CN112100829A - Sheath induced voltage and circulation calculation method of cable system laid along bridge - Google Patents

Sheath induced voltage and circulation calculation method of cable system laid along bridge Download PDF

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CN112100829A
CN112100829A CN202010894863.3A CN202010894863A CN112100829A CN 112100829 A CN112100829 A CN 112100829A CN 202010894863 A CN202010894863 A CN 202010894863A CN 112100829 A CN112100829 A CN 112100829A
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cable
matrix
bridge
admittance
voltage
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CN112100829B (en
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陈向荣
徐星
孟繁博
杜振东
刘燕平
周文俊
裘立峰
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Zhejiang University ZJU
Zhejiang Huayun Electric Power Engineering Design Consulting Co
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Zhejiang University ZJU
Zhejiang Huayun Electric Power Engineering Design Consulting Co
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Abstract

The invention discloses a sheath induced voltage and circulation calculation method of a cable system laid along with a bridge, which comprises the steps of deducing a unit length series impedance matrix and a unit length parallel admittance matrix of the cable system laid along with the bridge according to the geometric structure, basic material parameters and overhead transmission characteristics of the cable system laid along with the bridge, and determining a unit length node admittance matrix; deducing different-order matrixes and a cascade formula of the same-order matrix according to node constraint conditions, and calculating to obtain a complete node admittance matrix and a segmented node admittance matrix of the cable system along with the bridge; and calculating the induced voltage and the circulation distribution of the cable sheath and the return conductor section by section according to the boundary conditions and the node admittance matrix. The method can accurately describe the influence of the overhead laying environment on the cable system laid along with the bridge, simultaneously considers the influence of the line distribution parameters, can accurately calculate the induced voltage and the circulation distribution of the cable sheath and the return conductor, and provides a calculation basis for the grounding design of the cable system laid along with the bridge.

Description

Sheath induced voltage and circulation calculation method of cable system laid along bridge
Technical Field
The invention belongs to the technical field of power transmission, and particularly relates to a sheath induced voltage and circulation calculation method of a cable system laid along a bridge.
Background
With the proposal of the strategy of global energy Internet and the rapid development of marine energy engineering, the development scale of projects such as offshore wind power generation, marine oil and gas resource exploitation and the like is continuously enlarged, the offshore economy is rapidly developed, and the construction of an offshore power transmission system becomes one of the keys of the future power system development in China. The power cable is used as an important component of a new energy cross-sea power transmission system, has higher economical efficiency and operation and maintenance reliability compared with submarine cable laying through bridge laying, and becomes a preferred choice of the cross-sea power transmission system.
Different from the conventional underground cable, the corrosion effect of a grounding system of the power cable on a bridge reinforcing steel bar structure is usually considered when the power cable is laid through a sea-crossing bridge, so that the grounding point of the cable system laid along with the bridge is very limited, and the induced voltage on a cable metal sheath becomes a very critical problem in the grounding design. Meanwhile, due to the ground limitation of the cable system along with the bridge, a return conductor is often laid in a full bridge, and the circulating current generated on the cable sheath and the return conductor also becomes a key factor for inhibiting the electric energy transmission capacity of the cable along with the bridge.
In summary, the induced voltage and the circulation distribution of the sheath of the cable along with the bridge are very critical, and currently, researches on the problems are mainly focused on underground cable systems, while related researches on cable systems laid along with the bridge are rarely reported. In addition, the traditional cable sheath induced voltage and circulation calculation method has the following three problems: 1) lack of calculation formulas for electrical parameters of aerial cabling; 2) the influence of line distribution parameters is not considered, and the cable sheath circulating current is regarded as a fixed value; 3) the cable sheath induced voltage calculation model is too simple, and the calculation result is not accurate enough. Therefore, a method for calculating sheath induced voltage and circulating current of a cable system laid along with a bridge is needed to provide a calculation basis for the grounding design of the cable system along with the bridge.
