## 2 Reliability Model of the Components in the WS
### Multi-State Markov Modeling of Desalination Plants
The clear and product tank are water storage facilities with the following operational constraints:  
\[\left\{ \begin{array}{l}h_{t}^{\rm CWP}=h_{t-1}^{\rm CWP}+(q_{t}^{\rm in}\beta -q_{\rm RO,\it J}^{\rm in})\Delta t/A^{\rm CWP}\\ \\ h_{t}^{\rm PRP}=h_{t-1}^{\rm PRP}+(q_{\rm RO,\it J}^{\rm out}-q_{t}^{\rm out}) \Delta t/A^{\rm PRP}\\ \\ h^{\rm CWP,\rm min}\leq h_{t}^{\rm CWP}\leq h^{\rm CWP,\rm max}\\ \\ h^{\rm PRP,\rm min}\leq h_{t}^{\rm PRP}\leq h^{\rm PRP,\rm max}\\ \end{array} \right. \tag{4}\]  
where \(h_{t}^{\rm CWP}\) and \(h_{t}^{\rm PRP}\) are the water storage capacity of the clear tank and product tank; \(A^{\rm CWP}\) and \(A^{\rm PRP}\) are the bottom area of the clear tank and product tank; \(h^{\rm CWP,\rm max}\) and \(h^{\rm PRP,\rm max}\) are the maximum capacity of the clear tank and product tank; and \(h^{\rm CWP,\rm min}\) and \(h^{\rm PRP,\rm min}\) are the minimum capacity of the clear tank and product tank. The multi-state reliability model can be established by analyzing the failure mechanism of the desalination plant. The details are as follows:  
The desalination plant operates regularly when all its components work as usual, as shown in Figure 2. The consumed electricity power and the produced desalinated water are calculated using Equations (1)-(4). The desalination plant cannot operate normally if the water intake pump breaks. If the clear water tank fails, the RO desalination plant cannot desalinate without feed water. Suppose the high-pressure pump fails and cannot pressurize the water; the permeability pressure forces the permeated water to flow back to the feed water side through the membrane, and the RO unit cannot work correctly. If the product tank and after-treatment process fail, the desalination plant cannot produce water. Therefore, the RO desalination plant is in a fault shutdown state when any of the above components (referred to as component 1) fail. Suppose the pre-treatment unit fails; it will not be able to perform flocculation, precipitation, filtration, and the other processes of the treatment of seawater. The concentration of seawater in the RO unit will be higher; \(\Delta\pi\) will be bigger; \((p_{r,t}-\Delta\pi)\) will be smaller, which can be derived from (2) and which represents the desalination water flow rate \(q_{r,t}^{\rm out}\) through the RO unit is smaller. So, the RO desalination plant is in inefficient operation. In addition, the failure of the pre-treatment unit leads to an increase in the concentration \(c_{r}^{\rm in}\) of the feed seawater, which results in an increase in the power \(p_{r,t}^{\rm RO}\) consumed by the high-pressure pump. Therefore, in the case of the failure of any of the components mentioned above (denoted as component 2), the RO desalination plant will enter an inefficient operation state. Suppose the booster pump and energy recovery device fail; a reduction in RO recovery \(R\) will result in a reduction in desalinated water flow \(q_{r,t}^{\text{out}}\) through the RO unit, and the desalination plant will be in a state of inefficient operation. This component is identified as component 3. In the RO unit, over time, the RO membrane will gradually age, fouling and other phenomena will occur, and the permeation efficiency will be reduced, leading to a decrease in the permeation coefficient \(K_{r}\) of the RO membrane. The desalination plant will be in an inefficient operation state. RO membrane aging and the fouling phenomenon are referred to as component 4. Over time, the RO membrane will fail due to the immense pressure difference between the two sides, decreasing the number of effective RO membranes \(r\). The high-pressure pumps must increase electricity consumption to maintain the RO unit's water supply and ensure fresh water production. The desalination plant will be in an inefficient operation state. Therefore, the failure of the RO membrane is labeled as component 5, which will result in the inefficient operation of the RO unit.