**3.2 Neural network implementation**

In order to do not redo each time the finite element calculation when different problem geometry has to be studied, the authors had decided to implement a neural network solution to evaluate the self and mutual inductance matrix for any possible problem geometries. As input values the following geometrical parameters has been selected:

Artificial Intelligence Techniques Applied to

*D* [m]

2 [Ω\*m]

Table 4. Training EPL-MP problem geometries.

**3.3 Tested feed-forward architectures** 

EPL-MP problem geometries used to test the implemented NN

train the neural networks using the Levenberg-Marquardt method.

Fig. 19. Implemented feed-forward network architecture.

1 [Ω\*m]

Case No.

*d* [m]

Electromagnetic Interference Problems Between Power Lines and Metal Pipelines 269

1 310 800 900 850 900 16 490 1100 300 400 300 2 15 400 850 450 850 17 170 700 300 350 300 3 105 1100 550 550 550 18 150 700 500 500 500 4 350 900 500 800 500 19 240 500 80 750 80 5 250 800 150 150 150 20 125 800 300 600 300 6 60 800 500 900 500 21 420 100 550 20 550 7 340 400 600 150 600 22 75 400 350 700 350 8 65 400 650 350 650 23 105 1200 250 950 250 9 170 800 650 750 650 24 100 700 650 850 650 10 55 1000 900 400 900 25 85 400 140 160 140 11 40 200 600 800 600 26 300 400 900 100 900 12 115 800 800 800 800 27 145 300 350 900 350 13 120 900 750 350 750 28 15 500 140 700 140 14 135 400 180 500 180 29 100 1300 300 180 300 15 310 800 900 850 900 30 10 1000 200 750 200

In order to find the optimal neural network solution which will provide the most accurate results, the authors have implemented and tested different NN architectures. To test the implemented neural networks, the training database and a totally different data set that was not applied during the training process, were used. The error between the solutions provided by each implemented NN and the self and mutual impedance matrices are determined to identify the optimal architecture. Table 4 presents the randomly generated

To identify the optimal solution for each of the proposed neural networks, different feed-forward architectures with one output layer and two hidden layers were implemented (figure 19). Based on the experience gained after implementing the neural network for MVP calculation, the transfer function for the output layer has been chosen to be *purelin* (linear function) and *tansig* (hyperbolic tangent sigmoid function) for the hidden layers. The number of neurons was varied from 5 to 30 for the first hidden layer and from 5 to 20 for the second hidden layer. The performance evaluation function was set to *mse* (mean square error) and the descendent gradient with momentum weight learning rule was selected to

Case No.

*d* [m]

*D* [m]

2 [Ω\*m]

1 [Ω\*m]

3 [Ω\*m]

3 [Ω\*m]


During the pre-processing stage of the proposed neural network solutions implementation all the input parameters were automatically scaled by MatLab in the [-1,+1] range.

The outputs of the proposed neural network will be the impedance matrix elements. Considering the fact that the impedance matrix is a symmetrical matrix, the output values are the matrix elements above the main diagonal. For the proposed EPL-MP problem with one pipeline, three phase wires and one sky wire the matrix elements will be: *Z*<sup>11</sup> , *Z*<sup>12</sup> , *Z*<sup>13</sup> , *Z*<sup>14</sup> , *Z*<sup>15</sup> , *Z*<sup>22</sup> , *Z*<sup>23</sup> , *Z*<sup>24</sup> , *Z*<sup>25</sup> , *Z*<sup>33</sup> , *Z*<sup>34</sup> , *Z*<sup>35</sup> , *Z*<sup>44</sup> , *Z*<sup>45</sup> , *Z*<sup>55</sup> , where *Zii* represents the self impedance of conductor *i* and *Zij* the mutual impedance between conductor *i* and *j* ( *i* 1,3 represents EPL phase wires, 4 *i* represents EPL sky wire and 5 *i* represents MP).

After analysing in detail the impedance matrices for different EPL-MP problem geometries the authors concluded that in order to increase NN results accuracy and to reduce training time it will be better to implement different NN for the real and imaginary part of each impedance. Also for accuracy improvement were implemented 3 different networks for the impedance matrix elements: NN1 for all conductors self impedances ( *Z*<sup>11</sup> , *Z*<sup>22</sup> , *Z*<sup>33</sup> , *Z*<sup>44</sup> , *Z*<sup>55</sup> ), NN2 for the mutual impedances between the pipeline and the conductors( *Z*<sup>15</sup> , *Z*<sup>25</sup> , *Z*<sup>35</sup> , *Z*<sup>45</sup> ), NN3 for the other mutual impedance elements.


