# Stress Cone Effect Analysis and Optimum Design of Extra High-Voltage Cable Joint Using Neural Networks

Authors: Çağrı Köksal; Emrah Dokur; Aysel Ersoy
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The major problem in high-voltage cable accessories is the electric field stress caused by the high-voltage level at the cable junction. In this paper, a neural network and simulation-based optimal design for extra high-voltage (EHV) cable accessory is proposed to ensure the optimum electrical design. The different kinds of geometries of the EHV underground cable accessory are realized in the Comsol Multiphysics software program, and the electric field analyses are conducted based on the finite element method. The design of high-voltage underground cable accessories is examined both within and without the optimal stress cone geometry. A neural network-based model is developed according to the analysis results obtained in the simulation. Thanks to this model, the geometry of the stress cone is determined. In the simulation-based analysis, the electric field stress caused by the high-voltage level at critical points is controlled with an optimum joint design using neural networks. The proposed model will contribute to EHV cable joint design at different stress levels.


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## Body

## 1. Introduction

A higher capacity transmission path is required to satisfy the power demand of the society having a steady increase in electricity consumption. Installing high-voltage cables is an inevitable trend in metropolitan areas and densely populated areas.High-voltage underground cables are manufactured in limited length due to obstacles seen during production and transportation. High-voltage cable accessories and cables that have limited length are combined to ensure the continuity of the power line. With the rapid growth of installation of high-voltage underground cables, cable failure faults frequently occur in large cities due to the unqualified accessories installation and unusual external operating environment [1–3]. According to the statistics of cable failures in the last 10 years, the share of cable accessory failure is more than half of the total cable defect. Among the power cable failures in gird, more than 75% of cable failures occur in the conventional cable joint without considering the failures caused by the external destruction [4, 5]. The major problem in high-voltage cable accessories is the electric field stress caused by the high-voltage level at the cable junctions [6]. Generally, the high-voltage accessories are the most sensitive parts and are in danger of failure in the cable system [7–9]. The reliability of cable is related to its accessories such as joints and terminations since they have a much higher failure rate than the cable itself [10]. The weaknesses of cable accessories are correlated to the nonuniform distribution of their internal electric field [11].It is necessary to know the distribution of electric field intensity and especially its value in the places where it is the highest for high-voltage underground cables [12–14]. As the energy transmission system voltage increases, due to load conditions and development, the insulation must operate under greater electrical stresses. For this reason, using a larger-sized insulation material does not give good results as it will make the device heavier and increase the thermal impedance. It is necessary (i) to control the excessive electric field stresses that may occur on the high-voltage cable screen and (ii) to reduce the electric field stress in the critical region [15].The historical development of stress control is given in Figure1. The impedance stress control based on heat-shrinkable tubing and nonlinear stress control were used in these developments [16].Figure 1
Development of electric stress control.Originally, some researchers approximating and modelling elastic continua using discrete equivalent elastic bars in the early 1900s proposed the finite element method [17]. However, Courant presented and developed the modern finite element method in 1940. Piecewise polynomial interpolation over triangular subregions was used by Courant to research torsion problems. The next important step in the utilization of finite element methods was taken by Boeing in the 1950s by using triangular stress elements to model airplane wings. In 1960, Clough made the term “finite element” popular, and during the 1960s, investigators began to apply the finite element method to other areas of engineering, such as heat transfer and seepage flow problems. The first book about the finite element method was written in 1967 by Zienkiewicz and Cheung [18].In the open literature, there are many studies for electric field stress control in high-voltage underground cables [7, 19–22]. Yang et al. [12] conducted comparative analyses on the evaluation of contact resistance of simulation-based cable joints. Letvenuk et al. [23] examined the air gap effect for medium voltage cable accessories. They proposed a new material-based approach to control stress levels. Frobin et al. [24] presented a Comsol Multiphysics software-based general approach for the modelling of HVDC cable accessories. Hyperelastic material-based cable accessories were designed by Luo et al. [25]. Electric field distributions of stress cones were analysed using a material with nonlinear conductivity [26]. The electric field distribution inside the cable accessories can be improved by adjusting the curvature and axial length of the stress cone [27, 28]. Although there are empirical approaches to optimum design in the studies, it is seen that an AI-based general approach is not presented.The most important component of the joints the stress cone is used for the reduction of the maximum electric field stress, which is concentrated at the critical point where the screen ends during cable joining. This paper aims to compare and analyse the stress cone effect according to two different cases which are with stress cone and without stress cone structure firstly. Afterwards, an artificial neural network model was created according to the simulation results. Thus, the optimum stress cone was designed using this model. All analyses were performed by a simulation-based method for the 500 kV high-voltage accessories.The paper is organized as follows. The design of the EHV joint based on a neural network is presented in Section2. The geometry of the proposed model is given in Section 3. All analysis results are presented and discussed in Section 4. The conclusion is given in Section 5.


