New type side weir discharge coefficient simulation using three novel hybrid adaptive neuro-fuzzy inference systems
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Abstract
In many hydraulic structures, side weirs have a critical role. Accurately predicting the discharge coefficient is one of the most important stages in the side weir design process. In the present paper, a new high efficient side weir is investigated. To simulate the discharge coefficient of these side weirs, three novel soft computing methods are used. The process includes modeling the discharge coefficient with the hybrid Adaptive Neuro-Fuzzy Interface System (ANFIS) and three optimization algorithms, namely Differential Evaluation (ANFIS-DE), Genetic Algorithm (ANFIS-GA) and Particle Swarm Optimization (ANFIS-PSO). In addition, sensitivity analysis is done to find the most efficient input variables for modeling the discharge coefficient of these types of side weirs. According to the results, the ANFIS method has higher performance when using simpler input variables. In addition, the ANFIS-DE with RMSE of 0.077 has higher performance than the ANFIS-GA and ANFIS-PSO methods with RMSE of 0.079 and 0.096, respectively.
Keywords
ANFIS Differential evaluation Discharge coefficient Genetic algorithm Hybrid method Modified side weir Particle swarm optimizationIntroduction
The simplest side weirs have a rectangular shape. Many studies have been conducted to specify the characteristics of rectangular side weirs. However, besides the advantages of rectangular side weirs like easy construction, such side weirs have low efficiency (Bilhan et al. 2011). In case of overflow, there are two choices: first, the length of the rectangular side weir can be increased, and second, side wire efficiency can be increased. Increasing the side weir length leads to increasing the tributary channel—a choice that seems costly. The economical alternative is to increase the side weir efficiency. Modifying side weir shape could increase side weir efficiency between 1.5 and 4.5 times (Kumar and Pathak 1987; Cosar and Agaccioglu 2004; Ghodsian 2004; Emiroglu et al. 2010b; Aydin and Emiroglu 2013; Mirnaseri and Emadi 2013).
The ANFIS method, as a hybrid soft computing method of the Artificial Neural Network (ANN) and fuzzy logic knowledge, is widely used in various modeling (Khajeh et al. 2009; Talei et al. 2010; Petkovic et al. 2013a, b) and prediction (Dastorani et al. 2010; El-Shafie et al. 2011; Wahida Banu et al. 2011; Wu and Chau 2013) engineering problems.
In numerous studies, the ANFIS method has been successfully used to simulate the characteristics of side weirs. Dursun et al. (2012) applied ANFIS to predict the characteristics of semi-elliptical side weirs. Kisi et al. (2013) simulated the discharge coefficient of rectangular side weirs using ANFIS and concluded that ANFIS performs much better than linear and non-linear regression methods. Emiroglu and Kisi (2013) utilized ANFIS to simulate the discharge coefficient of trapezoidal side weirs and compared the results with the ANN method. The authors concluded that the ANFIS model could predict the trapezoidal side weir discharge coefficient with higher accuracy. Seyedian et al. (2014) used ANFIS to investigate the effect of changing the side weir length, flow depth and side weir height on the discharge coefficient.
The ability to predict the modified side weir discharge coefficient is significant in their design process. Soft computing methods are broadly used for the evaluation of modified side weir characteristics (Bilhan et al. 2010; Kisi et al. 2012; Onen 2014a, b).
Discharge coefficient of the modified side weir that is investigated in the present study is simulated in various studies. Zaji and Bonakdari (2014) compared the performance of Multi-Layer Perceptron Neural Network (MLPNN), Radial Basis Neural Network (RBNN), and linear and non-linear PSO based equations in modeling the modified side weir discharge coefficient. The authors concluded that the RBNN model has higher performance compare with the other regression methods in simulating the considered side weir discharge coefficient. Zaji et al. (2015) simulated the present side weir discharge coefficient using the PSO based RBNN method. The authors concluded that the PSO algorithm successfully improved the RBNN performance. Bonakdari et al. (2015) used ANFIS as sensitivity analyzer to find the most appropriated input variables in simulating the discharge coefficient of the present side weir. Zaji et al. (2016b) applied two types of Support Vector Machine (SVM) to simulate the considered modified side weir. In the first type, the radial basis kernel function is used and in the second type, the polynomial kernel function is employed. In addition, the sensitivity analysis on the input variables was done by examining six different input combinations. The results showed that both types of SVM method perform better when higher number of input variables are used and the radial basis kernel function performs better compare with the polynomial kernel function. Zaji et al. (2016a) compared the firefly based SVM with the simple SVM in simulating the present side weir discharge coefficient. The authors concluded that firefly optimization algorithm has a high role in accurate simulation of the side weir discharge coefficient. Shamshirband et al. (2016) uses the simple ANFIS in predicting the discharge coefficient of the present side weir. The authors concluded that ANFIS has a high capability in simulating the discharge coefficient of side weirs. Zaji and Bonakdari (2017) tried to find the optimum SVM method in simulating the considered side weir’s discharge coefficient. To do that, eight different SVM models with linear, polynomial, Gaussian, exponential, Laplacian, sigmoid, rational quadratic, and multiquadratic kernel functions and concluded that the polynomial kernel function performs better compare with the other kernels. Zaji et al. (2017) aim was to develop simple and practical equations to estimate the discharge coefficient of the considered side weir. The PSO algorithm is utilized to optimize the coefficients of the considered equations. The results showed that the side weir’ discharge coefficient could be modeled using simple and practical equations, accurately.
