Computation of energy dissipation across the type-A piano key weir by using gene expression programming technique

B

Length of side weir (⁠⁠)

B_b

Base length

B_i

Length of overhang portions at the inlet side

B_o

Length of overhang portions at the outlet side

CC

Coefficient of correlation

H

Specific energy at section ‘i’

H_r

Relative residual energy

H_L

Relative energy dissipation

g

Acceleration of gravity

H_t

Total energy head

y_t

Piezometric head

i

Represents the section

L

Total developed crest length

ME

Mean percentage error

MAE

Mean absolute error

MAPE

Mean absolute percentage error

N

Number of cycles

P

Weir height

Q

Discharge over the PKW

R²

Coefficient of determination

RSME

Root mean square error

S_i

Inlet key slope

S_o

Outlet key slope

V_t

Mean flow velocity

V_t²/2 g

Approach velocity head

W

Channel width/Width of PKW

W_i

Inlet key width

W_o

Outlet key width

Efficient computation of the energy dissipation over the hydraulic structures helps in designing the downstream dissipative systems and also helps in reducing downstream hazards during the flood season. Flooding is the most common natural disaster globally, and it has been increasing at an alarming rate over the last two decades. Extreme flooding is expected to become more common due to climate change (Chanson 2021). Flood control generally ensures that floods pass through and are released without causing damage to structural frames or their surroundings (Chanson 1994). The proper estimation of energy dissipation over the hydraulic structures can minimize the downstream expenses. Protection is an essential aspect of dam construction and exploration because an accident can often have significant consequences, to varying degrees, depending on the amount of water contained (Pinto 2017). During a flood or extreme hydrologic conditions, overtopping conditions may occur due to undersized spillways, culminating in dam overtopping. Therefore, spillways or weir structures must be designed to efficiently spill large amounts of water while maintaining high structural performance (Garg 2010). Consequently, increasing these structures’ flow release capability is critical for improving their protection.

Many researchers have shown their interest in the investigation of hydraulic jumps, including Rajaratnam (1990) and Hager et al. (1990), who expanded their research to include a jump with a control sill (Hager & Li 1992). Novak et al. (2010) examined the hydraulics of the jump and its implications for the submerged jump stilling basin and proposed that the morphology of the river bed usually determines the shape of the stilling basin. As a result, it is critical to understand the excavation required for its construction and operational purposes. It would be necessary to understand better the energy dissipation phenomenon downstream of the hydraulic structures to support the preceding statement. One of the most common solutions for increasing the spillway's discharging capacity is the installation of a labyrinth-style weir. Indeed, this form of weir will increase discharge while maintaining the same length as a traditional linear weir (Anderson & Tullis 2012; Leite Ribeiro et al. 2012). Energy dissipation at the base of the piano key weir (PKW) is an important phenomenon to consider when designing dissipative structures; special care should be taken to avoid undesirable effects such as scouring and cavitation (Silvestri et al. 2013a, 2013b).

Ho Ta Khanh et al. (2011a) presented the first experimental study with a stepped spillway, while a survey over the stilling basins has been examined by Troung Chi et al. (2006) and Pfister et al. (2017). Moreover, the general project-specific research and special studies on the energy dissipation of labyrinth and PKWs were investigated by Leite Ribeiro et al. (2007, 2011); Bieri et al. (2011); Ho Ta Khanh et al. (2011b); and Erpicum et al. (2013). In addition, Silvestri et al. (2013b) explored the energy dissipation over stepped chutes with a PKW crest and found that the low residual energy at the spillway toe increases with discharge and spillway length. The effects of the slope of the PKW on energy dissipation were examined by Al-Shukur & Al-Khafaji (2018). They concluded that the amount of energy dissipated decreases as the slope decreases. Recently, Eslinger & Crookston (2020) have conducted an experimental study to clarify the energy dissipation analysis at the base of type-A PKWs. In addition, Singh & Kumar (2022a, 2022b) presented the experimental investigation and the computational technique based on gene expression programming (GEP) to estimate the residual energy at the base of type-B PKW, respectively. They observed that the PKW's energy dissipation is not linear and is greater at a low head. Zounemat-Kermani & Mahdavi-Meymand (2019) used artificial intelligence data-driven models (adaptive network-based fuzzy inference system, ANFIS & multiple-layer perceptron neural network, MLPNN) embedded with several meta-heuristic algorithms (GA, PSO, FA & MFO) to simulate the passing flow over PKW, and compared the results. General results indicated that the ANFISs and MLPNNs could simulate the discharge coefficient of the PKW more accurately than empirical relations.

