DESIGN OF COMPOSITE IRRIGATION TRAPEZOIDAL CHANNEL USING OPTIMIZATION AND PAT

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DESIGN OF COMPOSITE IRRIGATION TRAPEZOIDAL CHANNEL USING OPTIMIZATION AND PAT Powered By Docstoc
					   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
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    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).
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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)

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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          
                                                                                   

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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.




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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:

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                                                 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

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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.


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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.

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