# Multi-Criteria Analysis of the “Lake Baikal—Irkutsk Reservoir” Operating Modes in a Changing Climate: Reliability, Resilience, Vulnerability

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Construction of a Forecast Hydrological Series

_{1}, x

_{2}, ⋯ x

_{n}, which is divided into two parts in r point (1 ≤ r ≤ n) and x

_{i}is distributed under law F

_{1}(x∕Θ

_{1}), i = 1, ⋯, r, x

_{i}is distributed under law F

_{2}(x∕Θ

_{2}), i = r + 1, ⋯, n and F

_{1}(x∕Θ

_{1}) ≠ F

_{2}(x∕Θ

_{2}). The task is that on the set sequence of observations x

_{1}, x

_{n}, and the known functions of thr distribution of F

_{1}and F

_{2}, it is necessary to draw a conclusion concerning a point of change of a condition of process.

_{1}(x ∕Θ

_{1}) and ρ

_{2}(x ∕Θ

_{2}), joint distribution of sample x

_{1}, ⋯, x

_{n}conditional in relation to parameters Θ

_{1}and Θ

_{2}and r having a point of change (1 ≤ r ≤ n), we will write down:

_{1}and Θ

_{2}, a posteriori distribution density of a point of change for the available observations x

_{1}, ⋯, x

_{n}, we will define from the general formula of total probability (Bayes formula):

#### 2.2. Water Resource Calculation Based on the Dispatch Schedule

#### 2.3. Water Resource Calculation by Optimization Methods

- (1)
- Are the DS parameters (coordinates) well chosen, i.e., a valid DS solution exists, but developer has not found it.
- (2)
- How “well” are water users’ requirements being met in the adopted hierarchy? Is it possible to select the DS parameters so as to improve the reliability indicators for conflicting requirements?
- (3)
- Is the DS management optimal or is it possible to achieve better results using other management tools? Does the reservoir “water capacity” of the and catchment area allow it to satisfy the water users’ requirements?

#### 2.3.1. Mathematical Model and Algorithm for Solving the Optimization Task

_{i}

_{ + 1}= W

_{i}+ P

_{i}− R

_{i}

**i**is the time interval,

**W**is the water volume in Lake Baikal at the interval beginning

_{i}**i**,

**P**is the volume (or average discharge) of “net” inflow (without losses for evaporation and seepage) to Lake Baikal in the interval

_{i}**i**,

**R**is the volume (or average discharge) of release to the Irkutsk reservoir downstream in interval

_{i}**i**.

_{i}= F

_{B}(W

_{i})—bathymetric dependence function of the Lake Baikal level

**Z**on the Lake Baikal volume;

_{i}^{u}

_{i}= F

_{u}(Z

_{i},R

_{i})—dependence function of the Irkutsk reservoir upstream level

**H**on the lake Baikal level and releases into the Irkutsk reservoir downstream;

^{u}_{i}^{d}

_{i}= F

_{d}(R

_{i})—dependence function of downstream level

**H**on the Irkutsk reservoir release into the downstream

^{d}_{i}_{i}= N(H

^{u}

_{i}, H

^{d}

_{i},R

_{i})—hydroelectric power dependence function on headwater level, tailwater and release into the Irkutsk reservoir downstream (difference on the head—H

^{u}

_{i−}H

^{d}

_{i})

- The Lake Baikal level should be in range (455.54, 457.5) m;
- The Lake Baikal level should be ≥456 m, [25];
- The Lake Baikal level should be ≤457 m, [25];
- The maximum release in winter should be less than 2500 m
^{3}/s; - The transport release during navigation should be more than 1500 m
^{3}/s; - The release for water supply should be in range of (1250, 1300) m
^{3}/s; - Flood control release should be less than 3200 m
^{3}/s; - Guaranteed winter power should be more than 347 MW;
- The Irkutsk reservoir upstream level for water intakes operation should be more than 454 m;
- The pressure on the dam for HPP operation should be more than 26 m;
- The Lake Baikal level on May 1 for normal fish spawning should be 456.15 m;
- The Lake Baikal level during September for normal fish spawning should be 457 m.

^{3}/s (criterion 7); (4) fisheries, 11, 12.

^{3}/s.

**T**years and

**M**intervals per year (in the article

**T**= 44,

**M**= 24):

_{11}+ P

_{11}− R

_{11}= W

_{12}, …, W

_{1m}+ P

_{1m}− R

_{1m}= W

_{1m+1}, …, W

_{1M}+ P

_{1M}− R

_{1M}= W

_{21},

……………………

W

_{t1}+ P

_{t1}− R

_{t1}= W

_{t2}, …, W

_{tm}+ P

_{tm}− R

_{tm}= W

_{tm+1}, …, W

_{tM}+ P

_{tM}− R

_{tM}= W

_{t+11},

……………………

W

_{T1}+ P

_{T1}− R

_{T1}= W

_{T2}, …, W

_{Tm}+ P

_{Tm}− R

_{Tm}= W

_{Tm+1}, …, W

_{TM}+ P

_{TM}− R

_{TM}= W

_{T+11}

**t = [1,T]**;

**m = [1,M]; W**and

_{tm}**W**are the initial and final reservoir water volumes;

_{tm}_{+1}**W**,

_{11}**W**specify starting and ending volumes. In accordance with the normative document [22],

_{T+11}**W**is accepted in Russia. This provides an overall water balance for the entire system (11).

