Statistical and Water Management Assessment of the Impact of Climate Change in the Reservoir Basin of the Volga–Kama Cascade on the Environmental Safety of the Lower Volga Ecosystem

Abstract

When managing water resources in order to provide water to consumers, a number of consequences arise related to the violation of the hydrological regime due to the regulation of flow by reservoirs. The second factor is possible climate change. These changes can negatively (or positively) affect the functioning of aquatic ecosystems. To reduce the impact on the environment, it is necessary to determine the nature and indicators of changes in the hydrological regime, calculate quantitative estimates of these indicators and ranges of acceptable values, and develop release rules that ensure compliance with these ranges with a given probability. To manage the water resources of the Volga and Kama Rivers, the main ecological task is to flood the floodplain meadows, to maintain the conditions of natural reproduction of fish on the Lower Volga, including the Volga River delta and the Volga–Akhtuba floodplain. In addition, it is necessary to meet with sufficient reliability the requirements of energy in the summer–autumn and winter low-water periods and water transport during the navigation period. The task of optimal management is to find such solutions in years of different water content that ensure the well-being of the main water users with a given probability and do not disturb the Lower Volga ecosystem. This article presents the research of the water resources state of the water resource system of the Volga and Kama river basins. A statistical analysis of the hydrological series of the observed inflow for 1916–2020 was performed, and the inflow change point (1979) was found by the Bayesian method of estimation. A statistically significant difference between the average inflow values of two series (1916–1978, 1979–2020) was proved using a two-sample Student’s test. The seasonal parameters of the reliability curves were calculated based on the three-parameter Kritsky and Menkel distribution. For these two series, water resource optimization calculations (using Excel Solver) were performed, and the reliability of fulfilling the requirements of water users was determined; for the series 1916–1978, an alternative solution was found in favor of fisheries, and an analysis of the results was also performed. The methodology used in the research allows finding trade-off solutions in the favor of different water users (ecology, agriculture and fisheries, water supply, hydropower, navigation, etc.) and is based on the use of multi-criteria optimization methods and the trade-offs theory. As a result of the research, new knowledge was obtained about the hydrological situation in the basin of the Volga–Kama reservoir cascade in connection with climate change.

1. Introduction

Since the beginning of the 21st century, the attention of many researchers has been increasingly attracted to the problems of climate change and the associated variability of water resources.

Understanding the relationship between climate change and the water content of a particular basin helps to plan the use of water resources more rationally. One of the methods of studying possible climate changes and related consequences is the analysis of river runoff in order to detect long-term patterns and trends caused by these changes. It is assumed that the trends in runoff changes identified as a result of the analysis can be somehow related to climate changes in the corresponding region.

Since the authors do not have much confidence in the results of climate change forecasts for various scenarios such as SSP585 (global models, MPI-ESM1–2-LR, HadGEM3-GC31-LL, CNRM-CM6-1, etc.), it was decided to analyze the long-term series of observed inflows (not a forecast) to determine the year when a possible change occurred, substantiate this event with statistical methods, and determine the quantitative indicators of the changes that have occurred and how they affected the water management situation in the river basin. Such an approach to research is given in [1,2,3,4,5].

An important task in this approach is the adequate identification of these trends in long-term hydrological series.

There are quite a lot of works devoted to the study of these problems in the basins of various rivers, in which a noticeable change in water content and its intra-annual redistribution have been observed in recent decades [6,7,8,9,10,11,12].

The Heihe River basin runoff analysis from 1958 to 2017 was carried out in [6]. The authors identified three intervals for this period—1958–1980, 1981–2003, and 2004–2017. In the first interval, the annual runoff showed a downward trend; in the second interval, fluctuations were observed in the annual runoff relative to a certain value; in the third interval, a clear upward trend was observed. The authors attributed these changes mainly to climate change.

The Weihe River basin runoff analysis from 1954 to 2014 was carried out in [7]. As in [11], the authors also identified three intervals—1954–1971, 1972–1993, and 1994–2014. However, in contrast to the Heihe River, for the Weikhe River, the annual runoff tended to increase in the first interval and decrease in the third. Fluctuations in the annual runoff relative to a certain value were also observed in the second interval.

Changes in the runoff of the Volga and the rivers of its basin were studied in [8,9,10,11]. It was found in [8,9] that long-term fluctuations in the Volga runoff contain two periods, which differ significantly in long-term average values. This difference is mainly due to changes in low-water runoff. The authors of [10] directly link the long-term variability of the Volga runoff with climate change.

In [11], the main trends in the annual and low-water runoff changes of the Volga basin rivers and the nature of the intra-annual distribution of the runoff were studied.

The authors emphasize that the “synchronism” of changes in low-water runoff and the scale of these changes are extraordinary and have no analogs in the 20th century. The increase in low-water runoff has led to an increase in water resources over the past 30–35 years, even in river basins where there has been a decrease in spring flood runoff.

The runoff of another large river in the European part of Russia was studied in [12]. The work also revealed the intra-annual redistribution of the river runoff of the Upper Don. It is noted that this is typical for modern climatic conditions, which are expressed by an increase in both the average annual air temperature and the air temperature of the cold period of the year, as well as an increase in the number of winter thaws.

In many rivers of Russia and the former USSR, disruptions of the hydrological regime stationarity are observed; they are mainly due to climatic changes and are of a complex nature [13,14,15]. There is a significant increase in the minimum runoff value in the Volga River basin, a noticeable decrease in the maximum water discharge values during spring floods in the Don River basin, and changes in the characteristics of the Central Asia rivers. An important conclusion for hydrological applications is that the stochastic model of the temporal variability of the studied runoff characteristic is no longer stationary and must be more complicated.

The hydrological system behavior in response to runoff changes is highly non-linear. The “threshold” behavior of the process appears as a result of a complex mechanism, the study of which is complicated by a significant lack of data on the conditions of surface runoff, moisture transfer in the unsaturated zone, and geofiltration processes and the uncertainty of future climate estimates. At the same time, applications require a fairly simple model that allows one to estimate the probabilities of a transition of a hydrological system to a new state.

