Estimating discharge coefficient of stepped spillways under nappe and skimming flow regime using data driven approaches

Author’s Accepted Manuscript Estimating Discharge Coefficient of Stepped Spillways under Nappe and Skimming Flow Regime using data driven approaches Kiyoumars Roushangar, Samira Akhgar, Farzin Salmasi www.elsevier.com/locate/flowmeasinst

PII: DOI: Reference:

S0955-5986(16)30136-4 https://doi.org/10.1016/j.flowmeasinst.2017.12.006 JFMI1390

To appear in: Flow Measurement and Instrumentation Received date: 7 September 2016 Revised date: 27 November 2017 Accepted date: 11 December 2017 Cite this article as: Kiyoumars Roushangar, Samira Akhgar and Farzin Salmasi, Estimating Discharge Coefficient of Stepped Spillways under Nappe and Skimming Flow Regime using data driven approaches, Flow Measurement and Instrumentation, https://doi.org/10.1016/j.flowmeasinst.2017.12.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Estimating Discharge Coefficient of Stepped Spillways under Nappe and Skimming Flow Regime using data driven approaches Kiyoumars Roushangar1*, Samira Akhgar2, Farzin Salmasi3 1

Associate Professor, Department of Civil Engineering, University of Tabriz, Tabriz, Iran

2

Ph.D Candidate, Department of Civil Engineering, University of Tabriz, Tabriz, Iran

3

Assistant Professor, Department of Water Engineering, Faculty of Agriculture, University of

Tabriz, Tabriz, Iran * Corresponding Author. Email address: [email protected] ; Tel: 00984133340081

Abstract

An important step toward spillways design always involves appropriate determination of the discharge coefficient. Since existing equations are incapable of estimating the discharge coefficient of stepped spillway accurately, other approaches such as data driven techniques can be employed as an alternative which are useful in modeling processes when the physical knowledge is limited. For this purpose, Gene Expression Programming (GEP) and Support Vector Machine (SVM) as data driven methods were applied for the modeling discharge coefficient of the stepped spillways using data derived from physical models and original experiments under nappe and skimming flow regimes. The input parameters included different dimensionless geometric and hydraulic parameters of napped and skimming regimes data. The obtained results indicated that the applied methods have a high capability in the modeling of discharge coefficient for the stepped spillways. The model comprised of four input parameters used for modeling of the discharge coefficient in nappe flow showed more accurate results (%RMSE=0.042, 0.328 and R2=0.966, 0.961 for the SVM and GEP models, respectively). In skimming flow data, the model with five input parameters produced RMSE=0.015, 0.357 and R2 =0.987, 0.979 in SVM and GEP, respectively. A comparison of the GEP and SVM results with the obtained results from the Multiple Linear Regression (MLR) showed that the performance of the linear model (i.e., MLR) was not suitable because the phenomenon of discharge coefficient is inherently complex and nonlinear. The results of sensitivity analysis for both proposed methods

showed that the Fr1 and H/y1 parameters in nappe flow and Fr1 and Re parameters in skimming flow had the most important influence on modeling of the stepped spillways discharge coefficient. The results also revealed that the SVM relatively surpassed the GEP in the modeling of discharge coefficient. Nevertheless, the utilization of GEP model is highly recommended due to the high performance and extracting of simple, explicit equation by it.

graphical abstract

Start

All supposed models

Run the SVM

Finding the percent of partitioning with maximum R2 and minimum RMSE

Finding the RBF kernel with maximum R2 for each model

Run the GEP

Finding the GEP Function with Minimum RMSE

Selecting the best SVM model

Selecting the best GEP model

Extracting the Cd equation of the best model

Sensitivity analysis of best models

Determining the effective parameters

End

Keywords: Discharge Coefficient, Stepped Spillway, Gene Expression Programming, Support Vector Machine, Multiple Linear Regression.

Introduction Spillways are one of the most important dam structures which should be resistant, stable and highperformance structures. Generally, the spillways are classified in terms of their most important feature that might be in conjunction with control structures, discharge channel or any other part of it. The ogee-type spillway generally presents lower energy loss than the other types, so its discharge coefficient values take higher magnitudes. Nonetheless, the body design of ogee spillways is simple. The flow discharge of these spillways is computed using the rectangular sharp-crested weir discharge equation as (USBR [1]):

Q

3 2 2 gC d LH d 2 3

(1)

