Analysis of discharge characteristics of a symmetrical stepped labyrinth side weir based on global sensitivity

As a new type of weir, the labyrinth weir can greatly increase the discharge for hydraulic structures with a limited overflow front (Karimi et al. 2019). Its excellent discharge capacity and low project cost have made it popular for reservoir reinforcement, landscape engineering, flood control engineering, and spillway reconstruction in recent years. Also, considering factors such as power supply guarantee and management cost in practical engineering, compared with the broad-crested weir controlled by gate control, the labyrinth weir can greatly reduce the operation and management cost. However, in practical engineering, the design parameters of labyrinth weir are generally estimated based on experience, and then verified by model experiment. The whole design process lasts for a long time. If the design parameters are not properly formulated, it will lead to repeated optimization and adjustment of engineering design and modeling experiments, resulting in a waste of time and cost (Yan 2023).

At present, there are mainly rectangular, triangular, circular, and other shapes for the crest shape of labyrinth weirs (Mahmoud et al. 2021). Alfatlawi et al. (2023) presented a symmetrical stepped labyrinth side weir. The discharge capacity of the weir can be increased by 15–35% using the larger length of the weir, small head angles, and steps for the triangular labyrinth side weir. The discharge coefficient (C_d) is an important hydraulic parameter of the structure. With the change in the number of steps, the water flows through the crest of the weir, there are both positive water flow and lateral water flow, which makes the overflow nappe collide with each other to produce secondary flow and vortex. Its discharge characteristics are more complex than the traditional weir type. Its discharge characteristics are more complex than the traditional labyrinth side weir. Thus, it is important to systematically study the discharge characteristics of the structure and reveal the interaction mechanism of hydraulic variables for the application of the structure in practical engineering.

Many scholars have studied the C_d for labyrinth side weirs through modeling experiments and numerical simulations (Ben Said & Ouamane 2022; Alfatlawi & Alkafaji 2023; Saffar et al. 2023; Zare et al. 2023). So far, some researchers have adopted soft computing methods to predict C_d for efficient calculation. Bijanvand et al. (2023) assessed the discharge in compound open using the soft computing method, and the most important parameters affecting the discharge were the relative roughness, the ratio in terms of flow dimensions, the relative depth, the convergence angle, and the relative radius. Simsek et al. (2023) developed a C_d prediction model for trapezoidal broad-crested weirs, and the Artificial Neural Network model outperformed M5Tree and the Support Vector Machines model. Roushangar et al. (2023) established a C_d prediction model using different flow conditions with radial gates. The values of R = 0.940, RMSE = 0.022, and R = 0.927, RMSE = 0.018 were obtained for KELM-GWO and Support Vector Machines models for flow under submerged and free conditions, respectively. Azma et al. (2023) developed the Gabion weir flow model, and while estimating the ratio of flow discharge, the CatBoost model achieves the best performance; when calculating the C_d, the performance of the XGBoost and CatBoost models is very close to each other. Emami et al. (2022) evaluated the Labyrinth Weir's C_d using the Walnut algorithm and support vector regression (Walnut-SVR), and the Walnut-SVR model has higher accuracy than other counterparts. Seyedian et al. (2023) state that Gaussian process regression (GPR) provides improved performance and robustness (RMSE = 0.009 and R² = 0.986) for a triangular labyrinth weir. Zaji et al. (2020) developed the neural network and genetic algorithm to evaluate the labyrinth side weir's C_d, and the Genetic Algorithm Radial Basis neural network model could still predict the C_d for this structure with the limited dataset. Zounemat-Kermani et al. (2019) predicted the C_d of the triangular arc labyrinth weir, and the Multi-Layer Perceptron Neural Network has the highest precision and generalization capability (R² = 0.999, RMSE = 0.00385, and bias coefficient bias < 0.0001). Akbari et al. (2019) developed a C_d calculation model for the Piano Key weir from 156 experimental datasets, and they showed that the GPR model significantly outperforms other intelligent models. Karami et al. (2018) demonstrated that the ELM model outperforms ANN and genetic programming (GP) in predicting the C_d for a triangular labyrinth weir.

