## 2 Materials and Methods
### Method for the Non-Stationary Hydrological Time Series Analysis and Recommendations
#### 2.3.2 Mathematical Model and Algorithm for Performing WRC  
Let us introduce the following notation:  
_i_--the number of the VKC reservoirs (_i_ = 1 \(\div\) 9);  
_j_--the number of seasons in a year: high water from April to June, SLW from July to November, WLW + FFD from December to March (_j_ = 1 \(\div\) 3);  
_\(\tau_{j}\)_--the number of days in a season (_\(\tau_{1}\)_ = 91; \(\tau_{2}\) = 153; \(\tau_{3}\) = 121);  
_T_--the number of years in the calculated long-term hydrological series, \(T\) = 105, from 1916 to 2020;  
_t_--water management year (water management year--from April to March of the next year) in the hydrological series (_t_ = 1 \(\div\) 106);  
_W\({}^{\text{NOL}}\)_, W\({}^{\text{IL}}\)_--the total volume of reservoirs of the VKC at normal operating level (NOL) and inactive level (IL), km\({}^{3}\);  
_W\({}^{\text{NOL}}_{i}\)_--the volume of the _i_-th reservoir of the VKC (_i_ = 1 \(\div\) 9) at NOL and IL, km\({}^{3}\). For each water management year t:  
_W\({}^{1}\)_--the total volume of water in the VKC reservoirs at the beginning of the flood after the pre-flood drawdown, km\({}^{3}\) (_W\({}^{1}\)\({}_{106}\)_ is the volume of the VKC after the pre-flood drawdown in 105; _W\({}^{1}\)\({}_{1}\)\({}_{1}\)_ = _W\({}^{1}\)\({}_{106}\)_ = 10,562 km\({}^{3}\));  
_B\({}^{1}\)_--the total volume of inflow to the VKC reservoirs during the flood, km\({}^{3}\);  
_R\({}^{1}\)_--the volume of release into the downstream of the Volgograd reservoir (outlet of the VKC) during HW, km\({}^{3}\);  
_W\({}^{2}\)_--the total volume of water in the VKC reservoirs at the beginning of the SLW, km\({}^{3}\);  
_B\({}^{2}\)_--the total volume of inflow to the reservoirs of the VKC during SLW, km\({}^{3}\);  
_R\({}^{2}\)_--the total volume of release into the downstream of the Volgograd reservoir (outlet of the VKC) during SLW, km\({}^{3}\);  
_Q\({}^{2}\)_--the average release discharge from the Volgograd reservoir (_Q\({}^{2}\)\({}_{t}\)_ = _R\({}^{2}\)\({}_{t}\)_ \(\times\) 1,000,000,000/86,400 \(\times\)\(\tau_{2}\)), m\({}^{3}\)/s;  
_W\({}^{3}\)_--the total volume of water in the reservoirs of the VKC at the beginning of the WLW, km\({}^{3}\);  
_W\({}^{3}\)_,t_--the volume of water in the _i_-th VKC reservoir (_i_ = 1 \(\div\) 9) at the beginning of the WLW, km\({}^{3}\);  
_Z\({}^{3}\)\({}_{h\gg,t,t}\)_--the level of the upstream (US) of the _i_-th VKC reservoir (_i_ = 1 \(\div\) 9) at the beginning of the WLW, m;  
_Z\({}^{3}\)\({}_{fin,t,t}\)_--the level of the upstream (US) of the _i_-th VKC reservoir (_i_ = 1 \(\div\) 9) at the end of the WLW, m;  
_B\({}^{3}\)_--the total volume of inflow to the VKC reservoirs during the WLW + FFD, km\({}^{3}\);  
_B\({}^{3}\)_,t_--the volume of inflow to the _i_-th WKC reservoir during the WLW + FFD (_B\({}_{3}\)_ = \(\Sigma\)B\({}_{3,i}\)), km\({}^{3}\);  
_R\({}^{3}\)_,t_--the volume of release into the downstream (DS) of the _i_-th VKC reservoir (_i_ = 1 \(\div\) 9) during the WLW + FFD, km\({}^{3}\);  
_Q\({}^{3}\)_,t_--the average discharge of the release to the DS of the Volgograd reservoir, during the WLW + FFD (_Q\({}_{3,i}\)_ = _R\({}_{3,i}\)_ \(\times\) 1,000,000,000/86,400 \(\times\)\(\tau_{3}\)), m\({}^{3}\)/c;  
_H\({}^{3}\)\({}_{hgg,t,i}\)_--level of the i-th VKC reservoir DS, considering the backwater from the downstream reservoir (_i_ = 1 \(\div\) 9) at the beginning of the WLW, m;  
_H\({}^{3}\)\({}_{fin,t,i}\)_--level of the _i_-th VKC reservoir DS, taking into account the backwater from the downstream reservoir (_i_ = 1 \(\div\) 9) at the end of the PFD, m;  
_N\({}^{3}\)\({}_{hgg,t,i}\)_--the HPP capacity of the _i_-th VKC reservoir (_i_ = 1 \(\div\) 9) at the beginning of the WLW, MW;  
_N\({}^{3}\)\({}_{fin,t,i}\)_--the HPP capacity of the _i_-th VKC reservoir (_i_ = 1 \(\div\) 9) at the end of the PFD, MW.