073  102 1.151  103 3.6265  103

0.056  10 0.16  104 63.20  109 4.0214  106

450 450 1000 1000

Case 2

CH-1 CH-2 CH-3

100.95 66.598 130.09

1.023 0.7579 0.3806

1.52  103 5.947  104 2.25  105

1.54  106 5.39  107 1.95  107

800 800 800

Table 2 Optimal chiller loading for Case 1 by LM or GM. Load (RT)

Chiller

RT

kW

Total kW

2610 (90%)

1 2 3 4

445.90 407.65 1000.0 756.5

345.43 298.07 693.8 519.99

1857.30

2320 (80%)

1 2 3 4

373.00 362.52 896.6 687.9

238.52 231.92 566.19 419.04

1455.66

2030 (70%)

1 2 3 4

326.79 333.09 721.6 648.5

194.50 203.93 398.30 381.39

1178.14

1740 (60%)

1 2 3 4

271.57 295.92 564.8 607.7

160.61 181.21 300.57 356.14

998.53

1450 (50%)

1 2 3 4

206.55 224.32 447.1 572.1

139.49 160.03 260.88 344.23

904.62

1160 (40%)

1 2 3 4

135.0 135.0 351.4 538.6

129.19 129.81 250.36 340.63

849.99

Table 3 Optimal chiller loading for Case 2. Load (RT)

LM

GM

Chiller

RT

kW

Total kW

RT

kW

Total kW

2160 (90%)

1 2 3

580.24 779.76 800

483.45 551.59 548.77

1583.81

580.24 779.76 800

483.45 551.59 548.77

1583.81

1920 (80%)

1 2 3

527.2 686.8 706

443.37 481.25 478.58

1403.20

527.2 686.8 706

443.37 481.25 478.58

1403.20

1680 (70%)

1 2 3

476.96 5.96 607.12

410.09 421.18 413.06

1244.32

476.96 5.96 607.12

410.09 421.18 413.06

1244.32

1440 (60%)

1 2 3

424.24 492.4 523.36

378.91 359.95 363.40

1102.26

424.24 492.4 523.36

378.91 359.95 363.40

1102.26

1200 (50%)

1 2 3

– – –

– – –

–

398.88 307.92 493.20

364.86 259.34 346.55

970.85

960 (40%)

1 2 3

– – –

– – –

–

240 240 480

280.22 221.70 339.52

841.44

1100

Y.-C. Chang et al. / Applied Energy 87 (2010) 1096–1101 0.8

40 chiller 1 chiller 2 chiller 3

0.75

35 30

Temperature(°C)

Lagrangian multiplier

0.7 0.65 0.6 PLR=0.3849 0.55 0.5

25 Tchwr Tchws Tcwr Tcws

20 15 10

0.45

5

0.4 0.4

0.45

0.5

0.55

0.6

0.65

0

0.7

50

PLR

150

250

300

350

400

450

Fig. 6. The chilled and cooling water temperatures of Chiller 3 for Case 2.

1

2600 90% 80% 70% 60% 50% 40%

2400 2200 2000

0.9 0.8 0.7 0.6

PLR

1800 1600

0.5 0.4

1400

0.3 1200

0.2

1000

0.1

800 5

10

15

20

25

0

30

50

100

150

Iteration

200

250

300

350

400

450

Hours

Fig. 4. Process of convergence with objective function for Case 1.

Fig. 7. PLR of Chiller 3 for Case 2.

500

1700 90% 80% 70% 60% 50% 40%

1500 1400

actual 2-order 3-order

450

Power consumption(kW)

1600

Power consumption(kW)

200

Hours

Fig. 3. Lagrangian multiplier for Case 2.

Power consumption(kW)

100

1300 1200 1100 1000

400

350

300

900

250

800 2

4

6

8

10

12

Iteration Fig. 5. Process of convergence with objective function for Case 2.

14

0

50

100

150

200

250

300

350

400

Hours Fig. 8. Actual kW and simulated kW of Chiller 3 for Case 2.

