# Uncertainty Impact on Water Management Analysis of Open Water Reservoir

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Monte Carlo Method

_{i}corresponding to the interval of initial uncertainty are generated. Value X

_{i}is considered as a random and independent value to the values X

_{i-1}and X

_{i+1}. This presumption allows the use of the normal probability distribution N(μ(X),σ(X)). Then, the input value X

_{i}is considered as the mean μ(X) and uncertainty size is defined as the standard deviation σ(X). Subsequently to each mean μ(X

_{i}), a cumulative distribution function F(X) of the standard normal probability distribution is created. The random number generator creates random numbers from the interval and then the position of the random value NX

_{i}is calculated.

_{i}, Nh

_{i}) is similar to the above mentioned theory. The difference is given by the construction of random points which requires building two independent Monte Carlo generators. Each generator will design a random position of point NX

_{i}(e.g., water level elevation Nh

_{i}) and will add a random value NY

_{i}to it (of the water volume in the reservoir NV

_{i}). For the line of the area–volume curve, the result is a random point coordinate (NV

_{i}, Nh

_{i}) of the line of the elevation–volume curve, see Figure 1.

#### 2.2. Reservoir Simulation Model and Reliability Assessment

_{i}is the water outflow from the reservoir, Q

_{i}is the water inflow into the reservoir for i = 1, …, k, Δt is the time step of the calculation (one month). O

_{i+1}is the water outflow from the reservoir in the subsequent time step in step i+1. If the sum in (1) and (2) is less than zero, the value O

_{i+1}will be substituted by the value of the reservoir outflow or water demand, called the required improved outflow O

_{p}. The required outflow O

_{p}is defined as the total outflow from the reservoir. In times of inflow water deficit, when the storage capacity is using for water supply, the required outflow consists of ecological outflow Q

_{ECO}and water consumption for water supply. The time course of the calculated sum simulates the course of the emptying of the reservoir storage volume by time steps i = 1, …, k. For i = 0, the initial condition of the solution needs to be entered after the sum value.

_{Z,max}empty storage capacity. These boundaries characterized the active conversation storage capacity, which is possible to use. From the argument calculation in (2), the actual emptying of the reservoir is obtained, called V’

_{Z,i+1}, which is then tested if it is in a given interval 〈0,V

_{Z,max}〉. If not, it is important to find the value O

_{i+1}, when the argument in the summation is equal to zero—then, a manipulated outflow is created, or a given argument is equal V

_{Z,max}—a failure or unsatisfactory state is created.

_{t,i}= 1 describes the state of the reservoir storage capacity in the satisfactory time step of the calculation. Z

_{t,i}= 0 describes the state of the reservoir storage capacity in the unsatisfactory (failure) time step of the calculation. The given reliability used in the paper is known as temporal reliability or time based reliability and can be calculated from Z

_{t,i}values. Each value Z

_{t,i}represents a month. The reliability R

_{T}is defined as Equation (4)

_{T}time-based reliability is calculated. These sets of simulation results are further statistically evaluated. Basic statistical analysis consists of a statistical histogram and statistical characteristic; (i) the mean value μ, (ii) dispersion D, (iii) standard deviation σ, (iv) variation coefficient C

_{v}and (v) excess coefficient C

_{a}.

## 3. Practical Application

_{a}was consequently divided among individual monthly values of evaporation according to the standard Reservoir storage capacity analysis (ČSN 75 2405, 2004) [27]. Bathymetric curves are determined using the GIS software and a DTM—Digital Terrain Model.

_{z}for 100% reliability. Furthermore, the reliability of the reservoir storage volume is analyzed, as well.

_{a}is 700 mm from the estimated water level altitude of approximately 460 m a.s.l. Improved outflow from the reservoir O

_{p}ranged between 0.6 and 0.8 (60% to 80%) of the reservoir yield. According McMahon and Adeloye [28], the yield is the controlled release from the reservoir system and is often expressed as a ration or percentage of the mean annual inflow to the reservoir. During calculation, many different possibilities of reservoir yield have tested. A yield interval from 0.6 to 0.8 is taken into account according to the best utilization of water inflow conditions. The input value of extended uncertainty of the storage volume is entered constantly for all parameters within the range of ±6% and ±9%. For all uncertainties, a uniform distribution is considered. The presented initial uncertainty evaluation is considered as more conservative, rising from uncertainty of measurement. The number of repetitions of random input parameter generation using the Monte Carlo method equaled 300. A total of 300 repetitions were done due to two reasons: first, better statistical evaluation; second, 300 repetitions is the best ratio between the value according to computation time and the accuracy of results. For these two reasons, the different number of repetitions was tested.

