Extraction of head-discharge relationship for submerged standard and modified piano key weirs using intelligent algorithms

Document Type : Research Article


1 Assistant Professor, Department of Civil Engineering, Faculty of Engineering, University of Maragheh, Maragheh, Iran.

2 Department of Civil Engineering, Ra,mhormoz Branch, Islamic Azad University,,Ra,hormoz, Iran

3 Univerdity Student

4 Uni Student



Introduction: Weirs are commonly used to measure flow, divert flow, and/or control flow water conduits. Although weirs are generally designed to run under free-flow conditions, they can become submerged under certain conditions. Weir submergence occurs when the downstream water level exceeds the crest elevation. Relative to traditional linear weirs (e.g., ogee crest), the use of nonlinear weirs to increase the flow capacity in discharge channels of limited width without much increasing the required Ho is becoming more common. Typical nonlinear weirs include labyrinth and piano key (PK) weirs which the second one has been considered in this paper. When deciding between a labyrinth and a PK weir for a channel application, the potential influence of submergence, along with the free-flow discharge capacity of both weirs, must be considered. Little works have been conducted on the submergence effects of PK weirs. This paper aims to describe an intelligent method to extract the head-discharge relationship in standard and modified Piano key weirs under submerged conditions. Support vector machine and gene expression programming intelligent algorithms were utilized to predict submerged head-discharge relationship using Dabling et al. (2014) experimental data. In the end, a comparison has been performed among SVM, GEP, and experimental based predictors using assessment criteria.
Methodology: Gathered data were from a laboratory flume measuring 0.93 m wide, 0.61 m deep, and 7.4 m long. A stilling well with a point gauge (readable to ±0.15 mm) was hydraulically connected to the flume sidewall at a distance of 4P times the weir height, (approximately 0.8 m) upstream of the weir for measuring the piezometric head level (ho and h*). A second stilling well with a point gauge connected to the flume 10P (approximately 2.0 m) downstream of the weir was used to measure the downstream piezometric head (hd). Both Ho and Hd were calculated by adding the velocity head (U2/2g) corresponding to the average cross-sectional velocity at the respective measurement locations. For the submergence investigation, variations in tailwater elevation were produced using an adjustable gate located 15P (approximately 3.0 m) downstream of the weir. A calibrated orifice meter located in the 305 mm diameter supply piping was used to accurately measure the weir discharge (± 0.2%). Repeating variables were opted as upstream flow velocity, Uu, downstream flow velocity, Ud, upstream flow depth relative to weir peak under free condition, ho, upstream flow depth relative to weir peak under submerged condition, h*, downstream flow depth relative to weir peak under submerged condition, hd, total free flow head relative to the weir crest, Ho, total submerged flow head relative to the weir crest, H*, weir height, P, gravity acceleration, g, specific mass of flow, ρ, water viscosity, µ, and water surface tension, σ. The dimensionless linear independent parameters were extracted as follow with omitting the viscosity and surface tension effects (Eq. 12)
Where Frd and Fru are downstream and upstream Froud numbers, respectively. Various combination of dimensionless parameters were examined to check the best performance of the SVM and the GEP algorithms to predict H^*/H_o using root mean square error, RMSE, determination coefficient, R2, mean normalized error, MNE, and Developed Discrepancy Ratio, DDR.
Results and Discussion: Two intelligent algorithms, i.e. SVM and GEP, were trained and tested using laboratory data from Dabling et al. (2014). The share of training and testing percent od measured data for the SVM were 60% and 40% and those of the GEP were 70% and 30%, respectively. Superior combination for the SVM to predict H^*/H_o included H_d/H_o andH_o/P for both PK-S and PK-M. The values of (RMSE, R2, MNE) for the train and the test phases for PK-S and PK-M were (0.008, 0.9996, 0.234), (0.002, 0.9989, 0.237), (0.098, 0.9833, 0.308) and (0.0918, 0.9899, 0.282), respectively. The opted dimensionless parameters for the GEP predictor included all four mentioned parameters in equation (1). The corresponding values of above mentioned criteria for train and test phases were calculated (0.0070, 0.9999, 0.3284), (0.0180, 0.9991, 0.3127), (0.0099, 0.9984, 0.2433) and (0.0097, 0.9998, 0.1825) respectively. A comparison was done between intelligent predictors and experimental equation extracted by Dabling et al., (2014). Their analogy was performed using standardized DDR values for every three predictors, i.e. ZDDR. The amount of the maximum values of ZDDR was obtained 8.517, 6.582, and 4.098 for the SVM, the GEP, and the experimental predictor, respectively. The results showed that the SVM intelligent algorithm has superiority than to the others.
Conclusion: Dispite the high values of experimental studies, the results showed that using artificial and intelligent algorithm is more practical to extract the hidden relationship among dependent and independent variables (Three-dimensional and complex flow over these weirs) for achieving high accuracy prediction. The results showed that the SVM and GEP as intelligent algorithms leads to very accurate results.


