Sensitivity analysis of the factors affecting the discharge capacity of side weirs in trapezoidal channels using extreme learning machines

Author’s Accepted Manuscript Sensitivity Analysis of the Factors Affecting the Discharge Capacity of Side Weirs in Trapezoidal Channels using Extreme Learning Machines Hamed Azimi, Hossein Bonakdari, Isa Ebtehaj www.elsevier.com/locate/flowmeasinst

PII: DOI: Reference:

S0955-5986(17)30054-7 http://dx.doi.org/10.1016/j.flowmeasinst.2017.02.005 JFMI1318

To appear in: Flow Measurement and Instrumentation Received date: 12 March 2016 Revised date: 16 November 2016 Accepted date: 5 February 2017 Cite this article as: Hamed Azimi, Hossein Bonakdari and Isa Ebtehaj, Sensitivity Analysis of the Factors Affecting the Discharge Capacity of Side Weirs in Trapezoidal Channels using Extreme Learning Machines, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2017.02.005 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.

Sensitivity Analysis of the Factors Affecting the Discharge Capacity of Side Weirs in Trapezoidal Channels using Extreme Learning Machines

Hamed Azimi1,2, Hossein Bonakdari1,2*, Isa Ebtehaj1,2 1

Department of Civil Engineering, Razi University, Kermanshah, Iran

2

Water and Wastewater Research Center, Razi University, Kermanshah, Iran

*

Corresponding author, Tel: +98 833 427 4537, fax: +98 833 428 3264,

[email protected]

Abstract Side weirs are installed on the side walls of main channels to control and regulate flow. In this study, sensitivity analysis is planned using Extreme Learning Machines (ELM) to recognize the factors affecting the discharge coefficient in trapezoidal channels. A total of 31 models with 1 to 5 parameters are developed. The input parameters are ratio of side weir length to trapezoidal channel bottom width (L/b), Froude number (Fr), ratio of side weir length to flow depth upstream of the side weir (L/y1), ratio of flow depth upstream of the side weir to the main channel bottom width (y1/b) and trapezoid channel side wall slope (m). Among the models with one input parameter, the model including Froude number modeled the discharge coefficient more accurately (MAPE=4.118, R2=0.835). Between models with two input parameters, the model using Fr and L/b produced MAPE and R2 values of 2.607 and 0.913 respectively. Moreover, among the models with four input parameters, the model containing Fr, L/b, L/y1 and y1/b was the most accurate (MAPE=2.916, R2=0.925).

Keywords: Side weir, Trapezoidal channel, Discharge capacity, Sensitivity analysis, Extreme learning machine (ELM).

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1. Introduction Hydraulic engineers use side weirs to measure, adjust and divert excess water in irrigationdrainage, surface runoff and wastewater collection networks. The most important side weir parameter is the discharge coefficient. Therefore, determining the discharge capacity of side weirs located in trapezoidal channels is of great importance. The flow within a main channel along a side weir is considered spatially varied with decreasing discharge. De Marchi [1] solved a dynamic equation of this type of flow in a rectangular channel along a side weir. The equation is presented for calculating the rectangular side weir discharge coefficient. Ackers [2], Collinge [3], Frazer [4], Subramanya and Awasthy [5], El-Khashab and Smith [6], and Ranga Raju et al. [7] also studied side weir flow in rectangular channels. In practice, trapezoidal channels along a side weir can be used for transporting water over farm fields and controlling drainage water. Ramamurthy et al. [8] were the first who studied the flow behavior in the vicinity of the side weir crest located in trapezoidal channels. By using the two-dimensional flow theory, a relationship to calculate the discharge coefficient was obtained. Hager [9] used one-dimensional flow to analyze the mechanism of flow passing through rectangular side weirs. Cheong [10] performed an experimental study on the flow over side weirs located in trapezoidal channels. By analyzing the experimental results, Cheong obtained an equation to calculate the discharge coefficient of this type of flow diversion structure. Cheong’s relationship is a function of the Froude number upstream of the side weir, which is used for trapezoidal and rectangular channels. Swamee et al. [11] employed De Marchi’s theory and obtained a relationship to calculate the discharge coefficient of the subcritical flow regime passing through a rectangular side weir. Their relationship is a function of side weir crest height to the flow head on the side weir. Jalili and Borghei [12] presented a relationship to estimate the discharge capacity of rectangular side weirs as a function of the Froude number and the ratio of side weir height to