_{a}is 4.087 m

^{3}·s

^{−1}.

## 4. Results

_{z}) is considered as the result of the calculation following statistical evaluation. The value 3σ(V

_{z}) subsequently shows the value of maximum uncertainty of the storage volume covering 99.97% of the volume occurrence probability in the observed set of realizations.

^{3}for the reservoir yield 0.7. Dead space is considered as 10% of the storage volume. Due to the uncertainties entered into the calculation, for the reservoir yield yield = 0.7 the resulting reliability is not R

_{T}= 100% as mentioned in Table 1 but only 99.90% for both input uncertainties ±6 % and ±9 %. In order to achieve 100% reliability for uncertainty ±6 %, we must decrease the required outflow, in particular, to yield = 0.693, and for uncertainty ±9 % even down to yield = 0.690. The decrease is determined by the randomness of input values, or input uncertainty which in a certain number of cases undervalues the series, thereby also causing the decrease of the value of reliability.

_{T}) is considered as the result of the calculation; the values 2σ(R

_{T}) and 3σ(R

_{T}) then describe the size of uncertainty occurring around the result of the calculation. The analysis has been done for the reservoir yield values of 0.7 to 0.78.

_{p}= 2.861 m

^{3}s

^{-1}for the reservoir yield = 0.7. For the mentioned outflow, the storage volume is calculated for 100% reliability of water outflow from the reservoir. The results can be interpreted as follows. The mean value of the storage volume is considered as the resulting value. During the check of the calculation correctness, the storage volume has calculated also for the deterministic solution. Its value is almost identical to the calculation in the stochastic solution. In the deterministic solution, the storage volume is V

_{z}= 44,127,380 m

^{3}. If, along with the results, we also consider the uncertainties entering the solution, the results will become markedly skewed. The storage volume with consideration of input uncertainties corresponding to the value 3σ∙μ(V

_{z}) can be presented this way. For the value of input uncertainty ±6%, the storage volume lies within the interval V

_{Z}∈〈42,281,741 m

^{3}; 45,942,059 m

^{3}〉 with the volume uncertainty being ±4.15%. For ±9% of input uncertainty, the volume range exceeds ±6% of the uncertainty interval. The storage volume ranges within V

_{Z}∈〈41,362,777 m

^{3}; 46,848,575 m

^{3}〉.

_{P}= 2.984 m

^{3}s

^{-1}as R

_{T}= 99.53% ± 0.18%, or in other words, it will lie within the interval R

_{T}∈〈99.35%; 99.71%〉; and R

_{T}= 99.54% ± 0.20% then works out for ±9% of the input uncertainty, or in other words, it lies within the interval R

_{T}∈〈99.34%; 99.74%〉.

## 5. Conclusions

_{T}≥ 99.5%, B–R

_{T}≥ 98.5%, C–R

_{T}≥ 97.5%, D–R

_{T}≥ 95%.

_{T}= 100.00% corresponds to the random ensembles with there being a 50.33% probability of zero failure month occurring, that is 151 random ensembles from 300 repetitions. Reliability R

_{T}= 99.87% corresponds to the random ensembles with there being a 19.00% probability of one failure month occurring, that is 57 of 300. Reliability R

_{T}= 99.74% corresponds to the random ensembles with there being a 30.67% probability of two failure months occurring, that is 92 of 300 repetitions. In general, it can be said that the reliability complying with the highest category of reservoir operational reliability according to Czech legislation Category A - R

_{T}≥ 99.5%, will be attained or exceeded with a 100.00% occurrence of probability, or in other words, all 300 random ensembles agree with this requirement. Interpretation of results as per Table 2 can be described in the following way. Reliability with the assumption 3σ (99.97% occurrence probability) for the same case, i.e., uncertainty ±9% and O

_{P}= 2.861 m

^{3}·s

^{−1}is R

_{T}= 99.90 ± 0.34% or R

_{T}∈〈99.56%; 100.00%〉. This interval will also comply with the whole range of Category A.

