Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System

Alexandria Engineering Journal (2017) xxx, xxx–xxx

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Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System Amir Hamzeh Haghiabi a,*, Abbas Parsaie a, Samad Ememgholizadeh b a b

Water Engineering Department, Lorestan University, Khorramabad, Iran Department of Water and Soil Engineering, Shahrood University of Technology, Shahrood, Semnan Province, Iran

Received 3 February 2016; revised 22 January 2017; accepted 7 May 2017

KEYWORDS Labyrinth weir; Gamma Test; Discharge coefficient; ANFIS; ANNs

Abstract In this paper, the discharge coefficient of triangular labyrinth weir was predicted using multi-layer perceptron (MLP) neural network and Adaptive Neuro Fuzzy Inference System (ANFIS). To this purpose, 223 related dataset were collected. The Gamma Test (GT) was carried out to obtain the most affective parameters on the discharge coefficient. The results of the GT indicated that the ratio of length of crest of weir to the main channel width Lw/Wmc, the ratio of length of one cycle to its width (Lc/Wc) and the ratio of total upstream head flow to the weir height H/P are the most important parameters. With regarding to the results of the GT, the structure of ANFIS model was designed. The results of ANFIS model with error indices including coefficient of determination value of 0.97 and root mean square error value of 0.03 was so suitable. Comparison the results of MLP with ANFIS model showed that both models has so suitable performance however the structure of ANFIS model is more optimal. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Abbreviations: ANFIS, Adaptive Neuro Fuzzy Inference System; Cd, discharge coefficient; CFD, computational fluid dynamic; g, gravitational acceleration; GT, Gamma Test; H, upstream total head; H/P, ratio of total upstram head flow to the weir height; Lc, length of one cycle; Lc/Wc, ratio of length of one cycle to its width; Lw, total length of crest of weir; Lw/Wmc, ratio of length of crest of weir to the main channel width; MLP, multi-layer perceptron neural network; P, weir height; PMF, probable maximum flood; Q, discharge capacity; R2, correlation of determination; RMSE, Root Mean Square Error; Wc, width of one cycle; WL, labyrinth weir; Wmc, main channel width * Corresponding author. E-mail addresses: [email protected] (A.H. Haghiabi), [email protected] (A. Parsaie), s_gholizadeh517@ Shahroodut.ac.ir (S. Ememgholizadeh). Peer review under responsibility of Faculty of Engineering, Alexandria University.

1. Introduction Nowadays, the regime of rivers have been changed due to effect of climate changes and probability of probable maximum flood (PMF) significantly increased [24,44]. Therefore, reanalyzing the safety factor of hydraulic structures constructed across the rivers is necessary [12,26]. Among hydraulic structures, weirs are the common structure widely used in most hydraulic engineering projects specifically in the dam projects. Due to increase the probability occurrence of PMF, to avoid the reduction of safety factor, the discharge capacity of weirs in dam projects should be revised [27]. To understand how

http://dx.doi.org/10.1016/j.aej.2017.05.005 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

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A.H. Haghiabi et al.

to increase the discharge capacity of weirs the conventional weir equation (Eq. (1)) should be considered. 2 pffiffiffiffiffi ð1Þ Q ¼ Cd 2gLH1:5 3 where Q is discharge capacity (m3/s), Cd is the discharge coefficient (), g is the gravitational acceleration (m2/s), H (m) is the upstream total head of flow and L (m) is the length of weir [17]. By attention to the Eq. (1), it is found that, there are four ways for increasing the discharge capacity of weirs. Firstly: increasing the width of weir, second: increasing the discharge coefficient [1,2,3,4,17,22,42,48]. Third: using the nonlinear weir instead of conventional linear weir, and fourth: increasing the head of flow over the weir. Increasing the width of weir due to high cost of executive works is not acceptable. Increasing the head of flow means that the elevation of weir’s crest should be reduced. This approach in addition to has high cost, causes of reduction of volume of dam’s lake. Increasing the discharge coefficient for the constructed projects is not practicable. Using the nonlinear weir instead of conventional linear weir is a rational approach [26,45,46,49]. In other word, as can see from Eq. (1), the discharge capacity of weir is a function of length of crest of weir and lieu to increase the width of weir, its length is increased at the same channel width. Fig. 1 shows a sketch of triangular labyrinth weir. Where P (m) is the weir height, Wmc (m) is the main channel width, Wc (m) is the width of one cycle and H (m) is the upstream total head of flow. Study of nonlinear weirs firstly was conducted by Taylor [47], he has conducted extensive studies on the labyrinth weirs with various form for the weir’s crest such as labyrinth, triangular, rectangular, and trapezoidal. He stated that the trapezoidal form of the crest is more efficient in compare with others. From Taylor [47] to now several studies have been conducted on the hydraulic of labyrinth weirs. Hay and Taylor [21] have proposed a monograph for designing the labyrinth weir. Houston [23] assessed the method that was proposed by Hay and Taylor [21] for practical purposes. They declared that using Hay and Taylor [21] approach for designing the labyrinth weirs cause of about 25 percent error in compare with measured data. Other investigators have tried to improve the performance of triangular labyrinth weirs.

