# An Improved NSGA-II Algorithm Based on Adaptive Weighting and Searching Strategy
## 4 Case Studies
### Parallel Chiller System  
The proposed algorithm is applied to the chiller load distribution problem for the sake of verifying the effectiveness of the presented algorithm in practical project problems [22]. The cooling capacity required for the central air conditioning system is provided by the chilled water system, which includes chilled water pumps and a chiller system, as shown in Figure 4. Generally, the chiller system is operated jointly by multiple chillers with different refrigeration performance to manufacture chilled water, and the different refrigeration performance determines that the energy consumption of each chiller is also different. A multi-chiller system is composed of two or more chillers connected by parallel or tandem piping to a common distribution system. The chiller system can adjust its own load size so that each chiller operates at the optimal operating point. The chiller can adjust its own load size to meet the basic end load demand so that the parallel chiller system can operate at optimal performance. In a refrigeration system, the system performs best when the total power of the chiller system is minimal and the end load demand is met. The problem of the load distribution of parallel chillers is to adjust the load distribution of the parallel chillers to achieve energy savings. At a certain wet bulb  
Figure 4: Chiller system diagram. temperature, the electrical power of a concentric chiller can be presented as an optimized function of the partial load rate [23] as follows:  
\[P_{i}=a_{i}+b_{i}\times PLR_{i}+c_{i}\times PLR_{i}^{2}+d_{i}\times PLR_{i}^{3} \tag{14}\]  
where \(P_{i}\) is the power of the \(i_{th}\) chiller, \(PLR_{i}\) is the partial load rate of the \(i_{th}\) chiller, and \(a_{i}\), \(b_{i}\), and \(c_{i}\) are the fitting coefficients of the energy consumption curve of the chiller. An optimal distribution problem for parallel chillers is to achieve the lowest energy consumption and the highest cooling capacity. The abstraction of the actual project into a mathematical problem could be represented as the formula below:  
\[\min\ P=\sum_{i=1}^{n}P_{i} \tag{15}\]  
\[\max\ Q=\sum_{i=1}^{n}PLR_{i}\times Q_{i} \tag{16}\]  
where \(P\) is the total power of multiple parallel chillers, and \(Q\) is the cooling capacity of the chiller. A chiller system with three chillers of 800 RT cooling capacity was selected for testing in Case 1 so as to confirm that the improved NSGA-II algorithm could be implemented to tackle the load distribution issues. To further verify the algorithm in solving the load distribution problem of parallel chillers, a cooling system which consists of six chillers was selected in Case 2, including four same chillers with a cooling capacity 1280 RT and 2 other chillers of 1250 RT. The purpose was to test whether the improved NSGA-II algorithm can search for optimal values in different multi-chiller systems. Through these two classical test cases, we verified the feasibility of the INSGA-II algorithm for solving practical problems. In the two test cases, the performance parameters of the chillers are shown in Table 6.  
We applied the proposed INSGA-II algorithm proposed in this paper to the parallel chiller's cooling capacity distribution problem. The improved NSGA-II algorithm was compared with the original NSGA-II, and other methods such as MOEA/D, IBEA and SPEA2 algorithms; the simulation and optimization results are shown as follows. In Figure 5, each point represents the optimal solution that satisfies both Equations (15) and (16).