Where, Q, g, Cd, L, and Hd denote the discharge, gravitational acceleration, discharge coefficient, width of spillway and height of water over the spillway crest, respectively. A stepped spillway has a stepped ogee-profile spillway instead of the traditional smooth ogee-profile spillway, where a series of drops are made in an invert direction from the vicinity of crest to toe (Roushangar et al., [2]). The stepped spillway showed the potential of reduction in construction dimensions of especial energy dissipaters due toits special shape; hence, construction cost and time of the project will be reduced greatly (Wright [3]). Flow over stepped spillway is divided into three different regimes including, nappe, skimming and transition (intermediate mode between the nappe and skimming modes) flow regimes. The nappe flow regime corresponds to the relatively lower discharges and the skimming flow corresponds to the higher discharges. Besides, transition flow regimes can be observed in the intermediate discharges. In the nappe flow regime, water undergoes a succession of free-falling

nappes. In the edge of each step, water becomes a jet of a free descent before it permeates the following step. The nappe and skimming flow regimes constraints have been presented by Rajaratnam [4]. Accurate estimation of the discharge coefficient of side weir and spillways is very important for the optimal hydraulic-economic design. Numerous empirical regression-based equations have been proposed so far to compute the discharge coefficient of spillways as listed in Table 1. Recently, data driven approaches [e.g. Artificial Neural Networks (ANNs), Adaptive Neuro-Fuzzy Inference System (ANFIS), Gene Expression Programming (GEP) and support vector machine (SVM)] have been implemented in wide range of water resources engineering criterion. The data driven approaches have also been applied for modeling of wet weather response in wastewater systems (Vojinovic et al. [5]) such as predicting the friction factor of open channel flow (Yuhong and Wenxin [6]), head loss on cascade weir (Haghiabi [7]), the velocity field in a curved open channel (Bonakdari et al. [8]), modeling of performance of detention dams (Parsaie et al. [9]) and scour at a bridge abutment (Azamathulla [10]). It has also been implemented in modeling of energy dissipation over stepped spillway (Roushangar et al. [2]); estimating of rectangular side weirs discharge coefficient (Uyumaz[11]), predicting daily and monthly suspended sediment load (Nourani and Andalib[12]), modeling DO (dissolved oxygen) concentration in the lakes and rivers (Bertone et al. [13]; Kisi et al.[14]) modeling scour depth at downstream of Grade-Control Structures (Roushangar et al. [15]), estimating scour depth at clear water condition (Najafzadeh et al. a,b [16,17]). In this study, data driven techniques [e.g. gene expressions programming (GEP) and support vector machine (SVM)], can be used to determine the discharge coefficient of stepped spillways as alternative methods. The literature review shows that there are mainly two groups of studies focusing on simulation of the discharge coefficient of spillways, including: i) empirical regression-based models and ii) data driven regression-based models. Among the first group (as listed in Table1), Subramanya and Awasthy [18] introduced equations to calculate the sharp crested side weir discharge coefficient by solving the basic differential equations of spatially varied flow in a horizontal rectangular channel. Ranga Raju et al. [19] confirmed the validity and accuracy of De-Marchi's equation in estimating the discharge

coefficient of the sharp-crested and broad-crested weirs. Hager [20] presented an empirical equation for the side weir (zero crest height) discharge coefficient. Swamee et al. [21] introduced the local flow intensity coefficient concept by using the sharp-crested weir without walls at both sides of channel. Singh et al. [22] showed that not only the upstream Froude number is effective on the discharge coefficient, but also the weir height and the upstream depth flow ratio are important factors. Borghei et al. [23] provided a relationship for the discharge coefficient on the basis of upstream Froude number under the subcritical flow regime. Borghei and Salehi [24] proposed an equation for the discharge coefficient which utilizes the Froude number, spillway height to water depth over spillway ratio, and spillway length to water depth over spillway ratio. Emiroglu et al. [25] investigated the side weir beak in vitro and presented two linear and nonlinear multivariate equations to estimate the discharge coefficient of side weir beak. Moghiseh and Esmaili [26] studied the discharge coefficient of variable-crested level side weir using a laboratory approach. Bagheri et al. [27] studied discharge coefficient of rectangular sharp-crested side weirs and with employing the dimensional analysis and all the experimental data; they also proposed an empirical equation to predict the discharge coefficient. Ameri et al. [28] proposed two equations for the discharge coefficient of compound triangular–rectangular sharp-crested side weirs with different apex angle in subcritical flow condition. Emiroglu and Ikinciogullari [29] determined the discharge capacity of rectangular side weirs using Schmidt’s approach.

Among the second group and in the data driven models application context, Khorchani and Blanpain [30] applied artificial neural network (ANN) to estimate side weir discharge coefficient. Also Ghobadian and Shafai Bajestan [31] and Honar and Tarazkar [32] optimized discharge coefficient of side weirs by utilizing ANNs. Parsaie et al. [33] predicted flow discharge in compound open channels using ANFIS.

Parsaie and Haghyabi [34] optimized side weir discharge coefficient using genetic algorithms and ANN. Roushangar et al. [35] predicted trapezoidal and rectangular side weirs discharge coefficient using machine learning methods (SVM-GA & GEP). Azamathulla et al [36] estimated the side weir

discharge coefficient by SVM technique. Also, Zaji [37] modified oblique side weirs discharge coefficient by SVM.

