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An efficient hybrid grey wolf optimization-based KELM approach for prediction of the discharge coefficient of submerged radial gates


Accurately prediction of discharge coefficient through radial gates is considered as a challenging hydraulic subject, particularly under highly submerged flow conditions. Incurring the advantages of kernel-depend extreme learning machine (KELM), this study offers a grey wolf optimization-based KELM (GWO-KELM) for effective prediction of discharge coefficient through submerged radial gates. Additionally, support vector machine (SVM) and Gaussian process regression (GPR) methods are also presented for comparative purposes. To build prediction models using GWO-KELM, GPR, and SVM, an extensive experimental database was established, consisting of 2125 data samples gathered by the US Bureau of Reclamation. From simulation results, it is observed that the proposed GWO-KELM approach with radial basis function and input parameters of the ratio of the downstream flow depth to the gate opening and submergence ratio provides the best performance with the correlation coefficient (R) of 0.983, the determination coefficient (DC) of 0.966 and the root-mean-squared error (RMSE) of 0.027. The obtained results showed that the proposed GWO-KELM with RBF kernel function gives better prediction accuracy than employed GPR and SVM approaches. Furthermore, the obtained results showed that the employed kernel-depend methods are capable of a statistically predicting the discharge coefficient under varied submergence conditions with satisfactory level of accuracy. Amon theme, proposed hybrid GWO-KELM method gave the most accurate results (R = 0.873, DC = 0.744, and RMSE = 0.035) for extremely highly submerged flow. Moreover, the results reflected that the employed kernel-depend methods give better predictions than the developed dimensionless formulas.

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Availability of data and materials

The data and materials that support the findings of this study are available on request from the corresponding author.


Q :

Flow discharge (m3/sec)

C d :

Discharge coefficient

y 1 :

Upstream flow depth

y 2 :

Flow depth at vena contracta (m)

y 3 :

Downstream flow depth (m)

L :

Channel width (m)

w :

Gate opening (m)

α :

Gate trunnion height above the invert (m)

θ :

Gate leaf angle from horizontal (°)


Reynolds number

y 1y 3/w :

Submergence ratio

y 1/R :

Upstream flow depth to the hydraulic radius ratio

y 3/R :

Downstream flow depth to the hydraulic radius ratio

y 3/w :

Downstream flow depth to the gate opening ratio

μ :

Water viscosity

ρ :

Water density


Gravitational constant (m/s2)


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Correspondence to Kiyoumars Roushangar.

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Roushangar, K., Shahnazi, S. & Sadaghiani, A.A. An efficient hybrid grey wolf optimization-based KELM approach for prediction of the discharge coefficient of submerged radial gates. Soft Comput 27, 3623–3640 (2023).

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  • Radial gates
  • Discharge coefficient
  • Kernel-depend extreme learning machine
  • Grey wolf optimization
  • Gaussian process regression
  • Support vector machine