# The Impact of the Uncertain Input Data of Multi-Purpose Reservoir Volumes under Hydrological Extremes

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- The first sub-objective was to develop a simulation-optimization model of the reservoir to determine the optimal storage volume of the reservoir under conditions of input data uncertainty (UNCE_RESERVOIR). The reservoir model is based on the balance equation of the reservoir and involves optimization using the grid method with the required temporal reliability.
- (b)
- The second sub-objective was to develop a simulation model for the transformation of uncertain flood discharges to determine the retention volume of the reservoir under conditions of input data uncertainty (TRANSFORM_WAVE). The model is based on the first order of the reservoir differential equation.
- (c)
- The main objective was to link the two models and analyze what effect the optimized reservoir storage volume will have on the transformation effect of the reservoir.

## 2. Background

_{T}, which is defined as the ratio of months without water failures, represented as water deficit, and the number of all months for a given time series [60,61,62]. Category A, with the highest priority, is for R

_{T}≥ 99.5% and category D, with the lowest priority, is for R

_{T}≥ 95.0%. A water deficit is identified when the storage volume of the reservoir is in an unsatisfactory state.

## 3. Case Study

_{A}is 3.28 m

^{3}s

^{−1}. The required water outflow from the reservoir O

_{R}is 2.5 m

^{3}s

^{−1}. The storage volume V

_{Z}is 44.056 million m

^{3}at an elevation of 464.45 m above sea level, the volume of inactive storage is 3.80 million m

^{3}, the retention volume V

_{R}is 8.337 million m

^{3}and the total volume of the reservoir V

_{TOT}is 56.193 million m

^{3}. The reservoir has two bottom outlets with diameters of 1800 mm. At the maximum water level, the flow rate of the bottom outlets is maximally 2 × 40 m

^{3}s

^{−1}. The emergency spillway of the reservoir is an uncontrolled crest structure at an elevation of 467.05 m above sea level, with a total length of 60.5 m (5 × 12.1 m) and a capacity of 180.5 m

^{3}s

^{−1}at the level of the uncontrollable retention volume. According to the Czech Technical Standard [12], the reservoir falls into category A, the highest priority, i.e., with a time reliability R

_{T}≥ 99.5%. The main purposes of the reservoir are storage of drinking water and flood protection.

_{1000}according to the classical regression without seasonality differentiation and with a time step of 360 min. Hydrological data were acquired from the Czech Hydrometeorological Institute.

^{−1}per 1000 m

^{2}[63].

## 4. Methodology

#### 4.1. Problem Formulation

#### 4.2. UNCE_RESERVOIR—Simulation-Optimization Model of the Reservoir for Determining the Storage Volume of the Reservoir

_{Z}in conditions of uncertainty. To determine the optimal storage volume of the reservoir V

_{Z}= f(O

_{R}, R

_{T}), which is a function of the required outflow O

_{R}and predetermined temporal reliability R

_{T}< 100%, repeated calculations determining the temporal reliability R

_{T}= f(O

_{R}, V

_{Z}) at the predetermined required outflow O

_{R}and storage volume V

_{Z}were used.

_{Z}. The criterion was temporal reliability according to the temporal reliability R

_{T}and the water management solution allowed water failures in the reservoir according to the categorization of the reservoir. The initial condition was a full reservoir at the beginning of the test period (a value of 0 on the left side of Equation (1) characterizes the full storage volume of the reservoir) and the boundary condition was a series of water inflows into the reservoir at the appropriate time steps. For each time step, a balance was made between the required outflow O

_{R}and the historical inflow of water into the reservoir Q. In addition, the limiting condition ∑(O

_{R}− Q) was tested, i.e., whether or not the reservoir was emptied at the end of each month (the V

_{Z}

_{,MAX}value on the right side of Equation (1) characterizes the empty storage volume of the reservoir). If it was emptied, a failure of the water outflow from the reservoir was judged to exist. This meant that in all months when the outflow of water O

_{i}was less than the required outflow O

_{R}, the reservoir failed to supply enough water to the distribution system. The total sum of all failure months according to Equation (2) was recorded and the temporal reliability R

_{T}was calculated (see Equation (3)).

_{T}= f(O

_{R}, V

_{Z}) is the modified balance equation of the reservoir in the sum form converted into the following inequality (Equation (1)) [65]:

_{i}is the water outflow from the reservoir (m

^{3}s

^{−1}) in a given month for i = 1, …, k; Q

_{i}is the inflow of water into the reservoir (m

^{3}s

^{−1}) in a given month for i = 1, …, k and Δt is the time step of the calculation of one month.