_{T}= 99.74% corresponding to the random solution with two failure months will occur with an 8.00% probability, that means 24 ensembles from 300 repetitions. Reliability R

_{T}= 99.62% corresponding to the occurrence of three failure months in the random solution will correspond to a 91.33% occurrence probability, or 274 of 300. For four failure months, the reliability is R

_{T}= 99.48%, corresponding to a 0.67% occurrence, which falls upon only two random solutions out of 300. Category A defined will be attained or exceeded with a 99.33% occurrence probability, or in other words, 298 random ensembles out of 300 will meet this requirement. On the contrary, a solution not complying with Category A but still complying with Category B (R

_{T}≥ 98.5%) will occur with a 0.67% probability—that means, only in two cases out of 300. As per Table 2 and again input uncertainty ±9% and O

_{P}= 2.943 m

^{3}·s

^{–1}is R

_{T}= 99.63 ± 0.11% or R

_{T}∈〈99.52%; 99.74%〉. This interval will comply with the whole range of Category A of hydraulic structure reliability.

_{T}∈〈98.93%; 99.65%〉; however, in yield = 0.76, there are 56% random ensembles, 168 cases of ensembles from 300, out of class B. Yield = 0.78 is out of class B for all random ensembles.

^{3}has been designed. If we consider the resulting storage volume corresponding to the value of 44.1 hm

^{3}and an uncertainty of ±2.7 hm

^{3}, then the reservoir flood protection volume may be up to a half affected by the uncertainty of the storage capacity design. This also relates to the design of the height of the top of the dam which can be approximately ±1.2 m of the total dam height. Figure 6 shows the connection of the uncertainty of the reservoir storage volume to all the design and operation parameters of the reservoir.

