19.2.1 Conventional Contour Correction Techniques for Electrode and Insulator Optimization

In 1975, Singer and Grafoner [4] published a methodology for optimizing electrodes and insulators designed for practical use. In their method, domains with constant field intensity were obtained for axi-symmetric arrangements by displacing contour points successively. Some more conventional methods of electric field optimization were reported in References [5–7].

In 1982, Misaki et al. [8] reported a method to get the optimum design of epoxy pole spacers used in sulphur hexafluoride (SF₆) gas-insulated cables. In their method an improved surface charge method using curved surface elements was employed for the computation of three-dimensional (3D) electric field distribution and the results of electric field computation were used for the optimization of insulator design. The optimum insulator design was performed automatically by correcting the insulator contour using a simple vector computation. In this method, the insulator contour was corrected in such a way that both tangential field intensity and normal field intensity became uniform along the new contour. If the insulator contour moves in the direction of the normal vector, then the tangential field intensity decreases, and if the insulator contour moves in the direction opposite to that of the normal vector, then the normal field intensity decreases on the insulator surface. Therefore, the direction in which correction of the insulator contour had to be carried out was decided by taking note of the normal stress on the insulator surface. The insulator contour was corrected in proportion to the normal Maxwell’s stress, which was obtained from both tangential and normal field intensities. A method for 3D electrode contour optimization was reported by Misaki et al. [9] in 1983, in which the electric field intensity and force on the electrode surface were computed, and then the electrode contour was moved in the opposite direction of the force and the contour displacement was done in proportion to the magnitude of the force.

In 1983, Grönewald [10] described an algorithm based on the computer-aided design (CAD) concept for optimizing electrode contour to obtain a given field distribution on the surface. They proposed a two-step procedure that strictly separated electric field calculation and geometrical contour corrections. At any point on the contour, the changes in curvature depending on the desired field were described with the help of relationships developed between field quantity and geometry. The new contour was found by a simultaneous displacement of all discretizing points on the contour.

Stih [11] in his paper described an iterative procedure for designing optimally stressed insulating system. The described procedure combined the approach of varying the distances between system elements and optimizing their contours. Axially symmetric electric fields were computed by an integral equation technique introducing cubic spline expansion for charge density distribution function. In an extension of the contour correction techniques reported in References [4,8,9], an additional step was introduced in this work in the form of smooth approximation of the corrected contour by circular arcs, which enabled relatively simple geometrical description of the optimal contours.

Abdel-Salam and Stanek [12,13] reported a method for optimizing the field stress on high-voltage insulators through the modification of their profile to obtain a uniform distribution of the tangential field along the insulator surface. An algorithm based on a modified charge simulation technique was developed for calculating the tangential field component along the insulator surface. After computation of the tangential field distribution for the non-optimized profile, the insulator profile was enlarged or reduced on going up or down along its axis. If the field distribution increased, when moving along the axis of the insulator, the profile radius was increased in that direction and if the field distribution decreased, then the profile radius was decreased. Exponential mathematical expressions, which defined a smooth enlargement/reduction of the profile radius, were used in this work.

Mosch et al. [14] published a method of optimization of large electrodes of ultra high voltage (UHV) testing equipment designed on the basis of inception of streamer and leader discharges. The stochastic nature of these two discharge processes leads to the well-known area effects, and also to time effects. Hence, the proposed model was related to the volume–time concept. The design procedure started with choosing the principle of field control and the main geometric parameters of the electrode. Then the electric field was calculated numerically and from the results of electric field computation the inception voltages V_iS, for streamer discharge, and V_iL, for leader discharge, were determined. The calculated values were compared to the maximum output voltage. If the calculated values were less, then the geometry of the electrode was improved and the cycle of computation was repeated till an optimum design was achieved. A similar study was reported by Kato et al. in Reference [15]. In this paper, they described a technique for the optimization of high-voltage electrode contour to make the electrical insulation performance highest by introducing the volume–time characteristics into the optimization procedure. Optimization technique was also investigated from the viewpoints of computation efficiency and accuracy.