_{t,i}= 1 describes the state of the V

_{Z}reservoir in the fault-free (satisfactory) time step of the calculation and Z

_{t,i}= 0 describes the state of the V

_{Z}reservoir in the faulty (unsatisfactory) time step of the calculation.

_{R}as a result of the outflow control is the probability that the actual outflow of water from the reservoir will not fall below the value of the improved outflow O

_{R}. In this case, the temporal reliability is applied according to the temporal reliability R

_{T}, which can be calculated from the values Z

_{t,i}according to Equation (2) [66]:

_{T}) is evaluated according to Equations (2) and (3). The solution is a variant in which the criterion coincides with the required value. In this variant, the selected parameter becomes the result of the solution. The task leads to an optimization in which the solved parameter is unknown and the criterion is the difference between the calculated and the required temporal reliability, which is minimized. The reservoir model uses a simple optimization method called the grid method, where parameters with a fixed step are selected at allowable intervals. The calculation also includes, among other things, water losses from the reservoir, specifically water losses by evaporation from the water surface and seepage of the dam body. The principle for introducing water losses from the reservoir into the solution is that the loss flows are counted using repeated simulation.

#### 4.3. TRANSFORM_WAVE—Reservoir Simulation Model for Determining the Retention Volume of the Reservoir

_{B}should be calculated according to Equation (4) and the emergency spillway capacity Q

_{ES}according to Equation (5):

^{2}), g is the gravity acceleration (m s

^{−2}), h

_{w}is the height of the water above the bottom outlets (m), m is the overflow coefficient (-), b is the width of the emergency spillway (m) and h

_{es}is the height of the water above the spillway (m).

#### 4.4. Monte Carlo Method for Applying Input Uncertainties to the Reservoir Simulation Model

_{i}values around the input value X resulting from the measurement completely randomly and independently of each other. The quantity X

_{i}is therefore random and independent of the previous and following values. The randomly generated quantities X

_{i}are the result of a number of mutually independent phenomena, which makes it possible to describe the input value with a corresponding normal probability distribution N(μ(X),σ(X)). The introduction of a normal probability distribution makes it possible to enter the vicinity of the resulting value of a random variable using the mean value μ(X) as the measured value and the standard deviations σ(X) as the standard uncertainty. Only the standard measurement uncertainty of type B u

_{B,X}was considered in the calculations. Finally, a simplification was introduced, whereby the standard uncertainty of measurement u

_{B,X}is deployed using the relative value of the coefficient of variation C

_{v}(X) (see Equation (10)) and the resulting standard deviation σ(X) is then calculated according to Equation (9).

_{t}(X), distribution curves F

_{t}(X) of the normal standardized probability distribution N(μ(X),σ(X)) are created for t = 1, 2, …, NE, where NE is the total number of elements (e.g., the total number of average monthly inflows or the total number of points from the flooded volume line). Using a pseudo-random number generator, generating random numbers from the interval ξ ∈ 〈0,1〉, random waveforms of a number of elements X

_{t}are repeatedly generated, which are referred to as random positions of NX

_{t,i}values, in the interval of specified uncertainty for i = 1, 2, …, NG, where NG is the total number of repetitions (generations). The general principle for generating random positions of input parameters can be found in previous studies [52,53]. This described procedure for generating random elements can be applied to all quantities entering the water management solution for the storage and retention volume of the reservoir, except the bathygraphic curves of the reservoir. In this case, a compilation of two independent Monte Carlo generators was required. Each generator constructs a random point position (e.g., water level height) with a second random point position added to it (e.g., the volume of water in the reservoir). Together, the random positions of two points then create a random point coordinate (e.g., a random point coordinate of a flooded volume line). A series of random points then form random lines of flooded volumes burdened with uncertainties. A symbolic depiction of the introduction of considered quantities burdened with uncertainties is shown Figure 4.

#### 4.5. Methods for Evaluation

#### 4.5.1. Mean Value

_{i}are elements of random selection and n is the number of elements of random selection.