The main hypothesis, as applied to the problem of substantiating a stochastic model in non-stationary conditions, is to describe the observed runoff fluctuations in the framework of semi-Markov models of random processes. In this case, in the observed series of runoff values, stationary sections are distinguished, within which fluctuations in water content are described by a model of a stationary random process (for example, a Markov one). The transition from one stationary state to another stationary state in this model depends only on the previous state; i.e., the state transition matrix that controls the process of changing states is Markovian.

In the static processing of sequences of this kind, the problem arises of determining the points in the time series at which the process state changes. To obtain statistically significant conclusions about the location of these points on the time axis, approaches based on the Fisher and Student homogeneity criteria, as well as on the analysis of difference integral curves (DICs) can be applied. In the case of the DIC analysis, it is possible to identify points corresponding to the change in high-water and low-water phases of water content, and the use of statistical criteria allows us to accept or reject the hypothesis of stationarity with respect to the averages or variances for the selected sections of the time series.

The approach used in the work is based on a statistical analysis of long-term hydrological series of observed inflows to the reservoirs of the Volga–Kama cascade (VKC), divided into four seasons: high water (April–June), summer–autumn low water (July–November), winter low water (December–February), pre-flood drawdown (March). The calculations use the water management year (from April of the current year to March of the next year).

The analysis of hydrological series was performed for the Volga–Kama cascade (VKC) as a whole and separately for eight out of nine large reservoirs of the cascade: Rybinskoye¹, Gorkovskoye², Cheboksarskoye³, Kuibyshevskoye⁷, Saratovskoye⁸, and Volgogradskoye⁹ on the Volga River, Kamskoye⁴ and Nizhnekamskoye⁶ on the Kama River (upper index is the reservoir number in the calculation scheme, Figure 1). The Votkinskoye⁵ reservoir was not considered due to the lack of significant inflow. The total useful volume of the nine VKC reservoirs is about 78 km³. The parameters of reservoirs are taken from the Basic Rules for the Use of Water Resources of Reservoirs on the Volga and Kama Rivers [16,17].

Climate change can significantly affect the water management situation in a river basin with cascade river flow management. In accordance with the Russian regulatory documents in the field of water relations, the parameters of hydraulic structures (HSs) during design, reconstruction, and operation significantly depend on the reliability of water resources in the basin. The tool for assessing the reliability of water is the water resources calculation (WRC).

WRCs are carried out in order to assess the amount and degree of development of water resources available for use under various conditions of water content of water bodies. WRCs are carried out in the design and reconstruction of hydraulic structures, preparation of a water management justification for the schemes for the integrated use and protection of water bodies (SKIOVO) [18], and development of the rules for the use of reservoirs water resources (PIVR) [19]. WRCs determine the ratio of available water resources (volumes of surface and groundwater available for long-term guaranteed use with the existing and projected composition of the water management complex) and estimated water consumption at the predicted level of economic development. Based on the WRCs, the sufficiency of water resources is assessed to meet the volumes established for water users’ allowable water resources withdrawal and the possibility of developing the water management complex in the near, planned, and long term.

In engineering hydrology, the concept of “reliability of a hydrological characteristic” is defined as the probability that the considered value of a hydrological characteristic can be exceeded among the totality of all its possible values. As an example, the set of values can be annual, seasonal, or monthly inflow in a long-term series [20].

The reliability of water yield in water resource calculations is understood as the probability of providing the consumer with water at the appropriate rate, expressed as a percentage of uninterrupted years, seasons, and months of water supply, depending on the duration of the entire calculation period [19].

Calculation of hydraulic structure parameters for water users of various sectors of the economy is carried out on the basis of the year of the corresponding reliability.

Table 1. Normative reliability levels for various sectors of the economy.

N	Sectors of the Economy	Years of the Normative Reliability Level by Runoff
1	Fisheries	75%
2	Agriculture (irrigation)	75%
3	Public utilities	95–99%
4	Sanitary release	95–99%
5	River transport	85–95%
6	Winter energy (capacity, MW)	90–95%
7	Dam parameters (headwater level—FCL, NOL, IL) depending on the HS class:	0.1–5%
8	1st class, 2nd class, 3rd class, 4th class	0.1%, 1%, 3%,5%
9	Flooding in the upstream and downstream of reservoirs *	0.01–5%
10	Limits and quotas for water intake and wastewater discharge	25%, 50%, 75%, 95%
11	Measures without considering the harmful effects of water (HEW)	25%, 50%, 75%, 95%
12	Measures to reduce HEW	0.01%, 1%, 5%, 10%

* High water 5% probability is the upper limit for non-floodplain flooding (floodplain should not be built up in the zone of 5% flood).

shows the normative flow rates (annual, seasonal, interval), which are used to substantiate water use indicators for various sectors of the economy.

The main indicator of the reliability of providing water users is the indicator of the calculated water yield. The estimated water yield reliability is defined as the probability of providing the consumer with water, expressed as a percentage of years without violation (seasons, months, decades), and is calculated by the formula m/(n + 1) where m is the number of years without violation of the water user requirements and n is the number of years in the long-term runoff series.

When developing SKIOVO and PIVR for water users of various economic sectors, the following standard reliability for ensuring normal water consumption by the number of years without violation should be met:

• Sanitary releases—97–99%;
• Water supply (drinking, household, industrial)—95–97%;
• Hydropower—85–95%;
• Navigation (maintenance of depths)—85–90%;
• Irrigation and agriculture—75–90%;
• Fisheries—75–90%.

The work is dedicated to the uninterrupted operation of the following main water VKC reservoirs users: energy, transport, agriculture and fisheries, ecology, and household services.

WRCs were performed by optimization methods for both runoff series—1916–1978 and 1979–2020. The management of VKC reservoirs in water resource calculations was performed on the basis of a search for optimal releases in total for the entire Volga–Kama cascade of reservoirs (release in the outlet—Volgogradskaya HPP) in different seasons and the total volumes of reservoir drawdown by the beginning of the flood.

