# A Novel Methodology for Multiple-Year Regulation of Reservoir Active Storage Capacity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Determination of yearly inflow values, which will provide the desired regulation with a predetermined level of probability of occurrence, and
- Reservoir capacity determination based on the maximum absolute cumulative deficit of water, which must be stored during monthly water surplus periods to allow for the coverage of demand during deficit months.

## 2. The Concept of Zero-Height Dam

_{in}, has been selected and the sequence of monthly demand rates, Q

_{d}, has been prescribed, then the cumulative inflow volume, V

_{in}, and the cumulative demand volume from an initial time, t

_{1}, up to a desired time, t, will be given by the relationships:

_{in}and V

_{d}will be given by the relationship:

_{in}and Q

_{d}, i.e.,:

_{in}(t) = Q

_{d}(t), the cumulative inflow volume, ΔV, (see Equation (3)) will have a local maximum or minimum value. More specifically, when Equation (4) has a negative derivative at Q

_{in}(t) = Q

_{d}(t), then ΔV has a local maximum value, and vice versa. Consequently, there exists at least one negative local minimum or at least one positive local maximum within the time interval under consideration when the derivative of Equation (4) is, respectively, negative or positive at the initial point of the time interval. This implies that there will be a water deficit in the interval with a negative minimum, which will be covered by the water stored in the reservoir. On the other hand, there will be a water surplus in the interval with a positive maximum; this surplus will be, partially or fully, stored in the reservoir to cover deficits. The above information is schematically shown in Figure 1.

_{in}(t) = Q

_{d}(t), hence the points at which Equation (4) becomes zero, and the intervals between successive equality between values of Q

_{in}(t) and Q

_{d}(t) where there is a surplus or deficit of water. The cumulative net inflow volume is computed from Equation (3) for t = t

_{2}, with t

_{1}, t

_{2}being the successive times at which Q

_{in}(t) = Q

_{d}(t) (see Equation (4) and Figure 1). The cumulative net inflow volume for the interval t

_{1}to t

_{2}can be computed from Equation (3). A shortcoming of this approach relates to the initial storage in the reservoir. To address the problem, Thomas and Fiering [9] introduced the Sequent Peak Algorithm, in which the analysis is performed with the selected record, for multiple-year regulation, concatenated with itself. This approach provides safe results when the total net cumulative inflow volume, ΔV

_{T}, is greater or equal to zero, where:

_{T}< 0, because the required volume will keep increasing if we concatenate the record one more time. This may be interpreted either as a poor choice of the site for the required demands or, more realistically, that the probability of appearance of the selected inflow data for more than two regulation periods in a row is negligible. In the latter case, it would be safe to accept the required storage capacity obtained from two regulation periods. Based on the above, we define as the minimum required active storage capacity the largest deficit computed during a period equal to twice the reservoir regulation period, as per the concept of the Sequent Peak Algorithm. This implies that the reservoir will have an active volume of water equal to zero at least once during the computation period.

_{T}> 0, V

_{T}= 0, and V

_{T}< 0, respectively.

_{T}decreases, the required storage, computed with the concatenation of data—as suggested by the Sequent Peak Algorithm—increases. More interesting is the case of V

_{T}< 0, where the record concatenated with itself twice gives even higher reservoir capacity in the third repeat of the cycle. This is shown only for information purposes, because the repeat of the cycle three times in a row may be considered highly unlikely. In Table A2 of Appendix A we present another example, in which a one-year regulation period of a proposed reservoir is examined with the time interval of the required information for demand and the selected information for inflow being equal to one month.

## 3. Inflow and Demand Data

#### 3.1. Site Information and Demand Data

^{2}. Actual discharge measurements are not available, but a time-series of monthly precipitation heights is available for the period of September 1974 to August 1995. These data have been collected near the hypothetical dam site. The precipitation data are given in Table A1 of Appendix A. It must be noted that neither the hydrology of the area nor the demand of water affects the applicability of the methodology proposed herein. Also, for analysis purposes, it is assumed that the surface run-off coefficient is equal to unity. Thus, the monthly precipitation data can be converted to inflow data (m

^{3}/month) for the proposed dam site after multiplication with 10

^{−3}× 1000 × 1000

^{2}= 10

^{6}.

