# Improving the Muskingum Flood Routing Method Using a Hybrid of Particle Swarm Optimization and Bat Algorithm

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

^{3}/s. Using the initial values of the Muskingum parameters as decision variables, CLA optimized the objective function to obtain the best parameter values.

#### 1.2. Innovation and Objectives

## 2. Methods

#### 2.1. Muskingum Model

^{3}); ${I}_{t}$ is the inflow (L

^{3}·T

^{−1}); ${O}_{t}$ is the outflow (L

^{3}·T

^{−1}); K is the storage time constant, which varies from 0 to 30 (T); and x is the weighting factor, which varies from 0 to 0.5. Previous studies have shown that the linear Muskingum model does not perform well for some rivers; thus, nonlinear Muskingum models have been suggested [33]:

^{3(1−m)}·T

^{m}). The current study uses a four-parameter nonlinear Muskingum model based on the model introduced by Easa [20], with a reported high flood routing ability:

#### 2.2. Bat Algorithm

- (1)
- All bats use echolocation to identify prey and obstacles based on received sound frequencies.
- (2)
- All bats fly randomly with the velocity (v
_{l}) at position (y_{l}), and the frequency, loudness and wavelength values are ${f}_{l}$, ${A}_{0}$ and $\lambda $, respectively. - (3)
- The loudness changes from a large positive (A
_{0}) to a small positive value (A_{min}).

_{l}) that varies from 0 to 1. The value 1 means that the pulsation rate has reached a maximum value, and 0 means that the pulsation rate has reached a minimum value. The velocity, frequency and position are updated based on the following equations [31]:

#### 2.3. Particle Swarm Optimization

_{1}and c

_{2}are the acceleration coefficients, r

_{1}and r

_{2}are the random numbers, $\mathsf{\Delta}t$ is the time step, ${x}_{id}^{n+1}$ is the new position, ${v}_{id}^{n+1}$ is the new velocity vector, and n is time index.

#### 2.4. Hybrid PSO and BA

_{1}and N

_{2}are equal, i.e., N/2. Figure 2 shows the performance of the HA.

- (1)
- The random parameters for both algorithms (PSO + BA) are initialized, and the initial populations for the two algorithms are considered.
- (2)
- The first initial values for the hydrological parameters (K, x, m and $\alpha $) are considered at the start of the algorithm.
- (3)
- The variation in storage is computed based on Equation (7). The initial outflow is the same as inflow.
- (4)
- The accumulated storage is computed based on Equation (8).
- (5)
- The outflow is computed based on Equation (6).
- (6)
- The time step is compared with the total flood time. If it is less than the total time, the algorithm goes to step 3; otherwise, the algorithm goes to the next level.
- (7)
- The objective function is computed for the two algorithms and all members that can be seen in the algorithms.
- (8)
- The velocity and position for the PSO algorithm are updated based on Equations (14) and (15), and the velocity, frequency and position are updated based on Equations (9)–(11).
- (9)
- The best particles migrate from the PSO algorithm to the BA, and there is a condition for BA similarity. In fact, the specific number of best members for each algorithm is known and is substituted for the worst solutions of the other algorithm.
- (10)
- The convergence criteria are considered. If the criteria are satisfied, the algorithm finishes; otherwise, the algorithm returns to the second step.

- (1)
- The sum of the squared deviations between observed and estimated discharges is considered the objective function and is computed based on the following equation:$$Minimize\left(SSQ\right)={{\displaystyle \sum _{t=1}^{n}\left({O}_{st}-{O}_{bt}\right)}}^{2}$$
- (2)
- The SAD between estimated and observed discharges is computed based on the following equation:$$Minimize\left(SAD\right)={\displaystyle \sum _{t=1}^{n}\left|{O}_{bt}-{O}_{st}\right|}$$
- (3)
- The mean absolute error (MARE) between estimated and observed discharges is computed based on the following equation:$$MARE=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\frac{\left|{O}_{bt}-{O}_{st}\right|}{{O}_{bt}}}$$
- (4)
- The error for peak discharge (EO) is computed based on the following equation:$$EO=\frac{\left|{O}_{peak,bt}-{O}_{peak,st}\right|}{{O}_{peak,bt}}$$
- (5)
- The error for peak time is computed based on the following equation:$$ET=\left|{T}_{peak,bt}-{T}_{peak,st}\right|$$

