CN109101463A - A kind of more precision method for solving of wide area multi-layered earth Green's function - Google Patents
A kind of more precision method for solving of wide area multi-layered earth Green's function Download PDFInfo
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Abstract
The present invention relates to research wide area the earth to the technology of effect on power system, more particularly to a kind of more precision method for solving of wide area multi-layered earth Green's function, comprising the following steps: step 1, by the definition of wide area multi-layered earth Green's function be wide area multi-layered earth unit point electric current source potential function；Step 2, the Green's function for solving wide area multi-layered earth, the mathematic(al) representation for obtaining Green's function is containing there are two Hankel to integrate kernel function；Step 3, the mathematic(al) representation that Green's function is solved using more accuracy computation library GMP.This method is not limited by the soil number of plies and soil parameters otherness, it avoids because numerical value singularity causes Green's function to solve failure, similarly it can ensure that the success for solving wide area slaty soil Green's function and accurate to avoid the precision limitation of double-precision variable in traditional algorithm.
Description
Technical field
It is big to the technical field of effect on power system more particularly to a kind of wide area stratiform the invention belongs to study wide area the earth
More precision method for solving of ground Green's function.
Background technique
Wide area the earth (Wide-Area Earth) refers to covering broad area area, depth and passes through Moho discontinuity from earth's surface
(land 20km~70km depth, ocean 7km~8km depth) is to the earth of outer mantle.Since wide area the earth is to electric system, length
There is highly important influence apart from oil-gas pipeline system and railway/Rail Transit System, so different industries are increasingly paid attention to
Research to wide area the earth.Electric system is the wide area energy source network based on the earthed system of underground, earthed system
Protecting field, place of working, lightning proof ground and antistatic etc. four big functions influenced deeply by wide area magnetotelluric parameter, and electric system is defeated
The zero sequence impedance of electric line and buried cable is also closely related with wide area magnetotelluric parameter.In recent years, wide area the earth is to electric power
System produces more extensive and far-reaching influence, so probing into the method for establishing wide area ground potential distributed model is very must
It wants.
Traditional grounding design is often carried out just for a substation in system, usually only needs to consider 10 times of ground connection
The earth parameter of net size.It is general than tradition ground connection model in terms of the particularity of wide area the earth is embodied in coverage area and depth
Big three to four orders of magnitude of size.Labyrinth the earth is a big difficulty of ground connection fining modeling.The ground connection of electric system at present
Equidistant four-electrode method measurement view is commonly used in design in soil resistivity, the soil that inverting obtains is based on horizontal multilayered structure, the number of plies
Generally within 7 layers.But to wide area the earth, geological magnetic prospecting can obtain accurate wide area the earth soil resistivity
Parameter.Therefore, the soil resistivity parameter of the model of wide area the earth has been provided with, and establishes so studying according to wide area soil model
Ground potential function model becomes very urgent.But the method for value solving of wide area multi-layered earth potential function is more complicated,
Obtain that exact value solution is extremely difficult, the currently used method or numerical solution for solving potential function.
Two major features are presented in wide area the earth: first is that the number of plies is more, generally in ten multilayers to tens of layers；Second is soil interlayer
Resistance parameter difference is big, as more than two orders of magnitude of adjacent interlayer resisitivity, the maximum thickness of stratiform wide area the earth with most
The ratio between small thickness is often beyond two orders of magnitude.Two above-mentioned features can enable wide area ground potential function, and there are serious numerical value
Singularity, at this moment numerical method only uses more complicated derivation algorithm for solving the problem of function will face solution failure
Accurate calculated result could be obtained.So in conclusion research can obtain the mostly smart of accurate wide area the earth Green's function
Spending calculation method has research significance.
Summary of the invention
The object of the present invention is to provide a kind of methods solved suitable for the more precision of wide area multi-layered earth Green's function, wide
The domain the earth number of plies is more, under the big feature of layer parameter difference, avoids because numerical value singularity causes Green's function to solve failure, protects
The success for solving wide area slaty soil Green's function and accurate is demonstrate,proved.
