Analytical and numerical capacitance values.
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.
- electrical insulation
- power quality monitoring
- capacitive divider
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 . 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 . 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  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 , 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 C1 and C2 (F) provides the voltage measurement. The capacitor C1 is a commercial capacitor used in electronic applications and is placed in an electronic board of the sensor. C2 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 C2 capacitor. The capacitive divider is designed in such way that C2 retains practically the totally line voltage. The voltage on C1 measured by the sensor is defined by:
where (V) is the total voltage. The approximation in Eq. (1) is valid when .
The maximum voltage concentrated in C1 is around 5 V. The line voltage in MV can be 1 or 35 kV according . Examples of voltage classifications between these values are: 4.16, 12.47, 13.2, 13.8, 24.94 and 34.5 kV.
The C2 design must meet some important requirements, such as, adequate dielectric strength to support pulse voltage up to 100 kV , 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 . 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 C2 and allows the creation of an analytical model. The internal electrode is at line potential () and the external electrode is in the earth potential (0 V). The internal and external electrode radii are, respectively, (mm) and (mm), the electrical permittivity of the medium is and the cylinder length is (mm). It should be noted that there is no region of electric flux dispersion through the upper horizontal line due to the presence of a guard’s electrode not shown in the figure. This figure also shows an equipotential line defined by the distance whose origin is the longitudinal axis of the cylinders. The equipotential line has a value exact in the region between the cylinders because the electric field in this region is constant and uniform while the equipotential in the hemispheres region has a value approximated because the electric field is non-uniform. As shown in , the equipotential surface tends toward a hemisphere and by heuristic approach this is a good approximation.
The analytical study demonstrates the optimum relation between the radii of the cylinders. The electric field is radial and is given by :
where (C) is the charge on the cylindrical part of the central electrode. The integral of the electric field between the electrodes provides the voltage between them, i.e.:
where, substituting to , being , it is evident that the maximum value of the electric field occurs on the surface of the internal electrode. This maximum field value is given by :
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 and . The first case refers to an extremely fine internal electrode, where the electric field on its surface is extremely high. The second case tends to an infinitesimal distance between the electrodes, which has to withstand voltage, . Therefore, it is clear that there is at least one minimum region between these two conditions, which is the optimal condition for the design of the sensor. Given , the minimum value of can be obtained by differentiating the maximum field with respect to and making this result equal to zero:
what provides the optimal condition:
where is the basis of natural logarithms. Therefore, the dimensioning of the electrodes should be such that the ratio of the radii of the internal and external electrode obeys the Eq. (7). Taking the condition of Eq. (7) into Eq. (5) it is obtained:
Figure 3 shows the behavior of the maximum electric field for an external electrode of of radius and different radii for the internal electrode when a voltage of 100 kV is applied between the electrodes. It is observed that the minimum region is obtained by Eq. (7), where . It is also noted in Figure 3 that the electric field variation around the minimum value is very small, such as small variations around the optimum relation expressed in Eq. (7) do not compromise the design. It should be noted that the critical region for the rupture of the dielectric occurs in the hemispheric region, however this region can be elongated, increase the radius of the center electrode, tending to a revolution ellipsoid in an attempt to reduce the electric field at the electrode surface. These variations could be evaluated through COMSOL Multiphysics software.
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 (C) is the charge on the hemispheric part of the central electrode. The integral of the electric field between the electrodes provides the voltage between them:
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 and making this result equal to zero, the minimum of this function occurs for the condition:
Figure 4 shows the electric field on the surfaces of the cylinder and the hemisphere, for different values of the ratio. It is observed that the minimum field of the hemisphere occurs for as showed by Eq. (13). The modification of the cylinder termination, passing from a flat half ellipsoid of revolution to a hemisphere and to a stretched half ellipsoid of revolution, aims to optimize the electric field on this region considering the cylindrical region as a reference. This can be done taken into account Equation 7. The dimensions of the ellipsoid could be obtained through FEM developed in the COMSOL Multiphysics software.
The capacitance between the electrodes can be easily calculated by integrating the electric flux along the spatial surface defined in Figure 3 resulting in:
This can also be expressed as:
where is the relative permittivity of the medium and the capacitance is obtained in pF for and in millimeters. For example, considering (acetal, dielectric strength >50 kV/mm, DIN 53481 ), , and (), is obtained.
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 , is the relative permittivity, (kV/mm) is the electric field and (C/m3) is the free charge density. The relation between electric field and the voltage is defined as:
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 as:
with units of V2/m2 for a one-dimensional domain, for example.
The function gives the voltage distribution that satisfies the governing partial equation when differentiated with respected to and equaled to zero. Thus, the change in the global function due to the change in the voltage of node
where the summation represents the contribution from all elements associated with or all the elements connected to node
where is the stiffness matrix whose terms are defined by geometric parameters of the nodes and material properties. is a column matrix of voltages, which are the unknowns to be solved. Additionally, the boundary values of voltage associated to some of the nodes are applied to this matrix. After the determination of the voltage distribution, the electric field can be obtained by numerically solving Eq. (18).
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 (inner radius) and (outer radius) in both cylindrical and hemispheric regions. Additionally, voltage boundary conditions were the same as the ones used for the analytical model. Electric field values were extracted along paths P1-P2 and P3-P4, 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 (%)|
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 () and setting the horizontal semi-axis () constant, the geometry of the cylinder termination shall change from a flat geometry up to a stretched geometry between points P5 (region next to transition from cylindrical region to the elliptic region) and P6 (region at the tip of centre electrode).
From the simulation of the electric field for different values of semi-axis , it is verified that for small values of semi-axis , the electric field is more concentrated on the edges and assumes large values in the transition region from cylinder region to elliptic region presenting irregular distribution, as shown in Figure 11. Similarly, an irregular distribution is observed for large values of semi-axis , but with more concentration of electric field at the tip of the central electrode. On the other hand, for values of semi-axis near the hemispheric geometry of radius , a better electric field distribution for the entire surface is observed with a minimum concentration. This result shows that the investigation for the optimum condition should be concentrated around the hemispheric geometry.
To do so, a normalized path between points P5′ 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 P5-P6, 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 P5′ while for the stretched geometry the concentration is observed at the tip (point P6′), as expected. Additionally, the hemispheric geometry produces a small electric field norm variation along the trajectory between points P5′ and P6′. 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 of the central electrode. Eleven cases were simulated with a step of 0.2 mm being five cases below 18.5 mm (tip flattening) and five cases above 18.5 mm (tip stretching), as shown in Figure 13. It is important to note the effect of geometric modification on electric field norm along normalized path P5′-P6′. As an example, the response curve for the flattening geometry has bigger values of electric field close to the point P5′ and lower values close to the point P6′. 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 P5′-P6′ 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 for this optimum condition is obtained through interpolation, leading to an optimum
A new parametric study was performed in order to confirm the best condition found. Three cases around the optimum semi-axis of 17.5 mm were simulated with a smaller step of 0.1 mm. Figure 15 shows these results indicating that the solution for is indeed the optimum solution since it ensures the lowest electric field norm when the whole normalized path P5′-P6′ 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 P5′-P6′ 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 were computed. The relative difference metric was calculated for the maximum value of each electric field norm distribution evidencing the optimum condition of 17.5 mm, as expected.
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 C2 capacitance variation for a wide-range of values of semi-axis covering the geometries presented in Figure 11. It is possible to see that the value of C2 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.
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 equals 17.5 mm.
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.
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.
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