Akbari, M., Salmasi, F., Arvanaghi, H., Karbasi, M. and Farsadizadeh, D. (2019). Application of Gaussian Process Regression Model to Predict Discharge Coefficient of Gated Piano Key Weir. Water Resources Management, 33, 3929-3947.
Azimi, H., Bonakdari, H. and Ebtehaj, I. (2019). Design of radial basis function‑based support vector regression in predicting the discharge coefficient of a side weir in a trapezoidal channel. Applied Water Science, 9, 78-90. 
Dabling M.R. (2014). Nonlinear weir hydraulics. (M.Sc. Thesis), Utah state university, Logan, Utah.
Dabling, M.R. and Tullis, B.P. (2012). Piano key weir submergence in channel applications. Journal of Hydraulic Engineering, ASCE, 138(7), 661-666.
Danandehmehr, A. and Majdzadeh Tabatabai, M.R. (2010). I Prediction of Daily Discharge Trend of River Flow Based on Genetic Programming. Journal of Water and Soil, 24(2), 325-333. (In Persian)
Fathizad, H., Safari, A., Bazgir, M. and Khosravi, Gh. (2017). Evaluation of SVM with Kernel method (linear, polynomial, and radial basis) and neural network for land use classification. Iranian Journal of Range and Desert Research, 23(4), 729-743. (In Persian)
Ferreira, C. (2001). Gene expression programming a new adaptive algorithm for solving problems. Complex Systems, 13(2), 87-129.
Fuladipanah, M., Majedi Asl, M. and Haghgooyi, A. (2020). Application of intelligent algorithm to model head-discharge relationship for submerged labyrinth and linear weirs. Journal of Hydraulics, DOI: 10.30482/JHYD.2020.232388.1461. (In Persian)
Ghobadian, R., Ghorbani, M.A. and Khalaj, M. (2013). Comparison of Performance of Dynamic Wave and Gen Expression Programming Methods to River flood routing. Journal of Water and Soil, 27(3), 592-602. (In Persian)
Granata, F., Nunno, F.D., Gargano, R. and Marinis, G. (2019). Equivalent Discharge Coefficient of Side Weirs in Circular Channel-A Lazy Machine Learning Approach. Water, 11(2406), 1-19.
Haghiabi, A.H., Parsaie, A. and Ememgholizadeh, S. (2018). Prediction of discharge coefficient of triangular labyrinth weirs using adaptive neuro fuzzy inference system. Alexandria Engineering Journal, 57, 1773-1782.
Henderson, F.M. (1966). Open chnnael flow. MacMilan Publication, New York, 576P.
Hu, J. and Zheng, K. (2015). A novel support vector regression for data set with outliers. Applied Soft Computing, 31, 405-411.
Kamaei Abbasi, B., Kamaei abbasi, S.R. and Heidarnejad, M. (2020). Experimental study of Discharge Coefficient in Two-Cycle Piano Key Weirs. Iranian Journal of Irrigation and Drainage, 13(73), 10-20. (In Persian)
Kumar, M., Sihag, P., Tiwari, N.K. and Ranjan, S. (2020). Experimental study and modelling discharge coefficient of trapezoidal and rectangular piano key weirs. Applied Water Science, 10, 43-52.
Kumar, S., Ahmad, Z., Mansoor, T. and Himanshu, S.K. (2012). Discharge Characteristics of Sharp Crested Weir of Curved Plan-form. Research Journal of Engineering Sciences, 1(4), 16-20.
Lempérière, F., Vigny, J.P. and Ouamane, A. (2011). General comments on Labyrinths and Piano Key Weirs: The past and present. Labyrinth and Piano Key Weirs-PKW, CRC press, London.
Majedi Asl, M. and Fuladipanah, M. (2018). Application of the Evolutionary Methods in Determining the Discharge Coefficient of Triangular Labyrinth Weirs. Journal of Water and Soil Science (Science and Technology of Agriculture and Natural Resources), 22(4), 279-290. (In Persian)