To achieve the above object, the technical solution adopted by the present invention is that: a kind of wide area multi-layered earth Green's function it is mostly smart
Spend method for solving, comprising the following steps:
Step 1, by the definition of wide area multi-layered earth Green's function be wide area multi-layered earth unit point electric current source current potential letter
Number；
Step 2, the Green's function for solving wide area multi-layered earth, the mathematic(al) representation for obtaining Green's function is containing there are two
Hankel integrates kernel function；
Step 3, the mathematic(al) representation that Green's function is solved using more accuracy computation library GMP.
In more precision method for solving of above-mentioned wide area multi-layered earth Green's function, the realization of step 2 includes following step
It is rapid:
Step 2.1 initially sets up the Poisson's equation that wide area the earth Green's function is met under cylindrical-coordinate system；
Step 2.2, the expression formula for obtaining Green's function are containing there are two Hankel to be asked to integrate kernel function；
Step 2.3 finds out two Hankel integral kernel functions to be asked according to boundary condition, introduces two boundary conditions,
The boundary condition of problem is that the current potential in slaty soil bed boundaries is continuous and normal direction electric current is continuous；Then Hankel to be asked integral
Kernel function tabular form are as follows:
In formula, Ai(λ) and Bi(λ) is that Hankel to be asked integrates kernel function, and λ is the integration variable introduced when solving；ziFor
The z of i-th layer of soil and i+1 layers of soil interface is to coordinate；TAiFor about AiRefraction function, XnIt is reflective function, TBiTo close
In BiRefraction function.
In more precision method for solving of above-mentioned wide area multi-layered earth Green's function, the realization of step 3 is including the use of more
Accuracy computation library GMP integrated arithmetic and elementary function, solves the corresponding A (λ) of any sampled value λ and B (λ).
The invention has the benefit that method of the invention is suitable for asking for the more precision of Green's function of wide area multi-layered earth
Solution；The characteristics of summarizing wide area multi-layered earth is beaten for the method for solving that selection is suitable for wide area multi-layered earth Green's function later
Lower basis；The definition and theory deduction process of the Green's function of wide area multi-layered earth are proposed, Green's function will be solved and be reduced to
The Hankel for solving Green's function integrates kernel function, the solution procedure for the Green's function that standardizes；It proposes and utilizes more accuracy computations
Library GMP solves Hankel and integrates kernel function.Solution is not limited by the soil number of plies and soil parameters otherness, is avoided because of number
Value singularity causes Green's function to solve failure, can similarly protect to avoid the precision limitation of double-precision variable in traditional algorithm
The success for solving wide area slaty soil Green's function and accurate is demonstrate,proved.
Detailed description of the invention
Fig. 1 is the program flow diagram of one embodiment of the invention；
Fig. 2 is the point electric current source schematic diagram in one embodiment of the invention wide area multi-layered earth；
Fig. 3 is the process schematic that the more accuracy computation library GMP of one embodiment of the invention calculate Green's function.
Specific embodiment
Embodiments of the present invention are described in detail with reference to the accompanying drawing.
The wide area the earth Green's function of the present embodiment refers to the potential function of wide area multi-layered earth unit point electric current source.This reality
It applies example to be achieved through the following technical solutions: a kind of more precision solution sides solved suitable for wide area multi-layered earth Green's function
Method.Firstly, starting with from the theoretical derivation method of wide area multi-layered earth Green's function, wide area multi-layered earth is derived with sign matrix
The Hankel of Green's function integrates kernel function, and sign matrix derivation method facilitates big using mathematical induction summary wide area stratiform
The Green's function analytic solutions on ground, it helps verify calculated result of more precision arithmetics about integral kernel function.Then, using more
Accuracy computation library GMP solves Hankel integral kernel it is possible to prevente effectively from its numerical value singularity causes to solve the problem of failing, realization
The more precision of wide area multi-layered earth Green's function solve.The limitation of such method nonfertilization degree.Wherein: design cycle is as shown in Figure 1:
1) wide area the earth Green's function is defined.The definition of wide area multi-layered earth Green's function is wide area multi-layered earth unit point
The potential function of current source；
2) Green's function of wide area multi-layered earth is solved.Initially setting up wide area the earth Green's function under cylindrical-coordinate system expires
The Poisson's equation of foot；The expression formula of Green's function is obtained later, and Green's function is containing there are two Hankel integral kernels to be asked at this time
Function；It needs to be found out according to boundary condition wait find a function, introduces two boundary conditions, the boundary condition of problem is slaty soil layer
The borderline current potential in face is continuous and normal direction electric current is continuous；It is closed expression wait find a function, although mathematical induction can be passed through
It finds out, but there are problems that inefficiency and data truncation, overflows；
3) two Hankel to be asked are solved using more accuracy computation library GMP and integrates kernel function, the four fundamental rules integrated using GMP
Operation and elementary function, so that it may easily calculate Green's function two Hankel to be asked and integrate kernel function, and solve not
It is limited by the soil number of plies and soil parameters otherness, it similarly can be to avoid 8 bytes of double-precision variable in traditional algorithm
Precision limitation realizes that more precision solve.