The literature survey by the authors showed that there were different studies about investigating various aspects of stepped spillway such as defining hydraulic parameters (e.g. chanson [38] and Roshan et al. [39]), aeration (e.g. Baylar et al. [40]) and modeling energy dissipation (e.g. Roushangar et al. [2]). However, to the best knowledge of the authors, no significant study has been carried out about the stepped spillway discharge coefficient. Thus, the present study tries to simulate stepped spillway discharge coefficient by utilizing physical models and original experimental data under nappe and skimming flow regimes. The most important parameters on the discharge coefficient were also investigated through GEP and SVM methods application. Also, the results of these methods were compared with obtained results from multiple linear regression (MLR).

Methodological structure Data Driven Modeling Data Driven Modeling (DDM) is an approach which analyzes the data about a specific problem to determine the possible relations in between the problem variables (input, internal and output variables) even without having an adequate knowledge of the physical behavior of the problem. In the other words, data-driven modeling is used to determine the relationship between the system’s inputs and outputs using a training data set, which is an indicative of all the behavior found within the system. Inanition, this method concentrates on computational intelligence (CI), artificial intelligence (AI), and machine learning (ML) techniques that can be used to construct various models for complementing or replacing physically based models. In these methods, when the model is trained, it can be tested using an independent data set to determine how well it can generalize to unseen data (Dimitri et al. [41]). Support Vector Machine (SVM)

Support Vector Machines (SVM) was introduced by Vapnik [42] and used in pattern recognition or classification of objects in special classes. This algorithm is one of the relatively new methods, which have shown to have an appropriate workability in classification issue compared to the traditional techniques such as perceptron neural networks. When using SVM method, the system is firstly trained by using a part of the data (training data); then, the obtained solution is evaluated using an independent data set (test data). SVM may include different kernel functions and other parameters that their setting manner can effectively influence the obtained outcomes. In SVM models, the choice of an appropriate kernel function is very important, which depends upon the type and the nature of the problem. Table 2 presents different types of kernel functions in SVM used to analyze in the current paper. Gene Expression Programming (GEP) In gene expression Programming (GEP) (Ferreira [43]), linear and simple chromosomes with a fixedlength are combined similar to the ones used in genetic algorithms. Additionally, the branch structures with different sizes and shapes are combined similar to the expression trees in genetic programming (GP). In GEP, all branch structures with various sizes and shapes are coded in the fixed-length linear chromosomes, and the system can now take all the evolutionary advantages due to their presence. Although the phenotype in GEP includes the same type of branch structures used in the GP, the branch structures derived by GEP (which are called as expression trees) represent the entire independent genomes (Koza [44]). Hence, notable point in the GEP is that the second evolutionary threshold, that is, "phenotype threshold" is passed and this means that during reproduction, only slightly modified genome is passed to the next generation. Consequently, the relatively heavy structures are not required for replication and mutation (Ferreira [45]). In GEP, by providing input data, a mathematical function is defined that include one chromosome and a few genes. Each gene consists of two parts, the head and the tail. GEP algorithm creates an initial chromosome randomly being displayed in the form of a mathematical function and then turns into a tree model (ET).

Multiple linear Regression Analysis MLR methods are the most common form of linear regression analysis. Application of these techniques is very common, so it is being widely used in predicting hydrology and hydraulic phenomena (Rajaee et al. [46]). As a predictive analysis, the MLR is used to explain the relationship between one continuous dependent variable and two or more independent variables as well as an error term (Snedecor and Cocheran [47]). The independent variables can be continuous or categorical. Every value of the independent variable x is associated with a value of the dependent variable y.

y i  0  1x i ,1  2 x i ,2  ... p x i , p   i

for i  1, 2, ...n

(2)

Where yi= predication of discharge coefficient value ith; x i,n= value of the nth predictor; β0= regression constant; βn=coefficient of nth predictor; n=total number of predictors; and εi=error term. The MLR technique is based on the model of least squares. In this model, the best-fitting line for an observed data set is being calculated by minimizing the sum of the squares of the predicted values deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Recommended literature for details of the MLR includes the study of Snedecor and Cocheran [47].

Materials and methods Experimental setup In order to prepare the required original data, several physical models of stepped spillways was considered with different geometries (different slopes and sizes). The stepped spillways were made of galvanized iron, wood and Plexiglas with slopes of 45, 25, 15 degrees, respectively. Ogee curve of steps upstream was designed as Y  1.68X

1.85

(for spillway as wide as 50 cm) and Y  2.29X

1.85

(for spillway, as wide as 25cm). In the continuation of the ogee-shaped curve, the steps were placed where the slope reaches to the required threshold (45, 25 or 15). Figure 1 shows the experimental setup used in this study (Roushangar et al. [2]). The average height of the physical models was considered 1m and 32 cm for the models with the width of 50cm and 25 cm, respectively. Table 3

shows all the laboratory models for the stepped spillway. Number of the steps is between 5 and 50; their height varied from 1cm to 17.2 cm as the flow discharge was between 2litr/s and 60litr/s.

