Chapter 26 - Application of machine learning models to side-weir discharge coefficient estimations in trapezoidal and rectangular open channels

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Abstract

Side weirs are a type of hydraulic structures used to divert or measure flows in open channels. In this chapter, two machine learning (ML) methods, named artificial neural network (ANN) and multigene genetic programming (MGGP), were applied to develop ML-based methods for estimating discharge coefficients of side weirs installed in rectangular and trapezoidal canals for subcritical flow. According to the literature, the latter ML method is the first time to be utilized for this purpose. For the comparative analysis, a large experimental dataset was collected from previous studies available in the literature. Unlike ANN, MGGP provides an explicit equation that incorporates all parameters that affect discharge coefficients of a side weir. Although the results indicate that ANN performs slightly better than the MGGP-based model for this application, the explicit estimation model developed by MGGP can be further used in numerical modeling in river engineering and open channel designs.

Introduction

In the parlance of open channels engineering, weirs are hydraulic structures widely used for various applications in water resource management, water supply, and irrigation (Niazkar and Afzali, 2018). Among different types of weirs, a side weir is commonly installed on a main channel sidewall not only to divert but also to regulate flow. To be more specific, the former use is considered in case of the shortage of enough space required for conveying the downstream water. Therefore, the diversion attempts to protect hydropower turbines, agricultural lands, or other hydraulic structures placed at the downstream of a main channel.

The key advantage of a weir is that the overflow height over a typical weir can be utilized as an index for flow measurement. In other words, the discharge flowing in the channel can be estimated using the water depth that overflows the weir. This particular feature makes it possible to measure flows in canals by installing a suitable weir. The exclusive challenge in using weirs as flow measurement structures is that each weir needs to be calibrated in advance of any operation. The calibration is nothing but predicting a discharge coefficient that relates flow to overflow depth.

According to the literature, some attempts has been made to estimate the discharge coefficient of side weirs in favor of improving the accuracy of calculating discharge through side weirs (Mirzaei and Sheibani, 2020; Zakwan and Khan, 2020). Singh et al. (1994) conducted an analytical-experimental analysis to investigate changes of discharge coefficients of rectangular side weirs under the subcritical flow condition. They reported that the corresponding discharge coefficient is inversely proportional to the Froude number. Furthermore, Keshavarzi and Ball (2014) carried out an experimental investigation to predict discharge coefficients of side weirs installed in rectangular and trapezoidal canals. They observed that the discharge coefficient is a function of the Froude number, the ratio of the crest height of the side weir to the flow depth at the upstream of the weir, and the wall slope of the main channel. In addition, Bagheri et al. (2014) studied discharge coefficients of rectangular side weirs experimentally. Their experiments aimed to determine impacts of hydraulic and geometric parameters on the corresponding discharge coefficient. Moreover, Ebtehaj et al. (2015a) explored the application of Gene Expression Programming (GEP) model to estimate discharge coefficients of rectangular side weirs. They developed an equation that incorporates hydraulic and geometric parameters in the calculation of the discharge coefficient, while their comparative analysis indicated that the GEP model performs better than those of other available methods. Similarly, Ebtehaj et al. (2015b) used Group Method of Data Handling (GMDH) to forecast discharge coefficients of side weirs. Their findings demonstrated that the GMDH yielded more accurate results than that of Artificial Neural Network (ANN). Also, Khoshbin et al. (2016) suggested a hybrid model to estimate the discharge coefficient of side weirs. The hybrid method combines three soft computing models: Adaptive Neuro-Fuzzy Inference System (ANFIS), Genetic Algorithm (GA) and the Singular Value Decomposition (SVD). In addition, Azimi et al. (2017a) employed Extreme Learning Machine (ELM) not only to predict discharge coefficients of side weirs on placed trapezoidal channels but also to delineate how each involving parameters may affect the results through a sensitivity analysis. Moreover, Azimi et al. (2017b) applied GEP to propose a relation for estimating discharge coefficients of side weirs located on trapezoidal canals under subcritical conditions. Furthermore, Azimi et al. (2019) compared six models based on Support Vector Machine for predicting discharge coefficients of weirs designed on a trapezoidal canal. Also, Zakwan and Khan (2020) applied Generalized Reduced Gradient (GRG) technique to estimate the discharge coefficient of a side weir. Additionally, Haghshenas and Vatankhah (2021) proposed a formula for semicircular side weirs using the height, approach Froude number and radius of side weirs. In addition, Maranzoni and Tomirotti (2021) employed three-dimensional computational fluid dynamics to understand flow characteristics of oblique side weirs. Despite of previous efforts for estimating discharge coefficients of side weirs, there is still a quest for further investigations on this endeavor, particularly with the emergence of new machine learning (ML) methods.

This chapter tends to investigate the application of a powerful ML named multigene genetic programming (MGGP) to estimate discharge coefficients of side weirs installed on rectangular and trapezoidal canals. For this application, a few sets of experimental results were collected from the literature. The results of the MGGP-based model were compared with that of ANN, while the observed data was considered as the benchmark solution for this comparative analysis.

Section snippets

Problem statement

A side weir can be used for flow measurement if it is calibrated. The calibration process of a typical weir is to determine a relationship between discharge and the water depth that flows over the weir. The relation commonly entails a coefficient, invariantly called the discharge coefficient, which cannot be measured directly in an experiment. The discharge coefficient can have a fixed value for a specific set of flow and channel geometry condition, while it may vary under different

Results and discussion

This study employs two ML methods, named ANN and MGGP, to estimate Cd of side weirs installed on rectangular and trapezoidal open channels. ANN, as a black box ML method, yielded a calibrated network that can be used for predictingCd, whereas MGGP achieved an explicit equation, which can be further implemented in any numerical model if required. The MGGP-based model for estimating Cd of side weirs is given in Eq. (7):Cd¯=12.87Wy1¯exp9.273Wy1¯0.07527exp9.215Wy1¯0.684m¯Wy1¯0.2241Fr¯+5.142Lb¯exp

Conclusion

This chapter explores the application of ANN and MGGP to develop new models for estimating discharge coefficients of side weirs placed on trapezoidal and rectangular canals under subcritical flow condition. Based on the current literature, it is the first time that MGGP has been used for this purpose. The ML-based prediction models are suggested as an alternative for the cumbersome process of calibration of side weirs. Although ANN develops a calibrated network that can be used for estimating

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