Disclosure of Invention
The invention aims to solve the problem of steady state calculation of a cable system laid along with a bridge and provides a method for calculating sheath induced voltage and circulation distribution. The grounding mode design of the cable system laid along the bridge provides basis for determining the length of the cable section and reducing the loss of the cable sheath and the return conductor.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method for calculating the sheath induced voltage and the circulation of a cable system laid along a bridge comprises the following steps:
step 1: according to the geometric structure and basic material parameters of the cable system laid along with the bridge, considering the influence of overhead laying, and determining a unit length series impedance matrix and a unit length parallel admittance matrix of the cable system laid along with the bridge;
step 2: determining a unit length node admittance matrix of the cable system along with the bridge by adopting the series decomposition of the matrix according to a transmission line telegraph equation;
and step 3: deducing a cascade formula of a same-order matrix and a different-order matrix according to node constraint conditions of the cable system, thereby obtaining a complete node admittance matrix and a segmented node admittance matrix of the cable system along with the bridge;
and 4, step 4: and calculating the induced voltage and the circulation distribution of the cable sheath and the return conductor along with the bridge section by section according to the boundary conditions and the node admittance matrix.
In an embodiment of the present invention, a specific process of determining the series impedance matrix per unit length in step 1 is as follows:
firstly, considering the influence of overhead laying on the parameter characteristics of the cable, the cable system along with the bridge is decomposed into two parts of the cable and an external medium,
then, determining a unit series impedance matrix of the cable part;
subsequently, determining a self-impedance part of a unit length series impedance matrix in the external medium;
thirdly, determining the mutual impedance part of the unit length series impedance matrix in the external medium;
and finally, obtaining a unit-length series impedance matrix of the cable system along with the bridge considering the influence of aerial laying.
In an embodiment of the present invention, a specific process of determining the parallel admittance matrix per unit length in step 1 is as follows:
firstly, considering the influence of aerial laying on the parameter characteristics of the cable, the cable system along with the bridge is decomposed into two parts of an inner conductor and an outer conductor of the cable,
then, determining a unit length parallel admittance matrix of the inner conductor part of the cable;
subsequently, determining a mutual admittance portion of the parallel admittance matrix per unit length of the outer conductor portion;
subsequently, determining a self-admittance portion of the parallel admittance matrix per unit length of the outer conductor portion;
and finally, obtaining a unit-length parallel admittance matrix of the cable system considering the overhead laying influence.
In an embodiment of the present invention, a specific process of determining the cascade formula of the same-order matrix and the different-order matrix in step 3 is as follows:
first, for two nodes of the same order, the admittance matrix Y1And Y2Respectively establishing a node voltage equation as follows:
Figure BDA0002658126870000031
Figure BDA0002658126870000032
wherein: y is1And Y2Node admittance matrixes of the cable section 1 and the cable section 2 are respectively provided, and the order numbers of the node admittance matrixes are the same; i iss1And Is2Current vectors of the head ends of the cable section 1 and the cable section 2 respectively flow from the head end to the tail end of the cable; i ism1And Im2The current vectors of the tail ends of the cable section 1 and the cable section 2 respectively flow from the tail end to the head end of the cable; u shapes1And Us2Voltage vectors to ground are respectively provided for the head ends of the cable section 1 and the cable section 2; u shapem1And Um2The voltage vectors of the tail ends of the cable segment 1 and the cable segment 2 to earth are respectively; a. the1、B1、C1And D1Is Y1The quarter-sub-matrix of (1); m1、N1、O1And P1Is Y2The quarter-sub-matrix of (1);
according to the node constraint condition, the currents flowing into and out of the middle nodes of the two cascaded cable sections are equal in magnitude and opposite in direction, and the following relational expression is obtained:
Im1＝-Is2 (11)
finally, the cascade expression of the same-order matrix is obtained as follows:
Figure BDA0002658126870000033
wherein: y iseq1The equivalent node admittance matrix after the cascade connection;
then, admittance matrix Y is applied to two different order nodesaAnd YbRespectively establishing a node voltage equation as follows:
Figure BDA0002658126870000034
Figure BDA0002658126870000035
wherein: y isaAnd YbNode admittance matrices, Y, for cable segment a and cable segment b, respectivelybOrder ratio Y ofa2 more; i issaAnd IsbCurrent vectors of the head ends of the cable section a and the cable section b respectively flow from the head end to the tail end of the cable; i ismaAnd ImbThe current vectors of the tail ends of the cable section a and the cable section b respectively flow from the tail end to the head end of the cable; u shapesaAnd UsbVoltage vectors to ground of the head ends of the cable segment a and the cable segment b are respectively; u shapemaAnd UmbVoltage vectors to earth at the tail ends of the cable segment a and the cable segment b respectively; a. the2、B2、C2And D2Is YaThe quarter-sub-matrix of (1); m2、N2、O2And P2Is YbThe quarter-sub-matrix of (1);
subsequently, a current transformation matrix T is introduced1Voltage transformation matrix T2Sum node admittance transformation matrix TAAnd TBThe following expression is obtained:
Figure BDA0002658126870000041
Figure BDA0002658126870000042
Figure BDA0002658126870000043
wherein:
Figure BDA0002658126870000044
Figure BDA0002658126870000045
thus obtaining:
Figure BDA0002658126870000046
wherein: y iseq2Is YbTransformed equivalent node admittance matrix, order and YaThe same; i'mbAnd U'mbRespectively converting the tail end current and voltage vectors of the cable section b;
finally, according to the equivalent matrix Y after transformationeq2And the cascade connection of matrixes of different orders can be completed by combining the formula of the cascade connection of matrixes of the same order.
