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A numerical simulation tool for multilayer grounding analysis integrated in an open-source CAD interface
Electrical Power and Energy Systems 45 (2013) 353–361
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A numerical simulation tool for multilayer grounding analysis integrated in an open-source CAD interface I. Colominas ⇑, J. Parı´s, D. Fernández, F. Navarrina, M. Casteleiro Group of Numerical Methods in Engineering, Department of Applied Mathematics, Civil Engineering School, University of La Coruña, Campus de Elviña, 15071 La Coruña, Spain
a r t i c l e
i n f o
Article history: Received 26 April 2011 Received in revised form 18 August 2012 Accepted 29 August 2012 Available online 29 October 2012 Keywords: Grounding Numerical analysis Convergence of numerical methods Computer aided analysis Computer graphics software Open-source platform
a b s t r a c t In this paper we present TOTBEM: a simulation tool for grounding systems based on the open-source platform SALOME. The package TOTBEM includes all the preprocessing, computing and postprocessing stages necessary to perform a complete earthing analysis. The kernel of TOTBEM is a numerical formulation based on the Boundary Element Method for uniform and stratified soil models proposed by the authors in the last years. Furthermore, in this work we show the main highlights of an efficient technique based on the Aitken d2-process in order to improve the rate of convergence of the involved series expansions in multilayer soil models. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Obtaining the distribution of potential levels of an earthing system has been one of the challenges of the electrical engineers and designers since the beginning of the large-scale harnessing of electricity. Thus, the grounded electrode dissipates the electrical currents generated during a fault condition in order to guarantee the safety of persons, to maintain the integrity and proper functioning of equipment and to assure the continuity of the electrical supply. Main parameters commonly used to characterize a grounding grid are the equivalent resistance of the system and the potential distribution on the earth surface. As general rules, the resistance should be low enough in order to produce the dissipation of the electrical current into the ground through the earthing electrode, while certain values of potential differences on the earth surface should be under some well-established safety limits [1,2]. Maxwell’s Electromagnetic Theory is the general framework for the statement of the mathematical model for the phenomenon of the electrical current dissipation into the ground. Notwithstanding the equations that govern this problem are known for a long time, the analysis of large earthing systems in practice presents some important complications: on the one hand, due to its specific ⇑ Corresponding author. E-mail address: [email protected] (I. Colominas). URLs: http://www.caminos.udc.es/gmni/ (J. Parı´s), http://www.caminos.udc.es/ gmni/ (D. Fernández), http://www.caminos.udc.es/gmni/ (F. Navarrina), http:// www.caminos.udc.es/gmni/ (M. Casteleiro). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.08.079
tridimensional geometry (a grid of bare conductors with a large ‘‘length/diameter’’ ratio), and on the other hand for the uncertainty in the electrical properties of the ground. These facts mean that the use of common and widespread numerical methods in engineering (FEM or FDM) is very costly (in computing time and memory storage) since it is required the discretization of the domain, i.e., the whole ground excluding the part occupied by the electrodes. Over recent decades a large number of methods to calculate and design these grounding systems has been proposed, generally based on the professional practice, on empirical work and on experimental data. Some of these techniques have resulted in computer methods for grounding analysis which have led to significant progress in this area [1,2]. However, some problems with the application of these methods in the analysis of real cases have been documented, for example, unrealistic results when the discretization level of the electrodes is risen, uncertainties in the margin of error, or high computational costs [3]. These topics were fully analyzed in the reference [4] where the authors explained from a rigorous point of view the anomalous asymptotic behavior of the classical grounding methods (identifying also the sources of error), taking as a starting point the numerical formulation for grounding analysis based on the Boundary Element Method proposed by the authors in the nineties [5]. This numerical approach is the framework from it is possible to develop high-accurate computer methods for grounding analysis in uniform and layered soil models [5–7]. Furthermore, in 2005, we propose a methodology also based in this BEM numerical formulation for the analysis of transferred earth potentials in earthing
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systems (the appearance of dangerous potential levels in metallic elements, structures or other conductors as a result of the energization of the grounding electrode during a fault condition [8]). The methodology was published in the reference [9] for uniform soil models, and in the reference [10] its extension to layered soil models. In this paper we present TOTBEM: a power system numerical simulation tool for grounding analysis. TOTBEM is based on the open-source software SALOME [11], and it allows to perform the preprocess of the model (geometry data of the grounding mesh, soil properties, etc.), the earthing grid analysis by using the Boundary Element approach developed by the authors, and the postprocess and visualization of results. The outline of the paper is as follows: after the introductory section, it is presented a summary of the fundamentals of the mathematical and numerical model for grounding analysis based on the BEM; Section 3 is devoted to present a technique to accelerate the convergence of the calculations recently developed by the authors [12] which has been implemented in TOTBEM; and next, we present and describe the TOTBEM platform and its functionality. Finally, main conclusions are stated. 2. Foundations of the mathematical model for the current dissipation problem through a grounding electrode Equations that govern the dissipation of the electrical current into a media through a grounded electrode are given by
divðrÞ ¼ 0;
r ¼ cgradðVÞ in E;
t
r nE ¼ 0 in CE ; V ¼ V C in C; V ! 0; if jxj ! 1
ð1Þ
As it can be seen, this model restricts the analysis to obtain the steady-state response; furthermore the potential is assumed constant on the conductors surface (so the internal resistivity of the electrodes is neglected). In (1), E denotes the earth, c its conductivity, CE its surface, nE its normal exterior unit field and C the surface of the electrodes of the grounding grid [5]. Solving this problem provides the current density r and the potential V at any point x when the grounded electrode is energized to a Ground Potential Rise (or GPR) VC with respect to remote earth. On the other hand, most safety parameters that characterize an earthing system should be obtained straight from V computed on CE and r on C [5,7]. An important point on the definition of the mathematical model for grounding analysis is the selection of the more appropriate soil model: Obviously, it is not possible to take into account all variations of the soil conductivity since it is not feasible or from an engineering point of view neither economic nor practical. The simplest model that has been proposed is the isotropic and homogeneous one, i.e., a ‘‘uniform soil model’’ (so an scalar conductivity c is introduced instead of conductivity tensor c [1,5]); the ‘‘layered models’’ represent the soil in a number of strata, each one defined by means of a scalar conductivity and thickness [1] (in practical grounding analysis, it should be enough to consider models with two or three strata to achieve accurate results). The set of Eq. (1) can be rewritten in terms of the next exterior problem if the soil is modeled by C layers with different scalar conductivities:
rc ¼ cc gradðV c Þ in Ec ; 1 6 c 6 C; rt1 nE ¼ 0 in CE ; V b ¼ V C in C; divðrc Þ ¼ 0;
V c ! 0 if jxj ! 1;
1 6 c 6 C;
rtc nc ¼ rtcþ1 nc in Cc ; 1 6 c 6 C 1;
ð2Þ
where b is the layer that contains the buried conductor, Ec is each stratum, cc is its conductivity, Vc is the potential at a point in layer
