Optimal Chiller Loading by Team Particle Swarm Algorithm for Reducing Energy Consumption

: Energy saving is an important issue for multiple-chiller systems. Optimal chiller loading (OCL) in multiple-chiller systems has been investigated with many optimization algorithms to save energy. Particle swarm optimization (PSO) algorithm has been successful in solving this problem in some cases, but not in all. This study innovatively added a team evolution to the original particle swarm optimization algorithm, called team particle swarm optimization (TPSO). The TPSO enhances the effectiveness of original particle swarm optimization to better solve the OCL problem. The TPSO algorithm is composed of two evolutions: particle evolution and team evolution. The partial load ratio (PLR) of each operating chiller and the on-off state of each chiller are the particle evolution parameters and team evolution parameters, respectively. To evaluate the performance of the proposed method, this paper adopts three case studies so the results generated from the proposed algorithm TPSO, the original particle swarm optimization (PSO) and other recently published algorithms can be compared. In these three case studies, the optimal results generated by using TPSO algorithm are the same as those by other compared algorithms. In case 1 under 5717 RT and 5334 RT cooling load, the results generated using the TPSO are lower than those by the original PSO in the amounts of 63.35 and 79.33 kW, respectively. The results indicated that the TPSO algorithm not only enabled the optimal solution in minimizing energy consumption, but also demonstrated the best stability when compared to other algorithms. In conclusion, the presented TPSO algorithm is an efﬁcient and promising new algorithm for solving the OCL problem.


Introduction
Air conditioning systems contributes considerably to the energy consumption in buildings-with their chiller systems being the main cause of energy consumption. Different operation strategies for the chiller systems result in significant differences in the energy used. An optimal chiller loading (OCL) method proposed by Chang [1], alongside many other algorithms, aimed to find the optimal on-off state and partial loading ratio (PLR) of each chiller to minimize the energy consumption. Chang [1] used the Lagrangian method to investigate the optimal loading ratio of each chiller to minimize the energy consumption at different cooling loads in two cases. The result showed that while the setup PLR value of the Lagrangian method saved much energy compared to the equal loading rate of traditional methods, the Lagrangian algorithm suffered a flaw in an inability to converge at low demands. Subsequently, Chang [2] proposed a genetic algorithm to overcome the flaw of the Lagrangian algorithm. However, as a result, the optimal energy consumption of genetic algorithm increased by about 0.4% compared to the Lagrangian Figure 1. Decoupled system of a multiple-chiller system [28].
The best solution is when the sum of energy consumption of each chiller is minimized while the load is satisfied. The energy consumption of a centrifugal chiller is a convex function of its PLR for a given wet-bulb temperature [2]: Power a b PLR c PLR d PLR = + * + * + * (1) where ai, bi, ci, di are coefficients of power curve of i th chiller. The objective of solving the OCL problem is to minimize the summation of energy consumption of each chiller, as shown in Equation (2) The first constraint of OCL problem is that the summation of cooling provided by each chiller should meet the system cooling load, as shown in Equation (3) where Qi = capacity of i th chiller, CL = system cooling load.
The other constraint is that the partial load of each operated chiller cannot be smaller than 30% [3], as shown in Equation (4) 0.3 i PLR ≥ (4) Figure 1. Decoupled system of a multiple-chiller system [28].
The best solution is when the sum of energy consumption of each chiller is minimized while the load is satisfied. The energy consumption of a centrifugal chiller is a convex function of its PLR for a given wet-bulb temperature [2]: where a i , b i , c i , d i are coefficients of power curve of i th chiller. The objective of solving the OCL problem is to minimize the summation of energy consumption of each chiller, as shown in Equation (2): The first constraint of OCL problem is that the summation of cooling provided by each chiller should meet the system cooling load, as shown in Equation (3) where Q i = capacity of ith chiller, CL = system cooling load. The other constraint is that the partial load of each operated chiller cannot be smaller than 30% [3], as shown in Equation (4)