The basic principle of electrode optimization method suggested by Liu and Sheng [16] was that the electrode was divided into two parts: the fixed part and the part to be optimized. The fixed part was simulated by the simulation charges and the second part by the optimization charges, in which the magnitudes and coordinates were assigned first. Normally, the magnitudes of the optimization charges were taken as optimization variables. In this method, the coordinates instead of the magnitudes of optimization charges were taken as the optimization variables. The reported results of optimization examples indicated that the use of the proposed method yielded good results. Liu et al. [17] also described an automatic procedure for the optimization of axi-symmetric electrodes using boundary element method. The criterion employed for the optimization of the shape of the electrodes was the minimization of the maximum field intensity on the surface of the electrodes. The proposed method was applicable in 3D fields too. Judge and Lopez-Roldan in Reference [18] discussed an approach to design components for high-voltage engineering based on the boundary element method in two and three dimensions. Optimization of the components was performed using an iterative method of refining a design and re-analyzing without having to resort to repeated prototype building and testing. Studies on the optimization of electrode contours using circular contour elements [19] and optimization of multi-electrode system [20] have also been reported in the literature.

Däumling and Singer [21] described a new algorithm based on the CAD concept for the optimization of insulator contours to get a given field distribution along the surface. Three optimization procedures with different targets were investigated: (1) uniform tangential field strength, (2) uniform resultant field strength and (3) uniform electrostatic pressure along the insulator contour. It was found that achieving a low tangential field strength component was not sufficient, and it was necessary to reduce the maximum value of the resultant field strength as far as possible, especially in the case of high air humidity.

Caminhas et al. [22] determined the optimum profile and location of shielding electrode used in high-voltage equipment with the aim to obtain a linear voltage distribution along the axis of symmetry. It was shown that the combined usage of the charge simulation method and the Broyden–Fletcher–Goldfarb–Shannon optimization technique was both efficient and simple. Very good attempt was made to review various optimization methods applicable to the specific case.

19.2.2 Optimization of High-Voltage System Elements

Optimization of field stresses in high-voltage bushings have been reported in References [23,24] and special techniques for electrode and insulator contour optimization, which were claimed to be efficient, have been reported in References [25–27]. Efficient techniques for electric field optimization using personal computers were reported in References [28,29]. Studies on 3D electric field optimization in high-voltage equipment was reported by Trinitis [30,31]. In Reference [31] optimal electric field strength distribution for 3D problems was achieved by utilizing a parametric CAD modelling system coupled to a 3D electric field calculation program. These two components were then linked to a numerical optimization algorithm. The package comprising the above-mentioned three components was then able to automatically optimize arbitrary 3D field problems in high-voltage engineering. Optimization of high-voltage insulators in 3D field configurations was reported in Reference [32].

High gradient insulators (HGI) consist of a periodic array of insulator and metal rings, which have been found to be more resilient to vacuum surface breakdown than homogeneous insulators of the same length. Studies based on calculations and experiments were reported by Leopold et al. [33] to understand the effect of geometry on the performance of well conditioned, flat surface HGI assemblies.

19.2.3 Soft-Computing Techniques for Electrode and Insulator Optimization

With the advent of artificial intelligence, it was realized that instead of iterative calculation of electric field and contour modification, electric field optimization could be done based on artificial neural network (ANN). Once the ANN is trained, optimum solution satisfying desired specification can be found without iterative calculation, resulting in high-speed method. ANN with supervised learning by error back-propagation method as used for high-voltage electrode optimization was reported by Chakravorti and Mukherjee [34]. It was found that the trained ANN can give results with mean absolute error (MAE) of about 1% in comparison with analytically obtained results. For electrode contour optimization, the results of electric field calculations for some pre-determined contours of an axi-symmetric electrode arrangement were used for training the ANN. Then the trained ANN was used to give the optimized electrode contour to obtain a desired field strength distribution on the electrode surface. Similar study on application of ANN in the design of toroidal electrodes was reported by Bhattacharya et al. [35]. Optimization of HV electrode systems by ANN with resilient propagation was carried out by Mukherjee et al. [36]. Two axi-symmetric examples were optimized: (1) the first one was the termination of a single-phase gas-insulated substation (GIS) bus and (2) the second one was a shield ring in a three-phase transformer. Charge simulation method was used for preparation of the training sets as well as for checking the test outputs from the ANN. The procedure of the electric field optimization method based on ANN as reported by Okubo et al. [37] comprised two ANNs. As the electric field strength on a high-voltage electrode surface has empirical relationship with the curvature and the gap length, one ANN, called ‘NN1’, was trained to learn this relationship. Another ANN, called ‘NN2’, was trained to learn the relationship between a set of curvature, gap length and electrode contour. Once the learning of the two ANNs were completed, the curvature, gap length and electrode contour for the optimized electrode configuration were obtained as output by giving the target electric field distribution as input into the trained ANN system.

