VIEWS: 3 PAGES: 9 CATEGORY: Business POSTED ON: 8/29/2013 Public Domain
INTERNATIONAL JOURNAL Technology (IJCIET), ISSN 0976 – 6308 (Print), International Journal of Civil Engineering andOF CIVIL ENGINEERING AND ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME TECHNOLOGY (IJCIET) ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) IJCIET Volume 4, Issue 4, July-August (2013), pp. 170-178 © IAEME: www.iaeme.com/ijciet.asp Journal Impact Factor (2013): 5.3277 (Calculated by GISI) © IAEME www.jifactor.com DESIGN OF COMPOSITE IRRIGATION TRAPEZOIDAL CHANNEL USING OPTIMIZATION AND PATTERN MATCHING BY ARTIFICIAL NEURAL NETWORK Mr. Faiz Mohd1, Prof. (Dr.) M. I. Khan2 and Dr. Raj Mohan Singh3 1 (Mechanical Engg. Dept., Integral University. Lucknow, India) 2 (Mechanical Engg. Dept., Integral University. Lucknow, India) 3 (Civil Engg. Deptt., MNNIT, Allahabad, India) ABSTRACT Irrigation channels are crucial for surface irrigation. Any effort to save the cost of construction or maximize the conveyance also serves to improve agricultural production. Apart from irrigation, these channels are the major conveyance systems for delivering water for various other purposes such as water supply, flood control, etc. The primary concern in the design of channels is to determine the optimum channel dimensions to carry the required discharge with the minimum costs of construction. The present work utilized ANN to address the flow variability in the optimal design of the channel sections. A set of flow values are generated and corresponding optimal sections (depth of flow, width of the channel and side slopes) are obtained by solution of the optimization formulation. The data thus generated is used to train and test the ANN model. The flow data is utilized as input and corresponding channel dimensions as output for the ANN model. Keywords: ANN model of trapezoidal channel, Channel design, Composite structures, Open channel, Optimization, 1. INTRODUCTION An optimal open channel cross section has channel dimensions for which the construction cost is minimum and the conveyance is maximum. In order to save costs, simple channels can be constructed with distinctly different materials for the bed and side slopes. To prevent seepage losses, for example the bed of a channel can be lined with concrete and the side slopes can be lined with rough rubble masonry and boulder pitching. The roughness along the wetted perimeter in such channels may be distinctly different from part to part of the perimeter. For channels having composite roughness, an equivalent uniform roughness coefficient is required to be used in the uniform flow formula. The equivalent roughness equation again incorporates the flow geometric elements and corresponding roughness coefficient values (Chow 1959). 170 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME The optimal design of channels has been of importance among researchers and hydraulic engineers (Guo and Hughes, 1984; Froehlich, 1994; Monadjemi, 1994; Das 2000; Swami et al., 2000; Jain et al., 2004; Bhattacharjya, 2004). Guo and Hughes (1984) designed optimal channel cross sections from the first principles of calculus. Loganathan (1991) presented optimality conditions for a parabolic channel cross section. Froehlich (1994) used the Langrange multiplier method to determine optimal channel cross sections incorporating limited flow top width and depth as additional constraints in his optimization formulation. Monadjemi (1994) used Langrange’s method of undetermined multipliers to find the best hydraulic cross sections for different channel shapes (triangular, trapezoidal, rectangular, round bottom triangular, etc.). Swamee et al. (2000) have proposed optimal open channel design considering seepage losses in the analysis. Bhattacharjya (2004) presents the findings of an investigation for optimal design of composite channels using genetic algorithm (GA). Some of the recent advances are available in Das (2007) and Bhattacharjya (2008). Most of the researchers used nonlinear optimization program (NLOP) to achieve the minimum cost design for a specified discharge. Present work incorporates variability in discharge using artificial neural network (ANN). The necessary data for training and testing is generated using solution of optimization formulation embedded with uniform flow considerations. 