The aim of this study is to model the discharge coefficient of a modified triangular side weir using three novel hybrid ANFIS methods, i.e., ANFIS-DE, ANFIS-GA and ANFIS-PSO. The models are investigated with eight different input combinations to find the most appropriated input variables. Training and validation of the ANFIS models were done using the experimental study of Borghei and Parvaneh (2011). The results of the most appropriate ANFIS model are compared with the previous findings of Borghei and Parvaneh (2011) and Emiroglu et al. (2010a).
Materials and methods
In the first part of this section, the experimental study by Borghei and Parvaneh (2011) is introduced. Then the optimization algorithms used namely DE, GA and PSO and ANFIS are presented. Finally, the statistics errors applied to find the performance of each model are presented.
Experimental study
Variations in the experimental parameters of the improved triangular side weir (Borghei and Parvaneh 2011)
θ/2 (^{°}) | L (m) | w (mm) | w/Y_{1} | Q_{1} (m^{3}/s) | F _{1} | Number of runs |
---|---|---|---|---|---|---|
30 | 0.3 0.4 | 50,75,100,150 50,75,100,150 | 0.46–0.83 | 0.019–0.030 | 0.19–0.96 | 40 |
45 | 0.3 0.4 0.6 | 50,75,100,150 50,75,100,150 50,100,150 | 0.46–0.83 | 0.019–0.030 | 0.19–0.96 | 55 |
60 | 0.3 0.4 0.6 | 50,75,100,150 50, 100,150 50, 100,150 | 0.46–0.83 | 0.019–0.030 | 0.19–0.96 | 50 |
70 | 0.3 0.4 0.6 | 50,75,100,150 50,75,100,150 50,100,150 | 0.46–0.83 | 0.019–0.030 | 0.19–0.96 | 55 |
Input variables
Input combinations
Input combinations | Input variables | ||||
---|---|---|---|---|---|
In#1 | w/b | y_{1}/b | L/b | sin(θ/2) | Fr _{1} |
In#2 | w/y_{1} | L/b | sin(θ/2) | Fr _{1} | |
In#3 | w/y_{1} | L × sin(θ/2)/b | Fr _{1} | ||
In#4 | w/y_{1} | L × Fr_{1}/b | sin(θ/2) | ||
In#5 | w × sin(θ/2)/y_{1} | L/b | Fr _{1} | ||
In#6 | w × Fr_{1}/y_{1} | L/b | sin(θ/2) | ||
In#7 | w × L/(b × y_{1}) | Fr_{1} × sin(θ/2) | |||
In#8 | w × L × Fr_{1} × sin(θ/2)/(b × y_{1}) |
According to Table 2, the input combinations from In#1 to In#8 have 5, 4, 3, 3, 3, 3, 2, and 1 variables. Therefore, by moving from the first input combination to the last, the number of input variables decreases, but their complexity increases. The goal of investigating input combinations is to identify the performance of the ANFIS-DE, ANFIS-GA and ANFIS-PSO methods with more, but simpler input variables (such as In#1) compared with the less, but more complex input variables (such as In#8).
Differential evolution
Genetic algorithm
Particle swarm optimization
Adaptive neuro-fuzzy interface system
Each node output represents the firing strength of a rule or weight.
The third layer is called the rule layer.
The outputs of this layer are called normalized firing strengths, or normalized weights.
In this paper, ANFIS works with three different evolutionary algorithms, DE, GA and PSO to adjust the parameters of the membership functions. The use of optimization techniques has the benefit of being less computationally expensive for a given network topology size. The membership functions investigated in this study are triangular-shaped.
