CN103427434B - Calculation method for harmonic current of direct-current side of hybrid bipolar direct-current transmission system - Google Patents

Abstract

The invention discloses a calculation method for harmonic current of a direct-current side of a hybrid bipolar direct-current transmission system. The calculation method includes the steps: (1) performing equivalent transformation to obtain an equivalent three-pulse voltage source corresponding to an LCC (line-commutated converter) and an equivalent passive circuit corresponding to an MMC (modular multilevel converter); (2) sectioning a direct-current transmission line and a ground wire thereof, and calculating an admittance matrix of each subsection to obtain an admittance matrix of the whole direct-current transmission line by a recurrence method; (3) building a systemic direct-current network according to the equivalent three-pulse voltage source, the equivalent passive circuit and the direct-current transmission line to obtain node voltage of the front end and the tail end of the direct-current transmission line according to the admittance matrixes; (4) acquiring node voltage of each subsection node of the direct-current transmission line by an inverse method according to the node voltage to obtain the harmonic current of the direct-current side. Calculation efficiency of a direct-current loop is remarkably improved on the premise of ensuring effectiveness.

Description

Method for calculating direct-current side harmonic current of hybrid bipolar direct-current transmission system

Technical Field

The invention belongs to the technical field of power transmission of power systems, and particularly relates to a method for calculating direct-current side harmonic current of a hybrid bipolar direct-current power transmission system.

Background

The main problem of the flexible direct current transmission applied to long-distance high-capacity overhead line transmission is the self-clearing problem of direct current side faults, and because no high-voltage high-capacity direct current breaker really applied in commercialization exists at present, the method of tripping an alternating current side switch is adopted for clearing the direct current side faults when the direct current side faults occur in the flexible direct current transmission system. According to the experience of ABB company in Caprivi Link engineering, the time from the beginning of the fault to the clearing of the fault and the restoration to the power level before the fault is 2-2.5s, and the fault time is about 0.5s similar to the traditional direct current transmission processing. To this end, chinese patent publication No. 102969732a proposes a hybrid bipolar dc transmission system in which an LCC (thyristor converter) is used for a rectification side, an MMC (modular multilevel converter) is used for an inversion side, and a diode valve that is unidirectionally conducted is connected in series at an outlet of a dc side of an inverter, as shown in fig. 1.

The hybrid direct-current power transmission structure formed by the LCC, the diode valve and the MMC has the main advantages that:

(1) the rectifier station adopts an LCC type converter, the technology is mature, the equipment cost is low, and the operation loss is small;

(2) the inverter station adopts an MMC type converter, so that the problem of phase commutation failure does not exist, and the problem of simultaneous phase commutation failure caused by multiple direct current feed-ins of a receiving end system can be solved; the reactive power source can be used as a reactive power source of a receiving end system, plays a role in voltage support, and is very beneficial to the safety and stability of the receiving end system;

(3) the diode valve has mature technology and low cost. Due to the fact that the diode valve conducting in the one-way mode is added, the hybrid direct-current system can effectively process temporary faults of the overhead direct-current line like a traditional direct-current transmission system, and the availability of the direct-current transmission line cannot be influenced by the temporary faults of the overhead direct-current line.

Therefore, the hybrid bipolar system structure of LCC plus diode valve plus MMC shown in fig. 1 is considered as a very competitive solution to the long-distance large-capacity power transmission problem in our country. Therefore, research on the design technology of the direct current transmission system with the above structure is urgently needed.

During normal operation, due to the nonlinear characteristic of the current converter, a large amount of node voltage and harmonic current in an audio frequency range can be generated on the direct-current transmission line. Through capacitive coupling, inductive coupling and resistive coupling, node voltages and harmonic currents can interfere with audio telephone loops in a telecommunication line, and call quality is reduced. A dc filter is generally added to the dc side of the dc transmission system to control the interference in the audio frequency range within an allowable level. In order to reasonably and effectively configure the direct current filter, each harmonic current at each point along the line is generally calculated theoretically according to the practical characteristics of engineering, and then converted into equivalent interference current for measurement. Therefore, the accuracy of the calculation of each harmonic current is directly related to the accuracy of the design of the filter, and the calculation of the harmonic current becomes the basis of the design of the direct current filter correspondingly. In view of various operating conditions of the system, an accurate and efficient method for calculating the harmonic current on the direct current side must be found.