The requirements of water users for the VKC reservoir operation mode conflict with each other. All requirements are formulated in relation to the accepted annual division into seasons in total for all VKC reservoirs. In winter, calculations were performed for each reservoir separately.

It is assumed that within the season the decision support team that forms the VKC reservoir operating modes (in Russia this is the Interdepartmental Working Group (IWG)), while satisfying the total seasonal requirements of water users over the cascade, is able to successfully satisfy the requirements for each reservoir on a daily basis.

The main objective of this work was to study the long-term (105 years, 1916–2020) hydrological series of annual and season inflow in the VKC basin, to identify statistically significant deviations of the flow from stationarity within this period, and to determine the reliability of water resources for the main water users in a changing climate.

To achieve this goal, the problems outlined in Figure 2 were solved.

2. Materials and Methods

2.1. Initial Data of Observed Inflow and Processing Methods

The authors of this work studied long-term hydrological series of observed inflow to the Volga–Kama cascade (VKC) reservoirs for the period from 1916 to 2020 with a total duration of 105 years.

The series of annual and seasonal inflow—high water (HW), summer–autumn low water (SLW), winter low water (WLW), and pre-flood drawdown (PFD) (decrease in storage) —were studied. Graphs of inflow data for the specified period for the Kuibyshevskoe reservoir and the cascade as a whole are shown in Figure 3 and Figure 4, respectively. The figures show linear trends and their equations for each graph.

For the hydrological series preliminary assessment and obtaining information about the central tendency and variability of the initial data for the cascade as a whole and for each reservoir, primary statistical analysis of the data was performed, both for annual inflow values and for its seasonal components. The existence of statistical dependencies in the intra-annual runoff structure was verified by calculating the pairwise correlation coefficients of the annual inflow and its seasonal components.

The method of difference integral curves [11,12,20] was used to identify patterns of fluctuations in seasonal runoff over long periods of time. The method is a modification of the cumulative anomaly curve method [6,7]. It is based on the calculation of the accumulated relative deviations of the runoff value from the average and makes it possible to single out periods in the general time series with runoff values close to the average annual runoff, greater or less than the average.

To construct difference integral curves (DICs), the value $f_n$ is calculated by the following formula:

$$f_n={\sum }_{i=1}^n\frac{W_i-\stackrel{-}{W}}{\stackrel{-}{W}}$$

(1)

where W_i is the runoff in the i-th interval of the period,$\stackrel{-}{W}$ is the average value of the runoff over the entire period, and f_n is the value of DIC in the n-th interval of the period.

Obviously, the average value $\frac{W_i-\stackrel{-}{W}}{\stackrel{-}{W}}$ for a period equal to the length of the time series must be equal to 0; therefore, f_n must take zero values at the beginning and end of the period on which the time series is considered. Sections of the integral difference curves with a positive derivative correspond to periods with runoff values greater than the average value for the entire hydrological series, and sections with a negative derivative correspond to periods with a runoff below the average. In areas with a derivative close to zero, the runoff slightly differs from the average value for the entire hydrological series.

2.2. Method for the Non-Stationary Hydrological Time Series Analysis and Recommendations

2.2.1. Estimation of the Change Point in Sequences of Random Variables of River Runoff

The main task of the non-stationary time series analysis is the task of detecting the point of change (CPD). To estimate the state change point, it is necessary to consider this point as a parameter of a non-stationary stochastic model and apply one of the estimation methods, for example, the Bayesian method.

Let us consider as an example one of the most common cases when the parameters of probabilistic models for homogeneous (stationary) sections are known. There is a sequence of random variables x₁, x₂,…, x_n, which is divided into two parts at the point r (1 ≤ r ≤ n), and x_i1 is distributed according to the law F₁ (x∕Θ₁),i = 1,∙ ∙ ∙, r, and x_i2 is distributed according to the law F₂ (x∕Θ₂), i = r + 1,…, n, and F₁ (x∕Θ₁) ≠ F₂ (x∕Θ₂).

Let us consider the case, following [21], when the distribution functions F₁ and F₂ are known, and it is necessary to draw a conclusion about the point of change in the state of the process.

Having set the a priori density ${\rho }_0\left(r\right)$ as a uniform distribution, we determine the a posteriori distribution density of a possible change point for the available observational data x₁,…, x_n based on the general formula for the total probability (Bayes formula):

$$\rho \left(\Theta \right|x)=\frac{\rho \left(\mathit{x}|\Theta \right)·\rho \left(\Theta \right)}{\tilde{\rho }\left(\mathit{x}\right)}$$

(2)

where ρ(x|Θ) is the likelihood of data x for a given value of the parameters Θ₁ and Θ₂. The method as applied to fluctuations in river flow is considered in detail in [22,23,24].

2.2.2. Bayesian Method for Determining the Design Characteristics of Runoff in Non-Stationary Conditions

To calculate runoff characteristics in the presence of violations of the homogeneity of time series due to climate change, the time series is divided into several homogeneous parts, each of which is characterized by its own set of parameters. The date of runoff transition from one homogeneous state to another is determined by the methods described in Section 2.2.1.

The determination of the design characteristics of river runoff in non-stationary conditions is based on the construction of a distribution law in the form of a sum of several distribution laws, weighted by the duration of conditionally stationary regimes. The calculated distribution curve for the entire time series is constructed as the sum of distribution laws with weights proportional to the sample lengths. The method is known as “composite distribution curves”.

To determine the calculated values for each of the homogeneous parts of the series, the distribution curve P_i(x) is determined. The composite distribution curve is calculated by the formula:

$$P\left(x\right)=1.0-{(1.0-P}_1\left(x\right))\frac{n_1}{N}-(1.0-P_2\left(x\right))\frac{n_2}{N}-(1.0-P_3\left(x\right))\frac{n_3}{N},$$

(3)

where N is the length of the entire time series and n_i are the lengths of the i-th homogeneous parts of the series.