^{3}/ha, i.e., 30 × 10

^{6}m

^{3}/month, 50% of the summer consumption for April, May, September, and October, and 0% during the remaining months, (b) water supply for an urban area of 500,000 residents with mean daily consumption of 0.25 m

^{3}per capita, i.e., 500,000 × 0.25 × 30 = 3.75 × 10

^{6}m

^{3}/month during the summer months and 50% of the above (i.e., 1.875 × 10

^{6}m

^{3}/month) during the rest of the year, and (c) demand for hydro-power generation, which was estimated to be 8.224 × 10

^{6}m

^{3}/month. The temporal distribution of the demands is given in Table 4. It is noted that the specific values utilized herein can be covered by the flow rates even during the driest year of the available record.

#### 3.2. Generation of Inflow Data

^{6}m

^{3}for a yearly inflow of 276.81 × 10

^{6}m

^{3}, corresponding to the year 1991–1992. On the other hand, the required storage capacity for the year 1984–1985 was equal to 191.14 × 10

^{6}m

^{3}, while the yearly inflow was 722.17 × 10

^{6}m

^{3}. This implies that it is the monthly variations that primarily determine required storage capacity, and to a lesser extent the yearly values. Using the data from the 21 years of record, the computation of storage capacity for 1-year regulation did not reveal any correlation between storage capacity and yearly inflows or their important statistics, i.e., normalized standard deviation, skewness, and kurtosis. The results are shown in Figure 2.

- We utilized the annual precipitation heights of the 21 hydrologic years and computed the average precipitation height for the combination of the 21 years per 1, 2, 3, 4, 5, and 6 years of the regulation period.
- Subsequently, we performed all possible permutations of the years in each combination and computed the required reservoir capacity, with the methodology of Section 2, to satisfy the prescribed demands.

^{6}m

^{3}. This limiting value has very small probabilities for regulation periods of 1 and 2 years and probabilities very close to zero for longer regulation periods. It can also be observed that the curves corresponding to r = 5 and r = 6 are practically the same; this leads to the conclusion that the reservoir capacity, as a function of the regulation period, will not increase for longer regulation periods.

- Twelve random integer numbers, R(I) with I = 1.12, were generated with values ranging from 1 to 21, which are the available hydrologic years of record.
- For each value of R(I), the corresponding month and its inflow were selected through a FORTRAN Do loop of the form:DO I = 1.12I1 = 12 × (R(I) − 1) + IF(I1) = MONTH (I1)END DOwhere MONTH(I1) contains, sequentially, the 12 × 21 = 252 values of inflow (see Table A1) for 21 consecutive years. This procedure ensures that a random monthly inflow (for each specific month) of the random year will be selected.
- For the year thus generated, the required reservoir capacity was computed.
- The procedure was applied for the generation of 1,000,000 random hydrologic years and the computation of the corresponding required reservoir capacities. The sample represents the 13.6 × 10
^{−9}% of all possible cases, which are equal to 21^{12}= 7,355,827,511,386,641.

^{6}m

^{3}, 186 × 10

^{6}m

^{3}, and 203 × 10

^{6}m

^{3}for 5%, 1%, and 0.1% probability of exceedance, respectively. The choice of the level of exceedance probability may be a matter of economic, social, and environmental factors; the lower the probability level the greater the reservoir storage capacity, hence the more expensive the project. These outcomes are necessary so that engineers may form alternative project budgets, which in turn are very significant to decision-makers for the final decision.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

No. | Hydrologic Year | Sep. | Oct. | Nov. | Dec. | Jan. | Feb. | Mar. | Apr. | May | Jun. | Jul. | Aug. | Year |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1974–1975 | 101.20 | 137.65 | 148.67 | 101.56 | 94.43 | 56.11 | 99.26 | 29.60 | 211.05 | 30.76 | 17.14 | 19.52 | 1046.96 |

