Prediction of aeration performance of different types of piano key weirs using different machine learning models

The design of hydraulic structures and sewerage supports aeration to ensure their good performance and environmental sustainability. In the process of water treatment, aeration enriches oxygen levels in the water and purifies water by removing impurities. Importantly, cavitation does not occur through aeration. Aeration supports hydraulic structure design and sewerage to ensure their good performance and environmental sustainability. Dissolved oxygen (DO) required for different species of aquatic organisms depends on species, habitat type, and specific ecological considerations. Crabs and oysters that live at the bottom need a minimum of 4 mg/L DO, while shallow-water fish require up to 4–15 mg/L DO for survival. The necessity of DO in aquatic organisms relies upon factors like temperature of water, salinity, nutrient concentration, as well as the presence of pollutants (Baylar & Bagatur 2000; Emiroglu & Baylar 2003). In general, aeration is the process that enhances the DO level in water bodies and is crucial for the existence of the aquatic life. It prevents the formation of anaerobic conditions (i.e., those that produce harmful gases like methane or hydrogen sulphide). It saves our environment and also plays a crucial role in combating climate change.

In hydraulic engineering, aeration serves to improve the flow characteristics of water by reducing its weight so that it can become more buoyant (Baylar et al. 2011). This prevents vapour pockets as well as reducing pressure fluctuation promoting efficiency, stability, and safety for hydraulic structures (Singh & Kumar 2022b). Hydraulic structures are built to decrease cavitation and increase oxygen content in flowing water (Rathinakumar et al. 2014) during extreme weather events such as heavy rainfall and droughts. Weirs and other water management structures can be used to manage water flow, reduce the risk of flooding, and maintain water levels during droughts. Properly managed weirs can also help mitigate the impacts of temperature increases on water quality by promoting aeration and maintaining oxygen levels. The last 10 years have seen growing interest in the study of aeration performance of various free flow structures such as triangular, rectangular, labyrinth, and PKWs for efficiency improvement and stability reasons. These studies aim to understand the flow dynamics of different types of weirs and how they affect the aeration and mixing of air in the water while the flow occurs over such types of weirs.

Although the water flows through the hydraulic structure for a short duration, the DO level increases significantly due to turbulence flow that facilitates the mixing of air with flowing water (Emiroglu & Baylar 2003; Guenther et al. 2013). However, these effects are transient, and the DO concentration returns to the normal level as the water flows in plane areas that are far away from the hydraulic structure (Wormleaton & Tsang 2000). This phenomenon mitigates greenhouse gas emissions and prevents the Earth's surface temperature from rising.

The efficient use of a weir for aeration in rivers or streams was initially shown by Gameson (1957). Weirs are known to stir the water and cause turbulence, which raises the concentration of DO. The work by Baylar & Bagatur (2000) was a landmark in the fields of hydrology and for improving the water quality. The effectiveness of the weir aeration depends on many factors such as the weir's size and placement, water flow rate, desired level of aeration, and the cost of equipment and maintenance (Baylar et al. 2001).

This study has several advantages as a continuation of earlier works on the impact of hydraulic structures on water quality. Experimental studies (Apted & Novak 1973; Ervine & Elsawy 1975; Gulliver & Rindels 1993; Baylar & Bagatur 2000; Baylar et al. 2001) examined the aeration capabilities of various types of weirs and found that the labyrinth weir exhibits a higher entrainment rate compared with the linear weir (Avery & Novak 1978; Nakasone 1987; Wormleaton & Soufiani 1998; Emiroglu & Baylar 2003). Many attempts have been made to estimate the ratio of air/water flow and specifically aeration performance of various hydraulic structures like weirs in detail (Bagatur & Sekerdag 2003; Baylar & Ozkan 2006; Baylar et al. 2010, 2011). From a range of PKW experiments, it was discovered that the water with low heads flowing down through these weirs is naturally aerated. In fluid dynamics, ‘nappe’ represents a sheet of water passing over in a structure such as a weir or a dam. The characteristics of the nappe while flowing is another important aspect to consider the design of weirs based on vibrations, the flow of nature that could be laminar or turbulent, fluctuations, or surging of the flow (Baylar et al. 2011; Jaiswal & Goel 2019; Jaiswal & Goel 2020).