_{11}= W_{T+11}**R**for the given inflow values

_{tm}**P**, fixed

_{tm}**W**, at which all water users’ requirements will be satisfied as much as possible. The number of intervals (or years) in which water users’ requirements (criteria) are violated is minimized in accordance with their priorities. Furthermore, in accordance with the standard [22], it is necessary to minimize the violation depth, the difference between the threshold and the criterion value, in case of failure. In this case, the optimization task becomes multi-criteria.

_{11}, W_{T+11}**K**of criteria is allowed, each of which is given in the form of system control parameter functions defined through the variables

**R**and

_{tm}**W**. These functions are reservoir volumes and water levels, levels of up- and downstream, releases, hydropower and electricity generation and others. For example, the criterion may be the square of release deviation

_{tm}**R**from allowable maximum

_{tm}**R**that does not lead to downstream flooding: if

_{max}**R**>

_{tm}**R**then (

_{max}**R**−

_{max}**R**)

_{tm}^{2}, else

**0.**

**Δ**) of deviation of

^{2}_{tmk}(W_{tm},R_{tm},W_{tm+}_{1}**k**th criterion threshold in

**i**th interval from the regulation result. In accordance with the significance of each requirement

**k**, a penalty factor

**C**is introduced. This is a weighting factor that determines the place of the requirement in the priorities hierarchy.

_{k}**W = {W**,

_{11}, …, W_{TM}}**R = {R**. The regulation quality is estimated using the objective function

_{11}, …, R_{TM}}**F(W,R)**defined as the sum of squared deviations from water users’ requirements for all intervals and all the criteria

**Δ**), taking into account the penalty factors:

^{2}_{tmk}(W_{tm},R_{tm},W_{tm+}_{1}_{k}

_{=[1,K]}C

_{k}× (Σ

_{t}

_{=[1,T]}Σ

_{m}

_{=[1,M]}Δ

^{2}

_{tmk}(W

_{tm},R

_{tm},W

_{tm+}

_{1}))

**K, T**and

**M.**

**W**and

_{tm}**R**:

_{tm}^{(0)}={R

^{(0)}

_{11}, …, R

^{(0)}

_{TM}}, W

^{(0)}={W

^{(0)}

_{11}, …, W

^{(0)}

_{TM}}

**F**

^{(0)}(W^{(0)}, R^{(0)}).**t**th two-year task: minimize objective function (OF):

_{2t-1,2t}(W,R) = Σ

_{k}

_{=[1,K]}C

_{k}× (Σ

_{m}

_{=[1,M]}Δ

^{2}

_{2t−1mk}(W

_{2t−1m},R

_{2t−1m},W

_{2t−1m+1})+

+Σ

_{m}

_{=[1,M]}Δ

^{2}

_{2tmk}(W

_{2tm},R

_{2tm},W

_{2tm+1}))

**W**and

**R**:

_{2t-11}+ P

_{2t-11}− R

_{2t-11}= W

_{2t-12};

……………………

W

_{2t-1m}+ P

_{2t-1m}− R

_{2t-1m}= W

_{2t-1m+1};

……………………

W

_{2t-1M}+ P

_{2t-1M}− R

_{2t-1M}= W

_{2t 1};

W

_{2t1}+ P

_{2t1}− R

_{2t1}= W

_{2t2};

……………………

W

_{2tm}+ P

_{2tm}− R

_{2tm}= W

_{2tm+1};

……………………

W

_{2tM}+ P

_{2tM}− R

_{2tM}= W

_{2t+1 1};

W

_{2t-11}= W

^{(0)}

_{2t-11}; W

_{2t+11}= W

^{(0)}

_{2t+11};

_{2t-1m}≥ 0, W

_{2tm}≥ 0, R

_{2t-1m}≥ 0 R

_{2tm}≥ 0, ∀m = [1,M]

**T*M**dimensional optimization task (11)–(13) splits into

**T/2**(for even

**T**and

**(T − 1)/2**for odd) subtasks of type (14)–(16), which can be solved independently from each other, hence one can use the multitask and multiprocessor computer modes. Such a decomposition of the optimization task into T/2 independently solvable subtasks allows using the multitasking mode of the operating system (Windows) and the multiprocessor mode of supercomputer, which can significantly reduce the optimization time (practically by several tens of times). These properties are implemented for the high-level programming language C

^{++}of a multiprocessor supercomputer, on which the next version of presented computing technology is supposed to be implemented.