To obtain a predictive runoff distribution, the Bayesian methodology is allowed to be used for solving statistical problems for two or three conditionally stationary periods. In contrast to the method of “composite distribution curves”, where the final solution considers only the weight coefficients of individual homogeneous populations, the Bayesian solution additionally considers the errors in sample estimates of distribution parameters.

With the Bayesian approach, for each of the homogeneous parts of the series, the parameter θ is determined, which can be considered as the mean value, variance, and asymmetry, as well as the relationship between them. The parameter θ is a random variable since it is calculated from a series y (sample) of limited length. The distribution of the parameter φ (θ, y) is called sample distribution and is characterized by parameters that have the meaning of errors.

The probability with which the process can be in one of the homogeneous states is assumed to be proportional to the duration of the corresponding period $\frac{n_1}{N}\mathrm{a}\mathrm{n}\mathrm{d}\frac{n_2}{N}$(if there are two such states), where $N=n_1+n_2$ is the total duration of the heterogeneous sample of observed runoff values. The distribution of the parameter p(θ) is the sum of two sample distributions of the parameter for homogeneous parts of the sample:

$$p\left(\theta |y\right)=\frac{n_1}{N}{\phi }_1\left(\theta ,y\right)+\frac{n_2}{N}{\phi }_2\left(\theta ,y\right),$$

(4)

where ${\phi }_i\left(\theta ,y\right)$ is the sample distribution of the parameter θ for the i-th observed conditionally stationary period of length $n_i$, $i=1,2$ with probabilities n_i/N. The sampling distribution of the parameter θ in most cases is a normal distribution and is given by two parameters: the mean value of the parameter and its standard deviation (error).

Determination of the calculated characteristics of the desired value x in non-stationary conditions within the framework of the Bayesian approach consists in calculating the predictive density $f\left(x|y\right)$, the probability distribution of the random variable x based on the total probability formula [20,25,26]:

$$f\left(x|y\right)=\int g(x│\theta ,y)\left|p\right(\theta │y)d\theta ,$$

(5)

where y is the available data (observed and predicted); θ is a parameter characterizing an inhomogeneous random process; g(x│θ, y) is the initial probabilistic runoff model suitable for describing runoff in a homogeneous period, for example, Pearson type III or Kritsky and Menkel distribution; and p(θ│y) is the distribution of the parameter θ, specified as a mixture of sample distributions of the parameter θ for each conditionally stationary state, weighted by the probabilities of occurrence of these states in the future (Formula (4)).

2.3. Water Resource Calculation (WRC) for a Given Inflow Series by the Optimization Method

2.3.1. Meaningful Statement of the Problem

The reliability for water users for two series (1916–1978 and 1979–2020) of inflow to the VKC reservoirs is calculated on the basis of the constructed mathematical model of WRCs by seasons (HW, SLW, WLW, PFD) using optimization methods. The mathematical model uses water balance equations: the volume of water in the reservoirs of the VKC at the end of the estimated season is equal to the volume of water at the beginning of the season plus the volume of inflow minus the volume of release from the Volgogradskaya HPP (outlet gate) [27,28]. The variables of the optimization model are the releases for the flood period, the releases for the summer–autumn low-water period, and the volume of water in the VKC reservoirs after the pre-flood drawdown (by the beginning of the flood—1 April). The sum with weight coefficients of normalized squares of deviations from the requirements of water users was used as the objective function (OF). Weight coefficients determine the significance of each water user in the process of functioning of the water management system (WMS). As weight coefficients for each group of equal water users, values are taken that differ from each other by an order of magnitude, which ensures the optimization principle: first, a solution is sought for the group with the highest priority, then the remaining resources are used to meet the requirements of water users with a lower priority, etc. At the same time, the optimal value of the objective function does not make any economic sense, it only guarantees, with the help of penalty factors, the fulfillment of the hierarchically ordered priorities of water users’ requirements in the optimal solution.

The following computational scheme is used:

• High water. By the beginning of the flood (after the pre-flood drawdown), the volume of water in the VKC reservoirs is added to the volume of inflow during the flood, and the release (variable) from the Volgogradskaya HPP is subtracted.
• SLW. To the volume of water in the reservoirs of the VKC after the flood, the volume of inflow during the SLW is added and the release from the Volgogradskaya HPP is subtracted.
• WLW + PFD. To the volume of water in the reservoirs of the VKC after SLW, the volume of inflow is added and the release during the WLW and PFD is subtracted. The resulting volume (variable) of filling the water reservoirs after the pre-flood drawdown is the initial volume for the flood season. The initial filling volume of the VKC in the first year of calculation and the volume after the calculations in the last year are constant and equal to 105.62 km³ (this is the volume of the VKC reservoirs after the maximum permitted pre-flood drawdown).
  The criteria for optimization are the following requirements of water users:
• The volume of release from the Volgograd hydroelectric power station (HPP) during the flood should have the following reliability:

  -

  Normal fishery release with a volume of 120 km^3—50%;

  -

  Reduced fishery release with a volume of 90 km^3—75%;

  -

  Sanitary release with a volume of 60 km^3—100%.

• The volume of accumulation of VKC reservoirs by the end of the flood should be in the range of 162–166 km³ (volume of VKC reservoirs at NOL) with 100% security.
• Water consumption in the downstream (DS) of the Volgogradskaya HPP in the interests of water transport during the navigation period must be ensured:

  -

  Normal transport release with a discharge of 5000 m³/s—90%;

  -

  Reduced transport release with a discharge of 4000 m³/s—100%.

• The total capacity of VKC HPP during the winter low water should be at least 2950 MW and have a reliability of at least 90%.

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 ÷ 9);

j—the number of seasons in a year: high water from April to June, SLW from July to November, WLW + PFD from December to March (j = 1 ÷ 3);

τ_j—the number of days in a season (τ₁ = 91; τ₂ = 153; τ₃ = 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 ÷ 106);

W^NOL, W^IL—the total volume of reservoirs of the VKC at normal operating level (NOL) and inactive level (IL), km³;

W_i^NOL, W_i^IL—the volume of the i-th reservoir of the VKC (i = 1 ÷ 9) at NOL and IL, km³.