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water depth at upstream of the side weir. Borghei et al. [13] performed an experimental study on the hydraulic behavior of side weirs located in rectangular channels in subcritical flow condition. An equation was provided to calculate the rectangular side weir discharge coefficient, which is a function of the hydraulic flow characteristics, side weir geometric parameters and main channel characteristics. Muslu et al. [14] examined the profile effects of side flow on side weir discharge in subcritical flow regime. Ramamurthy et al. [15] used the nonlinear partial least squares method and calculated the discharge coefficient of side weirs located on the side walls of rectangular and circular channels. A relationship was achieved as a function of the upstream Froude number of a side weir, the ratio of side weir length to the diameter or width of the main channel, and the ratio of side weir height to water depth upstream of the side weir. Venutelli [16] solved the governing equation for spatially varied flow with decreasing discharge. As well as an equation was introduced as function of the Froude number for rectangular side weirs. Emiroglu et al. [17] experimentally determined the discharge coefficient of labyrinth side weirs located in straight channels. The discharge coefficient equation was obtained a function of the geometric weir characteristics, and main channel and hydraulic flow parameters. Vatankhah [18] used the principles of specific energy and obtained an analytical solution to calculate the profile of free surface flow along a side weir located in a trapezoidal channel. Vatankhah’s solution is suitable for both subcritical and supercritical flow regimes. Parvaneh et al. [19] experimentally studied the hydraulic behavior of asymmetric labyrinth side weirs. By analyzing the experimental results, it was shown that the discharge coefficient of asymmetric labyrinth side weirs was 1.6 to 2.35 higher than symmetric labyrinth side weirs. Emiroglu et al. [20] studied the flow characteristics of trapezoidal labyrinth side weirs with one and two cycles. With the assumption of constant specific energy along the side weir, they proposed relationships to calculate the discharge coefficient of this type of side weirs. Azimi et al [21] proposed a relationship for estimation

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the discharge coefficient of the rectangular side weirs within circular channels for subcritical flow condition. In recent years, soft computing and artificial intelligence techniques have been used widely to predict various complex science problems. Many researchers have utilized different artificial neural network algorithms for pattern recognition of nonlinear hydrological and hydraulic phenomena. To model the runoff-precipitation process, several researchers including Cheng et al. [22], Rajurkar et al. [23] and Muttil and Chau [24] have used genetic algorithms and artificial neural networks. Bilhan et al. [25] predicted the discharge capacity of a side weir in a rectangular channel in subcritical flow condition using feedforward neural network (FFNN) and radial basis neural network (RBNN) algorithms. Ebtehaj and Bonakdari [26] modeled the sediment load transfer mechanism of flow inside circular channels in sewage networks using artificial neural networks. Bagheri et al. [27] predicted the discharge of side weirs located in rectangular channels in subcritical flow regime using artificial neural networks (ANNs). It was revealed that the most effective parameter on side weir discharge coefficient is the upstream Froude number. Zaji and Bonakdari [28] modeled the hydraulic discharge capacity of triangular side weirs using the multi-layer perceptron neural network (MLPNN), radial basis neural network (RBNN) and nonlinear particle swarm optimization (PSO). They showed that the discharge coefficient of this type of side weir is a function of the main channel geometry, weir and flow hydraulic parameters. Ebtehaj et al. [29] employed a group method of data handling type of neural network and predicted the discharge of side orifices in subcritical flow condition. The discharge coefficient is a function of the Froude number, the ratio of orifice height to width, the side orifice height to width, and flow depth upstream of the orifice to the width. Extreme Learning Machine (ELM) which is a learning algorithm for single-layer-feedforward neural networks (SLFNNs) is one of the newest soft computing method. The ELM has been