Method
The original particle swarm optimization (PSO) algorithm is proposed by Kennedy and Eberhart [29,30], and it moves the position of particles to find the optimal solution. The position of the particle is referred to the value of parameter. The progress of optimization is to move the particle position toward the personal best position and the best position of all particles. The number of iterations decides the end of evolution. The PSO evolution rule is showed as follows: where v i represents the velocity of particle i, k means the iteration number, w represents inertial weight, c 1 and c 2 are acceleration constants, r 1 and r 2 are two random values in the range of [0, 1], x i,k represents the current position of particle i, Pb i is the position of particle i with personal best solution and Gb is the position of the particle with best solution found thus far. The PSO algorithm and the improved versions have been successfully applied in several fields. However, the PSO algorithm was the best solution for the OCL problem in some cases but not all. In order to improve on the efficacy of the original PSO algorithm, a new concept of team and model best solutions are added to the PSO algorithm for evolution. The improved algorithm is named as team particle swarm optimization (TPSO). In TPSO, the particles are allocated to several teams. Particles in the same team have the same on-off state of each chiller. The TPSO algorithm is composed of two evolutions: particle evolution and team evolution. Particle evolution progresses on every evolutionary generation. Team evolution progresses after particle evolution has made a number of progressions. The partial load ratio (PLR) of each operating chiller and the on-off state of each chiller are particle evolution parameter and team evolution respectively. The particle evolution in the TPSO algorithm is different from the original particle evolution in the PSO, in that the progress of optimization is to move the particle position toward the personal best position, team best position, and model best position. The best team position is the best position of particles in the same team, and the model best position is the best position of all particles with the same on-off state as each chiller. The evolution formula of particle evolution of TPSO is presented as follows: where Tb i means the position of particle which has the best performance in the same team with particle i, Mb i means the position of particle which has the best performance in the same on-off state as each chiller with particle i thus far. The process of team evolution is: first, a random value is given as r 4 , and then r 4 is compared with the evolution threshold. If r 4 meets the conditions for change in the on-off state of chillers, the state of chillers in the team will change and the personal best position of each particle in this team, and best team position, will be reinitialized.
The evolution rule of team evolution is as follows    r 4 < Thr l , s i,k+1 = s i,k for each chiller i r 4 > Thr l and r 4 < Thr l , s i,k+1 = rand(0, 1) reinitialize ecah chilleri r 4 > Thr l , s i,k+1 = Gbs k for each chiller i s i = 1, chiller state is on 0, chiller state is off (10) where r 4 is the random value between 0 and 1, Thr l and Thr h are the threshold values for device state evolution. Gbs k means the chiller state of the best particle founded thus far. The progress of the TPSO evolution is presented in the flowchart of Figure 2, and the scudo code is shown in Figure 3. A case of six particles and two teams for a 3-chiller system is used to describe the evolution process. At step 1, the particles are assigned to teams in sequence, as shown in Table 1. Particle 1, 2, 3 are belong to team 1 and Particle 4,5,6 are belong to team 2. At step 2, the on-off states of chillers in each team are initialized, as shown in Table 2. The chiller on-off states of particles in team 1 are the same, chiller 1 is off, chiller 2 is on and chiller 3 is on. The on-off states of chiller 1, 2 and 3 of particles in team 2 are on, off, and on, respectively. At step 3, the parameter value, PLR, of each chiller are initialized to meet the system cooling load, as shown in Table 3. At step 4, the particle evolution for PLR of each chiller starts, as shown in Table 4. At step 5, the team evolution for on-off states of chillers starts, as shown in Table 5. The random value of r 4 for team1 is 0.3 at this evolution, therefore the on-off states of chiller 1, 2 and 3 of particles in team 1 are not changed; the r 4 for team2 is 0.5, therefore the on-off states and PLR of chiller 1, 2 and 3 of particles in team 2 are reinitialized.

Case Study
In this case study, three well-known cases of multiple-chiller systems, shown in 6, are utilized to compare the performance of the TPSO algorithm, the original PS recently published algorithms, i.e., (DE) [9], (IFOA) [18], (DCEDA) [20], and (EIWO Case 1 [22] includes four 1280 RT and two 1250 RT chillers; Case 2 [2] contains tw RT and two 450 RT chillers; Case 3 [2] consists of three 800 RT chillers. The perfor coefficients of the chillers are listed in Table 6.

Case Study
In this case study, three well-known cases of multiple-chiller systems, shown in 6, are utilized to compare the performance of the TPSO algorithm, the original PS recently published algorithms, i.e., (DE) [9], (IFOA) [18], (DCEDA) [20], and (EIWO Case 1 [22] includes four 1280 RT and two 1250 RT chillers; Case 2 [2] contains tw RT and two 450 RT chillers; Case 3 [2] consists of three 800 RT chillers. The perfor coefficients of the chillers are listed in Table 6.