2. DEVELOPMENT OF OPTIMAL COST METHODOLOGY 2.1. Problem Formulation In the present work following trapezoidal channel cross section is consider as shown in Fig.1, with side slopes of k1 :1 and k2 :1 (H:V). The Manning roughness coefficient values are n1 and n2 at the two sides and n3 at the bed of the channel. The bottom width, flow depth, and freeboard are w, h, and b, respectively. Figure 1 Composite channel cross section Let WA, WP and B, represent the wetted flow area, wetted perimeter, and flow top width, of the trapezoidal channel cross section, respectively. These hydraulic parameters can then be written as follows: h2 WA = wh + (k1 + k2 ) (1) 2 { WP = ( k12 + 1) + ( k2 + 1) h + w 0.5 2 0.5 } (2) B = w + ( k1 + k2 ) h (3) 171 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME Similarly, the total area TA, the total perimeter TP, and the total channel cross-section top width TW of the trapezoidal channel cross section accounting for freeboard can be written as follows: 2 TA = w ( h + y ) + ( k1 + k2 ) (h + y) (4) 2 { TP = ( k12 + 1) + ( k2 + 1) ( h + y ) + w .5 2 .5 } (5) TW = w + ( k1 + k2 )( h + y ) (6) Let l1 and l2 be the lengths of the side slopes excluding freeboard, l3 are the wetted bed width of the channel, AW1 and AW2 is the wetted portions of the cross-sectional areas corresponding to the two side slopes, and AW3 be the portion of the wetted cross-sectional area corresponding to the bed. These can be written as follows: 0.5 l1 = ( k12 + 1) h (7) 0.5 l2 = ( k2 + 1) h 2 (8) l3 = w (9) k1h 2 (10) AW1 = 2 k2 h 2 AW2 = (11) 2 AW3 = wh (12) 2.2 Equivalent uniform roughness coefficient For channels having composite roughness, an equivalent uniform roughness coefficient is required to be used in the uniform flow formula. The equivalent roughness equation again incorporates the flow geometric elements and corresponding roughness coefficient values (Chow 1959). The necessary optimality conditions must also be satisfied. In this study, Horton’s (1933) equation and Einstein’s (1934) equation is used for estimating the equivalent roughness. For a trapezoidal cross section, the expression for equivalent roughness is written as 2/3 l n1.5 + l2 n2 + l3 n3 1.5 1.5 nE = 1 1 (13) WP Putting the value of l1, l2, l3 and WP in the above equation we get k 2 + 1 0.5 n1.5 + k 2 + 1 0.5 n1.5 h + wn1.5 ( 1 ) 1 (2 ) 2 3 nE = (14) ( k 2 + 1)0.5 + ( k 2 + 1)0.5 h + w 1 2 172 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 2.3 Uniform flow formula for channels with composite roughness The Manning equation for uniform flow in a trapezoidal channel with composite roughness may be expressed as: 5 /3 Q nE W A 0 .5 = 2 /3 (15) z W P where Q = total discharge in the channel (m3/s); z = bed slope of the channel; nE = equivalent roughness coefficient of the channel cross section in wetted region; WA = wetted flow area (m2); and WP = wetted perimeter (m) Putting the values of nE, WA, WP in the above equation, the following expression for uniform flow in trapezoidal channel with composite roughness is obtained: 5/3 Q wh + ( k1 + k2 ) h 2 / 2 0.5 = 2/3 (16) z {( k 2 + 1) n + ( k 2 + 1) n h + wn1.5 1 0.5 1.5 1 2 0.5 1.5 2 3 } Eq. (16) represents the main constraint of this optimization formulation. Negative values of variables w and h are not possible, and negative values for k1 and k2 physically indicate channel sections with a closing crown, which are commonly known as Kind2 channels. In practice, design and construction of a Kind2 channel is undesirable. 