CN103969506A - Three-phase power cable harmonic loss computing method - Google Patents

Abstract

The invention discloses a three-phase power cable harmonic loss computing method which comprises the steps that (A), according to the actual structure and actual laying mode of cables, shielding layer current and magnetic field coupling conditions, a formula of three-phase power cable harmonic loss under current and voltage with a specific harmonic number is deduced; (A2), current density functions of all layers of a three-phase power cable under current with the specific harmonic number are calculated; (A3) the total harmonic loss of all layers of the three-phase power cable under the current and voltage with the specific harmonic number is calculated; (B) through the step (A), the total harmonic loss of all layers of the three-phase power cable under the current and voltage with various harmonic numbers is obtained; (C) the sum of the total harmonic loss, obtained in the step (B), of all layers of the three-phase power cable under the current and voltage with various harmonic numbers is solved to obtain the three-phase power cable harmonic loss. Estimation of the operation temperature and the service life of the power cable under harmonic is facilitated, model selection of the power cable under the condition of harmonic is facilitated, and technical support is provided for long-distance transmission project construction.

Description

A kind of three-phase electrical cable harmonic loss computing method

Technical field

The present invention relates to electric power quality analysis field, relate in particular to a kind of three-phase electrical cable harmonic loss computing method.

Background technology

Three-phase electrical cable is as the important medium in delivery of electrical energy process, and the problem that is subject to system harmonics impact becomes increasingly conspicuous.Carry out harmonic wave on the research of three-phase electrical cable impact in, due to factors such as the distributed capacitance that exists kelvin effect, approach effect and the insulating medium of wire to exist, increase very large difficulty all can to research.

Consult electric relevant criterion, specified by IEEE and IEC standard applicable to the power cable model under harmonic background is many.In IEEE cable model, extract harmonic current size and each harmonic containing ratio in the time of frequency analysis, by the correction to AC resistance under each harmonic, calculate respectively each harmonic loss, stack solves total three-phase electrical cable harmonic loss.In solution procedure, current algorithm is all with experimental formula correction AC resistance: R (h)=R _dc(0.187+0.532h ^0.5), although this algorithm consider to some extent for kelvin effect, ignored approach effect, do not consider the impact that bring loss in the magnetic field that intercouples between three-phase electrical cable.

IEC proposes with kelvin effect factor y in its standard IEC 60287-1 _s, approach effect factor y _previse industrial frequency AC resistance, thereby obtain the algorithm of electric parameter under harmonic wave, wherein, the kelvin effect factor the approach effect factor $y_p=y_s\·{\left(\frac{D_c}{s}\right)}^2\·[0.312{\left(\frac{D_c}{s}\right)}^2+\frac{1.8}{y_s+0.27}].$ Although the method has solved the impact of harmonic wave approach effect on three-phase electrical cable loss in IEEE cable model, the impact three-phase electrical cable system of laying and three-phase electrical cable screen layer electric current not being caused to loss is reasonably investigated.

Visible, in relevant international standard, there are some computing method to carry out correlation computations to three-phase electrical cable model under harmonic wave and three-phase electrical cable loss, but all there is certain limitation, main cause is:

(1) because three-phase electrical cable in existing computing method adopts overhead transmission line model, three-phase electrical cable is reduced to long straight conductor, do not set up rational harmonic-model for three-phase electrical cable self structure, calculate thereby directly have influence on three-phase electrical cable harmonic loss;

(2) owing in existing computing method, the electric parameter under harmonic wave being adopted to experimental formula correction, the impact of actual three-phase electrical cable and system of laying thereof is not paid attention to, affect computational accuracy.

Three-phase electrical cable because phase spacing is less, will produce the mutual coupling phenomenon in magnetic field in the time of actual motion, has increased running wastage, and particularly, in harmonic wave situation, this phenomenon is even more serious.Current all analysis means are usually ignored the problem of this aspect, and the actual system of laying of three-phase electrical cable is lacked to analysis, long distance powedr transmission cable loss is calculated and occur major defect.Therefore, in view of the harmfulness of harmonic loss to three-phase electrical cable operation and the shortage of present analysis means, need badly a kind of can be in conjunction with three-phase electrical cable practical structures, consider the three-phase electrical cable harmonic loss computing method of screen layer electric current and alternate laying state.

Summary of the invention

The object of this invention is to provide a kind of three-phase electrical cable harmonic loss computing method, can carry out comprehensive analytical calculation in conjunction with three-phase electrical cable practical structures and system of laying, consideration screen layer electric current and three-phase electrical cable phased magnetic field coupling situation, finally obtain three-phase electrical cable harmonic loss accurately.The present invention can be power cable running temperature and the estimation in life-span under harmonic wave and makes early-stage preparations, for power cable type selecting under harmonic environment is offered help, for long distance powedr transmission engineering construction provides technical support.

The present invention adopts following technical proposals:

A kind of three-phase electrical cable harmonic loss computing method, comprise the following steps:

A: under the prerequisite in conjunction with three-phase electrical cable practical structures and the actual system of laying of three-phase electrical cable, consideration screen layer electric current and three-phase electrical cable phased magnetic field coupling situation, three-phase electrical cable harmonic loss formula under derivation particular harmonic number of times electric current and voltage, the step of specifically deriving is as follows:

A1: the alternate spacing of determining the each phase of three-phase electrical cable according to the actual system of laying of three-phase electrical cable, induction electromotive force under calculating particular harmonic number of times electric current in each phase screen layer of unit length, then in conjunction with three-phase electrical cable screen layer two-terminal-grounding resistance, determine screen layer ground circuit and calculate the impedance of three-phase electrical cable screen layer and the impedance of three-phase electrical cable ground circuit according to the screen layer connected mode of three-phase electrical cable, final calculating obtains each phase power cable screen layer electric current under particular harmonic number of times electric current;

A2: calculate the function of current density of each layer of three-phase electrical cable under particular harmonic number of times electric current;