In Reference [38], ANN-based studies have been presented for the optimization of insulator contour in multiple dielectric systems, where the degree of field nonlinearity is more than that for single-dielectric configurations. ANN-aided contour optimization of axi-symmetric insulators in multi-dielectric arrangements had been carried out to obtain not only a uniform but also a complex electric stress distribution along the insulator surface. Multilayer feed-forward networks with error-back propagation as well as resilient propagation learning algorithms were employed.

As each ANN application requires a certain amount of training to achieve the desired accuracy, Chatterjee et al. [39] in their paper discussed the development of a self-organizing fuzzy inference system for designing optimized electrode contours. The process of optimization was accelerated using a fuzzy inference system that eliminated training time. Reliability of the proposed methodology was further improved by implementing an algorithm for automatic generation of the fuzzy rule base from input–output data sets.

Alotto et al. [40] gave an overview of several stochastic optimization strategies, namely, evolution strategies, genetic algorithms (GA) and simulated annealing (SA), as applied to electromagnetic optimization problems. The application of evolution strategy for optimal design of high-voltage bushing electrode in transformer was investigated in Reference [41]. Several regular circular arcs and straight lines were used to get the optimal shape of electrode. The aim of optimization was to obtain lower maximum field intensity and well-distributed electric stresses on the surface of electrode, greater reliability during operation and easy manufacturability in large-scale production.

Lahiri and Chakravorti [42] carried out the optimization of contours of 3D electrode–spacer arrangements used in gas-insulated transmission line (GIL) by ANN-aided GA. Input–output data set used for training the ANN was prepared by means of electric field calculations using surface charge simulation method varying several design parameters of each of the two 3D arrangements considered in their work. The trained ANN was coupled to a GA loop. At each exploratory move of the GA loop a new set of values of the design parameters were set and the corresponding maximum resultant field intensity along the insulator surface was obtained from the trained ANN instead of running a comprehensive field computation routine. This maximum resultant field intensity was returned to the GA loop as the value of the cost function to decide on the fact whether the loop has converged or not. In this way, a time-saving optimization technique had been developed to obtain an optimized field distribution along the insulator surfaces. In another paper, Lahiri and Chakravorti [43] used ANN-aided SA algorithm for the optimization of stress distribution on and around 3D electrode–spacer arrangements. By coupling the trained neural net with the annealing algorithm, the execution speed of the optimization routine was significantly increased to evaluate the optimum values for the design parameters of the electrode–spacer arrangements, because there is no need for the cost function calculation via the entire process for electric field calculation at every move of the optimization algorithm, which makes the optimization routine very fast.

Banerjee et al. [44] in their paper used support vector machine (SVM) for the optimization of electric field along the support insulators used in highvoltage systems. The SVM designed for insulator contour optimization was first trained with the results obtained from electric field computations for some predetermined contours of the arrangements. Then the trained SVM was used to provide the optimized insulator contour in such a way that the desired stress distribution was obtained on the insulator surface.