2.4. Minimization of Total Construction Cost Minimizing the total cost of channel construction is the overall objective. The objectives of minimizing (1) the total cross-sectional area; and (2) the lining cost can be considered as special cases. When the roughness along the perimeter is distinctly different, the optimal channel dimensions depend upon the objective of the optimization formulation. A simple cost function can be formulated that can address the various optimization objectives as mentioned above. In formulating the cost function, the following assumptions are made: 1. The cost of channel construction is divided into four parts, namely: (i) cost for cross-sectional area; (ii) & (iii) costs for each of two side slopes; and (iv) cost for bed width. 2. The channel side slope portion in the freeboard region is constructed with the same materials used in the wetted region. 3. The four cost units per unit running length of the channel are fixed and distinctly different from each other. With these assumptions, the following cost function, i.e., objective function can be written. Minimize: S = s1 w ( h + y ) + ( k1 + k2 ) . ( h + y ) / 2 + s2 ( k12 + 1) + s3 ( k 2 + 1) . ( h + y ) + s4 w 2 0.5 2 0.5 (17) where s1= per unit area cost for cross-sectional area; s2 = per unit length cost for side slope z1(H):1(V); s3 = per unit length cost for side slope z2(H):1(V); and s4 = per unit length cost for bed width. These equations are solved with the help of MATLAB using fmincon tool. 173 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 3. MODEL EVALUATION CRITERIA The performance of the developed model is evaluated based on some performance indices in both training and testing set. Varieties of performance evaluation criteria are available which could be used for evaluation and inter comparison of different models. Following performance indices are selected in this study to evaluation process. 3. 1.Correlation coefficient (R) The correlation coefficient measures the statistical correlation between the predicted and actual values. It is computed as: n ∑ ( Xai − X ai )( Xpi − X pi ) (18) R = i=1 n n ∑ ( Xai − X ai ) 2 ∑ ( Xpi − X pi ) 2 i=1 i=1 where Xai and Xpi are measured and computed values; Xai and Xpi are average values of Xai and Xpi values respectively; i represents index number and n is the total number of concentration observations. 3.2. Root mean square error (RMSE) Mean-squared error is the most commonly used measure of success of numeric prediction, and root mean-squared error is the square root of mean-squared-error; take to give it the same dimensions as the predicted values themselves. The root mean squared error (RMSE) is computed as: 1 n RMSE= ( ∑( Xai − Xpi)2 ) (19) n i =1 3.3. Normalized Error (NE) The NE, which is a measure of the methodology performance, is defined as: ∑ ( Xai − Xpi ) NE = (20) ∑ X ai 3.4. Standard Error of Estimates (SEE) The standard error of estimate (SEE) is an estimate of the mean deviation of the regression from observed data. It is defined as: n ∑ ( Xai − Xpi ) SEE = i =1 (21) (n − 2) 3.4. Model Efficiency (Nash–Sutcliffe Coefficient The model efficiency (MENash), an evaluation criterion proposed by Nash and Sutcliffe (1970), is employed to evaluate the performance of the developed model. It is defined as: 174 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME n ∑ ( Xa i − X pi ) 2 ME Nash = 1 .0 − i = 1 (22) n ∑ ( X ai − X ai ) 2 i=1 A value of 90% and above indicates very satisfactory performance; a value in the range of 80–90% indicates fairly good performance. 4. ARTIFICIAL NEURAL NETWORKS ANNs are good alternative candidates for traditional modeling techniques in the solution of large scale complex problems, such as pattern recognition, nonlinear modeling, classification and control (Tayfur and Singh, 2006). ANN is a broad term covering a large variety of network architecture, the most common of which is a multilayer perceptron. In applications, a single hidden layer-feed forward type of ANN is shown to have universal approximation ability and they are also relatively easier to train (Figure 2). There is no definite formula that can be used to calculate the number hidden layer(s) and number of nodes in the hidden layer(s) before the training starts, and usually determined by trial-and-error method. Present paper utilized Levenberg- Marquardt (LM) algorithm to optimize the weights