## 3. Case Studies

## 4. Results and Discussion

#### 4.1. Wilson Flood

#### 4.1.1. Sensitivity Analysis for Different Algorithms for the Wilson Flood

#### 4.1.2. Ten Random Results for Different Algorithms for Wilson Flood

#### 4.1.3. Discussion of the Wilson Flood Results

#### 4.2. Karahan Flood

#### 4.2.1. Discussion of the Karahan Results

#### 4.2.2. Ten Random Results for Karahan Flood

#### 4.3. Discussion of the Viessman and Lewis Flood Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Population Size | Objective Function | Maximum Frequency (Hz) | Objective Function | Minimum Frequency (Hz) | Objective Function | Maximum Loudness (dB) | Objective Function | c_{1} | Objective Function | c_{2} | Objective Function | w | Objective Function |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

20 | 5.123 | 3 | 5.254 | 1 | 5.565 | 0.20 | 4.999 | 1.6 | 5.145 | 1.6 | 5.011 | 0.3 | 5.133 |

40 | 4.789 | 5 | 4.884 | 2 | 4.987 | 0.40 | 4.845 | 1.8 | 4.933 | 1.8 | 4.987 | 0.5 | 4.654 |

60 | 4.234 | 7 | 4.233 | 3 | 4.234 | 0.60 | 4.234 | 2.0 | 4.234 | 2.0 | 4.234 | 0.70 | 4.235 |

80 | 4.312 | 9 | 4.679 | 4 | 4.789 | 0.80 | 4.565 | 2.2 | 4.555 | 2.2 | 4.445 | 0.90 | 4.512 |

Population Size | Objective Function | c_{1} | Objective Function | c_{2} | Objective Function | w | Objective Function |
---|---|---|---|---|---|---|---|

20 | 5.981 | 1.60 | 5.891 | 1.60 | 5.954 | 0.30 | 5.845 |

40 | 5.785 | 1.80 | 5.654 | 1.80 | 5.878 | 0.50 | 5.764 |

60 | 5.555 | 2.0 | 5.554 | 2.0 | 5.554 | 0.70 | 5.555 |

70 | 5.894 | 2.2 | 5.892 | 2.2 | 5.891 | 0.90 | 5.789 |

Population Size | Objective Function | Maximum Frequency (Hz) | Objective Function | Minimum Frequency (Hz) | Objective Function | Loudness (dB) | Objective Function |
---|---|---|---|---|---|---|---|

20 | 5.765 | 3 | 5.812 | 1 | 5.911 | 0.3 | 5.912 |

40 | 5.455 | 5 | 5.691 | 2 | 5.783 | 0.5 | 5.678 |

60 | 5.342 | 7 | 5.342 | 3 | 5.343 | 0.70 | 5.343 |

70 | 5.694 | 9 | 5.611 | 4 | 5.455 | 0.90 | 5.678 |

Run Number | HA | BA | PSO |
---|---|---|---|

1 | 4.234 | 5.342 | 5.555 |

2 | 4.233 | 5.348 | 5.555 |

3 | 4.234 | 5.342 | 5.555 |

4 | 4.234 | 5.342 | 5.555 |

5 | 4.234 | 5.342 | 5.559 |

6 | 4.233 | 5.342 | 5.560 |

7 | 4.234 | 5.342 | 5.555 |

8 | 4.234 | 5.342 | 5.555 |

9 | 4.234 | 5.342 | 5.555 |

10 | 4.234 | 5.342 | 5.555 |

Average | 4.234 | 5.342 | 5.555 |

Computational time | 20 s | 27 s | 25 s |

Variation coefficient | 0.00007 | 0.0003 | 0.0004 |

Method | SSQ | SAD | MARE | EO | ET |
---|---|---|---|---|---|

HA | 4.234 | 3.125 | 0.012 | 0.00111 | 0 |

PSO | 5.555 | 4.128 | 0.017 | 0.00251 | 0 |

BA | 5.342 | 4.117 | 0.015 | 0.00167 | 0 |

Method | SSQ | SAD | MARE | EO | ET |
---|---|---|---|---|---|

GA [40] (Three-parameter Muskingum) | 38.230 | 23.00 | 0.0912 | 0.0083 | 0 |

HS [40] (Three-parameter Muskingum) | 36.780 | 23.40 | 0.0878 | 0.0107 | 0 |

ICA [40] (Three-parameter Muskingum) | 36.801 | 23.46 | 0.0745 | 0.0105 | 0 |

BA (current research) (Three-parameter Muskingum) | 12.25 | 10.95 | 0.0215 | 0.0079 | 0 |