Many researchers have applied the GEP in the various fields of hydraulics to produce accurate estimates of the different hydraulics characteristics (Azamathulla et al. 2013, 2018; Karbasi & Azamathulla 2016). Karbasi & Azamathulla (2016) used GEP to predict the characteristics of a hydraulic jump over a rough bed. They compared it with the standard artificial intelligence (artificial neural network, ANN and support vector regression, SVR) techniques. They found that the artificial intelligence techniques indicated that the performance of these models is slightly better than the GEP model, but the application of the GEP model due to derivation of explicit equations is easier for practical purposes. Further, Azamathulla et al. (2018) used the GEP to predict the atmospheric temperature in Tabuk, Saudi Arabia. Abhash & Pandey (2020) and Singh & Kumar (2021) summarized the geometrical and hydraulic evaluation of PKWs over the last decade. Yazdi et al. (2021) investigated the effects of weir geometry on scouring development downstream of the PKW. According to them, the geometry of the weir and the discharge rate impact the scour characteristics. Zhao et al. (2020) discovered that the saltation height and length increase if the particle shape is not spherical. Moreover, Kumar et al. (2021) investigated sediment passage over type-A PKWs and observed that sediment passage is at a lesser rate as it goes through the intake key and speeds up at the key entrance. Plunging and impinging jets originating from the inlet and outflow keys were credited with forming the ridge and dip (Kumar & Ahmad 2022).

Studies have shown that local scour at the toe of PKWs placed in canals and rivers is also related to the energy dissipation of PKWs. Jüstrich et al. (2016) investigated the formation of scour holes and ridges caused by PKWs without scouring protection, concluding that the overall process is jet-induced scour. Pfister et al. (2017) conducted an experimental study to investigate the toe dug at the PKW and noticed that if the foundation on rock is not possible, then toe-scour occurring during flood discharges is relevant to weir stability. Shivashankar et al. (2022) describe the different methodologies for estimating the velocity phenomenon. This study proposed a hybrid generalized reduced gradient-genetic algorithm (hybrid GRG-GA) to assess the fall velocity. The hydraulic performance of the trapezoidal labyrinth-shaped stepped spillways was investigated by Ghaderi et al. (2020). The scour depth and volume of sediment removed at the toe of PKWs are determined primarily by sediment properties, discharge, residual energy, and tailwater depth. In addition, several experimental studies and machine learning algorithms/techniques were used in the prediction of the scour depth around submerged weirs, spur dikes, and a circular pier (Rashki Ghaleh Nou et al. 2019; Pandey et al. 2020a, 2020b; Birbal et al. 2021; Ghasempour et al. 2021; Pandey & Md Azamathulla 2021; Emadi et al. 2022; Singh et al. 2022). Pandey et al. (2021) suggested some critical points to Mohammad Najafzadeh and Ali Reza Kargar for their article on ‘gene-expression programming, evolutionary polynomial regression, and model tree to evaluate local scour depth at culvert outlets.’ Despite numerous significant experimental and computational investigations on energy dissipation across the labyrinth and PKWs, designers lack the knowledge to predict using traditional empirical models. As a result, new and precise approaches are still in high demand.

This article aims to create new equations based on gene expression programming to determine the relative residual energy downstream of the type-A PKW, which will aid hydraulic structure designers and engineers. During training, the measured flow and geometrical parameters are entered as input parameters into GEP, and the target parameter is the residual energy downstream of the PKW. The proposed GEP-based approach is compared to the conventional ones.