Table 3. Training EPL-MP problem geometries.

To train the proposed NN a training data base was created based on the impedance matrices obtained using FEM for different EPL-MP problem geometries. In order to create a useful training database approximately 5000 different EPL-MP problem geometries were simulated varying the EPL-MP separation distance from 0 m to 1000 m, the earth layers resistivity from 10 Ωm to 1000 Ωm and the middle layer width from 50 m to 1500 m. Table 3 presents some of the different EPL-MP problem geometries used to train the proposed NN.


Table 4. Training EPL-MP problem geometries.

268 Recurrent Neural Networks and Soft Computing

<sup>1</sup> - resistivity of middle layer earth (which varies between 30 Ωm and 1000 Ωm);

During the pre-processing stage of the proposed neural network solutions implementation

The outputs of the proposed neural network will be the impedance matrix elements. Considering the fact that the impedance matrix is a symmetrical matrix, the output values are the matrix elements above the main diagonal. For the proposed EPL-MP problem with one pipeline, three phase wires and one sky wire the matrix elements will be: *Z*<sup>11</sup> , *Z*<sup>12</sup> , *Z*<sup>13</sup> , *Z*<sup>14</sup> , *Z*<sup>15</sup> , *Z*<sup>22</sup> , *Z*<sup>23</sup> , *Z*<sup>24</sup> , *Z*<sup>25</sup> , *Z*<sup>33</sup> , *Z*<sup>34</sup> , *Z*<sup>35</sup> , *Z*<sup>44</sup> , *Z*<sup>45</sup> , *Z*<sup>55</sup> , where *Zii* represents the self impedance of conductor *i* and *Zij* the mutual impedance between conductor *i* and *j* ( *i* 1,3 represents EPL phase wires, 4 *i* represents EPL sky wire and 5 *i* represents MP).

After analysing in detail the impedance matrices for different EPL-MP problem geometries the authors concluded that in order to increase NN results accuracy and to reduce training time it will be better to implement different NN for the real and imaginary part of each impedance. Also for accuracy improvement were implemented 3 different networks for the impedance matrix elements: NN1 for all conductors self impedances ( *Z*<sup>11</sup> , *Z*<sup>22</sup> , *Z*<sup>33</sup> , *Z*<sup>44</sup> , *Z*<sup>55</sup> ), NN2 for the mutual impedances between the pipeline and the

conductors( *Z*<sup>15</sup> , *Z*<sup>25</sup> , *Z*<sup>35</sup> , *Z*<sup>45</sup> ), NN3 for the other mutual impedance elements.

3 [Ω\*m]

8 5 60 500 50 500 2301 20 550 30 250 30 373 100 60 500 750 500 2532 100 550 100 500 100 875 20 120 100 750 100 2914 5 1050 10 250 10 1231 500 120 100 30 100 3274 100 1050 500 1000 500 1391 0 240 50 10 50 3545 750 1050 30 750 30 1891 250 240 500 30 500 4320 750 1500 50 10 50 2134 0 550 50 250 50 4442 1000 1500 250 1000 250

To train the proposed NN a training data base was created based on the impedance matrices obtained using FEM for different EPL-MP problem geometries. In order to create a useful training database approximately 5000 different EPL-MP problem geometries were simulated varying the EPL-MP separation distance from 0 m to 1000 m, the earth layers resistivity from 10 Ωm to 1000 Ωm and the middle layer width from 50 m to 1500 m. Table 3 presents some of the different EPL-MP problem geometries used to train the proposed

Case No.

*d* [m]

*D* [m]

2 [Ω\*m]

1 [Ω\*m]

3 [Ω\*m]

1 [Ω\*m]

) - resistivity of left and right side earth layer (which varies as

<sup>1</sup> );

*d* - distance between EPL and MP (which varies between 0 m and 1000 m);

all the input parameters were automatically scaled by MatLab in the [-1,+1] range.

*D* - earth middle layer width (which varies between 50 m and 1100 m).

2 , 

Case No.

NN.

*d* [m]

*D* [m]

2 [Ω\*m]

Table 3. Training EPL-MP problem geometries.

<sup>3</sup> ( 2 3 

> In order to find the optimal neural network solution which will provide the most accurate results, the authors have implemented and tested different NN architectures. To test the implemented neural networks, the training database and a totally different data set that was not applied during the training process, were used. The error between the solutions provided by each implemented NN and the self and mutual impedance matrices are determined to identify the optimal architecture. Table 4 presents the randomly generated EPL-MP problem geometries used to test the implemented NN