Method is as follows:
I. Fig. 2 is the point electric current source schematic diagram in wide area multi-layered earth, and it is theoretical to carry out Simple Green Function on this basis
Derivation, the definition of wide area multi-layered earth Green's function is the current potential of wide area multi-layered earth unit point electric current source as shown in Figure 2
Function.In Fig. 2, ρiFor i-th layer of soil resistivity, ziFor i-th layer of soil and i+1 layers of soil interface z to coordinate, current source
Positioned at m layers.If the coordinate of point of observation is (r, z), point electric current source is located at m (1≤m≤n) layer.
II. the Green's function of wide area multi-layered earth is derived, wide area the earth Green's function meets Poisson side under cylindrical-coordinate system
Journey:
In formula: φ is the Green's function of wide area multi-layered earth；I is the size of point electric current source；M is that layer is compiled where point electric current source
Number；ρmFor active layer resistivity；δ is Dirac function；D is point source and point of observation distance.Use the separation of variable, i-th layer of lattice
Woods function phiiExpression formula is as follows:
In formula: λ is the integration variable introduced when solving；J0For first kind zero Bessel function；Ai(λ) and Bi(λ) be to
It quadratures kernel function.
III. the form that can be written as follow according to the convergence of boundary condition and numerical method, interlayer recurrence relation:
Wherein:
Ei=exp (2 λ (zi-z0)) (9)
For first floor the earth, A1And B1There is following relationship:
exp(2λz0)A1(λ)=B1(λ)+δ(m-1) (10)
Joint above formula can solve AiAnd Bi.The result shows that AiOr and BiIt is the function of λ:
Formula (11) and formula (12) show no matter active layer position, the intrinsic denominator term X of A and BnIt is identical.XnFor multinomial
Form shares 2n–1, and and z0It is unrelated.Molecule item T has 2n–2, with buried depth z0It is related.AiOr BiAlthough closed expression can
To be found out by mathematical induction, but the number of plies n of wide area multi-layered earth is greater than 10, so when analytically method changes into numerical methods of solving
Can there are problems that inefficiency.According to numerical method direct solution A and B, but formula (3) and (4) belong to singular equation, big in λ
The phenomenon that will appear data truncation when a certain critical value and overflowing.Due to the more general problem with grounding of the z of wide area multi-layered earth
To be big, therefore conventional method can not accurately solve wide area multi-layered earth Green's function.
IV. currently, computer can only approach number limitedly to store position.Infinite precision is a difficulty for computer
With the obstacle gone beyond, but people approach still through stronger and stronger hardware and software to this ability.The present embodiment
The more precision methods for solving wide area multi-layered earth Green's function are built upon famous more accuracy computation library GMP (The GNU
Multiple Precision Arithmetic Library) on.
GMP is a floating-point arithmetic library.It is support the signed integer of arbitrary accuracy, rational, floating number behaviour
Work and operation.Computer conditions permit does not have any restrictions for operable precision digit.GMP have one it is very rich
There is the interface of standard in function library, these libraries.GMP has to be widely applied very much, has all been obtained very in Mathematics Research and cryptography
Good application.For small-scale operand and extensive computation number, GMP is designed into most fast speed as far as possible.GMP is than it
He has the faster speed of service by large database concept.