19.2.4 Optimization of Switchgear Elements

Kitak et al. [45] described an algorithm for the design of medium-voltage switchgear insulation elements using numerical calculations on the basis of finite element method in connection with evolutionary optimization methods. Differential evolution and evolution strategy algorithms were used for optimization. The task of both optimization algorithms was to find an adequate capacitance of the voltage divider and the optimal distribution of electric field strength. The highlight of the work was simultaneous use of parametric representation in geometry, a novel mesh generator, numerical computation with finite element method (FEM) and a genetic optimization algorithm. The proposed methodology thus represented a generalized method of optimization for various objective functions of switchgear elements.

For accurate insulation design of the vacuum interrupter, an optimization technique was developed by Kato et al. [46] to improve the electrical insulation performance. Several design variables necessary for optimizing the electrode contour of the main contactor and centre shield in vacuum interrupter were evaluated. The electrode area effect on vacuum breakdown process as well as the electric field distribution was considered in the optimization process. Electrode contour optimization of arc quenching chamber of extra-high-voltage SF₆ circuit breaker had been carried out by Liu et al. [47] to obtain not only a uniform field but also a dynamic distribution of electric field strength along different contact surfaces. The variable interval GA was used to perform multivariable global optimization of the electrode contour of a single-break 550 kV SF₆ arc-quenching chamber.

Functionally graded material (FGM) spacer, the permittivity of which changes gradually, exhibits considerable reduction in the maximum electric field when compared to a conventional spacer with uniform permittivity. However, it is difficult to realize a gradual permittivity variation in the FGM spacer in real-life product processing due to its complicated shape. Thus, optimization processes were used by Ju et al. [48,49] to modify the shape of both the electrode and the FGM spacer on a commercial gas-insulated switchgear configuration to increase the possibility of real FGM insulator manufacturing. Modification of the spacer configuration was performed with the design of experiments. For a reliable and effective design process, the full factorial design method and the response surface methodology were employed in the development of an effective computational approach.

19.2.5 Optimization of Bushing Elements

Although it is entirely possible that the use of uniform field profile electrodes could achieve the goal of reducing the electric field in critical areas, the manufacturers of bushings often deemed that this was not an economically feasible solution. Hence, Monga et al. [50] used a basic bushing configuration as the starting point and the goal of the work was to design optimum grading hardware for the larger bushings. In this paper, the authors illustrated the use of electric field computation based on boundary element method to optimize the design of gas filled high-voltage composite bushings. The optimized design used both internal and external elements for electric stress grading at critical parts of the bushing.

Hesamzadeh et al. [51] proposed a methodology for finding the optimum electrical design of high-voltage condenser bushings using an improved GA. In this paper, the authors determined the optimal values of bushing design parameters to achieve well-distributed electric stress with the lowest possible maximum value and also a constant voltage drop between different layers of concentric conductive foils, which are isolated from each other, subject to practical and technological constraints. The proposed method was applied for optimal design of a 145 kV oil impregnated paper bushing and the performance of optimally designed bushing were satisfactorily verified under IEC 60137 tests.

19.2.6 User-Friendly Optimization Environment

Precise simulation and geometric optimization of the electric field distribution on electrodes and insulators are key aspects in the design and optimization process of high-voltage apparatus. Because these simulations and optimizations are computation intensive, an engineer working in industry demands a user-friendly working environment requiring as little knowledge as possible with regard to the computer specific aspects. From the user’s viewpoint, a user-friendly design and optimization environment should (1) require minimum interaction and knowledge about the simulation system and (2) provide a solution as quickly as possible. Trinitis [52] achieved the first requirement by a distributed design and optimization system based on a model-driven architecture and the second requirement was achieved by both parallelization of the simulation process and acceleration of the simulation process on each of the parallel computing nodes. By optimizing the simulation software with regard to the hardware it was running on, computation time for an overall optimization run was kept at an acceptable level.

19.3.1 Insulator Contour Optimization by Simultaneous Displacement

Considering two successive contour points on the surface of the insulator the tangential field intensity may be given by

Et=-ΔϕΔl≈ϕ1-ϕ2Δl⁢(19.3)

where:

• ϕ₁ and ϕ₂ are the potentials of the two successive contour points 1 and 2, respectively

• Δl is the distance between the contour points 1 and 2

If the actual value of the tangential field intensity does not match with the desired value, then either the potential difference between the points or the distance between the points need to be altered. Accordingly, there are two methods of displacement of contour points: (1) displacement keeping Δϕ constant and (2) displacement keeping Δl constant.