## 2. Neural Network-Based Optimal Design of EHV Joint

Insulating materials in the electric field are forced by the effect of this field. Electrical stress may cause surface discharge or breakdown of the insulator. To minimize the risk of superficial discharge or puncture and to get the most efficient result from electrode systems, the insulating material should be forced equally everywhere, and a uniform electric field distribution on the electrode surface should be provided.The electrical stress in critical areas of high-voltage cable joints has to be reduced at the design stage. Partial discharge (PD) can occur due to high electric field intensity in different critical areas of the cable joint. After the partial PD starts, treeing occurs, and then, the structure causing more PD appears. This will cause an acceleration in the aging procedure. Design failures, installation, and manufacturing process-related defects can cause PD to occur in the cable connection. Under 132 kV and above the HV level of joints, PD is very crucial at the design stage [19].In this study, the focus is on reducing the electric field in critical areas by using a stress cone. If the electric potential is known at every point in a region of space, the electric field can be derived from the potential. The electric field intensity is obtained from the negative gradient scalar potential. The relationship equation of electric fieldE⇀ and potential function V is given as follows:(1)E⇀=−∇V.The tangential component of the electrical field is continuous across the surfaces separating two different materials. Hence, the potential functionV is continuous everywhere in the domain. From the Ampere–Maxwell equation,(2)∇⋅J⟶+∂∇⋅D⟶∂t=0,where the conduction current density represents as J⇀ and D⇀ is the electric displacement.We assume the following as initial conditions:(3)∇⋅D⇀=0att=0.That is, att = 0, there is no free charge in the system. The electric displacement in linear isotropic materials is simply involved in the electric field as follows:(4)D⇀=εE⇀,where ε is the permittivity measured in farads per meter (F/m). The coefficient ε can be written as(5)ε=ε0εr,where ε0 (F/m) is the permittivity of free space and εr is the relative permittivity of the medium. The current density and the electric field are related to a complex relationship in nonlinear materials.(6)J⇀=σEE⇀,where a field-dependent conductivity represents as σE. In the nonconducting regions, such as in all regular points, from equations (l), (3), and (4), it follows that the Laplace equation for the potential holds(7)∇⋅−ε0εr∇V=0.In the nonlinear regions, that is, in all regular points, from equations (l)–(4) and (6), it follows that the dynamics of the electric potential is described by a nonlinear diffusion like equation [29].(8)∇⋅−σE⟶∇V=∂∂t∇⋅ε0εrE⟶∇V.By applying the backward Euler method, the problem is discretized, and (8) becomes the following equation:(9)∇⋅−σE⟶+ε0ε1Δt∇Vt=−ε0εr∇2Vt−ΔtΔt,where t is the time, and it is a suitable small time interval [30].A cross-sectional view of a typical 500 kV cable joint is shown in Figure2.Figure 2
Critical areas of the HV cable joint.The peak stress at the cable outer semicone cutting point can be controlled by the optimal stress cone curvature. (a) On the other hand, the peak stress inside the cable joint body can be regulated by the cage tip curvature, (b) which is one of the most important design parameters. The surface stress and surface flashover are affected by the distance between the cage and the stress cone. (c) The insulation thickness (d) between the joint cage and the outer ground layer needs to withstand the highest average stress. This area is similar to a parallel plate configuration. From the electrostatic model, it is clear that the most crucial path will be near the joint cage tip. The axial and radial electric fields can generally be calculated from equations (9) and (10) [20].(10)Eaxiall,r=−Un⋅Cn2πε1lr∂r∂l,Eradiall,r=Un⋅Cn2πε⋅1lr,where the length of each conductive layer is l, r is the distance between the conductive layer and screen cut, ε is the permittivity of the surrounding insulating material, and Un is the voltage between each layer.The electric field strengthE of each layer depends on the voltage U, length l, radial distance r, and permittivity ε as shown in the following equation:(11)El,r=Urnlnεrn∑n=1nlnrn+1/rn/lnεn.The electric field is one of the most important factors for the optimum sizing of EHV joints. Methods in which the electrode shape is iterated to achieve the desired area distribution [31–35] are not effective for many problems. Since the area must be calculated in every iteration, the calculation time is quite long. Besides being time-consuming, these methods cannot always provide a general solution.In this paper, such an optimization problem is solved using a multilayer feed-forward neural network algorithm (Figure3). The neural network is trained to obtain a stress cone profile that will provide the desired area distribution on the joint surface. The analysis data are divided into a 70 : 30 ratio after normalization. 70% of data are employed for training the model, and the remaining 30% data are utilized for testing the performance of the model. The proposed neural network consists of three hidden layers. In this study, 10 neurons are used in hidden layers. Also, the Levenberg-Marquardt learning function is used for training. All results of the neural network model are given in Section 4 detailed.Figure 3
The flowchart of neural network structure.Neural network-based models are run separately 100 times, and the results are presented statistically to eliminate the errors that may arise from the randomness of the model parameters. The performance of model is evaluated by considering the deviation of the predicted value from the target value. For this purpose, the root mean square error (RMSE), mean square error (MSE), and mean absolute error (MAE) are used. The performance metrics are mathematically represented in (12) through (14) as follows:(12)RMSE=∑i=1Nyi−y˜i2N,(13)MSE=∑i=1Nyi−y˜i2N,(14)MAE=∑i=1Nyi−y˜iN,where yi and yi˜ represent the observed value and predictive value of the model and N is the total number of data used for performance evaluation and comparison.