450

Y.-C. Chang et al. / Applied Energy 87 (2010) 1096–1101

Step 3. Adjust slightly outputs of the chillers with the coefficient @J x Þ and output of dependent unit under the condition that satð@Q i isfies the load demand to compute a new feasible solution. Step 4. Check if the new feasible solution meets all the limitation requirements. Step 5. If the change of objective function is less than tolerance (=0.01 kW), then end; or else, return to Step 2. Gradient search techniques always begin with a feasible solution, then follow the largest path of changing gradient of objective function to seek the optimal solution. In the process of establishing new operation point, objective function is monotonically changing, thus, the new solution is sought to be better than the former one. With enough iteration steps, optimal solution could be found. Therefore, this method can overcome the flaws of convergence on LM. Although high iteration steps are flaws to GM, the less number of chillers in an air-conditioning system and faster computer execution speed have overcome the flaw. 5. Examples and results The analysis of two case studies done is illustrated in Table 1. In Case 1, the subject is a hotel in Taipei, and the hotel has two 450 RT and two 1000 RT units. In Case 2, the subject is a semiconductor plant in Hsin Tsu Science-based Park, and the plant has three 800 RT units. Although the three units are identical in type and capacity in Case 2, extensive operating period and system structure have caused chilled water temperature and flow rate to be different, and have resulted in different coefficients. The testing software used is written in MatLab program and run on an Intel XP/2.40 GHz personal computer. Table 2 shows the result of EDCP in Case 1. GM and LM have the same results. Table 3 shows the result of EDCP in Case 2. Although the LM method came up with the global optimum, the problem of not being able to reach convergence occurs when demand is below 50%. For example, as shown in Table 2 for Case 2, the EDCP results by GM are 398.88 RT (PLR = 0.4986, partial load ratio), 307.92 RT (PLR = 0.3849) and 493.20 RT (PLR = 0.62) as cooling load is 1200 RT (50%). On the other hand, by LM, all three chillers’ Lagrangian multipliers must be the same from Eq. (6) when the EDCP results are evaluated. But the Lagrangian multiplier of Chiller 2 does not intersect with the optimal Lagrangian multiplier of Chiller 1 (horizontal line) at PLR = 0.3849 as shown in Fig. 3. So, the LM cannot reach convergence as demand is below 50%. Under the GM method, the problem of not being able to reach convergence does not occur. Figs. 4 and 5 show the process of convergence with objective function for these two cases. Although large iteration number is the weakness in GM, it does not become a limitation because the number of air conditioning units is low. The CPU times are only 1.58 and 1.36 s for these two cases. So, the presented study has a rapid timeframe and high accuracy. The data from Chiller 3 of Case 2 were tested from October 20, 2002 to November 9, 2002 to verify that the 3-order polynomial chiller model has significant advantages over the 2-order quadratic model. The data tested were the chillers’ power consumption, and the chilled and cooling water temperatures (Tchws, Tchwr, Tcws and Tcwr), which were recorded in an automatic monitoring system every 1 hour. Fig. 6 shows the chilled and cooling water temperatures of Chiller 3 and Fig. 7 is the PLR calculated from Tchws and Tchwr of Chiller 3. Fig. 8 displays the magnified diagram of the power consumption of Chiller 3. Usually, the 2-order and 3-order almost have the same errors between actual kW and simulated

1101

kW. Nevertheless, the 3-order model has better accuracy than 2-order model when the chillers are operated at high or low PLR, as shown in Fig. 8. Therefore, the 3-order polynomial chiller model has significant advantages over the 2-order quadratic model. 6. Conclusion With more traditional centralized air-conditioning systems (such as in office building, hotels and hospitals), the total cooling capacity is insubstantial, the number of units low, the operation method simple, and the EDCP is not well thought-out. As the requirement increases due to the development of semiconductor industry, however, cooling capacity and the number of units multiply. The EDCP of chiller unit has to be taken seriously. Each unit has to run with highest coefficient of performance, and the power consumption has to be minimized. The saving of power consumption on air conditioning could decrease the operating cost of the company, lessen carbon dioxide emissions, and ease greenhouse warming. This study utilizes GM to work out EDCP problem, and solves LM method’s problem of not being able to reach convergence at low demand. By utilizing GM, we can obtain rapid execution speed and high accuracy. It can be concluded then, that GM is a highly recommendable method.

Acknowledgement The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 91-2212-E-027-009. References [1] Hu SC, Chuah YK. Power consumption of semiconductor fabs in Taiwan area. Energy 2003;28:895–907. [2] Braun JE, Klein SA, Mitchell JW, Beckman WA. Applications of optimal control to chilled water systems without storage. ASHRAE Trans 1989;95:663–75. [3] Sun J, Reddy A. Optimal control of building HVAC&R systems using complete simulation-based sequential quadratic programming (CSB-SQP). Build Environ 2005;40(5):657–69. [4] Lu L, Cai WJ, Soh YC, Xie LH. Global optimization for overall HVAC systems – part I: problem formulation and analysis. Energy Convers Manage 2005; 46(7–8):999–1014. [5] Lu L, Cai WJ, Soh YC, Xie LH. Global optimization for overall HVAC systems – part II: problem solution and simulations. Energy Convers Manage 2005; 46(7–8):1015–28. [6] Ahn BC, Mitchell JW. Optimal control development for chilled water plants using a quadratic representation. Energy Build 2001;33(4):371–8. [7] Ma Z, Wang S, Xu X, Xiao F. A supervisory control strategy for building cooling water systems for practical and real time applications. Energy Convers Manage 2008;49(8):2324–36. [8] Yu FW, Chan KT. Optimization of water-cooled chiller system with load-based speed control. Appl Energy 2008;85:931–50. [9] Lee WS, Chen YT, Wu TH. Optimization for ice-storage air-conditioning system using particle swarm algorithm. Appl Energy 2009;86(9):1589–95. [10] Chang YC, Tu HC. An effective method for reducing power consumption– optimal chiller load distribution. In: Proceedings of the international conference on power con, power system technology, 2002, vol. 2; 2002. p. 1169–72. [11] ASHRAE Handbook CH. 38. Liquid chilling system. Atlanta (GA): ASHRAE; 2000. [12] Dossat Roy J. Principles of refrigeration. New York: Wiley; 1991. [13] ASHRAE Handbook CH. 40. Supervisory control strategies and optimization. Atlanta (GA): ASHRAE; 2000. [14] Levenhagen JI, Spethmann DH. HVAC controls and systems. New York: McGraw-Hill; 1993. [15] Winston WL. Operations research: applications and algorithms. Boston: Duxbury Press; 1987. [16] Wood AJ, Wollenberg BF. Power generation operation and control. New York: Wiley; 1984.