PSO (current research) (Three-parameter Muskingum) | 14.78 | 12.72 | 0.0325 | 0.0081 | 0 |

HA (current research) (Three-parameter Muskingum) | 8.215 | 6.515 | 0.0205 | 0.0043 | 0 |

Time | Inflow (cm) | Outflow (Observed-cm) | Hybrid Method (cm) | BA (cm) | PSO |
---|---|---|---|---|---|

0 | 22 | 22 | 22.0 | 22.0 | 22.0 |

6 | 23 | 21 | 22.0 | 23.0 | 23.0 |

12 | 35 | 21 | 21.0 | 22.5 | 23.5 |

18 | 71 | 26 | 25.0 | 25.0 | 26.0 |

24 | 103 | 34 | 34.0 | 35.0 | 35.5 |

30 | 111 | 44 | 43.5 | 44.0 | 44.0 |

36 | 109 | 55 | 54.0 | 55.0 | 55.5 |

42 | 100 | 66 | 66.0 | 67.0 | 68.0 |

48 | 86 | 75 | 74.0 | 74.0 | 75.0 |

54 | 71 | 82 | 81.5 | 82.0 | 83.0 |

60 | 59 | 85 | 85.0011 | 85.00251 | 85.00267 |

66 | 47 | 84 | 84.0 | 84.0 | 84.0 |

72 | 39 | 80 | 81.0 | 80.5 | 81.0 |

78 | 32 | 73 | 74.0 | 73.0 | 74.0 |

84 | 28 | 64 | 64.0 | 65.0 | 66.0 |

90 | 24 | 54 | 54.0 | 55.0 | 56.0 |

96 | 22 | 44 | 44.0 | 44.0 | 45.0 |

102 | 21 | 36 | 36.0 | 37.0 | 38.0 |

108 | 20 | 30 | 30.5 | 31.0 | 31.0 |

114 | 19 | 25 | 25.5 | 26.2 | 26.9 |

120 | 19 | 22 | 23.0 | 24.0 | 25.0 |

126 | 18 | 19 | 20.0 | 21.0 | 22.0 |

Method | K | x | m | $\mathit{\alpha}$ |
---|---|---|---|---|

HA | 0.164 | 0.2879 | 3.781 | 0.4678 |

BA | 0.152 | 0.2768 | 3.567 | 0.4567 |

PSO | 0.144 | 0.2645 | 3.123 | 0.3789 |

Method | SSQ | SAD | MARE | EO | ET |
---|---|---|---|---|---|

HA | 30,235 | 625 | 0.22 | 0.101 | 0 |

PSO | 32,119 | 697 | 0.25 | 0.109 | 0 |

BA | 31,112 | 676 | 0.24 | 0.108 | 0 |

Method | SSQ | SAD | MARE | EO | ET |
---|---|---|---|---|---|

GA [40] (Three-parameter Muskingum) | 35,123 | 1980 | 0.910 | 0.701 | 0 |

HS [40] (Three-parameter Muskingum) | 37,944 | 2161 | 0.924 | 0.798 | 0 |

ICA [40] (Three-parameter Muskingum) | 37,825 | 2054 | 0.914 | 0.787 | 0 |

BA (current research) (Three-parameter Muskingum) | 32,228 | 712 | 0.420 | 0.115 | 0 |

PSO (current research) (Three-parameter Muskingum) | 33,229 | 735 | 0.454 | 0.125 | 0 |

HA (current research) (Three-parameter Muskingum) | 31,125 | 697 | 0.254 | 0.105 | 0 |

Time | Inflow (cm) | Outflow (Observed-cm) | Hybrid Method (cm) | BA (cm) | PSO (cm) |
---|---|---|---|---|---|