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utilized in different engineering fields and applications [30-32] because of its robustness, generalization ability, fast learning rate and controllability [33]. Ebtehaj et al. [34] employed ELM to sediment transport in open channels modelling. The authors presented different nondimensional groups to find the best input combination for modeling by ELM. After that, the results of best input combination training and validated by ELM are compared with genetic programming, multi-layer perceptron neural network and gradient-based equation. The results demonstrated the higher level of accuracy of ELM in sediment transport prediction comparing with other approaches. Karami et al. [35] estimated discharge capacity of triangular labyrinth side weirs by ELM. The authors shown that use of ELM as training algorithm in feedforward neural network (FFNN) is superior to classical neural network which is trained by multi-layer perceptron. ELM has several significant and interesting features different from gradient-based algorithm which are used in classical FFNN: i) the training speed of FFNN by ELM is extremely fast while taking very long time in FFNN training by gradient-based neural networks even for simple problems, ii) the generalization performance of Elm is better than traditional learning algorithm for neural networks such as multilayer perceptron by considering weight magnitude in addition to minimum training error, iii) ELM can be employ for SLFNNs training with non-differentiable activation functions while classical neural network only work for differentiable activation functions and iv) ELM overcome the gradient-based algorithm such as overfitting, local minimum and improper learning rate [36]. Therefore, the ELM algorithm as a learning algorithm in FFNN seems much simpler than most traditional learning algorithms for FFNN. In this study, the side weir discharge coefficients within trapezoidal channels are predicted using Extreme learning machine (ELM). Extreme learning machine is a simple and fast single hidden layer feed forward neural network. Also, a sensitivity analysis to identify the affecting

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parameters to simulate the discharge coefficient by using the method is performed. Consequently, equations for the superior models were formulated as well.

2. Experimental setup In this study, Cheong’s [10] experimental results are applied to predict the discharge coefficient of a side weir located on a trapezoidal channel wall. Figure 1 represents Cheong’s [10] experimental model schematic. Table 1 shows Cheong’s [10] experimental measurement ranges for Froude number (Fr), flow depth upstream of the side weir (y1), bottom width of the trapezoidal channel (b), side weir length (L) and trapezoidal channel wall slope (m).

3. Methodology 3.1. Extreme Learning Machine (ELM) ELMs, introduced by Huang et al. [37], are powerful neural networks successfully employed in various engineering fields for solving nonlinear problems. In this study, ELM is applied to carry out sensitivity analysis of the discharge coefficient of trapezoidal channels. ELM has the structure of a single-layer feed forward neural network and the least squares generalization algorithm. According to Figure 2, similar to the multi-layer perceptron neural network, ELM contains three layers. The input variables of the considered problem are introduced to the ELM neural network via the input layer. The main calculation core of the ELM is the hidden layer. The information developed by the hidden layer is collected by the output layer where the final ELM results are prepared. The hidden layer weights (wij) are

6

determined randomly in the ELM training procedure. Thus, ELM should only determine the output layer weights (βjk) [36] Therefore, the computational cost of the ELM method is very low compared with other neural networks and this method is suitable for modeling applicable engineering problems. The number of input layer neurons is equal to the number of input variables of the considered problem, while the number of output layer neurons is equal to the number of output variables of the considered problem. In this study, the input variables are determined by sensitivity analysis and the output variable is the discharge coefficient (Cd). However, there is no certain way to find the right number of hidden layer neurons. In the present study, the number of hidden layer neurons is determined by trial and error. If l hidden layer neurons are selected, the input-to-hidden and hidden-to-output layer matrices are w and β as follows:

 w11 w12  w1l  w w22  w2l  w   21       wn1 wn 2 ... wnl  nl

 11 12  1i      2i     21 22        l1  l 2 ...  lm  lm

(1)

