Analytical and numerical capacitance values.

## Abstract

Thanks to the Smart Grid initiative, the focus for medium-voltage MV (13.8–34 kV) smart meters leveraged the development of sensors for distribution application. In order to be useful at power quality monitoring, the sensors needs to attend, at least, the International Electrotechnical Commission (IEC) 61000–4-30 and IEC 61000–4-7 standards with high-accuracy in terms of voltage (less than 0.1%), current (less than 1.0%) and measuring the waveform distortion data up to the 50th harmonic of 50 or 60 Hz alternating frequency. This kind of sensor is built with two capacitors connected in series. The first capacitor is a commercial electronic low-voltage device. One terminal of this capacitor is connected to the medium-voltage (MV) conductor. The second one, is connected to the other capacitor that is constructed using the own sensor packaging. This second capacitor has an electrode, that is connected with the first capacitor and the other terminal is connected to the ground. The voltage is measured between the terminals of the low voltage capacitor. The performance of this capacitor depends on the geometry and the materials used in the electrical insulation. This chapter describes the simulations and modeling of the capacitor electrodes using a finite-elements software, COMSOL Multiphysics, for modeling in order to optimize the performance of sensor in terms of electric field distribution.

### Keywords

- simulation
- electrical insulation
- sensors
- power quality monitoring
- medium-voltage
- capacitive divider

## 1. Introduction

One of the fundaments of the Smart Grid concept is that user safety should be ensured while monitoring, updating and continuously reliably distributing electricity grid by adding smart meters and monitoring systems to the power grid is obtained. This is necessary in order to ensure electronic communication between suppliers and consumers [1]. Smart grid monitoring systems require various types of sensors and transducers to monitor the grid conditions. After the Smart Grid initiative, the focus for medium-voltage (MV) smart meters leveraged the development of energy quality sensors for distribution application [2]. In order to be useful at power quality monitoring, the sensors needs to attend at least the International Electrotechnical Commission (IEC) 61000–4-30 and IEC 61000–4-7 standards [3, 4] with high-accuracy in terms of voltage (less than 0.1%), current (less than 1.0%) and measure waveform distortion data up to the 50th harmonic. In addition, this sensor must be easy to install and remove without disconnect the distribution network and it must monitor the grid for a period that may last longer than 1 week.

In this context, two power quality-monitoring technologies are prominent: wireless sensors [5] or optical fiber sensors [6, 7]. Wireless sensors have the advantage of not needing any physical medium to transmit the data to a remote measuring unit, but need to use batteries in order to keep the electronic circuits working. On the other hand, optical fiber sensors have the advantage of no need for electrical powering, but they need a physical link to the remote measuring unit.

Independently of the technologic choice, current and voltage waves have to be measured in the medium-voltage (MV) in order to obtain the power quality parameters. Particularly, for voltage measurements, a capacitive or a resistive circuit divider [8, 9, 10] can be used to obtain a voltage sample of the MV conductor. In this work, it is analyzed the capacitive case. The low capacitance of this circuit accumulates more than 99.9% of the total voltage (e.g. 13.8 or 34 kV) and is totally constructed using the own sensor packaging. In this capacitive circuit the first capacitor is a commercial electronic low-voltage (LV) device. One terminal of this capacitor is connected to the medium-voltage conductor. The second one, is connected to the other capacitor that is constructed using the own sensor packaging. This second capacitor has an electrode, that is connected with the first capacitor and the other terminal is connected to the ground.

A rigorous design is necessary for these sensors considering the safety aspects regarding to the technician activities and to some environmental effects that can influence their performance, such as, temperature, pressure and wind. Besides, it should be taken into account that the external elements in the proximities of the sensors can alter the electrical and magnetic field acting inside of them.

The current sensor for power quality measurements was not evaluated in this work since its operation is different from the voltage sensor and in general it does not affect the insulation properties of the power quality sensor. Traditional devices used as current sensor are current transformer or Rogowski coil [11], and they are connected direct to the MV conductor without ground connection.

This chapter is organized as follows. Section 2 describes the physical structure of the sensor studied in this work. Section 3 describes the analytical modeling of the sensor. Section 4 describes the finite-element method (FEM) review mainly focused in simulation of an electrostatic field and the sensor. In addition in this section it is presented the sensor simulation validation. Finally, Section 5 presents the conclusions.