Parameter Analysis
The optimal parameter values of TPSO in the OCL problem are discussed with two examples, Case 1 with cooling load 6096RT and Case 3 with 2160RT. The results are shown in Tables 7 and 8. The performance of the TPSO can be obtained by setting the following parameter values: the total particle number is 200, and the particle number of a team is 5, which means of 40 teams. The inertial weight is 0.4, and the acceleration constants are 1. The iteration numbers in particle and team evolutions are 500 and 30, respectively. Therefore, these parameter values are used for the following case studies.

Performance of TPSO
For testing the performance of TPSO, the results of Cases 1 to 3 under various cooling loads (40%, 50%, 60%, 70%, 75%, 80%, 85%, 90%) are presented in Tables 9-11, respectively. The results are carried out for 50 independent runs and the parameter values are shown in Table 12.      The difference between maximum and minimum values and standard deviations also shows zero to the third decimal. This information indicates the high stability of the TPSO algorithm under the chosen evolution parameters. The power consumptions of each cooling load under each case operated by TPSO algorithm with respect to the iteration number are presented in Figures 4-6. These results indicated that the TPSO algorithm also has been effective with convergence in each cooling load of each case. The difference between maximum and minimum values and standard deviations also shows zero to the third decimal. This information indicates the high stability of the TPSO algorithm under the chosen evolution parameters. The power consumptions of each cooling load under each case operated by TPSO algorithm with respect to the iteration number are presented in Figures 4-6. These results indicated that the TPSO algorithm also has been effective with convergence in each cooling load of each case.

Compare the TPSO Algorithm with Other Algorithms
To verify the effectiveness of the TPSO algorithm as the best solution, three case studies are adopted to compare the results of the proposed algorithm with original particle swarm optimization and other recently published algorithms, DE [9], DCEDA [20], EIWO [22] and IFOA [18]. The compared results for Case 1, Case 2, and Case 3 are shown in Tables 13-15, respectively.
In these three case studies, the results generated by using TPSO algorithm are the same with those by other algorithms. In Case 1, the compared algorithms are original PSO [22], DCEDA [20], EIWO [22] and IFOA [18] as shown on Table 13; In Case 2 and Case 3 the original PSO [3], DE [9], EIWO [22] and IFOA [18] are compared, as shown on Tables 14 and 15, respectively. In Case 1 under 5717 RT and 5334 RT cooling load, the results generated by using the TPSO are lower than those by the original PSO in the amounts of 63.35 and 79.33 kW, respectively. The comparison indicated that the TPSO algorithm outperformed the original PSO algorithm and all other algorithms presented in this paper.

Compare the TPSO Algorithm with Other Algorithms
To verify the effectiveness of the TPSO algorithm as the best solution, three case studies are adopted to compare the results of the proposed algorithm with original particle swarm optimization and other recently published algorithms, DE [9], DCEDA [20], EIWO [22] and IFOA [18]. The compared results for Case 1, Case 2, and Case 3 are shown in Tables 13-15, respectively.
In these three case studies, the results generated by using TPSO algorithm are the same with those by other algorithms. In Case 1, the compared algorithms are original PSO [22], DCEDA [20], EIWO [22] and IFOA [18] as shown on Table 13; In Case 2 and Case 3 the original PSO [3], DE [9], EIWO [22] and IFOA [18] are compared, as shown on Tables 14 and 15, respectively. In Case 1 under 5717 RT and 5334 RT cooling load, the results generated by using the TPSO are lower than those by the original PSO in the amounts of 63.35 and 79.33 kW, respectively. The comparison indicated that the TPSO algorithm out-performed the original PSO algorithm and all other algorithms presented in this paper.

Comparison of Stability
To compare the stability of TPSO witha number of other algorithms, the maximum, minimum, and average difference between maximum and minimum, and standard deviation of power consumption of chillers under various cooling loads in Cases 1 to 3 are presented in Tables 16-18. The algorithms selected for comparison are IFOA [18] and DCEDA [20]. The difference between the maximum and minimum values and standard deviations of the TPSO algorithm in all cases are shown in zero to the third decimal, as indicated in these three tables. It is demonstrative that the TPSO algorithm yields the best stability when compared with IFOA and DCEDA.