Fig. 1

in this regard the studies conducted by Ghodsian [13] could be mentioned. He studied the effect of rounding the top of side walls of weir on the increasing the discharge capacity. He stated that quarter rounding the side walls of weir has significant effect on increasing the performance of triangular labyrinth weir. Due to high cost of experiments researchers have attempted to use mathematical methods for identifying the hydraulic characteristics of labyrinth weirs. In field of mathematical modeling using the computational fluid dynamic (CFD) methods and soft computing techniques can be stated. In the field of CFD the Navier–Stokes equations coupled with turbulence models using numerical methods are solved [11,30,33,37]. Recently, for using the CFD techniques number of commercial softwares such as Fluent and Flow 3D and free open codes such as OpenFOAM have been provided. Using the CFD technique for modeling the flow over labyrinth weir was reported by Aydin [5]. Its notable that Aydin [5] has studied the hydraulic properties of the labyrinth side weir. Along the CFD modeling using the soft computing approaches have been reported for predicting the discharge capacity of labyrinth weirs specifically labyrinth side weir. Using the soft computing techniques such as artificial neural networks (ANN’s), Genetic Programming [35], Support Vector machine and M5 Model Tree, Group Method of Data Handling, Adaptive Neuro Fuzzy Inference System (ANFIS) for predicting the hydraulic characteristics have reported by many of investigators [6,7,8,9,14,15,28,29]. Based on the reports all soft computing techniques have so suitable performance for predicting the discharge coefficient of labyrinth side weir. As noted in literature, nonlinear weir is a rational approach to improve the discharge capacity of weirs. The main parameter related discharge capacity of WL is discharge coefficient. Therefore, estimation of this parameter is very important. Hence, in this paper, dimensional analysis technique is used for deriving the dimensionless parameters effectiveness the discharge coefficient of labyrinth weir. The Gamma Test (GT) is applied to define the most important parameters on the discharge coefficient. In the follow the ANFIS model as reliable soft computing technique is developed for predicting the triangular labyrinth weir discharge coefficient. During the ANFIS model development, the result of GT is considered for preparation of optimal

Sketch of triangular labyrinth weir.

Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

Prediction of discharge coefficient of triangular labyrinth weirs

3

structure of ANFIS model. At the end, to compare the performance of ANFIS model with other type of soft computing techniques, the multilayer perceptron (MLP) neural network as common type of ANNS was used.

where the vector X ¼ ðx1 ; . . . ; xM Þ is the input, confined to a closed bounded set C 2 RM and the scalar y is the corresponding output, without loss of generality. The only assumption made is that the relationship of the system is in the following form:

2. Method and materials

y ¼ fðx1 ; . . . ; xM Þ þ r

Discharge coefficient of flow measurement structures is one of the main parameters discussed for evaluating the efficiency of water engineering projects. Discharge coefficient of the triangular labyrinth weir is a proportional of hydraulic characteristics and weir geometric properties. W plan form is specific case of triangular labyrinth weirs which include only two cycles. The main geometrical and hydraulic parameters which have posed in related to the triangular plan form weirs are shown in Fig. 1. Formulation of the discharge coefficient and these influence parameters are presented in the Eq. (2) [10]. Cd ¼ fðH; Wmc ; Wc ; Lw ; Lc ; P; g; r; l; qÞ