Although the direct-current side harmonic current can be calculated by the time domain simulation technology at present, the time domain simulation needs to consume a large amount of calculation time and calculation resources. Considering that various wiring modes and power levels exist in practical engineering, the time domain simulation technology is not suitable for calculating the harmonic current on the direct current side.

Disclosure of Invention

The invention provides a method for calculating direct-current side harmonic current of a bipolar direct-current power transmission system, which can obviously improve the calculation efficiency of a direct-current loop on the premise of ensuring the calculation accuracy.

A method for calculating harmonic current on a direct current side of a hybrid bipolar direct current transmission system comprises the following steps:

(1) respectively carrying out equivalent transformation on the LCC and the MMC in the system according to the system operating condition and the system main loop parameter to obtain an equivalent triple-pulse voltage source corresponding to the LCC and an equivalent passive circuit corresponding to the MMC;

(2) carrying out segmentation processing on the direct current transmission line and the ground wire thereof, calculating an admittance matrix of each segment, and obtaining the admittance matrix of the whole direct current transmission line by a recurrence method;

(3) establishing a direct current network of the system according to the equivalent triple-pulse voltage source, the equivalent passive circuit and the direct current transmission line, and further calculating node voltages of the nodes at the first end and the last end of the direct current transmission line by using a node voltage analysis method after combining the admittance matrixes of the equivalent triple-pulse voltage source, the equivalent passive circuit and the direct current transmission line;

(4) obtaining the node voltage of each segmented node of the direct-current transmission line by a back-stepping method according to the node voltages of the nodes at the first end and the last end of the direct-current transmission line; and then the harmonic current of each section node of the direct current transmission line and the ground wire thereof is obtained.

In the step (1), the equivalent passive circuit is formed by connecting three sets of RLC (resistor-inductor-capacitor) links in parallel, the RLC links are sequentially formed by connecting a capacitor C, an inductor L and a resistor R in series from an input end to an output end, and the output end of the RLC links is grounded; wherein, C =2C₀/N，L=2L₀，R=NR₀(ii) a N is the cascade number of the bridge arm converter sub-modules of MMC, C₀For the capacitance in the converter submodule, L₀Bridge arm reactance, R, being MMC₀Is the on-resistance of the converter submodule.

In the step (2), the method for obtaining the admittance matrix of the whole direct current transmission line by the recursion method is as follows:

a. for the first sectional line L in the DC transmission line₁And a second segment line L₂Merging, calculating the merged line L according to the following formula₁₂Admittance matrix G of₁₂：

Wherein: l is₁₂=L₁+L₂，Y_s-1And Y_m-1Are respectively a line L₁Self-admittance and mutual admittance matrices, Y_s-2And Y_m-2Are respectively a line L₂The self-admittance matrix and the mutual admittance matrix of (a);

b. make the line L₁₂And a third segment line L₃Merging, calculating merged line L according to step a₁₃Admittance matrix G of₁₃；

c. And a, combining the lines section by section in sequence according to the steps a and b, calculating and storing the admittance matrix of the combined line each time, and finally obtaining the admittance matrix of the whole direct current transmission line.

In the step (4), the method for obtaining the voltage of each segmented node on the direct current transmission line by the inverse method is as follows:

a. splitting the direct current transmission line into a line L (1, n-1) and a line L (n-1, n), and calculating the node voltage U of the n-1 node in the direct current transmission line according to the following formula_n-1：

U_n-1＝[Y_s(1,n-1)+Y_s(n-1,n)]^-1[Y_m(1,n-1)U₁+Y_m(n-1,n)U_n]