Models implementation The data are derived from experimental results with physical models as listed in Table 3. In this study, the stepped spillway discharge coefficient is investigated for skimming and nappe flow regime. Considering above mentioned equation in Table 1 as well as the research carried out in the field of estimating the side weir discharge coefficient (Subramanya and Awasthy[18] El-Khashab [48]; Singh et al. [22]; Borghei et al. [23]), the independent parameters that must be considered are the function of hydraulic and geometric parameters:

C d  f (  ,  ,  , g ,Q , y 1 ,V 1 , H , B , Le )

(3)

Where C d ,  ,  ,  , g ,Q denote the equivalent discharge coefficient, density, dynamic viscosity, surface tension of water, gravitational acceleration, and discharge, respectively. y 1 ,V 1 stand for the water depth and flow velocity at spillway upstream, respectively. B , H, Le represent the width, height, and the effective width of spillway, respectively. Using the Buckingham theory, discharge coefficient will be a function of the following dimensionless parameters:

C d  (Fr1 ,

H Le B , , ,We , Re) y1 y1 y1

Where, Re 

(4)

V 1 y 1 and Fr1  V 1 are the Reynolds and Froude numbers at upstream of spillway,  gy 1

respectively. We 

 y 1V 12 is the Weber number. 

Considering that the minimum height of the water on the spillway was higher than 5mm, the effect of Weber number can be ignored (Novak and Cabelka [49]). Also, given that the discharge (Q) is a function of the flow velocity and the water depth, we can write:

C d  (Fr1 ,

H Le B , , , Re) y1 y1 y1

(5)

By using these dimensionless parameters, eight models (input configurations) were constructed as listed in Table 4. Performance criteria The actuarial measures considered for assessing the models’ performance werethe Determination Coefficient (R2), and the Root Mean square Error (RMSE), expressions for which are given as: N

R 2  1

 (C

mi

 C pi ) 2

 (C

mi

 C mi ) 2

i 1 N

i 1

(6)

N

RMSE 

 (C i 1

mi

 C pi ) 2

N (7)

Where, Cmi and Cpi represent the measured and predicting discharge coefficients, respectively. C mi and C pi denote the corresponding mean values, separately. N shows the number of data patterns. RMSE portrays the mean extent of the errors by ascribing more weight to major errors and ranges between the 0 and ∞ that lower amounts relate to a better model execution. The determination coefficient; R2 can be utilized to describe as the simulation precision of the model. The combined use of these indices will provide sufficient insight about the models’ accuracy Legates and McCabe, [50]). Models Development

The conducted experiments include 110 and 70 data patterns for the skimming and nappe flow regimes, respectively. To determine the best percentage of data partitioning for train and test stages, a trial and error method was employed by SVM method. Three modes percent of data partitioning for training and testing stages were applied including 75%–25% (i.e. 75% of data for training and 25% for testing stage) 70%–30% and 65%–30 % modes. Accordingly, 65%-35% data partitioning mode gives the most accurate results for nappe flow regime, while the skimming flow model produces the most accurate results by using 70% -30% mode. Nonetheless, different combinations of functions were evaluated as listed in Table5 for investigating the GEP parse tree. As presented in this table, it is clear that the second function set (F2), which comprises the (, , , ,

) function set, presents the

lowest error value (RMSE) among the other studied function sets for both the nappe and the skimming flow regimes. Selecting the type of kernel function is one of the most important steps in SVM developing. In the present study, four types of functions, including linear, polynomial, radial basis function (RBF) and sigmoid were tried and evaluated. The RBF approximates the scattered data with high accuracy and only challenge is to obtain the value of the kernel. The changes in kernel value of RBF function cause to change in results significantly. SVM Models Development In this study, two domains are considered for kernel values optionally. The first domain is [0.1-1] with step of 0.1 and the second domain is [1-10] with step of 1. The R2 results of testing stage showed that the second range has better results; hence, only the results of this range is presented in here. The R2 results of the different kernel in SVM method for nappe and Skimming data are shown in Figures 2 and 3, respectively. According to these figures, the kernel that has the largest R2 value will be considered as the superior kernel. For example, in Figure 2 the best kernel value in nappe data for MC3 model is 4 and for MC6 equal to 2. GEP models Development

In this study, four basic arithmetic operators (+, - , , /) as well as six basic mathematical functions

(

, 3 , x 2 , x 3 ,ln, e x ) were utilized in GEP function set. Moreover, each GEP model was evolved

until the fitness function remains unchanged for 10000 runs for each pre-defined number of gene (varying between 3- 4); then, the program was stopped. The model parameters and the size of the developed GEP models were then tuned (optimized) throughout refining (optimizing) the trained and fixed model as a starter. Table 6 lists various combinations of all genetic operators used in this study. Further details about GEP applications in hydraulic structures studies might be found in e.g. Roushangar et al. [2 & 51].

Results and discussion The present paper aims at assessing the capability of SVM and GEP methods for modeling stepped spillway discharge coefficient (Cd) and determining the most influential parameters on Cd under nappe and skimming flow regimes by using physical models and original experimental data.