For each water management year t:

W¹_t—the total volume of water in the VKC reservoirs at the beginning of the flood after the pre-flood drawdown, km³ (W¹₁₀₆ is the volume of the VKC after the pre-flood drawdown in 105; W¹₁ = W¹₁₀₆ = 10,562 km³);

B¹_t—the total volume of inflow to the VKC reservoirs during the flood, km³;

R¹_t—the volume of release into the downstream of the Volgograd reservoir (outlet of the VKC) during HW, km³;

W²_t—the total volume of water in the VKC reservoirs at the beginning of the SLW, km³;

B²_t—the total volume of inflow to the reservoirs of the VKC during SLW, km³;

R²_t—the total volume of release into the downstream of the Volgograd reservoir (outlet of the VKC) during SLW, km³;

Q²_t—the average release discharge from the Volgograd reservoir (Q²_t = R²_t × 1,000,000,000/86,400 × τ₂), m³/s;

W³_t—the total volume of water in the reservoirs of the VKC at the beginning of the WLW, km³;

W³_t,i—the volume of water in the i-th VKC reservoir (i = 1 ÷ 9) at the beginning of the WLW, km³;

Z³_beg,t,i—the level of the upstream (US) of the i-th VKC reservoir (i = 1 ÷ 9) at the beginning of the WLW, m;

Z³_fin,t,i—the level of the upstream (US) of the i-th VKC reservoir (i = 1 ÷ 9) at the end of the WLW, m;

B³_t—the total volume of inflow to the VKC reservoirs during the WLW + PFD, km³;

B³_t,i—the volume of inflow to the i-th WKC reservoir during the WLW + PFD (B₃ = ΣB_3,i), km³;

R³_t,i—the volume of release into the downstream (DS) of the i-th VKC reservoir (i = 1 ÷ 9) during the WLW + PFD, km³;

Q³_t,i—the average discharge of the release to the DS of the Volgograd reservoir, during the WLW + PFD (Q_3,i = R_3,i × 1,000,000,000/86,400 × τ₃), m³/c;

H³_beg,t,i—level of the i-th VKC reservoir DS, considering the backwater from the downstream reservoir (i = 1 ÷ 9) at the beginning of the WLW, m;

H³_fin,t,i—level of the i-th VKC reservoir DS, taking into account the backwater from the downstream reservoir (i = 1 ÷ 9) at the end of the PFD, m;

N³_begt,i—the HPP capacity of the i-th VKC reservoir (i = 1 ÷ 9) at the beginning of the WLW, MW;

N³_fin,t,i—the HPP capacity of the i-th VKC reservoir (i = 1 ÷ 9) at the end of the PFD, MW.

To calculate the water levels in the upstream and downstream of the VKC reservoirs and the power of the HPP, the following functions are used, the form of which is given in [16,17]:

Z^us_i = F_b(W_i)—the bathymetric function of the dependence of the upstream level of the i-th reservoir Z_i on the volume W_i;

Z^ds_i = F_d(Q_i,Z^ds_i)—the dependence function of the level of the downstream of the i-th reservoir Z^ds_i on the releases to the downstream Q_i and the upstream level of the low (i + 1)-th reservoir Z^ds_i (backwater from the downstream reservoir)

N_i = N(Z^us_i,Z^ds_i,Q_i)—the dependence function of HPP capacity of the i-th reservoir N_i on the upstream level Z^us, the downstream level Z^ds, and releases to the tailwater Q_i (dependence on the head Z^us_i − Z^ds_i)

For each water management year t (t = 1÷105), the water balance equations for the VKC reservoirs by seasons, starting from the high water, have the following form:

1. High Water: W²_t = W¹_t + B¹_t−R¹_t;

(6)

2. SLW: W³_t = W²_t + B²_t − R²_t;

(7)

Q²_t = R²_t × 1,000,000,000/(86,400 × τ₂);

(8)

The volume of water in each reservoir W³_t,i at the beginning of the WLW is proportional to the total volume of water in the VKC reservoirs W³_t at the beginning of the WLW, obtained in Equation (7). The coefficient of proportionality k_1,t is determined by the following equation:

W³_t = W^NOL × (1 − k_1,t) + W^IL × k_1,t;

(9)

The volume of water in each reservoir W³_{t + 1} (variable) at the beginning of the high water in the next (t + 1)-th year is proportional to the total volume of water in the VKC reservoirs at the beginning of the high water. The proportionality factor k₂ is determined by the equation:

W³_{t + 1} = W^NOL × (1 − k_2,t) + W^IL × k_2,t;

(10)

3. WLW + PFD: W³_t,i = W_i^NOL × (1 − k_1,t) + W_i^IL × k_1,t;

(11)

Z³_beg,t,i = F_b(W³_t,i);

(12)

Z³_fin,t,i = F_b(W_i^NOL × (1 − k_2,t) + W_i^IL × k_2,t );

(13)

R³_t,i = W³_t,i + B³_t,i—(W_i^NOL × (1 − k_2,t) + W_i ^IL × k_2,t);

(14)

Q³_t,I = R³_t,i × 1,000,000,000/(86,400 × τ₃);

(15)

H³_beg,t,i = F_d(Q³_t,i, Z³_beg,t,i);

(16)

H³_fin,t,i = F_d(Q³_t,i, Z³_fin,t,i);

(17)

N³_beg,t,i = N(Z³_beg,t,i,H³_beg,t,i,Q³_t,i);

(18)

N³_fin,t,i = N(Z³_fin,t,i,H³_fin,t,i,Q³_t,i);

(19)

Table 2. Requirements of water users of the VKC water management system.