#### 4.5.2. Variance and Standard Deviation

#### 4.5.3. Coefficient of Variation

_{v}(x) and is expressed as the ratio of the standard deviation and the mean value (Equation (10)):

#### 4.5.4. Coefficient of Variation

#### 4.5.5. Quantile

_{x}(x), the p-quantile x

_{p}is a value of a random variable X for which values less than x

_{p}occur only with probability p, i.e., for which the distribution function F

_{x}(x

_{p}) is equal to the probability p (Equation (12)):

## 5. Results and Discussion

#### 5.1. Storage Volume Modeling

_{R}reservoir for temporal reliability R

_{T}≥ 99.5% and existing V

_{Z}. The calculation was performed without input uncertainties, including consideration of water losses from the reservoir. As a result, the O

_{R}had to be reduced to achieve a satisfactory R

_{T}= f(O

_{R}, V

_{Z}), as shown in Table 1.

_{Z}was calculated deterministically and without input uncertainties, including the consideration of water losses from the reservoir for the calculated O

_{R}based on Table 1 and for R

_{T}≥ 99.5%, i.e., the calculation of V

_{Z}= f(O

_{R}, R

_{T}= 99.5%). The resulting optimized V

_{Z}should be close to the real reservoir volume. Based on the comparison of the optimized V

_{Z}with the real V

_{Z}, the O

_{R}was slightly changed. These values are given in Table 2 along with the O

_{R}value for the further calculations that follow.

_{R}needed to meet the significance of the reservoir with regard to the updated line of water inflow into the reservoir and the resulting V

_{Z}approaching the real V

_{Z}, the optimized storage volumes V

_{Z}of the reservoir for the whole range of water inflow into the reservoir and with input uncertainties were calculated and evaluated. Input uncertainties from the measurements were applied: (i) constantly for all inputs with sizes u

_{B}= ±1, ±2, ±3, ±5 and ±7% and (ii) differently for entered values according to the probable size of uncertainty for each input; specifically, ±3% for the inflow of water into the reservoir, ±5% for bathygraphic curves, ±4% for evaporation and ±3% for seepage through the reservoir body. The number of repetitions (generations) NG was always set to 300 repetitions.

_{B}= ±3%.

^{3}± 1.622 million m

^{3}, i.e., the result lay in the interval {42.527 million m

^{3}; 45.771 million m

^{3}}. To be on the safe side, it is desirable to work with the resulting upper interval, i.e., a higher storage volume, in a stochastic solution. Expressed by the 95% quantile, the resulting optimized volume was 45.628 million m

^{3}; compared to the upper quantile, this value of V

_{Z}is 0.31% lower. The relatively safe and therefore recommended final value of the storage volume from this analysis was 45.770 million m

^{3}. Based on the updated input series of water inflows into the reservoir and the introduction of input uncertainties, including consideration of water losses from the reservoir, it is therefore recommended that the existing storage volume of the Vír I reservoir be increased by up to 3.9%, specifically by 1.71 million m

^{3}.

_{Z}with the probable different input uncertainties.

_{Z}) compared to the existing V

_{Z}depending on the input uncertainties u

_{B}were expressed as percentages. The results are shown in Table 4.

_{Z}) demonstrate a steady increase depending on u

_{B}. Testing on another reservoir, Vranov (line three), showed that there are not always such steady increases in results but that these depend on the growing uncertainties of the input data. Thus, different reservoirs can react completely differently to input uncertainties.

_{Z}, which were sorted from minimum to maximum. Each overshoot probability curve corresponds to one type of input uncertainty setting. Furthermore, the 95% quantile (i.e., 5%) used for evaluation is marked on each curve.

_{Z}for all courses take on values just above 44 million m

^{3}, which is similar to the deterministic solution V

_{Z}= 44.056 million m

^{3}. This confirms the correctness of the random number generator. The courses of the individual curves are relatively symmetrical. The resulting mean values of the optimized storage volumes μ(V

_{Z}) in Table 3 increase in comparison with the current value of the storage volumes by percentage values from 0.10% (for u

_{B}= ±5%) to + 0.22% (for u

_{B}= ±3%). The average of deviations for all input uncertainties was about + 0.09%. The obtained results for the stochastic optimized storage volumes reach only slightly higher values than in the deterministic solution. These factors again confirm the correctness of the random number generator and the appropriateness of using the Monte Carlo method.