SVM Results

Tables 7 and 8 summarize the statistical criteria of all models based on the best kernel. The tables clearly show that MC4 and MC5 models acquire the highest accuracy in nappe and skimming flows, respectively, with the highest R2 and the lowest RMSE values. The Kernel values of MC4 and MC5 models are equal to 3 and 2, respectively. Figure 4.a displays the experimental and simulated discharge coefficient values of the best SVM model in nappe flow regime (MC4) for the training and testing stages. Figure 4.b illustrates the experimental vs. simulated discharge coefficient values of SVM in skimming flow regime.

GEP Results

The GEP results of all models in nappe and skimming data are listed in Tables 7 and 8 respectively. Similarly, the MC4 (with (Fr1 ,

H Le B H L B , , ) inputs) and the MC5 (with (Fr1 , , e , , Re) y1 y1 y1 y1 y1 y1

inputs)- based GEP models have the highest accuracy in both data series. Figures 5.a and 5.b present the experimental versus simulated discharge coefficient values in nappe and skimming flow regimes, respectively.

Nappe and Skimming GEP Formulation

Figure 6 shows the Expression Tree (ET) of GEP-based models with corresponding mathematical equation for discharge coefficient of nappe and skimming regimes, respectively. In this figure, d0, d1,

H L d2 and d3 in part (a) denote the B , Fr1 , e , and , respectively. Similarly, d0, d1, d2, d3 and d4 y1 y1 y1 H L in part (b) denote B , Fr1 , e , Re , and , respectively. The mathematical expression of these y1 y1 y1 ETs reads: MC4 Model (discharge coefficient of nappe flow regime)

Cd 

L 3B H  9.84*[ e  9.84]1  Fr1 (1  ) y1 y1 y1

(8)

MC5 Model (discharge coefficient of skimming flow regime)

C d  10* Re1*[ 

B Le 2 10.08 H *( )   1)]1  0.084 Re*[  0.084 Re 119.05]1 y1 y1 Re y1

B 2Le B [   Fr1  0.084]0.5 y1 y1 y1

(9)

As can be seen, GEP equations have a high degree of complexity, which might be ascribed to the nonlinearity of the relations between the flow characteristics, the geometry of stepped spillway and

channel, and the discharge coefficient. The results highlight the significance of an appropriate input selection process in assessing the model's precision and complexity.

MLR Results

The MLR results are presented in Table 9. According to this table, MC4 and MC5 are the best MLR models for nappe and skimming data, respectively. These models have the lowest RMSE value and the highest R2 value. According to Table 9, the R2 and %RMSE for MLR models are in ranges of 0.637 to 0.779 and 0.536 to 0.885, which is less accurate than the same values for GEP and SVM methods. So the performance of the linear model (i.e., MLR) was not suitable because of the inherent complexity and nonlinearity of the Cd phenomenon,

Sensitivity analysis of the GEP and SVM models MC4 and MC5 models were selected as the best models for modeling discharge coefficient of nappe and skimming flow regimes, respectively. The MC4 model includes four dimensionless parameters, i.e. (Fr1 ,

H Le B , , ) . Table 10 shows the results of sensitivity analysis of GEP and SVM models. y1 y1 y1

Based on this table, Fr1 ,

H B and Le have the highest impact on the discharge coefficient of the , y1 y1 y1

nappe flow regime, respectively. So Fr1 is the most important parameter affecting the discharge coefficient of stepped spillway under nappe the flow regime, while Le shows the lowest impact. In y1

case of MC5 model, which includes five dimensionless parameters, i.e. (Fr1 ,

sensitivity analysis shows that Fr1 , Re,

H B Le , , , Re) , y1 y1 y1

H Le B , and have the most impact on the discharge y1 y1 y1

coefficient of the skimming flow regime, respectively. Therefore, the Fr1 term seems to be the most important parameter affecting the Cd, whereas

B reveals the lowest impact (Table 10). As could be y1

anticipated, the ratio of width to water depth over spillway (

B ) shows the lowest effect on Cd for y1

both flow regimes.

Conclusions

The aim of this study was to evaluate GEP and SVM models for modeling discharge coefficient of stepped spillways under nappe and skimming flow regimes using original experimental data set. Also, the results of these methods were compared with the obtained results from multiple linear regression (MLR). The GEP, the SVM, and the MLR methods with eight input parameters were considered. For the GEP, four functions were assessed by different operators. The same data classified by the SVM method was also used for the GEP models. The functions were evaluated based on the lowest error and the function set including (, , , ,

) was selected as the best function set. Based on the

obtained results, discharge coefficient of stepped spillway in nappe flow is mainly affected by

(Fr1 ,

H Le B , , ) parameter set. Among these parameters, Fr1 is the most important parameter and y1 y1 y1

Le shows the lowest impact. Additionally, discharge coefficient of stepped spillway in skimming y1

Fr flow regime can be calculated by (Fr1 , H , Le , B , Re) dimensionless parameters, among which, 1 has y1 y1 y1

the most impact and

B has the lowest impact. As can be seen, the only apparent difference between y1

MC4 and MC5 models is Re that is probably related to the nature of skimming flow regime. According to the obtained results from the SVM and the GP methods as well as the sensitivity analysis of the best models, the discharge coefficient for the nappe flow regime can be calculated by using of the Froude number, stepped spillway height, channel width and depth of flow on stepped spillway, which is in fair agreement with the equation presented in Table 2. Furthermore, in the skimming flow regime, the discharge coefficient depends on the Froude number, stepped spillway height, Reynolds number, channel width and depth of flow on stepped spillway.

