# Design Flood Estimation Methods for Cascade Reservoirs Based on Copulas

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flood Regional Composition

#### 2.1. General Framework of Flood Regional Composition

_{p}represents flood characteristic variables (flood peak or volume) of site C with the corresponding design probability p. In the case of Figure 1a, when adequate observed flow data, i.e., the annual maximum peak discharge or flood volume series at site C is available, the general methodology of estimating design flood for site C is to derive the fitted theoretical distribution (e.g., P3 or LP3), representing the probability of z being exceeded based on the univariate distribution [3,6]). This conventional method is based on the assumption that the annual maximum flood series are statistically independent and identically distributed (i.i.d.) [5].

_{1}and A

_{2}denote the upper and lower reservoirs, respectively; B

_{1}and B

_{2}denote the interval basins between A

_{1}and A

_{2}, and A

_{2}and C, respectively. Let random variables X, Y

_{1}, Y

_{2}, Z

_{1}represent natural flood volume of the reservoir A

_{1}, interval basin B

_{1}, interval basin B

_{2}, and reservoir A

_{2}, respectively, and their corresponding values are x, y

_{1}, y

_{2}, and z

_{1}. Likewise, according to the principle of water balance, all of the combinations (x, y

_{1}, y

_{2}) should be subjected to the following equation [3,11]:

_{1}and A

_{2}operation.

#### 2.2. EFRC Method

_{p}with probability p occurs at downstream site C and the equivalent frequency flood volume x

_{p}occurs at upstream reservoir A site, then the corresponding flood volume at interval basin B is given by

_{p}, z

_{p}, x

_{p}] is an EFRC for single reservoir.

_{p}with probability p occurs at downstream site C and the equivalent frequency flood volume y

_{p}occurs at interval basin B, then the corresponding flood volume at upstream reservoir A site is given by

_{p}− y

_{p}, y

_{p}] is the other EFRC for single reservoir system. The EFRC method only considers two typical combinations, i.e., [x

_{p}, z

_{p}− x

_{p}] and [z

_{p}− y

_{p}, y

_{p}]. Actually, they are neither the MLRC, nor the worst regional composition. Generally speaking, if the flood characteristics of one sub-basin are closely linked with downstream site C, then the equivalent frequency flood is more likely to occur at this sub-basin.

_{p}follow the two-step procedure according to the equivalent frequency principle. At the first step, it is assumed that the equivalent frequency flood volume with downstream site C occurs at reservoir A

_{2}site, which is denoted as z

_{1p}. As a consequence, the corresponding flood volume y

_{2}at interval basin B

_{2}is given by

_{2}site occurs at reservoir A

_{1}site, which is denoted as x

_{p}. The corresponding flood volume y

_{1}at interval basin B

_{1}is given by

_{p}, z

_{1p}− x

_{p}, z

_{p}− z

_{1p}] is an EFRC for cascade reservoirs. Actually, there are four different EFRCs for the two cascade reservoirs. As the number of reservoirs (n) increases, the number of flood regional compositions (2

^{n}) will increase dramatically using the EFRC method.

## 3. Flood Regional Composition Methods Based on Copulas

#### 3.1. Joint Distribution Based on Copulas

_{xi}(x

_{i}) (i = 1, 2, …, n) be the marginal cumulative distribution functions (CDFs) of X

_{i}, the objective is to determine the multivariate distribution, which is denoted as H

_{x}

_{1,x2,…,xn}(x

_{1}, x

_{2}, …, x

_{n}) or simply H. Thus, the multivariate probability distribution H is expressed in terms of its margins and the associated dependence function, which is known as Sklar’s theorem:

_{xi}(x

_{i}) are continuous, and captures the essential features of the dependence among the random variables.

_{1}, y

_{1}), (x

_{2}, y

_{2}), …, (x

_{N}, y

_{N}), the Kendall’s tau $\tau $ can be computed as [23]:

_{i}− x

_{j}) (y

_{i}− y

_{j}) > 0; sign = −1, if (x

_{i}− x

_{j}) (y

_{i}− y

_{j}) < 0; i, j = 1, 2, …, N.

_{1}and θ

_{2}are the dependence parameters of copula, which can be estimated by the maximum likelihood method [36].

_{b}denote the test statistic and the bootstrapped test statistic, respectively. N is the number of bootstrap iterations.

#### 3.2. Conditional Expectation Regional Composition (CERC) Method

_{Y}

_{|X}(y|x) is the conditional CDF of Y given X = x, and P is the non-exceedance probability.