N	Criterion Content	Threshold	Reliability, %
1.	The volume of the VKC reservoirs should not exceed the volume at NOL (forbidden forcing), WNOL, km3	166, 74	100
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), WNOL, km3	[162, 74; 166, 74]	100
3.	Total winter HPP capacity of VKC reservoirs, MW	2945	90
4.	Normal transport release from the Volgograd HPP during the navigation period (high water, SLW), m3/s	5000	90
5.	Reduced transport release, m3/s	4000	100
6.	Normal fishery (agricultural) release from the Volgograd HPP, km3	120	50
7.	Reduced fishery (agricultural) release from the Volgograd HPP, km3	110	75
8.	Minimum ecological release ensuring the safety of the Lower Volga ecosystem, km3	60	100

shows the requirements of the main significant water users of the VKC water management system for 8 criteria.

The hierarchy of priorities of water users’ requirements in modern conditions is as follows: (1) no forcing, ecological release (criteria 1 and 8); (2) reservoir filling up to NOL, winter HPP capacity, reduced transport, reduced fishery (criteria 2, 3, 5, and 7); (3) normal transport, normal fishery (criteria 4, 6).

For each requirement of water users, an additive component of the objective function (criterion) is formed, which is represented by the square of the deviation of the requirement from the threshold value in case of its violation. This simplification reduces the number of disturbances in average flow years, but may increase the number of disturbances in dry years. However, this approach reduces the “depth” of the violation, which is sometimes a more important indicator. This approach makes it possible to exclude integrality and remain in the class of continuous piecewise differentiable functions.

Thus, the objective function (OF) is presented in the following form:

C = Σ_{j = 1 ÷ 8} μ_j × (Σ_{t = 1 ÷ T}C_j,t );

(20)

where μ_j are penalty coefficients that determine the hierarchy of priorities of water users’ requirements, t = 1÷T.

Criteria C_j,t are generally defined by the following formula:

C_j,t = if (X_j,t > Tr_j) then (X_j,t − Tr_j)² else 0;

(21)

where X_j,t is the desired value of the requirement, determined by the independent variables R¹_t, R²_t, and W¹_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.

The optimization problem for a given inflow series is formulated as follows:

Determine the positive values of the variables R¹_t, R²_t, and W¹_t, minimizing the OF (20) with balance constraints (6)–(19) and additional constraints on the variables: min (W¹_t) > 10,562 km³; min (W³_t) > W^IL; min (W³_t,9) > 1000 m³/s (sanitary winter); min (W³_t,4) > 600 m³/s (sanitary winter); max (W¹_t) < 145 km³ (maximum volume of reservoirs of the VKC by the beginning of the high water); W¹₁ = W¹_{T + 1} = 105,62 km³.

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.

3. Results and Discussion

3.1. The Data Analysis, Main Results

From the inflow data observed, the distributions’ main numerical characteristics of the annual and seasonal runoff of the cascade and all its reservoirs were obtained. In this paper, here and below, the results are given only for the Kuibyshevskoye reservoir and the cascade as a whole. The results are shown in

Table 3. Numerical characteristics of the inflow distributions for 1916–2020 (Kuibyshevskoye reservoir).

Statistics	Annual	HW	SLW	WLW	PFD
Mean (mln m3)	44,639	28,981	10,163	4010	1486
SEM	1081	744	396	175	85
Median	45,011	28,702	9444	3766	1295
Standard deviation σB = √DB	11,132	7657	4078	1801	879
Sample variance DB	123,926,692	58,637,086	16,629,466	3,244,917	772,173
Kurtosis	1.33	1.42	0.68	0.87	13.32
Skewness	0.50	0.67	0.83	0.96	3.03
Minimum (mln m3)	18,928	12,368	3121	1234	592
Maximum (mln m3)	84,211	57,698	23,288	10,149	6488
Sample size N	105	105	105	105	105
% annual runoff		65%	23%	9%	3%

and

Table 4. Numerical characteristics of the inflow distributions for 1916–2020 (VKC).

Statistics	Annual	HW	SLW	WLW	PFD
Mean (mln m3)	250,913	151,513	64,050	24,517	10,833
SEM	4493	2980	2005	960	552
Median	254,338	152,411	59,859	22,824	9797
Standard deviation σB = √DB	46,041	30,538	20,543	9838	5653
Sample variance DB	2,119,738,992	932,565,653	422,013,534	96,788,357	31,960,172
Kurtosis	0.34	−0.25	0.32	0.26	3.78
Skewness	0.0039	0.08	0.65	0.81	1.63
Minimum (mln m3)	141,759	83,444	28,271	10,486	3981
Maximum (mln m3)	374,650	234,826	130,728	54,563	35,438
Sample size N	105	105	105	105	105
% annual runoff		60%	26%	10%	4%

, respectively.

The distributions of the annual runoff and its seasonal components are not symmetrical, and at the same time, they are biased towards runoff larger values, since the skewness values for them are positive.

The variance of the annual runoff is not the sum of the variances of seasonal runoff; therefore, seasonal runoffs, strictly speaking, cannot be considered as independent random variables. At least some of them may be related by some kind of correlation.

In both cases, the main contribution to the annual inflow (more than 80%) comes from high water and summer–autumn low-water seasons.

Figure 6 shows scatterplots (correlation fields) for the following pairs of inflows to the Kubyshevskoye reservoir: Annual–HW, Annual–SLW, Annual–WLW, and HW–WLW. For each, trend lines and their equations are shown.

Figure 7 shows similar diagrams for the corresponding seasonal inflow pairs for the cascade as a whole.

The first three diagrams in both figures demonstrate the presence of sufficiently expressed statistical relationships between the respective series of inflows. The last diagram (HW–WLW) shows the lack of correlation between high-water inflow and winter low-water inflow.

A quantitative assessment of the correlation dependences of the annual and seasonal inflows of the Kuibyshevskoe reservoir (KR) and the VKC in the form of the pairwise correlation coefficients is given in

Table 5. The correlation coefficients—KR.

	Annual	HW	SLW	WLW	IIC
Annual	1	0.86	0.76	0.63	0.31
HW	0.86	1	0.35	0.23	0.01
SLW	0.76	0.35	1	0.72	0.35
WLW	0.63	0.23	0.72	1	0.56
PFD	0.31	0.01	0.35	0.56	1

and

Table 6. The correlation coefficients—VKC.

