CN104714136A - Method for calculating square wave carrying capacity of AC power cables - Google Patents

Abstract

The invention provides a method for calculating square wave carrying capacity of AC power cables. The method is characterized by comprising Fourier transform and temperature difference calculation caused by frequency doubling current, the size and period of square wave current can be reversely deduced from temperature constraint conditions of the cables through Fourier transform, and the method can be used for determining square wave carrying capacity in line transformation capacity expansion projects of the AC power cables, is scientific and reasonable, and has important practical significance and engineering value.

Description

A kind of computing method of ac power cable square wave current-carrying capacity

Technical field

Power cable field of the present invention, is a kind of computing method of ac power cable square wave current-carrying capacity, is applied to the determination of square wave current-carrying capacity in ac power cable track remodelling enlarging project.

Background technology

In recent years, China's Urbanization Progress speed is accelerated further, electricity need load constantly increases, objectively require that electrical network scale and transmission capacity keep sustainable development, but the current following problem of urban distribution network ubiquity: 1. electricity need load increases, alternating current circuit transport capacity is not enough, and line corridor is deficient; 2. urban distribution network structure is day by day tight, short-circuit current outstanding problem; 3. urban distribution network reactive voltage regulates difficulty increasingly, and voltage stabilization sex chromosome mosaicism can not be ignored.Have no so far about the reported in literature identical with the computing method content of ac power cable square wave current-carrying capacity of the present invention and useful application.

Summary of the invention

The object of the invention is, based on urban distribution network capacity-increasing transformation engineering, the determination for conventional AC power cable square wave current-carrying capacity provides a kind of scientific and reasonable, the computing method of the ac power cable square wave current-carrying capacity that practical value is high.

The object of the invention is to be realized by following technical scheme: a kind of computing method of ac power cable square wave current-carrying capacity, it is characterized in that, it comprises following content:

1) Fourier transform

Adopt Fourier transform square-wave AC current to be resolved into the superposition of serial sine-wave current, if the amplitude of square wave current and cycle are respectively I and T, launched according to Fourier transform formula, be expressed as follows:

(1) if square wave frequency is amplitude is 1, and its Fourier expansion formula is:

$f\left(x\right)=\frac{4}{\mathrm{\π}}[\sin x+\frac{1}{3}\sin 3x+...+\frac{1}{2k-1}\sin (2k-1)x+...]---\left(1\right)$

(2) if square wave frequency is amplitude is I, i.e. current-carrying capacity of cable, and its Fourier expansion formula is:

$I\left(x\right)=\frac{4I}{\mathrm{\π}}[\sin \frac{\mathrm{\π}}{l}x+\frac{1}{3}\sin \frac{3\mathrm{\π}}{l}x+...+\frac{1}{2k-1}\sin \frac{(2k-1)\mathrm{\π}}{l}x+...]---\left(2\right)$

In formula, I is current amplitude, then its effective value the i.e. effective value of current-carrying capacity of cable, its Fourier expansion formula is:

$I_{\mathrm{RMS}}\left(x\right)=\frac{4I}{\sqrt{2}\mathrm{\π}}[\sin \frac{\mathrm{\π}}{l}x+\frac{1}{3}\sin \frac{3\mathrm{\π}}{l}x+...+\frac{1}{2k-1}\sin \frac{(2k-1)\mathrm{\π}}{l}x+...]---\left(3\right)$

Amplitude and cycle are respectively to the square wave current of I and T, utilize formula (3) to be launched, obtain the effective value of serial sinusoidal current ripple

$I_{2k-1}=\frac{4I}{(2k-1)\sqrt{2}\mathrm{\π}}---\left(4\right)$

2) differential thermal calculation that causes of frequency multiplication electric current

The current-carrying capacity of ac power cable is calculated by (5) formula,

$I=\sqrt{\frac{({\mathrm{\θ}}_c-{\mathrm{\θ}}_0)-W_i(0.5T_1+T_3+T_4)}{R[T_1+(1+{\mathrm{\λ}}_1)(T_3+T_4)]}}---\left(5\right)$

(5), in formula, R is the AC resistance of cable conductor, λ ₁for protective metal shell loss ratio, W _ifor the dielectric loss of cable insulation, R, λ ₁, W _ithere is specific mathematical relation with the frequency f of each harmonic, be designated as the special parameter of cable, derive by (5) formula the temperature rise function that each frequency multiplication electric current causes on conductor, be designated as Δ θ=f (I, f, U ₀), wherein U ₀for cable phase voltage, (5) formula is arranged:

Δθ＝θ _c-θ ₀＝I ²R[T ₁+(1+λ ₁)(T ₃+T ₄)]+W _i(0.5T ₁+T ₃+T ₄) (6)

Further frequency multiplication electric current I _2k-1the temperature difference that conductor causes is