_{Y}

_{|X}(y|x) is the conditional density function, f

_{Y}

_{|X}(y|x) = d[F

_{Y}

_{|X}(y|x)]/dy.

_{X}(x) and F

_{Y}(y), respectively. Let U = F

_{X}(x) and V = F

_{Y}(y). Then, U and V are uniformly distributed random variables; and, u denotes a specific value of U, and v denotes a specific value of V. Using the copula function, the joint CDF F(x,y) can be expressed as F(x,y) = C(F

_{X}(x), F

_{Y}(y)) = C(u, v) [35].

_{Y}

_{|X}(y|x) and PDF f

_{Y}

_{|X}(y|x) can be expressed by Equations (15) and (16) using copula function, respectively.

_{1}, and Y

_{2}be random variables with marginal CDFs, $u={F}_{X}(x)$, $v={F}_{{Y}_{1}}({y}_{1})$, and $w={F}_{{Y}_{2}}({y}_{2})$, corresponding PDFs, ${f}_{X}(x)$, ${f}_{{Y}_{1}}({y}_{1})$, and ${f}_{{Y}_{2}}({y}_{2})$. $C(u,v)$ and $C(u,w)$ are the two-dimensional copula functions with PDFs $c(u,v)=$${\partial}^{2}C(u,v)/\partial u\partial v$ and $c(u,w)={\partial}^{2}C(u,w)/\partial u\partial w$. ${F}_{{Y}_{1}}{}^{-1}(v)$ and ${F}_{{Y}_{2}}{}^{-1}(w)$ are the inverse functions of CDF ${F}_{{Y}_{1}}({y}_{1})$ and ${F}_{{Y}_{2}}({y}_{2})$, respectively.

_{1}and Y

_{2}, respectively, which also can be calculated taking Equation (18) for reference. If $x$ can satisfy the water balance Equation (20), then the [$x$,$E({y}_{1}|x)$, $E({y}_{2}|x)$] is the CERC for cascade reservoirs system.

#### 3.3. Most Likely Regional Composition (MLRC) Method

_{x}(x) is the PDF of X.

_{p}. Substituting y = z

_{p}− x to Equation (21), then it leads to

_{p}− x) must be continuous and unimodal. In this case, the first order derivative equals to zero will reach the maximum value, and the following equation should be satisfied

_{p}− x) will be obtained.

_{1}, and Y

_{2}can be expressed, as follows [42,43]:

_{p}, the f(x, y

_{1}, y

_{2}) is maximized subjected to water balance constraint in Equation (2). Substituting y

_{2}= z

_{p}− x − y

_{1}to Equation (32) leads to

_{1}, z

_{p}− x − y

_{1}) that is equal to zero will reach the maximum value, and the following equation should be satisfied

_{1}, and Y

_{2}all follow the P3 distributions, taking Equations (25) to (30) for reference, then Equation (35) can also be rewritten as

_{1}, and Y

_{2}, respectively. The Newton iteration method is used to solve nonlinear Equation (36), and the MLRC for cascade reservoirs system (x, y

_{1}, z

_{p}

_{,}x, y

_{1}) will be obtained.

## 4. Case Study

^{2}. The mean annual rainfall, runoff depth, and annual average discharge are approximately 1460 mm, 876 mm, and 423 m

^{3}/s, respectively. The total length of the mainstream is 423 km with a hydraulic drop of 1430 m.

^{2}and 1220 km

^{2}, respectively. In this case study, design floods of the Gaobazhou reservoir site are estimated under the impact of Geheyan reservoir and Shuibuya-Geheyan cascade reservoirs operation, respectively; and, the results are compared with those of the natural condition to quantitatively evaluate the impact of upstream reservoirs.

#### 4.1. Natural River Flow Discharge Data Series

^{3}); Δt is the length of time interval (s); and, Q

_{S}is the stored water flow discharge (m

^{3}/s).

_{N}and Q

_{O}are the natural and observed river flow discharges at gauge station, respectively (m

^{3}/s); Q

_{R}is the reservoir stored water flow routed to the gauge station by the Muskingum method (m

^{3}/s); C

_{1}, C

_{2},and C

_{3}are routing coefficients that are defined in terms of Δt, K (storage-time constant), and x (weighting factor), and they can be estimated by the Least Square method. In this study, K and x are eaual to 2 h and 0.45, respectively.