Crookston & Tullis (2012) performed research on the labyrinth weir by examining four different ways that the nappe was ventilated: clinging, aerated, partially aerated, and drowned. Their examination presented the understanding of the processes involved in the creation of nappes. Moreover, the research emphasizes the substantial impact of fluctuating negative pressures beneath the overspill nappe, underscoring their capacity to trigger seismic tremors. In addition, researchers examined the consequences of intentionally increasing the oxygen content in the flow of the nappe (Leite Ribeiro et al. 2007). Usually, the size of the nappe depends on how high it falls and how much it spills over the top. When dealing with bigger runs or narrow outlet keys, there can be conflicts between water from lateral ridges, but these conflicts would not significantly affect the efficiency of the output if flow restrictions are kept constant (Machiels 2012).

In addition, Jaiswal & Goel (2020) applied various soft computing methods to forecast air movement around triangular weirs. They noticed that the Gaussian process and M5P techniques accurately predict the amount of oxygen added. Moreover, other approaches include adaptive neuro-fuzzy inference systems (ANFIS) and linear regression indicating PKW oxygenation (Komal et al. 2017).

In the specific field of hydraulic engineering, the scope of using soft computing techniques for the prediction of aeration efficiency for different hydraulic structures has been increased recently. The reason for their choice is that they are intelligent, fast, and precognitive. These methods find favour on account of their intelligence, speed computation, and predictive ability. If implemented, these models would reduce scale effects and simplify the processes involved in model building. In water resources management and hydraulic engineering, different tools of soft computing techniques have been used (Luxmi et al. 2022; Srinivas & Tiwari 2022). Machine learning and soft computing methodologies have recently received attention from more researchers regarding estimating aeration potentials for different weirs (Verma et al. 2022).

Baylar et al. (2011) investigated using gene expression programming (GEP) to estimate the oxygen transfer efficiency in stepped cascade aeration. Singh et al. (2021) attempted to analyse and assess labyrinth weir aeration efficiency by adopting diverse methods like the M5P model and support vector machine. Kumar et al. (2020) implemented a plunge jet analysis using artificial neural networks. Some of the other tools used for this purpose include generalized regression neural network, ANFIS, and multivariate adaptive regression splines and many others, especially for estimating oxygen transfer rates into water and lakes (Baylar et al. 2008). It is with proper recognition of these facts that studies by Baylar et al. (2009a, 2009b) have shown that it is feasible to apply a support vector machine for predicting the efficiency of aeration by plunging overfall jets from weirs. Other forms of machine learning techniques such as K-nearest neighbours were also tried in the process of aeration performance estimation used for predicting aeration performance using Parshall flumes, including random forest regression (Sangeeta et al. 2021). To derive these models, Aradhana et al. (2021) sought to identify the machine learning-based methodologies, which were adaptable for subsequent analysis of aeration capability of labyrinth weirs. Bansal et al. (2023) performed energy calculations on PKW with the help of extreme gradient boosting (XGBoost) and GEP. A recent literature review indicates that many machine leaning techniques have been applied to predict the performance of different hydraulic systems in terms of aeration. By promoting aeration, weirs help to improve water quality by encouraging the breakdown of organic materials through aerobic processes. This can reduce the levels of pollutants and prevent the growth of harmful algae, which thrive in low-oxygen environments.

ELM is a kind of feedforward neural network initially suggested by Huang et al. (2004, 2006). It is designed for simplicity and fast training. It has been demonstrated that the typical ELM with additive or Radial Basis Function (RBF) activation function (Huang et al. 2006) can approximate any function universally. These models have been applied to many problems, including classification, regression, and feature extractions (Rong et al. 2008; Huang et al. 2010; Wang et al. 2011; Lim et al. 2013). The ELM model's concept is to randomly assign hidden biases and input weights, which are solved analytically for the output weights, instead of using iterative optimization methods like backpropagation, gradient descent, etc. The model is applied to predict the aeration efficiency E for a given value of at different heights of PKWs and at temperature 33°C.

