Optimal chiller loading by differential evolution algorithm for reducing energy consumption

Energy and Buildings 43 (2011) 599–604

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Optimal chiller loading by differential evolution algorithm for reducing energy consumption Wen-Shing Lee a,∗ , Yi-Ting Chen a , Yucheng Kao b a b

Department of Energy and Refrigerating Air-Conditioning Engineering, National Taipei University of Technology, Taiwan Department of Information Management, Tatung University, Taiwan

a r t i c l e

i n f o

Article history: Received 30 April 2010 Received in revised form 5 October 2010 Accepted 18 October 2010 Keywords: Differential evolution algorithm Chiller loading Engineering optimization

a b s t r a c t This study employs differential evolution algorithm to solve the optimal chiller loading problem for reducing energy consumption. To testify the performance of the proposed method, the paper adopts two case studies to compare the results of the developed optimal model with those of the Lagrangian method, genetic algorithm and particle swarm algorithm. The result shows that the proposed differential evolution algorithm can find the optimal solution as the particle swarm algorithm can, but obtain better average solutions. Moreover, it outperforms the genetic algorithm in finding optimal solution and also overcomes the divergence problem caused by the Lagrangian method occurring at low demands. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The energy consumption of an air-conditioning system is huge in summer, especially in the subtropical region where the summer is hot and humid, causing peak load in electricity. In the air-conditioning system, the chiller system is one of the major equipment in energy consumption. Therefore, how to operate the chiller system for minimizing the energy consumption in different cooling loads becomes an important issue. Moreover, since the multi-chiller system consists of chillers with varying performance characteristics and capacities, the optimal combination of load ratio of each chiller becomes a valuable research topic. In the research of optimal chiller loading (OCL), Chang [1,2] used Lagrangian method and genetic algorithm to minimize the energy consumption at different cooling loads. The results showed that Lagrangian method could minimize the energy consumption, but it could not converge at low demands. Genetic algorithm overcame the convergence at low demands, but the minimum energy consumption increased about 0.4% on average compared with that of Lagrangian method. Chang [3–5] used simulated annealing method and gradient method to improve the convergence of Lagrangian method. Since the variables of the OCL problem are continuous, Ardakani [6] adopted continuous genetic algorithm and particle swarm algorithm to solve the problem, and Lee [7] also proposed a

∗ Corresponding author at: 1, Sec. 3, Chung-hsiao E. Rd., Taipei, Taiwan, ROC. Tel.: +886 2 2771 2171x3515; fax: +886 2 2731 4919. E-mail address: [email protected] (W.-S. Lee). 0378-7788/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2010.10.028

particle swarm algorithm to solve the optimal chiller loading problem. The results of these two studies showed that particle swarm algorithm could improve the performance of Genetic algorithm and Lagrangian method. In the optimization of air-conditioning system using simulation methods, Lee [8] proposed particle swarm algorithm for ice-storage air-conditioning system, and Kusiak [9] proposed data mining algorithms for the cooling output of an air handling unit (AHU). Lee [10] also developed a simplified model for evaluating chiller-system configurations. Differential evolution (DE) is a new population-based evolutionary computing algorithm and was proposed for optimization on a continuous domain. The differential evolution algorithm has been used as an efficient method of solving continuous parameters optimization problems. Babu and Angira [11] adopted DE to solve an optimum problem, which is a non-linear function with three local optimal solutions and one global optimal solution. The results indicated that the performance of DE was better than that of GA. Angira and Babu [12] attempted to use DE to solve two non-linear chemical engineering problems, indicating that the convergence speed of DE was faster than that of branch-and-bound algorithm in terms of CPU time. Karaboga and Okdem [13] also observed that the convergence speed of DE was obviously faster than genetic algorithms. Therefore, DE algorithm seems to be a promising approach for engineering optimization problems. This paper uses the differential evolution algorithm to minimize energy consumption in multi-chiller loading. The paper also adopts two case studies to compare the results of the proposed algorithm with the Lagrangian method, genetic algorithm and particle swarm algorithm.