## 2. Voltage sensor for power quality applications

The proposed case study was the modeling of a current and voltage sensor for MV applications to be applied in live lines for the evaluation of energy quality.

According to Figure 1, a capacitive divider represented by C_{1} and C_{2} (F) provides the voltage measurement. The capacitor C_{1} is a commercial capacitor used in electronic applications and is placed in an electronic board of the sensor. C_{2} is a capacitor formed by an electrode and a grounded pipe isolated by an insulation material (such as polymer, ceramic, glass or oil-impregnated paper) where a high-intensity electric field remains concentrated. To meet accuracy of less than 0.5% in the voltage measurements, it is necessary to connect the LV terminal of the sensor to the ground. Thus, it is not possible to use the parasitic capacitance as the C_{2} capacitor. The capacitive divider is designed in such way that C_{2} retains practically the totally line voltage. The voltage on C_{1} measured by the sensor is defined by:

where

The maximum voltage concentrated in C_{1} is around 5 V. The line voltage in MV can be 1 or 35 kV according [12]. Examples of voltage classifications between these values are: 4.16, 12.47, 13.2, 13.8, 24.94 and 34.5 kV.

The C_{2} design must meet some important requirements, such as, adequate dielectric strength to support pulse voltage up to 100 kV [13], homogeneous field electric around the electrode and absence of air bubbles near the electrode interface in order to reduce the growing of partial discharges in the sensor [14]. The growing of partial discharges in the insulation due to electric field concentration in a specific place and over time causes premature aging and breakdown of insulation systems [15, 16, 17, 18, 19].

The dimensions of the sensor and characteristics of the insulation material will be described in Section 3.

## 3. Capacitive analytical modeling for power quality sensor

The basic reference structure adopted for the voltage sensor is composed of two coaxial cylinders terminated in hemispheres, as shown in Figure 2. This type of geometry simplifies the practical construction of C_{2} and allows the creation of an analytical model. The internal electrode is at line potential (

The analytical study demonstrates the optimum relation between the radii of the cylinders. The electric field is radial and is given by [21]:

where

Substituting

where, substituting

It is emphasized that the maximum value of the electric field is of interest in the design of the sensor, since it determines the beginning of the insulation rupture process through the dielectric. An analysis of Eq. (5) shows that this maximum field tends to infinity for the boundary conditions

what provides the optimal condition:

where

Figure 3 shows the behavior of the maximum electric field for an external electrode of

As discussed earlier, the hemispheric region may be modified in order to attenuate the electric field at the tip surface of the central electrode. However, this makes analytical modeling difficult, and therefore the numerical calculations become necessary. For the purpose of analytical calculations, the hemispheric configuration for the electrodes (internal and external) is assumed, according to Figure 2. In the region between the hemispheres, the electric field is radial and could be described as follows:

where

Substituting

In a similar way to the case of the cylinder, the maximum field occurs on the surface of the electrode:

Differentiating Eq. (6) with respect to

Figure 4 shows the electric field on the surfaces of the cylinder and the hemisphere, for different values of the

The capacitance between the electrodes can be easily calculated by integrating the electric flux along the spatial surface defined in Figure 3 resulting in:

By inserting the condition of Eq. (7) into Eq. (14) one obtains:

This can also be expressed as:

where

It is observed that the minimum field of the hemisphere occurs for

## 4. FEM review and simulation validation

FEM is based on the solution of a boundary value problem composed of a governing equation and boundary conditions. The main idea behind this method is the division of the domain of interest in subdomains known as elements and the adoption of shape functions for the unknown variables, which are only solved for the nodes (element corners). Thus, instead of solving an analytical equation, these unknowns are determined by a set of algebraic equations and the results for regions other than the nodes can be obtained by interpolation. Proper domain discretization is crucial to ensure the accuracy of results since the mesh format must adequately reproduce the original geometry of the structure. An example of domain discretization can be seen in Figure 5.

This procedure reduces the generality of the mathematical framework, but enables the study of components of complex geometry. Many real world study cases involve the analysis of such problems, which are virtually impossible to be done by analytical methods.