and biases in the network. Figure 2. feed forward ANN architecture 4.1. Development of ANN Model The MATLAB-NEURAL NETWORKS TOOL BOX software from MATLAB (2007) version was used to perform the necessary computations. For the purpose of this research work, experimentation was performed with several network architectures to determine the most appropriate model. The optimum number of neurons in the hidden layer was determined by varying their number 175 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME starting with a minimum of 1 then increasing the network size in steps by adding1 neurons each time. The performance of the network during the training process was examined for each network size until no significant improvement was observed. As a result, the best-suited network architecture was determined to have: (a) One input neurons (Flow rate or discharge), (b) One hidden layer with 2 neurons, (c) Four output neuron. (Depth of flow, width of the channel and side slope z1,z2) and (d) Tangent hyperbolic function in the hidden layer Out of total 100 data sets, 75 are utilized for training and 25 for testing the ANN architecture. The progress of the training process was monitored by observing the mean sum squared error at every iteration of the back propagation process. The results are shown in Fig. 3. Figure 3. Variations of training error with number of epochs The best performing ANN architecture was obtained by experimentations (with trial and error by increasing or decreasing the number of neurons in hidden layer). 5. RESULTS AND DISCUSSION The work presented herein utilizes the ANN technique to model and prediction of depth of flow, width of the channel and side slopes for varying discharge. Most of the previous works obtained optimal section for a specified flow (discharge) value (Das, 2000; Jain et al., 2004; Bhattacharjya, 2007). Since flow is highly uncertain hydrological phenomenon, this methodology addresses the change in flow and corresponding change in optimal dimensions. In the present work, network with two hidden neurons, 1-2-4, was found to perform better. Out of 75 data sets some results obtained from ANN model training are shown in TABLE 1 and out of 25 data set some result obtained from ANN model testing are shown in TABLE 2. The final results are presented in TABLE 3. Results confirm that ANN model is able to perform to high degree of accuracy. However, the prediction error is more in testing as evident from the scatter plots for testing data. 176 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME Table1. Results obtained from ANN Model in training Disch. Actual Values ANN Model- Predicted Values Q Width Depth side side Cost Width Depth side side Cost slope slope slope slope (w) (h) (k1) (k2) (S) (w) (h) (k1) (k2) (S) 99.72 4.692 4.837 0.238 0.255 23.124 4.692 4.837 0.239 0.255 23.123 112.07 4.884 5.050 0.242 0.260 24.944 4.884 5.050 0.242 0.260 24.943 129.85 5.138 5.331 0.247 0.266 27.458 5.138 5.331 0.247 0.266 27.456 134.57 5.202 5.402 0.248 0.268 28.109 5.202 5.402 0.248 0.268 28.107 166.10 5.595 5.838 0.254 0.276 32.290 5.595 5.838 0.254 0.276 32.189 Table2. Results obtained from ANN Model in testing Disch. Actual Values ANN Model- Predicted Values Q Width Depth side side Cost Width Depth side side Cost slope slope slope slope (w) (h) (k1) (k2) (S) (w) (h) (k1) (k2) (S) 91.70 4.559 4.691 0.236 0.251 21.906 4.559 4.690 0.236 0.251 21.905 104.29 4.765 4.918 0.240 0.257 23.805 4.765 4.918 0.240 0.257 23.805 129.43 5.133 5.325 0.247 0.266 27.402 5.133 5.325 0.247 0.266 27.401 142.61 5.308 5.519 0.250 0.270 29.201 5.308 5.518 0.250 0.270 29.201 155.91 5.474 5.703 0.252 0.273 30.967 5.474 5.703 0.252 0.273 30.966 Table3. Training and testing results for ANN Model , 1-2-4 Training or Performance Evaluation statistics Testing NE RMSE R SSE MENash Training 8.91251E-05 0.0005 0.99 0.0002 0.99 Testing 8.78253E-05 0.0004 0.99 0.0004 0.99 The results show normalized error less than one percent and model efficiency (Nash and Sutcliffe, 1970) more than 98 percent in both training and testing. Thus, the initial results are encouraging and demonstrate the potential applicability of the methodology. An extensive evaluation of ANN architecture with observed data is also required for fully establishes the methodology. 