This research aims to develop a prediction equation for the relative residual energy downstream of a type-A PKW in rectangular horizontal channels under free-flow conditions. To this end, twelve different type-A PKW models were tested and assessed. The models’ configurations are as follows: magnification ratio L/W =5–6, where L is the total developed crest length, W is the width of the weir or channel, the height of the weir P varies from 0.15 to 0.20 m, relative width ratio W_i/W_o (1.0 ≤W_i/W_o≤ 1.5) where, W_i and W_o are the inlet and outlet key widths, respectively, and overhang portions (B_i=B_o) are alike for all models, where B_i and B_o are the inlet-outlet overhang portions, respectively (see Table 1). The discharges range is 0.005 m³/s ≤Q≤ 0.05 m³/s, head to weir ratio 0.085 ≤H_t/P≤ 0.85 for 0.20 m tall models and 0.11 ≤H_t/P≤ 0.88 for 0.15 m tall models, and the total head over crest is 0.0168 m ≤H_t≤ 0.167 m, other dimensionless ratios are 0.43 ≤(B_i/P=B_o/P) ≤0.95. A total of 240 tests have been conducted over the twelve different types of PKW, with 20 tests for each model.

The above relationship describes the relative residual energy ratio as a function of geometric and hydraulic factors. The ranges of various parameters and data collected in the present study are summarized in Table 1.

Gene expression programming (GEP) is the learning algorithm behind GeneXproTools. It learns explicitly about relationships between variables in data sets and then builds models to explain these relationships. Gene expression programming uses character linear chromosomes, made up of genes structurally organized in a head and a tail, first encoded by Ferreira (2001a). In GEP, chromosomes of various sizes and shapes can code in a simple graph (Ferreira 2001a, 2001b), and, like other evolutionary methods, GEP begins by randomizing early population chromosomes. It combines elements of genetic programming and genetic algorithms. The chromosomes function as a genome and are subject to mutation, transposition, root transposition, gene transposition, gene recombination, and one- and two-point recombination. The chromosomes encode the expression trees that are the aims of selection. In the GEP model, various fitness functions such as mean squared error (MSE), root mean squared error (RMSE), relative standard error (RSE), and root relative squared error (RRSE) can be used (Ferreira 2001a). The most advantageous chromosomes are likely to be passed down to future generations. Genetic operators perform the same acts with minor variations following selecting the best chromosomes.

The first step in choosing a GEP model is to select a fitness function. As a result, this study employs the RMSE function. The next step is to select the set of terminals and functions used to construct the chromosomes. The modeling process adopted in this study designates the relative residual energy (H₂/H₁) as the target value and the six independent parameters (H_t/P, L/W, W_i/W_o, S_i/S_o, B_i/B_o, and N) as input variables which are discussed in Equation (5). The basic operators (+, −, ×, ∕, ln, x², e^x, 1/x, ⁠, Avg. of 2) were used to develop the GEP model.

The functions were chosen based on their coherence to the quiddity of the problem in order to achieve an uncomplicated and sensible GEP model. The general sampling strategy included selecting 30 chromosomes, three genes, and eight different head sizes. Table 1 shows that 240 data points were used in modeling and distributed randomly for the training and testing data phases. Approximately 80% of the data is used for training, while the remaining 20% is used to test the current project. The GEP's training and testing data were chosen randomly from the original dataset.