0 | 154 | 102 | 102.0 | 102.0 | 102.0 |

6 | 150 | 140 | 139.23 | 138.23 | 154.2 |

12 | 219 | 169 | 170.21 | 171.24 | 152.1 |

18 | 182 | 190 | 185.12 | 183.24 | 179.4 |

24 | 182 | 209 | 202.34 | 200.11 | 190.9 |

30 | 192 | 218 | 212.23 | 198.23 | 185.4 |

36 | 165 | 210 | 207.11 | 192.32 | 186.9 |

42 | 150 | 194 | 192.12 | 189.23 | 180.2 |

48 | 128 | 172 | 170.21 | 169.24 | 164.1 |

54 | 168 | 149 | 147.21 | 146.74 | 143.7 |

60 | 260 | 136 | 137.21 | 139.23 | 152.8 |

66 | 471 | 228 | 219.21 | 212.23 | 196.3 |

72 | 717 | 303 | 300.11 | 298.21 | 267.3 |

78 | 1092 | 366 | 358.11 | 354.23 | 351.4 |

84 | 1145 | 456 | 436.32 | 426.73 | 431.8 |

90 | 600 | 615 | 612.21 | 623.24 | 617.4 |

96 | 365 | 830 | 830.101 | 830.108 | 830.109 |

102 | 277 | 969 | 894.12 | 879.12 | 836.70 |

108 | 227 | 665 | 665.101 | 665.108 | 665.109 |

114 | 187 | 519 | 519.21 | 523.12 | 549.10 |

120 | 161 | 444 | 435.68 | 424.32 | 416.90 |

126 | 143 | 321 | 315.23 | 312.11 | 305.0 |

132 | 126 | 208 | 210.21 | 212.21 | 221.40 |

138 | 115 | 176 | 169.21 | 166.24 | 163.38 |

144 | 102 | 148 | 142.12 | 139.23 | 131.20 |

150 | 93 | 125 | 119.21 | 115.67 | 110.0 |

156 | 88 | 114 | 109.21 | 100.21 | 96.40 |

162 | 82 | 106 | 110.21 | 112.11 | 89.20 |

168 | 76 | 97 | 92.21 | 89.23 | 82.70 |

174 | 73 | 89 | 82.12 | 79.43 | 76.30 |

180 | 70 | 81 | 80.23 | 78.12 | 73.00 |

186 | 67 | 76 | 79.14 | 75.12 | 69.80 |

192 | 63 | 71 | 70.14 | 70.11 | 66.7 |

198 | 59 | 66 | 70.23 | 69.12 | 62.40 |

Method | K | x | m | $\mathit{\alpha}$ |
---|---|---|---|---|

HA | 0.610 | 0.404 | 3.781 | 1.125 |

BA | 0.578 | 0.311 | 2.896 | 1.112 |

PSO | 0.578 | 0.309 | 2.789 | 1.105 |

Run Number | HA | BA | PSO |
---|---|---|---|

1 | 30,235 | 31,112 | 32,119 |

2 | 30,237 | 31,117 | 32,119 |

3 | 30,235 | 31,112 | 32,119 |

4 | 30,235 | 31,112 | 32,119 |

5 | 30,235 | 31,112 | 32,119 |

6 | 30,235 | 31,112 | 32,119 |

7 | 30,237 | 31,112 | 32,122 |

8 | 30,235 | 31,117 | 32,112 |

9 | 30,235 | 31,112 | 32,119 |

10 | 30,235 | 31,112 | 32,119 |

Average | 30,235.4 | 31,113 | 32,119 |

Computational time | 19 s | 23 s | 27 s |

Variation coefficient | 0.00002 | 0.00005 | 0.00007 |

Population Size | Objective Function | Maximum Frequency (Hz) | Objective Function | Minimum Frequency (Hz) | Objective Function | Maximum Loudness (dB) | Objective Function | c_{1} | Objective Function | c_{2} | Objective Function | w | Objective Function |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

20 | 34,231 | 3 | 33,278 | 1 | 32,278 | 0.20 | 31,124 | 1.6 | 32,112 | 1.6 | 32,114 | 0.3 | 31,127 |

40 | 32,278 | 5 | 32,211 | 2 | 31,112 | 0.40 | 30,298 | 1.8 | 31,214 | 1.8 | 31,289 | 0.5 | 31,119 |

60 | 30,235 | 7 | 30,235 | 3 | 30,235 | 0.60 | 30,235 | 2.0 | 30,235 | 2.0 | 30,235 | 0.70 | 30,235 |

