# Multi-Objective Parameter Estimation of Improved Muskingum Model by Wolf Pack Algorithm and Its Application in Upper Hanjiang River, China

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Area

#### 2.2. Muskingum Model

#### 2.2.1. Model Basic Principle

#### 2.2.2. Design of Objective Function

## 3. Methodology

#### 3.1. Wolf Pack Algorithm (WPA)

#### 3.2. Parameter Sensitivity Analysis

^{3}/s.

^{3}/s to 7891.6 m

^{3}/s, which indicates that the larger the step size, the finer the search, and the closer the flood forecast results are to the actual flood process.

^{3}/s only. When ${\varphi}_{1}$ is 0.8 or 0.9, the flood peak deviation takes the minimum, which indicated that the weight factor of Objective 1 focuses on the minimum deviation of flood peak.

^{3}/s to 7891.6 m

^{3}/s, and reaches the minimum value when ${\varphi}_{2}$ is 0.7, which indicated that the weight factor of Objective 2 focuses on the total error of the entire flood process.

## 4. Results and Discussion

^{3}/s, 197.1 m

^{3}/s and 865.8 m

^{3}/s, respectively. Moreover, the minimum deviations are 0.3 m

^{3}/s, 15 m

^{3}/s, and 402 m

^{3}/s, respectively, which indicate that the flood peak simulated by the WPA and PSO has obvious advantages over that of the TA. In the 20140909 flood simulation, flood peak deviations are 9.5 m

^{3}/s by the WPA and 197.1 m

^{3}/s by the PSO, respectively. Flood peak deviation simulated by the WPA is much smaller than that of the PSO, which indicates that the WPA can significantly improve peak forecast accuracy.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Reach | Height Difference (m) | River Length (km) | Gradient (%) |
---|---|---|---|

Ankang Reservoir–Ankang city (Reach 1) | 11.24 | 18.31 | 0.06 |

Ankang Reservoir–Shuhe Reservoir (Reach 2) | 61.00 | 109.00 | 0.06 |

Reach | Flood Event | Flood Peak (m^{3}/s) | Total Amount of Floods (10^{8} m^{3}) | Flood Peak Time (h) |
---|---|---|---|---|

Ankang Reservoir–Ankang city | 20100821 | 7260 | 173 | 1 |

20120707 | 8420 | 241 | 1 | |

20120901 | 7645 | 171 | 1 | |

Ankang Reservoir–Shuhe Reservoir | 20130722 | 7178 | 44 | 7 |

20140909 | 7607 | 67 | 6 | |

20140914 | 6637 | 52 | 7 |

W | S | δ (m^{3}/s) | Flood Peak Deviation (m^{3}/s) |
---|---|---|---|

20 | 100 | 7892.7 | 6.33 |

104 | 7892.4 | 6.31 | |

110 | 7891.9 | 6.29 | |

115 | 7891.7 | 6.29 | |

120 | 7891.6 | 6.29 | |

130 | 7891.6 | 6.29 |

W | S | δ (m^{3}/s) | Flood Peak Deviation (m^{3}/s) |
---|---|---|---|

20 | 120 | 7891.6 | 6.29 |

30 | 180 | 7891.4 | 6.20 |

40 | 240 | 7891.2 | 6.20 |

50 | 300 | 7891.0 | 6.12 |

60 | 360 | 7890.8 | 6.12 |

70 | 420 | 7890.8 | 6.12 |

${\mathit{\varphi}}_{\mathbf{1}}$ | ${\mathit{\varphi}}_{\mathbf{2}}$ | δ (m^{3}/s) | Flood Peak Deviation (m^{3}/s) |
---|---|---|---|

0.9 | 0.1 | 7916.9 | 6.16 |

0.8 | 0.2 | 7914.8 | 6.16 |

0.7 | 0.3 | 7912.9 | 6.18 |

0.6 | 0.4 | 7906.1 | 6.19 |

0.5 | 0.5 | 7901.0 | 6.19 |

0.4 | 0.6 | 7896.6 | 6.19 |

0.3 | 0.7 | 7891.1 | 6.20 |

0.2 | 0.8 | 7891.7 | 6.22 |

0.1 | 0.9 | 7891.6 | 6.26 |

Flood Event | 20100821 | 20120707 | 20120901 | 20130722 | 20140909 | 20140914 | |
---|---|---|---|---|---|---|---|