#### 5.2. Retention Volume Modeling

_{Z}values of the reservoir, Table 5 shows the specific V

_{Z}values of the optimal storage volumes in column three and the corresponding water heights in the reservoir h

_{Vz}in column two. It is clear that, with the increasing quantile (downwards in Table 5), the optimal value of V

_{Z}increases and therefore the height of water in the reservoir h

_{Vz}also increases. Since the emergency spillway of the reservoir is always considered at the same height h

_{VRC}(column four) in accordance with the chosen location, the increase of controllable retention V

_{RC}(column five) and h

_{VRC}is at the expense of the uncontrollable retention volume h

_{VRU}(column six) and V

_{RU}(column seven).

_{1000}with an input standard uncertainty u

_{B}= ±10% was selected, which was chosen as the minimum following [68], in which the reliability classes of hydrological data are given, including the probable variance of errors. The calculation was performed in the variant without pre-drainage of water from the reservoir. The number of uncertainties generated for the flood discharge was chosen to be 300 repetitions. The starting water level in the reservoir at the beginning of the flood discharge transformation solution was always the full storage volume V

_{Z}, i.e., the newly calculated height h

_{Vz}. Based on the above information on bottom outlets and the emergency spillway along with Equations (4) and (5), the outlet coefficient of the bottom outlets or the outflow coefficient μ and the overflow coefficient m were determined. Specifically, for the tested reservoir, the outflow coefficient μ = 0.435 (-) and overflow coefficient m = 0.407 (-). In the variant without pre-drainage, the level is kept at the level of the full storage volume and, as soon as this level is exceeded, the bottom outlets are opened to a harmless flow Q

_{NE}. When the emergency spillway is exceeded, the bottom outlets are smoothly closed and, after the flooding, they are smoothly opened to a water height of 0.5 m above the emergency spillway. The level after the flood is again kept at the level of the full storage volume.

_{Z}values. The generated courses of these flood discharges are shown here in red, the results of the transformed waves in blue and the courses of the water heights in the reservoir during the transformations in green.

_{Z}. The input uncertainty has a clear effect on flood discharges but also on the results of transformed floods and water heights in the reservoir.

#### 5.3. Summary of Results

_{VRU}, the height of the uncontrollable retention volume or the achieved peak of the height of the water in the reservoir during the transformations of flood discharges. These values include the expanded uncertainty of ±2σ. In Table 5, column seven then shows the results of the uncontrollable retention volume V

_{RU}, including the upper limit of the expanded uncertainty +2σ. Furthermore, column eight shows the total volume of the retention volume V

_{R}, including the upper limit of the expanded uncertainty +2σ (bold), and the total volume of the reservoir V

_{TOTAL}is shown in column nine. Finally, the last columns, 10 and 11, show the values of Q

_{1000}flood peaks, including the expanded uncertainty ±2σ, and the height between the uncontrollable retention volume (maximum water level in the reservoir) and the control maximum level CML (dam height), including the upper limit of the expanded uncertainty +2σ (bold).

_{Z}. Specifically, the first row shows the values for the current state of the reservoir according to [63] and the second row contains the results of the transformation of the flood discharge Q

_{1000}for the current state of V

_{Z}. The following rows show the results for selected quantiles of optimal V

_{Z}values.

_{VRC}level), the following decrease: (i) the controllable retention volume V

_{RC}, (ii) the height of the uncontrollable retention volume h

_{VRU}, (iii) the uncontrollable retention volume of the reservoir V

_{RN}, (iv) the retention volume of the reservoir V

_{R}and (v) the total reservoir volume V

_{TOTAL}. The decrease in h

_{VRU}has the effect of increasing the difference in size between the h

_{VRU}and CML (column 11), which is desirable for the solution. On the other hand, the peak flow Q

_{PEAK}increases. It should be noted that, although the values of volumes and heights decrease (columns five to nine), they are significantly higher (lower for column 11) than the actual values of the existing tested reservoir.

_{Z}(starting level) corresponds to a larger volume from the line of flooded volumes and, therefore, more water is captured in the initial flood step and subsequent steps than at the starting height of 63 m. As a result, the resulting retention volume and total reservoir volume decrease with increasing height V

_{Z}. In contrast, the difference between the h

_{VRU}and CML increases, which is a key parameter when designing or changing the functional volumes of a reservoir. This suggests a design for the safest solution in terms of optimal V

_{Z}, i.e., either the 95% quantile of V

_{Z}or the upper limit of V

_{Z}(+2σ). In these designs, the increases in the retention volume V

_{R}by 1.37 million m

^{3}and 1.09 million m

^{3}would correspond to increases of 16.5% and 13.1% compared to the actual V

_{R}. However, we must not forget that with this choice the peak flow of the transformed flood discharge increases. For example, for the transformation from the current state, the peak flow of the upper limit of the expanded uncertainty (+2σ) is 234.65 m

^{3}s

^{−1}and, for the transformation from the 95% quantile of V

_{Z}, the peak flow is 248.82 m

^{3}s

^{−1}, which is another key parameter in the design of functional volumes of a reservoir.