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Considering the suggestion of the manufacturer, the other constraint is that the partial load of each operating chiller cannot be less than 30%, as shown in Eq. (4): PLRi ≥ 0.3

(4)

3. Optimization method Differential evolution was proposed by Storn and Price [15]. Like genetic algorithms, it is a population-based algorithm using the similar operators: crossover, mutation and selection. The main difference in constructing better solutions is that genetic algorithms rely on crossover operator while DE relies on the mutation operator [13]. For readers reference, the procedure of DE for a D-dimension optimization problem [15] is simply described as follows: in the initialization stage, the number of the vector population is given. Each vector represents a trial solution in which the value of each dimension is given randomly. The initial vectors also are called the parent vectors of the first generation. To produce the parent vectors of the next generation, three operations, mutation, crossover and selection, are used. In the mutation operation, a mutated vector (v) for each parent vector (x) is generated by

vi,g+1 = xr1,g + F ∗ (xr2,g − xr3,g )

where g is the number of generation; r1 , r2 , r3 ∈ {1,2,. . .,NP} are randomly chosen integers and NP is the population size. These integers must be different from one another and also from the vector index i. F (>0) is a scaling factor which controls the effect of the differential value (xr2 ,g − xr3 ,g ). In the crossover operation, the parent vector is mixed with the mutated vector to produce a trial vector u,



Fig. 1. Decoupled system of a multiple chiller system [10].

uj,g+1 =

2. System description The following outline provides the basics of the typical decoupled system, as illustrated in detail in ASHRAE Handbook [14]. In the multiple decoupled chiller system, two or more chillers are connected by parallel to a common distribution system, as shown in Fig. 1. Each chiller can be operated at different capacities to meet different cooling demands, and this allows the chiller system to operate at its most efficient point. In a multiple chiller system, the best performance occurs when the sum of energy consumption of the chillers is minimized while the load demand is satisfied. The energy consumption of a centrifugal chiller (Power) is a convex function of its part load ratio (PLR) at a given wet-bulb temperature [2]: Poweri = ai + bi ∗ PLRi + ci ∗ PLR2i + di ∗ PLR3i

(1)

where ai , bi , ci , di are coefficients of power curve of the ith chiller. The objective function of optimum chiller loading problem is to minimize the sum of energy consumption of the chillers (J), as shown in Eq. (2): J = Min

l 

Poweri

(2)

i=1

where l refers to the total number of chillers. On the constraints about the chiller loading problems, first, the sum of cooling load of the chillers has to satisfy the system cooling load, as shown in Eq. (3): l 

PLRi ∗ Qi = CL

i=1

where Qi = capacity of the ith chiller, CL = system cooling load.

(5)

(3)

Vj,g+1 for j = nD , n + 1D , . . . , n + L − 1D Xj,g for all other j

(6)

where j = 1,2,. . .,. . .,D; n is the random integer between 1 to D; the acute brackets D denote the modulo function with modulus D. L signifies the number of components of trial vector u which will be equal to those of mutation vector v, and L is determined by the following pseudo code: L = 0; do { L = L + 1; } while (rand() < CR) AND (L < D); where CR is the crossover constant which takes value in the range [0,1] and rand() ∈ [0,1] is the randomly chosen value. In the selection operation, the trial vector is compared with the parent vector to determine which vector has the better fitness function and can be selected as the parent vector for the next generation. The evolution is continued until the maximum number of evolution generation is reached, and the parent vector with the best fitness function in the population is chosen as the best solution. 4. Case study The two case studies of multiple-chiller systems are illustrated in Table 1. In case 1, the subject is a semiconductor plant in Hsinchu Science-based Park, Taiwan, and the multiple-chiller system is composed of three 800RT units. In case 2, the subject is a hotel in Taipei, and the multiple-chiller system is composed of two 450RT and two 1000RT units [2]. The simulation program in use is written in visual basic and run on the personal computer. The procedure of DE used in optimum chiller loading for reducing energy consumption is shown in Fig. 2 and described as follows. The constraint, Eq. (3), is used to set up