Liang et al. (2006) suggested an online version of ELM. OS-ELM is a modification of ELM algorithms. Single hidden layer feedforward neural networks, such as the online sequential ELM model and ELM model, are well known for their ability to understand complex nonlinear functions. The critical difference between the two is that the OS-ELM is designed for online learning, while the ELM is intended for batch learning. All training data are fed once in ELM, while in OS-ELM, large training data are fed in modules. OS-ELM is used in case of memory constraints. OS-ELM is relatively fast in learning as ELM, while it has an advantage over ELM in learning from the newly arrived data. The hidden layer weights of OS-ELM are created at random and maintained as constant, whereas the adjustment of output weights takes place incrementally with the arrival of new data. This enables the model to acquire knowledge rapidly and adjust to shifting trends in the data. OS-ELM can use existing data to update the model, without having to train the whole dataset again, which can be faster and cheaper. OS-ELM can be used for various tasks such as recognizing images and sounds, analysing biological data, and controlling systems. The specifications of algorithms are outlined below. The OS-ELM algorithm's mathematical equations can be classified into two steps: the initialization step and the online learning step.

Initialization step:

• • A set of training data is fed to start the training process, involving unique and arbitrary samples such that ⁠.

• • Input parameters are assigned randomly, as in the ELM model, and the initial output matrix is computed:

• from the hidden layer

• • Calculate the initial output weight matrix from Equation (8):

  

• • Set and _, where represents the count of data chunks.

Online learning step:

• For distinct arbitrary samples, the given segment of new training data is considered.

• Determine the hidden layer output matrix similarly as described in Equation (6).

• Determine the output matrix by applying the following mathematical relation:

  

⁠

The algorithm used in OS-ELM is as follows:

Input:

• ⁠: hidden nodes count;

• ⁠: count of data chunks;

• The training data chunk is denoted by ⁠:

  

  

Outputs:

• 1. //Initialization;

• 2. for ⁠;

• 3. set input weight randomly;

• 4. set hidden layer bias randomly;

• 5. compute L₀ (the hidden layer output matrix);

• 6. determine (the output weight matrix);

• 7.

• 8. //Learning process;

• 9. while k is less than K⁠;

• 10. determine the output matrix of the hidden layer L_k+1; and

  

• 11. calculate the output weight matrix ⁠;

• 12. ⁠;

  

The incremental ELM is an extension of the ELM algorithm that is introduced (Huang & Chen 2008). It is designed to handle incremental learning. A limited number of hidden nodes are initially present in the model. The hidden layer's input weights and biases are chosen at random. The model is trained on the first batch of the data samples using the ELM training method. The new hidden nodes are added, and adjustments in the biases and the input weights of the already current hidden nodes have been updated incrementally. It uses simple linear least-square methods to calculate the output weight, considering the new data samples and previously learned data.

Step 1: Initially, let and H be the maximum hidden nodes. The predicted accuracy is predetermined to be ⁠. The output Y is set as the initial values for the residuals E, which represents the variance between the network's actual and desired outputs.

Step 2: Steps taken to train the model: While and ⁠.

• 1. ⁠, where H is the count of hidden nodes.

• 2. The newly expanded hidden layer neuron 's input weight and bias are generated at random.

• 3. Determine the output of the activation function for ⁠.

  

  (10)
• 4. Determine the output vector ⁠:

  

  (11)
• 5. Determine the output weight for as given in Equation (12):

  

  (12)
• 6. Once the new hidden node has been increased, the residual error is determined as follows:

  

  (13)

The output weight, as determined in Equation (12), decreases the error of the network quickly. Repeat the previous steps until the residuals are less than the expected error ⁠. The training process should be restarted when and the error is more than the anticipated error ⁠. Restarting the process of training is advised since it is typically because of the random calculation of the input weights (i.e.,⁠) and biases (i.e., ⁠). The user can check to see if the trained network satisfies the requirements using the provided testing set ⁠.

The aforementioned five metrics compare the degree of accuracy of the applied machine learning techniques. The section on simulation and results presents a comparison of the outcomes. The aforementioned three described models are trained and simulated for the given dataset, and results are presented in the following section.

Observation of water movement along the edge of the crest displays two discernible patterns. The initial pattern, closer to the upstream region, shows no signs of aeration. In contrast, the second pattern, located far from the sidewall crest, exhibits clear aeration. This aeration zone becomes more pronounced as the water discharge progresses downstream. As depicted in Figure 5, the flow over the PKW structure has a high degree of ventilation and a three-dimensional nature. Notably, areas of splashing and spraying are evident both inside the outlet keys and in the vicinity of the base of the structure (Singh & Kumar 2022b).

Concerning the trajectory of (head), there is a gradual expansion in the area where spraying and sprinkling occur. The planar jet starts far from the crest. In contrast, the region of air circulation undergoes a substantial increase as rises. This expansion is partly attributed to the rise in local velocity, which results in elevated and turbulent mixing and advection levels. This particular flow pattern enhances the aeration efficiency, as the intricate flow behaviour leads to an increase in air bubbles over a prolonged duration and distance (Eslinger & Crookston 2020).

