Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow
Abstract
:1. Introduction
2. Materials and Methods
2.1. Overview
2.2. Advanced Nonlinear Muskingum Model Considering Continuous Flow
2.3. Numerical Method for Parameter Estimation
2.4. Vision Correction Algorithm
- Generate initial solutions
- Calculate the fitness of solutions using the objective function
- Generate new solution
- Compare new solution with current worst solution
- Determine the replacement between two solutions
- Repeat steps 2–5 if iteration process is not finished.
3. Results
3.1. Application of Wilson Flood Data
3.2. Application of Flood Data by Wang et al. (2009)
3.3. Application of Flood Data for River Wye December in 1960
3.4. Application of Sutculer Flood Data
3.5. Application of the Flood Data for River Wyre October in 1982
3.6. Application for the Prediction in Daechung Flood Data
4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ANLMM-L | Advanced nonlinear Muskingum flood routing model considering continuous flow |
NLMM-L | Nonlinear Muskingum flood routing model incorporating lateral flow |
VCA | Vision correction algorithm |
SSQ | Sum of squares |
DR1 | Division rate 1 |
DR2 | Division rate 2 |
MTF | Modulation transfer function |
CF | Compression factor |
AR | Astigmatic rate |
RMSE | Root mean square error |
NSE | Nash-Sutcliffe efficiency |
KWM | Kinematic wave model |
LMM | Linear Muskingum method |
LMM-L | Linear Muskingum method incorporating lateral flow |
NLMM | Nonlinear Muskingum method |
AF | Astigmatic angle |
CG | Candidate glasses |
NFEs | Number of function evaluations |
CSA | Cuckoo search algorithm |
References
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Ranges of Parameters in ANLMM-L | K | χ | m | β | θ1 | θ2 |
---|---|---|---|---|---|---|
Wilson flood data | 0.01–1.00 | −0.50–0.50 | 1.00–3.00 | −0.10–0.10 | 0.00–1.00 | 0.00–1.00 |
Flood data by Wang et al. [10] | 0.01–1.00 | −1.50–1.50 | 1.00–3.00 | −3.00–3.00 | 0.00–1.00 | 0.00–1.00 |
Flood data for River Wye December in 1960 | 0.01–1.00 | −0.50–0.50 | 1.00–3.00 | −0.10–0.10 | 0.00–1.00 | 0.00–1.00 |
Sutculer flood data | 0.01–1.00 | −0.50–0.50 | 1.00–3.00 | −0.10–0.10 | 0.00–1.00 | 0.00–1.00 |
Flood data for River Wyre October in 1982 | 0.01–10.00 | −0.50–0.50 | 0.00–1.00 | −3.00–3.00 | 0.00–1.00 | 0.00–1.00 |
Daechung flood data | 0.01–100.00 | −0.50–0.50 | 1.00–10.00 | −3.00–3.00 | 0.00–1.00 | 0.00–1.00 |
Objective function f(x), x = (x1, x2, …, xd)T |
Generate initial glasses |
While (t < Max number of iterations) |
if (DR1 < rand) |
● Choose an existing glasses/generate new glasses |
if (DR2 < rand) |
● Determine global search direction |
end |
end |
if (new solution < current worst solution) |
● Replace new solution with current worst solution |
end |
● Find the current best solution |
End while |
Measures | LMM-L | NLMM | NLMM-L | ANLMM-L |
---|---|---|---|---|
SSQ | 56 s | 63 s | 64 s | 66 s |