In these equations, the weight that connects the ith input layer neuron to the jth hidden layer neuron is wij. In addition, the weight connecting the jth hidden layer neuron to the kth output layer neuron is βjk. The input and target matrices are shown as follows:  x11 x12  x1o  x x  x2 o  X   21 22       xn1 xn 2 ... xno  nQ

 y11 y12  y1o  y y  y2 o  Y   21 22       yn1 yn 2 ... yno  mQ

(2)

where Q is the number of input samples. If the final ELM result is shown as T = (t1, t2, …, tQ)m×Q, then tj is defined as:

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 l    i1 g wi xi  bi    i1  t1 j   l    t2 j   i 2 g wi xi  bi     tj    i1          t mj  m1  l   g w x  b  im i i i   m1 i 1

, (j = 1, 2, …, Q)

(3)

where g(x) is the hidden layer’s transfer function. In the present study, the sigmoid transfer function is applied. The summarized ELM form is:  g w1 x1  b1  g w2 x1  b1   g wl x1  bl    g w x  b  g w x  b   g w x  b   1 2 1 2 2 2 l 2 l  T Hβ  T where H           g w1 xQ  b1  g w2 xQ  b2   g wl xQ  bl 

(4) QL

If the number of hidden layer neurons is considered equal to the number of input samples, the training dataset simulation error is equal to zero. However, there are two defects. First, the ELM neural network is too large and the final model is not practical. Second, if the training prediction error is zero, overfitting will likely occur. Overfitting occurs when the training dataset prediction error is much lower than the testing dataset prediction error. Hence, by considering l
t J 1

j

 yj  

(ε > 0)

(5)

As mentioned before, in the training procedure w and b are developed randomly. β is obtained as follows: min H  T T

(6)



If H+ is the Moore-Penrose generalized inverse matrix of H, the solution to the tj equation is obtained as: ˆ  H  T T

(7)

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3.2. Discharge coefficient of side weirs Rectangular and triangular channels are special channel types with trapezoidal cross sections. The discharge coefficient of a rectangular side weir located in a rectangular channel is [38]:  Cd  f  Fr   

 u L L P , , , , , S0   gD b y1 y1 

(8)

where Fr is the Froude number upstream of the side weir, u is the mean flow velocity upstream of the side weir, g is gravity acceleration, D is the hydraulic diameter, L is the side weir length, b is the main channel width, y1 is the water depth upstream of the side weir, P is the side weir crest height,  is the flow deviation angle and S 0 is the channel bottom slope. El-Khashab [39] stated that the effect of the dimensionless parameter of side weir length (L/b) on the discharge coefficient includes flow deviation angle   as well. The  effects on the side weir discharge coefficient have not been studied in the past. In addition, Borghei et al. [13] reported that in subcritical flow regime, channel bed slope effects can be neglected. Therefore, the effective dimensionless parameters on the discharge coefficient can be written as follows:  Cd  f  Fr   

u L L P  , , , gD b y1 y1 

(9)

Cheong [10] investigated the discharge coefficient of side weirs located in trapezoidal channels and studied the side wall slope of the main channel (m). Hence the equation (9) can be expressed as:  Cd  f  Fr   

u L L P  , , , ,m gD b y1 y1 

(10)

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In this study, the combined effective parameters affecting the discharge coefficient of rectangular side weirs in trapezoidal channels are trained and tested using Cheong’s [10] experimental measurements. In Cheong’s [10] study, side weir crest height was considered zero (P=0.0). Moreover, to investigate the effective parameters on the discharge coefficient of side weirs located in trapezoidal channels, the dimensionless parameter of water depth upstream of a side weir on trapezoidal channel bottom width (y1/b) is considered in different model combinations. Therefore, equation (10) is written as follows:  Cd  f  Fr   

u L L y  , , , m, 1  b  gD b y1

(11)