In a specific embodiment of the present invention, the specific process of calculating the induced voltage and the circulating current distribution of the cable sheath and the return conductor along the bridge segment by segment according to the boundary conditions and the node admittance matrix in step 4 is as follows:
if the number of the nodes of the cable section is not changed, determining the distribution of the first end current and the tail end current according to the boundary condition, determining the vector of the tail end voltage and current according to the first end voltage and current and determining the vector of the head end voltage and current according to the tail end voltage and current respectively adopt the following expressions:
Figure BDA0002658126870000051
Figure BDA0002658126870000052
Figure BDA0002658126870000053
wherein: i'sIs head end current vector, I'mIs terminal current vector, U'sIs head end to ground voltage vector, U'mA terminal voltage vector to earth is adopted, Y 'is a node admittance matrix of the cable segment, and A', B ', C' and D 'are quarter submatrices of Y';
if the number of the nodes of the cable section changes, determining the voltage and current distribution by adopting the following steps:
firstly, introducing a voltage inverse transformation matrix T4There is the following node voltage equation:
Figure BDA0002658126870000054
Figure BDA0002658126870000055
wherein: u'sAnd U'mThe voltage vectors, U', of the head end and the tail end of the low-order cable segment, which are obtained according to the boundary condition and the formula (22) or the formula (23), are relative to the groundsAnd U ″)mFor the voltage vector of head end and tail end of high-order cable segment to be solvedsAnd I ″)mCurrent vectors of the head end and the tail end of the high-order cable section to be solved are obtained;
finally, the induced voltage and circulating current distributions with the bridge cable jacket and return conductor are obtained by combining formula (22), formula (23) and formula (24).
Compared with the prior art, the invention has the following beneficial effects:
(1) the method considers the influence of the overhead laying environment of the cable along with the bridge, deduces the calculation formula of the electrical parameters of the overhead cable, solves the problem that the existing calculation formula of the electrical parameters is not suitable for a cable system along with the bridge, and can accurately calculate the induced voltage and the circulation distribution of the cable along with the bridge;
(2) the invention considers the influence of the distribution parameters of the cable line along the bridge, divides the cable line into a plurality of infinitesimals, overcomes the defect that the sheath circulating current is regarded as a fixed value in the prior art, and ensures that the calculation results of the induced voltage and the circulating current distribution of the cable along the bridge are more accurate;
(3) the invention is based on the method of matrix cascade, deduces the cascade formula of the same-order matrix and different-order matrix, overcomes the defects of high order number and complex calculation process of the prior node admittance matrix, thereby achieving the purposes of quickly and accurately calculating the induced voltage and the circulation distribution of the cable along with the bridge;
drawings
FIG. 1 is a flow chart of the calculation method of the present invention.
Fig. 2 is a schematic diagram of a multi-conductor system structure of an on-bridge cable.
Fig. 3 is a schematic diagram of an equivalent circuit of a multi-conductor system of an on-bridge cable.
Fig. 4 is a schematic diagram of a typical on-bridge cable system configuration.
Fig. 5 is a schematic diagram of an equivalent circuit of a typical on-bridge cable system.
Fig. 6(a) is a graph of the induced voltage distribution calculation with the bridge cable sheath and return conductor.
Fig. 6(b) is a graph of the calculation result of the circulating current distribution with the sheath of the bridge cable and the return conductor.
Detailed Description
In order to describe the present invention more specifically, the following detailed description of the embodiments of the present invention is provided with reference to the accompanying drawings.