Wherein: n is the total number of the segmented nodes in the direct current transmission line, L (1, n-1) is a line from the first node to the n-1 node, L (n-1, n) is a line from the n-1 node to the n node, and U is a line from the n-1 node to the n node₁、U_nNode voltages of the nodes at the head end and the tail end of the direct current transmission line respectively, Y_s(1, n-1) and Y_m(1, n-1) are respectively the self-admittance matrix and the mutual admittance matrix of the line L (1, n-1), Y_s(n-1, n) and Y_m(n-1, n) are respectively a self-admittance matrix and a mutual admittance matrix of the line L (n-1, n);

b. then splitting a line L (1, n-1) in the direct current transmission line into the line L (1, n-2) and the line L (n-2, n-1), and calculating the node voltage U of the n-2 node in the split direct current transmission line according to the step a_n-2；

c. And c, sequentially splitting section by section according to the steps a and b, and simultaneously calculating the node voltage of each section node of the direct current transmission line.

In the step (4), the method for calculating the harmonic current of each segmented node of the direct current transmission line comprises the following steps:

$\left[\begin{array}{c}{I}_A\\ I_B\end{array}\right]=\left[\begin{array}{cc}{Y}_{s-\mathrm{AB}}& -Y_{m-\mathrm{AB}}\\ {-Y}_{m-\mathrm{AB}}& Y_{s-\mathrm{AB}}\end{array}\right]\left[\begin{array}{c}{U}_A\\ U_B\end{array}\right]$

wherein, U_AAnd U_BRespectively being any sectional line L in the direct current transmission line_ABNode voltages of first and last end nodes A and B, I_AAnd I_BHarmonic currents, Y, of node A and node B, respectively_s-ABAnd Y_m-ABRespectively a segmented line L_ABThe self-admittance matrix and the transadmittance matrix.

In the step (4), the method for obtaining the harmonic current on the ground wire is as follows:

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <msup> <mi>A</mi> <mo>&prime;</mo> </msup> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <msup> <mi>B</mi> <mo>&prime;</mo> </msup> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mo>[</mo> <mo>-</mo> <msubsup> <mi>Y</mi> <mrow> <mi>m</mi> <mo>-</mo> <mi>AB</mi> </mrow> <mo>&prime;</mo> </msubsup> <mo>]</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mi>A</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mi>B</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein, I_A'And I_B'Respectively, any one of the segment lines L of the ground wire_A'B'Harmonic currents, U, of first and last end nodes A' and B_AAnd U_BRespectively a segmented line L in a direct current transmission line_ABNode voltages of first and last end nodes A and B, segment line L_A'B'And a segment line L_ABCorresponding to, Y'_m-ABFor a sectional line L_ABAnd a segment line L_A'B'A coupled line transadmittance matrix.

The invention has the beneficial technical effects that:

(1) for a hybrid HVDC system of LCC plus a diode valve plus MMC, the invention fills the blank of the design research of a direct current main loop and can play a certain guiding role for the design of future engineering.

(2) In view of time-domain simulation time consumption and calculation resource consumption, the invention provides a direct-current side harmonic current calculation method based on an analytic method on the premise of ensuring effectiveness, so that the calculation efficiency can be improved, and the time spent on designing a direct-current main loop can be further reduced.

Drawings

Fig. 1 is a schematic structural diagram of a hybrid bipolar dc transmission system.

FIG. 2 is a schematic structural diagram of an MMC.

Fig. 3 (a) is an equivalent tri-ripple voltage source for LCC.

Fig. 3 (b) is an equivalent passive circuit of MMC.

Fig. 4 is a schematic diagram of an equivalent dc network of a hybrid bipolar dc transmission system.

Fig. 5 is a schematic structural diagram of the adopted dc filter.

Fig. 6 is a flowchart of calculating the harmonic current on the dc side.

Fig. 7 shows the calculated node voltage results of the number of times of the positive/negative dc transmission line along the line.

Fig. 8 shows the calculated harmonic current results of the times of the positive dc power transmission line along the line.

Fig. 9 is a schematic diagram of the calculated number harmonic distribution of the residual current portion along the line.

FIG. 10 is a schematic view of the distribution of the calculated equivalent interference current along the line.

Detailed Description

In order to more specifically describe the present invention, the following detailed description is provided for the technical solution of the present invention with reference to the accompanying drawings and the specific embodiments.