However, according to the current literature research, no scholars have proposed an intelligent calculation model for the C_d of the symmetrical stepped labyrinth side weir. Also, the C_d for the structure is affected by the slope of the steps, the height of the steps, the width of the steps, and the number of steps. In addition, many scholars have explored the impact of hydraulic parameters on the C_d through the sensitivity method. Samadi et al. (2022) researched the influence of crest height, cycle number, and plane weir configuration on the C_d for a labyrinth weir through the model experiments, and results show that under the same geometric parameters, the rectangular has a higher discharge for each head as compared to trapezoidal and triangular. Dogan & Kaya (2023) experimentally explored the influence of effective crest length on the C_d for the labyrinth side weir, and by reducing the upstream effective crest length by 1/3, 2/3, and 3/3, the discharge reduced about 10, 23, and 48%, respectively. It can be seen that the influence on C_d of hydraulic parameters has also been the focus of attention for many researchers. For the stepped labyrinth side weir, hydraulic parameters are obviously more than other labyrinth weirs. Furthermore, the stability of the model can be improved by reducing parameter uncertainties and analyzing the influence on Cd from parameter interactions. However, for the impact of hydraulic parameters on C_d, most scholars use local sensitivity studies to obtain the influence of individual parameters on C_d.

This paper is designed to address the discharge characteristics for the stepped labyrinth side weir in a subcritical flow regime. Firstly, the C_d calculation model is developed based on the experimental dataset and artificial intelligence algorithm. The dimensional analysis of the hydraulic parameters affecting the C_d is carried out to define the model input parameters and output parameters. The optimal intelligent model is synthesized by different evaluation indexes, and the model uncertainty is further analyzed. On this basis, the global sensitivity method is introduced to quantify the sensitivity coefficients between the dimensionless parameters and the discharge coefficients, and the insensitive parameters are removed. Finally, the variation of important parameters about the discharge coefficients is analyzed.

Considering that the dataset used is the experimental dataset of the physical model. The dimensional analysis shows that the model has six input parameters, while the extreme learning machine (ELM) has only one hidden layer, so its learning speed is faster. Additionally, in the process of using the ELM, the output weight of a Moore–Penrose generalized inverse matrix is a least squares problem, it is easy to overfit. Therefore, the kernel function is introduced to avoid randomly generating weights and thresholds. Secondly, ELM accuracy is very sensitive to the number of hidden neurons, which leads to a very time-consuming and labor-intensive process by manually tuning the parameters in the training process. So far, the Bayesian network optimization ELM has shown good prospects in many engineering fields (Olyaie et al. 2019; Quilty et al. 2023), and the ability of this method to solve hydraulic engineering problems has not been explored. Therefore, in this study, a Bayesian network is chosen to find the optimal hyperparameters for ELM automatically.

ELM as an improved neural network model using the least squares method for network development. ELM has a very fast learning speed because it is a simple untuned algorithm, which avoids the slow iterative learning process and easily falls into the local minimum (Luo & Zhang 2014). Compared with traditional intelligent algorithms, ELM has better scalability, generalization performance, lower computational complexity, and less manual intervention (Huang et al. 2011).

The kernel function mainly relies on Mercer's theorem, and kernel-based learning methods are widely used for their powerful generalization performance. The KELM introduces the concept of kernel matrix and adopts the kernel function matrix instead of the random matrix HH^T for the ELM model, which transforms the linearly indivisible data in the low-dimensional space into linearly divisible data, which not only preserves the speed of the ELM operation but also abandons the principle of randomly generating weights and biases of the ELM (Jalil-Masir et al. 2022). After determining the kernel function, KELM only needs to adjust the penalty factor, and the training results are stable and unchanged. The same accuracy can be obtained when the same samples are trained (Parida et al. 2021).

The Bayesian extreme learning machine (BELM) uses Bayesian networks for optimizing the output layer weights for ELM models, which helps to overcome the model falling into overfitting. The joint probability distribution is represented by encoding the independent relationships of random variables using a graph. Generally, Bayesian modeling is divided into two steps (Soria-Olivas et al. 2011; Olyaie et al. 2019)

• (1)
  The model parameter inference from the posterior distribution. In order to scale it, multiply the likelihood function by the prior distribution:

  

  (5)

  where P represents the probability distribution, w represents the parameter set, and D represents the dataset.
• (2)
  Calculate a new distribution for the input x_new and the output y_new. It is defined as the integration over the posterior distribution for parameter w.