Although the MLR model provided some reasonable results by employing regression characteristics, in comparison to the SVM and GEP models, the accuracy of the results weren’t high. This may be attributable to the linear nature of the MLR approach, whereas in the SVM and GEP models, nonlinear properties can help the models to detect and capture the nonlinear feature of the Cd phenomenon. The results revealed the capability of data-driven method (SVM and GEP) in modeling discharge coefficient of stepped spillway in nappe and skimming flow regime in terms of RMSE and R2. Furthermore, a comparison between the results of the SVM with the GEP models revealed the relatively better performance of the SVM model in modeling of the discharge coefficient. Nevertheless, due to the high performance the GEP method and extracting simple, explicit equation of it, utilizing this method is highly suggested in practical situations of designing stepped spillway. In general, since the data-driven method is sensitive to data; therefore, it is better to train each series of data at first.

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Figure 1.The experimental setup used in this study(Roushangar et al. [2])

MC1

MC3

0.94 0.935 0.93

0.945 0.94 0.935 0.93

0.945 0.94 0.935

3

4

5

6

7

8

1

9 10

2

3

0.95

4

5

6

7

8

1

9 10

MC6

MC5

3

4

0.93 0.925 0.92

0.92 0.9 0.88 0.86 0.84

8

1

9 10

MC7

3

Gamma(γ))

4

5

6

7

8

0.86 0.84

9

MC8

0.92 0.9 0.88 0.86 0.84 0.82

1 2 3 4 5 6 7 8 9 10

9 10

7

0.94

0.8 2

5

0.96

0.82

1

3

Gamma(γ))

0.9

0.8 1 2 3 4 5 6 7 8 9 10

7

0.88

0.82

0.915

6

0.92 R2 (Testing stage)

R2 (Testing stage)

0.94

5

0.94

0.94

0.935

0.93

Gamma(γ))

0.96 R2 (Testing stage)

2

Gamma(γ))

Gamma(γ))

0.945

0.94

R2 (Testing stage)

2

0.95

0.92

0.925

0.92 1

0.96

0.93

0.925

0.925

R2 (Testing stage)

0.945

0.95

0.95

R2 (Testing stage)

R2 (Testing stage)

0.95

MC4

0.97

0.955

0.955

0.955 R2 (Testing stage)

MC2

0.96

0.96

1 2 3 4 5 6 7 8 9 10

Gamma(γ))

Gamma(γ))

Gamma(γ))

Figure 2. The R2 results of eight models with different RBF Kernel using SVM in Nappe data

0.986

MC1

MC2

0.978 0.976 0.974

0.985

R2 (Testing stage)

0.98

0.98

0.97 1 2

3

4

5

6

7

8 9 10

0.983 0.981 0.979 0.977

MC6

0.99

0.97 0.96 0.95

0.97 0.96 0.95

0.96

0.94

0.94

0.955

0.93

0.93

1

2

3

4

5

6

7

Gamma(γ)

8

9 10

1

2

3

4

5

6

7

Gamma(γ)

8

9 10

R2 (Testing stage)

0.97 0.965

MC8 0.99

0.98 R2 (Testing stage)

R2 (Testing stage)

0.98

MC7

0.99

0.98

0.975

Gamma(γ)

1 2 3 4 5 6 7 8 9 10 Gamma(γ)

Gamma(γ)

MC5

1 2 3 4 5 6 7 8 9 10

0.975

1 2 3 4 5 6 7 8 9 10

0.985

0.98

0.975

0.975

Gamma(γ)

0.99

R2 (Testing stage)

0.985 R2 (Testing stage)

R2 (Testing stage)

0.985

0.982

0.972

R2 (Testing stage)

MC4

MC3

0.99

0.984

0.98 0.97 0.96 0.95 0.94 0.93

1

2

3

4

5

6

7

Gamma(γ)

8

9 10

1

2

3

4

5

6

7

Gamma(γ)

Figure 3. The R2 results of eight models with different RBF Kernel using SVM in Skimming data

8

9 10

SVM-Training

SVM-Testing 0.65

0.67

SVM Predicted Cd-Nappe

SVM Predicted Cd-Nappe

0.69

0.65 0.63 0.61 0.59 0.57 0.57 0.59 0.61 0.63 0.65 0.67 0.69

0.63

0.61

0.59

0.57 0.57

Experimental Cd

0.59

0.61

0.63

0.65

Experimental Cd

SVM-Training

0.68 0.66 0.64 0.62 0.6 0.58 0.58

0.6

0.62 0.64 0.66 0.68

SVM-Testing

0.7 SVM Predicted Cd-Skimming

SVM Predicted Cd-Skimming

0.7

(a)

0.68 0.66 0.64 0.62 0.6 0.58 0.58

0.7

0.6

0.62 0.64 0.66 0.68

0.7

Experimental Cd

Experimental Cd

(b) Figure4. Experimental vs. predicted discharge coefficient of stepped with SVM method in nappe: a) and, b) skimming flow regimes.