Ec, rc is its current density, Cc is the interface between Ec and Ec+1, and nc is the normal unit field to Cc [7]. In the following section of this paper we will present the improvement in the convergence for the case of two layered soil models (C = 2), although it is direct the extension of the analysis to other stratified models. On the other hand, by application of Green’s Identity and the ‘‘Method of Images’’ potential Vc(xc) at an arbitrary point xc 2 Ec can be expressed in an integral form [7] in terms of the leakage current density r(n) at any point n on the surface of the conductors C Eb (r = rtn, being n the normal exterior unit field to C):
V c ðxc Þ ¼
1
Z Z
4pcb
kbc ðxc ; nÞrðnÞdC;
8xc 2 Ec ;
ð3Þ
n2C
Furthermore, since most parameters that characterize an earthing system (mesh, touch, and contact voltages, for example) are obtained from the potential distribution computed on the earth surface CE, from now on we will consider c = 1 in the grounding analysis:
V 1 ðx1 Þ ¼
1 4pcb
Z Z
kb1 ðx1 ; nÞrðnÞdC;
8x1 2 CE ;
ð4Þ
n2C
Generally, kernel kb1(x1,n) is a series which terms correspond to the images introduced in the transformation of problem (2) to the integral form (3) [5,7,13]. The number of terms of these series can be finite, for example in the case of homogeneous and isotropic soil models (C = 1), or infinite, for example in two-layer soil models (C = 2). In the appendix section of the reference [10] can be found the explicit formulae of these integral kernels. It is well known the integral kernels are weakly singular and they depend on the distances from x1 to n and depend on the distances from x1 to all its images with respect to CE and with respect to Cc [1,14]. The kernels are also function of the thickness of the layer and the layer conductivities according to a ratio. In a twolayer soil model, this ratio j is:
j¼
c1 c2 c1 þ c2
ð5Þ
so the kernels should be expressed in the general form
kb1 ðx1 ; nÞ ¼
1 X ½n kb1 ðx1 ; nÞ; n¼0
½n
kb1 ðx1 ; nÞ ¼
wn ðjÞ ; rðx1 ; nn Þ
ð6Þ
r(x1, nn) represents the distance from x1 to images nn while the weighting coefficient wn(j) is a function that only depends on the ratio j and the thickness of the layer [7]. The substitution of the general form for the inner kernel (6) in (4) allows to express the computation of the potential at an arbitrary point x1 2 CE in terms of the contribution of each image as follows:
V 1 ðx1 Þ ¼
1 X ½n V 1 ðx1 Þ
ð7Þ
n¼0 ½n
and V 1 ðx1 Þ is the contribution in the potential computing due to the image n: ½n
V 1 ðx1 Þ ¼
1 4pcb
Z Z
½n
n2C
kb1 ðx1 ; nÞrðnÞdC:
ð8Þ
In the bibliography can be found some different computer methods for obtaining the potential contribution of each image given by (8) [1,2]. Specifically, the authors proposed a methodology based on the Boundary Element Method for the computational design of real earthing systems in layered soil models, which can be analyzed other related problems of industrial interest like the transferred earth potentials [4–7,9,10]. The rate of convergence of the different series that appear when the method of images is applied in the case of stratified soil models
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(e.g., (6) for the two-layer soil model) can be very low depending on the value of j, the ratio between the conductivities of the strata given by (5). This fact is particularly critical when jjj 1, i.e. in cases where there are significant differences between the conductivities of layers, which are commonly the most interesting ones in practical applications. On the other hand, it is important to remark that obtaining the distribution of potential levels on the earth surface is usually the most time-consuming process in earthing analysis since it may be necessary to compute potential values in an great number of points to rise high-accurate results: A good example of it can be found in the reference [7] in the study of the Santiago II substation grounding system where the total area analyzed is 78.000 m2 being necessary to compute potential values in near 100.000 points. So, it is required an accurate and low-cost computation of the potential: to achieve these goals the authors have developed a methodology to increase the convergence rate of the kernel series [12] which has been implemented in TOTBEM. In the following section, it is presented a summary of this technique. 3. Convergence improvement techniques for multiple layer soil models The development of this method is firstly based on the analysis of the upper bound of the absolute error in the computing of potential at an arbitrary point on the earth surface. For the sake of clarity, we will present this method in a particular and very known case: the calculation of potential due a point source of electrical current. The extension of this methodology to a general case of a grounding mesh is straightforward. Let be a point electrical current source of intensity I and located to a depth d from the earth surface (Fig. 1). Let consider a two-layer soil model with an upper stratum of thickness h and conductivity c1, and a lower one which conductivity is c2. The potential V1 on the surface can be obtained analytically depending on the position of the source [14,15]: point source in UPPER layer: d < h
V 1 ðrÞ ¼
1 I 1 I X jn pffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pdc1 ~r 2 þ 1 2pdc1 n¼1 ~2 ~ 1Þ2 r þ ð2nh
þ
1 I X jn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pdc1 n¼1 ~2 ~ þ 1Þ2 r þ ð2nh
1 I X ð1 jÞjn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pdc2 n¼0 ~2 ~ þ 1Þ2 r þ ð2nh
eN
I N j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; < 2pdc2 2 ~ 2 ~r þ ð2Nh þ 1Þ
ð9Þ
if d > h:
ð11Þ
So, the logarithm of the absolute error (eN) depends linearly with the number of images N, for a large value of N. This linear dependency can be clearly observed in Fig. 3 where it is shown the evolution of eN obtained from computed values of potential with the number of images for the case j = 0.998 when d > h for ratios ~ ¼ 0:25. The same conclusion can de derived for the case ~r ¼ 1 and h d < h, i.e. when the point current source is in the upper layer [12]. Now, the use of the Richardson’s deferred approach to the limit [16] with two different numbers of images in the series (N1, N2) allows to conclude the existence of geometric convergence eN2 ¼ eN1 jðN2 N1 Þ because jjj < 1. Although this formula could be used to compute extrapolated potential values, the use of Aitken d2-process [16] allows to obtain better results improving the rate of convergence of the series. Thus, an enhanced or extrapolated value of electrical potential (VE) can be computed as
VE ¼
point source in LOWER layer: d > h
V 1 ðrÞ ¼
In this paper and for the sake of brevity, we restrict this section considering a two-layer soil model such as j < 0 (i.e. when the conductivity of the lower layer is greater than the conductivity of the upper layer) and to the case d > h (the point source is in the lower layer); obviously the study can be straightforwardly extended to the other cases d < h and j > 0. Anyway, the negative values of j use to correspond to one of the most interesting cases in the practical grounding substation analysis: an upper layer less conductive than the lower layer (c1 < c2), e.g., clayey ground covered by a surface layer of gravel. In Fig. 2 it is represented the potential V1 at point ~r ¼ 1 in a 2-layer soil with j = 0.998 (c1 = 105 S/m and c2 = 102 S/m, for ~ ¼ h=d ¼ 0:25 example) due to a point current source at h (e.g., d = 1 m and h = 0.25 m). The intensity I is chosen equal to 2p dc2. Now, if we denote V1 the exact value of the potential and V N1 the approximated value of the potential computed by using the of the series in (10), the absolute error Nth first terms eN eN ¼ V 1 V N1 is upper bounded by
V N 1 V N3 V N 2 V N2
ð12Þ
V N1 þ V N3 2V N2
where V N1 , V N2 and V N3 are the computed values of the potential by using N1, N2 and N3 number of images (satisfying N1 < N2 < N3 and N3 N2 = N2 N1). The good agreement and quality of the extrapolated values for the potential obtained by using this simple and easy-to-use for-
ð10Þ 1.4
~ are dimensionless variables: ~r ¼ r=d and h ~ ¼ h=d. where ~r and h
Kappa = -0.998
Sum of the series
1.2
Aitken extrapolated values Computed values
1.0 0.8 0.6 0.4 0.2 0
10
20
30
40
50
Number of images (N)
Fig. 1. Scheme of a point current source (of intensity I) located to a depth d from the surface considering a 2-layer soil.