_{Z}values were also evaluated using quantiles, as was the case with the storage volume of the reservoir. Figure 10 shows the results of water height peaks in the reservoir during transformations of generated flood discharges burdened with input uncertainties u

_{B}= ±10% in the form of the probabilities of exceeding these peaks for selected optimal storage volumes (black and white shades) and for the current reservoir storage volume (brown shade).

_{Z}quantiles) lower peaks of the water level in the reservoir are achieved, i.e., the h

_{VRU}level or the maximum water level in the reservoir does not rise so high. At the same time, selected quantiles and the upper limit +2σ are marked on the plotted curves, similarly to the selection of optimized V

_{Z}. The complete results for the water height peaks in the reservoir with possible functional volumes are summarized in detail in Table 6, which is designed in the same form as Table 5. This table shows the results of flood discharge transformations for selected quantiles and the upper limit +2σ. Peak flow Q

_{PEAK}is expressed for the given variants in the form +2σ.

_{Z}at the expense of the size of the flood peak because, with such a solution, higher water height peaks in the reservoir during the transformation are achieved but, at the same time, there is a greater height between the h

_{VRU}and CML. In the case of large floods, which the Q

_{1000}undoubtedly is, the safety of the dam itself must be considered, i.e., elimination of the overflow of the CML.

## 6. Conclusions and Recommendations

- Input uncertainty significantly affects the results of V
_{Z}and V_{R}calculations. - To be on the safe side, it is appropriate to increase the values of either V
_{Z}or V_{R}in accordance with the calculated uncertainties. Specifically, the input uncertainties discussed here highlighted the need to increase the existing V_{Z}of the tested reservoir by up to 1.71 million m^{3}(3.9%) and the existing V_{R}by up to 1.37 million m^{3}(16.5%). - For a comprehensive determination of functional volumes, calculations of the transformation of the updated flood discharge burdened with uncertainty for selected optimal values of V
_{Z}were performed. These led to the determination of how an increase in V_{Z}can affect the transformation of the flood discharge and the change in the V_{R}of the reservoir. - Based on the above, Table 6 was created with solution options for V
_{Z}and V_{R}under conditions of uncertainty, including possible flood peaks and water height peaks in the reservoir. - The developed simulation-optimization (i) and simulation (ii) models of the reservoir, the methods used and the introduction of uncertainties on the input data proved their functionality in solving the functional volumes of the water in the reservoir.
- Uniqueness can be observed in the connection between the solutions of the functional volumes of the reservoir for input data under conditions of uncertainty.
- The source codes of both models are written in such a way as to maintain generality and thus can be quickly used to test other existing or planned reservoirs anywhere in the world, if suitable data are available.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Principle of the Klemeš graphical method [43].

**Figure 4.**Symbolic depiction of the introduction of considered quantities burdened with uncertainties.

**Figure 5.**Courses of filling and emptying of the optimized V

_{Z}of the tested reservoir for constant u

_{B}= ±1, ±2, ±3, ±5 and ±7% and for different u

_{B}.

**Figure 6.**Bar chart of the resulting optimized storage volumes μ(V

_{Z}) of the Vír I reservoir for ±2σ(V

_{Z}) and 95% quantiles of the tested input uncertainties u

_{B}.

**Figure 7.**Overshoot probability curves for the resulting optimized storage volumes of the tested reservoir for different sizes of input uncertainties.

**Figure 8.**Optimal V

_{Z}values selected from the overshoot probability curve of the obtained optimized storage volumes of the Vír I reservoir.

**Figure 9.**Results of the transformation of generated Q

_{1000}waves and water heights in the reservoir for the input u

_{B}= ±10% for the 75, 80, 85, 90, 95% quantiles and for the upper limit +2σ of the optimal V

_{Z}.

**Figure 10.**Lines indicating the probabilities of exceeding the calculated water height peaks in the reservoir during the transformation of the uncertain flood discharge Q

_{1000}for selected optimized storage volumes.

**Table 1.**Temporal reliability R

_{T}results for changing input O

_{R}for the updated data regarding water inflow into the reservoir.