_{T}may differ, as well.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Ministry of the Environment. Strategy for Adaptation to Climate Change within the Conditions of the Czech Republic. Available online: http://www.mzp.cz/cz/zmena_klimatu_adaptacni_strategie (accessed on 29 November 2016).
- Knight, F.H. Risk, Uncertainty, and Profit. Boston, Hart, Schaffner & Marx; Houghton Mifflin Company: Boston, MD, USA, 1921. [Google Scholar]
- WECC Doc. 19-1990: “Guidelines for Expression of the Uncertainty in Calibrations”. 1990. Available online: http://www.qcalibration.com/image/uncertainty.pdf (accessed on 29 November 2016).
- International Organization for Standardization. ISO Guide to Expression of Uncertainty in Measurement; International Organization for Standardization: Geneva, Switzerland, 1993. [Google Scholar]
- International Organization for Standardization. ISO GUM Suppl. 1 (DGUIDE 99998) Guide to the expression of uncertainty in measurement (GUM)—Supplement 1: Numerical Methods for the Propagation of Distributions; International Organization for Standardization: Geneva, Switzerland, 2004. [Google Scholar]
- Český Metrologický Institut. TPM 0051-93 Stanovení Nejistot při Měřeních, ÚNMZ–TPM; Český Metrologický Institut: Lesná, Czech Republic, 1993. [Google Scholar]
- Český Normalizační Institute. ČSN EN ISO 748–Hydrometrie–Měření Průtoku Kapalin v Otevřených Korytech–Metody Rychlostního Pole. Available online: https://csnonline.unmz.cz/Detailnormy.aspx?k=81080 (accessed on 3 February 2017).
- Beven, K.J.; Binley, A.M. The future of distributed models: Model calibration and uncertainty prediction. Hydrol. Processes
**1992**, 6, 279–298. [Google Scholar] [CrossRef] - Westerberg, I.K.; Wagener, T.; Coxon, G.; McMillan, H.K.; Castellarin, A.; Montanari, A.; Freer, J. Uncertainty in hydrological signatures for gauged and ungauged catchments. Water Resour. Res.
**2016**, 52, 1847–1865. [Google Scholar] [CrossRef] - Westerberg, I.K.; McMillan, H.K. Uncertainty in hydrological signatures. Hydrol. Earth Syst. Sci.
**2015**, 19, 3951–3968. [Google Scholar] [CrossRef] - Zhang, J.; Li, Y.; Huang, G.; Chen, X.; Bao, A. Assessment of parameter uncertainty in hydrological model using a Markov-Chain-Monte-Carlo-based multilevel-factorial-analysis method. J. Hydrol.
**2016**, 538, 471–486. [Google Scholar] [CrossRef] - Marton, D.; Starý, M.; Menšík, P. The influence of uncertainties in the calculation of mean monthly discharges on reservoir storage. J. Hydrol. Hydromech.
**2011**, 4, 228–237. [Google Scholar] [CrossRef] - Marton, D.; Starý, M.; Menšík, P. Water Management Solution of Reservoir Storage Function under Condition of Measurement Uncertainties in Hydrological Input Data. Available online: http://dx.doi.org/10.1016/j.proeng.2014.02.121 (accessed on 29 November 2016).
- Winter, T.C. Uncertainties in estimating the water balance of lakes Jawra. J. Am. Water Resour. As.
**1981**, 17, 82–115. [Google Scholar] [CrossRef] - LaBaugh, J.W.; Winter, T.C. The impact of uncertainties in hydrologic measurement on phosphorus budgets and empirical models for two Colorado reservoirs. Limnol. Oceanogr.
**1984**. [Google Scholar] [CrossRef] - Campos, J.N.B.; Souza Filho, F.A.; Lima, H.V.C. Risks and uncertainties in reservoir yield in highly variable intermittent rivers: Case of the Castanhão Reservoir in semi-arid Brazil. Hydrol. Sci. J.
**2014**, 59, 1184–1195. [Google Scholar] [CrossRef] - Kuria, F.W.; Vogel, R.M. A Global Reservoir Water Supply Yield Model with Uncertainty. Environ. Res. Lett.
**2014**. [Google Scholar] [CrossRef] - Sordo-Ward, Á.; Granados, I.; Martín-Carrasco, F.; Garrote, L. Impact of Hydrological Uncertainty on Water Management Decisions. Water Resour. Manage.
**2016**, 30, 5535. [Google Scholar] [CrossRef] - Oskoui, I.S.; Abdullah, R.; Montaseri, M. Multiple regression model using performance indices for storage capacity of a reservoir system in Johor catchment. Appl. Mech. Mater.
**2015**, 802, 563–568. [Google Scholar] [CrossRef] - Lu, D.; Zhang, G.; Webster, C.; Barbier, C. An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations. Water Resour. Res.
**2016**. [Google Scholar] [CrossRef] - Marton, D.; Starý, M.; Menšík, P. Analysis of the influence of input data uncertainties on determining the reliability of reservoir storage capacity. J. Hydrol. Hydromech.
**2015**, 4, 287–294. [Google Scholar] [CrossRef] - Starý, M. Reservoirs and Water Systems; Brno University of Technology: Brno, Czech Republic, 2006. [Google Scholar]
- Kritskiy, S.N.; Menkel, M.F. Water Management Computations (in Russian); GIMIZ: Leningrad, Russia, 1952. [Google Scholar]
- Klemes, V. Reliability estimates for a storage reservoir with seasonal input. J. Hydrol.
**1952**, 7, 198–216. [Google Scholar] [CrossRef] - Hashimoto, T.; Stedinder, J.R.; Loucks, D.P. Reliability, Resiliency, and Vulnerability Criteria for Water Resource System Performance Evaluation. Water Resour. Res.
**1982**, 18, 1. [Google Scholar] [CrossRef] - Enviromenta Ministry and Ministry of Agriculture of Czech Republic. Generel Území Chráněných pro Akumulaci Povrchových vod a Základní Zásady Využití Těchto Území. Available online: http://eagri.cz/public/web/file/133229/Generel_LAPV___vc._protokolu.pdf (accessed on 3 February 2017).
- Czech Technical Standard ČSN 75 2405 Reservoir Storage Capacity Analysis, ICS 93.160. Available online: http://seznamcsn.unmz.cz/Detailnormy.aspx?k=69792 (accessed on 3 February 2017).
- McMahon, T.A.; Adeloye, A.J. Water Resources Yield; Water Resource Publications: Littleton, CO, USA, 2005. [Google Scholar]

**Figure 1.**The principle of generating the uncertainty of input elements using the Monte Carlo method. Where V is the volume of water, h is height of the water level, μ(V) is the mean volume value, μ(h) is the mean of the water level, F(V) and F(h) are cumulative distribution functions and ξ is a random number ranged in the interval $\langle 0,1\rangle $.