Table 2

Functional set and operational parameters used in the GEP model

S. No. .	Description of parameter (1) .	Setting of parameter (2) .
1.	Function set	+, −, ×, ∕, ln, x2, ex, 1/x, ⁠, Avg. of 2
2.	No. of chromosomes	30
3.	Head size	8
4.	No. of genes	3
5.	Gene size	26
6.	Linking function	Addition
7.	Fitness function	RMSE
8.	Program size	41
9.	Literals	14
10.	Number of generations	1,15,310
11.	Constants per gene	10
12.	Data type	Floating-point
13.	Mutation	0.00138
14.	Inversion	0.00546
15.	Gene recombination rate	0.00277
16.	One-point recombination rate	0.00277
17.	Two-point recombination rate	0.00277
18.	Gene transposition rate	0.00277
19.	Insertion sequence (IS) transposition rate	0.00546
20.	Root insertion sequence (RIS) transposition rate	0.00546

S. No. .	Description of parameter (1) .	Setting of parameter (2) .
1.	Function set	+, −, ×, ∕, ln, x2, ex, 1/x, ⁠, Avg. of 2
2.	No. of chromosomes	30
3.	Head size	8
4.	No. of genes	3
5.	Gene size	26
6.	Linking function	Addition
7.	Fitness function	RMSE
8.	Program size	41
9.	Literals	14
10.	Number of generations	1,15,310
11.	Constants per gene	10
12.	Data type	Floating-point
13.	Mutation	0.00138
14.	Inversion	0.00546
15.	Gene recombination rate	0.00277
16.	One-point recombination rate	0.00277
17.	Two-point recombination rate	0.00277
18.	Gene transposition rate	0.00277
19.	Insertion sequence (IS) transposition rate	0.00546
20.	Root insertion sequence (RIS) transposition rate	0.00546

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In order to better predict or for an accurate estimation of the residual energy downstream of the PKW, the GEP technique was used to measure the downstream residual energy. The GEP approach demonstrates a highly nonlinear relationship between relative residual energy and the input parameters (H_t/P, L/W, W_i/W_o, S_i/S_o, B_i/B_o, and N) with high accuracy and relatively low errors. After multiple generations, the program was discontinued due to no progress in the fitness function value or the coefficient of determination. After 115,310 generations, there was no discernible difference. All of the parameters mentioned were chosen through trial and error in order to obtain the best model of the GEP in the form of an algebraic equation between output and input variables.

According to the author, the proposed equation of second-order exponential approximation may be adequate for conceptual designs and alternative analyses of the relative residual energy estimation. The R² value in the current approach for the two datasets (testing = 0.978 and training = 0.980) demonstrates the model's adequacy. GEP reported RMSE values of 0.032 and 0.029 for the training and testing data sets. ME, MAE, and MAPE values for training data were 3.26%, 0.025, and 5.30%, and for the testing, data were 4.21%, 0.016, and 5.40%, respectively, showing the performance and accuracy of the predicted model.

As a result, the developed GEP is a reliable method for predicting relative residual energy (H₂/H₁) with good generalization and no overtraining. The proposed GEP has a significant advantage over traditional regression-based models (traditional equations). Depending on the number of generations, the GEP model can predict the relative residual energy with high accuracy and in a short amount of time. It is capable of mapping the data into a high-dimensional feature space, where various methods are used to discover data relationships. The relationships are as diverse as the mapping. The constructed GEP model demonstrated excellent agreement with the observed values in terms of coefficient of determination (R²) and root mean square error (RMSE) for training and testing datasets.

Accurately estimating the relative energy dissipation or residual energy downstream of the weir structures is critical for hydraulic engineers and designers. This paper developed an empirical equation based on gene expression programming (GEP) to predict the relative residual energy downstream of the type-A PKW. The GEP approach produced a highly nonlinear relationship between relative residual energy and input parameters according to the proposed equation for relative residual energy. The GEP model's comparisons over the existing models show that the GEP models provide better predictions, and practitioners can use them as design approximations to assist researchers and design engineers in designing PKWs. The GEP model had the lowest R² (0.978 for training and 0.980 for testing), and root mean square error (RMSE) values of 0.032 and 0.029 for the training and testing datasets. The results suggest the efficiency of the GEP model and its potential use for practical applications within a similar range of non-dimensional parameters tested in this work. According to the findings of this study, the GEP model is more beneficial for any condition with no limitations. More research into the energy dissipation estimation of PKW is recommended. This could include other geometries like PKW types B, C, and D, or it could use turbid water to calculate the energy surplus.

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

The authors declare there is no conflict.