**R**,

^{(1)}_{2t−1m}, W^{(1)}_{2t−1m}**R**be the variable values that realize the minimum of the OF (14). The

^{(1)}_{2tm}, W^{(1)}_{2tm}, m = [1,M]**F**value does not increase compared to the approximation zero variable values

_{2t−1,2t}(W,R)**R**,

^{(0)}_{2t−1m}, W^{(0)}_{2t-1m}**R**. It follows that after solving all local subtasks (14)–(16),

^{(0)}_{2tm}, W^{(0)}_{2t1m}, m = [1,M]**t = [1,T/2]**, the value

**F**of the OF (12) for the found vectors,

^{(1)}(W^{(0,1)},R^{(1)})**W**, at least, will not increase compared to

^{(0,1)}= {W^{(0)}_{11}, …, W^{(0)}_{2t−11}, W^{(1)}_{2t−12,}…, W^{(1)}_{2t−1m}, …, W^{(1)}_{2t1}, …, W^{(1)}_{2tm}, …, W^{(0)}_{2t+11}, …, W^{(1)}_{TM}, …, W^{(0)}_{T+11}}; R^{(1)}= {R^{(1)}_{11}, …, R^{(1)}_{TM}}, t = [1,T/2], m = [1,M]**F**.

^{(0)}(W^{(0)},Q^{(0)})**W**,

^{(1)}_{2t1}**t = [1,T/2 − 1]**(even pass) are fixed and the odd ones become variables. The decomposition procedure is repeated, but with fixed even volumes.

_{2t,2t+1}(W,R) = Σ

_{k}

_{=[1,K]}C

_{k}× (Σ

_{m}

_{=[1,M]}Δ

^{2}

_{2tmk}(W

_{2tm},R

_{2tm},W

_{2tm+1}) +

Σ

_{m}

_{=[1,M]}Δ

^{2}

_{2t+1mk}(W

_{2t+1m},R

_{2t+1m},W

_{2t+1m+1}))

**W**and

**R**

_{2t1}+ P

_{2t1}− R

_{2t1}= W

_{2t2};

……………………

W

_{2tm}+ P

_{2tm}− R

_{2tm}= W

_{2tm+1};

……………………

W

_{2tM}+ P

_{2tM}− R

_{2tM}= W

_{2t+11};

W

_{2t+11}+ P

_{2t+11}− R

_{2t+11}= W

_{2t+12};

……………………

W

_{2t+1m}+ P

_{2t+1m}− R

_{2t+1m}= W

_{2t+1m+1};

……………………

W

_{2t+1M}+ P

_{2t+1M}− R

_{2t+1M}= W

_{2t+21};

_{2t1}= W

^{(1)}

_{2t1}; W

_{2t+21}= W

^{(1)}

_{2t+21};

W

_{2tm}≥ 0, W

_{2t+1m}≥ 0, R

_{2tm}≥ 0 R

_{2t+1m}≥ 0, ∀m = [1,M]

**R**,

^{(2)}_{2tm}, W^{(2)}_{2t+1m}**R**be the variable values that realize the minimum of OF (17). The

^{(2)}_{2tm}, W^{(2)}_{2t+1m}, m = [1,M]**F**value does not increase compared to the value for the previous pass variables

_{2t,2t+1}(W,R)**R**,

^{(1)}_{2tm}, W^{(1)}_{2t+1m}**R**. It follows that after solving all local subtasks (17)–(19),

^{(1)}_{2tm}, W^{(1)}_{2t+1m}, m = [1,M]**t = [1,T/2–1]**, the value

**F**of the OF (12) for the found vectors,

^{(2)}(W^{(1,2)},R^{(2)})**W**, at least, will not increase compared to

^{(0,1)}= { W^{(1)}_{11}, …, W^{(1)}_{2t−11}, W^{(2}_{2t−12,}…, W^{(2)}_{2t−1m}, …, W^{(2)}_{2t1}, …, W^{(2)}_{2tm}, …, W^{(1)}_{2t+11},…, W^{(2)}_{TM},…, W^{(1)}_{T+11}}; R^{(2)}= {R^{(2)}_{11},…,R^{(2)}_{TM}}, t= [1,T/2], m = [1,M]**F**.

^{(0)}(W^{(0)},Q^{(0)})#### 2.4. The Multi-Criteria Analysis Based on Performance Statistical Criteria

**X**} the time series data that is obtained as a result of water resource calculations (WRC) and X

_{t}**is the threshold value for this criterion [4, Chapter 9].**

^{T}**reliability**can be defined as the number of data in a satisfactory state, divided by the total number of data in the time series. If satisfactory values in a time series

**X**containing

_{t}**N**values (

**t**, time intervals) are equal to or exceed some threshold

**X**, then:

^{T}**X**] = [The number of time periods

**t**such that

**X**

_{t}**≥ X**]/

^{T}**N**

**or the**time series

**annual reliability**can be defined as the number of years in a satisfactory state divided by the total number of years in the time series. If

**N**the total number of years, then:

_{year}**X**] = [The number of years in which for all periods t within a year,

**X**

_{t}**≥**

**≥ X**]/

^{T}**N**

_{year}**Resilience**is defined as the satisfactory event occurrence probability in the time interval