**Figure 4.**Ensemble of reservoir filling considering water losses from the reservoir for the initial uncertainties of ±9% for 300 random numbers and for reservoir yield 0.70, the histogram of failure months and detail with zero, one or two failure months. The vertical axis is the reservoir storage volume Vz (m

^{3}) and the horizontal axis is the time step T (month).

**Figure 5.**Ensemble of reservoir filling considering water losses from the reservoir for the initial uncertainties of ±9% for 300 random numbers and for the reservoir yield 0.72, 0.74, 0.76 a 0.78 and histograms on the right side show the distribution of numbers of failure months.

Uncertainty | ±6% | ±9% | |||||
---|---|---|---|---|---|---|---|

yield | O_{P} (m^{3}·s^{-1}) | μ(V_{z}) (m^{3}) | 3 σ(V_{z}) (m^{3}) | 3 σ(V_{z}) (%) | μ(V_{z}) (m^{3}) | 3 σ(V_{z}) (m^{3}) | 3 σ(V_{z}) (%) |

0.6 | 2.453 | 27 764 638 | ±638 831 | ±2.30 | 27 823 438 | ±903 776 | ±3.25 |

0.7 | 2.861 | 44 111 900 | ±1 830 159 | ±4.15 | 44 105 676 | ±2 742 899 | ±6.22 |

0.8 | 3.270 | 68 148 752 | ±1 831 426 | ±2.69 | 68 148 040 | ±2 756 372 | ±4.05 |

**Table 2.**Calculation of temporal reliability of water outflow from the reservoir considering the input uncertainties.

Uncertainty | ±6% | ±9% | |||||
---|---|---|---|---|---|---|---|

Yield | O_{P} (m^{3}·s^{−1}) | μ(R_{T}) (%) | 2 σ(R_{T}) (%) | 3 σ(R_{T}) (%) | μ(R_{T}) (%) | 2 σ(R_{T}) (%) | 3 σ(R_{T}) (%) |

0.690 | 2.819 | 100.00 | ±0.00 | ±0.00 | 100.00 | ±0.00 | ±0.00 |

0.693 | 2.833 | 100.00 | ±0.00 | ±0.00 | 99.99 | ±0.05 | ±0.07 |

0.70 | 2.861 | 99.90 | ±0.21 | ±0.32 | 99.90 | ±0.23 | ±0.34 |

0.72 | 2.943 | 99.62 | ±0.03 | ±0.04 | 99.63 | ±0.07 | ±0.11 |

0.73 | 2.984 | 99.53 | ±0.12 | ±0.18 | 99.54 | ±0.13 | ±0.20 |

0.74 | 3.025 | 99.30 | ±0.18 | ±0.26 | 99.29 | ±0.24 | ±0.36 |

0.76 | 3.107 | 98.49 | ±0.14 | ±0.22 | 98.52 | ±0.21 | ±0.31 |

0.78 | 3.188 | 98.06 | ±0.22 | ±0.33 | 98.02 | ±0.25 | ±0.38 |

© 2017 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 ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Marton, D.; Paseka, S.
Uncertainty Impact on Water Management Analysis of Open Water Reservoir. *Environments* **2017**, *4*, 10.
https://doi.org/10.3390/environments4010010

**AMA Style**

Marton D, Paseka S.
Uncertainty Impact on Water Management Analysis of Open Water Reservoir. *Environments*. 2017; 4(1):10.
https://doi.org/10.3390/environments4010010

**Chicago/Turabian Style**

Marton, Daniel, and Stanislav Paseka.
2017. "Uncertainty Impact on Water Management Analysis of Open Water Reservoir" *Environments* 4, no. 1: 10.
https://doi.org/10.3390/environments4010010