CN111929348B - Method for calculating surface potential of direct current grounding electrode in horizontal multilayer structure earth environment - Google Patents

Method for calculating surface potential of direct current grounding electrode in horizontal multilayer structure earth environment Download PDF

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CN111929348B
CN111929348B CN202011115177.8A CN202011115177A CN111929348B CN 111929348 B CN111929348 B CN 111929348B CN 202011115177 A CN202011115177 A CN 202011115177A CN 111929348 B CN111929348 B CN 111929348B
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CN111929348A (en
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徐碧川
万华
潘卓洪
李唐兵
王鹏
刘玉婷
童军心
王华云
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State Grid Corp of China SGCC
Electric Power Research Institute of State Grid Jiangxi Electric Power Co Ltd
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Abstract

The invention discloses a method for calculating the surface potential of a direct current grounding electrode in a horizontal multilayer structure earth environment. The method simplifies the Green function step of solving the DC grounding extremely fine earth model, does not need matrix decomposition operation, and has high solving efficiency; the linear filtering method has the outstanding advantages of intuition, simplicity and quickness, is very suitable for solving a fine geodetic model, and has great advantages for forming a software algorithm.

Description

Method for calculating surface potential of direct current grounding electrode in horizontal multilayer structure earth environment
Technical Field
The invention belongs to the field of research on influence analysis of a high-voltage direct-current transmission technology on the surrounding environment, and particularly relates to a numerical calculation method for solving a horizontal multilayer structure direct-current grounding extremely fine earth model by using a linear filtering method under two conditions of considering the embedding depth of a direct-current grounding electrode (based on a sign-algebraic equation method) and not considering the embedding depth (based on a transmission line model method).
Background
The resistivity model of the earth in the DC grounding electrode address is closely related to the problems of the grounding performance of the DC grounding electrode, the DC magnetic bias of the transformer and the like, and an accurate earth resistivity model needs to be obtained by inverting the measured data after the on-site exploration. The soil generally has heterogeneity and can be treated approximately as a horizontally stratified soil structure of several layers. Due to the difference in resistivity of the soil on both sides of the interface, the current field and the electric field change at the layered soil interface. In the traditional power system grounding calculation, because the area occupied by the transformer substation grounding grid is smaller, the model scale of the horizontal multilayer structure ground used in the grounding calculation is smaller, and the traditional theoretical and numerical methods can meet the solving requirement.
The long-distance direct current transmission project has a transmission distance of hundreds to three thousand kilometers, the current in the ground of direct current transmission has great propagation depth, the ground potential of a direct current grounding electrode relates to a large-scale ground surface coverage area, the used horizontal multi-layer structure ground far exceeds the existing traditional model in both layering number and depth coverage range, and the soil model numerical calculation process is complicated and the theoretical difficulty is high. And when the numerical values of the soil resistivities of the two adjacent layers are greatly different and the layer with large/small thickness exists, the numerical value singularity phenomenon exists when the fine horizontal layering earth model is solved by the pure numerical method. With the increase of the complexity of the direct current grounding electrode fine earth model, the traditional theoretical method cannot meet the solving precision and speed of the fine horizontal layered earth model calculation, so that a new theoretical method and a digital calculation means are required to solve the direct current grounding electrode earth surface potential in the fine horizontal multilayer structure earth environment.