## 3. Geometry of the Proposed Model

The geometries of the high-voltage underground cable accessories have been realized in the Comsol Multiphysics software program, and the electric field analyses have been conducted based on the finite element method (FEM). For electric field simulations using FEM, COMSOL Multiphysics software is a complete platform for simulating variations in the power cables. For the FEM analysis, an object is divided into equal parts, and based on the equations, each element’s behavior is forecasted. Thanks to this method, all individual behaviors are analysed and added, thus obtaining the object’s total behavior resulting from the sum of each element. With the sum of the elements, a mesh is obtained, which can have variations that result in greater accurate and computational cost, depending on the configuration used. There are several ways to determine each element, which are subdivided for mathematical evaluation. In this paper, the triangular shapes on the boundary are used for mesh. The number of finite element and boundary element is designed as 53066 and 2083, respectively. The computational time of the analyses is three seconds. The operations that will be applied when a problem is solved with a finite element package software COMSOL are given as follows:(i)
The problem is geometrically plot(ii)
Materials and properties of the region are defined for the geometry(iii)
Boundary conditions for the problem are identified(iv)
The geometry is divided into smaller subregions, and equations for these subregions are defined according to the boundary conditions by the software(v)
The mathematical model is solved by the package software by considering all the stated conditionsStress cone design in joint geometry for 500 kV high-voltage underground cables analysed in Comsol Multiphysics program. The design of high-voltage underground cable accessories has been examined for different stress cone geometries and without stress cones. In addition, different conditions for joint design have been analysed comparatively. The joint used in the simulation is an ethylene propylene diene monomer (EPDM) filled flat connection for 500 kV crosslinked polyethylene (XLPE) cables. Figure4 shows the joint geometry and part of the joint.Figure 4
The cable joint geometry. (1) Stress cone; (2) middle electrode; (3) cable insulation (XLPE); (4) joint insulation (EPDM); (5) connector; (6) conductor.Information about the cable used in the simulation is given in Table1.Table 1
500 kV HV cable information.
Cable dimensionsSize (mm2)2500Diameter (mm)63,5Thickness of insulation (mm)30Thickness of conductive screen (mm)1,6The properties of the cable insulation system consisting of insulation materials used in the joint design are presented in Table2.Table 2
Material parameters for 500 kV EHV cable joint [26].
MaterialRelative permittivityConductivity (S/m)Cable core15.7 × 107Semiconductive stress cone5010XLPE2.310–15Nonlinear material4.3σ = f(E)EPDM2.610–15Semiconducting middle electrode5010These materials have been selected to keep the maximum electric field stress under control. The AC voltage test specification states that the cable joints should be tested at 2.5 U0 for 15 minutes with no breakdown. In this paper, 1550 kV impulse voltage has been applied in the joint. The 3D voltage profile of the proposed model is given in Figure5.Figure 5
3D view of the cable joint and voltage profile.