**t + 1**after a failure event in the interval

**t**. The resilience is calculated as follows:

**X**] = [number of times a satisfactory value follows an unsatisfactory value]/[number of times an unsatisfactory value occurred]

**Vulnerability**is defined as the average deviation failure time series values

**X**from a threshold value

_{t}**X**(average “depth” of failures). The vulnerability is calculated as follows:

^{T}**X**] = [sum of positive values of (

**X**−

^{T}**X**)]/

_{t}[number of failures occurred]

**Or the normalized vulnerability**is defined as the average deviation fraction from the maximum deviation (maximum “depth” of failures). The normalized vulnerability is calculated as follows:

**X**] = [sum of positive values of (

**X**)/Max{(

^{T}− Xt**X**) |

^{T}− X_{t}**t**-failures}]//[number of failure times occurred]

## 3. Results

#### 3.1. The Result of Predicted Hydrological Series Construction

#### 3.2. The Water Resource Calculations Results Using Dispatch Schedule

#### 3.3. The Water Resource Calculation Result Based on the Optimization Methods

#### 3.4. The Multi-Criteria Analysis of the Water Resource Calculations Results

#### 3.4.1. Analysis of the Water Resource Calculations Results Using DS

^{3}/s, it can be assumed that the DS 1988 made it possible to form releases with standard reliability for almost all requirements.

^{3/}s), 8, 11 and 12. Criteria two and three were added in 2001 [25], so the DS 1988 also does not provide the required water levels.

**Conclusion:**It is necessary to rebuild DSs at least once every 10 years, taking into account the changed hydrological situation, current requirements and the water users’ priorities hierarchy.

#### 3.4.2. Analysis of the Water Resource Calculation Results Using Optimization Methods

^{3}/s), which is violated when using DS, and the fishery requirements are “poorly” satisfied, which will further lead to a decrease in the omul population in low-water dry years.

**Conclusion:**The water capacity of the Lake Baikal basin (catchment area) is sufficient to satisfy practically all water users’ requirements, except for fishery, with normative reliability, even in dry periods. It is only necessary to formulate such release rules that implement the optimization approach. The use of an optimization approach for the reservoir operation mode formation significantly increases the reliability, resilience and vulnerability of management. The constructed Irkutsk hydroelectric power station violates the fishery requirements, especially in dry years, and even optimization does not allow them to be provided with sufficient reliability.

#### 3.4.3. Comparison of WRC Results Obtained Using DS and Optimization Methods

#### 3.4.4. Formation of Reservoir Operation Modes Based on Optimization Methods

**P**of inflow to a reservoir with a duration of

**T**years. It is necessary to make a decision on the reservoir operating modes in the current

**(T + 1)**th year.

**m**estimated time intervals (pentads, decades, months) in accordance with the formed hydrological observed inflow. For the Lake Baikal, every month from May to October is divided into three conditional decades (the last one can contain 11 days), and the rest of the intervals are monthly (24 intervals in total). For the last interval of

**m**year of

**T**, the reservoir volume

**W**

^{f}**, observed at the end of the interval is known. This volume is taken as equal to the initial volume of**

_{TM}**W**

^{b}**reservoirs in the first estimated time interval of the**

_{T+11}**(T + 1)**th year. Below we describe an algorithm finding an optimal solution.

- We construct a predicted long-term hydrological series of inflows
**P**for the last 40–50 years of observed inflows with a duration of^{f}**T**years, using the method described in Section 2.1 and Section 3.1 or other methods [16].^{f}