A3: ask for respectively cable core loss, insulation course loss and the screen layer loss of each layer of three-phase electrical cable under particular harmonic number of times, again cable core loss, insulation course loss and screen layer loss are sued for peace, finally calculated the harmonic wave total losses of each layer of three-phase electrical cable under particular harmonic number of times electric current and voltage;

B: detect by actual checkout equipment, obtain electric current and the voltage of the each phase of three-phase electrical cable, then by FFT resolver, analyze the size and the relative harmonic content that obtain individual harmonic current and harmonic voltage; Then the electric current and voltage value under each harmonic is carried out to computing according to steps A, obtain respectively each phase power cable screen layer electric current under individual harmonic current by steps A 1, try to achieve respectively the function of current density of each layer of three-phase electrical cable under individual harmonic current by steps A 2, try to achieve respectively the harmonic wave total losses of each layer of three-phase electrical cable under individual harmonic current and voltage by steps A 3;

C: the processing of suing for peace of the harmonic wave total losses to each layer of the three-phase electrical cable obtaining in step B under individual harmonic current and voltage, what finally obtain is three-phase electrical cable harmonic loss with value.

In described steps A 1, the induction electromotive force under particular harmonic number of times electric current in each phase screen layer of unit length comprises the induction electromotive force of the each phase screen layer generation of three-phase cable core electric current to unit length and the induction electromotive force that screen layer electric current produces each phase screen layer of unit length;

The induction electromotive force that three-phase cable core electric current produces in the A of unit length phase screen layer P for:

The induction electromotive force that three-phase cable core electric current produces in the B of unit length phase screen layer P for:

${\stackrel{.}{e}}_{\mathrm{SB}}=2\mathrm{\ωI}\×{10}^{-7}[\frac{\sqrt{3}}{2}\ln \frac{\mathrm{mS}}{R}-j\frac{1}{2}\ln \frac{S}{m\·R}];$

The induction electromotive force that three-phase cable core electric current produces in the C of unit length phase screen layer P for:

${\stackrel{.}{e}}_{\mathrm{SC}}=2\mathrm{\ωI}\×{10}^{-7}[-\frac{\sqrt{3}}{2}\ln \frac{\mathrm{mS}}{R}-j\frac{1}{2}\ln \frac{n^{2}s}{m\·R}];$

Wherein, j is imaginary unit; ω=2 π f, f represents harmonic frequency; P is the conductor that is parallel to cable core A, B, C, as virtual screen layer; the magnetic flux producing in P for three-phase cable core; I is three-phase cable core electric current; Three-phase electrical cable A, C spaced apart are S; The n that A, B spaced apart are S times, i.e. nS; The m that B, C spaced apart are S times, i.e. mS; R is the radius of P;

The induction electromotive force that screen layer electric current produces unit length screen layer is:

${\stackrel{.}{e}}_{\mathrm{SA}}^{\′}={\stackrel{.}{I}}_{\mathrm{SB}}\·{\mathrm{jM}}_{\mathrm{AB}}+{\stackrel{\·}{I}}_{\mathrm{SC}}\·{\mathrm{jM}}_{\mathrm{AC}};$

${\stackrel{.}{e}}_{\mathrm{SB}}^{\′}={\stackrel{.}{I}}_{\mathrm{SA}}\·{\mathrm{jM}}_{\mathrm{AB}}+{\stackrel{.}{I}}_{\mathrm{SC}}\·{\mathrm{jM}}_{\mathrm{BC}};$

${\stackrel{.}{e}}_{\mathrm{SC}}^{\′}={\stackrel{.}{I}}_{\mathrm{SA}}\·{\mathrm{jM}}_{\mathrm{AC}}+{\stackrel{.}{I}}_{\mathrm{SB}}\·{\mathrm{jM}}_{\mathrm{BC}};$

Wherein, for the induction electromotive force that in three-phase electrical cable, B, C phase screen layer electric current produce in unit length A phase screen layer; the induction electromotive force producing in unit length B phase screen layer for three-phase electrical cable A, C phase screen layer electric current; the induction electromotive force producing in unit length C phase screen layer for three-phase electrical cable A, B phase screen layer electric current; represent respectively three-phase electrical cable A, B, C phase screen layer electric current; M _aB, M _bC, M _cArespectively C and the mutual inductance of A phase screen layer under B and the mutual inductance of C phase screen layer under three-phase electrical cable A and the mutual inductance of B phase screen layer, unit length, unit length under representation unit length; J is imaginary unit;

Determine screen layer ground circuit according to the screen layer connected mode of three-phase electrical cable, in conjunction with three-phase electrical cable screen layer two-terminal-grounding resistance, and calculate the impedance of three-phase electrical cable screen layer, the impedance of three-phase electrical cable ground circuit and induction electromotive force; Induction electromotive force, the impedance of three-phase electrical cable screen layer and the impedance of three-phase electrical cable ground circuit are multiplied by line length by institute's calculated value under unit length and are obtained;

Final calculating obtains each phase power cable screen layer electric current under particular harmonic number of times electric current.

In described steps A 2, utilize the boundary condition Simultaneous Equations of vector magnetic potential A on power cable actual physics layer, introduce bessel function, adopt separation of variable solving equation, in conjunction with boundary condition, obtain the result of vector magnetic potential A, then pass through relational expression: can obtain the function of current density J of each layer of three-phase electrical cable under particular harmonic number of times electric current is:

$J=\underset{n=1}{\overset{\∞}{\mathrm{\Σ}}}{J}_n\left(r\right)\·{\mathrm{\Θ}}_n\left(\mathrm{\θ}\right);$

Wherein, modifying factor ${\mathrm{\Θ}}_n\left(\mathrm{\θ}\right)=\underset{k=1}{\overset{m}{\mathrm{\Σ}}}\frac{I_k}{I}\·\frac{1}{a_k^n}\cos n(\mathrm{\θ}+{\mathrm{\α}}_k),$ M represents the cable number of phases of studied cable outside; b _krepresent the spacing of k phase cable and the cable of studying, definition a _k=b _k/ b ₁; I represents to study the cable core electric current of cable; I _krepresent the in addition cable core electric current of k phase cable of institute's cable of studying; α _krepresent the locus of k phase cable; R represents utmost point footpath, and θ represents polar angle.