2 | 1975–1976 | 0.00 | 49.12 | 222.86 | 262.69 | 133.90 | 134.41 | 27.73 | 150.33 | 38.18 | 29.46 | 59.35 | 0.00 | 1108.04 |

3 | 1976–1977 | 13.47 | 137.51 | 285.60 | 201.90 | 103.80 | 90.33 | 13.25 | 20.02 | 8.86 | 0.00 | 18.22 | 3.82 | 896.77 |

4 | 1977–1978 | 64.25 | 0.00 | 264.85 | 142.84 | 189.65 | 197.15 | 115.39 | 139.31 | 50.78 | 24.42 | 0.00 | 0.00 | 1188.64 |

5 | 1978–1979 | 72.32 | 100.91 | 217.60 | 143.34 | 238.42 | 92.56 | 66.27 | 154.43 | 90.04 | 14.77 | 9.51 | 46.75 | 1246.91 |

6 | 1979–1980 | 0.00 | 225.09 | 305.98 | 194.99 | 164.66 | 91.12 | 74.84 | 128.00 | 144.71 | 41.92 | 0.00 | 0.00 | 1371.31 |

7 | 1980–1981 | 20.82 | 191.89 | 151.41 | 231.72 | 178.35 | 146.29 | 39.18 | 87.80 | 49.70 | 26.58 | 35.73 | 19.16 | 1178.63 |

8 | 1981–1982 | 27.52 | 65.69 | 196.21 | 297.20 | 15.13 | 82.19 | 120.58 | 165.16 | 56.11 | 10.80 | 0.00 | 2.16 | 1038.74 |

9 | 1982–1983 | 51.86 | 154.14 | 222.57 | 294.10 | 46.53 | 103.58 | 56.18 | 18.01 | 0.00 | 46.10 | 96.52 | 33.13 | 1122.73 |

10 | 1983–1984 | 19.45 | 36.02 | 213.21 | 241.30 | 206.01 | 193.04 | 102.79 | 204.57 | 0.00 | 0.00 | 0.00 | 11.16 | 1227.54 |

11 | 1984–1985 | 10.08 | 0.00 | 83.27 | 118.85 | 206.01 | 157.75 | 87.16 | 59.06 | 0.00 | 0.00 | 0.00 | 0.00 | 722.17 |

12 | 1985–1986 | 0.00 | 34.57 | 176.47 | 156.31 | 179.35 | 113.09 | 113.09 | 18.73 | 51.86 | 7.92 | 0.00 | 0.00 | 851.39 |

13 | 1986–1987 | 0.00 | 169.99 | 27.37 | 203.84 | 155.58 | 166.39 | 127.49 | 56.90 | 10.08 | 0.00 | 0.00 | 38.90 | 956.56 |

14 | 1987–1988 | 54.02 | 99.40 | 239.14 | 133.98 | 107.32 | 146.22 | 107.32 | 33.13 | 38.18 | 0.00 | 0.00 | 0.00 | 958.72 |

15 | 1988–1989 | 11.52 | 94.36 | 198.80 | 105.88 | 0.00 | 24.49 | 28.81 | 69.87 | 42.50 | 0.00 | 0.00 | 0.00 | 576.24 |

16 | 1989–1990 | 0.00 | 121.73 | 90.04 | 110.93 | 0.00 | 57.98 | 0.00 | 36.02 | 27.37 | 0.00 | 0.00 | 13.69 | 457.75 |

17 | 1990–1991 | 38.18 | 83.55 | 65.55 | 307.57 | 41.78 | 60.00 | 71.53 | 87.52 | 67.71 | 11.52 | 24.49 | 6.41 | 865.80 |

18 | 1991–1992 | 0.00 | 39.90 | 46.46 | 33.49 | 7.49 | 2.88 | 29.10 | 68.79 | 10.88 | 7.20 | 30.61 | 0.00 | 276.81 |

19 | 1992–1993 | 9.65 | 9.87 | 26.15 | 39.76 | 9.87 | 99.26 | 64.47 | 31.19 | 72.25 | 18.08 | 0.00 | 0.00 | 380.53 |