ð2Þ

In which Lw: total length of the weir, Lc: length of one cycle, g: gravitational acceleration, r is Surface tension and q is Density of flow. Using the dimensional analysis such as p theorem involved parameters on the discharge coefficient is derived as Eq. (3). It is notable that flow in the channel is turbulence and investigators try to remove the effect of surface tension, therefore, the Reynolds and Weber numbers can be negligible [15,16].   H Lw Lc H Cd ¼ f ; ð3Þ ; ; P Wmc Wc Wc Developing soft computing models are based on the data set, therefore, for estimation of discharge coefficient of triangular labyrinth weir, 223 dataset were collected from Kumar et al. [25], Ghodsian [13]. Summary of the collected data are given in Table 1. 2.1. Gamma Test (GT)

where f represents a smooth function and r denotes an indeterminable part, which may be due to real noise lack of functional determination in the assumed input/output relationship. The Gamma Test is used to return a data-derived estimate for VarðrÞ without knowing the underlying function f, just directly from the data. The estimate of the model’s output variance called the Gamma statistic and represented by C cannot be accounted for by a smooth data model. The Gamma Test is derived from the Delta function of the input vectors: dM ðkÞ ¼

M 1 X jxN½i;k  xi j2 M i¼1

ð6Þ

where xN½i;k denotes the index of the kth nearest neighbour to xi, and |.| denotes Euclidean distance. Thus dM ðkÞ is the mean square distance to the kth nearest neighbour. The corresponding Gamma function of the output values is: cM ðkÞ ¼

M  2 1 X yN½i;k  yi 2M i¼1

ð7Þ

The Gamma Test computes the mean-squared kth nearest neighbour distances dðkÞ, ð1 6 k 6 kMax Þ and the corresponding cðpÞ2 . In order to compute C the best line is constructed for the p points ðdM ðkÞ; cM ðkÞÞ, and the vertical intercept, C is returned as the gamma value. The regression line slope is also returned to show the complexity of the model f. The Vratio is the standardized results by considering C=VarðyÞ. It returns a scale invariant noise estimate which normally lies between zero and one [18,19,32]. 2.2. Artificial neural networks (ANNs)

The Gamma Test is used to examine the relationship between inputs and outputs in numerical data-set without a need to construct the prediction model. The Gamma Test is used for estimating the variance of the output before modelling, even though the model is unknown. This error variance estimate presents a target Mean Squared Error that any smooth nonlinear function should attain on unseen data. Suppose we have a set of observed data represented by: ððx1 ; . . . ; xM Þ; yÞ ¼ ðx; yÞ

Table 1

ð5Þ

ð4Þ

ANN is a nonlinear mathematical model that is able to simulate many complexes mathematical that relate the inputs and outputs. Multilayer Perceptron (MLP) networks are common types of ANN that are widely used in researches. To use MLP model, definition of appropriate functions, weights and bias should be considered. Due to the nature of the problem, different activity functions in neurons can be used. An ANN maybe has one or more hidden layers. Fig. 2 demonstrates MLP neural network consisting of inputs layer, hidden layer

Summary of collected dataset related to triangular labyrinth weir.

Parameters range

Min

Max

Avg

STDEV

Weir length Channel width Cycle width Cycle length Weir height Total head Discharge coefficient

0.245 0.245 0.123 0.123 0.092 0.007 0.148

1.200 0.300 0.280 1.082 0.170 0.145 0.906

0.475 0.271 0.213 0.373 0.110 0.046 0.595

0.282 0.019 0.075 0.263 0.024 0.024 0.172

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A.H. Haghiabi et al.

Fig. 2

Sketch of multi-layer ANN architecture.

(layers) and outputs layer. As shown in Fig. 2 wi is the weight and bi is the bias for each neuron. Weight and biases’ values will be assigned progressively and corrected during training process comparing the predicted outputs with known outputs. Such networks are often trained using back propagation algorithm. In the present study, ANN was trained by Levenberg– Marquardt technique because this technique is more powerful and faster than the conventional gradient descent technique [7,20,34,43].

where A1; A2 and B1; B2 are the MFs for inputs x and y; respectively; p1; q1; r1 and p2; q2; r2 are the parameters of the output function. ANFIS architecture is presented in Fig. 3 in the first layer, all the inputs variable gave the grade membership with membership function and in layer 2, all the membership grade will be multiplies together and in the layer 3, all the grade of member will be normalized and the layer 4 in this layer contribution of all the rule will be compute. And in the last layer output variable will be compute as weighted average of grade membership [7,8,31,36,39,41].

