## 2 Models for Various Types of Pumped Storage
### Constraints on Reservoir Capacity of Pumped Storage  
During practical operation, PS must adhere to reservoir capacity constraints, which are addressed in this paper through electric energy balance [21]. For the PS of the daily regulation mode, to ensure the normal operation of pumped storage in the next scheduling cycle, the initial reservoir capacity and final reservoir capacity of the PS shall be equal [28]. In other words, the daily pumping and generation electric energies should be equal. \[\sum_{t=1}^{T}\sum_{n=1}^{N}P_{\text{Pump},t,n}\eta_{\text{Pump},n}=\sum_{t=1 }^{T}\sum_{n=1}^{N}P_{\text{Gen},t,n} \tag{12}\]  
where \(\eta_{\text{Pump},n}\) represents the pumping efficiency of the \(n\)-th PSU, and \(T\) represents the total time period of the day-ahead optimization scheduling. In this paper, the day is segmented into \(T=24\) periods, with each period lasting for 1 h.  
The energy in a PS is continuous and depends on the pumping and generating powers from previous moments [2]. \[E_{t+1}=E_{t}+\sum_{n=1}^{N}\left(P_{\text{Pump},t,n}\eta_{\text{Pump},n}-P_{ \text{Gen},t,n}\right) \tag{13}\]  
\[E_{1}=\beta E_{\text{max}} \tag{14}\]  
\[E_{\text{max}}=\sum_{n=1}^{N}P_{\text{Storage},n}T_{\text{Genmax}} \tag{15}\]  
where \(E_{t}\) represents the level of the electric energy of a PS at time \(t\); \(E_{1}\) represents the initial electric energy of a PS; \(E_{\text{max}}\) represents the maximum electric energy of a PS; \(\beta\) representsthe ratio of the initial electric energy \(E_{1}\) to the maximum electric energy \(E_{\max}\); and \(T_{\rm{Gemax}}\) represents the maximum power generation hours of a PS, that is, for a full water pumped storage, the maximum hours it can generate electricity if it generates electricity at its maximum power. Any type of PS must satisfy the above constraints of Equation (12) to Equation (15). At any time, the electric energy level of a PS must be between its maximum and minimum values. For pure FS-PS or pure VS-PS, the pumping electric energy must not exceed the remaining electric energy of the PS, and the generation electric energy must not exceed the available electric energy of the PS [21]. \[\sum_{n=1}^{N}P_{\rm{Pump},t,n}\eta\eta_{\rm{Pump},n}\leq E_{\max}-E_{t} \tag{16}\]  
\[\sum_{n=1}^{N}P_{\rm{Gen},t,n}\leq E_{t}-E_{\min} \tag{17}\]  
where \(E_{\min}\) represents the minimum value of the electric energy of PS, which can be taken as a value greater than 0 and less than \(E_{\max}\). For the hybrid PS that includes both FS-PSUs and VS-PSUs, the pumping electric energy and the generation electric energy also need to satisfy Equations (16) and (17) since the electric energy constraints of FS-PSUs and VS-PSUs are the same. For pure T-PS, since the PSUs can pump and generate electricity simultaneously, the upper and lower limits of the energy constraints can be expressed as the following:  
\[\sum_{n=1}^{N}\left(P_{\rm{Pump},t,n}\eta\eta_{\rm{Pump},n}-P_{\rm{Gen},t,n} \right)\leq E_{\max}-E_{t} \tag{18}\]  
\[\sum_{n=1}^{N}\left(P_{\rm{Gen},t,n}-P_{\rm{Pump},t,n}\eta_{\rm{Pump},n} \right)\leq E_{t}-E_{\min} \tag{19}\]  
For the hybrid PS that includes both FS-PSUs and T-PSUs, the electric energy must be constrained separately.