  

  (6)

The linear modeling follows

The Sobol method is used as a global sensitivity analysis method, which can quantitatively give the parameter sensitivity of the model (Sobol 1990). Sobol's method is centered on decomposing the overall variance in the objective function into the variance for individual parameters and the variance between combinations of parameters, and the sensitivity coefficient is obtained. More importantly, this method can simultaneously reflect the direct and interactive effects of parameters. Compared with other methods, the Sobol method has higher stability. The specific calculation principle can be found in our previous research (Shanshan et al. 2023).

Tables 1 and 2 show the evaluation indicators for all the models in the training and testing phases. The smaller the values of MAPE, RMSE, and bias, the higher the values of R², WIA, NSE, and VAF, indicating that the model has less error and better computing efficiency and accuracy. The ELM model has the worst performance in all evaluation indicators. The KELM has slightly better performance indicators than the ELM. Among them, in the training phase, all evaluation indicators are better than the ELM. In the testing phase, the values of WIA and VAF are very close to those of the ELM model, which indicates that the prediction abilities of the ELM and KELM models are close to each other. For the BELM model, compared with KELM, the MAPE is reduced by about 8.95 and 50.77% in the training and testing phases, respectively, which indicates that the hyperparameters obtained through Bayesian network optimization are suitable for the model, and improved modeling accuracy. The value of VAF is significantly better than that of KELM and ELM, indicating that the prediction ability and generalization ability of BELM are higher than those of KELM and ELM.

Tables 3 and 4 show uncertainty analysis results for all models. The mean error values for all the models are negative, indicating that all the models are underestimated in the calculation process of C_d of the symmetric stepped labyrinth side weir, but in the training and testing phases, the average error values are very smaller, and it can be considered that the models are more stable. In addition, the BELM has the smallest average error and the narrowest confidence bandwidth at the same confidence level. Meanwhile, it can be seen that the BELM has the smallest range of uncertainty widths in different phases, and the range of the 95% confidence intervals is also significantly smaller than that of the ELM and the KELM.

The flow characterization of this structure is complicated due to the interaction of multiple factors affecting the symmetric stepped labyrinth side weir. In this study, the discharge coefficient estimation model was developed by ELM and Bayesian Network to comprehensively evaluate the stability and generalization ability from multiple perspectives. Also, uncertainty and sensitivity analyses of the model are analyzed, and the relationship between the variations of the geometric parameters. The following conclusions are obtained:

• (1)

  The model performance of BELM in evaluating the C_d for the symmetric stepped labyrinth side weir is better than that of ELM and KELM, which can be used as the discharge coefficient prediction model for this structure. In the testing phase, Bias = 0.0006, R² = 0.981, with the smallest average error and 95% confidence interval range [−0.055, 0.040], indicating that the model has robust performance, data consistency, and high confidence.

• (2)

  The global sensitivity indicates that the first-order sensitivity coefficient S₁ for F_r, L/B, L/y₁, p/y₁, b_s/a_s, and b_s/L are 0.35503, 0.0276, 0.00242, 0.54203, 0.05110, and 0.00100, respectively. Global sensitivity coefficient S_T is 0.36082, 0.03467, 0.01622, 0.55653, 0.05822, and 0.00100, respectively. Therefore, the influence of each dimensionless parameter on the C_d is p/y₁ > F_r > b_s/a_s > L/B > L/y₁ > b_s/L.

• (3)

  C_d increases with increasing p/y₁. In particular, the discharge coefficient is larger when the number of steps N = 4. Also, the smaller internal head angle θ = 45° corresponds to a larger value of C_d, and the tendency of increasing C_d decreases as the internal angle increases.

In summary, the dataset for this study is small, and only flow characteristics under subcritical flow are analyzed. Secondly, this paper is carried out under the condition of clear water and does not involve the change of flow characteristics of the structure when the flow contains sediment. In future research, it can be further studied.

This study was supported by the National Natural Science Foundation of China (Grant Nos. 52079122, 52379080).

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

The authors declare there is no conflict.