19.3.2 Electrode and Insulator Contour Correction with Approximation of Corrected Contour

Stih [11] proposed a procedure for optimization of electrode and insulation system combining the approach of varying the distances between contour points and then approximating the corrected contour by circular arcs. The procedure was applied for optimizing axi-symmetric configurations. In the proposed approach, the target was to attain uniform distribution of normal field intensity on the electrode surface and uniform distribution of tangential field intensity on the insulator surface. The contour correction principle, as presented in Reference [11] is described below.

As shown in Figure 19.6a if the point 1 moves to 1′ in the normal direction while the potential V_A of the electrode is kept constant, then the normal field intensity E_A will increase to E_A′, such that

EA=VA-VBl12 andEA=VA-VBl1′2⁢(19.17)

where:

• E_A′ > E_A as l_{1′ 2} < l₁₂

On the other hand, if the point P on the insulator surface moves to the P′, as shown in Figure 19.6b, along the equipotential line, that is, normal to the electric field intensity, then the tangential field intensity decreases from E_tP to E_tP’ such that

EtP=VO-VPlOP and EtP′=VO-VPlOP′⁢(19.18)

where:

• E_OP’ < E_OP as l_OP’ > l_OP

The quantum of displacement of contour points was controlled by moving the contour points in proportion to the difference between the actual and desired values of field intensity. Let l→Ai be the vector that defines the ith contour point on the actual contour, while l→ci be the vector that defines the ith contour point on the corrected contour. Then these two vectors were related as

l→ci=l→Ai(1+ΔEiEiu^ni)⁢(19.19)

where:

• u^ni is the unit vector normal to the electrode or to the electric field intensity at the ith contour point on the surface of electrode or insulator, as the case may be

In Equation 19.19,

ΔEi=Edi-EAiforΔEmx<f1EdiandΔEi=Edi-EAiEmxf1forΔEmx>f1Edi⁢(19.20)

where:

ΔEmx=max{|Edi-EAi|} i=1,…,N

where:

• E_di is the desired value of electric field intensity

• E_Ai is the value of electric field intensity on the actual contour at the ith contour point

• N is the number of contour points for which correction is carried out

• f₁ is a factor that limits the quantum of displacement of the contour points, as Equations 19.17 and 19.18 are valid only for small displacements. Typically the value of f₁ was chosen as 0.1

In the next step of the optimization procedure [11], smooth approximation of the corrected contour was done by circular arcs in the following way. For smooth approximation of a particular set of points H = [P_i,(x_i, y_i)], i = 1,…,N by a circular arc, the conditions that are to be satisfied are as follows: (1) the centre of the circle should be on a given line y = m_h x + c_h, (2) the circle passes through the given point P_j(x_j,y_j) and (3) the circle is the best approximation of the given set of points. With reference to Figure 19.7, it may be written that

y0=mhx0+ch⁢(19.21)

and(xj-x0)2+(yj-y0)2=r2⁢(19.22)

where:

• r is the radius

• (x₀,y₀) are the coordinates of the centre of the circular arc

Considering the conditions given by Equations 19.21 and 19.22, for best approximation the minimum is to be found of the function G, which is given below

G=∑i=1N((xi-x0)2+(yi-y0)2-r)2⁢(19.23)

For this purpose, the minimum of the function F is to be determined, where F is given as follows:

F=G+α(y0-mhx0-ch)+β[(xj-x0)2+(yj-y0)2-r2]⁢(19.24)

The conditions to be satisfied for minimum F are

∂F∂x0=0, ∂F∂y0=0, ∂F∂r=0, ∂F∂α=0, ∂F∂β=0⁢(19.25)

From these conditions a system of equations for the unknown variables x₀, y₀, r, α and β was obtained, which in turn was reduced to a non-linear equation for x₀ that was solved numerically.