	Annual	HW	SLW	WLW	PFD
Annual	1	0.75	0.75	0.64	0.24
HW	0.75	1	0.19	0.1	−0.12
SLW	0.75	0.19	1	0.69	0.23
WLW	0.64	0.1	0.69	1	0.45
PFD	0.24	-0.12	0.23	0.45	1

, respectively.

The tables show that in both cases, the strongest correlation is between annual inflow and high water, summer–autumn low water, and winter low water, as well as between SLW and WLW. Correlation in other pairs can be characterized as weak and very weak.

Figure 8 shows the difference integral curves (DICs) for the annual runoff and its intra-annual components of the Kubyshevskoye reservoir and the cascade as a whole. The curves were built in accordance with expression (1) for each hydrological series. The curves obtained again demonstrate a high degree of synchronism in the long-term changes in the low-water components of the annual runoff.

Based on the difference integral curves in Figure 8, constructed for the VKC, it was concluded that for all seasons of the water management year, with the exception of the high water period, the original runoff series is not stationary and can be divided into two parts, approximately at the turn of 1973–1979, depending on the intra-annual seasons.

The year 1979 was calculated as the most characteristic year of the annual runoff “turning point” as it was described in Section 2.2.1, and the initial hydrological series of the annual inflow of the VKC 1916–2020 was divided into two series: 1916–1978 and 1979–2020.

The distributions’ numerical characteristics of inflow values for these two hydrological series are given in

Table 7. The distributions’ numerical characteristics for the 1916–1978 series.

Statistics	Annual	HW	SLW	WLW	PFD
Mean (mln m3)	234,688	150,750	56,310	18,552	9077
SEM	5627	4139	2263	727	581
Median	237,862	150,467	53,481	17,789	7267
Standard deviation σB = √ DB	44,662	32,849	17,960	5767	4610
Sample variance DB	1,994,655,640	1,079,041,861	322,556,913	33,255,047	21,252,457
Kurtosis	0.33	−0.34	0.31	2.28	0.54
Skewness	0.15	0.15	0.74	1.33	1.20
Minimum (mln m3)	141,759	83,444	28,271	10,486	3981
Maximum (mln m3)	360,820	234,826	106,056	38,481	22,003
Sample size N	63	63	63	63	63
% annual runoff		64%	24%	8%	4%

and

Table 8. The distributions’ numerical characteristics for the 1979–2020 series.

Statistics	Annual	HW	SLW	WLW	PFD
Mean (mln m3)	275,250	152,656	75,661	33,465	13,468
SEM	5668	4174	2897	1180	940
Median	269,833	154,065	71,498	31,784	11,403
Standard deviation σB = √ DB	36,734	27,048	18,776	7650	6089
Sample variance DB	1,349,366,302	731,577,879	352,551,273	58,528,168	37,077,255
Kurtosis	1.23	−0.14	0.49	0.73	4.36
Skewness	0.46	−0.04	0.92	1.10	1.98
Minimum (mln m3)	197,877	91,738	46,836	22,824	6222
Maximum (mln m3)	374,650	208,240	130,728	54,563	35,438
Sample sizeN	42	42	42	42	42
% annual runoff		55%	27%	12%	5%

, respectively.

The calculation results show that, indeed, at the specified year intervals for all intra-year periods of the water management year, with the exception of the high water period, the inflow average values for the corresponding periods are different, increasing noticeably in the 1979–2020 series compared to the 1916–1978 series. In addition, the share of high water in the annual runoff decreased from 64% to 55%; i.e., there was a redistribution of intra-annual inflow in favor of low-water runoff due to flood runoff.

Using a two-sample Student’s test, the hypotheses of a significant difference in the average inflow values given in Table 5 and Table 6 for the series 1916–1978 and 1979–2020 were confirmed for the corresponding seasons of the water management year.

3.2. Analysis of Time Series of Inflow to the Volga–Kama Cascade Reservoirs

To assess the changes that have occurred, all the studied inflow series (flood inflow, summer low-water inflow, winter low-water inflow, pre-flood drawdown) were divided into two parts in accordance with the selected state change point, and each part was processed by applying the Kritsky–Menkel distribution. The calculation results are presented in Figure 9 by the reliability graphs of inflow and in

Table 9. Comparison of inflow volumes for two long-time series, different runoff reliability years, and different seasons.

The Long-Term Series of Inflow, Seasons	Inflow Volume for Years of Different Reliability by Flow, Cubic km
	2%	25%	50%	75%	95%	98%
Year, series 1916–1978	331	264	233	203	163	148
Year, series 1979–2020	367	300	272	248	217	206
% volume change	10.9%	13.6%	16.7%	22.2%	33.1%	39.2%
High water, series 1916–1978	221	172	150	128	98.6	87.3
High water, series 1979–2020	208	171	153	135	108	96.5
% volume change	−5.9%	−0.6%	2.0%	5.5%	9.5%	10.5%
Summer–Autumn Low Water, series 1916–1978	101	67	54.1	43.4	31.1	26.9
Summer–Autumn Low Water, series 1979–2020	124	86.1	73	62.5	50.7	46.6
% volume change	22.8%	28.5%	34.9%	44.0%	63.0%	73.2%
Winter Low Water, series 1916–1978	33.4	21.4	17.4	14.5	11.5	10.5
Winter Low Water, series 1979–2020	53.3	37.6	32.3	28.2	23.6	22.1
% volume change	59.6%	75.7%	85.6%	94.5%	105.2%	110.5%
Pre-flood Drawdown, series 1916–1978	21.3	11.5	8.22	5.81	3.41	2.7
Pre-flood Drawdown, series 1979–2020	30	15.7	12	9.43	6.9	6.13
% volume change	40.8%	36.5%	46.0%	62.3%	102.3%	127.0%

and Figure 10 in the form of distribution quantiles.