In described steps A 3,

By the Joule law of differential form is carried out to integration, cable core loss and the screen layer loss of calculating unit length cable under particular harmonic number of times electric current are

$P_i=\frac{\mathrm{\π}}{g_i}\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}{\mathrm{\φ}}_M\·{\∫J}_n\left(r\right)\·J_n^{*}\left(r\right)\·\mathrm{rdr};$ Wherein, i=1,3, P ₁represent cable core loss, P ₂represent screen layer loss,

m represents to study the cable external cable number of phases; a _k=b _k/ b ₁, b _krepresent the spacing of k phase cable and the cable of studying; represent i phase cable conductor current phase; represent J _n(r) conjugate;

Under particular harmonic number of times voltage, the insulation course loss W of unit length cable is:

wherein, γ is insulating material conductivity, and U is harmonic voltage value, r ₁for cable core radius, r ₂for screen layer external radius.

The present invention is according to vector magnetic potential Poisson equation and Laplace's equation, in conjunction with the actual system of laying of power cable, consider kelvin effect, the impact of approach effect on three-phase electrical cable loss under harmonic wave, utilize bessel function and separation of variable processing vector magnetic potential to set up three-phase electrical cable harmonic loss formula under particular harmonic number of times, finally calculate three-phase electrical cable harmonic loss according to three-phase electrical cable actual parameter.The present invention, in the time that the actual system of laying of research three-phase electrical cable affects loss, provides introducing modifying factor according to derivation result, can accurately and easily analyze three-phase electrical cable and even polyphase electric power cable harmonic loss.The present invention can be power cable running temperature and the estimation in life-span under harmonic wave and makes early-stage preparations, for power cable type selecting under harmonic environment is offered help, for long distance powedr transmission engineering construction provides technical support.

Brief description of the drawings

Fig. 1 is process flow diagram of the present invention;

Fig. 2 is screen layer induction electromotive force computation model schematic diagram;

Fig. 3 is three-phase electrical cable screen layer circulation equivalent circuit diagram in the present invention;

Fig. 4 is multi-phase cable space distribution schematic diagram in the present invention.

Embodiment

As shown in Figure 1, three-phase electrical cable harmonic loss computing method of the present invention, comprise the following steps:

A: under the prerequisite in conjunction with three-phase electrical cable practical structures and the actual system of laying of three-phase electrical cable, consideration screen layer electric current and three-phase electrical cable phased magnetic field coupling situation, three-phase electrical cable harmonic loss formula under derivation particular harmonic number of times electric current and voltage, the step of specifically deriving is as follows:

A1: the alternate spacing of determining the each phase of three-phase electrical cable according to the actual system of laying of three-phase electrical cable, induction electromotive force under calculating particular harmonic number of times electric current in each phase screen layer of unit length, then in conjunction with three-phase electrical cable screen layer two-terminal-grounding resistance, determine screen layer ground circuit and calculate the impedance of three-phase electrical cable screen layer and the impedance of three-phase electrical cable ground circuit according to the screen layer connected mode of three-phase electrical cable, final calculating obtains each phase power cable screen layer electric current under particular harmonic number of times electric current;

Determine the alternate spacing of the each phase of three-phase electrical cable according to the actual system of laying of three-phase electrical cable, the induction electromotive force under calculating particular harmonic number of times electric current in each phase screen layer of unit length; Induction electromotive force under particular harmonic number of times electric current in each phase screen layer of unit length comprises the induction electromotive force of the each phase screen layer generation of three-phase cable core electric current to unit length and the induction electromotive force that screen layer electric current produces each phase screen layer of unit length;

Consider the arrangement mode of three-phase electrical cable, in conjunction with Fig. 2, by asking for the magnetic flux in screen layer and then trying to achieve induction electromotive force size.As shown in Figure 2, in three-phase electrical cable, the spaced apart of A, C two-phase is S; The n that in three-phase electrical cable, the spaced apart of A, B two-phase is S times, i.e. nS; The m that in three-phase electrical cable, the spaced apart of B, C two-phase is S times, i.e. mS.For convenience of calculating, using P as virtual screen layer; Screen layer P is the conductor that is parallel to three-phase electrical cable cable core A, B, C, the virtual location of the positional representation phase cable shield of screen layer P in Fig. 2, R is the radius of screen layer P, the spacing of A, B, C three-phase and screen layer P is continued to use the definition mode of three spaced apart in above-mentioned three-phase electrical cable, is respectively D, lD, kD.Consider soil magnetic permeability and permeability of vacuum approximately equal, can obtain the magnetic flux that A, B, C three-phase cable core electric current produce in the screen layer P of unit length and be:

Wherein, represent respectively the magnetic flux that A, B, C three-phase cable core electric current produce in the screen layer P of unit length, represent respectively A, B, C three-phase cable core electric current.The total magnetic flux that three-phase cable core electric current produces in the screen layer P of unit length for:

In the time that the suffered cable core electric current of the A phase screen layer that solves unit length produces influence of magnetic field, can judge the center superposition of cable core A and screen layer P.Now, D=R, lD=nS, kD=S, obtains:

Consider three-phase cable core electric current symmetry, order ? ${\stackrel{.}{I}}_C=(-\frac{1}{2}+j\frac{\sqrt{3}}{2})I,$ Substitution formula (3), obtains:

According to induction electromotive force, definition can obtain the induction electromotive force that three-phase cable core electric current produces in the A of unit length phase screen layer P shown in (5):

Wherein, j is imaginary unit, ω=2 π f, and f represents harmonic frequency.