^{f}= { P

^{f}

_{tm}| t = [1,T

^{f}], m = [1,M]}

- 2.
**The algorithm is built on the principle of mathematical induction.**The basis of induction is as follows: the calculation for the first interval**m = 1**, year**t = 1**(corresponds to**T + 1**year of the original series**P**). The reservoir volume at the beginning of the**W**^{b}interval is known and is equal to_{11}**W**. For the first interval, we set a hierarchy of priorities for water users’ requirements, which are actual for this calculation. Based on the mid-term interval inflow forecast, we determine the value^{f}_{TM}**P**and replace^{f*}_{11}**P**нa^{f}_{11}**P**(if the forecast is deeper, then replace the inflow in^{f*}_{11}, P^{f}_{11}:= P^{f*}_{11}**P**for other intervals^{f}**m>1**). After that we solve the optimization task using the hydrological series**P**with the replaced predicted inflow for given priorities, with a given^{f}**W**by the algorithm given in Section 2.3.1. Variables that are optimized during the calculation are the releases^{b}_{11}**R**, where_{tm}**t = [1,T**. The found solution,^{f}], m = [1,M]**R**is taken as the desired (or as the sought for) one_{11}**R**. After the solution implementation in the management process, the actual reservoir volume at the end of the first interval_{T+11}**W**, is determined (real^{f}_{T+11}**R**may differ from^{*}_{T+11}**R**) which is taken as the initial volume in the second interval_{11}**W**. The predicted inflow^{b}_{T+12}= W^{f}_{T+11}**P**is replaced by the actual observed^{f*}_{11}**P**and is stored instead of^{f**}_{11}**P**.^{f}_{11}- 3.
**The induction step**. Assume that the reservoir operating modes for the interval**m > 1**are formed and the initial volume**W**is determined. The search algorithm of releases^{b}_{T+1m+1}**R**for the interval_{T+1m+1}**m+1**is similar to step two, i.e., for the interval**m+1**, we set a hierarchy of priorities for water users’ requirements, which are actual for this calculation. Now, based on the short-term interval inflow forecast, we determine the value**P**and replace^{f*}_{1m+1}**P**нa^{f}_{1m+1}**P**(if the forecast is deeper, then replace the inflow in^{f*}_{1m+1}, P^{f}_{1m+1}:= P^{f*}_{1m+1}**P**for other intervals >^{f}**m**+**1**). Then we solve the optimization task using the hydrological series**P**with the replaced predicted inflow for given priorities with a given^{f}**W**by the algorithm given in Section 2.3.1. Variables that are optimized during the calculation are releases^{b}_{1m+1}**R**where_{tm},**t = [1,T**. The found solution^{f}], m = [1,M]**R**is taken as the desired one_{1m+1}**R**. After solution implementation in the management process, the actual reservoir volume at the end of the first interval_{T+1m+1}**W**, is determined (real^{f}_{T+1m+1}**R**may differ from^{*}_{T+1m+1}**R**), which is taken as the initial volume in the second interval_{1m+1}**W**. The predicted inflow^{b}_{T+1m+2}= W^{f}_{T+1m1}**P**is replaced by the actual observed^{f*}_{1m+1}**P**and is stored instead of^{f**}_{1m+1}**P**. Step three is repeated until^{f}_{1m+1}**m = M**.- 4.
- If necessary, the daily value release in the interval m is determined, so that the average release for the interval is
**R**._{T+1m} - 5.
- The initial series
**P**is supplemented by the observed inflow data for the year**T + 1**. The steps 1–5 are repeated for the new series**P**and the next year**T + 2**, etc.

## 4. Discussion

**INRI**) and the integrated normalized vulnerability index (

**INVI**).

**The integrated normalized reliability index (INRI)**is defined as the sum of failure for all criteria, divided by the number of years in the time series. It is determined by the formula

**:**

**INRI**=

**Σ**

_{k}**(1—Reliability**

_{= [1,K]}**[**

^{k}**X**])

**[**

^{k}**X**] is the reliabilty (in fractions) for the

**k**th criterion, K is the number of criteria.

**INRI**can be determined for both annual and interval reliability, but it is better to use the annual

**INRI**, since there are normative values for the reliability of the criteria for it. This normative reliability can be used as an estimated

**T**threshold for

_{INRI}**INRI**by defining it as follows:

**T**=

_{INRI}**Σ**

_{k}_{= [1,K]}(1 −

**T**)

^{k}**T**is the normative security for the

^{k}**k**th criterion expressed in fractions.

**INRI**characterizes the control components (a set of criteria, release rule and inflow time series) as follows:

**0 ≤ INRI ≤ K**; the closer the

**INRI**value is to

**T**, the better the control components are combined, if the

_{INRI}**INRI ≤ T**value, then the release rule can be used for management. Although it is impossible to set the absolute

_{INRI}**INRI**indicator value, which characterizes the release rule quality for a given set of water users’ requirements and inflow time series, this indicator allows you to form different management options for different criteria and inflow time series, to compare them with the

**T**, in order to evaluate and thus to choose the best solution. Table 2 shows the annual

_{INRI}**INRI**values for eighth performed WRC.

**INRI = T**.

_{INRI}**The integrated normalized vulnerability index (INVI)**is defined as the sum of the normalized vulnerabilities for all criteria and is determined by the formula:

**INVI**=

**Σ**

_{k}**Vulnerability_N**

_{= [1,K]}**[**

^{k}**X**])

**[**

^{k}**X**] is the normalized vulnerability for the kth criterion.

**INVI**characterizes the control components (a set of criteria, release rule and inflow time series) as follows:

**0 ≤ INVI ≤ K,**when the average deviation is equal to the maximum, then

**INVI**shows the number of criteria without violations (failure), indicator T

**=**

_{INVI}**INVI/K**gives the fraction of the average value faliure from the maximum for the entire set of criteria. The indicator T

**makes it possible to estimate the average depth of failure, depending on the maximum for the entire set of criteria, therefore, frpm knowing the maximum depth of failure for the criterion, one can decide on the significance of this failure and depending on the specific requirement, neglect such failure.**

_{INVI}**INRI**, it is impossible to set the absolute

**INVI**indicator value, which characterizes the release rule quality for a given set of water users’ requirements and inflow time series, however, this indicator allows you to compare different release rule options for different criteria and inflow time series, and the

**T**value makes it possible to estimate the depth of failures and, at small values, to neglect them. Table 3 shows the

_{INVI}**INVI**and

**T**values for the performed eighth WRC.

_{INVI}**INVI**for the time series 2020–2064 average is generally extremely low. The fraction of the cumulative average failure depth is 0.11. Figure 22 shows the calculation result comparison (normalized vulnerability) for the series 2020–2064 average for DS and optimization methods. The management quality by optimization methods is obvious.