The data show a significant increase in annual inflow, summer–autumn low water, winter low water, and pre-flood drawdown. The flood volume has not changed much. In a year with 2% reliability, the annual inflow for the 1979–2020 series increased by 10,9% compared to the 1916–1978 series, the inflow of summer–autumn low water increased by 22,8%, the inflow of winter low water increased by 59,6%, and inflow in March (pre-flood drawdown) increased by 40,8%; in a year with 25% reliability, the annual inflow increased by 13,6%, the inflow of summer–autumn low water increased by 28,5%, the inflow of winter low water increased by 75,7%, and the inflow in March increased by 36,5%; in a year with 50% reliability, the annual inflow increased by 16,7%, the inflow of summer–autumn low water increased by 34,6%, the inflow of winter low water increased by 85,6%, and the inflow in March increased by 46%; in a year with 75% reliability, the annual inflow increased by 22,2%, the inflow of summer–autumn low water increased by 44%, the inflow of winter low water increased by 94,5%, and the inflow in March increased by 62,3%; in a year with 95% reliability, the annual inflow increased by 33,1%, the inflow of summer–autumn low water increased by 63%, the inflow of winter low water increased by 105,2%, and the inflow in March increased by 102,3%; in a year with 98% reliability, the annual inflow increased by 39,2%, the inflow of summer–autumn low water increased by 73,2%, the inflow of winter low water increased by 10,5%, and the inflow in March increased by 127%.

A particularly large increase in inflow occurs in dry and extremely dry years (during the winter low-water season and in March, the inflow more than doubled). The physical reasons for the increase in runoff in the region of the VKC reservoir basin are most likely associated with warming and, as a result, increased precipitation and inflow to rivers in the region of the Arctic Ocean seas (Barents Sea, White Sea, Kara Sea, Laptev Sea, East Siberian Sea, and Chukchi Sea). All these seas border the territory of Russia in the north and affect the inflow of the basins of the Upper and Middle Volga and Kama rivers, but the authors did not conduct studies on the relationship of these processes. In the study, the authors answered only two questions: has the inflow changed with sufficient reliability and how did these changes affect the water management situation in the VKC reservoir basin?

3.3. Analysis of Results of Water Management Optimization Calculations

In order to determine the impact (positive or negative) of climate change on the water management situation in the basin of the VKC reservoirs, an water management optimization calculation was performed for the inflow series 1916–1978 and 1979–2020 according to the mathematical model, (6)–(20), presented in Section 2.3.2. Optimization was performed in the Microsoft Excel software using the generalized reduced gradient (GRG) method built into the Solver Add-in.

The reliability for eight water user requirements (

Table 10. Reliability for water users’ requirements and the depth of violations for the inflow series 1916–1978 and 1979–2020.

The Long-Term Series	Indicators	1. No Forcing, Cubic km	2. Filling to NOL, Cubic km	3. Winter Power, MW	4. Normal Transport cub.m/s	5. Reduced Transport, cub.m/s	6. Normal Fishery, cub.m/s	7. Reduced Fishery, cub.m/s	8. Ecological Release, cub.m/s
1916–1978	Reliability	98%	98%	100%	85%	95%	30%	70%	93%
1979–2020	Reliability	100%	100%	100%	98%	100%	63%	93%	100%
	Threshold	100%	100%	90%	90%	100%	50%	75%	100%
1916–1978	Depth of violation	1.54	1.78	0	730	114	27	28	20
1979–2020	Depth of violation	0.00	0.11	0	9	0	23	5	0

) for the inflow series 1916–1978 and 1979–2020, obtained as a result of performing water resource optimization calculation, is shown in Table 10 and Figure 11.

Analysis of the results of water resource optimization calculation showed that the requirements of water users obtained from the hydrological series 1916–1978 are violated for almost all criteria, except for criterion 3 (winter HPP capacity). The requirements for normal fishery (agricultural) releases are especially strongly violated (by 20%), which significantly affects the ecological situation in the Lower Volga basin due to the disturbance of the fauna and flora balance.

The water resource optimization calculation according to the hydrological series 1916–1978 provides the standard reliability for all water users’ requirements. At the same time, the violation depth is much lower than that for the 1916–1978 series.

This shows that the climate change that has occurred has significantly improved the water management and environmental situation in the VKC reservoir basin. Unfortunately, this fact is not natural in the case of possible climate change. Research carried out for the Lower Don basin showed a significant deterioration in the fulfillment of fishery requirements. Over the last 29 years, the normal fishery release from the Tsimlyanskoye reservoir has never been carried out.

To clarify the conflict of interest (icing on the cake) between the normal fishery release and the rest of the water users’ requirements [29,30] for the series 1916–1978 (element of multi-criteria analysis), a water resource optimization calculation was performed with penalty coefficients in favor of fishery release (agriculture) and ecology (criteria 6, 7, and 8). The calculation results are shown in Figure 12.

An analysis of the results shows that attempts to bring the requirements of the fishery (agriculture) and ecology to the normative ones significantly worsen the indicators of energy and transport reliability (criteria 1, 2, 4, and 5). This is unacceptable for the functioning of other sectors of the economy.

4. Conclusions

As a result of the analysis of long-term data on the inflow to the reservoirs, it was possible to find out that there were significant changes in the hydrological regime in the period 1979–2020. The non-stationary nature of the oscillations was revealed and was investigated by the methods considered above.

The results obtained allow us to conclude that in the last approximately 45 years there has been a noticeable increase in inflow to the VKC reservoirs and a change in the structure of river inflow with its redistribution in favor of low-water inflows. Due to this, the inflows to the SLW, WLW, and PFD increased significantly, while the inflow remained practically unchanged during the flood. A similar result was obtained in [9] for a number of rivers in the Volga basin.

According to the research results, due to a significant change in the hydrological regime in the basin of the VKC reservoirs, it is necessary to consider the parameters of the water regime proven in the article when designing newly built hydraulic structures and reconstructing existing hydraulic structures for the following industries: fisheries, agriculture (irrigation), public utilities, sanitary release, river transport, and winter energy (capacity). Since the flood inflow has not changed much, flood prevention measures in the upstream and downstream of reservoirs do not need to be adjusted.