Improving the Muskingum Flood Routing Method Using a Hybrid of Particle Swarm Optimization and Bat Algorithm
Abstract
:1. Introduction
1.1. Background
1.2. Innovation and Objectives
2. Methods
2.1. Muskingum Model
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 (vl) at position (yl), and the frequency, loudness and wavelength values are , and , respectively.
- (3)
- The loudness changes from a large positive (A0) to a small positive value (Amin).
2.3. Particle Swarm Optimization
2.4. Hybrid PSO and BA
- (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 ) 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:
- (2)
- The SAD between estimated and observed discharges is computed based on the following equation:
- (3)
- The mean absolute error (MARE) between estimated and observed discharges is computed based on the following equation:
- (4)
- The error for peak discharge (EO) is computed based on the following equation:
- (5)
- The error for peak time is computed based on the following equation:
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 | c1 | Objective Function | c2 | 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 | c1 | Objective Function | c2 | 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 | |
---|---|---|---|---|
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 | |
---|---|---|---|---|
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 | c1 | Objective Function | c2 | 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 | c1 | Objective Function | c2 | 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 |
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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
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 StyleEhteram, 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