## 4 Case Study and Discussion
### Probability Distribution of Chiller Plant State  
In this study, the chiller plant is assumed to include 2\(\sim\)6 chillers totally. The failure rate of constant-speed chillers is assumed to be 0.0001/h and the failure rate of variable-speed chillers is assumed to be 0.000125/h. Both their repair rates are assumed to be 0.002/h [8; 43].  
The following step is to generate the steady probability distribution of a chiller plant under various numbers of variable-speed and constant-speed chillers. The chiller plant consists of one or two variable-speed chillers as mentioned. Referring to Equation (7), the probability distribution of variable-speed chillers can be obtained accordingly. Table 2 shows the steady probability distribution of various states of variable-speed chillers. The probability of state 0 is 0.9412 and the probability of state 1 is 0.0588, when only one variable-speed chiller is employed. The probabilities of state 0, 1, and 2 are 0.8828, 0.1103, and 0.0069, respectively, when two variable-speed chillers are employed. Referring to Equation (6), the probability distribution of constant-speed chillers can be obtained accordingly. Table 3 shows the steady probability distribution of various states for a different number of constant-speed chillers. It can be observed that the probability of state 0 decreases while the number of constant-speed chillers increases.  
Referring to Equation (5), the probability distribution of the chiller plant can be obtained accordingly. Table 4 shows an example of the probability distribution of the chiller plant, which consists of two variable-speed chillers and five constant-speed chillers. This chiller plant gets 18 states totally (i.e., three states of variable-speed chillers and six states of constant-speed chillers). The probability of the chiller plant under the state 0 of variable-speed chillers and state 0 of constant-speed chillers is 0.67481, while the probability of the chiller plant under state 2 of variable-speed chillers and state 5 of constant-speed chillers is 0.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Probability Distribution of Chiller Plant**} & \multicolumn{4}{c}{**State of Variable-Speed Chillers**} \\ \cline{2-5}  & **0** & **0** & **1** & **2** \\ \hline \multirow{4}{*}{State of constant-speed chillers} & 0 & 0.67481 & 0.08431 & 0.00527 \\  & 1 & 0.16870 & 0.02108 & 0.00132 \\ \cline{1-1}  & 2 & 0.03372 & 0.00421 & 0.00026 \\ \cline{1-1}  & 3 & 0.00503 & 0.00063 & 0.00004 \\ \cline{1-1}  & 4 & 0.00053 & 0.00007 & 0 \\ \cline{1-1}  & 5 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: An example of the probability distribution of a chiller plant.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**State**} & \multicolumn{2}{c}{**Variable-Speed Chillers**} \\ \cline{2-4}  & **1** & **2** \\ \hline
0 & 0.9412 & 0.8828 \\
1 & 0.0588 & 0.1103 \\
2 & - & 0.0069 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Probability distribution of steady states of variable-speed chillers.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**State**} & \multicolumn{4}{c}{**Number of Constant-Speed Chillers**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** & **5** \\ \hline
0 & 0.9524 & 0.9050 & 0.8578 & 0.8109 & 0.7644 \\
1 & 0.0476 & 0.0905 & 0.1278 & 0.1622 & 0.1911 \\
2 & - & 0.0045 & 0.0129 & 0.0243 & 0.0382 \\
3 & - & - & 0.0006 & 0.0024 & 0.0057 \\
4 & - & - & - & 0.0001 & 0.0006 \\
5 & - & - & - & - & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Probability distribution of steady states of constant-speed chillers.