4. Results and discussion In this study, model accuracy is evaluated using the statistical indices of correlation coefficient (R2) and mean absolute percentage error (MAPE):

n

n

R  2

n

n

i 1

MAPE 

C 

i 1

Cd Pr edicted i Cd Observed i  i1 Cd Pr edicted i i1 Cd Observed i n

    C 

d Pr edicted i

2

n

i 1

n

  n C 

d Pr edicted i

2

n

i 1



2

    C 

d Observed i

2

n

i 1

   2

(12)

d Observed i

1 n Cd Observedi Cd Pr edicted i  100  n i1 Cd Observedi

(13)

Where Cd Observed i and Cd Pr edicted i are the experimental discharge coefficient and predicted by the extreme learning machine (ELM), respectively, and n is the number of experimental measurements predicted by ELM. In this study, the 5 parameters presented in Table 1 were combined and 31 different models were introduced. Five models with one input (models 1 to 5), ten models with two inputs (models 6 to 15), ten models with three inputs (models 16 to 25), five models with four inputs (models 26 to 30) and one model with five inputs (model 31) are defined. In addition, 50% of data were used to train and 50% to test the ELM models. In Figure 3, the input combinations for each different model are presented. 10

4.1 Sensitivity analysis For sensitivity analysis, 31 models are defined in Figure 3, and the statistical indices of correlation coefficient and mean absolute percentage error are used. The R2 and MAPE values after training each model are calculated and shown in Figure 4. Models 1 to 5 with one input each include the Froude number (Fr), ratio of side weir length to main channel bottom width (L/b), ratio of side weir length to flow depth upstream of the side weir (L/y1), trapezoidal channel wall slope (m) and ratio of the depth upstream of the side weir to the trapezoidal channel bottom width (y1/b). As seen from Figure 4, the minimum and maximum MAPE values for model training are for model 1 (MAPE=4.118) and model 4 (MAPE=9.636); the highest correlation coefficient is for model 1 (0.835). Therefore, among the models with one input parameter, the Froude number is the most effective parameter on the discharge coefficient of a side weir located in a trapezoidal channel. Models 6 to 15 are combinations of two different inputs, and model 6 has the lowest MAPE (2.607) and R2 of 0.913. Model 6 is a combination of Froude number (Fr) and ratio of side weir length to the bottom width of the main channel (L/b). Also among the models with two different inputs each, after Model 6, the MAPE and R2 values of model 9 are 2.784 and 0.927 respectively. Among models 16 to 25 with combinations of three different inputs, model 18 and model 20 had the lowest MAPE and highest R2 values (2.282 and 2.426; 0.949 and 0.945 respectively). In this study, among models 26 to 30 with combinations of four different inputs (Froude number, ratio of side weir length to bottom width of the main channel, ratio of side weir to depth upstream of the side weir and ratio of depth upstream of the side weir to the bottom width of a trapezoidal channel), model 27 has the lowest mean absolute percentage error (MAPE=2.916) and the

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highest correlation coefficient (R2=0.925). Model 31 with a combination of five dimensionless parameters (Fr, L/b, L/y1, m and y1/b) has MAPE and R2 values of 3.308 and 0.879 respectively.

Figure 5 presents the comparison between R2 and MAPE for model testing and sensitivity analysis of extreme learning machines (ELMs). According to Figure 5, R2 and MAPE changes pattern is to test the models 1 to 31 similar to mentioned statistical indices changes to train the ELM. Among the models with one dimensionless parameter input (models 1 to 5), model 1 has the lowest MAPE value of 4.571 and maximum R2 value of 0.629. From the models with a combination of two different inputs (models 6 to 15), model 9 with Fr and y1/b inputs and MAPE=3.307 and R2=0.831 showed the highest accuracy in predicting the side weir discharge coefficient. In data testing, among models with a combination of three input parameters (models 16 to 25), model 18 has the lowest mean absolute percentage error (MAPE=2.814) and model 20 has the highest correlation coefficient (R2=0.869). Also in the data testing mode, among models with four dimensionless parameters (models 26 to 30), models 26, 27 and 28 show similar results. The MAPE values for models 26, 27 and 28 are 3.402, 3.308 and 3.454, and R2=0.84, 0.831 and 0.844 respectively. For model 31 with a combination of five different inputs, the calculated MAPE and R2 values are 3.624 and 0.855, respectively.