As shown in figure 3, mainly having following a few class function libraries: high-precision symbol integer arithmetic function inside GMP function library
(mpz type)；High-precision rational number operation function (mpq type)；High-precision floating-point operation function (mpf type).Make writing
Before program with the library GMP, it should cover the header file in program.In the library GMP, all statements using GMP are all wrapped
It is contained among header file<gmp.h>, it supports the compiling of C language and C Plus Plus.It should be noted that GMP needs will in program
<stdio.h>can just come into force after including, so must be before addition<gmp.h>comprising just can be with into<stdio.h>
It operates normally, such as:
#include<stdio.h>
#include<gmp.h>
The present embodiment utilizes the GMP arithmetic integrated and elementary function, so that it may intuitive and convenient ground calculating formula (3), formula
(4) and formula (10) to solve the corresponding A (λ) of any sampled value λ and B (λ) avoids the precision in traditional algorithm from limiting.If
Computer conditions permit, the operable precision digit of GMP do not have any restrictions.In view of the balance of precision, speed and memory, originally
Embodiment recommends the numerical method direct solution that AB is carried out using the GMP of 1024 bytes.It should be noted that limited storage position
Although GMP cannot solve AB the phenomenon that (overflow and truncation still have) λ → ∞ the case where, formula (11) and formula (12) are in λ
Expression formula when → ∞ be can be derived by by analytic method, as long as therefore more precision methods apply when can cover type (2) integral
Calculate the AB valid interval needed.
When it is implemented, a kind of method solved suitable for the more precision of wide area multi-layered earth Green's function, first defines
Wide area multi-layered earth Green's function；Then start with from the theoretical derivation method of wide area multi-layered earth Green's function, with symbol square
The Hankel that battle array derives wide area multi-layered earth Green's function integrates kernel function；Finally, being solved using more accuracy computation library GMP
Hankel integral kernel realizes that the more precision of wide area multi-layered earth Green's function solve:
A. wide area the earth Green's function is defined.The definition of wide area multi-layered earth Green's function is wide area multi-layered earth unit point
The potential function of current source.
B. the Green's function of wide area multi-layered earth is solved.Initially setting up wide area the earth Green's function under cylindrical-coordinate system expires
The Poisson's equation of foot；The expression formula of Green's function is obtained later, and Green's function is containing there are two Hankel integral kernels to be asked at this time
Function；Two Hankel integral kernel function needs to be asked are found out according to boundary condition, introduce two boundary conditions, the side of problem
Boundary's condition is that the current potential in slaty soil bed boundaries is continuous and normal direction electric current is continuous；It is closed expression wait find a function, although
It can be found out by mathematical induction, but there are problems that inefficiency and data truncation, overflow.
C. Hankel integral kernel is solved using more accuracy computation library GMP, utilizes the GMP arithmetic integrated and elementary letter
Number, so that it may easily calculate Green's function and wait finding a function, and solve not by the limit of the soil number of plies and soil parameters otherness
System similarly can realize that more precision solve to avoid the precision limitation of double-precision variable in traditional algorithm.
Method is as follows:
1. the definition of wide area multi-layered earth Green's function is the electricity of wide area multi-layered earth unit point electric current source as shown in Figure 2
Bit function, when solving Green's function, column write the Poisson's equation that wide area the earth Green's function is met under cylindrical-coordinate system first；
By solving equation the expression formula of available Green's function, discovery Green's function expression formula is containing there are two Hankel integral kernel letters
Number will just convert the Solve problems of Green's function for the solution to Hankel integral kernel function.
2. more precision methods that the present embodiment solves wide area multi-layered earth Green's function are built upon famous more precision meters
It calculates on the GMP of library.Utilize the GMP arithmetic integrated and elementary function, so that it may easily calculating formula (3), formula (4) and formula
(10), to solve the corresponding A (λ) of any sampled value λ and B (λ), the precision of double-precision variable in traditional algorithm is avoided to limit.
It should be understood that the part that this specification does not elaborate belongs to the prior art.