## 4. Simulation Results and Effect of Stress Cone Design

During cable joint production, stress control is concentrated in areas where the cable screens are peeled off. Design is important for these regions. The contour plot of the voltage profile in the joint is given in Figure6. Electric field stress control is carried out by means of the middle electrode in the connector area and the stress cone.Figure 6
Voltage profile of the cable joint.The joint designs without stress cones are shown in Figure7. Here, the critical regions must be determined from the electric field distribution in the direction of the z component. The electric field distribution in the region indicated by the red line is given in Figures 7 and 8.Figure 7
Critical region of the joint without stress cone. (a) Mesh grid. (b) Electric field density.
(a)(b)Figure 8
Electric fieldz component of the joint without stress cone.It is observed that the electric field value reaches approximately 34 kV/mm in critical regions. Reducing this value in critical areas is essential for component safety. In this case, along with the deformation of the insulation, failure will occur as a result of partial discharge and breakdown. Even if increasing the size of the insulation material allows the electric field to be reduced in these areas, this is not an effective approach. Therefore, an optimal stress cone design can provide stress control in this region. The joint structure with stress cone analysed is shown in Figure9.Figure 9
Critical region of the joint with stress cone. (a) Electric field density. (b) Mesh of the cable joint.
(a)(b)Using the stress cone, the highest stress can be seen to be approximately 21 kV/mm at the critical region of the joint (Figure10).Figure 10
Electric fieldz component of the joint with stress cone.Stress cone is the most important component of the joint, which reduces the maximum electric field stress that condenses at the critical point where the screen ends during cable jointing, thus preventing partial discharge and insulation problems. In the analyses, the electric field value in the critical region has been reduced using the neural network-based optimal stress cone design. Additionally, the effect of joint geometry on the tangential electric field strength in material interfaces is shown in the analysis.While determining the optimum stress cone design, the effect of the distance between the stress cone and the middle electrode, which is one of the most important parameters, on the electric field distribution in critical regions was investigated (Table3). The variation of the electric field values (in different regions as shown in Figure 11) according to the distance between the middle electrode and the stress cone was examined.Table 3
Electric field changes according to different geometries.
Joint numberStress cone-middle electrode (cm)Electric fields (kV/mm)ABCDz = 0EFGr = 31 cmr = 67 cmr = 91 cm1249.35523.46612.47019.98530.11435.6930.2092229.34723.44312.50419.99830.09235.7050.2113209.33523.41012.49320.00830.08735.7240.2104189.33323.36212.63920.02830.07935.7510.2115169.32123.30412.56220.06030.06935.7880.2146149.29923.22612.59720.10230.07335.8290.2147129.27723.11112.72720.15930.05335.9000.2178109.24522.95112.74020.24330.02636.0040.221989.22722.71512.90520.35629.96836.1540.2261069.16522.37512.93820.52929.92436.3740.2341149.08321.81812.95020.81929.80836.7310.2471229.62420.76513.14821.33629.64337.3780.272Figure 11