The hybrid bipolar dc power transmission system according to the present embodiment is shown in fig. 1, and the system parameters are shown in table 1:

TABLE 1

The dc filter DCF configuration adopted by the system is shown in fig. 5, and the parameters thereof are shown in table 2:

TABLE 2

As shown in fig. 6, the dc-side harmonic current of the bipolar dc transmission system is calculated according to the following method:

(1) and performing equivalent transformation on the LCC in the system according to the system operating condition and the main loop parameter to obtain an equivalent triple-pulse voltage source corresponding to the LCC.

As shown in FIG. 3 (a), the equivalent triple-ripple voltage source consists of two inductors L₄～L₅A capacitor C4 and two voltage sources Y1-Y2; inductor L₄And an inductance L₅In series connection with a central node and a capacitor C₄Connected to a capacitor C₄The other end of the first and second electrodes is grounded; wherein: according to engineering experience, the capacitance C can be adjusted₄The size of (A) is set to 10-20 nF; inductor L₄～L₅The inductance value of (A) and the voltage values of the voltage sources Y1-Y2 can be calculated by a voltage source of a 3-pulse node of high-voltage direct-current transmission (J) through literature (Wang Feng, Su Guang, Huang Ying, Li Xiao Ling, Xue Zheng, Xue, etc.)]High voltage technology, 2009, 35 (10): 2586 + 2590).

The results of the three-ripple voltage source calculation at some harmonic times in fig. 4 are shown in table 3.

TABLE 3

(2) As shown in fig. 2, which is a schematic structural diagram of an MMC, according to a main loop parameter of a system, performing equivalent transformation on the MMC in the system to obtain an equivalent passive circuit corresponding to the MMC.

As shown in fig. 3 (b), the equivalent passive circuit is formed by connecting three sets of RLC links in parallel, wherein the RLC links are formed by connecting a capacitor C, an inductor L and a resistor R in series from an input end to an output end, and the output end of the RLC link is grounded; wherein, C =2C₀/N，L=2L₀，R=NR₀(ii) a N is the cascade number of the bridge arm converter sub-modules of MMC, C₀For the capacitance in the converter submodule, L₀Bridge arm reactance, R, being MMC₀Is the on-resistance of the converter submodule.

In the present embodiment, each equivalent capacitor C =241.48 μ F, and each equivalent inductor L =40 mH; a conservative estimate, taking the on-state voltage drop of 0.01 Ω for each submodule, then R =2.5 Ω.

(3) Carrying out segmentation processing on a direct current transmission line and a ground wire of the system to obtain series impedance, a parallel admittance matrix, harmonic frequency and segmentation information of the direct current transmission line and the ground wire in unit length; by adopting hyperbolic cosecant and hyperbolic cosecant function Roron series expansion, an admittance matrix equation of each section of the direct current transmission line and the ground wire is defined as follows, and an admittance matrix of each section is further solved;

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$n=\mathrm{0,1,2}...$

<math> <mrow> <msub> <mi>Y</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>csch</mi> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> </mrow> </math>

<math> <mrow> <mo>=</mo> <msup> <mrow> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>7</mn> <mn>360</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>-</mo> <mfrac> <mn>31</mn> <mn>15120</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>5</mn> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mrow> <mo>(</mo> <mn>2</mn> <mi>n</mi> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </math>

<math> <mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>[</mo> <mi>E</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>7</mn> <mn>360</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>4</mn> </msup> <mo>-</mo> <mfrac> <mn>31</mn> <mn>15120</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>6</mn> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mrow> <mo>(</mo> <mn>2</mn> <mi>n</mi> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>]</mo> </mrow> </math>

<math> <mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>[</mo> <mi>E</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> <mrow> <mo>(</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>&CenterDot;</mo> <mi>Y</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>7</mn> <mn>360</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>&CenterDot;</mo> <mi>Y</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mn>31</mn> <mn>15120</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>&CenterDot;</mo> <mi>Y</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mrow> <mo>(</mo> <mn>2</mn> <mi>n</mi> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>&CenterDot;</mo> <mi>Y</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>l</mi> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>]</mo> </mrow> </math>