The calculation of the direct current grounding pole fine earth model can be divided into two scenes: the method comprises the following steps of common grounding calculation and earth surface potential calculation of direct current magnetic bias, and inversion calculation of earth resistivity exploration data. Both involve a core algorithm, which is how to solve the algorithm of forming the space potential distribution by the known point current source, namely the green function problem of the fine earth model. The Green function analytic solution for the fine earth model can be written as follows:
Figure 8866DEST_PATH_IMAGE001
formula (1) belongs to the generalized infinite integral, wherein: f and F are respectively a Hankel (Hankel) integral function and an integral kernel function;rand
Figure 555385DEST_PATH_IMAGE002
position parameters and integral variables respectively; j. the design is a square0Is a zero order bessel function of the first kind. Due to the complexity of the integral kernel function F, there is not necessarily an analytical solution for F. Therefore, the numerical calculation of the formula (1) becomes a basic problem in the fields of geology and grounding, and the current solving method mainly comprises the following four steps:
a numerical integration method such as a sinbose integration method and a Bessel function zero point interval integration method;
the classical mirror image method is only suitable for the condition that the Hankel integral kernel function f can be expanded into exponential series;
the linear filtering method belongs to a general method in the field of geology, and is less applied to the grounding of a power system;
the complex mirror image method is introduced from high-frequency computing electromagnetism and is widely applied to grounding computation of a power system at present.
The numerical integration method and the classical mirror image method have the problems of large calculated amount, low convergence rate and the like, and are rarely applied at present. Although the linear filtering method has a great deal of research results in the geological industry, the complex mirror image method is still used for most of the power system grounding calculation. With the development of a numerical calculation method, researchers have studied and used an improved complex mirror image method to solve the problem of a singular constant current field, and the accuracy of the algorithm is improved to a certain extent. In view of the above, aiming at the above limitations encountered in the current calculation method, the present invention provides a numerical calculation method for solving a horizontal multilayer structure dc ground pole fine earth model by a linear filtering method under two conditions of considering the dc ground pole burying depth (based on a sign-algebraic equation method) and not considering the burying depth (based on a transmission line model method), and makes the method become a good supplement of a complex mirror image method, so as to provide a new idea and a new means for calculating the dc ground pole surface potential in the fine horizontal multilayer structure earth environment.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a horizontal layered direct current grounding electrode grounding model solving method starting from a linear filtering method under the two conditions of considering the direct current grounding electrode embedding depth (based on a sign-algebraic equation method) and not considering the embedding depth (based on a transmission line model method),
the present invention employs the following solution. A high-efficiency fine geodetic model solving scheme is formed based on a linear filtering method, and the existing filtering weight coefficient is used for converting the Hankel infinite integral into a finite term summation form, so that the direct-current grounding extremely fine geodetic model is solved.
A method for calculating the earth surface potential of a direct current grounding electrode in a horizontal multilayer structure earth environment comprises the following specific steps:
step 1, under the condition of considering the buried depth, order
Figure 899516DEST_PATH_IMAGE003
And (3) rewriting the Hankel transformation of the Green function analytic solution of the fine earth model in the formula (1) into the following steps by using a linear filtering method:
Figure 847881DEST_PATH_IMAGE004
in the formula, F and F are respectively a Hankel (Hankel) integral function and an integral kernel function;
Figure 19099DEST_PATH_IMAGE005
representing integral variables, e x The parameters of the position are represented by,
Figure 52914DEST_PATH_IMAGE006
；J0representing a first class of zero order bessel functions.
Step 2, e in the formula (2) x y–According to the interval
Figure 200736DEST_PATH_IMAGE007
Sampling, sampling interval
Figure 144553DEST_PATH_IMAGE007
Less than 0.5 timesfThe cut-off frequency, at which the sampled discrete signal can be restored to a continuous signal without distortion, can be expressed as equation (2):
Figure 221093DEST_PATH_IMAGE008
in the formula:H n is a filter coefficient; f is an integral kernel function in formula (1);
Figure 240739DEST_PATH_IMAGE007
n to be the position of the sampling point(s),nindicating the number of sample points.
Step 3, the analytic solution form of the horizontal multilayer earth Green function (potential function of the unit point current source) is
Figure 428138DEST_PATH_IMAGE009
Figure 351095DEST_PATH_IMAGE010
Figure 598537DEST_PATH_IMAGE011
Indicating that the point current source is at the secondiThe green's function of the layer of soil,ρ i resistivity of the source layer (omega ∙ m), z-z0To observe the horizontal distance from the point to the point current source, in equation (5),
Figure 465998DEST_PATH_IMAGE012
is a function of Kronecker-delta, if and only ifi=sTime of flight
Figure 955623DEST_PATH_IMAGE012
(i– s)=1Otherwise
Figure 733087DEST_PATH_IMAGE012
(i–s)=0；AAndBall relate to integral variables
Figure 151430DEST_PATH_IMAGE013
The function to be solved of (a) is,
Figure 381554DEST_PATH_IMAGE013
is an integral variable.
