## 3 Implementation of Robust Optimal Design for Chiller Plant Optimization
### Optimization of the Configuration of Chiller Plant  
In the previous design methods, optimization is often based on a given total cooling capacity (e.g., the cooling capacity corresponding to 50 unmet hours in the typical sample year). In this paper, the chiller plant configuration is optimized by comparing a series of total cooling capacities within a certain searching range. The optimized chiller plant could operate within a higher efficient region and have sufficient capacity to supply the cooling. By considering that chillers are manufactured in discrete size, trials on various total cooling capacities and discrete number/size of chillers are carried out to determine the optimum chiller plant configuration. As mentioned above, the optimum chiller plant is selected based on the expected total cost. The most important part of total cost is the operating cost, which is mainly determined by the distribution of cooling load and energy efficiency of the chiller plant (i.e., COP). As shown in Equation (15), COP is mainly related to part load ratio (PLR) when the operating parameters, such as condensing temperature and evaporating temperature, are set to be constant. \[COP_{i}=D_{0}+D_{1}\cdot PLR_{i}+D_{2}\cdot PLR_{i}^{2}+D_{3}\cdot PLR_{i}^{3} \tag{15}\]  
where, D\({}_{0}\)-D\({}_{3}\) are the coefficients that can be fitted according to the chiller catalog or onsite measurement data. As shown in Equation (16), PLR is the ratio of operating cooling load (\(CL_{opt}\)) to the operating cooling capacity of chillers (\(CL_{cc}\)). Where, \(CL_{rate}\) is chiller rated capacity. \(N_{qp}\) represents the number of chillers in operation. \[PLR=\frac{CL_{opt}}{CL_{cc}}=\frac{CL_{opt}}{N_{qp}\cdot CL_{rate}} \tag{16}\]  
According to Equation (16), when the cooling load is given, the value of PLR is actually determined by the number of chillers in operation and the rated capacity. If more chillers are equipped (i.e., with larger flexibility to control the number of chillers in operation according to the actual cooling load to maximize the PLR), the actual operating PLR could be higher under proper sequence control strategies. Meanwhile, equipping more chillers indicates that the rated capacity of the individual chiller is smaller, which leads to the lower rated COP. Theoretically, an optimal value of chiller number exists when the highest operating COP is achieved. As shown in Figure 6[42], once the operating chiller number approaches to a certain value, the operating COP can then achieve a higher level because the impact of increasing PLR is larger than that of decreasing rated COP. When the operating chiller number continually increases, the operating COP becomes lower because its PLR maintains at a nearly constant value and the rated COP reduces accordingly. Figure 7 presents the implementation of simulation trials on each total cooling capacity to determine the optimum number and capacity of chillers. It contains two parts: (1) the trials under the condition that only one variable-speed chiller is employed; and (2) the trials under the condition that two variable-speed chillers are employed. For the trials under the condition that only one variable-speed chiller is employed, the numbers of constant-speed chillers are calculated starting from one until the operation cost starts to increase. In other words, at least two chillers are employed for system maintenance and reliability consideration. The option corresponding to the lowest total cost is selected from different cooling capacities. Equation (17) describes the sizing problem of a constant-speed chiller when only one variable-speed chiller is used. \[ \begin{array}{rl}\text{find}&C_{1},\,n_{1},\,C_{2}\\ \text{minimize}&TC_{all}(C_{1},\,n_{1},\,C_{2})\\ \text{constraint}&n_{1}C_{1}+C_{2}=CL_{T}\\ &C_{1}>C_{2},\,n_{1}\geq 1\end{array}  \tag{17}\]  
where \(C_{1}\) represents capacity of constant-speed chiller, \(C_{2}\) represents capacity of variable-speed chiller, \(TC\) represents total cost, and \(CL_{T}\) represents total cooling capacity.