Fig. 2. Potential value on the ground surface at point ~r ¼ 1 versus the number of images. The values of the potential obtained by using the formula (12) based on the Aitken acceleration are also represented in this graphic.
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Absolute Error (log-scale)
0,0160
Absolute Error (log-scale)
0,1
0,0155
0,0150
0,0145
0,01 100
102
104
106
108
110
Number of computed images (N)
kappa = -0.998
1E-3
Bound of the absolute error Computed absolute error 1E-4 0
200
400
600
800
1000
Number of computed images (N) Fig. 3. Absolute error eN in the potential computing on the ground surface at point ~r ¼ 1 versus the Number of Images. The upper bound of the absolute error predicted by expression (11) is also represented in this graphic.
-2
3000
10
2250
10
-3
kappa = -0.998 -4
Relative error Computed images
1500
10
Relative error (log-scale)
Number of computed images (N)
-5
10
750
-6
10
0 0
10
20
30
40
50
Number of images with Aitken extrapolation Fig. 4. Number of images (on the left vertical scale) and Relative Error (on the right vertical scale) versus the number of images (on the horizontal axis) necessary for obtaining the improved potential values by using the Aitken acceleration formula (12).
6500
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
6000 5500 5000 4500 kappa = -0.998 4000 Relative error Computed images
3500
Relative error (log-scale)
Number of computed images (N)
mula can be seen in Fig. 2 even when a few number of terms of the series is used. In Fig. 4 it is shown the equivalence between the number of images if no extrapolation is used (N, on the left vertical scale) versus the number of images (on the horizontal axis) needed to obtain the same value of the potential if it is used the Aitken extrapolation formula (12). On the right vertical scale, it is shown the relative error in the electrical potential: for example, computing 10 terms of the series and using Aitken-formula produce the same value of the potential as computing N = 1000 images if extrapolation is not used (moreover, the relative error obtained is <5 104). The Aitken d2-process can also be applied to the extrapolated values of the potential since the logarithm of the error depends linearly with the number of images used for a large number of terms of the series (see in the Fig. 4 the log-error in the right-hand side vertical axis versus the number of images with Aitken extrapolation). We denote as ‘‘Aitken-Aitken extrapolation’’ this double application of the Aitken d2 extrapolation process. Fig. 5 shows the effect of this technique: for example, computing the potential with 10 images and using Aitken-formula twice produces the same result as computing N = 2950 images without extrapolation (the error
3000 2500 2000 1500 0
10
20
30
40
50
Number of images with Aitken-Aitken extrapolation Fig. 5. Number of images if no extrapolation is used (on the left vertical scale) and Relative Error (on the right vertical scale) versus the number of images (on the horizontal axis) necessary for obtaining the improved potential values by using the Aitken acceleration formula (12) twice.
obtained is <3 106). As we can see, with these techniques it is possible to obtain spectacular improvements in the computations of the potential. The authors have verified that these same conclusions of the acceleration of convergence are obtained by computing in different points on the ground surface and with different characteristics (upper-layer thickness, conductivities, depth of the point source, etc.). We have also achieved a very relevant improvement in the rate of convergence of the series used for computing the electrical potential with no-significant differences; indeed, the speed-up of the series (that is, the ratio between the number of images required to compute without using any extrapolation technique and the Number of images required if an Aitken-extrapolation formula is applied) is basically function on the maximum relative error established as target, and do not depend on the geometrical or other characteristics of the model. The previous extrapolation method can be applied to improve the convergence in the computing of the series resulting from the application of the ‘‘method of images’’ in many of the different methods proposed in the bibliography for grounding analysis in stratified soil models. This Aitken d2-extrapolation process has
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Fig. 6. TOTBEM: Toolbox for preprocessing and input data.
been applied to the Boundary Element Method numerical formulation for earthing analysis developed by the authors in the last years [4–7,9,10,13] and it has been implemented in the TOTBEM platform. 4. Description of the grounding simulation tool TOTBEM In the previous section and in the cited references we have presented the fundamentals of the numerical model based on the
Boundary Element Method for grounding analysis. Next we present its implementation in a computing and visualization platform. TOTBEM is an integrated package which essentially contains a set of modules to perform the preprocessing tasks, the grounding grid computing and the postprocessing tasks. TOTBEM has been developed on the SALOME platform [11]. SALOME is an opensource software for pre and postprocessing in numerical simulation. It allows to include external applications, tools and add-ons, incorporates a graphic interface for preprocessing, includes
Fig. 7. TOTBEM: Example of the input data for vertical rods.
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Fig. 8. TOTBEM Computing module: Visualization of the execution console during a grounding analysis.
Fig. 9. TOTBEM Postprocessing module: Example of visualization of a 3d view of potential computed on the ground surface.
postprocessing tools based on Visualization Toolkit libraries (VTK libraries), and in addition it allows to add specific extra functions for the pre and postprocessing stages. The SALOME platform includes general tools and modules for the geometry, meshing and postprocessing stages of a numerical simulation. The TOTBEM module developed by the authors includes specific tools and applications for preprocessing and data input (geometry, mesh and data definition for the numerical model), for the computation and analysis by means of the numerical model based on the BEM and for the postprocessing of results in order to easily visualize and study grounding grids installations by numerical simulation. 4.1. Preprocessing and data input module The working environment has been developed in order to facilitate the input of data using a specific graphic interface: the name
of the project, the number of layers, the thickness of each layer, their conductivities, the depth of the grounding mesh and the number of vertical rods and their length. The geometrical disposition of the electrodes in the grounding system can be introduced one by one or by using an ‘‘orthogrid’’ option, which automatically generates a mesh of orthogonal electrodes. Different options to add or erase electrodes, or modify their geometric characteristics are also available. Obviously the user can also define all environmental variables (point of view, perspective, zoom, colors, etc.), all of them in a very friendly working platform (Figs. 6 and 7). Furthermore, all the input data can be also loaded from two external files containing the geometry of the mesh of electrodes and the specific data of the problem (conductivities, number of layers, etc.), in case the mesh has been previously planned and design with other CAD applications or in the case of previous analysis developed with TOTBEM.