A succinct overview of FEM for the simulation of an electrostatic field is presented next considering a stationary solving method since the voltage boundary condition chosen has a constant value. A more detailed explanation can be found in [24, 25, 26, 27, 28]. The voltage distribution in a dielectric of arbitrary geometry is described by the following differential equation:

where ^{3}) is the free charge density. The relation between electric field and the voltage

Substituting Eq. (18) into Eq. (17) and considering a homogenous dielectric with

Eq. (19) has to be transformed into an energy functional form that relates directly to the energy of the system in order to be used in FEM. This function can be written for an element

with units of V^{2}/m^{2} for a one-dimensional domain, for example.

The function gives the voltage distribution that satisfies the governing partial equation when differentiated with respected to *i* is:

where the summation represents the contribution from all elements associated with *i*. These derivatives are equaled to zero resulting in a group of simultaneous equations arranged in matrix form as:

where

As it can be noticed, the assumptions made in the original analytical problem for the FEM strongly affect the results. Thereby, the results provided by the simulation have to be validated to ensure their accuracy. To do so, these results are compared to the ones of an analytical model derived for specific conditions as previously presented. Analytical results of electric field from Eqs. (4) and (11) are taken for, respectively, the cylindrical and hemispheric regions as references for the simulation whose parameters of interest are the order of the shape functions and the mesh refinement.

It is worth to mention that the dielectric geometry has a longitudinal axis-symmetry, which implies that the unknown values do not change along the azimuthal axis and thus only a transversal plan needs to be modeled in FEM and the rest of the solution can be extrapolated. Figure 6 shows the structure modeled around its axis of symmetry for _{1}-P_{2} and P_{3}-P_{4}, respectively, in the cylindrical and hemispheric regions.

As a default, COMSOL uses a second order shape function in order to improve the results’ accuracy. This choice seems to be adequate as the analytical results showed that the radial variation of the electric field fallows a quadratic pattern. However, a mesh convergence study is still needed in order to minimize domain discretization effects on the results. In this procedure, the mesh is successively refined and the values of interest are compared to a reference. Figure 7 shows the meshes evaluated.

Figures 8 and 9 show the comparison of analytical results given by Eqs. (4) and (11) with numerical results for three mesh refinements using quadratic shape functions, respectively, for the cylindrical and hemispheric regions. The electric field results obtained by the simulations presented good agreement with the analytical results indicating that quadratic shape functions are indeed a good choice. In addition, it can be seen that mesh refinement led to a better solution field as the relative error computed using the analytical values as reference decreased. Besides, the duration of the simulations for the three cases did not increase significantly with the last case taking about 1 s. The computer used was a workstation with an Intel(R) Core™ i7–4790 3.60 GHz CPU and 16 GB of RAM.

Finally, another way to validate the simulations is to compute capacitance values from the simulations by numerical integration of the electrical flux along the inner surface in which 100 kV was applied. These values are compared with the analytical one described by Eq. (14) as shown in Table 1.

Numerical (pF) | Analytical (pF) | Difference (%) | |
---|---|---|---|

Coarse mesh | 16.812 | 0.500 | |

Normal mesh | 16.810 | 16.728 | 0.488 |

Fine mesh | 16.809 | 0.482 |

Since the model related to the fine mesh provided electric field distributions that best agreed with the analytical results in both cylindrical and hemispheric regions of the voltage sensor, and get a capacitance value that closest matched with Eq. (14), this model was successfully validated and will be used in the case study presented below.

## 5. A case study using COMSOL

As explained in Section 3, the modification of the cylinder termination of the central electrode, passing from a flat half ellipsoid of revolution to a stretched half ellipsoid of revolution, aims to optimize the electric field on this region. Therefore, the optimum geometry of the ellipsoid could be obtained through a parametric study, as shown in Figure 10. Varying only the vertical semi-axis (_{5} (region next to transition from cylindrical region to the elliptic region) and P_{6} (region at the tip of centre electrode).

From the simulation of the electric field for different values of semi-axis

To do so, a normalized path between points P_{5}′ and F′ is used in order to provide a better visualization for the distribution of electric field norm for different geometries of the central electrode, as shown in Figure 10. As an illustration of this procedure, considering two distinct geometries having different lengths and shapes of original paths P_{5}-P_{6}, such as the flat and stretched cases of Figure 11, their electric filed can be directly compared thought the path normalization proposed. This is performed in Figure 12 where the flat geometry shows an electric field concentration close to point P_{5}′ while for the stretched geometry the concentration is observed at the tip (point P_{6}′), as expected. Additionally, the hemispheric geometry produces a small electric field norm variation along the trajectory between points P_{5}′ and P_{6}′. This emphasizes that the best geometry for the central electrode in terms of electric field distribution must be close to the hemispheric geometry.