GEP-Training

GEP-Testing 0.65

0.67

GEP Predicted Cd-Nappe

GEP Predicted Cd-Nappe

0.69

0.65 0.63 0.61 0.59 0.57 0.57 0.59 0.61 0.63 0.65 0.67 0.69 Experimental Cd

0.63

0.61

0.59

0.57 0.57

0.59

0.61

0.63

0.65

Experimental Cd

GEP-Testing

GEP-Training 0.7 GEP Predicted Cd-Skimming

GEP Predicted Cd-Skimming

0.7 0.68 0.66 0.64 0.62 0.6 0.58 0.58

(a)

0.6

0.62 0.64 0.66 0.68

Experimental Cd

0.7

0.68 0.66 0.64 0.62 0.6 0.58 0.58

0.6

0.62 0.64 0.66 0.68

0.7

Experimental Cd

(b) Figure5. Experimental vs. predicted discharge coefficient of stepped spillway with GEP method in nappe: a) and, b) skimming flow regime

Figure6.Expression Tree (ET) and corresponding equation s of the best model for nappe (part (a)) and skimming (part (b)) flow regimes for GEP formulation

Table 1. Discharge coefficient regression equations for rectangular side weirs No.

source

1

Subramanya and Awasthy [13]

2

Ranga Raju et al. [14]

3

Hager [15]

4

Swamee et al.[16]

5

Singh et al. [17]

6

Borghei et al. [18]

7

Borghei and Salehi [19]

8

Emiroglu et al. [20]

9

Bagheri et al. [22]

10

Ameri et al. [23]

Discharge coefficient regression equations Cd

Remarks

3Fr12  0.611 1  2  Fr12

0  H  0.6

C d  0.81  0.6Fr1 C d  0.485   44.7H  C d  0.485    49H  y 1   

0.2  H  0.5

2  Fr 2  3Fr

2 1 2 1

6.67

 y H   1  y1  

H 0 6.67

C d  0.33  0.0 / 018Fr1  0.49( H

   

0.15

y1

0  H  0.6 )

C d  0.55  0.47Fr1

C d  0.82  0.38Fr1  0.22(H

y1

)  0.08(B/ b)

H 12.69 b    0.158( ) 0.59  0.836  ( 0.035  0.39( y ) B 1  Cd   b 0.42   2.125 3.018  0.049( )  0.244 Fr ) 1   y1  

5.366

C d  1.423Fr10.138  0.744(y1 / B) 0.083  0.723( y 1 / H ) 0.088 0.182(B/ b)0.241

C d =0.5123-0.390 Fr1 + 0. 4264(w / y 1 )4.324  0.1514(B/ y 1 )0.202

apex angle = 60°

C d =0.5433-0.402 Fr1 + 0. 3399(w / y 1 )1.115  0.0116(B/ y 1 )0.412

apex angle =90°

b= channel width; w = weighted crest height

Table2. Kernel Functions Kernel type

Function

Linear Polynomial

Kernel parameter

-

K( , )= K( , )=

RBF

K(

Sigmoid

K(

, ,

d

) = exp() = tanh(- α(

,

)

γ

)+c

α,c

Table 3. Hydraulic models of stepped spillway made for this study. physical model made of flume width equal 50 cm (stepped spillway height =1m) Number of steps Number of models 5,10,15,20,35,50 6 (h=w)* -

physical model made of flume width equal 25 cm (stepped spillway height =32cm) Number of steps Number of models 3,5,10,15,35 5(h=w) 5,10,15 3 (h  w )

-

5,10,15,30

4 (h

Slope of stepped spillway

All number of physical model

45 25

11 3

15

4

w )

* Step height is equal to its width.

Table 4. Eight models using dimensionless parameters Models

Input Parameters

Models

MC1

(Fr1 )

MC5

MC2

MC3

MC4

H ) y1 H L (Fr1 , , e ) y1 y1 H L B (Fr1 , , e , ) y1 y1 y1 (Fr1 ,

MC6

MC7

MC8

Input Parameters

H Le B , , , Re) y1 y1 y1 H L (Fr1 , , e , Re) y1 y1 H B (Fr1 , , , Re) y1 y1 H (Fr1 , , Re) y1

(Fr1 ,

Table5. Result of combination of basic functions at GEP method Functions F1 F2 F3 F4