In the time of cable core B and screen layer P center superposition, kD=R, D=S, lD=mS, the induction electromotive force that three-phase cable core electric current produces in the B of unit length phase screen layer P shown in (6):

${\stackrel{.}{e}}_{\mathrm{SB}}=2\mathrm{\ωI}\×{10}^{-7}[\frac{\sqrt{3}}{2}\ln \frac{\mathrm{mS}}{R}-j\frac{1}{2}\ln \frac{S}{m\·R}]---\left(6\right);$

In like manner, the induction electromotive force that three-phase cable core electric current produces in the C of unit length phase screen layer P shown in (7):

${\stackrel{.}{e}}_{\mathrm{SC}}=2\mathrm{\ωI}\×{10}^{-7}[-\frac{\sqrt{3}}{2}\ln \frac{\mathrm{mS}}{R}-j\frac{1}{2}\ln \frac{n^{2}s}{m\·R}]---\left(7\right);$

Because a certain phase screen layer electric current can produce induction electromotive force in other phase screen layers, in conjunction with the mutual inductance between each phase screen layer, the induction electromotive force that can produce in unit length screen layer in the hope of screen layer electric current, the induction electromotive force that screen layer electric current produces unit length screen layer is suc as formula shown in (8), (9), (10).

${\stackrel{.}{e}}_{\mathrm{SA}}^{\′}={\stackrel{.}{I}}_{\mathrm{SB}}\·{\mathrm{jM}}_{\mathrm{AB}}+{\stackrel{\·}{I}}_{\mathrm{SC}}\·{\mathrm{jM}}_{\mathrm{AC}}---\left(8\right);$

${\stackrel{.}{e}}_{\mathrm{SB}}^{\′}={\stackrel{.}{I}}_{\mathrm{SA}}\·{\mathrm{jM}}_{\mathrm{AB}}+{\stackrel{.}{I}}_{\mathrm{SC}}\·{\mathrm{jM}}_{\mathrm{BC}}---\left(9\right);$

${\stackrel{.}{e}}_{\mathrm{SC}}^{\′}={\stackrel{.}{I}}_{\mathrm{SA}}\·{\mathrm{jM}}_{\mathrm{AC}}+{\stackrel{.}{I}}_{\mathrm{SB}}\·{\mathrm{jM}}_{\mathrm{BC}}---\left(10\right);$

Wherein, for the induction electromotive force that in three-phase electrical cable, B, C phase screen layer electric current produce in unit length A phase screen layer; the induction electromotive force producing in unit length B phase screen layer for three-phase electrical cable A, C phase screen layer electric current; the induction electromotive force producing in unit length C phase screen layer for three-phase electrical cable A, B phase screen layer electric current; represent respectively three-phase electrical cable A, B, C phase screen layer electric current; M _aB, M _bC, M _cArespectively C and the mutual inductance of A phase screen layer under B and the mutual inductance of C phase screen layer under three-phase electrical cable A and the mutual inductance of B phase screen layer, unit length, unit length under representation unit length.Introduce the equivalent depth D of Carson formula _e, can obtain under unit length in three-phase electrical cable two-phase screen layer mutual inductance M arbitrarily:

$M=2\mathrm{\ω}\×{10}^{-7}\ln \frac{D_e}{S}---\left(11\right);$

$D_e=659.33\sqrt{\frac{\mathrm{\ρ}}{f}}---\left(12\right);$

Wherein, ρ is soil resistivity, and f is harmonic frequency, and ρ and f all can be by using actual measurement device measuring to draw; S is two-phase screen layer spacing.

Determine screen layer ground circuit according to the screen layer connected mode of three-phase electrical cable, calculate particular harmonic number of times screen layer electric current; Three-phase electrical cable screen layer circulation equivalent circuit diagram as shown in Figure 3.

In long distance powedr transmission, screen layer often adopts the cross interconnected mode of connection of segmentation, screen layer internal induction electromotive force is asked in segmentation, determine the associated loop parameter in screen layer loop according to the concrete connected mode of screen layer, the associated loop parameter in screen layer loop comprises cable shield layer impedance, cable shield two-terminal-grounding resistance and ground circuit impedance.As shown in Figure 3, R+jX represents cable shield layer impedance, R ₁, R ₂represent screen layer two-terminal-grounding resistance, Re represents ground circuit impedance, represent respectively the induction electromotive force that three-phase cable core electric current produces in A, B, C threephase cable screen layer, E _sAthe induction electromotive force that in ' expression three-phase electrical cable, B, C phase screen layer electric current produce in A phase screen layer, E _sBthe induction electromotive force that in ' expression three-phase electrical cable, A, C phase screen layer electric current produce in B phase screen layer, E _sCthe induction electromotive force that in ' expression three-phase electrical cable, A, B phase screen layer electric current produce in C phase screen layer.In Fig. 3, induction electromotive force, cable shield layer impedance, ground circuit impedance are multiplied by line length by institute's calculated value under unit length and are obtained.

Write system of equations according to Fig. 3 row:

$\left[\begin{array}{cccccc}{R}_A& R_B& R_B& -X& -X_1& -X_2\\ R_B& R_A& R_B& -X_1& -X& {-X}_1\\ R_B& R_B& R_A& -X_2& -X_1& -X\\ X& X_1& X_2& R_A& R_B& R_B\\ X_1& X& X_1& R_B& R_A& R_B\\ X_2& X_1& X& R_B& R_B& R_A\end{array}\right]\left[\begin{array}{c}{I}_{\mathrm{SAr}}\\ I_{\mathrm{SBr}}\\ I_{\mathrm{SCr}}\\ I_{\mathrm{SAi}}\\ I_{\mathrm{SBi}}\\ I_{\mathrm{SCi}}\end{array}\right]=\left[\begin{array}{c}{E}_{\mathrm{SAr}}\\ E_{\mathrm{SBr}}\\ E_{\mathrm{SCr}}\\ E_{\mathrm{SAi}}\\ E_{\mathrm{SBi}}\\ E_{\mathrm{SCi}}\end{array}\right]---\left(13\right)$

Wherein, R _a=R ₁+ R ₂+ R _e+ R, R _b=R ₁+ R ₂+ R _e, X, X ₁, X ₂represent screen layer induction reactance and mutual inductance, I _sAr, I _sBr, I _sCr, E _sAr, E _sBr, E _sCrrepresent the real part of screen layer three phase circulations, induction electromotive force, I _sAi, I _sBi, I _sCi, E _sAi, E _sBi, E _sCirepresent the imaginary part of screen layer three phase circulations, induction electromotive force.

