Prediction of inlet-to-outlet width ratio of type-A piano key weir using fuzzy neural network (FNN)

P

PKW's height

L

Overall development length

W

Weir's width

H_t

Cumulative head over the weir

W_i

Width of inlet key

W_o

Width of outlet key

N

Cycle number

S_i

Slope of inlet key

S_o

Slope of outlet key

B_i

Length of inlet key

B_o

Length of outlet key

C_DL

Coefficient of discharge along developed crest length

Q

Discharge flow of PKW

E_L

Relative energy dissipation

E₁ or E₂

Energy dissipation in sections 1 and 2

MLPNN

Multi-level perceptron neural network

CWR

Crop water requirement

CIR

Crop irrigation requirement

EC

Electrical conductivity

GCM

General circulation model

NSE

Nash-Sutcliffe efficiency model

CFD

Computational fluid dynamics

ANFIS

Adaptive Neuro-fuzzy Inference System

PSO

Particle swarm optimization

GA

Genetic algorithm

MFO

Moth flame optimization

FA

Factor analysis

SVR

Support vector machine

ANN

Artificial Neural Network

x

Input for the FNN model

Weights associated with i^th layer in FNN model

e

Epsilon

Outputs associated with i^th layer in FNN model

Partial differentiation

Piano key weir (PKW) is a spillway that is used in water management at dams. In dam engineering, there are height and space restrictions; at that given condition, PKW is a very effective management solution and is nonlinear in design. As the name suggests, it resembles piano keys, a structure where keys are grouped in a stepped manner to serve the purpose. When PKW is compared with traditional linear spillways, it has much better discharge capacity, energy dissipation, and lower material utilization. This is a very cutting-edge structure used to manage and regulate water flows in dams. PKW is an upgraded form of labyrinth weirs, which were first researched by and gained notoriety as a more effective and economical weir (Lempérière & Ouamane 2003). A recent project in France that uses PKW to boost the water release capacity of the country's current suction dams (Laugier et al. 2017) and as a promising solution for dam weir on derailed structures in Vietnam (Khanh 2017) is rapidly increasing. Since 2011, significant knowledge exchange during geographical conferences has guided the compilation of reference material summarizing the current state of PKW technology (Erpicum et al. 2011, 2013, 2017). All around the world, more than 35 PKWs have been successfully built, including in the UK, Algeria, South Africa, France, Switzerland, Vietnam, India, Australia, and Sri Lanka (Crookston et al. 2019).

The first PKWs were built in 2006 at the Goulours Dam in France and more recently at the Hazelmere Dam in South Africa. They significantly increased the possibility of reservoir and spillway release. An upcoming PKW implementation at Tzaneen Dam is in progress (van Deventer et al. 2015). Notably, recent interest in PKW implementations extends to interior weir applications, complementing the primary focus on utilizing PKWs as lateral weirs for runoff release (Karimi et al. 2018). Over the past two decades, detailed summaries of PKW geometry and hydraulic behaviour have been presented by Abhash & Pandey (2020) and Singh & Kumar (2021). Weir altitude (P), relative advancement L/W ratio (crest length), (H_t/P) ratio, alveolar width (W_i/W_o), and cycle number (N) for a constant channel width are geometric parameters that have a significant impact on head-discharge efficiency and energy dissipation (where L is the overall development length, W is weir's width, H_t is cumulative head over the weir, and P is weir altitude). PKW behaviour is further influenced by the type of crest, lengths of overhangs, upstream apex walls, and raising the crest via a parapet wall (Anderson & Tullis 2013). Numerous investigations have explored scientific hypotheses, case studies, and original research on PKW energy loss and labyrinths (Bieri et al. 2011; Ribeiro et al. 2011; Khanh 2013). Silvestri et al. (2013) observed that in PKWs with stepped chutes, residual energy was low at the spillway toe and increased with release and spillway length. Al-Shukur & Al-Khafaji (2018) investigated how the PKW slope affected energy loss and showed that the proportion of energy dissipation decreased as the slope decreased. Computational techniques employing AI models such as MLPNN, ANFIS, PSO, GA, MFO, and FA have been used to define PKW hydraulic behaviour based on geometrical parameters (Zounemat-Kermani & Mahdavi-Meymand 2019). Karbasi & Azamathulla (2016) used GEP for hydraulic jump aspects over a rough bed, comparing its performance with conventional AI approaches SVR and ANN. Many studies have tried to estimate how deep scour can occur along dikes, weirs, and piers that are partially submerged in water (Karbasi & Azamathulla 2016; Pandey et al. 2020; Birbal et al. 2021; Emadi et al. 2022; Singh et al. 2022). Therefore, more new ideas and rivals are needed to improve PKW technology. Comprehensive reviews on PKWs for discharge measurement (Bhukya et al. 2022) and energy dissipation (Silvestri et al. 2013; R. Eslinger & Crookston, 2020; Singh & Kumar, 2022a, 2022b, 2022c, 2023a, 2023b) have been presented, emphasizing the need for robust experimental and computational analyses to better understand PKW hydraulic properties.