Definition

RMSE 0.211

{, , , }

}

0.031

, 3 , x 2 , x 3}

0.248

, 3 , x 2 , x 3 ,ln, e x }

0.419

{, , , , {, , , , {, , , ,

Table6. Parameters of GEP models used in present study Parameter

Description of parameter

Setting of parameter

(, , , ,

p1

Function set

p2

Chromosomes

30

p3

Head size

8,9

p4

Number of genes

2,3

, 3 , x 2 , x 3 ,ln, e x )

p5

Linking function

Addition

p6

Fitness function error type

Root Mean square error (RMSE)

p7

Mutation rate

0.044

p8

Inversion rate

0.2

p9

One-point recombination rate

0.2

p10

Two-point recombination rate

0.2

p11

Gene recombination rate

0.1

p12

Gene transposition rate

0.1

Table 7. Results of performance criteria for eight models utilizing GEP and SVM methods in nappe data Training Stage Nappe Models MC1 MC2 MC3 MC4 MC5 MC6 MC7 MC8

Testing Stage

Methods

R2

%RMSE

R2

%RMSE

RBF Kernel(γ)

SVM GEP SVM GEP SVM GEP SVM GEP SVM GEP SVM GEP SVM GEP SVM GEP

0.969 0.954 0.967 0.960 0.981 0.967 0.989 0.983 0.949 0.955 0.974 0.969 0.980 0.969 0.981 0.967

0.303 0.369 0.323 0.366 0.325 0.287 0.043 0.356 0.328 0.394 0.262 0.310 0.424 0.489 0.177 0.322

0.957 0.947 0.956 0.946 0.951 0.937 0.966 0.961 0.947 0.937 0.939 0.928 0.929 0.919 0.938 0.911

0.367 0.372 0.363 0.375 0.201 0.405 0.042 0.328 0.375 0.405 0.206 0.438 0.153 0.437 0.490 0.553

8 2 4 3 1 2 1 3 -

Table8. Results of performance criteria for eight models utilizing GEP and SVM methods in Skimming data

Skimming Models MC1 MC2 MC3 MC4

Training Stage Methods SVM GEP SVM GEP SVM GEP SVM GEP

R2

%RMSE

R2

%RMSE

0.974 0.974 0.973 0.976 0.979 0.976 0.978 0.973

0.378 0.377 0.379 0.353 0.059 0.352 0.039 0.382

0.985 0.965 0.986 0.966 0.985 0.965 0.984 0.974

0.450 0.3589 0.451 0.565 0.283 0.364 0.281 0.373

Testing Stage RBF Kernel(γ) 3 7 7 5 -

SVM GEP SVM GEP SVM GEP SVM GEP

MC5 MC6 MC7 MC8

0.990 0.990 0.985 0.980 0.988 0.982 0.986 0.981

0.018 0.224 0.214 0.293 0.103 0.262 0.221 0.332

0.987 0.979 0.984 0.974 0.986 0.972 0.982 0.972

0.015 0.357 0.296 0.367 0.122 0.357 0.429 0.591

2 1 1 2 -

Table9. Results of performance criteria for eight models utilizing MLR method for Nappe and Skimming data Data Nappe Skimming

Performance criteria 2

R %RMSE R2 %RMSE

MC1

MC2

MC3

MC4

MC5

MC6

MC7

MC8

0.652 0.833 0.653 0.837

0.654 0.824 0.657 0.816

0.657 0.816 0.667 0.812

0.785 0.703 0.637 0.865

0.696 0.802 0.779 0.536

0.663 0.826 0.697 0.805

0.672 0.811 0.649 0.885

0.684 0.801 0.671 0.885

Table10. Sensitivity analyses of the best models Omitted variable

Model

R2 %RMSE Nappe flow regime (MC4 model)

GEP

0.486

0.852

SVM

0.362

0.885

GEP

0.567

0.745

SVM

0.683

0.739

GEP

0.699

0.337

SVM

0.871

0.329

GEP

0.711

0.636

SVM

0.608

0.646

(Fr1 )

(

(

(

H ) y1

Le ) y1

B ) y1

Skimming flow regime (Mc5 model) GEP

0.405

0.658

SVM

0.587

0.647

GEP

0.838

0.388

SVM

0.843

0.366

GEP

0.727

0.498

SVM

0.847

0.476

(Fr1 )

(

(

B ) y1 Le ) y1 H ) y1

GEP

0.649

0.316

SVM

0.772

0.385

(Re)

GEP

0.538

0.523

(

SVM

0.678

0.591

Highlights



The aim of this study is to apply data driven methods [Gene Expression Programming (GEP) and Support Vector Machine (SVM)] for estimating discharge coefficient of stepped spillways under Nappe and skimming flow regime by using original experimental.

   

We applied two kinds of data including the nappe and skimming regimes data as models input-output variables. For both of GEP and SVM techniques 8 same models were run and according to performance criteria ranked. And finally, we selected the best model for all kind of data to discharge coefficient of stepped spillways. For nappe flow the Fr1 and H/y1 parameters and for skimming flow Fr1 and Re parameters had the most important influence in modeling stepped spillways discharge coefficient.