Solve formula (13) and can obtain each phase power cable screen layer electric current under particular harmonic number of times electric current.

A2: calculate the function of current density of each layer of three-phase electrical cable under particular harmonic number of times electric current.

Because the curl of vector magnetic potential A is magnetic induction density B, magnetic induction density B size direction can magnetic reaction fields distribution situation, therefore function of current density and vector magnetic potential exist relation of equal quantity, asking for of function of current density can be converted into asking for of vector magnetic potential.Because vector magnetic potential meets Laplace's equation formula (14) at insulation course and cable perimeter, in copper core and screen layer, meet Poisson equation formula (15), the cylindrical coordinates form that therefore can obtain vector magnetic potential is:

$\frac{{\∂}^{2}A}{{\∂r}^2}+\frac{1}{r}\·\frac{\∂A}{\∂r}+\frac{1}{r^2}\·\frac{{\∂}^{2}A}{{\∂\mathrm{\θ}}^2}=0---\left(14\right);$

$\frac{{\∂}^{2}A}{{\∂r}^2}+\frac{1}{r}\·\frac{\∂A}{\∂r}+\frac{1}{r^2}\·\frac{{\∂}^{2}A}{{\∂\mathrm{\θ}}^2}=-m^{2}A=0---\left(15\right);$

Wherein, m=j ω μ g, r represents utmost point footpath, and θ represents polar angle, and μ represents magnetic permeability, and g represents conductivity.

Consider that vector magnetic potential A (is cable core, insulation course, metal screen layer at power cable actual physics layer, other structures are ignored) on boundary condition Simultaneous Equations, introduce bessel function, adopt separation of variable solving equation, in conjunction with boundary condition, can obtain the result of vector magnetic potential A, suc as formula (17) to shown in formula (20).Then pass through relational expression: can obtain function of current density J.As shown in Figure 4, for polyphase electric power cable, consider the impact of actual system of laying on current phasor function, propose modifying factor Θ _n(θ):

${\mathrm{\Θ}}_n\left(0\right)=\underset{k=1}{\overset{m}{\mathrm{\Σ}}}\frac{I_k}{I}\·\frac{1}{a_k^n}\cos n(\mathrm{\θ}+{\mathrm{\α}}_k)---\left(16\right);$

Synthesizing map 4 and formula (16), m represents the cable number of phases of studied cable outside; b _krepresent the spacing of k phase cable and the cable of studying, definition a _k=b _k/ b ₁; I represents to study the cable core electric current of cable; I _krepresent the in addition cable core electric current of k phase cable of institute's cable of studying; α _krepresent the locus of k phase cable.

Use r ₁, r ₂, r ₃represent respectively the radius of cable core, insulation course, screen layer, only consider I ₁research each Physical layer magnetic field of cable is intercoupled, carries out solving of vector magnetic potential, can obtain:

Work as r ₁<r≤r ₂, the expression formula of vector magnetic potential is

$A=A_0^{\&prime;}\ln r+\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}[A_n^{\&prime;}\&CenterDot;r^{-n}+B_n^{\&prime;}\&CenterDot;r^n]\cos \left(\mathrm{n\&theta;}\right)---\left(17\right);$

As r≤r ₁, the expression formula of vector magnetic potential is $A(r,\mathrm{\&theta;})=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{A}_n{I}_n\left(m_{1}r\right)\cos \left(\mathrm{n\&theta;}\right)---\left(18\right);$

Work as r ₂≤ r≤r ₃, the expression formula of vector magnetic potential is

$A(r,\mathrm{\&theta;})=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}[B_n{I}_n\left(m_{2}r\right)+C_n{K}_n\left(m_{2}r\right)]\cos \left(\mathrm{n\&theta;}\right)---\left(19\right);$

Work as r ₃<r, the expression formula of vector magnetic potential is

$A=-\frac{{\mathrm{\&mu;}}_{0}I}{2\mathrm{\&pi;}}[\ln \left(b\right)-\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{1}{n}{\left(\frac{r}{b}\right)}^n\cos \left(\mathrm{n\&theta;}\right)]+C_0^{\&prime;}\ln \left(r\right)+\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{C}_n^{\&prime;}\&CenterDot;r^{-n}\&CenterDot;\cos \left(\mathrm{n\&theta;}\right)---\left(20\right);$

Wherein, x ₁=m ₁r ₁, x ₂=m ₂r ₂, x ₃=m ₂r ₃,

$B_0=\frac{{\mathrm{\&mu;}}_0{I}_c}{2\mathrm{\&pi;}{m}_2{r}_3}\&CenterDot;\frac{r_3{K}_1\left(x_3\right)-r_2{K}_1\left(x_2\right)}{D_0}-\frac{{\mathrm{\&mu;}}_0{I}_s}{2\mathrm{\&pi;}{m}_2{r}_3}\&CenterDot;\frac{K_1\left(x_2\right)}{D_0}$

$C_0=\frac{{\mathrm{\&mu;}}_0{I}_c}{2\mathrm{\&pi;}{m}_2{r}_3}\&CenterDot;\frac{r_3{I}_1\left(x_2\right)-r_3{I}_1\left(x_3\right)}{D_0}+\frac{{\mathrm{\&mu;}}_0{I}_s}{2\mathrm{\&pi;}{m}_2{r}_3}\&CenterDot;\frac{I_1\left(x_2\right)}{D_0},D_0=K_1\left(x_2\right)I_1\left(x_3\right)-I_1\left(x_2\right)K_1\left(x_3\right),$