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Fig. 10. TOTBEM Postprocessing module: Example of visualization of the potential isolines on the ground surface.
Fig. 11. TOTBEM Postprocessing module: Example of visualization of a 3d view of potential clipped by a plane selected by the user.
The preprocessing module of TOTBEM has been developed by using Python scripts in order to define the specific interface and user functions since it is the native interface language of the SALOME platform (together with C++ libraries). Some specific operations like the computation of the orthogonal grids have been stated by using C++ functions. The preprocessing module has been designed to introduce the data of the problem and to create plain text files with the information required by the computing module.
If these data files already exist (from an existing project or from other different application) they can be also uploaded with the TOTBEM module. Obviously the preprocessing module is also linked to the SALOME geometry module in order to use its native functions (geometry visualization, geometry modification, . . . ). The computational mesh of each problem is directly defined (and saved in plain text files if it is necessary) by the TOTBEM module once the geometry is defined and just before the numerical analysis starts.
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Fig. 12. TOTBEM Postprocessing module: Example of visualization of the potential clipped by a plane selected by the user and the potential distribution.
4.2. Computing module The computing module is the kernel of TOTBEM. It was originally programmed in ForTran and included in the Salome platform as an external application. This external routine is called from the TOTBEM graphical interface and it starts by reading the data of the plain text files generated (or uploaded) in the preprocessing stage. Then it solves the numerical model for the grounding grid analysis. This routine is an efficient implementation of the numerical approach based on the Boundary Element Method for the grounding analysis in the case of layered soil models and it includes techniques for accelerating the calculations (Section 3). The analysis program developed in ForTran produces at the end a VTK file that contains the geometry and the potential distribution on the earth surface according to the parameters stated in the preprocessing stage. This VTK file is immediately converted to a MED file by using the conversion functions included with the SALOME platform. MED files are the standard formats used by the postprocessing module of SALOME, which natively allows to represent contour lines, 3D surfaces, clipping planes among other interesting visualization tools. The real-time evolution of the computing program can be followed in an execution console where main parameters of the grounding analysis are shown (Fig. 8). 4.3. Postprocessing and results visualization module Once the computing stage has finished, the postprocess module can upload the results files in VTK and MED formats. Both formats are very widespread and open-source, so the visualization of results can be also exported to other postprocessing platforms as Paraview [17] and Open Cascade [18], among others. The TOTBEM module and the SALOME platform include specific options for the visualization of output data for grounding analysis: 3-d views of the potential distribution on the earth surface, isopotential lines on the ground surface, potential values along a specific line, equivalent resistance of the grounding system, the total fault current derived through the grounding grid, etc. Figs. 9–12 show some examples of the postprocess module. Some of these functions are
natively included in the SALOME interface and others are given by the computing module in the analysis console during the execution. The TOTBEM platform based on the open-source software SALOME can be downloaded as a Ubuntu Live-DVD version from the web page of the authors: http://caminos.udc.es/gmni/. 5. Conclusions In this paper, the foundations of the mathematical model for the physical phenomenon of the electrical current dissipation through a earthing electrode into the ground has been revised, and the main highlights of a numerical formulation for grounding analysis in uniform and two-layer soil models have been pointed out. Furthermore, it has been presented a numerical simulation tool for grounding analysis (TOTBEM) based on the open-source software SALOME. TOTBEM is a CAD system that allows to analize and design grounding systems in a friendly environment including the preprocess, computing and postprocess stages. The kernel of the system is a powerful and efficient computing module which implements a complete and well-founded boundary element numerical approach. Acknowledgments This work has been partially supported by the ‘‘Ministerio de Educación y Ciencia’’ (Grants #DPI2009-14546-C02-01 and #DPI2010-16496) and by R&D projects of the Xunta de Galicia (Grants CN2011/002, PGDIT09 MDS00718PR and PGDIT09 REM005118PR) cofinanced with FEDER funds. References [1] IEEE Std.80. IEEE Guide for safety in AC substation grounding, New York, 2000. [2] Sverak JG. Progress in step and touch voltage equations of ANSI/IEEE Std 80. Historical perspective. IEEE Trans Power Del 1998;13(3):762–7. [3] Garrett DL, Pruitt JG. Problems encountered with the average potential method of analyzing substation grounding systems. IEEE Trans Power Appl Syst 1985;104(12):3586–96.
I. Colominas et al. / Electrical Power and Energy Systems 45 (2013) 353–361 [4] Navarrina F, Colominas I, Casteleiro M. Why do computer methods for grounding produce anomalous results? IEEE Trans Power Del 2003;18(4):1192–202. [5] Colominas I, Navarrina F, Casteleiro M. A boundary element numerical approach for grounding grid computation. Comput Method Appl Mech Eng 1999;174:73–90. [6] Colominas I, Gómez-Calviño J, Navarrina F, Casteleiro M. Computer analysis of earthing systems in horizontally and vertically layered soils. Electr Power Syst Res 2001;59:149–56. [7] Colominas I, Navarrina F, Casteleiro M. A numerical formulation for grounding analysis in stratified soils. IEEE Trans Power Del 2002;17(2):587–95. [8] IEEE Std. 142. IEEE Recommended practice for grounding of industrial and commercial power systems, New York, 2007. [9] Colominas I, Navarrina F, Casteleiro M. Analysis of transferred earth potentials in grounding systems: a BEM numerical approach. IEEE Trans Power Del 2005;20(1):339–45. [10] Colominas I, Navarrina F, Casteleiro M. Numerical simulation of transferred potentials in earthing grids considering layered soil models. IEEE Trans Power Del 2007;22(3):1514–22.