Further, a 61-point version of guptaasma-Singh is used in step 2 to solve equation (3):
Figure 674870DEST_PATH_IMAGE014
wherein: log (log)10(
Figure 306839DEST_PATH_IMAGE015
n )=–5.199 138 304 + 0.116 638 304 n，n=1, 2, …, 61；H n The values are shown in Table 1.
Figure 630504DEST_PATH_IMAGE016
Figure 347925DEST_PATH_IMAGE017
Figure 444931DEST_PATH_IMAGE018
Further, in step 3, horizontalmSimplest universal hankel integral kernel function meter for stratum soilThe calculation method comprises the following steps:
a. establishing a symbolic equation:
Figure 665828DEST_PATH_IMAGE019
x and Y are both sparse matrixes/vectors, and the vector C = [ to be solved ]A 1, B 1, …, A m , B m ]T. X is 2m
Figure 691553DEST_PATH_IMAGE020
2mDimension matrix, orderi = 1, …, m-1, having:
Figure 427428DEST_PATH_IMAGE021
Figure 62546DEST_PATH_IMAGE022
Figure 137950DEST_PATH_IMAGE023
Figure 600155DEST_PATH_IMAGE024
Figure 931648DEST_PATH_IMAGE025
Figure 606343DEST_PATH_IMAGE026
whereinK i Is undergroundiThe reflection coefficient of the boundary is determined,
Figure 67411DEST_PATH_IMAGE027
. Y is 2mDimension column vector, specific form and source layer numbersIt is related.s(ii) =1, case where a non-zero element of Y is represented by formula (14);s=mnon-zero elements of YThe extract is represented by formula (15); the non-zero element for the other cases Y is taken as formula (16):
Figure 434939DEST_PATH_IMAGE028
Figure 207723DEST_PATH_IMAGE029
Figure 325589DEST_PATH_IMAGE030
b. will be described in detail
Figure 765798DEST_PATH_IMAGE015
n Solving the substituted symbol equation (6)ABThen using formula (5) to obtain the correspondingf i (
Figure 38647DEST_PATH_IMAGE015
n )；
c. According to the formula (4) to obtain
Figure 705252DEST_PATH_IMAGE031
。
Further, in step 3, by combining the engineering practice of the dc magnetic biasing problem, when a large-scale earth surface potential distribution is solved, the dc grounding electrode is simplified into a point source and the buried depth of the dc grounding electrode is ignored, that is, the grounding electrode is simplified into an earth surface point current source, and at this time, the hankel integral kernel function can be solved by adopting a transmission line model, so that the effect of simplifying analysis is achieved.
Solving earth surface potential from transmission line modelVThe calculation formula of (2) is as follows:
Figure 220285DEST_PATH_IMAGE032
wherein the content of the first and second substances,rthe distance between the earth surface observation point and the point current source is taken as the distance; j. the design is a square0Is a first class zero order Bessel function;
Figure 390366DEST_PATH_IMAGE033
is an integral variable;R 1as a function of the form:
Figure 834117DEST_PATH_IMAGE034
wherein R is obtained by the following recursion formula1：
Figure 722438DEST_PATH_IMAGE035
Figure 41162DEST_PATH_IMAGE036
Figure 331329DEST_PATH_IMAGE037
Wherein the content of the first and second substances,
Figure 211560DEST_PATH_IMAGE038
in order to be the angular frequency of the frequency,
Figure 321599DEST_PATH_IMAGE039
are respectively the firstiThe thickness, resistivity and permeability of the layer soil,
Figure 178434DEST_PATH_IMAGE033
i is as followsiThe wave number of the layer soil is,r i is as followsiThe wave impedance of the layer of soil,R 1in order to represent the apparent impedance,R i （i = 2, …,
Figure 588687DEST_PATH_IMAGE040
) In order to be a function of the impedance coefficient,mthe total number of layers of soil.