As shown in fig. 1, a method for calculating induced voltage and circulating current of a sheath of a cable system laid along a bridge includes the following steps:
(1) according to the geometric structure and the basic material parameters of the cable system laid along with the bridge, considering the influence of overhead laying, determining a unit length series impedance matrix and a unit length parallel admittance matrix of the cable system laid along with the bridge:
the cable system under study is composed of a three-phase power cable and 1 return bare conductor, and the structure of the cable system is shown in fig. 2. Firstly, determining a unit length series impedance matrix of a cable system along with a bridge, considering the influence of aerial laying on the parameter characteristics of the cable, and decomposing the cable system along with the bridge into two parts, namely a cable and an external medium, wherein the unit length series impedance matrix expression is as follows:
Figure BDA0002658126870000061
wherein: zcBeing a unit series impedance matrix, Z, of a cable section in an overhead cable systemoIs built on stiltsA unit series impedance matrix of an external dielectric portion in the cable system;
then, a unit series impedance matrix for the cable section is determined, using the following expression:
Figure BDA0002658126870000071
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, z iscciCore-earth loop self-impedance, z, for cable icsiIs the mutual impedance between the core-ground loop and the sheath-ground loop of cable i, zssiSelf-impedance of sheath-earth loop for cable i, zc44Is the self-impedance of the return conductor-earth return;
subsequently, the self-impedance portion of the unit length series impedance matrix in the external medium is determined, using the following expression:
Figure BDA0002658126870000072
Figure BDA0002658126870000073
Figure BDA0002658126870000074
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, z isoiiFor series self-impedance to ground, z, of cable i in the outer dielectric parto44Is the series self-impedance to ground of the return conductor in the external dielectric part, j is an imaginary unit, omega is an angular frequency, lambda is an integral independent variable, mu0The magnetic conductivity of air, h is the height of the cable from the ground, R is the radius of an outer insulating layer of the cable or the radius of a return conductor, and sigma is the conductivity of soil;
furthermore, the transimpedance portion of the single-length series impedance matrix in the external medium is determined using the following expression:
Figure BDA0002658126870000075
Figure BDA0002658126870000081
Figure BDA0002658126870000082
wherein: i and k are integers representing different cables, 1, 2 and 3 represent A, B and a C-phase cable, 4 represents a return conductor, and z representsoikIs the mutual impedance to ground, z, between cables i and k in the outer dielectric partoi4Is the mutual impedance to ground between the cable i and the return conductor of the external dielectric part, j is an imaginary number unit, omega is an angular frequency, lambda is an integral independent variable, mu0Is the permeability of air, hiHeight from ground h of cable ikHeight of cable k from ground, dikIs the horizontal separation of conductors i and k, σ is the conductivity of the soil;
and finally, synthesizing the derivation process to obtain a series impedance matrix of unit length of the cable system along with the bridge considering the influence of overhead laying.
Then, determining a unit-length parallel admittance matrix of the cable system along with the bridge, considering the influence of aerial laying on the parameter characteristics of the cable, decomposing the aerial power system shown in fig. 2 into two parts, namely an inner conductor and an outer conductor of the cable, and then, obtaining a unit-length parallel impedance matrix expression as follows:
Figure BDA0002658126870000083
wherein: y iscFor single-unit parallel admittance matrices, Y, of inner conductor portions of cables in overhead cable systemsoBeing units of external conductor parts in overhead cable systemsParallel admittance matrixes;
then, a unit parallel admittance matrix of the inner conductor portion of the cable is determined, using the following expression:
Figure BDA0002658126870000084
Figure BDA0002658126870000085
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, yciIs the admittance between the core and the sheath of the cable i in the conductor part in the cable, j is an imaginary number unit, omega is an angular frequency,0in order to have a dielectric constant in a vacuum,inis the relative dielectric constant, R, of the inner insulation layer of the cable1Radius of the cable core, R2Radius of the cable inner insulation layer;
subsequently, the transimpedance portion of the unit parallel admittance matrix of the outer conductor portion is determined, using the following expression:
Figure BDA0002658126870000091
Figure BDA0002658126870000092
Figure BDA0002658126870000093
wherein: i and j are integers representing different cables, 1, 2 and 3 representing A, B and a C-phase cable, 4 representing a return conductor, yoijIs the mutual admittance between the sheaths of cable i and cable j in the outer conductor part, yoi4Is the mutual admittance between the jacket of the cable i and the return conductor in the outer conductor portion, j being the imaginary unit, omega being the angular frequency,0in order to have a dielectric constant in a vacuum,exis the relative dielectric constant, R, of the inner insulation layer of the cable3Is the radius of the cable sheath, R4Radius of the outer insulation layer of the cable, hiHeight from ground h of cable i4To return the conductor to ground height, dikIs the horizontal spacing between cable i and cable k;
subsequently, the self-impedance portion of the unit parallel admittance matrix of the outer conductor portion is determined, using the following expression:
Figure BDA0002658126870000094
Figure BDA0002658126870000095
Figure BDA0002658126870000096
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, yogiFor the ground admittance of the sheath of cable i in the outer conductor part, yog4For the admittance of the return conductor to ground in the outer conductor part, yoijFor mutual admittance, y, between cable i and cable j sheath in the outer conductor partoi4Is the mutual admittance between the jacket of the outer conductor part cable i and the return conductor, j is the imaginary unit, ω is the angular frequency,0in order to have a dielectric constant in a vacuum,exis the relative dielectric constant, R, of the inner insulation layer of the cable3Is the cable sheath radius, R4Radius of the outer insulation layer of the cable, hiHeight from ground h of cable i4For height of return conductor above ground, RhIs the return conductor radius;
and finally, synthesizing the derivation process to obtain the unit-length parallel admittance matrix of the cable system considering the overhead laying influence.
