Development of ANN model for discharge prediction and optimal design of sharp-crested triangular plan form weir for maximum discharge using linked ANN–optimization model

Labyrinth weirs, which are similar to triangular plan form weirs, having one or multiple folds and of various plan forms viz., triangular, trapezoidal, rectangular etc., are also used in practice. Taylor (1968) used the same head over the crest to calculate the discharges of various types of labyrinth weirs and the corresponding normal weirs. He further calculated the magnification ratio, which is the ratio of discharge over a labyrinth weir to the discharge over a normal weir. Hay & Taylor (1969, 1970) developed a computer model to evaluate the performance of labyrinth weirs. They found that the efficiency of a triangular plan form weir is better than that of a trapezoidal plan form weir. Tullis et al. (2007) studied submerged labyrinth weirs of various geometries and developed head–discharge relationships. Wormleaton & Soufiani (1998) and Wormleaton & Tsang (2000) respectively studied the aeration performances of triangular and trapezoidal plan form labyrinth weirs. They reported that the aeration efficiency of a labyrinth weir is better than that of a normal weir of equivalent length. Similar to triangular plan form weirs, many researchers have studied the discharge characteristics of skew/oblique weirs of equivalent crest length. Some of the selected studies are reviewed below.

Swamee (1988) proposed a full-range equation for discharge coefficient using the experimental data of Kandaswamy & Rouse (1957).

Tuyen (2006) studied the discharge characteristics of sharp-crested, broad-crested and trapezoidal skew weirs with weir angle θ = 45°. Based on the experimental results, it was concluded that the discharge over a skew weir is higher than that of a normal weir of identical geometry. Emiroglu et al. (2010) used an adaptive neuro-fuzzy technique to predict the discharge capacity of a labyrinth side weir. Ayaz & Mansoor (2018) developed an ANN model for estimation of the discharge coefficient of an oblique sharp-crested weir.

The literature reviewed above establishes the fact that a substantial amount of work has been done to develop methodologies for prediction of the discharge coefficient of weirs and efforts are still continuing to improve it. Many researchers have proposed various approaches to address this problem. Most of them have proposed empirical equations for calculation of the discharge coefficient C_d with h, w and θ as independent variables. Empirical equations generally have certain limitations in incorporating nonlinearity. On the other hand, the use of artificial neural networks (ANNs) has become popular in various areas of water resources engineering because of their capability of fitting complex nonlinear input–output relationships. ANNs are also known as universal approximators. Hornik et al. (1989) reported that any standard multilayer feed-forward network with a sufficient number of hidden units can approximate any measurable function to any degree of accuracy. In this study, ANNs have been used to fit the nonlinear relation between h, w and θ or L and therefore, to predict the discharge coefficient of a sharp-crested triangular plan form weir.

Measurement of discharge plays an important role in the design and control of open channels and various flow diversion structures. Weirs are the most widely used discharge measuring devices in open channels. In order to measure the real-time data of discharge, first of all a weir needs to be calibrated by identifying its discharge coefficient (C_d). The problem statement of this study is to develop an ANN model to improve the predicted discharge coefficient of a sharp-crested triangular plan form weir which is already predicted by Kumar et al. (2011). Since the non-dimensional input and output parameters have been used in the proposed ANN model, therefore it will be applicable to all sharp-crested triangular plan form weirs which are used to measure real-time data. The discharge coefficient predicted by the proposed ANN model will work for all. A significant improvement can be observed in the prediction results. Also, a novel approach has been developed in this study by linking the developed ANN model with the optimization model to find the optimal design parameters of a weir which give the maximum discharge. The following are the highlights of this study:

• 1.

  The developed ANN model shows significant improvement in the estimation of discharge coefficient and reduces the prediction error in C_d by 64%.

• 2.

  The sensitivity analysis results show that the developed ANN model appears to be robust even for large random errors up to 10% in the input parameters.

• 3.

  A novel approach of linked ANN–optimization model has been developed to find the optimal values of design parameters (L, w and h) of a weir for which the discharge Q is maximum.

The experimental data collected by Kumar et al. (2011) has been used in this study. They conducted experiments in a horizontal rectangular channel of length 12.0 m, depth 0.41 m and width 0.28 m. The plan view of the sharp-crested triangular plan form weir is shown in Figure 1 and the sectional elevation about the section x–x is shown in Figure 2. The channel was constructed of concrete material. A triangular plan form weir made up of mild steel plates was installed at a distance of 11.0 m from the head of the weir at a desired angle θ and weir height w. Discharge Q was allowed into the channel with the help of a supply valve and the depth of flow was measured, when the condition became steady. A point gauge of accuracy ±0.1 mm was used to measure the depth of flow. In order to minimize free surface disturbances, wave suppressors and grid walls were provided at the upstream end of the channel. The experiments were conducted on various sharp-crested triangular plan form weirs having vertex angles = 30°, 60°, 90°, 120°, 150° and 180°. For each vertex angle, many sets of experimental data were recorded by varying the discharge. A total number of 123 sets of experimental data were recorded corresponding to 123 different combinations of weir height, flow depth and weir angle. Ranges of experimental parameters of the triangular plan form weir are summarized in Table 1.