As per existing research, hydraulic structures' effectiveness in aeration is impacted by factors that include the temperature of the flowing water, water quality, depth of the tailwater, height of the drop, and the discharge of water. It was seen that the weir's aeration performance is highly affected by the drop height as well as the discharge rate across it (Singh & Kumar 2022a).

The experimental data collected by Singh & Kumar (2022a) for types A, B, and C are used to train ELM, OS-ELM, and I-ELM models. The ELM and its variant are used as powerful tools for regression tasks. Here, regression establishes the relationship between the output and the corresponding input dataset. Aeration efficiency (E) depends on many factors, as explained in Section 1; the objective of applying different machine models is to examine how accurately the machine models establish a relation between the output E and inputs ⁠. The dataset contains three features and one output. The features affect the aeration efficiency predominantly as given in Equation (2) and are also compatible with ELM and its variants. There are two groupings within the entire dataset: (1) training dataset and (2) testing dataset. Eighty percent of the overall dataset is the training dataset, which is chosen at random, with the remaining 20% going toward testing.

The results demonstrate the predicted outcome of different models and their correlation with varying input parameters with several graphs. The findings based on the various parametric studies over type A PKW and its correlation with input parameters are presented in the following subsection.

After analysing the overall pattern of data points on the graphs plotted between actual and predicted aeration efficiency, it is found that the data points are clustered diagonally in the case of ELM and I-ELM, suggesting that the predictions nearly reflect the actual values. Hence, ELM and I-ELM predict the results more precisely and accurately. The results deviate from the accurate values, and therefore, the absolute error is maximum in the case of OS-ELM as the input data are not streaming time series data.

In Section 3.1, three machine learning techniques – ELM, OS-ELM, and I-ELM – are provided for comparison in terms of prediction accuracy. Then their performances are evaluated using five metrics described in Section 2.6, which are MSE, RMSE, MAE, CC, and NSE. The accuracy of the prediction is inversely proportional to MSE, RMSE, and MAE, while CC and NSE are directly proportional to the accuracy of the models in prediction aeration efficiency.

MAE, MSE, and RMSE values are the minimum for I-ELM and the maximum for OS-ELM. The value of CC and NSE is the maximum and close to unity for I-ELM compared with the other two models, as shown in Table 2. Hence, the prediction made by I-ELM is nearer to the actual value of aeration efficiency in all three cases of PKWs. The results given by OS-ELM deviate the most from the actual value, hence the least efficient model for the given dataset and conditions. The accuracy of the ELM lies between the remaining two. The complete dataset is available to feed at once in the models and does not change rapidly with time. Hence, ELM is more efficient than OS-ELM, which is more suitable when the data arrive sequentially. It is observed that for the available complete dataset, ELM and I-ELM perform better than OS-ELM. I-ELM can process data in batch and incremental modes that can handle vast data. It provides better results for the conditions in this research work. The model can be placed as OS-ELM, ELM, and I-ELM to increase their performance, as concluded in Table 2.

Accurately assessing the aeration performance of the weir structures is critical for hydraulic engineers and designers because aeration is the most effective way to enhance oxygen content in flowing water and prevent the anaerobic conditions in the water bodies, which are critical for the survival of aquatic life. To this end, the dataset obtained experimentally to calculate the aeration efficiency of different types of PKWs is used to train different extreme machine learning models and their variants like OS-ELM and I-ELM. Hyperparameters of all the applied models are correctly tuned. The major portion of the dataset is allocated for training the model, while the residual portion is utilized for testing the learning of different models. The prediction efficiency of three models, ELM, OS-ELM, and I-ELM, is compared while calculating the aeration efficiency of different hydraulic structures of type A, B, and C PKWs. It is found that for a complete available dataset that does not change rapidly with time, I-ELM performs the best, whereas, for these conditions, the prediction made in OS-ELM deviates the most from the actual value. The presented work can be extended to establish a well-defined relation between the input and output variables where the input dataset is unorganized. The performance of the models can be further evaluated under different data types, conditions, and problems from different domains.

The authors express their gratitude to the faculty, staff, and technical team of Delhi Technological University's Department of Civil Engineering for their help and assistance in conducting this research.

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

The authors declare there is no conflict.