20 | 1993–1994 | 2.59 | 14.19 | 159.98 | 121.01 | 100.55 | 142.40 | 32.63 | 66.77 | 49.05 | 0.00 | 29.17 | 15.49 | 733.84 |

21 | 1994–1995 | 0.00 | 54.02 | 131.09 | 130.01 | 164.95 | 72.03 | 108.41 | 12.61 | 24.85 | 0.00 | 7.92 | 7.92 | 713.82 |

**Table A2.**Calculation of the required storage capacity, V (m

^{3}), for hydrologic year 1991–1992, with the lowest yearly inflow of 276.811 × 10

^{6}m

^{3}.

No. | Month | Inflow (×10^{6}) | Demand (×10^{6}) | Volume Deficit (−)/Surplus (+) | Cumulative Volume Deficit, V (×10^{6}) |
---|---|---|---|---|---|

0 | 0.000 | ||||

1 | S | 0.000 | 25.099 | −25.099 | −25.099 |

2 | O | 39.905 | 25.099 | 14.806 | −10.293 |

3 | N | 46.459 | 10.099 | 36.360 | 0.000 |

4 | D | 33.494 | 10.099 | 23.395 | 0.000 |

5 | J | 7.491 | 10.099 | −2.608 | −2.608 |

6 | F | 2.881 | 10.099 | −7.218 | -9.825 |

7 | M | 29.100 | 10.099 | 19.001 | 0.000 |

8 | A | 68.789 | 25.099 | 43.690 | 0.000 |

9 | M | 10.877 | 25.099 | −14.222 | −14.222 |

10 | J | 7.203 | 41.974 | −34.771 | −48.993 |

11 | J | 30.613 | 41.974 | −11.361 | −60.354 |

12 | A | 0.000 | 41.974 | −41.974 | −102.328 |

13 | S | 0.000 | 25.099 | −25.099 | −127.427 |

14 | O | 39.905 | 25.099 | 14.806 | −112.621 |

15 | N | 46.459 | 10.099 | 36.360 | −76.261 |

16 | D | 33.494 | 10.099 | 23.395 | −52.866 |

17 | J | 7.491 | 10.099 | −2.608 | −55.473 |

18 | F | 2.881 | 10.099 | −7.218 | −62.691 |

19 | M | 29.100 | 10.099 | 19.001 | −43.690 |

20 | A | 68.789 | 25.099 | 43.690 | 0.000 |

21 | M | 10.877 | 25.099 | −14.222 | −14.222 |

22 | J | 7.203 | 41.974 | −34.771 | −48.993 |

23 | J | 30.613 | 41.974 | −11.361 | −60.354 |

24 | A | 0.000 | 41.974 | −41.974 | −102.328 |

## References

- Yang, H.; Haynes, M.; Winzenread, S.; Okada, K. The History of Dams. 1999. Available online: https://watershed.ucdavis.edu/shed/lund/dams/Dam_History_Page/History.htm (accessed on 2 August 2018).
- Gupta, A. The World’s Oldest Dams Still in Use. Water Technology. 2013. Available online: https://www.water-technology.net/features/feature-the-worlds-oldest-dams-still-in-use/ (accessed on 2 August 2018).
- FAO. 2017. Available online: http://www.fao.org/nr/water/aquastat/dams/index.stm (accessed on 2 August 2018).
- Lehner, B.; Liermann, C.R.; Revenga, C.; Vörösmarty, C.; Fekete, B.; Crouzet, P.; Döll, P.; Endejan, M.; Frenken, K.; Magome, J.; et al. High-resolution mapping of the world’s reservoirs and dams for sustainable river-flow management. Front. Ecol. Environ.
**2011**, 9, 494–502. [Google Scholar] [CrossRef] - Rippl, W. Capacity of storage reservoirs for water supply. Minutes Proc. Inst. Civ. Eng.
**1883**, 71, 270–278. [Google Scholar] - American Society of Civil Engineers (ASCE). Hydrology Handbook, 2nd ed.; ASCE: Reston, VA, USA, 1996. [Google Scholar]
- Bharali, B. Estimation of Reservoir Storage Capacity by using Residual Mass Curve. J. Civ. Eng. Environ. Technol.