(2) Determining a unit length node admittance matrix of the cable system along with the bridge by adopting the series decomposition of the matrix according to a transmission line telegraph equation:
according to the theory of multi-conductor analysis, the cable system along with the bridge can be equivalent to an equivalent circuit diagram combining series impedance and parallel admittance, the structure of which is shown in fig. 3, and the frequency domain telegraph equation of the cable system is as follows:
Figure BDA0002658126870000101
then, the differential equation is solved, and the following node voltage equation is obtained by combining the boundary conditions:
Figure BDA0002658126870000102
Figure BDA0002658126870000103
wherein: z is the series impedance matrix per unit length, Y is the parallel admittance matrix per unit length, l is the cable length, I1Is a head-end current vector, I2Is a terminal current vector, U1Is a head end voltage vector, U2Is the terminal voltage vector and is the propagation constant matrix;
finally, solving the node admittance matrix by using the series expansion of the matrix
Figure BDA0002658126870000104
Figure BDA0002658126870000105
Figure BDA0002658126870000106
Wherein: b isnIs Bernoulli number, [ m ═ 0%]When m is 0, 1 is selected,the rest is 0 and B 01 is ═ 1; the number of stages is 0<||l||<Convergence at pi, i.e. ensuring 0<||(ZY)l2||<π。
Finally, combining the unit length series impedance matrix and the parallel admittance matrix obtained in the step 1, obtaining a node admittance matrix along with the unit length of the bridge cable or a node admittance matrix of a fixed length section through the derivation process;
(3) deducing a cascade formula of a same-order matrix and a different-order matrix according to node constraint conditions of the cable system, thereby obtaining a complete node admittance matrix and a segmented node admittance matrix of the bridge-following cable system:
the two ends of the cable system under study are grounded at single end, the middle part is grounded by cross interconnection, the ground points are electrically connected through a return conductor, and finally the grounding is realized through a bridge grounding device, and the structure of the cable system is shown in fig. 4. Fig. 5 is a schematic node segmentation diagram, and the whole on-bridge cable system is divided into 15 segments, such as a power internal resistance, a grounding box, a single-ended grounding cable segment, a cross-connection connector, and an equivalent load resistance segment, to finally form 16 node groups. The invention takes nodes 3, 4 and 5 as examples to illustrate the cascade of matrixes of the same order, and the node voltage equation is as follows:
Figure BDA0002658126870000111
Figure BDA0002658126870000112
I4a＝-I4b (26)
wherein: i is3aIs the current vector of node 3, I4aAnd I4bIs the current vector of node 4, I5bFor the current vector of node 5, subscript a is defined as the current direction to the right, subscript b is defined as the current direction to the left, U3、U4And U5Voltage to ground vector, Y, for node 3, node 4 and node 5, respectively34Between nodes 3 and 4Nodal admittance matrix, Y, of a cable segment45Node admittance matrix for a cable segment between nodes 4 and 5, A34、B34、C34And D34Is Y34The quarter-sub-matrix of (1); m45、N45、O45And P45Is Y45The quarter-sub-matrix of (1);
according to the fact that the currents flowing into and out of the nodes 4 are equal in magnitude and opposite in direction, the cascade expression of the same-order matrix is finally determined as follows:
Figure BDA0002658126870000113
wherein: y is35eqThe equivalent matrix after the cascade connection is obtained;
then, for the typical cabled cable system shown in fig. 5, the cascading of matrices of different orders is illustrated by taking nodes 1, 2 and 15 as examples, and the node voltage equation is as follows:
Figure BDA0002658126870000121
Figure BDA0002658126870000122