Artificial neural networks (ANNs) are computational models inspired by the central nervous system: particularly, the brain. ANNs are used as nonlinear statistical data modeling tools to fit the complex relationship between inputs and outputs. The artificial neuron is the basic processing unit of an ANN model. ANNs usually comprise two or more layers of processing units. Each neuron of a layer is connected to all the neurons of the next adjacent layer with associated weights. Input layer neurons receive the information and transmit it in the forward direction to the next layers through weighted connections. The hidden layer transforms the inputs with the help of nonlinear activation functions, such as the sigmoid function, into an alternative bounded space. This process continues until the output layer is reached. Such networks are known as feed-forward neural networks (FFNN). A detailed description of the origin and basic concepts of ANN and neural computing can be found in Zurada (1990) and Schalkoff (1997). The typical structure of a feed-forward neural network comprises an input layer, an output layer, and a hidden layer, as shown in Figure 3. The number of neurons in the input and output layers depends upon the problem being trained. The number of hidden layers in the network depends on the complexity of the problem, and it may be increased for complex problems to fit the input–output pattern. Nørgaard et al. (2000) reported in their study that the performance of complex problems can be improved by using a greater number of hidden layers. ANNs with at least one hidden layer are also known as universal function approximators. Hornik et al. (1989) reported that any standard multilayer feed-forward network with a sufficient number of hidden units can approximate any measurable function to any degree of accuracy.

The process of fixing the appropriate weights associated with the connections is known as training (Rumelhart et al. 1986) and it is necessary for accurate prediction with ANN models. The training of ANNs is comprised of two steps, namely, the feed-forward step and the back-propagation step.

In this step, the weights are initialized to start the training and a back-propagation algorithm is used to minimize the total error function. In the minimization process, the total error computed for the training data set at the output layer is back-propagated through the network and, consequently, the associated weights between the connections of the artificial neurons are suitably adjusted. The Levenberg–Marquardt back-propagation algorithm is the most effective and widely used learning technique in the training of artificial neural networks (Hagan & Menhaj 1994). A short description of the working principle of the Levenberg–Marquardt algorithm is given below.

For larger values of μ, the LM algorithm behaves like the gradient descent algorithm. In the training of ANN models, the LM algorithm starts with larger μ value to ensure global convergence and then quickly shifts towards the Gauss–Newton algorithm for faster local convergence by reducing the μ value after each successful step. The LM algorithm is considered as robust and the fastest converging algorithm for the training of ANN models. A detailed description of the applications of the LM algorithm in neural networks can be found in Hagan et al. (1996) and Hagan & Menhaj (1994).

The literature suggests that the relationship of discharge as a function of weir height (w), weir length (L) and flow depth (h) is nonlinear in nature. ANN models with one hidden layer are known as universal approximators as they can fit any nonlinear relationship. The universal approximating capability of ANN models has been utilized in this study to estimate the discharge coefficient of a sharp-crested triangular plan form weir. A feed-forward back-propagation ANN model has been developed to estimate the C_d value. Different values of weir heights (w), weir lengths (L) and flow depths (h) are used as input while the corresponding discharge coefficients (C_d) are used as output of the ANN model. This model consists of three neurons in the input layer and one neuron in the output layer. All the input parameters of the ANN model have been non-dimensionalized by dividing them by the channel width (B). A schematic representation of the ANN model is shown in Figure 4.

The performance results of the developed ANN model are shown in Figures 5–10. Values of performance parameters for training, validation and testing of the ANN model are summarized in Table 2. Figures 5–7 represent the regression plots for training, validation, and testing of the ANN model. Figure 8 shows the regression plots all together. In these plots, the values of predicted discharge coefficients are plotted against the actual discharge coefficients along with a line of agreement or line of fit. It was observed that the predicted discharge coefficients are quite close to the actual values. Values of the correlation coefficient R for these regression plots are close to 1. The prediction error in C_d of the ANN model stays within the range of ±2.5%. Kumar et al. (2011) reported the prediction error for the same experimental data set within the range of ±7%. The proposed ANN model reduces the prediction error in C_d by 64%.

Statistical parameters like mean squared error (MSE) and correlation coefficient (R) give only the average values and do not provide any information about error distribution. In order to show the error distribution, an error histogram has been used. An error histogram with 20 bins is depicted in Figure 9 to show the error distribution in the predicted discharge coefficients of the ANN model. From the error histogram, it is quite clear that most of the data points fall into the bins of lesser error ranges. A bell-shape error distribution can be observed in this plot. Mean squared error (MSE) with the number of epochs for the ANN model is shown in Figure 10. The best validation performance was obtained at epoch 5 and the corresponding MSE value was found to be equal to 9.6829 × 10⁻⁵. The performance results presented above demonstrate the capability and practical applicability of the proposed ANN model. Furthermore, it can be concluded that the performance of the developed ANN model is reasonably good and therefore can be efficiently utilized to predict the discharge coefficient of the sharp-crested triangular plan form weir.

