Table 9 provides the statistical indicators of the employed kernel-depend models during the testing phase. It can be seen that under free-flow conditions, using the ratio of the downstream flow depth to the gate opening (y3/w) as the only input parameter provided the desired prediction accuracy while using only the ratio of the downstream flow depth to the radial gate radius (y3/r) as input did not yield the appropriate prediction accuracy. This can show the importance of the gate opening on the prediction process of the discharge coefficient under free-flow conditions. According to SI values, when comparing F(II) and F(III), adding the ratio of the gate opening to the radial gate radius (w/r) improves the average prediction accuracy of employed KELM-GWO and GPR approaches by 4 and 7%, respectively. It can be seen that the parameter y1/w is the most influential parameter for the prediction of the discharge coefficient of radial gates under free-flow conditions. Taking into consideration the statistical indices, KELM-GWO-Polynomial and GPR-Exponential models generated the most accurate results with model F(IV), where y1/w, w/r, and α-w/r were utilized as the input parameters. As mentioned in the previous section, under free-flow conditions, the comparison results of the employed kernel-depend methods revealed that the hybridized KELM-GWO-Polynomial with model F(IV) as input combination outperformed the SVM and GPR standalone models with R = 0.927, SI = 0.023, BIAS = −0.005946, and RMSE = 0.018. It should be noticed that the prediction of discharge coefficient under field conditions for multiple gates with different openings is complicated. It may become more intractable if one gate is free-flowing and another in the transition zone.
Flow condition . | Method . | Model . | Performance criteria . | |||
---|---|---|---|---|---|---|
Testing . | ||||||
R . | SI . | BIAS . | RMSE . | |||
Free flow | KELM-GWO | F(I) | 0.305 | 0.057 | −0.005410 | 0.044 |
F(II) | 0.866 | 0.028 | 0.002324 | 0.022 | ||
F(III) | 0.907 | 0.026 | 0.008926 | 0.021 | ||
F(IV) | 0.927 | 0.023 | −0.005946 | 0.018 | ||
GPR | F(I) | 0.549 | 0.050 | 0.000421 | 0.038 | |
F(II) | 0.896 | 0.025 | −0.003127 | 0.019 | ||
F(III) | 0.908 | 0.024 | −0.002365 | 0.018 | ||
F(IV) | 0.911 | 0.023 | −0.002162 | 0.018 | ||
SVM | F(I) | 0.442 | 0.052 | −0.004215 | 0.040 | |
F(II) | 0.870 | 0.028 | 0.002492 | 0.022 | ||
F(III) | 0.880 | 0.028 | −0.004524 | 0.021 | ||
F(IV) | 0.896 | 0.027 | −0.001713 | 0.021 | ||
Submerged flow | KELM-GWO | S(I) | 0.657 | 0.276 | 0.080632 | 0.214 |
S(II) | 0.680 | 0.254 | 0.031145 | 0.197 | ||
S(III) | 0.767 | 0.255 | 0.035795 | 0.197 | ||
S(IV) | 0.839 | 0.185 | −0.027117 | 0.144 | ||
GPR | S(I) | 0.727 | 0.213 | −0.000803 | 0.165 | |
S(II) | 0.828 | 0.176 | 0.012242 | 0.136 | ||
S(III) | 0.880 | 0.148 | 0.006356 | 0.115 | ||
S(IV) | 0.944 | 0.102 | 0.009989 | 0.079 | ||
SVM | S(I) | 0.864 | 0.179 | −0.017431 | 0.139 | |
S(II) | 0.822 | 0.183 | 0.010703 | 0.141 | ||
S(III) | 0.897 | 0.162 | −0.015473 | 0.125 | ||
S(IV) | 0.940 | 0.106 | −0.000572 | 0.022 |
Flow condition . | Method . | Model . | Performance criteria . | |||
---|---|---|---|---|---|---|
Testing . | ||||||
R . | SI . | BIAS . | RMSE . | |||
Free flow | KELM-GWO | F(I) | 0.305 | 0.057 | −0.005410 | 0.044 |
F(II) | 0.866 | 0.028 | 0.002324 | 0.022 | ||
F(III) | 0.907 | 0.026 | 0.008926 | 0.021 | ||
F(IV) | 0.927 | 0.023 | −0.005946 | 0.018 | ||
GPR | F(I) | 0.549 | 0.050 | 0.000421 | 0.038 | |
F(II) | 0.896 | 0.025 | −0.003127 | 0.019 | ||
F(III) | 0.908 | 0.024 | −0.002365 | 0.018 | ||
F(IV) | 0.911 | 0.023 | −0.002162 | 0.018 | ||
SVM | F(I) | 0.442 | 0.052 | −0.004215 | 0.040 | |
F(II) | 0.870 | 0.028 | 0.002492 | 0.022 | ||
F(III) | 0.880 | 0.028 | −0.004524 | 0.021 | ||
F(IV) | 0.896 | 0.027 | −0.001713 | 0.021 | ||
Submerged flow | KELM-GWO | S(I) | 0.657 | 0.276 | 0.080632 | 0.214 |
S(II) | 0.680 | 0.254 | 0.031145 | 0.197 | ||
S(III) | 0.767 | 0.255 | 0.035795 | 0.197 | ||
S(IV) | 0.839 | 0.185 | −0.027117 | 0.144 | ||
GPR | S(I) | 0.727 | 0.213 | −0.000803 | 0.165 | |
S(II) | 0.828 | 0.176 | 0.012242 | 0.136 | ||
S(III) | 0.880 | 0.148 | 0.006356 | 0.115 | ||
S(IV) | 0.944 | 0.102 | 0.009989 | 0.079 | ||
SVM | S(I) | 0.864 | 0.179 | −0.017431 | 0.139 | |
S(II) | 0.822 | 0.183 | 0.010703 | 0.141 | ||
S(III) | 0.897 | 0.162 | −0.015473 | 0.125 | ||
S(IV) | 0.940 | 0.106 | −0.000572 | 0.022 |
Italic values indicate superior results.