$\mathrm{\Δ}{\mathrm{\θ}}_{2k-1}={\mathrm{\θ}}_c-{\mathrm{\θ}}_0=I_{2k-1}^{2}R[T_1+(1+{\mathrm{\λ}}_1)(T_3+T_4)]+W_i(0.5T_1+T_3+T_4)---\left(7\right)$

So the total temperature difference that each frequency multiplication electric current causes is

$\mathrm{\Δ}{\mathrm{\θ}}_T=\underset{k=1}{\mathrm{\Σ}}\mathrm{\Δ}{\mathrm{\θ}}_{2k-1}---\left(8\right)$

For specific cable, its maximum temperature difference allowed is fixing numerical value, can be designated as Δ θ ' _t, so cable temperature constraint condition is

Δθ _T≤Δθ′ _T(9)

For square wave current, adjust its size I and cycle T, above-mentioned constraint condition is met, namely can determine the square wave current-carrying capacity of ac power cable.

The present invention is a kind of computing method of ac power cable square wave current-carrying capacity, adopt Fourier transform, by the temperature constraint of cable, anti-size and the cycle pushing away square wave current, can be applicable to the determination of square wave current-carrying capacity in ac power cable track remodelling enlarging project, have scientific and reasonable, important practical significance and construction value.

Accompanying drawing explanation

Fig. 1 is square wave current figure.

Embodiment

Be further described below in conjunction with the computing method of the drawings and specific embodiments to a kind of ac power cable square wave current-carrying capacity of the present invention:

With reference to Fig. 1, have in a kind of computing method of ac power cable square wave current-carrying capacity:

1) Fourier transform

Adopt Fourier transform square-wave AC current to be resolved into the superposition of serial sine-wave current, if the amplitude of square wave current and cycle are respectively I and T, launched according to Fourier transform formula, be expressed as follows:

(1) if square wave frequency is amplitude is 1, and its Fourier expansion formula is:

$f\left(x\right)=\frac{4}{\mathrm{\π}}[\sin x+\frac{1}{3}\sin 3x+...+\frac{1}{2k-1}\sin (2k-1)x+...]---\left(1\right)$

(2) if square wave frequency is amplitude is I, i.e. current-carrying capacity of cable, and its Fourier expansion formula is:

$I\left(x\right)=\frac{4I}{\mathrm{\π}}[\sin \frac{\mathrm{\π}}{l}x+\frac{1}{3}\sin \frac{3\mathrm{\π}}{l}x+...+\frac{1}{2k-1}\sin \frac{(2k-1)\mathrm{\π}}{l}x+...]---\left(2\right)$

In formula, I is current amplitude, then its effective value the i.e. effective value of current-carrying capacity of cable, its Fourier expansion formula is:

$I_{\mathrm{RMS}}\left(x\right)=\frac{4I}{\sqrt{2}\mathrm{\π}}[\sin \frac{\mathrm{\π}}{l}x+\frac{1}{3}\sin \frac{3\mathrm{\π}}{l}x+...+\frac{1}{2k-1}\sin \frac{(2k-1)\mathrm{\π}}{l}x+...]---\left(3\right)$

Amplitude and cycle are respectively to the square wave current of I and T, utilize formula (3) to be launched, obtain the effective value of serial sinusoidal current ripple

$I_{2k-1}=\frac{4I}{(2k-1)\sqrt{2}\mathrm{\π}}---\left(4\right)$

2) differential thermal calculation that causes of frequency multiplication electric current

The current-carrying capacity of ac power cable is calculated by (5) formula,

$I=\sqrt{\frac{({\mathrm{\θ}}_c-{\mathrm{\θ}}_0)-W_i(0.5T_1+T_3+T_4)}{R[T_1+(1+{\mathrm{\λ}}_1)(T_3+T_4)]}}---\left(5\right)$

(5), in formula, R is the AC resistance of cable conductor, λ ₁for protective metal shell loss ratio, W _ifor the dielectric loss of cable insulation, R, λ ₁, W _ithere is specific mathematical relation with the frequency f of each harmonic, be designated as the special parameter of cable, derive by (5) formula the temperature rise function that each frequency multiplication electric current causes on conductor, be designated as Δ θ=f (I, f, U ₀), wherein U ₀for cable phase voltage, (5) formula is arranged:

Δθ＝θ _c-θ ₀＝I ²R[T ₁+(1+λ ₁)(T ₃+T ₄)]+W _i(0.5T ₁+T ₃+T ₄)(6)

Further frequency multiplication electric current I _2k-1the temperature difference that conductor causes is

$\mathrm{\Δ}{\mathrm{\θ}}_{2k-1}={\mathrm{\θ}}_c-{\mathrm{\θ}}_0=I_{2k-1}^{2}R[T_1+(1+{\mathrm{\λ}}_1)(T_3+T_4)]+W_i(0.5T_1+T_3+T_4)---\left(7\right)$

So the total temperature difference that each frequency multiplication electric current causes is

$\mathrm{\Δ}{\mathrm{\θ}}_T=\underset{k=1}{\mathrm{\Σ}}\mathrm{\Δ}{\mathrm{\θ}}_{2k-1}---\left(8\right)$

For specific cable, its maximum temperature difference allowed is fixing numerical value, can be designated as Δ θ ' _t, so cable temperature constraint condition is

Δθ _T≤Δθ′ _T(9)

For square wave current, adjust its size I and cycle T, above-mentioned constraint condition is met, namely can determine the square wave current-carrying capacity of ac power cable.

