## 2 Reliability Model of the Components in the WS
### Multi-State Markov Modeling of Desalination Plants  
The RO desalination method is the simplest, most economical, and most efficient of the existing desalination technologies, and its popularity has been increasing in recent years [16; 17]. In this paper, the RO desalination method is taken as an example to model the desalination plant. The RO desalination system uses the selective permeability of semi-permeable membranes to realize seawater desalination. The basic process flow is shown in Figure 2. The feed seawater is firstly sent into the pre-treatment unit by the intake pump, and it then enters into the clean water tank after flocculation, sedimentation, filtration, etc. The treated seawater is then pressurized by the high-pressure pump of the RO desalination device to force the water molecules to enter into the fresh water side through semi-permeable membranes to realize the separation of salt water. Finally, the product water is formed after water quality adjustment and other after-treatment processes. The water intake pump is one of the major energy consuming devices in RO desalination plants [2]. Assuming that the hydrostatic head of the water intake pump is fixed, the operating constraint is as follows:  
\[\left\{ \begin{array}{l}P_{t}^{\text{in}}=\frac{\rho g}{\eta^{\text{in}}}q_{t} ^{\text{in}}H_{t}^{\text{in}}\\ q^{\text{in,min}}\phi_{t}^{\text{in}}\leq q_{t}^{\text{in}}\leq q^{\text{in, max}}\phi_{t}^{\text{in}}\end{array} \right. \tag{1}\]  
where \(P_{t}^{\text{in}}\), \(q_{t}^{\text{in}}\), and \(H_{t}^{\text{in}}\) represent the power consumption, water flow, and hydrostatic head of the water intake pump at time \(t\), respectively; \(\rho\) is the density of seawater; \(g\) is the acceleration of gravity; and \(\eta^{\text{in}}\) is the efficiency of the pump. In this paper, \(\phi\) always represents the working state of the equipment, which is a 0-1 variable (take 1 to indicate the operation), and will not be repeated below. The RO device is composed of high-pressure pumps, booster pumps, energy recovery devices, and several RO membranes. The inlet and outlet volumes of the RO units and the power they consume satisfy the following (2):  
\[\left\{ \begin{array}{l}q_{r,t}^{\text{out}}=K_{r}A_{r}(p_{r,t}-\Delta\pi)\\ q_{r,t}^{\text{in}}=q_{r,t}^{\text{out}}/R\\ P_{r,t}^{\text{RO}}=q_{r,t}^{\text{out}}[a_{r}c_{r}^{\text{in}}\frac{2-R}{2(1 -R)}+b_{r}(p_{r,t}-\Delta\pi)]\\ \Delta p_{r}^{\text{min}}\phi_{r,t}\leq p_{r,t}-\Delta\pi\leq\Delta p_{r}^{ \text{max}}\phi_{r,t}\end{array} \right. \tag{2}\]  
Figure 2: Process flow diagram of RO desalination plant. where \(q_{r,t}^{\rm in}\) and \(q_{r,t}^{\rm out}\) are the water inlet and outlet of the RO unit; \(P_{r,t}^{\rm RO}\) is the power consumed by the RO unit; \(p_{r,t}\) is the working pressure of the RO unit; \(K_{r}\), \(A_{r}\), and \(\Delta\pi\) are the permeability coefficients of the osmosis membrane, the permeable area, and the difference in permeability pressure across the surface area of the membrane, respectively; and \(R\) is the recovery rate of the RO. Assuming that the number of RO membranes of the RO unit is \(r\), the total water inlet and outlet of the RO unit, the power consumed by the high-pressure pump, as well as the booster pump, satisfy the following relationship:  
\[\left\{ \begin{array}{l}q_{\rm RO,\it J}^{\rm out}=\sum\limits_{r\in\rm RO}q_ {r,t}^{\rm out}\\ \\ q_{\rm RO,\it J}^{\rm in}=\sum\limits_{r\in\rm RO}q_{r,t}^{\rm in}\\ \\ p_{\rm RO,\it J}^{\rm RO}=\sum\limits_{r\in\rm RO}P_{r,t}^{\rm RO}\\ \\ q_{\rm RO,\it J}^{\rm in}\leq q_{t}^{\rm in}\beta\end{array} \right. \tag{3}\]  
where \(q_{\rm RO,\it J}^{\rm in}\), \(q_{\rm RO,\it J}^{\rm out}\) and \(P_{\rm RO,\it J}^{\rm RO}\) represent the total inlet water volume of the RO unit, the total outlet water volume, and the total power consumed by the high-pressure pump and the booster pump, respectively; \(\beta\) refers to the ratio of the amount of seawater after pre-treatment to the amount of feed water.