2.3. Adaptive Neuro fuzzy Inference Systems (ANFIS) 3. Results and discussions Adaptive Neuro fuzzy Inference Systems (ANFIS) is a powerful tool for modeling of complex system based on input and output data. ANFIS are realized by an appropriate combination of neural and fuzzy systems. This combination enables to use both the numeric power of intelligent systems. In fuzzy systems, different fuzzification and defuzzification strategies with different rule was considered for inputs parameter. For doing the effect of fuzzy logic on the inputs data, three stages should be considered. One- selecting the membership function for each inputs variable. In this stage maybe considered a Gaussian function for each of inputs variables. Fig. 3 shows a fuzzy reasoning process. For simplicity illustrating, a fuzzy system with two inputs variable and one output was considered. Suppose that the rule base containing two fuzzy if-then rules. Rule 1 : if x is A1 and y is B1 then f1 ¼ p1 x þ q1 y þ r1 Rule 2 : if x is A2 and y is B2 then f2 ¼ p2 x þ q2 y þ r2

Designing the optimal structure for ANFIS requires to define most effective parameters. To this purpose, in this study GT was applied. To use GT the win-Gamma software was utilized. After definition of most important parameters for predicting the discharge coefficient of triangular labyrinth weir, designing the structure of ANFIS model is considered. In this approach, more membership functions are assigned to the most important input parameters. The performance of applied soft computing methods were assessed by comparison their results with observed data. To this purpose, standard statistical indices such as correlation of determination (R2), Root Mean Square Error (RMSE) and developed discrepancy ratio (DDR) that were applied by Parsaie et al. [38,40] were calculated. It is noticeable that these error indices provide an average value for error and doesn’t give any more information about error distribution, concentration and etc., hereupon, it is advisable that in addition to calculate the error indices, the performance of these methods and measured data should be plotted.   Predicted Value DDR ¼ 1 ð8Þ Observed Value 3.1. Result of Gamma Test

Fig. 3

ANFIS model structure.

In this study, to define the most affective parameters on discharge coefficient five scenarios were considered. In the follow these scenarios were analyzed using the GT. At the first scenario, all the input variables were used for the GT and in the next scenarios, one of the input variables was remove and again the GT was performed. Results of each scenario are given in Table 2. The GT parameters including gamma, gradient, Standard error, and V-ratio were chosen as criteria for defining the

Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

Prediction of discharge coefficient of triangular labyrinth weirs Table 2

5

Results of gamma test.

Row

Inputs

Absent

Gammas

Gradient

Standard error

V-ratio

1 2 3 4 5

H/P, Lw/Wmc, Lc/Wc, H/Wc Lw/Wmc, Lw/Wc, H/Wc H/P, Lc/Wc, H/Wc H/P, Lw/Wmc, H/Wc H/P, Lw/Wmc, Lc/Wc

– H/P Lw/Wmc Lc/Wc H/Wc

0.0019 0.0074 0.0033 0.0033 0.0020

0.1557 0.2218 0.2915 0.1758 0.1619

0. 0014 0.0013 0.0027 0.0011 0.0015

0.0075 0.0295 0.0133 0.0135 0.0083

most important parameters. The scenario which has minimum value for the GT parameters shows the most influenced input variables on the output. The variation of V-ratio is between the 0 and 1. This point is notable that whatever this factor attends to zero, shows that related scenario could accurately predict the output. Revirewing Table 2, it is declared that the scenario number (1) which involved the total input variables has minimum values for the GT parameters. Table 2 shows that removing the parameters H/P, Lw/Wmc, and Lc/Wc causes to significantly increase the Gamma value, hence, it was found that these parameters are the most important parameters on discharge coefficient. Variations of gamma values along the standard error values for the dataset based on the all input variables is given in the shown in Fig. 4. Fig. 4 shows that the curves of standard error and gamma have flat trend after point 180. It means that the for modeling the Cd with regrading the collected dataset qualification of 180 dataset is enough. 3.2. ANFIS development Designing the ANFIS model for predicting the discharge coefficient of WL is considered in this section. Developing the ANFIS model is based on the dataset. This means first stage of ANFIS model is data preparation. To this purpose, collected dataset were randomly separated into two groups as training and testing. Training and testing dataset are used for model calibration and validation, respectively. Based on