O_{R} (m^{3} s^{−1}) | >>> | R_{T} (%) |
---|---|---|

2.5 | 98.776 | |

2.4 | 99.028 | |

2.3 | 99.533 | |

2.31 | 99.404 |

**Table 2.**O

_{R}results for varying input O

_{R}and R

_{T}≥ 99.5% for the updated data regarding water inflow into the reservoir.

O_{R} (m^{3} s^{−1}) | >>> | V_{Z} (m^{3}) |
---|---|---|

2.3 | 43,657,000 | |

2.31 | 44,371,700 | |

2.305 | 44,069,000 |

(m^{3}) | u_{B} = ±0% | u_{B} = ±1% | u_{B} = ±2% | u_{B} = ±3% | u_{B} = ±5% | u_{B} = ±7% | u_{B} Different |
---|---|---|---|---|---|---|---|

μ(Vz) | 44,069,000 | 44,098,652 | 44,121,960 | 44,154,544 | 44,010,168 | 44,078,572 | 44,148,504 |

±2σ(Vz) | 0 | 545,346 | 1,137,567 | 1,627,581 | 2,574,596 | 3,958,923 | 1,621,724 |

Vz_{bottom 2σ}_{(Vz)} | 44,069,000 | 43,553,306 | 42,984,393 | 42,526,963 | 41,435,572 | 40,119,649 | 42,526,780 |

Vz^{upper 2σ}^{(Vz)} | 44,069,000 | 44,643,998 | 45,259,527 | 45,782,125 | 46,584,764 | 48,037,495 | 45,770,228 |

95%_{quant. Vz} | 44,069,000 | 44,673,900 | 45,114,800 | 45,496,700 | 46,569,400 | 47,560,700 | 45,628,200 |

**Table 4.**Percentage increases (%) of the resulting variances ±2σ(V

_{Z}) of the optimized reservoir volume depending on the input uncertainties u

_{B}for the tested reservoir and another reservoir.

u_{B} = ±1% | u_{B} = ±2% | u_{B} = ±3% | u_{B} = ±5% | u_{B} = ±7% | u_{B} Different | |
---|---|---|---|---|---|---|

Vír | 1.24 | 2.58 | 3.69 | 5.84 | 8.99 | 3.68 |

Vranov | 8.04 | 8.35 | 9.14 | 10.89 | 13.42 | 9.20 |

**Table 5.**The calculated volumes of water in the reservoir and the corresponding heights of water in the reservoir for selected optimal V

_{Z}values (m

^{3}), including the peak size of the transformed flood discharges Q

_{1000}.

h_{Vz}(m) | Optimal V_{Z} (m ^{3}) | h_{VRC}(m) | V_{RC}(m ^{3}) | h_{VRU}(m) | V_{RU}(m ^{3}) | V_{R}(m ^{3}) | V_{TOTAL}(m ^{3}) | Q_{PEAK}(m ^{3} s^{−1}) | Height to CML (m) | |
---|---|---|---|---|---|---|---|---|---|---|

Current state | 63.00 | 44,056,000 | 65.60 | 5,286,000 | 67.00 | 3,051,000 | 8,337,000 | 56,193,000 | - | 2.00 |

Calculation for the current state | 63.00 | 44,056,000 | 65.60 | 5,286,000 | 67.80 ± 0.64 | 5,002,000 + 1,554,000 | 10,288,000 11,842,000 | 58,144,000 59,698,000 | 172.11 ± 62.54 | 1.20 0.56 |

75% quantile V_{Z} | 63.32 | 44,682,300 | 65.60 | 4,659,700 | 67.80 ± 0.61 | 4,999,000 + 1,499,000 | 9,658,700 11,157,700 | 58,141,000 59,640,000 | 177.91 ± 61.36 | 1.20 0.58 |

80% quantile V_{Z} | 63.41 | 44,858,400 | 65.60 | 4,483,600 | 67.80 ± 0.60 | 4,990,000 + 1,471,000 | 9,473,600 10,944,600 | 58,132,000 59,603,000 | 179.68 ± 61.56 | 1.20 0.60 |

85% quantile V_{Z} | 63.47 | 44,984,800 | 65.60 | 4,357,200 | 67.78 ± 0.60 | 4,950,000 + 1,458,000 | 9,307,200 10,765,200 | 58,092,000 59,550,000 | 181.12 ± 61.54 | 1.22 0.62 |