The above equation is valid only for θ ranging from 30° to 180° and h/w ranging from 0 to 0.7. The scatter plot of actual and predicted discharge coefficients using Kumar et al.’s model is shown in Figure 11. It is observed that the prediction error in the discharge coefficient stays within the range of ±7% from the line of agreement. The proposed ANN model reduces the prediction error in C_d by 64%. The results of the ANN model and Kumar et al.'s model are compared in Figure 12. The R value using Equation (14) was found to be equal to 0.9617, whereas using the ANN model it is equal to 0.9927. Also, the MSE between the actual and the predicted discharge coefficients using Kumar et al.'s approach was found to be equal to 3.695 × 10⁻⁴ whereas using the ANN model it is equal to 7.155 × 10⁻⁵. A comparison between the statistical parameters of both the models is summarized in Table 3. Based on the comparison data, it can be concluded that the performance of the ANN model is much better than that proposed by Kumar et al. (2011).

Sensitivity refers to how the output of a neural network is influenced by perturbations in its input parameters. In addition to this, sensitivity is also important in assessing the robustness of an ANN model against noise in the input parameters. In order to know the relative influence of each of the input parameters as well as the robustness of ANN models, sensitivity analysis has been conducted in this study. Perturbations in input parameters are carried out by embedding uniformly distributed random noise within the range of ±10% of the actual values. The ANN model comprises three input parameters, namely, L, w, and h. In this analysis, only one input parameter has been perturbed at a time. Regression plots of the ANN model for perturbation in the values of weir length (L), weir height (w) and flow depth (h) are shown in Figures 13–15 respectively. The prediction errors in discharge coefficients stay within the ranges of ±9%, ±5% and ±3% for perturbations within the range of ±10% in the weir height, flow depth and weir length respectively. Values of performance parameters R and MSE for perturbed responses of the ANN model are summarized in Table 4. Based on these results, it can be concluded that weir height (w) is the most and weir length (L) is the least sensitive parameter in the developed ANN model. Also, the ANN model appears to be robust even for significantly large perturbations of ±10% in input parameters.

The discharge Q in Equation (16) is a function of L, w, h and C_d. In order to calculate C_d, the optimization model is linked externally with the ANN model developed in this study. The ANN model is trained to find the discharge coefficient C_d. The linked ANN–optimization model starts with an initial guess of decision variables L, w and h. This initial guess then serves as an input to the ANN model, which is externally linked with the optimization model. The output of the ANN model is C_d, which then serves as an input to the optimization model along with the other input parameters L, w and h. These input parameters are then used to calculate the discharge Q in order to minimize the objective function mentioned in Equation (16). The optimal values of design parameters L, w and h corresponding to maximum discharge Q are then determined. The flow chart of the linked ANN–optimization model is shown in Figure 16. The plot of objective function with iteration and the resulting optimal values of the design parameters of weir are shown in Figure 17. The linked ANN–optimization terminates after 23 iterations and the minimum value of the objective function obtained is equal to 0.996504. The corresponding optimal values of design parameters obtained are L = 1.082, w = 0.095 and h = 0.073 with h/w ratio equal to 0.7684. The optimal value of weir length L = 1.082 for which the discharge Q is maximum corresponds to the weir angle of 30°. This optimal value of weir length L corresponding to maximum discharge is in agreement with the findings of Kumar et al. (2011) in which they reported that the discharge carrying capacity of a triangular plan form weir is increased by increased weir length L.

The ANN model has been developed in this study to estimate the discharge coefficient of a sharp-crested triangular plan form weir. The performance of the developed ANN model has been evaluated using the Levenberg–Marquardt algorithm and the results were compared with Kumar et al.'s model. The developed ANN model shows significant improvement in the estimation of discharge coefficient and reduces the prediction error in C_d by 64%. Also, the sensitivity analysis of the developed ANN model has been carried out by perturbing the input parameters and the weir height was found to be the most sensitive parameter. The ANN model appears to be robust even for a large random error level up to 10% in the input parameters. Furthermore, the linked ANN–optimization model has been developed to find the optimal design parameters of a weir for maximum discharge. Based on the performance results and sensitivity analysis of the developed ANN model, it can be concluded that the ANN model can be efficiently utilized to predict the discharge coefficient of a sharp-crested triangular plan form weir. Also, the linked ANN–optimization model can be utilized for optimal design of a weir for maximum discharge.

On behalf of all the authors, the corresponding author states that there is no conflict of interest.

All relevant data are included in the paper or its Supplementary Information.