The present invention is a kind of computing method of ac power cable square wave current-carrying capacity, adopt Fourier transform, by the temperature constraint of cable, anti-size and the cycle pushing away square wave current, can be applicable to the determination of square wave current-carrying capacity in ac power cable track remodelling enlarging project, have scientific and reasonable same, important practical significance and construction value.

Claims (1)

1. computing method for ac power cable square wave current-carrying capacity, is characterized in that, it has in comprising:

1) Fourier transform

Adopt Fourier transform square-wave AC current to be resolved into the superposition of serial sine-wave current, if the amplitude of square wave current and cycle are respectively I and T, launched according to Fourier transform formula, be expressed as follows:

(1) if square wave frequency is amplitude is 1, and its Fourier expansion formula is:

$f\left(x\right)=\frac{4}{\mathrm{\π}}[\sin x+\frac{1}{3}\sin 3x+...+\frac{1}{2k-1}\sin (2k-1)x+...]---\left(1\right)$

(2) if square wave frequency is amplitude is I, i.e. current-carrying capacity of cable, and its Fourier expansion formula is:

$I\left(x\right)=\frac{4I}{\mathrm{\π}}[\sin \frac{\mathrm{\π}}{l}x+\frac{1}{3}\sin \frac{3\mathrm{\π}}{l}x+...+\frac{1}{2k-1}\sin \frac{(2k-1)\mathrm{\π}}{l}x+...]---\left(2\right)$

In formula, I is current amplitude, then its effective value the i.e. effective value of current-carrying capacity of cable, its Fourier expansion formula is:

$I_{\mathrm{RMS}}\left(x\right)=\frac{4I}{\sqrt{2}\mathrm{\π}}[\sin \frac{\mathrm{\π}}{l}x+\frac{1}{3}\sin \frac{3\mathrm{\π}}{l}x+...+\frac{1}{2k-1}\sin \frac{(2k-1)\mathrm{\π}}{l}x+...]---\left(3\right)$

Amplitude and cycle are respectively to the square wave current of I and T, utilize formula (3) to be launched, obtain the effective value of serial sinusoidal current ripple

$I_{2k-1}=\frac{4I}{(2k-1)\sqrt{2}\mathrm{\π}}---\left(4\right)$

2) differential thermal calculation that causes of frequency multiplication electric current

The current-carrying capacity of ac power cable is calculated by (5) formula,

$I=\sqrt{\frac{({\mathrm{\θ}}_c-{\mathrm{\θ}}_0)-W_i({0.5T}_1+T_3+T_4)}{R[T_1+(1+{\mathrm{\λ}}_1)(T_3+T_4)]}}---\left(5\right)$

(5), in formula, R is the AC resistance of cable conductor, λ ₁for protective metal shell loss ratio, W _ifor the dielectric loss of cable insulation, R, λ ₁, W _ithere is specific mathematical relation with the frequency f of each harmonic, be designated as the special parameter of cable, derive by (5) formula the temperature rise function that each frequency multiplication electric current causes on conductor, be designated as Δ θ=f (I, f, U ₀), wherein U ₀for cable phase voltage, (5) formula is arranged:

Δθ＝θ _c-θ ₀＝I ²R[T ₁+(1+λ ₁)(T ₃+T ₄)]+W _i(0.5T ₁+T ₃+T ₄) (6)

Further frequency multiplication electric current I _2k-1the temperature difference that conductor causes is

${\mathrm{\Δ\θ}}_{2k-1}={\mathrm{\θ}}_c-{\mathrm{\θ}}_0=I_{2k-1}^{2}R[T_1+(1+{\mathrm{\λ}}_1)(T_3+T_4)]+W_i(0.5T_1+T_3+T_4)---\left(7\right)$

So the total temperature difference that each frequency multiplication electric current causes is

${\mathrm{\Δ\θ}}_T=\underset{k=1}{\mathrm{\Σ}}{\mathrm{\Δ\θ}}_{2k-1}---\left(8\right)$

For specific cable, its maximum temperature difference allowed is fixing numerical value, can be designated as Δ θ ' _t, so cable temperature constraint condition is

Δθ _T≤Δθ′ _T(9)

For square wave current, adjust its size I and cycle T, above-mentioned constraint condition is met, namely can determine the square wave current-carrying capacity of ac power cable.