wherein: i is1aIs the current vector of node 1, I'2aAnd I2bIs the current vector of node 2, I'15bFor the current vector at node 15, subscript a is defined as the current direction to the right, subscript b is defined as the current direction to the left, U1、U2And U'2And U'15Voltage vectors to ground of the node 1, the node 2 and the node 15 respectively, the absence of superscript indicates that the corresponding vector order is 4 orders, the presence of superscript' ″ indicates that the corresponding vector order is 5 orders, and Y12An 8-order node admittance matrix, Y, for a cable segment between node 1 and node 22_15An admittance matrix of 10 th order nodes, A, for a cable segment between node 2 and node 1512、B12、C12And D12Is Y12One fourth ofA sub-matrix; m'2_15、N′2_15、O′2_15And P'2_15Is Y2_15The quarter-sub-matrix of (1);
then, according to corresponding node constraint conditions, introducing a current transformation matrix T1Voltage transformation matrix T2Sum node admittance transformation matrix TAAnd TBThe following expression is obtained:
Figure BDA0002658126870000123
namely, it is
Figure BDA0002658126870000124
Figure BDA0002658126870000125
Figure BDA0002658126870000126
Wherein:
Figure BDA0002658126870000127
Figure BDA0002658126870000131
then, according to the equivalent matrix Y after transformation2_15eqAnd the cascade connection of matrixes of different orders can be completed by combining the formula of the cascade connection of matrixes of the same order.
Finally, combining the node admittance matrix of the unit length of the cable along with the bridge obtained in the step 2 or the node admittance matrix of the fixed length section, and utilizing the cascade connection of the same-order matrix and the cascade connection formula of different-order matrixes to obtain the full-line node admittance matrix and the segmented node admittance matrix of the cable along with the bridge system section by section;
(4) calculating the induced voltage and the circulation distribution of the cable sheath and the return conductor along with the bridge section by section according to the boundary conditions and the node admittance matrix:
if the number of the nodes of the cable section is not changed, determining the distribution of the first end current and the tail end current according to the boundary condition, determining the vector of the tail end voltage and current according to the first end voltage and current and determining the vector of the head end voltage and current according to the tail end voltage and current respectively adopt the following expressions:
Figure BDA0002658126870000132
Figure BDA0002658126870000133
Figure BDA0002658126870000134
wherein: i issIs a head-end current vector, ImIs a terminal current vector, UsIs a head end voltage vector, UmIs the terminal voltage vector, YsmNodal admittance matrix for cable segmentation, Asm、Bsm、CsmAnd DsmIs YsmThe quarter-sub-matrix of (1);
if the number of nodes changes in the cable segment, the method for determining the voltage and current vectors of the cable is described by taking the node 2 and the node 15 in fig. 5 as an example, and the following expression is adopted:
firstly, introducing a voltage inverse transformation matrix T4The following node voltage equation is obtained:
Figure BDA0002658126870000135
Figure BDA0002658126870000136
Figure BDA0002658126870000141
wherein: u shape2sAnd U15mThe 4 th order head end and tail end to ground voltage vectors, U ', at node 2 and node 15, respectively, found from the boundary conditions and equations (37) and (38)'2sAnd U'15m5-order head and tail-end to ground voltage vectors, I ', at nodes 2 and 15, respectively, to be sought'2sAnd l'15m5 th order head and tail current vectors at node 2 and node 15, respectively, to be solved;
combining the full-line node admittance matrix and the segmented node admittance matrix of the cable system along with the bridge obtained in the step 3, firstly obtaining current vectors at the nodes 1 and 16 according to a formula (36) and a boundary voltage condition, then obtaining voltage and current vectors at the nodes 2 and 15 according to a formula (37) and a formula (38), then obtaining voltage and current vectors at the nodes 3, 4, 13 and 14 according to a voltage and current solving formula (39) with variable node number, and finally obtaining voltage and current distribution of a cable section sheath and a return conductor along lines by combining the segmented node admittance matrix of the cable system along with the bridge and the formula (36).