$n=\mathrm{0,1,2}...$

<math> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mi>Z</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> <mi>Y</mi> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> </msqrt> </mrow> </math>

wherein: z (omega) and Y (omega) are respectively a unit length series impedance and a parallel admittance matrix of the direct current transmission line or the ground wire with respect to the frequency, and Y (omega)_s(omega) and Y_m(omega) is respectively a self-admittance matrix and a mutual admittance matrix of each section of the direct current transmission line or the ground line, omega =2 pi f, f is the harmonic frequency, l is the length of each section of the direct current transmission line or the ground line, B_nIs Bernoulli number, and n is a series term.

After obtaining the admittance matrix of each section of the direct current transmission line, obtaining the admittance matrix of the whole direct current transmission line by a recurrence method, wherein the method comprises the following steps:

a. make the first sectional line L in the DC transmission line₁And a second segment line L₂Merging, calculating the merged line L according to the following formula₁₂Admittance matrix G of₁₂：

Wherein: l is₁₂=L₁+L₂，Y_s-1And Y_m-1Are respectively a line L₁Self-admittance and mutual admittance matrices, Y_s-2And Y_m-2Are respectively a line L₂The self-admittance matrix and the mutual admittance matrix of (a);

b. make the line L₁₂And a third segment line L₃Merging, calculating merged line L according to step a₁₃Admittance matrix G of₁₃；

c. And a, combining the lines section by section in sequence according to the steps a and b, calculating and storing the admittance matrix of the combined line each time, and finally obtaining the admittance matrix of the whole direct current transmission line.

Through calculation, the admittance matrix of the whole direct current line can be obtained as follows (taking 100Hz (2 times frequency) and 600Hz (12 times frequency) as examples):

100Hz：

$Y_L\left(2\right)=\left[\begin{array}{cccc}0.0003-0.0003i& 0.0002+0.0008i& -0.0001+0.0032i& -0.0001-0.0010i\\ 0.0002+0.0008i& 0.0003-0.0003i& -0.0001-0.0010i& -0.0001+0.0032i\\ -0.0001+0.0032i& -0.0001-0.0010i& 0.0003-0.0003i& 0.0002+0.0008i\\ -0.0001-0.0010i& -0.0001+0.0032i& 0.0002+0.0008i& 0.0003-0.0003i\end{array}\right]$

600Hz：

$Y_L\left(12\right)=\left[\begin{array}{cccc}0.0020+0.0006i& 0.0018+0.0008i& 0.0015+0.0028i& 0.0015-0.0012i\\ 0.0018+0.0008i& 0.0020+0.0006i& 0.0015-0.0012i& 0.0015+0.0028i\\ 0.0015+0.0028i& 0.0015-0.0012i& 0.0020+0.0006i& 0.0018+0.0008i\\ 0.0015-0.0012i& 0.0015+0.0028i& 0.0018+0.0008i& 0.0020+0.0006i\end{array}\right]$

(4) establishing a direct current network of the system according to the equivalent triple-pulse voltage source, the equivalent passive circuit and the direct current transmission line, as shown in fig. 4; and then after combining the three pulsating voltage sources, the equivalent passive circuit and the respective admittance matrixes of the direct current transmission line, calculating the node voltage of the nodes at the head end and the tail end of the direct current transmission line in the direct current network according to a node voltage analysis method disclosed in the literature (Hanzhen Zhen kingdom. electric power system analysis [ M ]. Hangzhou, Zhejiang university press, 2013, 5 th edition).

The voltages of the nodes at the first end and the last end of the line are obtained through calculation and are shown in table 4 (taking 100Hz (frequency multiplication by 2), 600Hz (frequency multiplication by 12) and 1200Hz (frequency multiplication by 24) as examples).