90% quantile V_{Z} | 63.64 | 45,310,700 | 65.60 | 4,031,300 | 67.71 ± 0.55 | 4,770,000 + 1,336,000 | 8,801,300 10,137,300 | 57,912,000 59,248,000 | 185.86 ± 61.43 | 1.29 0.75 |

95% quantile V_{Z} | 63.79 | 45,628,200 | 65.60 | 3,713,800 | 67.70 ± 0.51 | 4,755,000 + 1,239,000 | 8,468,800 9,707,800 | 57,897,000 59,136,000 | 188.41 ± 60.41 | 1.30 0.79 |

Upper limit V_{Z} (+2σ) | 63.90 | 45,770,228 | 65.60 | 3,571,772 | 67.67 ± 0.49 | 4,676,000 + 1,181,000 | 8,247,772 9,428,772 | 57,818,000 58,999,000 | 190.87 ± 59.59 | 1.33 0.85 |

**Table 6.**The calculated volumes of water in the reservoir and their corresponding heights for selected optimal V

_{Z}values (m

^{3}), including the sizes of peaks of transformed flood discharges Q

_{1000}for selected quantiles and the upper limit +2σ.

h_{Vz}(m) | Optimal V_{Z}(m ^{3}) | h_{VRC}(m) | V_{RC}(m ^{3}) | Selected Quantiles and V_{RU} (+2σ) | h_{VRU}(m) | V_{RU}(m ^{3}) | V_{R}(m ^{3}) | V_{TOTAL}(m ^{3}) | Upper limit (+2σ) Q_{PEAK}(m ^{3} s^{−1}) | Height to CML (m) | |
---|---|---|---|---|---|---|---|---|---|---|---|

For the current state | 63.00 | 44,056,000 | 65.60 | 5,286,000 | 75% quan. V_{RU} | 68.02 | 5,533,000 | 10,819,000 | 58,675,000 | 234.65 | 0.98 |

80% quan. V_{RU} | 68.07 | 5,655,000 | 10,941,000 | 58,797,000 | 0.93 | ||||||

85% quan. V_{RU} | 68.15 | 5,857,000 | 11,143,000 | 58,999,000 | 0.85 | ||||||

90% quan. V_{RU} | 68.26 | 6,118,000 | 11,404,000 | 59,260,000 | 0.74 | ||||||

95% quan. V_{RU} | 68.43 | 6,532,000 | 11,818,000 | 59,674,000 | 0.57 | ||||||

up. l. V_{RU} (+2σ) | 68.44 | 6,556,000 | 11,842,000 | 59,698,000 | 0.56 | ||||||

75% quantile V_{Z} | 63.32 | 44,682,300 | 65.60 | 4,659,700 | 75% quan. V_{RU} | 68.04 | 5,582,000 | 10,241,700 | 58,724,000 | 239.27 | 0.96 |

80% quan. V_{RU} | 68.08 | 5,679,000 | 10,338,700 | 58,821,000 | 0.92 | ||||||

85% quan. V_{RU} | 68.17 | 5,898,000 | 10,557,700 | 59,040,000 | 0.83 | ||||||

90% quan. V_{RU} | 68.26 | 6,118,000 | 10,777,700 | 59,260,000 | 0.74 | ||||||

95% quan. V_{RU} | 68.36 | 6,362,000 | 11,021,700 | 59,504,000 | 0.64 | ||||||

up. l. V_{RU} (+2σ) | 68.42 | 6,498,000 | 11,157,700 | 59,640,000 | 0.58 | ||||||

80% quantile V_{Z} | 63.41 | 44,858,400 | 65.60 | 4,483,600 | 75% quan. V_{RU} | 68.03 | 5,557,000 | 10,040,600 | 58,699,000 | 241.24 | 0.97 |

80% quan. V_{RU} | 68.11 | 5,752,000 | 10,235,600 | 58,894,000 | 0.89 | ||||||

85% quan. V_{RU} | 68.15 | 5,857,000 | 10,340,600 | 58,999,000 | 0.85 | ||||||

90% quan. V_{RU} | 68.24 | 6,069,000 | 10,552,600 | 59,211,000 | 0.76 | ||||||

95% quan. V_{RU} | 68.33 | 6,288,000 | 10,771,600 | 59,430,000 | 0.67 | ||||||

up. l. V_{RU} (+2σ) | 68.40 | 6,461,000 | 10,944,600 | 59,603,000 | 0.60 | ||||||