_{sg}is estimated by

_{s}and Q

_{g}are the natural river flows of Shuibuya and Geheyan dam sites, respectively. Q

_{s}

_{~g}represents the natural river flow discharge at the Shuibuya dam site that was routed to the Geheyan dam site by the Muskingum routing method [44,45].

#### 4.2. Operation Rules of Shuibuya and Geheyan Reservoirs

- (1)
- If the inflow discharge is less than or equal to the design peak flow with probability 5% (the return period is 20 years), then the reservoir water level is controlled within 397 m (a.m.s.l.);
- (2)
- If the inflow discharge is greater than the design peak discharge with a probability of 5% (the return period is 20 years), but less than the spillway capacity, then the reservoir outflow equals to the inflow; otherwise, the reservoir outflow equals to the spillway capacity.

- (1)
- If the inflow discharge is less than or equal to 11,000 m
^{3}/s, then the outflow of reservoir equals to the inflow; - (2)
- If the inflow discharge is larger than 11,000 m
^{3}/s and reservoir water level is less than 200.0 m (a.m.s.l.), then the outflow of reservoir is controlled within 11,000 m^{3}/s; - (3)
- When the reservoir water level has reached 200.0 m (a.m.s.l.), if the inflow discharge is less than or equal to 13,000 m
^{3}/s, then the outflow of reservoir is equal to the inflow; if the inflow discharge is larger than 13,000 m^{3}/s and the reservoir water level is less than 203.0 m (a.m.s.l.), then the outflow of reservoir is controlled within 13,000 m^{3}/s; and, - (4)
- When the reservoir water level has reached 203.0 m (a.m.s.l.), if the inflow discharge is larger than 13,000 m
^{3}/s and less than the spillway capacity, then the outflow of reservoir is equal to the inflow; otherwise, the outflow of the reservoir is equal to the spillway capacity.

#### 4.3. Estimation of Marginal Distributions

_{3}) at Shuibuya, Geheyan, and Gaobazhou reservoir sites, Shui-Ge and Ge-Gao inter-basins, and natural peak flow (Q

_{m}) at Gaobazhou reservoir site are all i.i.d. random variables and are assumed to follow P3 distributions. The parameters of these six P3 distributions are estimated by the curve-fitting method [3], and the results are listed in Table 2. A Chi-Square Goodness-of-fit test is performed to test the assumption, H

_{0}, that the flood magnitudes follow the P3 distribution. Table 2 shows that the assumption could not be rejected at the 5% significance level.

#### 4.4. Determinations of Joint Distributions

_{3}at Geheyan reservoir site and Ge-Gao inter-basin. To study the impact of Shuibuya-Geheyan cascade reservoirs, CERC method needs (b) bivariate distributions of W

_{3}at Shuibuya reservoir site and Shui-Ge inter-basin, and (c) bivariate distributions of W

_{3}at the Shuibuya reservoir site and Ge-Gao inter-basin. While MLRC method needs, the (d) trivariate distribution of W

_{3}at the Shuibuya reservoir site, Shui-G and Ge-Gao inter-basins. In this case study, these four (one for single reservoir and three for cascade reservoirs) joint distributions are used and are constructed using the GH copula functions, respectively.

_{ei}) can be computed by the Gringorten plotting-position formula [23]:

_{m,n}is the number of occurrences of the combinations of x

_{i}and y

_{i}.

_{m,n,r}is the number of occurrences of the combinations of x

_{i}, y

_{j}, and z

_{k}.

#### 4.5. The Impact of Geheyan Reservoir Operation

_{N}represents the peak discharge under natural condition; Q

_{E}is the peak discharge estimated by one of these methods (i.e., EFRC, CFRC, and MLRC) under the impact of upstream reservoir operation.

^{3}/s to 5900 m

^{3}/s. In general, the differences among the methods are not large. For return periods between 20 and 1000 years, the maximum variation of reduction rate among these three methods is about 3.3%. However, the design values of EFRC method are systematically smaller than those of the two methods based on copulas and thus yield an underestimation, which may lead to an increased risk in the hydrological design.