TABLE 4

(5) According to the node voltages of the nodes at the first end and the last end of the direct current transmission line, the node voltage of each segmented node of the direct current transmission line is obtained through a back-stepping method, and the method comprises the following steps:

a. splitting the direct current transmission line into a line L (1, n-1) and a line L (n-1, n), and calculating the node voltage U of the n-1 node of the direct current transmission line according to the following formula_n-1：

U_n-1＝[Y_s(1,n-1)+Y_s(n-1,n)]^-1[Y_m(1,n-1)U₁+Y_m(n-1,n)U_n]

Wherein: n is the total number of the segmented nodes in the direct current transmission line, L (1, n-1) is a line from the first node to the n-1 node, L (n-1, n) is a line from the n-1 node to the n node, and U is a line from the n-1 node to the n node₁、U_nNode voltages of the nodes at the head end and the tail end of the direct current transmission line respectively, Y_s(1, n-1) and Y_m(1, n-1) are respectively the self-admittance matrix and the mutual admittance matrix of the line L (1, n-1), Y_s(n-1, n) and Y_m(n-1, n) are respectively a self-admittance matrix and a mutual admittance matrix of the line L (n-1, n);

b. then splitting a line L (1, n-1) in the direct current transmission line into the line L (1, n-2) and the line L (n-2, n-1), and calculating the node voltage U of the n-2 node in the split direct current transmission line according to the step a_n-2；

c. And c, sequentially splitting section by section according to the steps a and b, and simultaneously calculating the node voltage of each section node of the direct current transmission line.

The results of the calculations of the node voltages along the line are shown in fig. 7 (taking 600Hz (12 times doubled) and 1200Hz (24 times doubled) as examples).

(6) After the node voltage of each segmented node is obtained, the harmonic current of the nodes at the head end and the tail end of each segmented line of the direct-current transmission line is further obtained, and the method comprises the following steps:

$\left[\begin{array}{c}{I}_A\\ I_B\end{array}\right]=\left[\begin{array}{cc}{Y}_{s-\mathrm{AB}}& -Y_{m-\mathrm{AB}}\\ {-Y}_{m-\mathrm{AB}}& Y_{s-\mathrm{AB}}\end{array}\right]\left[\begin{array}{c}{U}_A\\ U_B\end{array}\right]$

wherein, U_AAnd U_BRespectively being any sectional line L in the direct current transmission line_ABNode voltages of first and last segment nodes A and B, I_AAnd I_BHarmonic currents, Y, of node A and node B, respectively_s-ABAnd Y_m-ABRespectively a segmented line L_ABThe self-admittance matrix and the transadmittance matrix.

The method for solving the harmonic current on the ground wire is as follows:

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <msup> <mi>A</mi> <mo>&prime;</mo> </msup> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <msup> <mi>B</mi> <mo>&prime;</mo> </msup> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mo>[</mo> <mo>-</mo> <msubsup> <mi>Y</mi> <mrow> <mi>m</mi> <mo>-</mo> <mi>AB</mi> </mrow> <mo>&prime;</mo> </msubsup> <mo>]</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mi>A</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mi>B</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein, I_A'And I_B'Respectively, any one of the segment lines L of the ground wire_A'B'Harmonic currents, U, of first and last end nodes A' and B_AAnd U_BRespectively a segmented line L in a direct current transmission line_ABNode voltages of first and last end nodes A and B, segment line L_A'B'And direct current transmissionSegmented line L in an electric line_ABCorresponding to, Y'_m-ABFor a sectional line L_ABAnd a segment line L_A'B'A coupled line transadmittance matrix.

Y′_m-ABIs determined by the following method: for any segment line L_ABAnd a segment line L_A'B'Assuming that the corresponding node voltage equation of the undeleted ground line is:

wherein, U_A′And U_B′Respectively a segment line L in the earth line_A'B'Node voltages, Y ' of first and last two-terminal nodes A ' and B '_s-ABAnd Y'_m-ABRespectively, segment lines L without ground lines erased_ABThe self-admittance matrix and the transadmittance matrix.

Since the ground is ideally grounded, the above equation can be rewritten as:

thus, a segmented line L is obtained_ABAnd a segment line L_A'B'Coupled line transadmittance matrix Y'_m-AB。

In order to verify the effectiveness of the method, the calculation results of the harmonic current under partial harmonic times are shown in table 5, and the comparison between the calculation results of the simulation software and the calculation results of the analytic algorithm of the invention is included:

TABLE 5

Compared with the calculation result obtained by adopting the time domain simulation technology, the calculation result of the invention is found to be more ideal. Therefore, the method can bring higher calculation efficiency and can also guarantee higher effectiveness.