RMSE | 60 s | 67 s | 68 s | 70 s |
NSE | 68 s | 70 s | 72 s | 73 s |
Time (h) | Input (m3/s) | Output (m3/s) | LMM-L (O’Donnell [4]) | NLMM (Karahan [17]) | NLMM-L (Karahan [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|---|
0 | 22 | 22 | 22 | 22 | 22.00 | 22.00 |
6 | 23 | 21 | 22.1 | 22 | 21.71 | 21.57 |
12 | 35 | 21 | 21.7 | 22.4 | 22.02 | 21.67 |
18 | 71 | 26 | 22.6 | 26.6 | 26.08 | 25.46 |
24 | 103 | 34 | 30.7 | 34.5 | 33.51 | 34.59 |
30 | 111 | 44 | 44.7 | 44.2 | 42.83 | 43.73 |
36 | 109 | 55 | 58.1 | 56.9 | 55.44 | 54.59 |
42 | 100 | 66 | 68.9 | 68.1 | 66.67 | 66.01 |
48 | 86 | 75 | 76.1 | 77.1 | 75.77 | 75.52 |
54 | 71 | 82 | 79.2 | 83.3 | 82.12 | 82.16 |
60 | 59 | 85 | 78.5 | 85.9 | 84.78 | 85.04 |
66 | 47 | 84 | 75.6 | 84.5 | 83.42 | 84.00 |
72 | 39 | 80 | 70.7 | 80.6 | 79.44 | 79.62 |
78 | 32 | 73 | 65.1 | 73.7 | 72.48 | 72.63 |
84 | 28 | 64 | 59.1 | 65.4 | 64.08 | 63.80 |
90 | 24 | 54 | 53.4 | 56 | 54.58 | 54.31 |
96 | 22 | 44 | 47.9 | 46.7 | 45.22 | 44.80 |
102 | 21 | 36 | 43.1 | 37.7 | 36.34 | 36.25 |
108 | 20 | 30 | 38.9 | 30.5 | 29.21 | 29.45 |
114 | 19 | 25 | 35.4 | 25.2 | 24.21 | 24.63 |
120 | 19 | 22 | 32.3 | 21.7 | 20.96 | 21.39 |
126 | 18 | 19 | 29.9 | 20 | 19.41 | 19.81 |
SSQ | - | - | 815.68 | 36.77 | 9.82 | 4.54 |
RMSE | - | - | 6.232327 | 1.330234 | 0.683938 | 0.464948 |
NSE | - | - | 0.965417 | 0.998424 | 0.999584 | 0.999808 |
Time (12 h) | Input (m3/s) | Output (m3/s) | LMM (Wang et al. [10]) | NLMM (Geem [7]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|---|
1 | 261 | 228 | 228 | 228 | 228.00 | 228.00 |
2 | 389 | 300 | 305.19 | 303.8 | 299.74 | 300.92 |
3 | 462 | 382 | 382 | 382.3 | 382.57 | 381.51 |
4 | 505 | 444 | 442.7 | 442.4 | 442.76 | 443.15 |
5 | 525 | 490 | 483.6 | 482.4 | 482.16 | 482.69 |
6 | 543 | 513 | 513 | 511.2 | 509.89 | 510.09 |
7 | 556 | 528 | 534.29 | 532.3 | 530.72 | 530.66 |
8 | 567 | 543 | 550.44 | 548.5 | 546.77 | 546.62 |
9 | 577 | 553 | 563.53 | 561.7 | 559.96 | 559.77 |
10 | 583 | 564 | 573.16 | 571.6 | 569.94 | 569.80 |
11 | 587 | 573 | 580.02 | 578.7 | 577.07 | 576.95 |
12 | 595 | 581 | 587.32 | 586.2 | 584.39 | 584.22 |
13 | 597 | 588 | 592.14 | 591.2 | 589.68 | 589.60 |
14 | 597 | 594 | 594.59 | 593.9 | 592.34 | 592.30 |
15 | 589 | 592 | 592.02 | 591.8 | 590.33 | 590.34 |
16 | 556 | 584 | 574.89 | 575.7 | 574.68 | 574.86 |
17 | 538 | 566 | 556.85 | 558.5 | 556.41 | 556.23 |
18 | 516 | 550 | 536.93 | 539 | 537.43 | 537.13 |
19 | 486 | 520 | 512.18 | 514.8 | 513.47 | 513.35 |
20 | 505 | 504 | 507.96 | 509.6 | 507.07 | 506.51 |
21 | 477 | 483 | 493.22 | 494.9 | 494.86 | 494.95 |
22 | 429 | 461 | 462.34 | 464.8 | 464.39 | 464.94 |
23 | 379 | 420 | 421.87 | 425.1 | 423.97 | 424.15 |
24 | 320 | 368 | 372.34 | 376.1 | 375.05 | 375.07 |
25 | 263 | 318 | 318.97 | 322.4 | 321.35 | 321.35 |
26 | 220 | 271 | 270.39 | 272.5 | 271.42 | 271.40 |
27 | 182 | 234 | 226.99 | 227.5 | 226.94 | 227.09 |