Kumar et al. (2021) reviewed and understood the change in extreme circumstances to develop effective, durable strategies. The document identifies regions in India prone to droughts and floods to prepare in advance and reduce risk. The author offers a solid approach for assessing risks and boosting the resilience of essential infrastructure. The proposed approach first assesses the potential impact of extreme weather on critical infrastructure and then develops strategies that can withstand climate-related challenges. Tiwari et al. (2024) explore the relationship between climate change and water quality by focusing on the electrical conductivity in the Narmada basin. AI techniques can be used to predict water quality. The author develops 10 AI models to predict the EC levels at the Sandia station in the Narmada basin. This information can be used to manage the region's drinking water, irrigation, and other uses. Model 8 performs exceptionally well among all models, with an R² value of 0.889. Poonia et al. (2021a, 2021b, 2021c) addressed the lack of comprehensive analysis of the joint dependence of drought duration and its effect across Indian river basins. There is a research gap that hinders the development of effective drought strategies. The author suggests modelling the joint dependence structure of drought features using a bivariate copula-based technique. The author aims to identify patterns and hotspots of severe and prolonged drought events across India. Poonia et al. (2021a, 2021b, 2021c) aim to address the research gap related to the unpredictability of climate model projections and scenarios in assessing CWR and CIR in the Eastern region of Himalayan, specifically in Sikkim. Das et al. (2020) provide a thorough analysis of how climate change affects a Sikkim crop field, particularly rice, wheat, and maize. They use calibrated AquaCrop simulations based on historical data to project future crop yields under different emission scenarios and GCMs.

Additionally, they add possibility theory to examine uncertainty with GCMs. Jamal et al. (2024) aim to address the challenge and land loss in states like West Bengal and India. Existing systems for protecting riverbanks include different materials and methods to determine stone size and effectiveness under different temporal variations. The author identifies a lack of research finding the most effective stone for safeguarding against bank failure. The author suggests a way to test how a riverbank reacts to different sizes of stones using a physical model. Poonia & Tiwari (2020) aim to satisfy the demand for rainfall-runoff modelling in Madhya Pradesh's Hoshangabad Narmada River watershed. This area often experiences floods, so it is important to have accurate runoff estimation for good flood management. The author suggests using ANN-based models to simulate water flow in the Hoshangabad area. Different ways of measuring how well the models work is used, such as how close they are to the real data, how much error they have, and how much they improve the results. Kantharia et al. (2024) want to find out how well they can predict the amount of water that flows out of the Damanganga River daily by using information about the rain and the soil. The study's objective is to fill the gap by creating a model using ANFIS that is customized for the Damanganga basin, taking into account soil moisture at various depths as input variables. We measure how well the model fits the data by using two numbers: the R² and the NSE.

Although there is growing awareness of the urban heat island (UHI) phenomenon and its consequences, Sharma et al. (2023) discovered a scarcity of in-depth investigations on this topic, especially in regions with hot semi-arid climates, such as India. The author's model intends to fill the current knowledge gaps on UHI effects, and it employs remote sensing and geospatial analysis methods to examine how surface temperatures and UHI effects vary throughout the year, especially considering the influence of land use and water body cooling. Baudhanwala et al. (2024) discuss the necessity for accurate rainfall prediction models in regions like South Gujarat, India. Severe rainstorms are happening more often in the area, and this causes problems for water, farming, buildings, and people's lives. The writer utilizes four distinct machine learning (ML) techniques: support vector regression (SVR), multiple linear regression (MLR), decision tree (DT), and random forest (RF). The research suggests that a width ratio between 1.25 and 1.30 provides the best results in terms of hydraulic efficiency, as indicated by the highest discharge coefficient. Beyond this range, energy dissipation increases, yielding diminishing returns. This research offers valuable insights for PKW design and highlights the applicability of FNNs in hydraulic engineering, ultimately enhancing civil engineering and water resource management practices.

A real-world dataset and the suggested strategy were used to demonstrate the contribution. An overview of the contribution would be as follows:

• The dataset utilized in this research is experimentally obtained and used to evaluate proposed methodologies; it is not pre-defined. The dataset utilized in this work was obtained through experimentation. Unlike traditional methods, which rely on pre-defined datasets, this study offers a unique dataset, enhancing the data's validity.