Fig. 8 shows the calculated harmonic current results of the times of the positive dc power transmission line along the line.

For a direct-current transmission line with two ground wires and two direct-current transmission line structures, the residual current along the line can be calculated according to the following formula:

<math> <mrow> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mi>res</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mi>P</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mi>W</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>I</mi> <mo>&CenterDot;</mo> </mover> <mi>V</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math>

wherein,is the nth harmonic component of the residual current at x along the line distance from the rectifying side,is the n-th harmonic current on the positive pole direct current transmission line,is the n-th harmonic current on the negative pole direct current transmission line,andis the nth harmonic current on the two ground lines.

Fig. 9 is a diagram showing the distribution of the calculated number harmonics of the residual current portion along the line.

For a direct-current transmission line with a two-ground wire and two-direct-current transmission line structure, the equivalent interference current I along the line_eq(x) Can be calculated according to the following formula:

<math> <mrow> <msub> <mi>I</mi> <mi>eq</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <munderover> <mi>&Sigma;</mi> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>50</mn> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>res</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>*</mo> <msub> <mi>H</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </msqrt> </mrow> </math>

wherein, P_nFor n-th harmonic noise weighting coefficients, H_nIs the coupling coefficient.

Fig. 10 is a calculation result of the equivalent interference current along the line.

Claims (4)

1. A method for calculating harmonic current on a direct current side of a hybrid bipolar direct current transmission system comprises the following steps:

(1) respectively carrying out equivalent transformation on the LCC and the MMC in the system according to the system operating condition and the system main loop parameter to obtain an equivalent triple-pulse voltage source corresponding to the LCC and an equivalent passive circuit corresponding to the MMC;

(2) the method comprises the following steps of carrying out segmentation processing on a direct current transmission line and a ground wire thereof, calculating an admittance matrix of each segment, and further obtaining the admittance matrix of the whole direct current transmission line by a recurrence method, wherein the specific implementation method comprises the following steps:

A1. for the first sectional line L in the DC transmission line₁And a second segment line L₂Merging, calculating the merged line L according to the following formula₁₂Admittance matrix G of₁₂：

$G_{12}=[Y_{11}-Y_{12}{Y}_{22}^{-1}{Y}_{21}]$

$\begin{array}{cccc}{Y}_{11}=\left[\begin{array}{cc}{Y}_{s-1}& 0\\ 0& Y_{s-2}\end{array}\right]& Y_{12}=\left[\begin{array}{c}-Y_{m-1}\\ -Y_{m-2}\end{array}\right]& Y_{21}=\left[\begin{array}{cc}-Y_{m-1}& -Y_{m-2}\end{array}\right]& Y_{22}=Y_{s-1}+Y_{s-2}\end{array}$

Wherein: l is₁₂＝L₁+L₂，Y_s-1And Y_m-1Are respectively a line L₁Self-admittance and mutual admittance matrices, Y_s-2And Y_m-2Are respectively a line L₂The self-admittance matrix and the mutual admittance matrix of (a);

A2. make the line L₁₂And a third segment line L₃Merging is carried out, and the merged line L is calculated according to the step A1₁₃Admittance matrix G of₁₃；

A3. Sequentially combining section by section according to the steps A1 and A2, simultaneously calculating and storing the admittance matrix of the combined line each time, and finally obtaining the admittance matrix of the whole direct current transmission line;

(3) establishing a direct current network of the system according to the equivalent triple-pulse voltage source, the equivalent passive circuit and the direct current transmission line, and further calculating node voltages of the nodes at the first end and the last end of the direct current transmission line by using a node voltage analysis method after combining the admittance matrixes of the equivalent triple-pulse voltage source, the equivalent passive circuit and the direct current transmission line;