Fig. 6(a) and (b) are induced voltage and circulating current profiles of a cable jacket and a return conductor, respectively. Firstly, considering the influence of overhead laying on the parameter characteristics of the cable, decomposing a cable system along with a bridge into two parts of the cable and an external medium to obtain a series impedance matrix and a parallel admittance matrix in unit length; solving the node admittance matrix by using the series expansion of the matrix to obtain the node admittance matrix along with the unit length of the bridge cable or the node admittance matrix of a fixed length section; then, the same-order matrix cascade and different-order matrix cascade formulas are used for solving section by section to obtain a full-line node admittance matrix and a subsection node admittance matrix of the cable system along with the bridge; and finally, solving a formula according to the boundary conditions and the voltage and the current to obtain the induced voltage and the circulating current distribution of the cable sheath and the return conductor. The simulation method curve is obtained by establishing a corresponding simulation model in the electromagnetic transient simulation software PSCAD and calculating, the three-phase power supply model adopts a 220kV alternating current power supply model carried by the software, the load adopts an equivalent impedance model, the cable and the return conductor are simulated by inputting an equivalent RLC resistance inductance capacitance network in the PSCAD by calculating the frequency characteristic under 50 Hz. As can be seen from fig. 6 a, the distribution laws of the distribution curves of the induced voltages of the cable sheath and the return conductor obtained by the two methods are basically consistent, the maximum induced voltage at the ungrounded portion of the single-ended grounding section of the bridge cable, the maximum induced voltage at the cross-connection portion of the cross-connection portion, the linear rising trend of the induced voltage at the single-ended grounding portion, the high and low trend of the induced voltage distribution of the cross-connection middle unit, the low and high characteristics of the return conductor, and the actual operation law is met. The induced voltage peak values calculated by the two methods are basically the same, and the error is only 2%, which shows that the calculation method can accurately calculate the induced voltage distribution of the cable system along with the bridge. As can be seen from fig. 6 b, the distribution laws of the circulation distribution curves of the cable sheath and the return conductor obtained by the two methods are also basically consistent, the circulation at the ungrounded portion of the single-ended grounding section of the bridge cable is 0, the circulation at the cross interconnection of the cross interconnection sections is the largest, the circulation at the single-ended grounding section is in a linear descending trend, the circulation distribution of the cross interconnection intermediate unit is in a trend of high at two ends and low at the middle, the circulation of the return conductor is in a characteristic of high at the middle and low at two ends, and the circulation of the return conductor at different grounding modes is basically unchanged and accords with the actual operation law. The induced voltage peak values calculated by the two methods are basically the same, and the error is only 2.5A, which shows that the calculation method can accurately calculate the circulation distribution of the cable system along with the bridge. In conclusion, the calculation method can accurately calculate the induced voltage and the circulation distribution of the sheath and the return conductor of the cable system along with the bridge.

Claims (5)

1. A method for calculating the sheath induced voltage and the circulation of a cable system laid along a bridge comprises the following steps:
step 1: according to the geometric structure and basic material parameters of the cable system laid along with the bridge, considering the influence of overhead laying, and determining a unit length series impedance matrix and a unit length parallel admittance matrix of the cable system laid along with the bridge;
step 2: determining a unit length node admittance matrix of the cable system along with the bridge by adopting the series decomposition of the matrix according to a transmission line telegraph equation;
and step 3: deducing a cascade formula of a same-order matrix and a different-order matrix according to node constraint conditions of the cable system, thereby obtaining a complete node admittance matrix and a segmented node admittance matrix of the cable system along with the bridge;
and 4, step 4: and calculating the induced voltage and the circulation distribution of the cable sheath and the return conductor along with the bridge section by section according to the boundary conditions and the node admittance matrix.
2. The method for calculating the sheath induced voltage and the circulating current of the cable system laid along the bridge according to claim 1, wherein the method comprises the following steps: the specific process of determining the unit-length series impedance matrix in the step 1 is as follows:
firstly, considering the influence of aerial laying on the parameter characteristics of the cable, and decomposing the cable system along with the bridge into two parts of the cable and an external medium, the series impedance matrix expression of the unit length is as follows:
Figure FDA0002658126860000011
wherein: zcBeing a unit series impedance matrix, Z, of a cable section in an overhead cable systemoA unit series impedance matrix for an external dielectric portion of an overhead cable system;
then, a unit series impedance matrix for the cable section is determined, using the following expression:
Figure FDA0002658126860000012
wherein: i is an integer, i is 1,2 and 3 represent A, B and C-phase cables, respectively, i is 4 represents the return conductor, zcciCore-earth loop self-impedance, z, for cable icsiIs the mutual impedance between the core-ground loop and the sheath-ground loop of cable i, zssiSelf-impedance of sheath-earth loop for cable i, zc44Is the self-impedance of the return conductor-earth return;
subsequently, the self-impedance portion of the unit length series impedance matrix in the external medium is determined, using the following expression:
Figure FDA0002658126860000021
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, z isoiiFor series self-impedance to ground, z, of cable i in the outer dielectric parto44A series self-impedance to ground for the return conductor in the outer dielectric portion;
furthermore, the transimpedance portion of the single-length series impedance matrix in the external medium is determined using the following expression:
Figure FDA0002658126860000022
wherein: i and j are integers representing different cables, 1, 2 and 3 representing A, B and a C-phase cable, 4 representing a return conductor, zoijIs the mutual impedance to ground, z, between cables i and j in the outer dielectric partoi4Is the mutual impedance to ground between the outer dielectric part cable i and the return conductor;
and finally, obtaining a unit-length series impedance matrix of the cable system along with the bridge considering the influence of aerial laying.
