## 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
Thus, the objective function (OF) is presented in the following form:  
\[2.\ \text{SLW:}\ W^{3}_{\ t}=W^{2}_{\ t}+B^{2}_{\ t}-R^{2}_{\ t}; \tag{7}\]4  
where \(\mu_{j}\) are penalty coefficients that determine the hierarchy of priorities of water users' requirements, \(t=1\div T\). Criteria \(C_{j,t}\) are generally defined by the following formula:  
\[2.\ \text{SLW:}\ W^{3}_{\ t}=W^{2}_{\ t}+B^{2}_{\ t}-R^{2}_{\ t}; \tag{7}\]5  
where \(X_{j,t}\) is the desired value of the requirement, determined by the independent variables \(R^{1}t\), \(R^{2}t\), and \(W^{1}t\); \(T_{j}\) is the threshold value. The optimization approach used during water resource calculation (WRC) is to search for the \"best\" releases from the VKC reservoirs, which ensures the maximum satisfaction of the hierarchically ordered requirements of water users. Optimization calculations were carried out for two inflow series: 1916-1978 and 1979-2020. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**N**} & **Criterion Content** & **Threshold** & **Reliability, \%** \\ \hline \multirow{4}{*}{**1.**} & The volume of the VKC reservoirs should not exceed the volume at NOL (forbidden forcing), \(\text{W}^{\text{NOL}}\), \(\text{km}^{3}\) & \multirow{2}{*}{166, 74} & \multirow{2}{*}{100} \\ \cline{3-3} \cline{5-5}  & By the end of the flood, the volume of the VKC reservoirs should be close to the volume at the NOL (be within the specified range), \(\text{W}^{\text{NOL}}\), \(\text{km}^{3}\) & & \\ \hline \multirow{4}{*}{**2.**} & By the end of the flood, the volume of the VKC reservoirs should be close to the volume at the NOL (be within the specified range), \(\text{W}^{\text{NOL}}\), \(\text{km}^{3}\) & & \\ \hline \multirow{4}{*}{**3.**} & Total winter HPP capacity of VKC reservoirs, MW & \multirow{2}{*}{2945} & \multirow{2}{*}{90} \\ \cline{3-3} \cline{5-5}  & Normal transport release from the Volgograd HPP during the navigation period (high water, SLW), \(\text{m}^{3}/\text{s}\) & & \\ \hline \multirow{4}{*}{**5.**} & Reduced transport release, \(\text{m}^{3}/\text{s}\) & \multirow{2}{*}{4000} & \multirow{2}{*}{100} \\ \cline{3-3} \cline{5-5}  & Normal fishery (agricultural) & & \\ \cline{3-3} \cline{5-5}  & release from the Volgograd HPP, \(\text{km}^{3}\) & & \\ \hline \multirow{4}{*}{**7.**} & Reduced fishery (agricultural) & \multirow{2}{*}{110} & \multirow{2}{*}{75} \\ \cline{3-3} \cline{5-5}  & release from the Volgograd HPP, \(\text{km}^{3}\) & & \\ \hline \multirow{4}{*}{**8.**} & Minimum ecological release ensuring the safety of the Lower Volga ecosystem, \(\text{km}^{3}\) & \multirow{2}{*}{60} & \multirow{2}{*}{100} \\ \cline{3-3} \cline{5-5}  & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Requirements of water users of the VKC water management system. The optimization problem for a given inflow series is formulated as follows:  
Determine the positive values of the variables \(R^{1}{}_{t}\), \(R^{2}{}_{t}\), and \(W^{1}{}_{t}\), minimizing the OF (20) with balance constraints (6)-(19) and additional constraints on the variables: min (\(W^{1}{}_{t}\)) \(>\) 10,562 km\({}^{3}\); min (\(W^{3}{}_{t}\)) \(>\)\(W^{IL}\); min (\(W^{3}{}_{t}\),9) \(>\) 1000 m\({}^{3}\)/s (sanitary winter); min (\(W^{3}{}_{t}\),4) \(>\) 600 m\({}^{3}\)/s (sanitary winter); max (\(W^{1}{}_{t}\)) \(<\) 145 km\({}^{3}\) (maximum volume of reservoirs of the VKC by the beginning of the high water); \(W^{1}{}_{t}\) = \(W^{1}{}_{T+1}\) = 105,62 km\({}^{3}\). To perform optimization calculations, in addition to the data given in Section 2.1 (Figure 4), a series of inflows for winter low water and pre-flood drawdown for each reservoir were used (Figure 5). Optimization was performed in the Microsoft Excel software with the Solver Add-in (the dimension of the problem is less than 200 independent variables), using the generalized reduced gradient methods. The results of WRCs and their discussion for two series are given in Section 3.3.