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Table 1 The coefficients of each chiller [2]. System

Chiller

ai

bi

Case 1

CH-1 CH-2 CH-3

100.95 66.598 130.09

Case 2

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

104.09 −67.15 384.71 541.63

ci

di

−973.43 −380.58 14.377

818.61 606.34 304.5

−430.13 −2174.53 1151.42 −3626.5

166.57 1177.79 −779.13 413.48

Capacity (RT)

788.55 275.95 99.8

800 800 800

512.53 1456.53 −63.2 4021.41

450 450 1000 1000

the maximum and minimum load for each chiller and to calculate the load of the last chiller. -

Start

Set up the parameter values

Initialize the vectors

produce mutated vectors

produce trial vectors by crossover

select the parent vectors

Step 1. Set up the parameter values of DE. Step 2. Initialize the vectors. Step 3. Produce mutated vectors for each parent vectors. Step 4. Produce trial vectors by crossover. Step 5. If the trial vector satisfied the constraint, Eq. (4), then select the parent vectors for next generation by comparing the fitness function. If not, using the original parent vector as the parent vectors for next generation. - Step 6. If the iteration number is greater than the set up value, then go to step 7. If not, go back to step 3. - Step 7. Output results. 5. The investigation of the value of DE parameters The testing result of best solution and average solution for the scaling factor and the crossover constant of DE, given the population is 10 and maximum iteration number is 100, is shown in Table 2. The scaling factor is tested from 0.1 to 0.9 and the crossover constant is tested from 0.4 to 0.9 using four examples. Each combination of these parameters has the same performance in the minimum fitness value. Moreover, the result of average solution value on different crossover constants in every scaling factor is shown in Table 3. Of the four examples, the scaling factor 0.7 has reached the minimum average value. Therefore, the scaling factor 0.7 is adopted for further analysis. Then, we pay the attention on the crossover constant analysis with the scaling factor equal to 0.7. As shown in Table 2, the four examples indicate that the value 0.8 has the best performance and, as a result, is adopted for further analysis. 6. The convergence test of DE

NO

Iteration number greater than set up value

YES Output Fig. 2. The flow chart of differential evolution optimization.

The ratio of iteration value to minimum value of fitness function

The convergence performance of DE, based on vectors 10, iterations 100, scaling factor 0.7 and crossover constant 0.8, is shown in Fig. 3. For each test case, the ratio of fitness function value to the final best value can converge in 1% before the 40th iteration. This

1.047 1.040 1.033

1440RT

1920RT

2030RT

2320RT

1.026 1.019 1.012 1.005 0.998

0

10

20

30

40

50

60

70

iteration Fig. 3. Fitness function against iteration.

80

90

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Table 2 The result of the best solution testing and the average solution for the scaling factor and the crossover constant of DE. F

CR

Example 1

Example 2

Example 3

Example 4

Case l

Case l

Case 2

Case 2

1440 (RT)

1920 (RT)

2030 (RT)

2320 (RT)

Min (kW)

Ave (kW)

Min (kW)

Ave (kW)

Min (kW)

Ave (kW)

Min (kW)

Ave (kW)