the results of GT, 180 dataset (80 percent) of total dataset were considered for the training and remains (20 percent) for testing. The main advantage of ANFIS model in compare with MLP model is utility in stage of designing its structure. This utility is related to assigning the number of the membership function to the inputs variable based on their influence on the output. This utility of ANFIS model leads to develop a model that has optimal structure. Optimal structure for soft computing causes of increasing the reliability of results of model, because each parameter which are more influence, can get more membership function. In this study the results of GT was used to develop an optimal structure. The structural of the ANFIS which has best performance are given in Table 3. As shown in Table 3, the Gaussian function (guassmf) was considered for the membership function and weight average (wtaver) approach was considered for defuzzification method. As can see from Table 3, during the process of development of ANFIS model, H/P, Lw/Wmc, and Lc/Wc as most affective input variables have gotten more membership function in compare with others. The performances of ANFIS model in training and testing stages are shown in Figs. 5 and 6. Another index that was utilized for evaluating the performance of ANFIS model is developed discrepancy ratio (DDR). The DDR model shows the properties of model in term of over or lower estimation. As shown in Figs. 5 and 6, histogram of DDR and distribution of errors also have been plotted. The histogram of error shows that the values of error focuses around the zero and their histogram are almost symmetrical.

0.025

0.04

0.020

0.035

Gamma

0.025 0.010

0.02

0.005

0.015

Standard Error

0.03 0.015

0.01

0.000

0.005 -0.005 20

40

60

80

100 120 140 Unique Data Points

Gamma

Fig. 4

160

180

200

220

Standard Error

The variation of Gamma static and standard error with unique data points.

Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

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A.H. Haghiabi et al. Table 3

The performances and summery of the ANFIS structure.

Parameter

Nmf

MF

And method

OR method

Defuzz method

Agg method

Type

R2

H/P Lw/Wmc Lw/Wc H/Wc

3 6 4 4

Gaussmf Gaussmf Gaussmf Gaussmf

Prod Prod Prod Prod

Max Max Max Max

Wtaver Wtaver Wtaver Wtaver

Max Max Max Max

Sugeno

Train

Test

Train

Test

0.99

0.97

0.030

0.045

RMSE

Note: Nmf: number of membership function, MF: membership function.

Fig. 5

The performance of ANFIS model during the training stage.

In overall, as shown in Figs. 5 and 6 the capability of ANFIS model is suitable for predicting the values of the discharge coefficient of triangular labyrinth weir. 3.3. ANNs development Developing the MLP model as common type of soft computing techniques is based on the dataset, therefore, collected data set divided into two groups as training and testing dataset. Data selection for preparation of MLP model carried out by randomly approach. 80 percent of the total data set was considered for training, remains (20 percent) for testing. It is notable that the same data that had used for development of ANFIS model was used for development of MLP. Similar to

ANFIS model, for development of MLP model, the dimensionless parameters which presented in Eq. (3) were desired as input parameters and discharge coefficient was considered as model output. Designing the structure of MLP model is more based on the designer experience and trial and error process whereas recommendation of investigators who conducted similar research is useful. In this paper the recommendation of Parsaie and Haghiabi [34] was used. Preparation of the MLP model is included the number of the hidden layer(s), number of the neurons in each hidden layer, defining the suitable transfer function for the neurons of hidden layer(s), defining the suitable transfer function for output layer and then training algorithm. To achieve an optimal structure for the MLP model, firstly the one hidden layer was considered and in the

Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

Prediction of discharge coefficient of triangular labyrinth weirs

Fig. 6

The performance of ANFIS model during the testing stage.

follow, the number of the neurons in the hidden layer was increased one by one. Various type of transfer functions such as log-sigmoid (logsig), tan-sigmoid (tansig), linear (purelin) and etc. were tested for choosing the suitable transfer function. This process continues to obtain a model with suitable performance. All the stage of MLP preparation was conducted in the environment of Matlab software. Table 4 presents a summary of trial and error process which was conducted during the MLP model. As presented in Table 4, scenarios numbers 4 and 7 has suitable performance for predicting the discharge coefficient

Table 4

7

and at the end the model number seven is selected because adding the hidden layer as second hidden layer causes of stability of results of model. As presented in Table 4, scenarios number seven has two hidden layers which first and second hidden layer include seven and three neurons respectively. The tansig and purelin were considered as transfer functions. The structure of developed MLP model is shown in Fig. 7. It is notable that the Levenberg–Marquardt technique was used for MLP model learning. The performances of developed MLP model in training and testing stages are shown in Figs. 8 and 9. In these figures the results of MLP model were plotted along

The performance and summary of the MLP model during the development stage.