Therefore, this work has been focused on parametric studies around the hemispheric geometry with variations in the value of the semi-axis _{5}′-P_{6}′. As an example, the response curve for the flattening geometry has bigger values of electric field close to the point P_{5}′ and lower values close to the point P_{6}′. Also, these curves indicate the existence of a case whose maximum electric field value along E′-F′ path is the smaller one among the other curves, making it a candidate for the optimum condition.

Figure 14 presents the maximum electric field norm along the normalized path P_{5}′-P_{6}′ for each case presented in Figure 13. The existence of an optimum condition is evidenced by the trend in the results provided by the simulated cases. The exact value of the semi-axis *value* of 17.5 mm.

A new parametric study was performed in order to confirm the best condition found. Three cases around the optimum semi-axis _{5}′-P_{6}′ is considered. Another form of analysis is to compute a relative percent difference taking the electric filed norm from the cylindrical region, which is constant for any modifications in the geometry of the hemispheric region, as a reference (5.447 kV/mm). As it can be seen, again the optimum condition curve presents smaller values than the other curves when the whole normalized path P_{5}′-P_{6}′ is considered.

This conclusion is supported by an extension of the last parametric study, as shown in the Figure 16, where 21 different values around 17.5 mm with a step of 0.1 mm for the semi-axis

Another conclusion is that, for this optimum geometry, a reduction of approximately 1.5% in the maximum electric field is achieved when compared to the hemispheric geometry. This improvement is strategic to ensure the safe operation of the sensor since partial discharge is a localized phenomenon influenced by electric field concentration. In a real scenario, because of imperfections in the dielectric due to the manufacturing process, such as air bubbles, a lightning impulse of 100 kV can lead to local discharges in points of high-electric field concentration, which will cause the failure of the sensor.

Finally, Figure 17 shows the C_{2} capacitance variation for a wide-range of values of semi-axis _{2} for the optimum condition is very close to the ones shown in Table 1 since the geometry of the optimum condition is very close to a hemisphere. In addition, the range of capacitance values obtained is compatible to the design of the capacitive divider.

## 6. Conclusion

A rigorous capacitor design is necessary for MV sensors when capacitive divider is used to obtain a voltage sample. This is due to safety aspects regarding to the technician activities and some environmental effects that can influence their performance, such as temperature, pressure and wind. In addition, external elements in the vicinity of the sensor can alter the electric field acting inside of it.

The basic structure adopted for the voltage sensor is composed of two-coaxial cylinders terminated in a hemisphere, which simplifies the practical construction of the capacitor. Although, this work demonstrated that the geometry of the electrode termination should be different of a hemisphere in order to minimize the electric field distribution in this region.

This conclusion was based in finite-element studies developed in COMSOL software. The first one, considered a hemispheric electrode termination of 18.4 mm of radius and was compared to an analytical model for validation purposes. Next, a parametric study was developed, in which the termination was changed from a flat to a stretched geometry, to obtain the optimum condition. The electric field distribution along the termination for this condition was compared to a reference value extracted from the cylindrical region (5.447 kV/mm). The result is that the optimum geometry is slightly flatter than a hemisphere having a semi-axis

Additionally, the electric field distributions of the optimum and the hemispheric geometries were compared evidencing a magnitude reduction of approximately 1.5%. This improvement is strategic to ensure the safe operation of the sensor since partial discharge is a localized phenomenon influenced by electric field concentration. In a real scenario, because of imperfections in the dielectric due to manufacturing process imprecision, such as air bubbles, a lightning impulse of 100 kV can lead to local discharges in points of high electric field concentration, which will cause the failure of the sensor.

## Acknowledgments

The authors wish to thank their colleague Celio Fonseca Barbosa.

This work was funded by FINEP (Brazilian Innovation Agency) grant number 0115002800 0407/14.

CNPq (National Counsel of Technological and Scientific Development) sponsors the author Joao B. Rosolem under scholarship DT.

## Conflict of interest

The authors declare no conflict of interest.