Compared with the prior art, the invention has the following beneficial effects: the Green function step of solving the direct current grounding extremely fine earth model is simplified, matrix decomposition operation is not needed, and the method has high solving efficiency; the linear filtering method has the outstanding advantages of intuition, simplicity and quickness, is very suitable for solving a fine geodetic model, and has great advantages for forming a software algorithm.
Drawings
Fig. 1 is a schematic diagram of a point current source in a horizontal multi-layer structure earth ground.
Fig. 2 is a schematic diagram of a transmission line model.
Detailed Description
It should be noted that, in the present application, the embodiments and features of the embodiments may be combined with each other, and the present invention will be described in detail with reference to the accompanying drawings and embodiments.
For better understanding of the present invention, the following examples are provided to further illustrate the present invention, and the examples described are only a part of the present invention, but the present invention is not limited to the following examples. Various changes or modifications may be effected therein by one skilled in the art and such equivalents are intended to be within the scope of the invention as defined by the claims appended hereto.
The invention provides a method for calculating the earth surface potential of a direct current grounding electrode in a horizontal multilayer structure earth environment, which comprises the following four steps:
step 1, under the condition of considering the buried depth, order
Figure 374240DEST_PATH_IMAGE041
And rewriting the Hankel transformation of the Green function analytic solution of the fine earth model into the following steps by using a linear filtering method:
Figure 237154DEST_PATH_IMAGE042
step 2, taking e in formula (22) x y–According to the interval
Figure 258200DEST_PATH_IMAGE015
Sampling, if the sampling interval
Figure 21494DEST_PATH_IMAGE015
If the f cutoff frequency is less than 0.5, the sampled discrete signal can be restored to a continuous signal without distortion, so equation (22) can be:
Figure 243528DEST_PATH_IMAGE043
H n is a filter coefficient; f is an integral kernel function in formula (1);
Figure 328159DEST_PATH_IMAGE015
n the position of the sampling point.
Step 2a, further, using a 61-point version of guptaasma-Singh to solve equation (23) in step 2:
Figure 293841DEST_PATH_IMAGE044
wherein log10(
Figure 911642DEST_PATH_IMAGE015
n )=–5.199 138 304 + 0.116 638 304 n，n=1, 2, …, 61；H n The values are shown in Table 1.
Step 3, calculating the level by using a Hankel integral kernel functionmLayer soil, using a set of well-established horizontal 4-layer earth models as the subject of validation. The resistivity of 4 layers of soil is 235
Figure 38998DEST_PATH_IMAGE045
m、5900
Figure 876504DEST_PATH_IMAGE045
m、141,000
Figure 114718DEST_PATH_IMAGE045
m and 120
Figure 587025DEST_PATH_IMAGE045
m, the thickness is respectively 30m, 1km and 50km, and a surface point current source of 1kA is taken as an origin. As shown in FIG. 1, the analytic solution form of the horizontal multi-layer earth Green's function (potential function of a single-point current source) is
Figure 885283DEST_PATH_IMAGE046
Figure 210085DEST_PATH_IMAGE047
In the formula (I), the compound is shown in the specification,
Figure 251990DEST_PATH_IMAGE048
is a function of Kronecker-delta, if and only ifi=sTime of flight
Figure 204903DEST_PATH_IMAGE048
(i–s) =1, otherwise
Figure 461613DEST_PATH_IMAGE048
(i–s)=0；AAndBare all the functions to be solved,
Figure 149077DEST_PATH_IMAGE033
is an integral variable.