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[11] CEA and EDF and OpenCascade, Salome: the open source integration platform for numerical simulation, 2010.. [12] Colominas I, Parı´s J, Navarrina F, Casteleiro M. ‘‘Convergence acceleration of computer methods for grounding analysis in stratified soils. In: IOP conf. series: materials science and engineering, 10, 012079, 2010. . [13] Colominas I. A CAD system of grounding grids for electrical installations: a numerical approach based on the boundary element integral method. Ph.D. thesis, Civil Engrg. School, Universidade da Coruña, March 1995. . [14] Tagg GF. Earth resistances. New York: Pitman Pub. Co.; 1964. [15] Sunde ED. Earth conduction effects in transmission systems. New York: McMillan; 1968. [16] Isaacson E, Keller HB. Analysis of numerical methods. Mineola (NY): Dover Pub.; 1994. [17] Kitware et al., Paraview, [www.paraview.org], 2010. [18] Open Cascade S.A.S, Open CASCADE Technology, 3D modeling & numerical simulation, [www.opencascade.com], 2010.
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A numerical simulation tool for multilayer grounding analysis integrated in an open-source CAD interface I. Colominas ⇑, J. Parı´s, D. Fernández, F. Navarrina, M. Casteleiro Group of Numerical Methods in Engineering, Department of Applied Mathematics, Civil Engineering School, University of La Coruña, Campus de Elviña, 15071 La Coruña, Spain
a r t i c l e
i n f o
Article history: Received 26 April 2011 Received in revised form 18 August 2012 Accepted 29 August 2012 Available online 29 October 2012 Keywords: Grounding Numerical analysis Convergence of numerical methods Computer aided analysis Computer graphics software Open-source platform
a b s t r a c t In this paper we present TOTBEM: a simulation tool for grounding systems based on the open-source platform SALOME. The package TOTBEM includes all the preprocessing, computing and postprocessing stages necessary to perform a complete earthing analysis. The kernel of TOTBEM is a numerical formulation based on the Boundary Element Method for uniform and stratified soil models proposed by the authors in the last years. Furthermore, in this work we show the main highlights of an efficient technique based on the Aitken d2-process in order to improve the rate of convergence of the involved series expansions in multilayer soil models. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Obtaining the distribution of potential levels of an earthing system has been one of the challenges of the electrical engineers and designers since the beginning of the large-scale harnessing of electricity. Thus, the grounded electrode dissipates the electrical currents generated during a fault condition in order to guarantee the safety of persons, to maintain the integrity and proper functioning of equipment and to assure the continuity of the electrical supply. Main parameters commonly used to characterize a grounding grid are the equivalent resistance of the system and the potential distribution on the earth surface. As general rules, the resistance should be low enough in order to produce the dissipation of the electrical current into the ground through the earthing electrode, while certain values of potential differences on the earth surface should be under some well-established safety limits [1,2]. Maxwell’s Electromagnetic Theory is the general framework for the statement of the mathematical model for the phenomenon of the electrical current dissipation into the ground. Notwithstanding the equations that govern this problem are known for a long time, the analysis of large earthing systems in practice presents some important complications: on the one hand, due to its specific ⇑ Corresponding author. E-mail address: [email protected] (I. Colominas). URLs: http://www.caminos.udc.es/gmni/ (J. Parı´s), http://www.caminos.udc.es/ gmni/ (D. Fernández), http://www.caminos.udc.es/gmni/ (F. Navarrina), http:// www.caminos.udc.es/gmni/ (M. Casteleiro). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.08.079
tridimensional geometry (a grid of bare conductors with a large ‘‘length/diameter’’ ratio), and on the other hand for the uncertainty in the electrical properties of the ground. These facts mean that the use of common and widespread numerical methods in engineering (FEM or FDM) is very costly (in computing time and memory storage) since it is required the discretization of the domain, i.e., the whole ground excluding the part occupied by the electrodes. Over recent decades a large number of methods to calculate and design these grounding systems has been proposed, generally based on the professional practice, on empirical work and on experimental data. Some of these techniques have resulted in computer methods for grounding analysis which have led to significant progress in this area [1,2]. However, some problems with the application of these methods in the analysis of real cases have been documented, for example, unrealistic results when the discretization level of the electrodes is risen, uncertainties in the margin of error, or high computational costs [3]. These topics were fully analyzed in the reference [4] where the authors explained from a rigorous point of view the anomalous asymptotic behavior of the classical grounding methods (identifying also the sources of error), taking as a starting point the numerical formulation for grounding analysis based on the Boundary Element Method proposed by the authors in the nineties [5]. This numerical approach is the framework from it is possible to develop high-accurate computer methods for grounding analysis in uniform and layered soil models [5–7]. Furthermore, in 2005, we propose a methodology also based in this BEM numerical formulation for the analysis of transferred earth potentials in earthing
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systems (the appearance of dangerous potential levels in metallic elements, structures or other conductors as a result of the energization of the grounding electrode during a fault condition [8]). The methodology was published in the reference [9] for uniform soil models, and in the reference [10] its extension to layered soil models. In this paper we present TOTBEM: a power system numerical simulation tool for grounding analysis. TOTBEM is based on the open-source software SALOME [11], and it allows to perform the preprocess of the model (geometry data of the grounding mesh, soil properties, etc.), the earthing grid analysis by using the Boundary Element approach developed by the authors, and the postprocess and visualization of results. The outline of the paper is as follows: after the introductory section, it is presented a summary of the fundamentals of the mathematical and numerical model for grounding analysis based on the BEM; Section 3 is devoted to present a technique to accelerate the convergence of the calculations recently developed by the authors [12] which has been implemented in TOTBEM; and next, we present and describe the TOTBEM platform and its functionality. Finally, main conclusions are stated. 2. Foundations of the mathematical model for the current dissipation problem through a grounding electrode Equations that govern the dissipation of the electrical current into a media through a grounded electrode are given by
divðrÞ ¼ 0;
r ¼ cgradðVÞ in E;
t
r nE ¼ 0 in CE ; V ¼ V C in C; V ! 0; if jxj ! 1
ð1Þ
As it can be seen, this model restricts the analysis to obtain the steady-state response; furthermore the potential is assumed constant on the conductors surface (so the internal resistivity of the electrodes is neglected). In (1), E denotes the earth, c its conductivity, CE its surface, nE its normal exterior unit field and C the surface of the electrodes of the grounding grid [5]. Solving this problem provides the current density r and the potential V at any point x when the grounded electrode is energized to a Ground Potential Rise (or GPR) VC with respect to remote earth. On the other hand, most safety parameters that characterize an earthing system should be obtained straight from V computed on CE and r on C [5,7]. An important point on the definition of the mathematical model for grounding analysis is the selection of the more appropriate soil model: Obviously, it is not possible to take into account all variations of the soil conductivity since it is not feasible or from an engineering point of view neither economic nor practical. The simplest model that has been proposed is the isotropic and homogeneous one, i.e., a ‘‘uniform soil model’’ (so an scalar conductivity c is introduced instead of conductivity tensor c [1,5]); the ‘‘layered models’’ represent the soil in a number of strata, each one defined by means of a scalar conductivity and thickness [1] (in practical grounding analysis, it should be enough to consider models with two or three strata to achieve accurate results). The set of Eq. (1) can be rewritten in terms of the next exterior problem if the soil is modeled by C layers with different scalar conductivities:
rc ¼ cc gradðV c Þ in Ec ; 1 6 c 6 C; rt1 nE ¼ 0 in CE ; V b ¼ V C in C; divðrc Þ ¼ 0;
V c ! 0 if jxj ! 1;
1 6 c 6 C;
rtc nc ¼ rtcþ1 nc in Cc ; 1 6 c 6 C 1;
ð2Þ
where b is the layer that contains the buried conductor, Ec is each stratum, cc is its conductivity, Vc is the potential at a point in layer
Ec, rc is its current density, Cc is the interface between Ec and Ec+1, and nc is the normal unit field to Cc [7]. In the following section of this paper we will present the improvement in the convergence for the case of two layered soil models (C = 2), although it is direct the extension of the analysis to other stratified models. On the other hand, by application of Green’s Identity and the ‘‘Method of Images’’ potential Vc(xc) at an arbitrary point xc 2 Ec can be expressed in an integral form [7] in terms of the leakage current density r(n) at any point n on the surface of the conductors C Eb (r = rtn, being n the normal exterior unit field to C):
V c ðxc Þ ¼
1
Z Z
4pcb
kbc ðxc ; nÞrðnÞdC;
8xc 2 Ec ;
ð3Þ
n2C
Furthermore, since most parameters that characterize an earthing system (mesh, touch, and contact voltages, for example) are obtained from the potential distribution computed on the earth surface CE, from now on we will consider c = 1 in the grounding analysis:
V 1 ðx1 Þ ¼
1 4pcb
Z Z
kb1 ðx1 ; nÞrðnÞdC;
8x1 2 CE ;
ð4Þ
n2C
Generally, kernel kb1(x1,n) is a series which terms correspond to the images introduced in the transformation of problem (2) to the integral form (3) [5,7,13]. The number of terms of these series can be finite, for example in the case of homogeneous and isotropic soil models (C = 1), or infinite, for example in two-layer soil models (C = 2). In the appendix section of the reference [10] can be found the explicit formulae of these integral kernels. It is well known the integral kernels are weakly singular and they depend on the distances from x1 to n and depend on the distances from x1 to all its images with respect to CE and with respect to Cc [1,14]. The kernels are also function of the thickness of the layer and the layer conductivities according to a ratio. In a twolayer soil model, this ratio j is:
j¼
c1 c2 c1 þ c2
ð5Þ
so the kernels should be expressed in the general form
kb1 ðx1 ; nÞ ¼
1 X ½n kb1 ðx1 ; nÞ; n¼0
½n
kb1 ðx1 ; nÞ ¼
wn ðjÞ ; rðx1 ; nn Þ
ð6Þ
r(x1, nn) represents the distance from x1 to images nn while the weighting coefficient wn(j) is a function that only depends on the ratio j and the thickness of the layer [7]. The substitution of the general form for the inner kernel (6) in (4) allows to express the computation of the potential at an arbitrary point x1 2 CE in terms of the contribution of each image as follows:
V 1 ðx1 Þ ¼
1 X ½n V 1 ðx1 Þ
ð7Þ
n¼0 ½n
and V 1 ðx1 Þ is the contribution in the potential computing due to the image n: ½n
V 1 ðx1 Þ ¼
1 4pcb
Z Z
½n
n2C
kb1 ðx1 ; nÞrðnÞdC:
ð8Þ
In the bibliography can be found some different computer methods for obtaining the potential contribution of each image given by (8) [1,2]. Specifically, the authors proposed a methodology based on the Boundary Element Method for the computational design of real earthing systems in layered soil models, which can be analyzed other related problems of industrial interest like the transferred earth potentials [4–7,9,10]. The rate of convergence of the different series that appear when the method of images is applied in the case of stratified soil models
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(e.g., (6) for the two-layer soil model) can be very low depending on the value of j, the ratio between the conductivities of the strata given by (5). This fact is particularly critical when jjj 1, i.e. in cases where there are significant differences between the conductivities of layers, which are commonly the most interesting ones in practical applications. On the other hand, it is important to remark that obtaining the distribution of potential levels on the earth surface is usually the most time-consuming process in earthing analysis since it may be necessary to compute potential values in an great number of points to rise high-accurate results: A good example of it can be found in the reference [7] in the study of the Santiago II substation grounding system where the total area analyzed is 78.000 m2 being necessary to compute potential values in near 100.000 points. So, it is required an accurate and low-cost computation of the potential: to achieve these goals the authors have developed a methodology to increase the convergence rate of the kernel series [12] which has been implemented in TOTBEM. In the following section, it is presented a summary of this technique. 3. Convergence improvement techniques for multiple layer soil models The development of this method is firstly based on the analysis of the upper bound of the absolute error in the computing of potential at an arbitrary point on the earth surface. For the sake of clarity, we will present this method in a particular and very known case: the calculation of potential due a point source of electrical current. The extension of this methodology to a general case of a grounding mesh is straightforward. Let be a point electrical current source of intensity I and located to a depth d from the earth surface (Fig. 1). Let consider a two-layer soil model with an upper stratum of thickness h and conductivity c1, and a lower one which conductivity is c2. The potential V1 on the surface can be obtained analytically depending on the position of the source [14,15]: point source in UPPER layer: d < h
V 1 ðrÞ ¼
1 I 1 I X jn pffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pdc1 ~r 2 þ 1 2pdc1 n¼1 ~2 ~ 1Þ2 r þ ð2nh
þ
1 I X jn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pdc1 n¼1 ~2 ~ þ 1Þ2 r þ ð2nh
1 I X ð1 jÞjn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pdc2 n¼0 ~2 ~ þ 1Þ2 r þ ð2nh
eN
I N j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; < 2pdc2 2 ~ 2 ~r þ ð2Nh þ 1Þ
ð9Þ
if d > h:
ð11Þ
So, the logarithm of the absolute error (eN) depends linearly with the number of images N, for a large value of N. This linear dependency can be clearly observed in Fig. 3 where it is shown the evolution of eN obtained from computed values of potential with the number of images for the case j = 0.998 when d > h for ratios ~ ¼ 0:25. The same conclusion can de derived for the case ~r ¼ 1 and h d < h, i.e. when the point current source is in the upper layer [12]. Now, the use of the Richardson’s deferred approach to the limit [16] with two different numbers of images in the series (N1, N2) allows to conclude the existence of geometric convergence eN2 ¼ eN1 jðN2 N1 Þ because jjj < 1. Although this formula could be used to compute extrapolated potential values, the use of Aitken d2-process [16] allows to obtain better results improving the rate of convergence of the series. Thus, an enhanced or extrapolated value of electrical potential (VE) can be computed as
VE ¼
point source in LOWER layer: d > h
V 1 ðrÞ ¼
In this paper and for the sake of brevity, we restrict this section considering a two-layer soil model such as j < 0 (i.e. when the conductivity of the lower layer is greater than the conductivity of the upper layer) and to the case d > h (the point source is in the lower layer); obviously the study can be straightforwardly extended to the other cases d < h and j > 0. Anyway, the negative values of j use to correspond to one of the most interesting cases in the practical grounding substation analysis: an upper layer less conductive than the lower layer (c1 < c2), e.g., clayey ground covered by a surface layer of gravel. In Fig. 2 it is represented the potential V1 at point ~r ¼ 1 in a 2-layer soil with j = 0.998 (c1 = 105 S/m and c2 = 102 S/m, for ~ ¼ h=d ¼ 0:25 example) due to a point current source at h (e.g., d = 1 m and h = 0.25 m). The intensity I is chosen equal to 2p dc2. Now, if we denote V1 the exact value of the potential and V N1 the approximated value of the potential computed by using the of the series in (10), the absolute error Nth first terms eN eN ¼ V 1 V N1 is upper bounded by
V N 1 V N3 V N 2 V N2
ð12Þ
V N1 þ V N3 2V N2
where V N1 , V N2 and V N3 are the computed values of the potential by using N1, N2 and N3 number of images (satisfying N1 < N2 < N3 and N3 N2 = N2 N1). The good agreement and quality of the extrapolated values for the potential obtained by using this simple and easy-to-use for-
ð10Þ 1.4
~ are dimensionless variables: ~r ¼ r=d and h ~ ¼ h=d. where ~r and h
Kappa = -0.998
Sum of the series
1.2
Aitken extrapolated values Computed values
1.0 0.8 0.6 0.4 0.2 0
10
20
30
40
50
Number of images (N)
Fig. 1. Scheme of a point current source (of intensity I) located to a depth d from the surface considering a 2-layer soil.