28 | 167 | 193 | 197.2 | 195.7 | 194.92 | 195.13 |
29 | 152 | 178 | 174.87 | 172.6 | 172.46 | 172.76 |
SSQ | - | - | 1086.84 | 979.96 | 917.06 | 909.35 |
RMSE | - | - | 6.121869 | 5.820949 | 5.623120 | 5.599733 |
NSE | - | - | 0.998180 | 0.998354 | 0.998464 | 0.998477 |
Time (h) | Input (m3/s) | Output (m3/s) | LMM-L (O’Donnell [4]) | NLMM (Karahan et al. [17]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|---|
0 | 154 | 102 | 102 | 154 | 102.00 | 102.00 |
6 | 150 | 140 | 116 | 154 | 149.50 | 146.52 |
12 | 219 | 169 | 120 | 152 | 156.59 | 155.74 |
18 | 182 | 190 | 147 | 181 | 191.40 | 194.41 |
24 | 182 | 209 | 158 | 191 | 200.79 | 194.19 |
30 | 192 | 218 | 165 | 185 | 195.14 | 196.05 |
36 | 165 | 210 | 176 | 187 | 197.46 | 198.35 |
42 | 150 | 194 | 178 | 179 | 188.48 | 186.83 |
48 | 128 | 172 | 176 | 162 | 170.80 | 172.12 |
54 | 168 | 149 | 164 | 141 | 148.10 | 150.37 |
60 | 260 | 136 | 160 | 154 | 162.59 | 167.56 |
66 | 471 | 228 | 167 | 198 | 210.36 | 216.61 |
72 | 717 | 303 | 218 | 264 | 281.58 | 294.27 |
78 | 1092 | 366 | 303 | 344 | 367.75 | 378.29 |
84 | 1145 | 456 | 484 | 416 | 447.65 | 461.17 |
90 | 600 | 615 | 690 | 599 | 629.57 | 612.03 |
96 | 365 | 830 | 700 | 871 | 892.78 | 862.51 |
102 | 277 | 969 | 642 | 834 | 859.01 | 884.60 |
108 | 227 | 665 | 572 | 689 | 719.30 | 737.54 |
114 | 187 | 519 | 505 | 535 | 567.50 | 565.33 |
120 | 161 | 444 | 442 | 397 | 427.85 | 414.97 |
126 | 143 | 321 | 386 | 283 | 308.86 | 297.45 |
132 | 126 | 208 | 338 | 202 | 220.90 | 216.14 |
138 | 115 | 176 | 296 | 152 | 163.64 | 164.43 |
144 | 102 | 148 | 260 | 124 | 131.90 | 134.94 |
150 | 93 | 125 | 228 | 106 | 111.93 | 114.46 |
156 | 88 | 114 | 201 | 94 | 99.28 | 101.24 |
162 | 82 | 106 | 179 | 88 | 92.90 | 94.00 |
168 | 76 | 97 | 160 | 82 | 86.14 | 86.94 |
174 | 73 | 89 | 144 | 75 | 79.34 | 80.13 |
180 | 70 | 81 | 130 | 73 | 76.46 | 76.87 |
186 | 67 | 76 | 118 | 69 | 73.13 | 73.54 |
192 | 63 | 71 | 109 | 66 | 69.85 | 70.23 |
198 | 59 | 66 | 100 | 62 | 65.09 | 65.60 |
SSQ | - | - | 251,802 | 37,944.15 | 25,915.27 | 20,494.98 |
RMSE | - | - | 87.351953 | 33.900478 | 28.023846 | 24.921077 |
NSE | - | - | 0.892983 | 0.983882 | 0.988986 | 0.991290 |
Time (h) | Input (m3/s) | Output (m3/s) | KWM (Karahan and Gurarslan [13]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|
0 | 7.53 | 7 | 7.00 | 7.00 | 7.00 |
1 | 9.06 | 8 | 7.62 | 7.24 | 7.26 |
2 | 28 | 23 | 9.98 | 9.00 | 9.01 |
3 | 79.8 | 25 | 29.16 | 27.35 | 27.35 |
4 | 64.3 | 75 | 73.78 | 74.84 | 74.81 |
5 | 38.2 | 60 | 63.01 | 61.57 | 61.59 |
6 | 41.4 | 40 | 41.98 | 37.40 | 37.41 |
7 | 41.3 | 41 | 41.25 | 39.63 | 39.62 |
8 | 33.8 | 41 | 40.48 | 39.47 | 39.47 |
9 | 32 | 32 | 34.64 | 32.57 | 32.58 |
10 | 29 | 30 | 32.13 | 30.68 | 30.68 |
11 | 35 | 34 | 29.52 | 28.00 | 28.00 |
12 | 63.1 | 35 | 36.16 | 33.93 | 33.93 |
13 | 110 | 60 | 62.98 | 60.62 | 60.62 |
14 | 170 | 105 | 108.45 | 105.25 | 105.25 |
15 | 216 | 160 | 166.29 | 162.06 | 162.07 |