Further, for step 3, the solution is performed in consideration of the burial depth, and the specific steps are as follows:
step 3a, establishing a symbolic equation of the following form:
Figure 119308DEST_PATH_IMAGE049
wherein X and Y are both sparse matrices/vectors, and the vector C = [ is to be solved ]A 1, B 1, …, A m , B m ]T. X is
Figure 67672DEST_PATH_IMAGE050
Dimension matrix, orderi = 1, …, m-1, having:
Figure 471846DEST_PATH_IMAGE051
Figure 646607DEST_PATH_IMAGE052
Figure 30315DEST_PATH_IMAGE053
Figure 597300DEST_PATH_IMAGE054
Figure 673840DEST_PATH_IMAGE055
Figure 460531DEST_PATH_IMAGE056
whereinK i Is underground in FIG. 1iThe reflection coefficient of the boundary is determined,
Figure 647930DEST_PATH_IMAGE057
. Y is 2mDimension column vector, specific form and source layer numbersIt is related.s(ii) =1, case where a non-zero element of Y is represented by formula (34);s=mthe non-zero element of Y is represented by formula (35); the non-zero elements for the other cases Y are taken as formula (36):
Figure 69421DEST_PATH_IMAGE058
Figure 51284DEST_PATH_IMAGE059
Figure 59691DEST_PATH_IMAGE060
step 3b, mixing
Figure 50781DEST_PATH_IMAGE015
n Solving the substituted symbol equation (6)ABThen using formula (5) to obtain the correspondingf i (
Figure 326779DEST_PATH_IMAGE015
n )；
Step 3c, obtaining the compound according to the formula (25)
Figure 604177DEST_PATH_IMAGE061
As shown in Table 3 below
Figure 834301DEST_PATH_IMAGE062
And 4, when the large-range ground surface potential distribution is solved, simplifying the direct current grounding electrode into a point source and neglecting the buried depth of the direct current grounding electrode, namely, the grounding electrode in the figure 1 is considered to be simplified into a ground surface point current source, and at the moment, the Hankel integral kernel function can be solved by adopting the transmission line model shown in the figure 2, so that the effect of simplifying analysis is achieved. Method for solving earth surface potential in FIG. 1 from model of FIG. 2VThe calculation formula of (2) is as follows:
Figure 409508DEST_PATH_IMAGE063
wherein the content of the first and second substances,rthe distance between the earth surface observation point and the point current source is taken as the distance; j. the design is a square0Is a first class zero order Bessel function;
Figure 25166DEST_PATH_IMAGE033
is an integral variable;R 1as a function of the form:
Figure 879989DEST_PATH_IMAGE064
r1 can be derived from the following recursion equation:
Figure 361524DEST_PATH_IMAGE065
Figure 428837DEST_PATH_IMAGE066
Figure 915313DEST_PATH_IMAGE067
wherein the content of the first and second substances,
Figure 941038DEST_PATH_IMAGE068
in order to be the angular frequency of the frequency,
Figure 644290DEST_PATH_IMAGE069
and
Figure 515294DEST_PATH_IMAGE070
are respectively the firstiThe thickness, resistivity and permeability of the layer soil,
Figure 980910DEST_PATH_IMAGE033
i is as followsiThe wave number of the layer soil is,r i is as followsiThe wave impedance of the layer of soil,R 1in order to represent the apparent impedance,R i （i = 2, …,
Figure 177536DEST_PATH_IMAGE071
) In order to be a function of the impedance coefficient,mthe total number of layers of soil.
Potential of earth's surfaceVThe calculation results are shown in Table 4
Figure 243450DEST_PATH_IMAGE072
Details not described in this specification are within the skill of the art that are well known to those skilled in the art. It is within the spirit of the invention that conventional alternatives according to the prior art be made within the spirit of the invention.