0.1

0.4 0.6 0.8 0.9

993.6 993.6 993.6 993.6

1026.2 1034.1 1014.2 1015.4

1403.2 1403.2 1403.2 1403.2

1403.4 1404.1 1403.9 1404.2

1178.2 1178.5 1179.3 1179.0

1189.6 1187.2 1192.9 1193.1

1455.7 1455.7 1456.4 1456.5

1458.1 1458.8 1466.2 1465.4

0.3

0.4 0.6 0.8 0.9

993.6 993.6 993.6 993.6

1002.8 1002.8 1022.5 1015.3

1403.2 1403.2 1403.2 1403.2

1403.2 1403.2 1403.2 1403.2

1178.1 1178.1 1178.1 1178.1

1178.3 1178.8 1179.2 1181.9

1455.7 1455.7 1455.7 1455.7

1455.8 1455.8 1456.4 1456.5

0.5

0.4 0.6 0.8 0.9

993.6 993.6 993.6 993.6

1013.4 1022.8 1002.8 1013.6

1403.2 1403.2 1403.2 1403.2

1403.2 1403.2 1403.2 1403.2

1178.1 1178.1 1178.1 1178.1

1178.2 1178.1 1178.1 1178.2

1455.7 1455.7 1455.7 1455.7

1455.7 1455.7 1455.7 1455.7

0.7

0.4 0.6 0.8 0.9

993.6 993.6 993.6 993.6

1002.8 1012.0 993.6 1002.8

1403.2 1403.2 1403.2 1403.2

1403.2 1403.2 1403.2 1403.2

1178.1 1178.1 1178.1 1178.1

1178.1 1178.1 1178.1 1178.1

1455.7 1455.7 1455.7 1455.7

1455.7 1455.7 1455.7 1455.7

0.9

0.4 0.6 0.8 0.9

993.6 993.6 993.6 993.6

1013.7 1012.0 1012.0 1002.8

1403.2 1403.2 1403.2 1403.2

1403.2 1403.2 1403.2 1403.2

1178.1 1178.1 1178.1 1178.1

1178.3 1178.1 1178.2 1178.1

1455.7 1455.7 1455.7 1455.7

1455.7 1455.7 1456.4 1455.7

Table 3 The result of average solution value on different crossover constant constants in every scaling factor. F

0.1 0.3 0.5 0.7 0.9

Example 1

Example 2

Example 3

Example 4

Case l

Case l

Case 2

Case 2

1440 (RT)

1920 (RT)

2030 (RT)

2320 (RT)

(kW)

(kW)

(kW)

(kW)

1022.5 1010.9 1013.2 1002.8 1010.1

1403.9 1403.2 1403.2 1403.2 1403.2

1190.7 1179.5 1178.2 1178.1 1178.2

1462.1 1456.1 1455.7 1455.7 1455.9

Table 4 The comparison of optimum results of case 1 of Lagrangian method, genetic algorithm, PSO algorithm and DE algorithm. Load (PLR)

Chiller

LGM PLR

GA Total kW Min (A)

PSO

(D − A)/A (%)

DE

PLR

Total kW Min (B)

PLR

Total kW Min (C)

PLR

Total kW Min (D)

(D − B)/B (%)

(D − Q/C (%)

2160 (90%)

1 2 3

0.73 1583.81 0.97 1.00

0.81 0.93 0.96

1590.96

0.73 0.97 1.00

1583.81

0.73 0.97 1.00

1583.81

0.00

−0.45

0.00

1920 (80%)

1 2 3

0.66 1403.20 0.86 0.88

0.70 0.80 0.90

1406.02

0.66 0.86 0.88

1403.20

0.66 0.86 0.88

1403.20

0.00

−0.20

0.00

1680 (70%)

1 2 3

0.60 1244.32 0.75 0.76

0.69 0.68 0.73

1250.06

0.60 0.74 0.76

1244.32

0.60 0.74 0.76

1244.32

0.00

−0.46

0.00

1440 (60%)

1 2 3

0.53 1102.26 0.62 0.65

0.52 0.74 0.54

1107.75

0.00 0.89 0.91

993.60

0.00 0.89 0.91

993.60

−9.86

−10.30

0.00

1200 (50%)

1 2 3

0.49 0.44 0.57

971.21

0.00 0.74 0.76

832.33

0.00 0.74 0.76

832.33

−14.30

0.00

– –

1 2 3

0.31 0.32 0.58

842.18

0.00 0.57 0.63

692.25

0.00 0.57 0.63

692.25

−17.80

0.00

– –

960 (40%)

–

W.-S. Lee et al. / Energy and Buildings 43 (2011) 599–604

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Table 5 The comparison of average results of case 1 of PSO algorithm and DE algorithm. Load (PLR)