TA | k | 1.50 | 1.50 | 1.50 | 1.00 | 1.00 | 1.00 |

x | 0.10 | 0.10 | 0.10 | 0.20 | 0.20 | 0.20 | |

PSO | k | 0.21 | 1.18 | 0.88 | 1.48 | 1.37 | 1.17 |

x | −4.17 | 0.23 | −0.43 | 0.23 | −0.42 | 0.11 | |

WPA | k | 0.58 | 1.23 | 0.86 | 1.46 | 1.56 | 1.20 |

x | −0.88 | 0.24 | −0.42 | 0.22 | −0.68 | 0.12 |

Time | Inflow (m^{3}/s) | Outflow (m^{3}/s) | Relative Error (100%) | |||||
---|---|---|---|---|---|---|---|---|

Observed | TA | WPA | PSO | TA | WPA | PSO | ||

0 | 1315 | 2405.47 | 1196.50 | 2405.47 | 2405.47 | 50.26% | 0.00% | 0.00% |

6 | 3105 | 2682.43 | 1585.65 | 2956.10 | 2682.43 | 40.89% | 10.20% | 2.73% |

12 | 4671 | 3783.79 | 3076.53 | 3784.10 | 3783.79 | 18.69% | 0.01% | 5.22% |

18 | 4830 | 4351.65 | 4358.84 | 4141.88 | 4351.65 | 0.17% | 4.82% | 5.96% |

24 | 4916 | 5026.62 | 4813.45 | 4400.83 | 5026.62 | 4.24% | 12.45% | 11.93% |

30 | 5998 | 4813.56 | 4914.02 | 5102.37 | 4813.56 | 2.09% | 6.00% | 5.96% |

36 | 6652 | 5010.76 | 5867.91 | 5709.84 | 5010.76 | 17.11% | 13.95% | 14.74% |

42 | 8318 | 6846.56 | 6631.47 | 6836.78 | 6846.56 | 3.14% | 0.14% | 0.00% |

48 | 8505 | 7217.19 | 8082.39 | 7395.13 | 7217.19 | 11.99% | 2.47% | 4.20% |

54 | 8251 | 7607.34 | 8473.15 | 7616.87 | 7607.34 | 11.38% | 0.13% | 2.59% |

60 | 6247 | 7101.53 | 8221.21 | 6816.21 | 7101.53 | 15.77% | 4.02% | 0.01% |

66 | 6120 | 5917.64 | 6654.63 | 6574.14 | 5917.64 | 12.45% | 11.09% | 13.25% |

72 | 5649 | 5839.00 | 6438.00 | 6196.58 | 5839.00 | 10.26% | 6.12% | 7.21% |

78 | 3390 | 5446.03 | 5613.68 | 4897.68 | 5446.03 | 3.08% | 10.07% | 7.84% |

84 | 2390 | 3789.75 | 3409.38 | 3925.44 | 3789.75 | 10.04% | 3.58% | 3.53% |

Floods | 20100821 | 20120707 | 20120901 | ||||||
---|---|---|---|---|---|---|---|---|---|

Method | TA | PSO | WPA | TA | PSO | WPA | TA | PSO | WPA |

δ (m^{3}/s) | 13,960 | 7892 | 7865 | 11,404 | 10,606 | 10,607 | 197,269 | 5529 | 5528 |

Flood peak deviation (m^{3}/s) | 34.3 | 6.5 | 0.5 | 33 | 51.6 | 51.4 | 87.76 | 52.1 | 51.7 |

Flood peak time transmission error (h) | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

Floods | 20130722 | 20140909 | 20140914 | ||||||
---|---|---|---|---|---|---|---|---|---|

Method | TA | PSO | WPA | TA | PSO | WPA | TA | PSO | WPA |

δ (m^{3}/s) | 4666 | 2221 | 2162 | 9141 | 4422 | 4278 | 4389 | 2697 | 2678 |

Flood peak deviation (m^{3}/s) | 402.0 | 19.5 | 0.3 | 865.8 | 197.1 | 9.5 | 409.0 | 15.0 | 8.6 |

Flood peak time transmission error (h) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Bai, T.; Wei, J.; Yang, W.; Huang, Q.
Multi-Objective Parameter Estimation of Improved Muskingum Model by Wolf Pack Algorithm and Its Application in Upper Hanjiang River, China. *Water* **2018**, *10*, 1415.
https://doi.org/10.3390/w10101415

**AMA Style**

Bai T, Wei J, Yang W, Huang Q.
Multi-Objective Parameter Estimation of Improved Muskingum Model by Wolf Pack Algorithm and Its Application in Upper Hanjiang River, China. *Water*. 2018; 10(10):1415.
https://doi.org/10.3390/w10101415

**Chicago/Turabian Style**

Bai, Tao, Jian Wei, Wangwang Yang, and Qiang Huang.
2018. "Multi-Objective Parameter Estimation of Improved Muskingum Model by Wolf Pack Algorithm and Its Application in Upper Hanjiang River, China" *Water* 10, no. 10: 1415.
https://doi.org/10.3390/w10101415