Claims (3)

1. A method for calculating the earth surface potential of a direct current grounding electrode in a horizontal multilayer structure earth environment is characterized by comprising the following specific steps:
step 1, under the condition of considering the buried depth, order
Figure 528912DEST_PATH_IMAGE001
Resolving the Green function of the fine earth model by using a linear filtering method
Figure 242790DEST_PATH_IMAGE002
The hankel transformation is rewritten as:
Figure 429052DEST_PATH_IMAGE003
in the formula, F and F are a Hankel integral function and an integral kernel function respectively;
Figure 496365DEST_PATH_IMAGE004
representing integral variables, e x The parameters of the position are represented by,x, y
Figure 107475DEST_PATH_IMAGE005
；J0representing a first class of zero order Bessel functions;
step 2, e in the formula (2) x y–According to the interval
Figure 834997DEST_PATH_IMAGE006
Sampling, sampling interval
Figure 774134DEST_PATH_IMAGE006
Less than 0.5 times of the cut-off frequency, the discrete signal obtained by sampling can be recovered to a continuous signal without distortion, so equation (2) can be:
Figure 769772DEST_PATH_IMAGE007
in the formula:H n is a filter coefficient; f is an integral kernel function in formula (1);
Figure 845175DEST_PATH_IMAGE006
n to be the position of the sampling point(s),na serial number representing a sampling point;
step 3, the analytic solution form of the horizontal multilayer geodesic Green function is
Figure 510643DEST_PATH_IMAGE008
Figure 61710DEST_PATH_IMAGE009
Wherein
Figure 709641DEST_PATH_IMAGE010
Indicating that the point current source is at the secondiThe green's function of the layer of soil,ρ i is the source layer resistivity, z-z0To observe the horizontal distance of the point-to-point current source,
Figure 905130DEST_PATH_IMAGE011
is a function of Kronecker-delta,s representsSource layer numbering，If and only ifi=sTime of flight
Figure 866133DEST_PATH_IMAGE011
(i–s) =1, otherwise
Figure 779862DEST_PATH_IMAGE011
(i–s)=0；
Figure 992669DEST_PATH_IMAGE012
And
Figure 806779DEST_PATH_IMAGE013
are all about
Figure 938683DEST_PATH_IMAGE014
The function to be solved of (a) is,
Figure 605288DEST_PATH_IMAGE014
is an integral variable.
2. The method for calculating the earth surface potential of the direct current grounding electrode in the horizontal multilayer structure earth environment according to claim 1, wherein a 61-point version of Guptaasma-Singh is used in the step 2 to solve the formula (3):
Figure 762731DEST_PATH_IMAGE015
wherein: log (log)10(
Figure 962506DEST_PATH_IMAGE006
n ) = –5.199 138 304 + 0.116 638 304 n，n=1, 2, …, 61。
3. The method for calculating the earth surface potential of the direct current grounding electrode in the horizontal multilayer structure earth environment according to claim 1, wherein the engineering practice of the direct current magnetic biasing problem is combined in step 3, when the large-range earth surface potential distribution is solved, the direct current grounding electrode is simplified into a point source, the buried depth of the direct current grounding electrode is ignored, namely the grounding electrode is considered to be simplified into an earth surface point current source, and at the moment, the Hankel integral kernel function is solved by adopting a transmission line model;
solving earth surface potential from transmission line modelVThe calculation formula of (2) is as follows:
Figure 406257DEST_PATH_IMAGE016
wherein the content of the first and second substances,rthe distance between the earth surface observation point and the point current source is taken as the distance; j. the design is a square0Is a first class zero order Bessel function;
Figure 294578DEST_PATH_IMAGE014
is an integral variable;R 1as a function of the form:
Figure 973821DEST_PATH_IMAGE017
wherein, the formula is obtained by the following recursionR 1：
Figure 998409DEST_PATH_IMAGE018
Figure 846017DEST_PATH_IMAGE019
Figure 80689DEST_PATH_IMAGE020
Wherein the content of the first and second substances, h i 、
Figure 173410DEST_PATH_IMAGE021
are respectively the firstiThe thickness and resistivity of the layer of soil, R 1in order to represent the apparent impedance,R i in order to be a function of the impedance coefficient,i = 2, …,
Figure 318084DEST_PATH_IMAGE022
，mthe total number of layers of soil.
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