$A_n=\frac{{\mathrm{\&mu;}}_{0}I}{\mathrm{\&pi;}{m}_1{r}_1}{\left(\frac{r_1}{b}\right)}^n\&CenterDot;\frac{1}{I_{n-1}\left(x_1\right)}{Z}_n,A_n^{\&prime;}=-\frac{{\mathrm{\&mu;}}_{0}I}{2\mathrm{\&pi;}}\&CenterDot;\frac{r_1^{2n}}{n{b}^n}\&CenterDot;\frac{I_{n+1}\left(x_1\right)}{I_{n-1}\left(x_1\right)}\&CenterDot;Z_n,$

$B_n=\frac{{\mathrm{\&mu;}}_{0}I}{\mathrm{\&pi;}{m}_2{r}_3}{\left(\frac{r_3}{b}\right)}^n\&CenterDot;\frac{K_{n+1}\left(x_2\right)-{\mathrm{\&Delta;}}_n{K}_{n-1}\left(x_2\right)}{D_n^{\&prime;}},$ $B_n^{\&prime;}=\frac{{\mathrm{\&mu;}}_{0}I}{2\mathrm{\&pi;}}\&CenterDot;\frac{1}{{\mathrm{nb}}^n}{Z}_n,$

$C_n=\frac{{\mathrm{\&mu;}}_{0}I}{\mathrm{\&pi;}{m}_2{r}_3}{\left(\frac{r_3}{b}\right)}^n\frac{I_{n+1}\left(x_2\right)-{\mathrm{\&Delta;}}_n{I}_{n-1}\left(x_2\right)}{D_n^{\&prime;}}$

$C_n^{\&prime;}=-\frac{{\mathrm{\&mu;}}_{0}I}{2\mathrm{\&pi;}}\&CenterDot;\frac{r_3^{2n}}{{\mathrm{nb}}^n}\&CenterDot;\frac{1}{D_n^{\&prime;}}\{K_{n+1}\left(x_2\right)I_{n+1}\left(x_3\right)-I_{n+1}\left(x_2\right)K_{n+1}\left(x_3\right)-{\mathrm{\&Delta;}}_n[K_{n-1}\left(x_2\right)I_{n+1}\left(x_3\right)-I_{n-1}\left(x_2\right)K_{n+1}\left(x_3\right)\left]\right\}$

W _n＝K _n+1(x ₂)I _n-1(x ₂)-I _n+1(x ₂)K _n-1(x ₂)， ${\mathrm{\&Delta;}}_n=\frac{I_{n+1}\left(x_1\right)}{I_{n-1}\left(x_1\right)}{\left(\frac{r_1}{r_2}\right)}^{2n},$ E _n=K _n-1(x ₂) I _n-1(x ₃)-I _n-1(x ₂) K _n-1(x ₃), D _n=K _n+1(x ₂) I _n-1(x ₃)-I _n+1(x ₂) K _n-1(x ₃), i _c, I _srepresent respectively cable core electric current, the screen layer electric current of research phase cable; B is cable core spacing; μ ₀represent permeability of vacuum; m _i=j ω μ ₀g _i, g _irepresent the conductivity of each Physical layer, i=1,2,3 are respectively cable core, insulation course, metal screen layer; I _n(x) be first kind distortion bessel function; K _n(x) represent n rank Equations of The Second Kind distortion bessel function; D ' _nfor D _nfirst order derivative.

According to the result of vector magnetic potential A, in conjunction with can obtain function of current density, its form is r represents utmost point footpath, and θ represents polar angle; Consider the modifying factor Θ exerting an influence because of system of laying _n(θ), can obtain the function of current density J of each layer of three-phase electrical cable under particular harmonic number of times electric current:

$J=\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{J}_n\left(r\right)\&CenterDot;{\mathrm{\&Theta;}}_n\left(\mathrm{\&theta;}\right)---\left(21\right)$

A3: ask for respectively cable core loss, insulation course loss and the screen layer loss of each layer of three-phase electrical cable under particular harmonic number of times, again cable core loss, insulation course loss and screen layer loss are sued for peace, finally calculated the harmonic wave total losses of each layer of three-phase electrical cable under particular harmonic number of times electric current and voltage;

Calculate by the each layer of superimposed mode of loss for the harmonic wave total losses under particular harmonic number of times, ask for respectively cable core loss, insulation course loss, screen layer loss and sued for peace again.Cable core loss, screen layer loss adopt the method calculating of the Joule law of differential form being carried out to integration.Cable core loss and the screen layer loss of particular harmonic number of times current unit length cables are:

$P_i=\frac{1}{g_i}\underset{0\&RightArrow;2\mathrm{\&pi;}}{\&Integral;\&Integral;}{\left|J(r,\mathrm{\&theta;})\right|}^2\&CenterDot;\mathrm{rdrd\&theta;}---\left(22\right);$

Wherein, i=1,3, P ₁represent cable core loss, P ₂represent screen layer loss.Convolution (21), abbreviation formula (22):

$P_i=\frac{\mathrm{\&pi;}}{g_i}\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&phi;}}_M\&CenterDot;{\&Integral;J}_n\left(r\right)\&CenterDot;J_n^{*}\left(r\right)\&CenterDot;\mathrm{rdr}---\left(23\right)$

Wherein,

m represents to study the cable external cable number of phases; a _k=b _k/ b ₁, b _krepresent the spacing of k phase cable and the cable of studying; represent i phase cable conductor current phase; represent J _n(r) conjugate.