## 5. Conclusions

- The dispatch schedule of 1988 does not give reliable results when performing water resource calculations on modern hydrological series, in comparison with the inflow series on the basis of which it is built. This suggests, on the one hand, that the inflow genesis has changed (there was an intra-annual change in runoff), on the other hand, over the past 44 years, the average annual inflow has decreased significantly (by 13%). The average annual inflow for the 1932–1976 series was 63.7 km
^{3}, and for the 1976–2020 series-55.7 km^{3}. Calculations have shown that if a low-water forecast is implemented, reliability, resilience and vulnerability of water users will significantly deteriorate when using DS 1988. For normal operation of the Irkutsk reservoir, it is necessary to develop a new dispatch schedule that would consider modern hydrology (last 20–30 years), modern requirements of water users and modern priorities (such attempts were undertaken in 2004, 2007 and 2013, however, new DSs were not approved due to the conflict between the water users’ requirements during these years). - As shown by multi-criteria analysis, water resource calculations based on optimization methods give results in terms of reliability, resilience and vulnerability, much better than when using DS. However, in almost all studies, optimization is used only for strategic planning of measures to improve the water resource situation in the river basin. Therefore, the authors have developed a mathematical model, an algorithm and computer technique for the formation of the reservoir optimal trade-off operation modes in real-time (for the next time interval) based on optimization methods. The data for calculating releases are: reservoir volume at the beginning of the interval, short-term (for the next interval) and long-term hydrological forecast, and given hierarchically ordered requirements of water users. For the implementation of the computer technique, a unique optimization algorithm was developed that allows for quick solving of complex nonlinear tasks of large dimensions. This approach will significantly improve the parameters of the reliability of management decisions in comparison with DS.
- Methods for a comprehensive assessment of the developed rules for reservoir management (release rule) are proposed, based on an integrated analysis of the set water users’ requirements (criteria) when assessing the management reliability.
- As a discussion, a long-term forecasting method based on a fairly good correlation between precipitation in a particular region, and the 11-year cycle of the Schwab solar activity is proposed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lotov, A.V.; Ryabikov, A.I.; Buber, A.L. Pareto frontier visualization in the development of release rules for hydro-electrical power stations. Sci. Tech. Inf. Process.
**2014**, 41, 314–324. [Google Scholar] [CrossRef] - Buber, A.; Lotov, A.; Ryabikov, A. Pareto frontier visualization in multi-objective water resources control rules development problem for the Baikal Lake and the Angara River cascade. IFAC PapersOnLine
**2019**, 52, 9–16. [Google Scholar] [CrossRef] - Lotov, A.V.; Riabikov, A.I.; Buber, A.L. A multi-criteria decision-making procedure with an inherited set of starting points of local optimization of the scalarizing functions. Sci. Tech. Inf. Process.
**2019**, 46, 328–336. [Google Scholar] [CrossRef] - Loucks, D.P.; van Beek, E.; Stedinger, J.R.; Dijkman, J.P.M.; Villars, M.T. Water resources systems planning and management: An introduction to methods. Models Appl.
**2005**, 2, 8–30. [Google Scholar] - Hashimoto, T.; Stedinger, J.R.; Loucks, D.P. Reliability, resiliency, and vulnerability criteria for water resource system performance evaluation. Water Resour. Res.
**1982**, 18, 14–20. [Google Scholar] [CrossRef][Green Version] - Kjeldsen, T.R.; Rosbjerg, D. Choice of reliability, resilience and vulnerability estimators for risk assessments of water resources systems (Choix d’estimateurs de fiabilité, de résilience et de vulnérabilité pour les analyses de risque de systèmes de ressources en eau). Hydrol. Sci. J.
**2004**, 49. [Google Scholar] [CrossRef] - Li, M.-H.; Tseng, K.-J.; Tung, C.-P.; Shih, D.-S.; Liu, T.-M. Assessing water resources vulnerability and resilience of southern Taiwan to climate change. Terr. Atmos. Ocean. Sci.