Further, the smooth approximation of complex electrode and insulator contours by several circular arcs is based on the fact that two circular arcs are joined smoothly at a point J1 or J2 if the centres of the two circles lie on the line passing through the point J1 or J2, as shown in Figure 19.8.

The optimization process was started by choosing an initial contour from not only electrical considerations but also considering mechanical, thermal and other technical constraints. The electric field distribution was computed numerically using integral equation technique. The distances between the contour points were changed using the following rule: if the maximum field intensity is lower than the allowable value then the distances are decreased and vice versa. The stopping criteria of the optimizations steps were

EmaxEmin<fuandEa11ow>Emax>feEa11ow⁢(19.26)

where:

• E_max, E_min and E_allow are maximum, minimum and allowable values of electric field intensities, respectively

• f_u is the field uniformity factor

• f_e is the chosen limit of maximum field intensity with respect to allowable field intensity. Typical range of values of f_u and f_e were 1 < f_u < 1.1 and 0.9 < f_e < 1, respectively

Stih [11] applied the methodology for optimization of the shielding electrode of a transformer winding. The initial geometry of the system is shown in Figure 19.9, where the shielding electrode is chosen as semicircle. The optimized final geometry of the shielding electrode is shown in Figure 19.10, which was obtained for f_u = 1.06 and f_e = 0.99 [11].

19.3.3 Parametric Optimization of Insulator Profile

Abdel-Salam and Stanek [12] described a method to obtain uniform tangential field intensity along high-voltage insulator surface by modifying the profile.

In this algorithm actual value of tangential field intensity was used to correct the mathematical expression of the insulator contour to achieve the final optimized profile. In order to search for a profile for which tangential field intensity is uniform along the profile, the following procedure was adopted:

1. Tangential field intensity values were computed along the non-optimized profile numerically. If the tangential field intensity increases while moving along the axis of the insulator, then the profile radius needs to be increased in that direction.

2. In order to get a smooth change in the profile radius, pre-defined mathematical expression for the profile was used in this work. While formulating the profile expression, care was taken, so that the contact angle at the insulator–electrode junction remained 90°. In other words, the profile expression must satisfy the right angle contact criterion for the insulator with electrodes. This is necessary to avoid excessive field concentration at the contact points.

As shown in Figure 19.11, the profile radius was proposed to have an exponential variation as given below.

r=R0 for0≤z≤xh100r=R0+DR[e0.693[z-(xh/100)][100/(100-2x)h]-1] forxh100<z≤(1-xh100)hr=R0+DR for(1−xh100)h<z≤h⁢(19.27)

where:

• x% of the height h was kept vertical to maintain the right angle contact criterion at both the live and ground electrode ends

Typically x was chosen to be 5%. DR was the largest value of enlargement in profile radius through exponential increase along the z-direction from the ground end to live end. As a result, the radius of the insulator at the ground end was R₀ and that at the live end was R₀ + DR.

N number of contour points were chosen along the insulator profile and the tangential field intensities were computed at these points by numerical field computation. Then the value of DR was changed iteratively to the final value DR_F to achieve an acceptable degree of uniformity in tangential field intensity along the optimized insulator profile, which is depicted in Figure 19.11. Figure 19.12 shows the tangential field distribution for non-optimized as well as for the optimized profiles [12].

19.4.1 ANN-Based Optimization of Electrode Contour

Chakravorti and Mukherjee [34] applied the inverse logic in optimizing the electrode shape to obtain a uniform normal field intensity along the end profile of a parallel disc electrode arrangement. In this approach, the ANN was trained with the electric field intensities as input patterns and the geometric dimensions as the output patterns. Then the trained ANN was used to predict the geometric dimensions of the electrode to get a uniform field intensity along the electrode end profile. Chakravorti and Mukherjee [34] optimized the end profile of an axi-symmetric electrode arrangement in the form of parallel discs using the above-mentioned inverse logic. For the purpose of training the ANN, input–output patterns need to be generated. These training patterns were generated by numerical field computations carried out for pre-defined electrode shapes for which the end profile was taken to be circular, as shown in Figure 19.13.