#### 4.6. Impact of Shuibuya-Geheyan Cascade Reservoirs Operation

^{2}, constituting 7.8% of the control area of the Gaobazhou reservoir. In addition, the design floods of the Geheyan and Gaobazhou reservoirs are both based on the Changyang hydrological gauge station. The P3 parameters for annual maximum W

_{3}series for Geheyan and Gaobazhou reservoir are quite similar with scale and shape parameters almost equal to each other (Table 2). Therefore, their design floods almost change by ratio and the proportions of Ge-Gao interval basin are approximately constant (8.3%) and independent of the design frequency under the equivalent frequency assumption. Results of the MLRC method exhibit that when flood with return period exceeding 20 years at Gaobazhou reservoir site occurs, the proportions of Shuibuya reservoir site range from 57.4% to 61.1%, which is smaller than its proportion of area (69.4%); the proportions of Shui-Ge inter-basin range from 29.2% to 31.8%, which is larger than its proportion of area (22.8%); the proportions of Ge-Gao inter-basin range from 9.7% to 10.8%, which is larger than its proportion of area (7.8%). It is also shown that the floods at the Gaobazhou reservoir site are mainly dependent on the floods at Shuibuya reservoir site, and the proportion of the Shuibuya reservoir site decreases gradually with the increase of the design flood magnitude at the Gaobazhou reservoir site.

^{3}/s to 6920 m

^{3}/s. For return periods between 20- and 1000-year, the maximum variation of reduction rate among these three methods is about 3.0%.

## 5. Discussion

_{sby}, P

_{ghy}, and P

_{gbz}represent the exceedance probability of 3-day flood volume for Shuibuya, Geheyan and Gaobazhou reservoir sites, respectively. The straight lines show the conditions when exceedance probability of three-day flood volume for Geheyan and Gaobazhou, Shuibuya and Geheyan are identical, respectively. It is seen that the equivalent frequency floods at Geheyan and Gaobazhou reservoir sites are more likely to occur in comparing with the Shuibuya and Geheyan reservoir sites. This difference is due to that the area proportion of Ge-Gao inter-basin (7.8%) is smaller than that of Shui-Ge inter-basin (24.7%). The difference of results for these methods will be increased as the proportion of the inter-basin area increases.

^{3}/s, 7060 m

^{3}/s, and 1070 m

^{3}/s corresponding with return periods of 20-year, 200-year, and 1000-year, respectively. While the reduction curve increases continuously for the cascade reservoirs and the peak discharge reductions are 3580 m

^{3}/s, 7820 m

^{3}/s, and 9980 m

^{3}/s corresponding with 20-, 200-, and 1000-year return periods, respectively.

## 6. Conclusions

- (1)
- Design peak discharges at the Gaobazhou reservoir site have been reduced significantly due to the impact of upstream reservoirs operation when compared with those in the natural condition. Moreover, the impact of Shuibuya-Geheyan cascade reservoirs is greater than that of only Geheyan reservoir being taken into consideration.
- (2)
- The comparison between the current EFRC method and the proposed CERC and MLRC methods that were based on copulas demonstrates that the EFRC method not taking the actual dependence of floods occurred at various sub-basins into account; as a consequence, it yields an under-or overestimation of the risk that is associated with a given event in hydrological design. Moreover, when the control area of the inter basin is relatively small, the EFRC may lead to unreasonable results due to the perfect correlation assumption between flood events occurred at sub-basin and downstream site. The new methods with stronger statistical basis overcome the main drawback of EFRC method and can better capture the actual spatial correlation of flood events that occurred at various sub-basins, and the estimated design flood values are more reasonable than the currently used EFRC method.
- (3)
- The EFRC method can only be applied step by step to complex cascade reservoirs system, which is not only difficult to implement, but also subjective to some degree. As the number of cascade reservoirs (n) increases, the number of EFRCs (2
^{n}) will increase dramatically. Therefore, the MLRC method is recommended for design flood estimation in the complex cascade reservoirs since its composition is unique and easily implementable.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch diagrams of natural condition and flood control system. (

**a**) Natural condition; (

**b**) Single reservoir; and, (

**c**) Cascade reservoirs.

**Figure 3.**Plots of empirical and theoretical values for four joint cumulative distribution functions (CDFs). Note: Order represents number of ordered pair, ranked in the ascending order in terms of theoretical joint CDF, respectively. (