**2017**, 28, 67–81. [Google Scholar] [CrossRef][Green Version] - Meng, C.; Wang, X.; Li, Y. An optimization model for water management based on water resources and environmental carrying capacities: A case study of the Yinma River Basin, Northeast China. Water
**2018**, 10, 565. [Google Scholar] [CrossRef][Green Version] - Tan, Y.; Dong, Z.; Xiong, C.; Zhong, Z.; Hou, L. An optimal allocation model for large complex water resources system considering water supply and ecological needs. Water
**2019**, 11, 843. [Google Scholar] [CrossRef][Green Version] - Chen, M.; Dong, Z.; Jia, W.; Ni, X.; Yao, H. Multi-objective joint optimal operation of reservoir system and analysis of objectives competition mechanism: A case study in the Upper Reach of the Yangtze River. Water
**2019**, 11, 2542. [Google Scholar] [CrossRef][Green Version] - Jahandideh-Tehrani, M.; Bozorg-Haddad, O.; Loáiciga, H.A. A review of applications of animal-inspired evolutionary algorithms in reservoir operation modelling. Water Environ. J.
**2020**, 35, 628–646. [Google Scholar] [CrossRef] - Tehrani, M.J.; Bozorg-Haddad, O.; Loáiciga, H.A. Application of non-animal-inspired evolutionary algorithms to reservoir operation: An overview. Environ. Monit. Assess.
**2019**, 191, 439. [Google Scholar] [CrossRef] [PubMed] - Buber, A.L.; Buber, V.B. Regulation of reservoir operation modes in water management calculations based on multicriteria analysis methods and trade-offs theory (using the water management complex Lake Baikal-Irkutsk Reservoir as an example)—Water resources of Russia: Current state and management. In Proceedings of the All-Russian Scientific and Practical Conference, Sochi, Russia, 8–14 October 2018; pp. 120–127. [Google Scholar]
- Buber, A.L.; Buber, V.B. Optimal control of the operating modes of the complex “Lake Baikal-Irkutsk Reservoir” based on the methods of multicriteria analysis. Reservoirs of the Russian Federation: Modern environmental problems, status, management. In Proceedings of the All-Russian Scientific and Practical Conference, Sochi, Russia, 23–29 September 2019; pp. 56–63. [Google Scholar]
- Ilich, N.; Basistha, A. Importance of multiple time step optimization in river basin planning and management: A case study of Damodar River basin in India. Hydrol. Sci. J.
**2021**, 66, 809–825. [Google Scholar] [CrossRef] - Laurenz, L.; Lüdecke, H.J.; Lüning, S. Influence of solar activity changes on European rainfall. J. Atmos. Sol. Terr. Phys.
**2019**, 185, 29–42. [Google Scholar] [CrossRef] - Bolgov, M.V.; Trubetskova, M.D.; Kharlamov, M.A. On estimates of the statistical characteristics of rainfall in the Moscow region. Meteorol. Hydrol.
**2020**, 7, 77–85. [Google Scholar] - Bolgov, M.V.; Korobkina, E.A.; Osipova, N.V.; Filippova, I.A. On estimates of the maximum runoff of the river Iya, taking into account the extreme flood of 2019. Meteorol. Hydrol.
**2020**, 11, 53–63. [Google Scholar] - Bolgov, M.V.; Korobkina, E.A.; Filippova, I.A. Bayesian forecast of minimum runoff in non-stationary conditions taking into account possible climate changes. Meteorol. Hydrol.
**2016**, 7, 72–81. [Google Scholar] - Smith, A.F.M. A Bayesian approach to inference about a change—Point in a sequence of random variables. Biometrika
**1975**, 62, 407–416. [Google Scholar] [CrossRef] - Adams, R.P.; MacKay, D.J. Bayesian online changepoint detection. arXiv
**2007**, arXiv:0710.3742. Available online: https://arxiv.org/abs/0710.3742 (accessed on 10 October 2001). - The Ministry of Natural Resources and Environment of the Russian Federation. Guidelines for the Development of Rules for the Use of Reservoirs; 26 January 2011 N 17; The Ministry of Natural Resources and Environment of the Russian Federation: Moscow, Russia, 2011. Available online: https://docs.cntd.ru/document/902260669 (accessed on 27 June 2011).
- Castelletti, A.; Pianosi, F.; Soncini-Sessa, R. Water reservoir control under economic, social and environmental constraints. Automatica
**2008**, 44, 1595–1607. [Google Scholar] [CrossRef] - Ministry of Land Reclamation and Water Economy RSFSR. Basic Rules for the Use of Water Resources in the Angara HPP Cascade Reservoirs (Irkutsk, Bratsk, and Ust-Ilimsk); Ministry of Land Reclamation and Water Economy RSFSR: Moscow, Russia, 1988; p. 64.
- Decree of the Government of the Russian Federation, 26 March 2001 N 234: On the Limiting Values of the Water Level in Lake Baikal in the Implementation of Economic and Other Activities. Available online: http://base.garant.ru/2157975/#ixzz6naRbcnXT (accessed on 2 October 2001).
- Bolgov, M.V.; Buber, A.L.; Korobkina, E.; Lyubushin, A.; Filippova, I.A. Lake Baikal: Extreme level as a rare hydrological event. Water Resour.
**2017**, 44, 522–536. [Google Scholar] [CrossRef] - Abasov, N.V.; Bolgov, M.; Nikitin, V.M.; Osipchuk, E.N. Level regime regulation in Lake Baikal. Water Resour.
**2017**, 44, 537–546. [Google Scholar] [CrossRef]