Because insulation course internal current is made up of conduction current and displacement current two parts, and the electric field force that displacement current produces does not become heat to particle institute work, but become the kinetic energy of particle, so the differential form of Joule law now and be false, thereby can not use cable core, metal screen layer internal loss computing method.Consider that the insulation course between cable core and screen layer is equivalent to be added with harmonic voltage, therefore can obtain the insulation course loss W of unit length cable under particular harmonic number of times voltage:

$W=\frac{2\mathrm{\&pi;\&gamma;}}{\ln \frac{r_2}{r_1}}{U}^2---\left(24\right);$

In formula, γ is insulating material conductivity, and U is harmonic voltage value, r ₁for cable core radius, r ₂for screen layer external radius.

Through type (23), (24) solve the each phase loss under unit length, by cable core loss, screen layer loss, the superimposed unit length cable harmonic wave total losses that obtain of insulation course loss, be multiplied by line length, can learn harmonic wave total losses under particular harmonic number of times.

B: detect by actual checkout equipment, obtain electric current and the voltage of the each phase of three-phase electrical cable, then by FFT resolver, analyze the size and the relative harmonic content that obtain individual harmonic current and harmonic voltage.FFT resolver can change the voltage signal of the each phase of three-phase electrical cable of input and current signal into voltage and electric current under each harmonic, and uses amplitude secondary indication relative harmonic content.Electric current and voltage value under each harmonic is carried out to computing according to steps A, obtain respectively each phase power cable screen layer electric current under individual harmonic current by steps A 1, try to achieve respectively the function of current density of each layer of three-phase electrical cable under individual harmonic current by steps A 2, try to achieve respectively the harmonic wave total losses of each layer of three-phase electrical cable under individual harmonic current and voltage by steps A 3.

C: the processing of suing for peace of the harmonic wave total losses to each layer of the three-phase electrical cable obtaining in step B under each harmonic, what finally obtain is three-phase electrical cable harmonic loss with value.

Below in conjunction with specific embodiment, the present invention will be further elaborated:

Embodiment:

Certain power cable model is: 8.7/10kV YJV-50, and, design parameter is as shown in table 1 below.

Table 1 power cable parameter list

Table1power cable parameter

The cross interconnected two sides earth of 1km power cable screen layer, every 333.3m screen layer Commutating connect once.Soil direct-burried, threephase cable triangular arranged, spacing is 30mm.Soil resistivity is 100 Ω m, and two sides earth resistance is 1 Ω.5 subharmonic current 20A, 5 subharmonic voltage 200V, 7 subharmonic current 15A, 7 subharmonic voltage 100V.

As shown in step B, for 5 subharmonic, first according to steps A 1, try to achieve formula (13) matrix of coefficients and induction electromotive force according to formula (5)-(12), solve that formula (13) obtains A, B, C phase shielded layer electric current is 0.493+0.0383j A;

According to steps A 2, through type (17)-(20) solve the distribution of vector magnetic potential A;

According to steps A 3, solve 5 subharmonic losses in unit length cable core, screen layer, insulation course according to formula (23), (24), result is as follows: cable core loss 0.1236W/m, screen layer loss 0.0185W/m, insulation course loss 5.3286 × 10 ^-9w/m.Combined circuit length, obtaining harmonic wave total losses under 5 subharmonic is 1421.11W.In like manner can obtain harmonic wave total losses under 7 subharmonic is 870.25W;

According to step C, to harmonic wave total losses summation under each harmonic, can obtain harmonic loss in this situation is 2291.36W.

Claims (4)

1. three-phase electrical cable harmonic loss computing method, is characterized in that, comprise the following steps:

A: under the prerequisite in conjunction with three-phase electrical cable practical structures and the actual system of laying of three-phase electrical cable, consideration screen layer electric current and three-phase electrical cable phased magnetic field coupling situation, three-phase electrical cable harmonic loss formula under derivation particular harmonic number of times electric current and voltage, the step of specifically deriving is as follows:

A1: the alternate spacing of determining the each phase of three-phase electrical cable according to the actual system of laying of three-phase electrical cable, induction electromotive force under calculating particular harmonic number of times electric current in each phase screen layer of unit length, then in conjunction with three-phase electrical cable screen layer two-terminal-grounding resistance, determine screen layer ground circuit and calculate the impedance of three-phase electrical cable screen layer and the impedance of three-phase electrical cable ground circuit according to the screen layer connected mode of three-phase electrical cable, final calculating obtains each phase power cable screen layer electric current under particular harmonic number of times electric current;

A2: calculate the function of current density of each layer of three-phase electrical cable under particular harmonic number of times electric current;

A3: ask for respectively cable core loss, insulation course loss and the screen layer loss of each layer of three-phase electrical cable under particular harmonic number of times, again cable core loss, insulation course loss and screen layer loss are sued for peace, finally calculated the harmonic wave total losses of each layer of three-phase electrical cable under particular harmonic number of times electric current and voltage;

B: detect by actual checkout equipment, obtain electric current and the voltage of the each phase of three-phase electrical cable, then by FFT resolver, analyze the size and the relative harmonic content that obtain individual harmonic current and harmonic voltage; Then the electric current and voltage value under each harmonic is carried out to computing according to steps A, obtain respectively each phase power cable screen layer electric current under individual harmonic current by steps A 1, try to achieve respectively the function of current density of each layer of three-phase electrical cable under individual harmonic current by steps A 2, try to achieve respectively the harmonic wave total losses of each layer of three-phase electrical cable under individual harmonic current and voltage by steps A 3;

C: the processing of suing for peace of the harmonic wave total losses to each layer of the three-phase electrical cable obtaining in step B under individual harmonic current and voltage, what finally obtain is three-phase electrical cable harmonic loss with value.