**a**) Geheyan and Ge-Gao inter-basin; (

**b**) Shuibuya and Shui-Ge inter-basin; (

**c**) Shuibuya and Ge-Gao inter-basin; and, (

**d**) Shuibuya, Shui-Ge, and Ge-Gao inter-basins.

**Figure 4.**Design inflow hydrographs at two sub-basins for single reservoir. (

**a**) Geheyan basin; and, (

**b**) Ge-Gao inter-basin.

**Figure 5.**Design inflow hydrographs at three sub-basins for cascade reservoirs. (

**a**) Shuibuya basin; (

**b**) Shui-Ge inter-basin; and, (

**c**) Ge-Gao inter-basin.

**Figure 6.**Occurrence probability of equivalent frequency floods for sub-basin and downstream site. Note: P

_{sby}, P

_{ghy}, and P

_{gbz}represent the exceedance probability of three-day flood volume for Shuibuya, Geheyan and Gaobazhou reservoir site, respectively. The straight lines show the conditions when exceedance probability of three-day flood volume for Geheyan and Gaobazhou, Shuibuya and Geheyan are identical, respectively. (

**a**) Geheyan and Gaobazhou reservoir sites; and (

**b**) Shuibuya and Geheyan reservoir sites.

**Figure 7.**Comparison of design peak discharges at Gaobazhou reservoir site in the natural condition and those of under the influence of upstream reservoirs operation.

**Figure 8.**Design peak discharges reduction at the Gaobazhou reservoir site under the influence of upstream reservoirs operation.

Reservoir | Shuibuya | Geheyan | Gaobazhou |
---|---|---|---|

Type of dam | Face rock fill dam | Gravity arch dam | Gravity dam |

Discharging capacity (m^{3}/s) | 18,320 | 23,900 | 22,650 |

Control area (km^{2}) | 10,860 | 14,430 | 15,650 |

Total storage (10^{8} m^{3}) | 45.8 | 37.7 | 4.85 |

Flood control storage (10^{8} m^{3}) | 5.0 | 5.0 | — |

Flood limited water level (a.m.s.l.) | 391.8 | 192.2 | — |

Normal water level (a.m.s.l.) | 400.0 | 200.0 | 80.0 |

Design flood water level (a.m.s.l.) | 402.25 | 203.14 | 80.0 |

Regulation ability | multi-year | annual | Daily |

Variable | Parameter | Chi-Square Statistics, χ^{2} | χ_{0.05} | ||
---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | |||

W_{3} at Shuibuya reservoir site | 2.30 | 0.43 | 2.65 | 0.738 | 3.841 |

W_{3} at Shui-Ge inter-basin | 1.16 | 0.56 | 1.03 | 1.256 | 3.841 |

W_{3} at Geheyan reservoir site | 1.85 | 0.25 | 3.70 | 1.646 | 3.841 |

W_{3} at Ge-Gao inter-basin | 1.16 | 2.64 | 0.35 | 3.305 | 3.841 |

W_{3} at Gaobazhou reservoir site | 1.85 | 0.23 | 4.03 | 2.033 | 3.841 |

Q_{m} at Gaobazhou reservoir site | 2.78 | 0.0005 | 2773 | 1.341 | 3.841 |

Variables | τ | θ/[θ_{1}, θ_{2}] | p-Value |
---|---|---|---|

(a) W_{3} at Geheyan reservoir site and Ge-Gao inter-basin | 0.583 | 2.40 | 0.825 |

(b) W_{3} at Shuibuya reservoir site and Shui-Ge inter-basin | 0.468 | 1.88 | 0.507 |

(c) W_{3} at Shuibuya reservoir site and Ge-Gao inter-basin | 0.766 | 4.28 | 0.324 |

(d) W_{3} at Shuibuya reservoir site, Shui-Ge and Ge-Gao inter-basins | — | [2.11, 4.28] | 0.623 |

Method | Return Period | 1000 | 500 | 200 | 100 | 50 | 20 |
---|---|---|---|---|---|---|---|

Gaobazhou (10^{8} m^{3}) | 42.89 | 39.57 | 35.14 | 31.74 | 28.30 | 23.65 | |

EFRC | Geheyan reservoir site | 91.7 | 91.7 | 91.7 | 91.7 | 91.7 | 91.7 |

Ge-Gao inter-basin | 8.3 | 8.3 | 8.3 | 8.3 | 8.3 | 8.3 | |

CERC | Geheyan reservoir site | 89.8 | 90.0 | 90.2 | 90.5 | 90.7 | 91.1 |

Ge-Gao inter-basin | 10.2 | 10.0 | 9.8 | 9.5 | 9.3 | 8.9 | |

MLRC | Geheyan reservoir site | 89.0 | 89.1 | 89.3 | 89.4 | 89.6 | 89.9 |

Ge-Gao inter-basin | 11.0 | 10.9 | 10.7 | 10.6 | 10.4 | 10.1 |

**Table 5.**Peak discharges at Gaobazhou reservoir site with different return periods under the impact of Geheyan reservoir operation (m

^{3}/s).