85% quantile V_{Z} | 63.47 | 44,984,800 | 65.60 | 4,357,200 | 75% quan. V_{RU} | 68.02 | 5,533,000 | 9,890,200 | 58,675,000 | 242.66 | 0.98 |

80% quan. V_{RU} | 68.08 | 5,679,000 | 10,036,200 | 58,821,000 | 0.92 | ||||||

85% quan. V_{RU} | 68.16 | 5,874,000 | 10,231,200 | 59,016,000 | 0.84 | ||||||

90% quan. V_{RU} | 68.20 | 5,972,000 | 10,329,200 | 59,114,000 | 0.80 | ||||||

95% quan. V_{RU} | 68.30 | 6,215,000 | 10,572,200 | 59,357,000 | 0.70 | ||||||

up. l. V_{RU} (+2σ) | 68.38 | 6,408,000 | 10,765,200 | 59,550,000 | 0.62 | ||||||

90% quantile V_{Z} | 63.64 | 45,310,700 | 65.60 | 4,031,300 | 75% quan. V_{RU} | 67.90 | 5,241,000 | 9,272,300 | 58,383,000 | 246.29 | 1.10 |

80% quan. V_{RU} | 67.98 | 5,436,000 | 9,467,300 | 58,578,000 | 1.02 | ||||||

85% quan. V_{RU} | 68.02 | 5,533,000 | 9,564,300 | 58,675,000 | 0.98 | ||||||

90% quan. V_{RU} | 68.11 | 5,752,000 | 9,783,300 | 58,894,000 | 0.89 | ||||||

95% quan. V_{RU} | 68.17 | 5,898,000 | 9,929,300 | 59,040,000 | 0.83 | ||||||

up. l. V_{RU} (+2σ) | 68.25 | 6,106,000 | 10,137,300 | 59,248,000 | 0.75 | ||||||

95% quantile V_{Z} | 63.79 | 45,628,200 | 65.60 | 3,713,800 | 75% quan. V_{RU} | 67.90 | 5,241,000 | 8,954,800 | 58,383,000 | 248.82 | 1.10 |

80% quan. V_{RU} | 67.95 | 5,362,000 | 9,075,800 | 58,504,000 | 1.05 | ||||||

85% quan. V_{RU} | 67.99 | 5,460,000 | 9,173,800 | 58,602,000 | 1.01 | ||||||

90% quan. V_{RU} | 68.05 | 5,606,000 | 9,319,800 | 58,748,000 | 0.95 | ||||||

95% quan. V_{RU} | 68.12 | 5,777,000 | 9,490,800 | 58,919,000 | 0.88 | ||||||

up. l. V_{RU} (+2σ) | 68.21 | 5,994,000 | 9,707,800 | 59,136,000 | 0.79 | ||||||

Upper limit V_{Z} (+2σ) | 63.90 | 45,770,228 | 65.60 | 3,571,772 | 75% quan. V_{RU} | 67.85 | 5,118,000 | 8,689,772 | 58,260,000 | 250.46 | 1.15 |

80% quan. V_{RU} | 67.89 | 5,216,000 | 8,787,772 | 58,358,000 | 1.11 | ||||||

85% quan. V_{RU} | 67.93 | 5,313,000 | 8,884,772 | 58,455,000 | 1.07 | ||||||

90% quan. V_{RU} | 68.00 | 5,484,000 | 9,055,772 | 58,626,000 | 1.00 | ||||||

95% quan. V_{RU} | 68.07 | 5,655,000 | 9,226,772 | 58,797,000 | 0.93 | ||||||

up. l. V_{RU} (+2σ) | 68.15 | 5,857,000 | 9,428,772 | 58,999,000 | 0.85 |

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Paseka, S.; Marton, D.
The Impact of the Uncertain Input Data of Multi-Purpose Reservoir Volumes under Hydrological Extremes. *Water* **2021**, *13*, 1389.
https://doi.org/10.3390/w13101389

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Paseka S, Marton D.
The Impact of the Uncertain Input Data of Multi-Purpose Reservoir Volumes under Hydrological Extremes. *Water*. 2021; 13(10):1389.
https://doi.org/10.3390/w13101389

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Paseka, Stanislav, and Daniel Marton.
2021. "The Impact of the Uncertain Input Data of Multi-Purpose Reservoir Volumes under Hydrological Extremes" *Water* 13, no. 10: 1389.
https://doi.org/10.3390/w13101389