2. three-phase electrical cable harmonic loss computing method according to claim 1, it is characterized in that: in described steps A 1, the induction electromotive force under particular harmonic number of times electric current in each phase screen layer of unit length comprises the induction electromotive force of the each phase screen layer generation of three-phase cable core electric current to unit length and the induction electromotive force that screen layer electric current produces each phase screen layer of unit length;

The induction electromotive force that three-phase cable core electric current produces in the A of unit length phase screen layer P for:

The induction electromotive force that three-phase cable core electric current produces in the B of unit length phase screen layer P for:

${\stackrel{.}{e}}_{\mathrm{SB}}=2\mathrm{\&omega;I}\&times;{10}^{-7}[\frac{\sqrt{3}}{2}\ln \frac{\mathrm{mS}}{R}-j\frac{1}{2}\ln \frac{S}{m\&CenterDot;R}];$

The induction electromotive force that three-phase cable core electric current produces in the C of unit length phase screen layer P for:

${\stackrel{.}{e}}_{\mathrm{SC}}=2\mathrm{\&omega;I}\&times;{10}^{-7}[-\frac{\sqrt{3}}{2}\ln \frac{\mathrm{mS}}{R}-j\frac{1}{2}\ln \frac{n^{2}s}{m\&CenterDot;R}];$

Wherein, j is imaginary unit; ω=2 π f, f represents harmonic frequency; P is the conductor that is parallel to cable core A, B, C, as virtual screen layer; the magnetic flux producing in P for three-phase cable core; I is three-phase cable core electric current; Three-phase electrical cable A, C spaced apart are S; The n that A, B spaced apart are S times, i.e. nS; The m that B, C spaced apart are S times, i.e. mS; R is the radius of P;

The induction electromotive force that screen layer electric current produces unit length screen layer is:

${\stackrel{.}{e}}_{\mathrm{SA}}^{\&prime;}={\stackrel{.}{I}}_{\mathrm{SB}}\&CenterDot;{\mathrm{jM}}_{\mathrm{AB}}+{\stackrel{\&CenterDot;}{I}}_{\mathrm{SC}}\&CenterDot;{\mathrm{jM}}_{\mathrm{AC}};$

${\stackrel{.}{e}}_{\mathrm{SB}}^{\&prime;}={\stackrel{.}{I}}_{\mathrm{SA}}\&CenterDot;{\mathrm{jM}}_{\mathrm{AB}}+{\stackrel{.}{I}}_{\mathrm{SC}}\&CenterDot;{\mathrm{jM}}_{\mathrm{BC}};$

${\stackrel{.}{e}}_{\mathrm{SC}}^{\&prime;}={\stackrel{.}{I}}_{\mathrm{SA}}\&CenterDot;{\mathrm{jM}}_{\mathrm{AC}}+{\stackrel{.}{I}}_{\mathrm{SB}}\&CenterDot;{\mathrm{jM}}_{\mathrm{BC}};$

Wherein, for the induction electromotive force that in three-phase electrical cable, B, C phase screen layer electric current produce in unit length A phase screen layer; the induction electromotive force producing in unit length B phase screen layer for three-phase electrical cable A, C phase screen layer electric current; the induction electromotive force producing in unit length C phase screen layer for three-phase electrical cable A, B phase screen layer electric current; represent respectively three-phase electrical cable A, B, C phase screen layer electric current; M _aB, M _bC, M _cArespectively C and the mutual inductance of A phase screen layer under B and the mutual inductance of C phase screen layer under three-phase electrical cable A and the mutual inductance of B phase screen layer, unit length, unit length under representation unit length; J is imaginary unit;

Determine screen layer ground circuit according to the screen layer connected mode of three-phase electrical cable, in conjunction with three-phase electrical cable screen layer two-terminal-grounding resistance, and calculate the impedance of three-phase electrical cable screen layer, the impedance of three-phase electrical cable ground circuit and induction electromotive force; Induction electromotive force, the impedance of three-phase electrical cable screen layer and the impedance of three-phase electrical cable ground circuit are multiplied by line length by institute's calculated value under unit length and are obtained;

Final calculating obtains each phase power cable screen layer electric current under particular harmonic number of times electric current.

3. three-phase electrical cable harmonic loss computing method according to claim 2, it is characterized in that: in described steps A 2, utilize the boundary condition Simultaneous Equations of vector magnetic potential A on power cable actual physics layer, introduce bessel function, adopt separation of variable solving equation, in conjunction with boundary condition, obtain the result of vector magnetic potential A, then pass through relational expression: can obtain the function of current density J of each layer of three-phase electrical cable under particular harmonic number of times electric current is:

$J=\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{J}_n\left(r\right)\&CenterDot;{\mathrm{\&Theta;}}_n\left(\mathrm{\&theta;}\right);$

Wherein, modifying factor ${\mathrm{\&Theta;}}_n\left(\mathrm{\&theta;}\right)=\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{I_k}{I}\&CenterDot;\frac{1}{a_k^n}\cos n(\mathrm{\&theta;}+{\mathrm{\&alpha;}}_k),$ M represents the cable number of phases of studied cable outside; b _krepresent the spacing of k phase cable and the cable of studying, definition a _k=b _k/ b ₁; I represents to study the cable core electric current of cable; I _krepresent the in addition cable core electric current of k phase cable of institute's cable of studying; α _krepresent the locus of k phase cable; R represents utmost point footpath, and θ represents polar angle.

4. three-phase electrical cable harmonic loss computing method according to claim 3, is characterized in that: in described steps A 3,

By the Joule law of differential form is carried out to integration, cable core loss and the screen layer loss of calculating unit length cable under particular harmonic number of times electric current are

$P_i=\frac{\mathrm{\&pi;}}{g_i}\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&phi;}}_M\&CenterDot;{\&Integral;J}_n\left(r\right)\&CenterDot;J_n^{*}\left(r\right)\&CenterDot;\mathrm{rdr};$ Wherein, i=1,3, P ₁represent cable core loss, P ₂represent screen layer loss,

m represents to study the cable external cable number of phases; a _k=b _k/ b ₁, b _krepresent the spacing of k phase cable and the cable of studying; represent i phase cable conductor current phase; represent J _n(r) conjugate;

Under particular harmonic number of times voltage, the insulation course loss W of unit length cable is:

wherein, γ is insulating material conductivity, and U is harmonic voltage value, r ₁for cable core radius, r ₂for screen layer external radius.