4.2. Superior models

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Subsequently, according to the data training sensitivity analysis, the superior models are introduced for different input combinations. The discharge coefficient of a side weir located in a trapezoidal channel (Cd) can be calculated with the following equation: T

  1 Cd     OutW  1  exp InW  InV  BHI 

(14)

where InW is the input variable matrix which varies from 1 to 5 input combination, input weights (InW ), bias of hidden layer (BHI) and output weights (OutW) are the coefficients matrices and based on the InW and their values are presented as follows.

4.1.1. One input parameter Among the models with one input parameter, model 1 was the best as seen in Figure 6-a, which shows a comparison between the observed and discharge coefficient predicted by the model. Model 1 is a function of the Froude number (Fr), which indicates the influence of the Froude number on the discharge coefficient of a side weir located in a trapezoidal channel. The coefficients matrices for Equation (14) for the superior equation with one input parameter are calculated as follows:

InV  Fr 

0.43 0.63  BHI    0 .6    0.18

 3.41e4   2.85e3  OutW   3.18e4    3.71e3 

0.57   0.78  InW   0.58    0 

(15)

4.1.2. Two input parameters From the models with two-input combinations, model 6 is considered the best. Figure 6-b displays the comparison of the observed discharge coefficient and that predicted by model 6. This model is a function of the Froude number (Fr) and ratio of side weir length to bottom

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width of the main channel (L/b). Equation (14) based on coefficient of equation (16) is introduced as the discharge coefficient equation for models with two inputs:

 Fr  InV   L    b 

0.59  0.47    0.68    0.51  0.95  BHI    0.63   0.8    0.44  0.45    0.95 

 387.47   549.72     97.6     667.5   1.41e3  OutW     717.9   1.31e3     900.57   601.89     203.96 

0.18 0.78    0 .4  0.64  0.32 InW    0.07 0.67  0.89  0.38   0.83

0.65   0.21 0.93   0.84  0.19   0.31  0.78   0.18   0.18    0.77 

(16)

4.1.3. Three input parameters Among the models with three inputs, model 18 has the lowest mean absolute percentage error value and the highest correlation coefficient (Figure 6-c). This model is a combination of three dimensionless parameters: Froude number (Fr), ratio of side weir length to bottom width of the main channel (L/b) and ratio of the depth upstream of the side weir to the trapezoidal channel bottom width (y1/b). Thus, equation (17) is considered to calculate the coefficient matrices for the discharge flow (equation 14) for models with three inputs:

   Fr    L InV    b  y   1 b 

0.95 0.59   0.05   0.08  0.1  BHI    0.48 0.65   0.78 0.23   0.22

  29.51  43.71      41.03     18.09    122.47  OutW     29.52    19.88     28.58   1.09     81.05 

4.1.4. Four input parameters 14

 0.86 0.79  0.92  0.18 0.09 InW   0.84 0.96  0.43  0.41   0.31

0.81 0.97   0.34 0.22   0.34  0.66   0.94 0.78  0 .2 0.56  (17)  0 .2  0 .1  0 .9 0.85  0.4  0.55   0.08 0.49   0.71 0.82 

Figure 6-d displays the results of the superior model with four inputs (model 27) in predicting the discharge coefficient. Model 27 estimated the discharge coefficient with good accuracy. This model is a function of Fr, L/b, L/y1, m and y1/b, and the discharge coefficient equation for model 27 is written as: 0.52  0.99 0.04  1.54   0.98 0.23  5   0.55  0.79 0.26       Fr   0 .2   2.39   0.31 0.47  0.49 L         0.28  4.94   0.27  0.91  0.19 b       0.11 0.17 InV   L  BHI  0.46 OutW    9.53 InW   0.43   0.7 0.03 0.95   8 .6   0.27  y1  0.02  4.72   1 0.24  0.06 y       1   0.36 0.03  1   15.28   0.72 b  0.39  0.42   0.94 0.88 0.55      0.35  3.91   0.75  0.03  0.14