6. CONCLUSIONS The optimization formulation based ANN methodology is presented in this work for optimal design of irrigation channels. The performance evaluations show high predictive accuracy in both training and testing data. This certainly establishes the application of the methodology. The study based upon limited flow and channel section constraints. An extensive evaluation is required with increased number of flow constraints (e.g., steady and unsteady; critical and super critical flows etc.) and channel stability criteria. 177 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME REFERENCES [1] Chow, V. T. (1959). Open channel hydraulics, McGraw-Hill, Singapore. [2] Guo, C. Y., and Hughes, W. C. (1984). ‘‘Optimal channel cross section with freeboard.’’ J. Irrig. Drain. Eng., 110(3), 304–314. [3] Froehlich, D. C. (1994). “Width and depth constrained best trapezoidal section.” J. Irrig. Drain. Eng., 120(4), 828–835. [4] Monadjemi, P. (1994). “General Formation of Best Hydraulic Channel Section.” J. Irrig. Drain. Eng., 120 (1), 27–35. [5] Das, A. (2000). ‘‘Optimal channel cross section with composite roughness.’’ J. Irrig. Drain. Eng., 126(1), 68–72. [6] Swamee, P. K., Mishra, G. C., and Chahar, B. R. (2000). “Design of minimum seepage loss canal sections.” J. Irrig. Drain. Eng., 126(1), 28–32. [7] Jain, A., Bhattacharjya, R. K., and Srinivasulu, S. (2004). “Optimal design of composite channel using genetic algorithm.” J. Irrig. Drain. Eng., 130(4), 286–295. [8] Bhattacharjya, Rajib Kumar (2004) “Optimal Design of Unit Hydrograph Using Probability Distribution Functions and Genetic Algorithms.” Sadhana, Academy Proceedings in Engineering Sciences, 29(5), 499-508. [9] Loganathan, G. V. (1991). “Optimal design of parabolic canals.” J. Irrig. Drain. Eng., 117(5), 716–735. [10] Das, A. (2007). “Flooding probabilities constrained optimal design of trapezoidal channels.” J. Irrig. Drain. Eng., 133(1), 53–60. [11] Bhattacharjya, R. K., and Satish, M. G. (2008). “Discussion of ‘Flooding probability constrained optimal design of trapezoidal channels’ by Amlan Das.” J. Irrig.Drain. Eng., [12] Horton, R. E. (1933). “Separate roughness coefficients for channel bottom and sides.” Eng. News-Rec., 111(22), 652–653. [13] Einstein, H. A. (1934). ‘‘Der hydraulische oder profile-radius [The hydraulic or cross-section radius].’’ Schweizerische Bauzeitung, Zurich, 103(8), 89–91 (in German). [14] Nash, J. E., Sutcliffe, J. V. (1970). River flow forecasting through conceptual models. Part 1- A: Discussion principles, Journal of Hydrology, 10, 282–290. [15] Tayfur G, Singh V.P. (2006). ANN and Fuzzy Logic Models for Simulating Event-Based Rainfall-Runoff, J. Hyd. Engg. ASCE, 132(12), 1321-1330. [16] Bishop, C.M. (1995). Neural Networks for Pattern Recognition, Oxford University Press, India. [17] Singh, R. M., Datta, B., and Jain, A. (2004). Identification of Unknown Groundwater Pollution Sources Using Artificial Neural Network, Journal of Water Resources Planning and Management, ASCE, 130(6), 506-514. [18] Dr. Salim T. Yousif, “New Model of Cfrp-Confined Circular Concrete Columns: ANN Approach” International Journal of Civil Engineering & Technology (IJCIET), Volume 4, Issue 3, 2013, pp. 98 - 110, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316, [19] B. M. Kharat and Prof. V. A. Kulkarni, “Automatic Multichannel Drip Irrigation”, International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 41 - 49, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316. [20] Dr. Laith Khalid Al- Hadithy, Dr. Khalil Ibrahim Aziz and Mohammed Kh. M. Al-Fahdawi, “Flexural Behavior of Composite Reinforced Concrete T-Beams Cast in Steel Channels with Horizontal Transverse Bars as Shear Connectors”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4, Issue 2, 2013, pp. 215 - 230, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316. 178