(4) obtaining the node voltage of each segmented node of the direct-current transmission line by a back-stepping method according to the node voltages of the nodes at the first end and the last end of the direct-current transmission line; further, the harmonic current of each segmented node of the direct current transmission line and the ground wire of the direct current transmission line is obtained;

the specific implementation process of solving the voltage of each segmented node on the direct-current transmission line by the inverse method is as follows:

B1. splitting the direct current transmission line into a line L (1, n-1) and a line L (n-1, n), and calculating the node voltage U of the n-1 node in the direct current transmission line according to the following formula_n-1：

U_n-1＝[Y_s(1，n-1)+Y_s(n-1，n)]^-1[Y_m(1，n-1)U₁+Y_m(n-1，n)U_n]

Wherein: n isFor the total number of the segmented nodes in the direct current transmission line, L (1, n-1) is a line from a first node to an n-1 node, L (n-1, n) is a line from the n-1 node to the n node, and U₁、U_nNode voltages of the nodes at the head end and the tail end of the direct current transmission line respectively, Y_s(1, n-1) and Y_m(1, n-1) are respectively the self-admittance matrix and the mutual admittance matrix of the line L (1, n-1), Y_s(n-1, n) and Y_m(n-1, n) are respectively a self-admittance matrix and a mutual admittance matrix of the line L (n-1, n);

B2. then splitting a line L (1, n-1) in the direct current transmission line into the line L (1, n-2) and the line L (n-2, n-1), and calculating the node voltage U of the n-2 node in the split direct current transmission line according to the step B1_n-2；

B3. And D, sequentially splitting the direct current transmission line section by section according to the steps B1 and B2, and simultaneously calculating the node voltage of each section node of the direct current transmission line.

2. The method according to claim 1, wherein in step (1), the equivalent passive circuit is formed by connecting three sets of RLC links in parallel, wherein the RLC links are formed by connecting a capacitor C, an inductor L and a resistor R in series from an input end to an output end, and the output end of the RLC link is grounded; wherein C is 2C₀/N，L＝2L₀，R＝NR₀(ii) a N is the cascade number of the bridge arm converter sub-modules of MMC, C₀For the capacitance in the converter submodule, L₀Bridge arm reactance, R, being MMC₀Is the on-resistance of the converter submodule.

3. The method for calculating the harmonic current on the direct current side of the hybrid bipolar direct current transmission system according to claim 1, wherein in the step (4), the method for calculating the harmonic current of each segmented node of the direct current transmission line is as follows:

$\left[\begin{array}{c}{I}_A\\ I_B\end{array}\right]=\left[\begin{array}{cc}{Y}_{s-\mathrm{AB}}& -Y_{m-\mathrm{AB}}\\ -Y_{m-\mathrm{AB}}& Y_{s-\mathrm{AB}}\end{array}\right]\left[\begin{array}{c}{U}_A\\ U_B\end{array}\right]$

wherein, U_AAnd U_BRespectively being any sectional line L in the direct current transmission line_ABNode voltages of first and last end nodes A and B, I_AAnd I_BHarmonic currents, Y, of node A and node B, respectively_s-ABAnd Y_m-ABRespectively a segmented line L_ABThe self-admittance matrix and the transadmittance matrix.

4. The method for calculating the harmonic current on the dc side of the hybrid bipolar dc transmission system according to claim 1, wherein in the step (4), the harmonic current on the ground line is obtained by:

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <msup> <mi>A</mi> <mo>&prime;</mo> </msup> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <msup> <mi>B</mi> <mo>&prime;</mo> </msup> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mo>[</mo> <mo>-</mo> <msubsup> <mi>Y</mi> <mrow> <mi>m</mi> <mo>-</mo> <mi>AB</mi> </mrow> <mo>&prime;</mo> </msubsup> <mo>]</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mi>A</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mi>B</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein, I_A′And I_B′Respectively, any one of the segment lines L of the ground wire_A′B′Harmonic currents, U, of first and last end nodes A' and B_AAnd U_BRespectively a segmented line L in a direct current transmission line_ABNode voltages of first and last end nodes A and B, segment line L_A′B′And a segment line L_ABCorresponding to, Y'_m-ABFor a sectional line L_ABAnd segmentationLine L_A′B′A coupled line transadmittance matrix.