The joint regions used in the data set for the neural network model.In Figure11, A is the outer stress cone, curvature, side electrode, or semiconductive deflector. B shows the distance between the stress cone and the middle electrode. Middle electrode, EHV deflector, Faraday cage, or inner stress cone are presented by C. Stress cone and cable insulation interface are indicated with D. As it can be seen from Table 3, the electric fields in different regions are determined by increasing the electrode length by 2 cm intervals for each design scenario. While the electric field in different regions constitutes the input vector of artificial neural networks, the output vector is the distance between the stress cone and the middle electrode.The neural network-based model was run 100 times, and the best, worst, average, and standard deviation (Std) values of the error performance criteria are shown in Table4. After the training and testing are completed, the artificial neural network gives the optimal distance to the desired electric field values.Table 4
Comparison of model training and test results with 100 independent runs.
TrainingTestRMSEMSEMAERMSEMSEMAEBest1.57e−032.46e−061.07e−031.41e−021.98e−041.12e−02Worst8.09e−026.55e−036.68e−022.14e−014.58e−022.13e−01Mean1.48e−024.20e−041.10e−021.07e−011.31e−029.25e−02Std.1.42e−029.58e−041.24e−024.09e−029.27e−033.71e−02Considering the permissible electric field levels in critical regions for a 500 kV joint, the joint number 5 is obtained in the model created in the neural network. The created model will contribute to the decision-making at different voltage levels.Silicone rubber semiconductor and conductive components designed according to the geometric stress control method have been designed with iterative steps until all desired ambient conditions. It has been predefined to achieve the desired final accessory design using the neural network model.Electric field distribution was obtained with the optimal model blue line (joint number 5) and joint number 12; that is, the green line for different regions is also presented in Figure12.Figure 12
The joint regions used in the data set for the neural network model.The electric field distribution on the EPDM surface is shown in Figure13 for both scenarios (joint numbers 5 and 12).Figure 13
Electric field variations in different regions for joint numbers-5 and 12.It is seen that the electric field distribution in Figure12 for the optimal model (joint number 5) that was found by the neural network model in the region where EPDM is located remains within the allowable limits as stated in the literature [36].


## 5. Conclusion

The optimal joint design has become increasingly important with the widespread use of cables at extra high-voltage levels. One of the most important components in the joint design is the creation of an optimal stress cone structure. Especially, determining and decreasing the area of the highest electric field on the joint is an important factor in terms of partial discharges.In this paper, the electric field stress created by the high-voltage level at critical points has been kept under control with the appropriate joint design via simulation and neural network-based optimization. The use of stress cone configuration reduces the maximum electric field strength inside the EHV joint. The use of the stress cone has been successful in controlling the maximum electric field in the joint design. However, the distance of the stress cone to the middle electrode is an important problem in the design. A neural network-based model is used to solve this problem. In this way, the electric fields in different critical regions are optimized with the neural network-based stress cone design. Model performance with 12 different scenarios is shown comparatively in this paper. In the results obtained, choosing the distance between the stress cone and the middle electrode as 16 cm is the most appropriate value for a 500 kV joint.This initial phase of the study will continue with the experimental verification of the optimal design.



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