0.15  0.98   0.92   0.66  0.07  (18)  1   0.77    0.59  0.34   0.76 

4.1.5. Five input parameters The comparison of the discharge coefficient calculated for model 31 with the observed results is shown in Figure 6-e. This model is a combination of 5 input parameters, and the side weir discharge coefficient was predicted with acceptable accuracy (MAPE=3.308 and R2=0.879). Equation (19) was obtained for model 31 to calculate the discharge coefficient of a side weir located on the side wall of a trapezoidal channel: 0.05  0.8 0.56  1.63   0.08  0.6       Fr  0.86  0.14 0.82 0.69  1.63   0.83   0.49  0.14   0.94  0.87 0.22  0.4 L       b  0.42  0.28   0.61  0.21  0.26  0.63 L      0.49 0.69 0.91 InV    BHI  0.69 OutW    0.93 InW    0.2  y1  0.11  0.7 0.23 0.33  0.08   0.11   0.64  0.46   0.69  0.93 m 0.42 0.35         y1   0.12  0.53 0.5 0.89   3.13  0.52        b 0.45 3.01 0.96  0.11  0.56 0.71      0.88  0.16   0.09 0.09 0.43  0.59

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 0.53  0.45  0.51    0.86  0.75  (19)   0.55   0.75    0.7   0.94   0.09 

Conclusion Engineers use side weirs as flow control structures in irrigation/drainage networks, municipal sewage disposal systems and for other hydraulic purposes. In this study, extreme learning machines (ELMs) were used to simulate the discharge coefficient of a rectangular side weir in a trapezoidal channel. According to the effective parameters on the discharge coefficient (Fr, L/b, L/y1, m, y1/b), 31 different models were used to estimate the discharge coefficient. For sensitivity analysis, various parameters affecting the discharge coefficient were employed, and 5 models with one parameter, 10 models with two parameters, 10 models with three parameters, 5 models with four parameters and 1 model with five input parameters were generated. In training and testing modes, among the models with one input dimensionless parameter, model 1 (a function of Fr) showed the lowest and highest error and correlation coefficient values, respectively. In training mode, a model with two input parameters (Froude number and ratio of side weir to rectangular channel bottom width) exhibited the lowest MAPE (2.607) and highest R2 (0.913). In testing mode, among the models with combinations of three input parameters, the model with Fr, L/b and y1/b produced the lowest error value (MAPE=2.814). Also in testing mode, among models with four input parameters, the model that predicted the discharge coefficient based on Fr, L/b, L/y1 and y1/b achieved MAPE of 3.308. Consequently, equations for the superior models were formulated as well. References [1] J. van der Geer, J.A.J. Hanraads, R.A. Lupton, The art of writing a scientific article, J. Sci. Commun. 163 (2010) 51–59. [1] G. De Marchi, Saggio di teoria del funzionamento degli stramazzi laterali, L’Energia electrica Milan. 11 (1934) 849-860 (in Italian).

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Table 1. The range of effective parameters on the discharge coefficient Parameter

Numbers

Range

Fr

64

0.24-0.988

y1 (cm)

64

2.8-22.14

b (cm)

64

34-67

L (cm)

64

27.7-97

m

64

0-2

21

Figure 1. Cheong’s [10] experimental model schematic

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Figure 2. ELM structure

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Figure 3. Effective parameters of each model for sensitivity analysis

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Figure 4. Comparison of the MAPE and R2 values in model training for sensitivity analysis

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Figure 5. Comparison of R2 and MAPE in model testing for sensitivity analysis

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Figure 6. Comparison of observed and predicted discharge coefficients: (a) model 1, (b) model 6, (c) model 18, (d) model 27, (e) model 31 27

Hightlights    

Factors affecting the discharge coefficient in trapezoidal channels are investigated Extreme Learning Machines (ELM) is used Different kinds of ELM models with different input combinations are investigated Equations for the superior models are formulated

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