## 3 Implementation of Robust Optimal Design for Chiller Plant Optimization
### Quantification of Probability Distribution of State for Chiller Plant  
For the reliability assessment of multi-state systems, the Markov method is widely adopted [40]. In this paper, steady probability distribution of various states (e.g., fault state and fault-free state) for a chiller plant is generated by adopting the Markov method. Once probability distribution for each state is determined, the available capacity of a chiller plant can be estimated accordingly. The detailed assumptions of using the Markov method can be found in Reference [40]. The life-cycle of each component in this study contains operating period, maintenance period, and failure period. As shown in Equation (3), the MTTF (mean time to failure, 1/\(\lambda\)) represents the operating time. As shown in Equation (4), the MTTR (mean time to repair, 1/\(\mu\)) is usually employed to represent the sum of maintenance time and failure time. Usually, the failure rate (\(\lambda\)) and the repair rate (\(\mu\)) are employed as the major parameters for conducting the reliability assessment. \[MTTF=\frac{1}{\lambda} \tag{3}\]  
\[MTTR=\frac{1}{\mu} \tag{4}\]  
Each chiller can be in two states: normal state (0) and failure state (1), which respectively represent when a chiller operates normally and abnormally. A chiller plant is assumed to consist of \(n_{1}\) constant-speed chillers and \(n_{2}\) variable-speed chillers. The constant-speed chillers have (\(n_{1}+1\)) states (i.e., each state includes several reliability scenarios) by accounting the reliability of constant-speed chillers as shown in Figure 5[41]. As mentioned above, the number of variable-speed chillers is no more than two. As a result, the variable-speed chillers have three possible states (i.e., no chillers fail, one chiller fail, and two chillers fail). Totally, the chiller plant has 3(\(n_{1}+1\)) states, as shown in Equations (5-7). When only one variable-speed chiller is employed, \(p_{2,n2}\) is equal to 0. \[(p_{0,n_{1}}+p_{1,n_{1}}+\ldots+p_{n_{1}-1,n_{1}}+p_{n_{1},n_{1}})\cdot(p_{0,n _{2}}+p_{1,n_{2}}+p_{2,n_{2}})=1 \tag{5}\]  
\[p_{0,n_{1}}+p_{1,n_{1}}+\ldots+p_{n_{1}-1,n_{1}}+p_{n_{1},n_{1}}=1 \tag{6}\]  
\[p_{0,n_{2}}+p_{1,n_{2}}+p_{2,n_{2}}=1 \tag{7}\]  
The states of \(n_{1}\) constant-speed chillers and the possible transitions among these different states are shown in Figure 5. There are totally (\(n_{1}+1\)) states, where the state 0 means that no constant-speed chillers failed and the state k means that there are \(k\) (\(1\leq k\leq n\)) constant-speed chillers in failure. The chiller plant may transfer from one state to another due to failure and repair actions at a given  
Figure 5: States of \(n_{1}\) constant-speed chillers and possible transitions. period. As indicated by the arrows from left to right, the failure rate \(\lambda_{1}\) equals the probability from one state in the left to the next adjacent state in the right. As indicated by the arrows from right to left, the repair rate \(\mu\) equals the probability from one state in the right to the next adjacent state in the left. Herein, the term of \"adjacent state\" is used to describe two neighbor states. For example, \"k + 1\" state and \"k - 1\" state are two \"adjacent states\" of \"k\" state. As shown in Equation (8), the transition density matrix \(A\) represents the transition probability between any two adjacent states, which is corresponding to the failure and repair rates of constant-speed chillers. As shown in Equation (9), vector \(P(\text{t})\) represents the probabilities of various states of constant-speed chillers at time \(t\), which can be calculated by the derivation based on Equations (10) and (11). As shown in Equation (12), when the transition time accesses the infinity, the vector \(P(\infty)\) will be constant. The probabilities of steady states are calculated by Equations (13) and (14). \[A=\left[