**2015**, 2, 15–18. [Google Scholar] - Clark, J.W.; Viessman, W., Jr.; Hammer, M.J. Water Supply and Pollution Control, 3rd ed.; Harper & Row Publishers: New York, NY, USA, 1977. [Google Scholar]
- Thomas, H.A.; Fiering, M.B. Mathematical Synthesis of Streamflow Sequences for the Analysis of River Basins by Simulations. In Design of Water Resource Systems; Harvard University Press: Cambridge, UK, 1962; pp. 459–493. [Google Scholar]
- Şen, Z. A mathematical model of monthly flow sequences. Hydrol. Sci. J.
**1978**, 23, 223–229. [Google Scholar] [CrossRef] - Langousis, A.; Koutsoyiannis, D. A stochastic methodology for generation of seasonal time series reproducing overyear scaling behaviour. J. Hydrol.
**2006**, 322, 138–154. [Google Scholar] [CrossRef] - Oliveira, B.; Maia, R. Stochastic Generation of Streamflow Time Series. J. Hydrol. Eng.
**2018**, 23, 04018043. [Google Scholar] [CrossRef]

**Figure 2.**Correlations of reservoir capacity V to: (

**a**) yearly inflow; (

**b**) standard deviation of monthly mean inflows/monthly mean; (

**c**) skewness of monthly mean inflows; (

**d**) kurtosis of monthly mean inflows.

**Figure 3.**Probability of exceedance for any given reservoir capacity and for regulation periods from 1 to 6 years.

**Figure 4.**Probability of exceedance for any given reservoir capacity with both methods and for a regulation period of (

**a**) 1 year; (

**b**) 2 years; (

**c**) 3 years; (

**d**) 4 years; (

**e**) 5 years; (

**f**) 6, 8, and 10 years.

Year | Inflow (m^{3}/s) | Demand (m^{3}/s) | Volume Deficit (−)/Surplus (+) ^{1} | Cumulative Volume Deficit |
---|---|---|---|---|

0 | 0 | |||

1 | 90 | 120 | −30 | −30 |

2 | 200 | 120 | 80 | 0 |

3 | 90 | 120 | −30 | −30 |

1 | 90 | 120 | −30 | −60 ^{2} |

2 | 200 | 120 | 80 | 0 |

3 | 90 | 120 | −30 | −30 |

^{1}Volumes have been divided by 12 months × 30 days/month × 24 h/day × 3600 s/h = 31.104 × 10

^{6}s.

^{2}Required storage: 60 × 31.104 × 10

^{6}= 1.866 × 10

^{9}m

^{3}.

Year | Inflow (m^{3}/s) | Demand (m^{3}/s) | Volume Deficit (−)/Surplus (+) ^{1} | Cumulative Volume Deficit |
---|---|---|---|---|

0 | 0 | |||

1 | 90 | 120 | −30 | −30 |

2 | 200 | 120 | 80 | 0 |

3 | 70 | 120 | −50 | −50 |

1 | 90 | 120 | −30 | −80 ^{2} |

2 | 200 | 120 | 80 | 0 |

3 | 70 | 120 | −50 | −50 |

^{1}Volumes have been divided by 12 months × 30 days/month × 24 h/day × 3600 s/h = 31.104 × 10

^{6}s.

^{2}Required storage: 80 × 31.104 × 10

^{6}= 2.488 × 10

^{9}m

^{3}.