Row

N-H-L

F-HL&TF

S-HL&TF

R2a

MSEa

RMSE

1 2 3 4 5 6 7

1 1 1 1 1 1 2

5-purelin 9-logsig 13-purelin 7-tansig 9-tansig 13-tansig 7-tansig

– – – – – – 3-tansig

0.91 0.93 0.93 0.99 0.99 0.99 0.99

0.003 0.002 0.002 0.0001 0.001 0.001 0.0001

0.053 0.05 0.05 0.01 0.01 0.01 0.01

a

R2b

MSEb

RMSEb

0.83 0.92 0.89 0.99 0.99 0.99 0.99

0.004 0.002 0.003 0.0001 0.0001 0.0001 0.0001

0.06 0.04 0.06 0.01 0.01 0.01 0.01

Note: N-H-L: number of hidden layer(s), F-HL&TF: first hidden layer and transfer function, S-HL&TF: second hidden layer and transfer function. a Error indices of MLP during the training stage. b Error indices of MLP during the testing stage.

Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

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A.H. Haghiabi et al.

Fig. 7

The structure of developed MLP model.

Fig. 8

The performance of MLP model during the training stage.

Fig. 9

The performance of MLP model during the training stage.

Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005

Prediction of discharge coefficient of triangular labyrinth weirs and versus the observed data. To evaluate the performance of MLP model the distribution of error along the dataset and histogram of DDR index are plotted. As seem from these figures the error distribution focused around the zero for the training, validation and testing concentrated around the zero. Compression of the results of the MLP model and ANFIS with regarding the histogram of error found that the results of ANFIS model in the testing stage are more accurate. 4. Conclusion Modeling the discharge coefficient of flow measurement structure specifically weirs are one of the main parameter for water management in hydro system projects. Recently, nonlinear crest have been proposed to improve the hydraulic efficiency of weirs. The main factor for evaluating the discharge capacity of nonlinear weirs is discharge coefficient. Nowadays, by advancing the soft computing techniques in most area of engineering specifically hydraulic engineering, hydraulic researchers have tried to improve the accuracy of their studies. To develop an optimal structure for soft computing models, it is better to attention more to most influenced parameters in designing procedure. Several mathematical approaches such as Gamma Test have been proposed to this purpose. Using this technique the most important parameters are defined and researchers can more attention to them in stages development process. This point causes of increasing the reliability of developed models. In this study, the ANFIS model was utilized for predicting the discharge coefficient the triangular labyrinth weir. Using the Gamma Test, most influences parameters on discharge coefficient of triangular labyrinth weir including ratio of length of crest of weir to the main channel width, the ratio of length of one cycle to its width and the ratio of total upstram head flow to the weir height was characterized. Results of GT were considered to assign more membership function to most influenced parameters in staged of development of ANFIS model. The results of this study showed that applying the Gamma Test on the involved parameters of discharge coefficient of triangular labyrinth weir causes to design optimal structure for ANFIS model. References [1] A. Abdollahi, A. Kabiri-Samani, K. Asghari, H. Atoof, S. Bagheri, Numerical modeling of flow field around the labyrinth side-weirs in the presence of guide vanes, ISH J. Hydraulic Eng. 1–9 (2016), http://dx.doi.org/10.1080/09715010.2016.1239555. [2] R.V. Ali, Discussion of ‘‘hydraulic characteristics of flow over sinusoidal sharp-crested weirs” by Zahra Oreizi, Manouchehr Heidarpour, and Sara Bagheri (2016), http://dx.doi.org/10.1061/ (ASCE)IR.1943-4774.0000941. [3] R.V. Ali, R. Vatankhah Ali, Discussion of ‘‘stage-discharge models for concrete orifices: impact on estimating detention basin drawdown time” by W.T. Barlow and D. Brandes, 2016, doi: http://dx.doi.org/10.1061/(ASCE)IR.1943-4774.0001102. [4] R.V. Ali, F. Velayati, M. Azimi, Discussion of ‘‘discharge characteristics of a trapezoidal labyrinth side weir with one and two cycles in subcritical flow” by M. Emin Emiroglu, M. Cihan Aydin, and Nihat Kaya, 2015, doi: http://dx.doi.org/10.1061/ (ASCE)IR.1943-4774.0000866. [5] M.C. Aydin, CFD simulation of free-surface flow over triangular labyrinth side weir, Adv. Eng. Softw. 45 (1) (2012) 159–166, http://dx.doi.org/10.1016/j.advengsoft.2011.09.006.

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Please cite this article in press as: A.H. Haghiabi et al., Prediction of discharge coefficient of triangular labyrinth weirs using Adaptive Neuro Fuzzy Inference System, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.005