Although being described in conjunction with the accompanying a specific embodiment of the invention above, those of ordinary skill in the art should
Understand, these are merely examples, various deformation or modification can be made to these embodiments, without departing from original of the invention
Reason and essence.The scope of the present invention is only limited by the claims that follow.
Claims (3)
1. a kind of more precision method for solving of wide area multi-layered earth Green's function, characterized in that the following steps are included:
Step 1, by the definition of wide area multi-layered earth Green's function be wide area multi-layered earth unit point electric current source potential function；
Step 2, the Green's function for solving wide area multi-layered earth, the mathematic(al) representation for obtaining Green's function is containing there are two Hankel
Integrate kernel function；
Step 3, the mathematic(al) representation that Green's function is solved using more accuracy computation library GMP.
2. more precision method for solving of wide area multi-layered earth Green's function as described in claim 1, characterized in that step 2
Realize the following steps are included:
Step 2.1 initially sets up the Poisson's equation that wide area the earth Green's function is met under cylindrical-coordinate system；
Step 2.2, the expression formula for obtaining Green's function are containing there are two Hankel to be asked to integrate kernel function；
Step 2.3 finds out two Hankel integral kernel functions to be asked according to boundary condition, introduces two boundary conditions, problem
Boundary condition be that current potential in slaty soil bed boundaries is continuous and normal direction electric current is continuous；Then Hankel integral kernel letter to be asked
Number tabular form are as follows:
In formula, Ai(λ) and Bi(λ) is that Hankel to be asked integrates kernel function, and λ is the integration variable introduced when solving；ziIt is i-th
The z of layer soil and i+1 layers of soil interface is to coordinate；TAiFor about AiRefraction function, XnIt is reflective function, TBiFor about Bi
Refraction function.
3. more precision method for solving of wide area multi-layered earth Green's function as described in claim 1, characterized in that step 3
It realizes including the use of more accuracy computation library GMP arithmetic integrated and elementary function, solves the corresponding A (λ) of any sampled value λ
With B (λ).
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| CN110850231A (en) * | 2019-11-07 | 2020-02-28 | 中铁电气化局集团有限公司 | Ground grid safety real-time analysis method and system applied to railway traction substation |
| CN111723491A (en) * | 2020-06-29 | 2020-09-29 | 重庆大学 | Grounding parameter acquisition method based on arbitrary layering soil Green function |
| CN111753247A (en) * | 2020-06-29 | 2020-10-09 | 重庆大学 | Method for acquiring horizontal layered soil grounding parameters by using segmented sampling |
| CN111929348A (en) * | 2020-10-19 | 2020-11-13 | 国网江西省电力有限公司电力科学研究院 | Method for calculating surface potential of direct current grounding electrode in horizontal multilayer structure earth environment |
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Cited By (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110850231A (en) * | 2019-11-07 | 2020-02-28 | 中铁电气化局集团有限公司 | Ground grid safety real-time analysis method and system applied to railway traction substation |
| CN111723491A (en) * | 2020-06-29 | 2020-09-29 | 重庆大学 | Grounding parameter acquisition method based on arbitrary layering soil Green function |
| CN111753247A (en) * | 2020-06-29 | 2020-10-09 | 重庆大学 | Method for acquiring horizontal layered soil grounding parameters by using segmented sampling |
| CN111753247B (en) * | 2020-06-29 | 2023-08-18 | 重庆大学 | Method for acquiring horizontal layered soil grounding parameters by using segmented sampling |
| CN111723491B (en) * | 2020-06-29 | 2024-03-19 | 重庆大学 | Grounding parameter acquisition method based on arbitrary layered soil green function |
| CN111929348A (en) * | 2020-10-19 | 2020-11-13 | 国网江西省电力有限公司电力科学研究院 | Method for calculating surface potential of direct current grounding electrode in horizontal multilayer structure earth environment |
| CN111929348B (en) * | 2020-10-19 | 2021-03-02 | 国网江西省电力有限公司电力科学研究院 | Method for calculating surface potential of direct current grounding electrode in horizontal multilayer structure earth environment |
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