Year | Inflow (m^{3}/s) | Demand (m^{3}/s) | Volume Deficit (−)/Surplus (+) ^{1} | Cumulative Volume Deficit |
---|---|---|---|---|

0 | 0 | |||

1 | 90 | 120 | −30 | −30 |

2 | 200 | 120 | 80 | 0 |

3 | 50 | 120 | −70 | −70 |

1 | 90 | 120 | −30 | −100 ^{2} |

2 | 200 | 120 | 80 | −20 |

3 | 50 | 120 | −70 | −90 |

1 | 90 | 120 | −30 | −120 |

2 | 200 | 120 | 80 | −40 |

3 | 50 | 120 | −70 | −110 |

^{1}Volumes have been divided by 12 months × 30 days/month × 24 h/day × 3600 s/h = 31.104 ×10

^{6}s.

^{2}Required storage: 100 × 31.104 × 10

^{6}= 3.11 × 10

^{9}m

^{3.}

Month | Irrigation | Water Supply | Hydro-Power | Total |
---|---|---|---|---|

September | 15 | 1.875 | 8.224 | 25.099 |

October | 15 | 1.875 | 8.224 | 25.099 |

November | 0 | 1.875 | 8.224 | 10.099 |

December | 0 | 1.875 | 8.224 | 10.099 |

January | 0 | 1.875 | 8.224 | 10.099 |

February | 0 | 1.875 | 8.224 | 10.099 |

March | 0 | 1.875 | 8.224 | 10.099 |

April | 15 | 1.875 | 8.224 | 25.099 |

May | 15 | 1.875 | 8.224 | 25.099 |

June | 30 | 3.75 | 8.224 | 41.974 |

July | 30 | 3.75 | 8.224 | 41.974 |

August | 30 | 3.75 | 8.224 | 41.974 |

Years, r, of Regulation Period | Number of Combinations of 21 Years Per r Years | Total Number of Permutations | Maximum Reservoir Storage Capacity from All Computed Required Values (×10^{6} m^{3}) | Sequence of Years ^{1} for Each Regulation Period Corresponding to Maximum Reservoir Storage Capacity |
---|---|---|---|---|

1 | 21 | 21 | 191.14 | 11 |

2 | 210 | 420 | 191.15 | 11–18 or 18–11 |

3 | 1330 | 7980 | 184.45 | 11–18–20 or 20–12–11 |

4 | 5985 | 143,640 | 184.45 | 1–11–18–20 or 20–12–11–18 |

5 | 20,349 | 2,441,880 | 184.45 | 1–2–11–18–20 or 20–2–1–11–18 |

6 | 54,264 | 39,070,080 | 184.45 | 1–2–3–11–18–20 and 3–4–5–11–18–20 or 20–3–2–1–11–18 and 20–5–4–3–11–18 |

^{1}Each number in this column corresponds to the enumeration of years, given in Table A1 of Appendix A.

Years, r of Regulation Period | Maximum Reservoir Capacity from All Computed Required Values (×10^{6} m^{3}) | Enumeration of Month Where Reservoir Capacity Becomes Zero | Size of Random Sample |
---|---|---|---|

1 | 277.87 | 24 | 1,000,000 |

2 | 233.71 | 14 | 1,000,000 |

3 | 244.01 | 26 | 1,000,000 |

4 | 253.38 | 61 | 1,000,000 |

5 | 271.74 | 36 | 1,000,000 |

6 | 278.88 | 73 | 1,000,000 |

8 | 278.22 | 74 | 1,000,000 |

10 | 301.34 | 50 | 1,621,958 |

© 2018 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**

Yannopoulos, P.C.; Demetracopoulos, A.C.
A Novel Methodology for Multiple-Year Regulation of Reservoir Active Storage Capacity. *Water* **2018**, *10*, 1254.
https://doi.org/10.3390/w10091254

**AMA Style**

Yannopoulos PC, Demetracopoulos AC.
A Novel Methodology for Multiple-Year Regulation of Reservoir Active Storage Capacity. *Water*. 2018; 10(9):1254.
https://doi.org/10.3390/w10091254

**Chicago/Turabian Style**

Yannopoulos, Panayotis C., and Alexander C. Demetracopoulos.
2018. "A Novel Methodology for Multiple-Year Regulation of Reservoir Active Storage Capacity" *Water* 10, no. 9: 1254.
https://doi.org/10.3390/w10091254