16 | 131 | 206 | 206.02 | 204.11 | 204.11 |
17 | 101 | 128 | 136.31 | 127.58 | 127.61 |
18 | 65 | 97 | 104.68 | 97.14 | 97.10 |
19 | 62.4 | 61 | 67.14 | 63.33 | 63.32 |
20 | 53.8 | 60 | 63.35 | 59.78 | 59.76 |
21 | 36.3 | 50 | 53.18 | 51.51 | 51.50 |
22 | 29.6 | 33 | 37.84 | 35.16 | 35.16 |
23 | 25 | 27 | 30.34 | 28.49 | 28.48 |
24 | 21.3 | 23 | 25.11 | 24.03 | 24.03 |
25 | 19.6 | 19 | 21.68 | 20.49 | 20.49 |
26 | 18 | 18 | 19.71 | 18.81 | 18.81 |
27 | 17.3 | 17 | 18.1 | 17.29 | 17.29 |
28 | 17 | 17 | 17.39 | 16.60 | 16.60 |
29 | 16 | 17 | 16.97 | 16.29 | 16.29 |
SSQ | - | - | 532.62 | 281.11 | 280.95 |
RMSE | - | - | 4.213541 | 3.062067 | 3.060222 |
NSE | - | - | 0.992271 | 0.995918 | 0.995923 |
Time (h) | Input (m3/s) | Output (m3/s) | LMM-L (O’Donnell [4]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|
0 | 2.6 | 8.3 | 8.3 | 8.3 | 8.30 |
1 | 4.2 | 9 | 8.2 | 8.51 | 8.52 |
2 | 12.3 | 9.9 | 8.1 | 8.79 | 9.94 |
3 | 25.4 | 10.2 | 12.7 | 10.94 | 12.74 |
4 | 24.1 | 18.9 | 27.9 | 20.28 | 19.71 |
5 | 20.3 | 35.9 | 39.9 | 37.54 | 35.73 |
6 | 23.3 | 51.8 | 45.7 | 49.07 | 48.87 |
7 | 27.7 | 59.4 | 52.2 | 55.11 | 55.95 |
8 | 27.7 | 63.3 | 61.4 | 62.5 | 62.74 |
9 | 26.9 | 69.6 | 68.9 | 71.44 | 71.35 |
10 | 24.8 | 76.7 | 74.7 | 78.03 | 77.95 |
11 | 26.9 | 82 | 77.2 | 82.07 | 82.67 |
12 | 33.7 | 85.3 | 79.8 | 83.72 | 85.27 |
13 | 33.9 | 89 | 87.8 | 87.43 | 88.11 |
14 | 27.8 | 94.6 | 95.5 | 95.49 | 94.74 |
15 | 20.8 | 98.8 | 97.7 | 100.88 | 99.90 |
16 | 15.6 | 98 | 94.4 | 99.29 | 98.87 |
17 | 11.9 | 91.8 | 87.9 | 92.06 | 92.05 |
18 | 9.5 | 82.3 | 79.8 | 82.22 | 82.36 |
19 | 7.8 | 72 | 71.5 | 71.75 | 71.88 |
20 | 6.5 | 61.9 | 63.6 | 61.94 | 61.93 |
21 | 5.8 | 53 | 56.1 | 53.12 | 53.10 |
22 | 5.0 | 45.6 | 49.6 | 45.47 | 45.37 |
23 | 4.8 | 39.2 | 43.7 | 39.14 | 39.04 |
24 | 4.5 | 33.8 | 38.8 | 33.76 | 33.65 |
25 | 4.1 | 29.3 | 34.6 | 29.55 | 29.39 |
26 | 3.7 | 26.2 | 30.9 | 26.12 | 25.96 |
27 | 3.4 | 23.5 | 27.7 | 23.2 | 23.08 |
28 | 3.2 | 21.2 | 24.8 | 20.67 | 20.59 |
29 | 2.9 | 19.2 | 22.3 | 18.52 | 18.44 |
30 | 2.8 | 17.7 | 20.1 | 16.71 | 16.68 |
31 | 2.6 | 16.4 | 18.2 | 15.12 | 15.09 |
SSQ | - | - | 468.84 | 53.66 | 40.16 |
RMSE | - | - | 3.790780 | 1.263563 | 1.120320 |
NSE | - | - | 0.989570 | 0.998842 | 0.999090 |
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Lee, E.H.; Lee, H.M.; Kim, J.H. Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow. Water 2018, 10, 760. https://doi.org/10.3390/w10060760
Lee EH, Lee HM, Kim JH. Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow. Water. 2018; 10(6):760. https://doi.org/10.3390/w10060760
Chicago/Turabian StyleLee, Eui Hoon, Ho Min Lee, and Joong Hoon Kim. 2018. "Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow" Water 10, no. 6: 760. https://doi.org/10.3390/w10060760
APA StyleLee, E. H., Lee, H. M., & Kim, J. H. (2018). Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow. Water, 10(6), 760. https://doi.org/10.3390/w10060760