Chiller

PSO

(B − A)/A (%)

DE

PLR (Min)

Total kW

PLR (Min)

Min

Max

Ave (A)

Stdev

Total kW Min

Max

Ave (B)

Stdev

2160 (90%)

1 2 3

0.73 0.97 1.00

1583.81

1583.81

1583.81

0.00

0.73 0.97 1.00

1583.81

1583.81

1583.81

0.00

0.00

1920 (80%)

1 2 3

0.66 0.86 0.88

1403.20

1403.20

1403.20

0.00

0.66 0.86 0.88

1403.20

1403.20

1403.20

0.00

0.00

1680 (70%)

1 2 3

0.60 0.74 0.76

1244.32

1244.32

1244.32

0.00

0.60 0.74 0.76

1244.32

1244.32

1244.32

0.00

0.00

1440 (60%)

1 2 3

0.00 0.89 0.91

993.60

1102.26

1004.47

32.60

0.00 0.89 0.91

993.60

993.60

993.60

0.00

−1.08

1200 (50%)

1 2 3

0.00 0.74 0.76

832.33

898.90

838.98

19.97

0.00 0.74 0.76

832.33

832.33

832.33

0.00

−0.79

960 (40%)

1 2 3

0.00 0.57 0.63

692.25

749.33

709.37

26.15

0.00 0.57 0.63

692.25

692.25

692.25

0.00

−2.41

shows the DE algorithm with the given parameter value set gives a good performance in convergence. 7. The performance of DE The parameters of DE algorithm are set up as follows: The number of the vectors is 10, the maximum iteration number 100, the scaling factor 0.7, and the crossover constant 0.8. The optimum results, calculated 10 times by DE in case 1, are compared with those by the Lagrangian method, genetic algorithm [2], and PSO algorithm [7], respectively, as shown in Table 4. The population of GA and PSO are 100 and 10, and the generation number of GA and

PSO is 100. The minimum energy consumptions calculated by DE are the same as those calculated by PSO in each different demand load. When the partial load ratio is greater than 60%, the minimum energy consumption calculated by DE equals the value by Lagrangian method and is better than that of genetic algorithm by 0.2–0.46%. When the partial load ratio equals 60%, the DE is better than Lagrangian method by about 9.9%, and is better than genetic algorithm by 10.3%. This is because a chiller can be stopped from operating in order to increase the overall efficiency in the optimum model [7]. When the cooling load is less than 60%, the Lagrangian method cannot converge, and DE is better than genetic algorithm by between 14.3% and 17.8%. Furthermore, the average value of

Table 6 The comparison of optimum results of case 2 of Lagrangian method, genetic algorithm, PSO algorithm and DE algorithm. Load (PLR)

Chiller

LGM

GA

PSO

(D − A)/A (%)

DE

PLR

Total kW Min (A)

PLR

Total kW Min (B)

PLR

Total kW Min (C)

PLR

Total kW Min (D)

(D − B)/B (%)

(D − C)/C (%)

2610 (90%)

1 2 3 4

0.99 0.91 1.00 0.76

1857.30

0.99 0.95 1.00 0.74

1862.18

0.99 0.91 1.00 0.76

1857.30

0.99 0.91 1.00 0.76

1857.30

0.00

−0.26

0.00

2320 (80%)

1 2 3 4

0.83 0.81 0.90 0.69

1455.66

0.86 0.81 0.88 0.69

1457.23

0.83 0.81 0.90 0.69

1455.66

0.83 0.81 0.90 0.69

1455.66

0.00

−0.11

0.00

2030 (70%)

1 2 3 4

0.73 0.74 0.72 0.65

1178.14

0.66 0.76 0.76 0.64

1183.80

0.73 0.74 0.72 0.65

1178.14

0.73 0.74 0.72 0.65

1178.14

0.00

−0.48

0.00

1740 (60%)