Performance evaluation
Results
Investigation of input combinations
ANFIS-DE performance evaluation for various input combinations
ANFIS-DE | Training dataset | Testing dataset | ||||
---|---|---|---|---|---|---|
RMSE | MAE | %δ | RMSE | MAE | %δ | |
In#1 | 0.057 | 0.042 | 6.432 | 0.077 | 0.055 | 7.725 |
In#2 | 0.057 | 0.042 | 6.417 | 0.078 | 0.056 | 7.794 |
In#3 | 0.119 | 0.098 | 15.270 | 0.130 | 0.104 | 14.469 |
In#4 | 0.068 | 0.051 | 7.758 | 0.115 | 0.086 | 11.333 |
In#5 | 0.080 | 0.062 | 9.474 | 0.091 | 0.071 | 10.323 |
In#6 | 0.065 | 0.046 | 7.007 | 0.090 | 0.063 | 8.820 |
In#7 | 0.105 | 0.083 | 12.364 | 0.128 | 0.095 | 14.309 |
In#8 | 0.123 | 0.103 | 15.067 | 0.143 | 0.125 | 16.043 |
ANFIS-GA performance evaluation for various input combinations
ANFIS-GA | Training dataset | Testing dataset | ||||
---|---|---|---|---|---|---|
RMSE | MAE | %δ | RMSE | MAE | %δ | |
In#1 | 0.040 | 0.032 | 4.927 | 0.079 | 0.055 | 7.453 |
In#2 | 0.041 | 0.033 | 5.008 | 0.092 | 0.056 | 7.929 |
In#3 | 0.069 | 0.055 | 8.471 | 0.156 | 0.129 | 15.947 |
In#4 | 0.044 | 0.034 | 5.192 | 0.114 | 0.095 | 11.919 |
In#5 | 0.067 | 0.051 | 7.777 | 0.363 | 0.149 | 20.279 |
In#6 | 0.056 | 0.041 | 6.252 | 0.665 | 0.395 | 110.944 |
In#7 | 0.089 | 0.070 | 10.556 | 0.145 | 0.111 | 17.726 |
In#8 | 0.111 | 0.088 | 13.382 | 0.190 | 0.136 | 16.877 |
ANFIS-PSO performance evaluation for various input combinations
ANFIS-PSO | Training dataset | Testing dataset | ||||
---|---|---|---|---|---|---|
RMSE | MAE | %δ | RMSE | MAE | %δ | |
In#1 | 0.030 | 0.023 | 3.444 | 0.096 | 0.072 | 10.551 |
In#2 | 0.028 | 0.021 | 3.132 | 0.240 | 0.181 | 21.064 |
In#3 | 0.079 | 0.064 | 9.656 | 0.215 | 0.174 | 19.971 |
In#4 | 0.035 | 0.027 | 4.046 | 0.098 | 0.082 | 10.727 |
In#5 | 0.046 | 0.034 | 5.099 | 0.124 | 0.095 | 14.811 |
In#6 | 0.051 | 0.035 | 5.252 | 0.108 | 0.087 | 11.550 |
In#7 | 0.069 | 0.053 | 7.997 | 0.191 | 0.138 | 21.777 |
In#8 | 0.107 | 0.086 | 13.023 | 0.184 | 0.135 | 16.875 |
Residual comparison between the hybrid ANFIS models
Comparison of the best hybrid ANFIS models with previous studies
Conclusion
In this study, three different hybrid ANFIS methods, namely ANFIS-DE, ANFIS-GA and ANFIS-PSO were used to evaluate the discharge coefficient of a modified triangular side weir. To evaluate the most appropriate hybrid ANFIS model, eight different input combinations were tested on the models. The first input combination consists of five simple input variables. By moving from the first input combination to the last, the number of variables decreases and their complexity increases. The results show that all of the considered methods performed significantly better when using the first input combination. Subsequently, the results of the three hybrid ANFIS models with the first input combination were compared with each other using standard deviation. According to the results, the difference between the training and testing errors in the ANFIS-GA and ANFIS-PSO models is very high and these models’ results are not reliable to use in practical situations. Finally, the results of the ANFIS-DE method with the first input combination of w/b, y_{1}/b, L/b, sin(θ/2), and Fr_{1} was compared with previous works by Borghei and Parvaneh (2011) and Emiroglu et al. (2010a). The results indicate that ANFIS-DE performs much better compared with the previous equations. Therefore, it could be conceded that separate studies are required for each modified side weir and the results for the triangular side weir cannot be used as a modified triangular side weir discharge coefficient equation.
Notes
References
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