Return Period | Q_{N} under Natural Condition | Q_{E} under the Impact of Geheyan Reservoir | |||||
---|---|---|---|---|---|---|---|

EFRC | RR (%) | CFRC | RR (%) | MLRC | RR (%) | ||

1000 | 24,300 | 23,020 | 5.3 | 23,280 | 4.2 | 23,380 | 3.8 |

500 | 22,700 | 15,380 | 32.2 | 15,890 | 30.0 | 16,130 | 28.9 |

200 | 20,500 | 13,110 | 36.0 | 13,490 | 34.2 | 13,730 | 33.0 |

100 | 18,800 | 12,900 | 31.4 | 13,200 | 29.8 | 13,440 | 28.5 |

50 | 17,100 | 12,700 | 25.7 | 12,900 | 24.6 | 13,140 | 23.2 |

20 | 14,700 | 12,420 | 15.5 | 12,530 | 14.8 | 12,740 | 13.3 |

Method | Return Period | 1000 | 500 | 200 | 100 | 50 | 20 |
---|---|---|---|---|---|---|---|

Gaobazhou (10^{8} m^{3}) | 42.89 | 39.57 | 35.14 | 31.74 | 28.30 | 23.65 | |

EFRC | Shuibuya reservoir site | 59.1 | 59.4 | 60.0 | 60.4 | 61.0 | 62.0 |

Shui-Ge inter-basin | 32.6 | 32.3 | 31.7 | 31.3 | 30.7 | 29.7 | |

Ge-Gao inter-basin | 8.3 | 8.3 | 8.3 | 8.3 | 8.3 | 8.3 | |

CERC | Shuibuya reservoir site | 59.8 | 60.3 | 61.0 | 61.8 | 62.8 | 64.2 |

Shui-Ge inter-basin | 29.6 | 29.2 | 28.2 | 27.5 | 26.8 | 25.8 | |

Ge-Gao inter-basin | 10.6 | 10.5 | 10.8 | 10.7 | 10.4 | 10.0 | |

MLRC | Shuibuya reservoir site | 57.4 | 57.9 | 58.6 | 59.2 | 59.9 | 61.1 |

Shui-Ge inter-basin | 31.8 | 31.5 | 31.0 | 30.6 | 30.0 | 29.2 | |

Ge-Gao inter-basin | 10.8 | 10.6 | 10.4 | 10.2 | 10.1 | 9.7 |

**Table 7.**Design peak discharges at Gaobazhou site with different return periods under the impact of Shuibuya-Geheyan cascade reservoirs operation (m

^{3}/s).

Return Period | Q_{N} under Natural Condition | Q_{E} under the Impact of Shuibuya-Geheyan Cascade Reservoirs | |||||
---|---|---|---|---|---|---|---|

EFRC | RR (%) | CFRC | RR (%) | MLRC | RR (%) | ||

1000 | 24,300 | 13,980 | 42.5 | 14,320 | 41.1 | 14,350 | 40.9 |

500 | 22,700 | 13,370 | 41.1 | 14,010 | 38.3 | 14,060 | 38.1 |

200 | 20,500 | 12,750 | 37.8 | 12,660 | 38.2 | 12,710 | 38.0 |

100 | 18,800 | 11,880 | 36.8 | 12,120 | 35.5 | 12,090 | 35.7 |

50 | 17,100 | 11,640 | 31.9 | 11,810 | 30.9 | 11,780 | 31.1 |

20 | 14,700 | 11,160 | 24.1 | 11,120 | 24.4 | 11,200 | 23.8 |

© 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**

Guo, S.; Muhammad, R.; Liu, Z.; Xiong, F.; Yin, J.
Design Flood Estimation Methods for Cascade Reservoirs Based on Copulas. *Water* **2018**, *10*, 560.
https://doi.org/10.3390/w10050560

**AMA Style**

Guo S, Muhammad R, Liu Z, Xiong F, Yin J.
Design Flood Estimation Methods for Cascade Reservoirs Based on Copulas. *Water*. 2018; 10(5):560.
https://doi.org/10.3390/w10050560

**Chicago/Turabian Style**

Guo, Shenglian, Rizwan Muhammad, Zhangjun Liu, Feng Xiong, and Jiabo Yin.
2018. "Design Flood Estimation Methods for Cascade Reservoirs Based on Copulas" *Water* 10, no. 5: 560.
https://doi.org/10.3390/w10050560