**Figure 5.**Chronological graph of the inflow average annual values of the Lake Baikal (

**a**) and posterior density (Bayesian estimate) of the change point in the inflow time series to Lake Baikal (

**b**).

**Figure 6.**Long-term series of observed and forecast inflows. (

**a**) 1932–1976 and 1976–2020; (

**b**) 2020–2064.

**Figure 7.**Releases to the Irkutsk reservoir downstream obtained as a result of WRC by DS for series 1932–1976 and 1976–2020 (

**a**) for series 2020–2064 average and 2020–2064 low water (

**b**).

**Figure 8.**Water level of the Baikal obtained as a result of WRC by DS for series 1932–1976 and 1976–2020 (

**a**) for series 2020–2064 average and 2020-2064 low water (

**b**).

**Figure 9.**Releases to the Irkutsk reservoir downstream obtained as a result of WRC using optimization methods for series 1932–1976 and 1976–2020 (

**a**) for series 2020–2064 average and 2020–2064 low water (

**b**).

**Figure 10.**Baikal water level obtained as a result of WRC using optimization methods for series 1932–1976 and 1976–2020 (

**a**) for series 2020–2064 average and 2020–2064 low water (

**b**).

**Figure 11.**Annual (

**a**) and interval (

**b**) indicators of reliability when WRC uses DS for different time series.

**Figure 13.**Vulnerability for different time series (DS) for criteria 1, 2, 3, 9, 10, 11, 12 (

**a**) for criteria 4, 5, 6, 7, 8, (

**b**).

**Figure 14.**Annual (

**a**) and interval (

**b**) indicators of reliability when WRC use optimization for different time series.

**Figure 15.**The resilience (

**a**) normalized vulnerability (

**b**) for different time series (optimization methods).

**Figure 16.**The vulnerability for different time series (optimization methods) for criteria 1, 2, 3, 9, 10, 11, 12 (

**a**) for criteria 4, 5, 6, 7, 8, (

**b**).

**Figure 17.**Annual reliability indicators when WRC use DS and optimization (

**a**) series 1932–1976 (

**b**) 1976–2020.

**Figure 18.**Interval resilience indicators when WRC use DS and optimization (

**a**) series 1932–1976 (

**b**) 1976–2020.

**Figure 20.**Interval vulnerability when WRC use DS and optimization (series 1976–2020) for criteria 1, 2, 3, 9, 10, 11, 12 (

**a**) for criteria 4, 5, 6, 7, 8, (

**b**).

**Figure 21.**Interval vulnerability when WRC use DS and optimization (

**a**) series 1932–1976 (

**b**) series 1976–2020.

**Figure 22.**Normalized vulnerability when WRC use DS and optimization for the time series 2020–2064 average.

**Table 1.**Comparison by the sum of failures of the WRC using DS and optimization on different time series of inflow.

Time Series of Inflow | WRC Using DS | WRC Using Optimization | Improvement% |

Annual Reliability (Failure) | |||

1932–1976 | 92 | 57 | 38% |

1976–2020 | 121 | 74 | 39% |

2020–2064 average | 83 | 48 | 42% |

2020–2064 low water | 117 | 63 | 46% |

Interval reliability (failure) | |||

1932–1976 | 321 | 200 | 38% |

1976–2020 | 431 | 249 | 42% |

2020–2064 average | 261 | 142 | 46% |

2020–2064 low water | 387 | 169 | 56% |

Time Series of Inflow | WRC Using DS | WRC Using Optimization | Improvement% |
---|---|---|---|

1932–1976 | 1.96 | 1.53 | 22% |

1976–2020 | 2.78 | 1.91 | 31% |

2020–2064 average | 1.98 | 1.33 | 33% |

2020–2064 low water | 2.80 | 1.47 | 48% |

T_{INRI} | 1.33 | 1.33 |

Time Series of Inflow | WRC Using DS | WRC Using Optimization | Improvement % | T_{INVI}(DS) | T_{INVI}(Opt) |
---|---|---|---|---|---|

1932–1976 | 3.83 | 3.37 | 12% | 0.32 | 0.28 |

1976–2020 | 6.70 | 2.28 | 66% | 0.56 | 0.19 |

2020–2064 average | 5.04 | 1.27 | 75% | 0.42 | 0.11 |

2020–2064 low water | 5.78 | 3.29 | 43% | 0.48 | 0.27 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Buber, A.; Bolgov, M. Multi-Criteria Analysis of the “Lake Baikal—Irkutsk Reservoir” Operating Modes in a Changing Climate: Reliability, Resilience, Vulnerability. *Water* **2021**, *13*, 2879.
https://doi.org/10.3390/w13202879

**AMA Style**

Buber A, Bolgov M. Multi-Criteria Analysis of the “Lake Baikal—Irkutsk Reservoir” Operating Modes in a Changing Climate: Reliability, Resilience, Vulnerability. *Water*. 2021; 13(20):2879.
https://doi.org/10.3390/w13202879

**Chicago/Turabian Style**

Buber, Alexander, and Mikhail Bolgov. 2021. "Multi-Criteria Analysis of the “Lake Baikal—Irkutsk Reservoir” Operating Modes in a Changing Climate: Reliability, Resilience, Vulnerability" *Water* 13, no. 20: 2879.
https://doi.org/10.3390/w13202879