80 | 31,112 | 9 | 31,265 | 4 | 31,112 | 0.80 | 30,236 | 2.2 | 31,112 | 2.2 | 31,112 | 0.90 | 30,254 |

Method | SSQ | SAD | MARE | EO | ET |
---|---|---|---|---|---|

HA (4PMM) | 45,225 | 998.24 | 0.794 | 0.111 | 0 |

PSO (4PMM) | 55,124 | 1012.22 | 0.812 | 0.209 | 0 |

BA (4PMM) | 47,224 | 1001.14 | 0.798 | 0.118 | 0 |

HA (3PMM) | 48,225 | 1002.23 | 0.812 | 0.115 | 0 |

PSO (3PMM) | 56,712 | 1014.45 | 0.867 | 0.288 | 0 |

BA (3PMM) | 49,112 | 1009.23 | 0.724 | 0.202 | 0 |

WA (3PMM) | 73,312 | 1037.25 | 0.994 | 0.488 | 0 |

Population Size | Objective Function | Maximum Frequency (Hz) | Objective Function | Minimum Frequency (Hz) | Objective Function | Maximum Loudness (dB) | Objective Function | c_{1} | Objective Function | c_{2} | Objective Function | w | Objective Function |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

20 | 47,229 | 3 | 47,312 | 1 | 49,278 | 0.20 | 46,124 | 1.6 | 47,119 | 1.6 | 48,124 | 0.3 | 48,119 |

40 | 46,214 | 5 | 47,001 | 2 | 47,112 | 0.40 | 45,298 | 1.8 | 46,224 | 1.8 | 47,211 | 0.5 | 47,015 |

60 | 45,225 | 7 | 45,225 | 3 | 45,225 | 0.60 | 45,225 | 2.0 | 45,225 | 2.0 | 45,225 | 0.70 | 45,225 |

80 | 49,112 | 9 | 45,287 | 4 | 48,112 | 0.80 | 47,119 | 2.2 | 46,179 | 2.2 | 46,117 | 0.90 | 46,119 |

Run Number | HA | PSO | BA |
---|---|---|---|

1 | 45,225 | 55,124 | 47,224 |

2 | 45,226 | 55,124 | 47,226 |

3 | 45,225 | 55,127 | 47,224 |

4 | 45,225 | 55,124 | 47,224 |

5 | 45,225 | 55,124 | 47,224 |

6 | 45,225 | 55,124 | 47,224 |

7 | 45,225 | 55,124 | 47,224 |

8 | 45,225 | 55,124 | 47,224 |

9 | 45,225 | 55,124 | 47,224 |

10 | 45,225 | 55,124 | 47,224 |

Average | 45,225 | 31113 | 47,224 |

Computational time | 15 s | 17 s | 19 s |

Variation coefficient | 0.000004 | 0.000006 | 0.00005 |

© 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**

Ehteram, M.; Binti Othman, F.; Mundher Yaseen, Z.; Abdulmohsin Afan, H.; Falah Allawi, M.; Bt. Abdul Malek, M.; Najah Ahmed, A.; Shahid, S.; P. Singh, V.; El-Shafie, A.
Improving the Muskingum Flood Routing Method Using a Hybrid of Particle Swarm Optimization and Bat Algorithm. *Water* **2018**, *10*, 807.
https://doi.org/10.3390/w10060807

**AMA Style**

Ehteram M, Binti Othman F, Mundher Yaseen Z, Abdulmohsin Afan H, Falah Allawi M, Bt. Abdul Malek M, Najah Ahmed A, Shahid S, P. Singh V, El-Shafie A.
Improving the Muskingum Flood Routing Method Using a Hybrid of Particle Swarm Optimization and Bat Algorithm. *Water*. 2018; 10(6):807.
https://doi.org/10.3390/w10060807

**Chicago/Turabian Style**

Ehteram, Mohammad, Faridah Binti Othman, Zaher Mundher Yaseen, Haitham Abdulmohsin Afan, Mohammed Falah Allawi, Marlinda Bt. Abdul Malek, Ali Najah Ahmed, Shamsuddin Shahid, Vijay P. Singh, and Ahmed El-Shafie.
2018. "Improving the Muskingum Flood Routing Method Using a Hybrid of Particle Swarm Optimization and Bat Algorithm" *Water* 10, no. 6: 807.
https://doi.org/10.3390/w10060807