For the purpose of generating the training data, radius of parallel discs and the separation distance between the discs were kept constant at r_d and 30% of r_d, respectively. N_C numbers of different contours were obtained by varying the radius of end profile (r_e) in steps. For each of these N_C contours, electric field intensities were calculated at (N_P + 2) number of points on the electrode surface, out of which two were on the parallel surface and the rest N_P were on the end profile. The electric field intensities at the (N_P + 2) points in each contour were then given as input pattern vector to the ANN, while the coordinates of the N_P points on the end profile were given as output pattern vector. The N_P points on the end profile were taken at fixed z-coordinates for all the circular end profiles having pre-defined radii. Hence, only the r-coordinates of the N_P points were given as output pattern vector. The ANN is thus made up of (N_P + 2) and N_P neurons in the input and output layers, respectively, and there were N_C number of input–output patterns. For preparing the training set, electric field computations were carried out considering the potential of the live electrode as 0.1 kV, where 100 V represented the percent potential difference between the two electrodes.

For electrode contour optimization, for every set of input pattern the output pattern was known. Hence, for this problem, ANN with supervised learning was implemented involving multilayer feed-forward network with error-back propagation [34]. After completion of the training process, an optimized end profile was determined as a test case with the desired field intensity value of 390 V/m in the critical domain between B and C on the end profile, as shown in Figure 19.13. For this optimized end profile, as shown in Figure 19.14, electric field computations were carried out to determine the electric field intensities along the end profile to find out the deviation of the test results from the desired value of 390 V/m. MAE in field results along profiles obtained from ANN as output with respect to the desired values was found to be ≈3% [34]. The variation of actual electric field intensity distribution for a desired uniform field intensity of 390 V/m is shown in Figure 19.15 [34].

19.4.2 ANN-Based Optimization of Insulator Contour

Using an approach similar to that discussed in Section 19.4.1, Bhattacharya et al. [38] reported ANN-based contour optimization of axi-symmetric insulators in multi-dielectric arrangements to obtain not only uniform but also complex electric field intensity distributions along the insulator surface. For the purpose of generating the training data, electric field computations were carried for conical support insulators having linear contour, as shown in Figure 19.16.

In electric field computations h, h₁ and h₂ were kept constant. Typically h₁ and h₂ were taken as 10% of h. N_R numbers of different values of R₂ were taken when R₁ was kept constant. Similarly, N_R numbers of different values of R₁ were taken when R₂ was kept constant. Therefore, there were 2N_R numbers of contours for generating the training data. Tangential stresses were calculated at N_P numbers of points on all the 2N_R numbers of pre-defined insulator surfaces. For each of the N_P points, the z-coordinate was kept fixed on all the pre-defined contours. For every one of these 2N_R contours, N_P tangential field intensities were given to the ANN as input pattern vector and the corresponding r-coordinates of the N_P points were given as output pattern vector during training. Therefore, the ANN had N_P neurons each for input and output layers and 2N_R numbers of input–output training patterns. Multi-layer feed-forward network with error-back propagation as well as resilient propagation were employed for supervised learning of ANN.

After completion of successful training, a normalized uniform tangential field intensity of 0.1 V/cm was desired along the insulator surface, as shown in Figure 19.17. N_P values of uniform tangential field intensities at the N_P specified points were fed to the trained ANN as inputs and the corresponding r-coordinates of the N_P points were obtained as outputs from the trained ANN. These r-coordinates, as obtained from the trained ANN as outputs, along with their respective z-coordinates were then plotted to get the optimized insulator profile, which is shown in Figure 19.18 [38]. Subsequently, with the help of numerical field computation the actual values of tangential field intensities at these N_P points on the optimized insulator contour were determined. MAE between the actual and desired tangential field intensities was found to be ≈2% [38]. The actual and desired tangential field intensity distributions along the optimized insulator profile of Figure 19.18 are shown in Figure 19.17.

Answers:

1) c;

2) b;

3) d;

4) a;

5) d;

6) d;

7) b;

8) d;

9) a;

10) a