• A new model named FNN is proposed; this method has never been used before. The research void in hydraulic behaviour pattern prediction for PKWs will be filled by proposing FNN.

• A thorough derivation of the tuning of the hyperparameters suggested in FNN is presented in this article. A crucial part of building a model is tuning; the contribution offers a systematic and precise way to adjust the parameters to the specifications. The derivation is novel and contributes to improving the model's performance.

The organization of the article is as follows: Section 2.1 talks about the experimental set-up and dataset used in the process, Section 2.2 shows the methodology used in the article and also the tuning of hyperparameters, then further in Section 3, experimental results are discussed and in Section 4, conclusion is given.

This article proposes a novel FNN. The biological neurons inspire neurons in ML. Pattern recognition and prediction are required in many situations where automation is challenging. Neural networks (NNs) were developed to address these issues, as they can automate the process and provide accurate data predictions. NN processes the input along with the hyperparameters. A fuzzy system is a logic that deals with binary values, divides the data based on membership function, and gives values as a degree of truth. Equations are given a more human touch through the use of fuzzy logic.

Comparably, the function for the fourth layer is displayed in Equation (4). It accepts input as and handles weights as to produce an output of ⁠.

This layer is the production layer, which combines all the components of the fuzzification layer. This particular layer has no variable weights; hence, updating the parameters is impossible.

In Equation (45), the parameter is from Equation (35). is the parameter related to layers 2 and 3 of the FNN as seen in Figure 3, is the partial differentiation of the sigmoid function utilized in layer 3, and are the parameters related to fuzzy logic, and x is the input provided to the FNN model.

Equation (57) is similar to Equation (54) only L in Equation (57) is used for the variables of the last layers.

The dataset is collected experimentally, preprocessed, and passed to the FNN model. The method used to validate data is splitting the data into two categories, i.e. training and testing data. Both datasets are chosen randomly and divided in a ratio of 4:1. The training data is used to train the model by tuning the hyperparameters. Then, the model is validated using the testing data. This method allows the judgement of the model's performance to be balanced.

Furthermore, the data in Figure 8 reveals that W_i/W_o = 1.25 achieves notably higher discharge efficiency than W_i/W_o = 1.3 for H_t/P < 0.35. However, for the range 0.35 < H_t/P < 0.44, W_i/W_o = 1.3 surpasses W_i/W_o = 1.25 regarding discharge efficiency. Notably, W_i/W_o = 1.4 stands out as the configuration with the highest discharge capacity within the 0.44 < H_t/P < 0.81 range. In particular, PKW setups with W_i/W_o values of 1.25 and 1.3 demonstrate a remarkable 7–17% increase in efficiency compared with W_i/W_o = 1.0 and an approximately 8–13% gain over W_i/W_o = 2.0. It's worth noting that the FNN model excels in precision, as evidenced by its RMSE of 0.0206 and MAE of 0.0150, as shown in Figure 6.

To precisely compute the optimal inlet-to-outlet key width ratio of PKW and the influence of the different width ratios on energy dissipation, the experimental data from a previous study by Singh & Kumar (2023a, 2023b) were used to create the FNN algorithm-based model in this research. According to the proposed model for predicting the key's width ratio, the approaches resulted in a highly nonlinear relationship between the width ratio and input parameters, with promising prediction results. The hydraulic behaviours measured in the process are discharge flow over the PKW, coefficient of discharge along developed crest length (C_DL) and relative energy dissipation (E_L). The peak efficiency occurs at a width ratio (W_i/W_o) of approximately 1.2755–1.28, with a 7–17% efficiency advantage over W_i/W_o= 1 and 8–13% over W_i/W_o = 2.0. The energy losses over the weir decreases as the W_i/W_o ratio increases, with the highest relative energy dissipation corresponding to the lowest width ratio (i.e. E_L = 0.8093 or 80.93%, the corresponding W_i/W_o = 1) and the lowest energy loss for the highest width ratio (i.e. E_L = 0.5818 or 58.18% the related W_i/W_o = 2.0). This means the energy dissipation across the weir for W_i/W_o = 2.0 indicates 12–23% less energy dissipation than W_i/W_o = 1. The performance of the algorithm is measured based on the RMSE value and MAE value. More studies are needed on prediction and various soft computing techniques. Furthermore, doing an experimental investigation or CFD simulation while accounting for scaling effects is possible.

The faculty, staff, and technical team of the Hydraulic Laboratory of the Civil Engineering Department are much appreciated by the authors for their support, and they are also beseeched with sympathy.

This study has not received any funding.

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

The authors declare there is no conflict.