Fig. 2. Potential value on the ground surface at point ~r ¼ 1 versus the number of images. The values of the potential obtained by using the formula (12) based on the Aitken acceleration are also represented in this graphic.
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Absolute Error (log-scale)
0,0160
Absolute Error (log-scale)
0,1
0,0155
0,0150
0,0145
0,01 100
102
104
106
108
110
Number of computed images (N)
kappa = -0.998
1E-3
Bound of the absolute error Computed absolute error 1E-4 0
200
400
600
800
1000
Number of computed images (N) Fig. 3. Absolute error eN in the potential computing on the ground surface at point ~r ¼ 1 versus the Number of Images. The upper bound of the absolute error predicted by expression (11) is also represented in this graphic.
-2
3000
10
2250
10
-3
kappa = -0.998 -4
Relative error Computed images
1500
10
Relative error (log-scale)
Number of computed images (N)
-5
10
750
-6
10
0 0
10
20
30
40
50
Number of images with Aitken extrapolation Fig. 4. Number of images (on the left vertical scale) and Relative Error (on the right vertical scale) versus the number of images (on the horizontal axis) necessary for obtaining the improved potential values by using the Aitken acceleration formula (12).
6500
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
6000 5500 5000 4500 kappa = -0.998 4000 Relative error Computed images
3500
Relative error (log-scale)
Number of computed images (N)
mula can be seen in Fig. 2 even when a few number of terms of the series is used. In Fig. 4 it is shown the equivalence between the number of images if no extrapolation is used (N, on the left vertical scale) versus the number of images (on the horizontal axis) needed to obtain the same value of the potential if it is used the Aitken extrapolation formula (12). On the right vertical scale, it is shown the relative error in the electrical potential: for example, computing 10 terms of the series and using Aitken-formula produce the same value of the potential as computing N = 1000 images if extrapolation is not used (moreover, the relative error obtained is <5 104). The Aitken d2-process can also be applied to the extrapolated values of the potential since the logarithm of the error depends linearly with the number of images used for a large number of terms of the series (see in the Fig. 4 the log-error in the right-hand side vertical axis versus the number of images with Aitken extrapolation). We denote as ‘‘Aitken-Aitken extrapolation’’ this double application of the Aitken d2 extrapolation process. Fig. 5 shows the effect of this technique: for example, computing the potential with 10 images and using Aitken-formula twice produces the same result as computing N = 2950 images without extrapolation (the error
3000 2500 2000 1500 0
10
20
30
40
50
Number of images with Aitken-Aitken extrapolation Fig. 5. Number of images if no extrapolation is used (on the left vertical scale) and Relative Error (on the right vertical scale) versus the number of images (on the horizontal axis) necessary for obtaining the improved potential values by using the Aitken acceleration formula (12) twice.
obtained is <3 106). As we can see, with these techniques it is possible to obtain spectacular improvements in the computations of the potential. The authors have verified that these same conclusions of the acceleration of convergence are obtained by computing in different points on the ground surface and with different characteristics (upper-layer thickness, conductivities, depth of the point source, etc.). We have also achieved a very relevant improvement in the rate of convergence of the series used for computing the electrical potential with no-significant differences; indeed, the speed-up of the series (that is, the ratio between the number of images required to compute without using any extrapolation technique and the Number of images required if an Aitken-extrapolation formula is applied) is basically function on the maximum relative error established as target, and do not depend on the geometrical or other characteristics of the model. The previous extrapolation method can be applied to improve the convergence in the computing of the series resulting from the application of the ‘‘method of images’’ in many of the different methods proposed in the bibliography for grounding analysis in stratified soil models. This Aitken d2-extrapolation process has
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Fig. 6. TOTBEM: Toolbox for preprocessing and input data.
been applied to the Boundary Element Method numerical formulation for earthing analysis developed by the authors in the last years [4–7,9,10,13] and it has been implemented in the TOTBEM platform. 4. Description of the grounding simulation tool TOTBEM In the previous section and in the cited references we have presented the fundamentals of the numerical model based on the
Boundary Element Method for grounding analysis. Next we present its implementation in a computing and visualization platform. TOTBEM is an integrated package which essentially contains a set of modules to perform the preprocessing tasks, the grounding grid computing and the postprocessing tasks. TOTBEM has been developed on the SALOME platform [11]. SALOME is an opensource software for pre and postprocessing in numerical simulation. It allows to include external applications, tools and add-ons, incorporates a graphic interface for preprocessing, includes
Fig. 7. TOTBEM: Example of the input data for vertical rods.
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Fig. 8. TOTBEM Computing module: Visualization of the execution console during a grounding analysis.