Mehri, Y., Esmaeili, S., Soltani, J., Saneie, M. and Rostami, M. (2018). Evaluation of SVM and nonlinear regression models for predicting the discharge coefficient of side piano key weirs in irrigation and drainage networks. Iranian Journal of Irrigation and Drainage, 12(70), 994-1003. (In Persian).
Norouzi, R., Daneshfaraz, R. and Ghaderi, A. (2019). Investigation of discharge coefficient of trapezoidal labyrinth weirs using artificial neural networks and support vector machines. Applied Water Science, 9(148), 1-10.
Novak, G., Kozelj, D., Steinman, F. and Bajcar, T. (2013) Study of flow at side weir in narrow flume using visualization techniques. Flow Measurment Instrument, 29, 45-51.
Olyaie, E., Heydari, M., Banejad, H., and Chau, K.W. (2019a). A laboratory investigation on the potential of computational intelligence approaches to estimate the discharge coefficient of piano key weir. Journal of Rehabilitation in Civil Engineering, 7(1), 42-61.
Olyaie, E., Banejad, H., and Heydari, M. (2019b). Estimating discharge coefficient of PK-weir under subcritical conditions based on high-accuracy machine learning approaches. Iranian Journal of Science and Technology, Transactions of Civil Engineering, 43(1), 89-101.
Parsaie, A., Haghiabi, A.H., Emamgholizadeh, S. and Azamathulla, H.M. (2019). Prediction of discharge coefficient of combined weir-gate using ANN, ANFIS and SVM. International journal of Hydrology Science and Technology, 9(4), 412-430.
Roushangar, K., Majedi Asl, M. and Alami, M.T. (2018). Experimental Evaluation of Hydraulic Performance of Modified Piano Key Weirs. Journal of water and soil science, 28(3), 93-104. (In Persian)  
Roushangar, K., Alami, M.T., Shiri, J. and Majedi Asl, M. (2017). Determining discharge coefficient of labyrinth and arced labyrinth weirs using support vector machine. Journal of Hydrology Research, 49(3), 924-938.
Safarzadeh, A. and Norouzi, B. (2018). Experimental and Numerical Study on Hydraulics of the Piano-key Weirs with Modified Keys. Dam and Hydroelectric Power plant, 5(16), 36-47. (In Persian)
Safarzadeh, A., Khayat Rostami, S. and Khayat Rostami, B. (2019). Investigation on the effects of water head on discharge distribution and streamlines pattern over the Asymmetric Piano key weirs. Journal of Hydraulics, 14(1), 1-17. (In Persian)
Salarijazi, M., Ghorbani, K., Sohrabian, E. and Abdolhosseini, M. (2016). Prediction of Daily Stream-flow Using Data Driven Models. Iranian Journal of Irrigation and Drainage, 4(10), 479-488. (In Persian)
Solgi, A., Zarei, H. and Golabi, M.R. (2017). Performance assessment of gene expression programming model using data preprocessing methods to modeling river flow. Journal of Water and Soil Conservation, 24(2): 185-201. (In Persian)
Zhou, Q., Zhou, H., Zhou, Q., Yang, F., Luo, L. and Li, T. (2015). Structural damage detection based on posteriori probability support vector machine and Dempster-Shafer evidence theory. Appllied Soft Computing, 36, 368–374.
Zounemat-Kermani, M. and Mahdavi-Meymand, A. (2019). Hybrid meta-heuristics artificial intelligence models in simulating discharge passing the piano key weirs. Journal of hydrology, 569, 12-21.
Villemonte, D. (1947). Submerged weir discharge studies. Engineering News Record, 866 p.