1 2 3 4

0.60 0.66 0.56 0.61

998.53

0.60 0.70 0.57 0.59

1001.62

0.60 0.66 0.56 0.61

998.53

0.60 0.66 0.56 0.61

998.53

0.00

−0.31

0.00

1450 (50%)

1 2 3 4

0.46 0.50 0.45 0.57

904.62

0.60 0.36 0.44 0.58

907.72

0.61 0.00 0.57 0.61

820.07

0.61 0.00 0.57 0.61

820.07

−9.35

−9.66

0.00

1160 (40%)

1 2 3 4

0.30 0.30 0.35 0.54

849.99

0.33 0.32 0.32 0.54

856.30

0.00 0.00 0.56 0.60

651.07

0.00 0.00 0.56 0.60

651.07

−23.40

−23.97

0.00

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W.-S. Lee et al. / Energy and Buildings 43 (2011) 599–604

Table 7 The comparison of average results of case 2 of PSO algorithm and DE algorithm. Load (PLR)

Chiller

PSO PLR (Min)

(B − A)/A (%)

DE Total kW

PLR (Min)

Min

Max

Ave (A)

Stdev

Total kW Min

Max

Ave (B)

Stdev

2610 (90%)

1 2 3 4

0.99 0.91 1.00 0.76

1857.30

1857.45

1857.43

0.04

0.99 0.91 1.00 0.76

1857.30

1858.57

1857.43

0.40

0.00

2320 (80%)

1 2 3 4

0.83 0.81 0.90 0.69

1455.66

1522.42

1462.34

20.03

0.83 0.81 0.90 0.69

1455.66

1455.66

1455.66

0.00

−0.46

2030 (70%)

1 2 3 4

0.73 0.74 0.72 0.65

1178.14

1178.14

1178.14

0.00

0.73 0.74 0.72 0.65

1178.14

1178.14

1178.14

0.00

0.00

1740 (60%)

1 2 3 4

0.60 0.66 0.56 0.61

998.53

1013.43

1005.36

5.71

0.60 0.66 0.56 0.61

998.53

1009.20

1000.21

3.66

−0.51

1450 (50%)

1 2 3 4

0.61 0.00 0.57 0.61

820.07

847.53

826.52

10.88

0.61 0.00 0.57 0.61

820.07

821.28

820.19

0.38

−0.77

1160 (40%)

1 2 3 4

0.00 0.00 0.56 0.60

651.07

691.19

667.12

19.65

0.00 0.00 0.56 0.60

651.07

655.63

651.53

1.44

−2.34

energy consumption calculated by DE is the same as the minimum value, and three loads out of six indicate that it is better than that of PSO by 0.8–2.4%, as shown in Table 5. The optimum results calculated 10 times by DE in case 2 are compared with those by Lagrangian method, genetic algorithm [2], and PSO algorithm [7] as shown in Table 6. When the partial load ratio is greater than 50%, the minimum energy consumption calculated by DE equals those by the Lagrangian method and by PSO algorithm, and is better than those by genetic algorithm between 0.11% and 0.48%. When the partial load ratio is equal to or less than 50%, DE equals PSO, outperforming Lagrangian method between 9.35% and 23.4%, and genetic algorithm between 9.66% and 23.97%. Moreover, four loads out of six indicate that the average value of energy consumption calculated by DE is better than that of PSO by 0.5–2.3%, as shown in Table 7. 8. Conclusion This paper uses differential evolution algorithm to develop an optimization model for optimal multi-chiller loading problems. The energy consumption is considered as the objective function, while the loading ratio of each chiller is considered as the optimum parameter. Two multi-chiller systems including three chillers and four chillers are used as case studies. The results show that the proposed differential evolution algorithm, with the scaling factor 0.7 and the cross over factor 0.8, can find the optimal solution as the particle swarm algorithm can, but obtain better average solutions. Moreover, it outperforms the genetic algorithm in finding optimal solution and also overcomes the divergence problem caused by the Lagrangian method occurring at low demands. From the above comparison, we conclude that using differential evolution algorithm can improve the calculation performance in energy saving problems of multi-chiller systems.

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