Fig. 9. TOTBEM Postprocessing module: Example of visualization of a 3d view of potential computed on the ground surface.
postprocessing tools based on Visualization Toolkit libraries (VTK libraries), and in addition it allows to add specific extra functions for the pre and postprocessing stages. The SALOME platform includes general tools and modules for the geometry, meshing and postprocessing stages of a numerical simulation. The TOTBEM module developed by the authors includes specific tools and applications for preprocessing and data input (geometry, mesh and data definition for the numerical model), for the computation and analysis by means of the numerical model based on the BEM and for the postprocessing of results in order to easily visualize and study grounding grids installations by numerical simulation. 4.1. Preprocessing and data input module The working environment has been developed in order to facilitate the input of data using a specific graphic interface: the name
of the project, the number of layers, the thickness of each layer, their conductivities, the depth of the grounding mesh and the number of vertical rods and their length. The geometrical disposition of the electrodes in the grounding system can be introduced one by one or by using an ‘‘orthogrid’’ option, which automatically generates a mesh of orthogonal electrodes. Different options to add or erase electrodes, or modify their geometric characteristics are also available. Obviously the user can also define all environmental variables (point of view, perspective, zoom, colors, etc.), all of them in a very friendly working platform (Figs. 6 and 7). Furthermore, all the input data can be also loaded from two external files containing the geometry of the mesh of electrodes and the specific data of the problem (conductivities, number of layers, etc.), in case the mesh has been previously planned and design with other CAD applications or in the case of previous analysis developed with TOTBEM.
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359
Fig. 10. TOTBEM Postprocessing module: Example of visualization of the potential isolines on the ground surface.
Fig. 11. TOTBEM Postprocessing module: Example of visualization of a 3d view of potential clipped by a plane selected by the user.
The preprocessing module of TOTBEM has been developed by using Python scripts in order to define the specific interface and user functions since it is the native interface language of the SALOME platform (together with C++ libraries). Some specific operations like the computation of the orthogonal grids have been stated by using C++ functions. The preprocessing module has been designed to introduce the data of the problem and to create plain text files with the information required by the computing module.
If these data files already exist (from an existing project or from other different application) they can be also uploaded with the TOTBEM module. Obviously the preprocessing module is also linked to the SALOME geometry module in order to use its native functions (geometry visualization, geometry modification, . . . ). The computational mesh of each problem is directly defined (and saved in plain text files if it is necessary) by the TOTBEM module once the geometry is defined and just before the numerical analysis starts.
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Fig. 12. TOTBEM Postprocessing module: Example of visualization of the potential clipped by a plane selected by the user and the potential distribution.
4.2. Computing module The computing module is the kernel of TOTBEM. It was originally programmed in ForTran and included in the Salome platform as an external application. This external routine is called from the TOTBEM graphical interface and it starts by reading the data of the plain text files generated (or uploaded) in the preprocessing stage. Then it solves the numerical model for the grounding grid analysis. This routine is an efficient implementation of the numerical approach based on the Boundary Element Method for the grounding analysis in the case of layered soil models and it includes techniques for accelerating the calculations (Section 3). The analysis program developed in ForTran produces at the end a VTK file that contains the geometry and the potential distribution on the earth surface according to the parameters stated in the preprocessing stage. This VTK file is immediately converted to a MED file by using the conversion functions included with the SALOME platform. MED files are the standard formats used by the postprocessing module of SALOME, which natively allows to represent contour lines, 3D surfaces, clipping planes among other interesting visualization tools. The real-time evolution of the computing program can be followed in an execution console where main parameters of the grounding analysis are shown (Fig. 8). 4.3. Postprocessing and results visualization module Once the computing stage has finished, the postprocess module can upload the results files in VTK and MED formats. Both formats are very widespread and open-source, so the visualization of results can be also exported to other postprocessing platforms as Paraview [17] and Open Cascade [18], among others. The TOTBEM module and the SALOME platform include specific options for the visualization of output data for grounding analysis: 3-d views of the potential distribution on the earth surface, isopotential lines on the ground surface, potential values along a specific line, equivalent resistance of the grounding system, the total fault current derived through the grounding grid, etc. Figs. 9–12 show some examples of the postprocess module. Some of these functions are
natively included in the SALOME interface and others are given by the computing module in the analysis console during the execution. The TOTBEM platform based on the open-source software SALOME can be downloaded as a Ubuntu Live-DVD version from the web page of the authors: http://caminos.udc.es/gmni/. 5. Conclusions In this paper, the foundations of the mathematical model for the physical phenomenon of the electrical current dissipation through a earthing electrode into the ground has been revised, and the main highlights of a numerical formulation for grounding analysis in uniform and two-layer soil models have been pointed out. Furthermore, it has been presented a numerical simulation tool for grounding analysis (TOTBEM) based on the open-source software SALOME. TOTBEM is a CAD system that allows to analize and design grounding systems in a friendly environment including the preprocess, computing and postprocess stages. The kernel of the system is a powerful and efficient computing module which implements a complete and well-founded boundary element numerical approach. Acknowledgments This work has been partially supported by the ‘‘Ministerio de Educación y Ciencia’’ (Grants #DPI2009-14546-C02-01 and #DPI2010-16496) and by R&D projects of the Xunta de Galicia (Grants CN2011/002, PGDIT09 MDS00718PR and PGDIT09 REM005118PR) cofinanced with FEDER funds. References [1] IEEE Std.80. IEEE Guide for safety in AC substation grounding, New York, 2000. [2] Sverak JG. Progress in step and touch voltage equations of ANSI/IEEE Std 80. Historical perspective. IEEE Trans Power Del 1998;13(3):762–7. [3] Garrett DL, Pruitt JG. Problems encountered with the average potential method of analyzing substation grounding systems. IEEE Trans Power Appl Syst 1985;104(12):3586–96.
I. Colominas et al. / Electrical Power and Energy Systems 45 (2013) 353–361 [4] Navarrina F, Colominas I, Casteleiro M. Why do computer methods for grounding produce anomalous results? IEEE Trans Power Del 2003;18(4):1192–202. [5] Colominas I, Navarrina F, Casteleiro M. A boundary element numerical approach for grounding grid computation. Comput Method Appl Mech Eng 1999;174:73–90. [6] Colominas I, Gómez-Calviño J, Navarrina F, Casteleiro M. Computer analysis of earthing systems in horizontally and vertically layered soils. Electr Power Syst Res 2001;59:149–56. [7] Colominas I, Navarrina F, Casteleiro M. A numerical formulation for grounding analysis in stratified soils. IEEE Trans Power Del 2002;17(2):587–95. [8] IEEE Std. 142. IEEE Recommended practice for grounding of industrial and commercial power systems, New York, 2007. [9] Colominas I, Navarrina F, Casteleiro M. Analysis of transferred earth potentials in grounding systems: a BEM numerical approach. IEEE Trans Power Del 2005;20(1):339–45. [10] Colominas I, Navarrina F, Casteleiro M. Numerical simulation of transferred potentials in earthing grids considering layered soil models. IEEE Trans Power Del 2007;22(3):1514–22.
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[11] CEA and EDF and OpenCascade, Salome: the open source integration platform for numerical simulation, 2010.