3. The method for calculating the sheath induced voltage and the circulating current of the cable system laid along the bridge according to claim 1, wherein the method comprises the following steps: the specific process of determining the unit length parallel admittance matrix in the step 1 is as follows:
firstly, considering the influence of aerial laying on the parameter characteristics of the cable, and decomposing the cable system along with the bridge into two parts of an inner conductor and an outer conductor of the cable, the parallel impedance matrix expression of the unit length is as follows:
Figure FDA0002658126860000023
wherein: y iscFor connecting admittance matrices, Y, in parallel per unit length of inner conductor portions of cables in overhead cable systemsoAn admittance matrix is connected in parallel for the unit length of the external conductor part in the overhead cable system;
then, a unit length parallel admittance matrix of the inner conductor portion of the cable is determined, using the following expression:
Figure FDA0002658126860000031
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, yciIs admittance between a cable core and a sheath of a cable i in the conductor part in the cable;
subsequently, the transadmittance portion of the parallel admittance matrix per unit length of the outer conductor portion is determined, using the following expression:
Figure FDA0002658126860000032
wherein: i and j are integers representing different cables, 1, 2 and 3 representing A, B and a C-phase cable, 4 representing a return conductor, yoijIs the mutual admittance between the sheaths of cable i and cable j in the outer conductor part, yoi4Is the mutual admittance between the jacket of the cable i in the outer conductor portion and the return conductor;
subsequently, the self-admittance sections of the parallel admittance matrix per unit length of the outer conductor portions are determined, using the following expression:
Figure FDA0002658126860000033
wherein: i is an integer, i is 1, 2 and 3 represent A, B and C-phase cable respectively, i is 4 represents return conductor, yogiFor the ground admittance of the sheath of cable i in the outer conductor part, yog4For the admittance of the return conductor to ground in the outer conductor part, yoijFor mutual admittance, y, between cable i and cable j sheath in the outer conductor partoi4Is the mutual admittance between the sheath of the outer conductor part cable i and the return conductor;
and finally, obtaining a unit-length parallel admittance matrix of the cable system considering the overhead laying influence.
4. The method for calculating the sheath induced voltage and the circulating current of the cable system laid along the bridge according to claim 1, wherein the method comprises the following steps: the specific process of determining the cascade formula of the same-order matrix and the different-order matrix in the step 3 is as follows:
first, for two nodes of the same order, the admittance matrix Y1And Y2Respectively establishing a node voltage equation as follows:
Figure FDA0002658126860000041
Figure FDA0002658126860000042
wherein: y is1And Y2Node admittance matrixes of the cable section 1 and the cable section 2 are respectively provided, and the order numbers of the node admittance matrixes are the same; i iss1And Is2Current vectors of the head ends of the cable section 1 and the cable section 2 respectively flow from the head end to the tail end of the cable; i ism1And Im2The current vectors of the tail ends of the cable section 1 and the cable section 2 respectively flow from the tail end to the head end of the cable; u shapes1And Us2Voltage vectors to ground are respectively provided for the head ends of the cable section 1 and the cable section 2; u shapem1And Um2The voltage vectors of the tail ends of the cable segment 1 and the cable segment 2 to earth are respectively; a. the1、B1、C1And D1Is Y1The quarter-sub-matrix of (1); m1、N1、O1And P1Is Y2The quarter-sub-matrix of (1);
according to the node constraint condition, the currents flowing into and out of the middle nodes of the two cascaded cable sections are equal in magnitude and opposite in direction, and the following relational expression is obtained:
Im1＝-Is2 (11)
finally, the cascade expression of the same-order matrix is obtained as follows:
Figure FDA0002658126860000043
wherein: y iseq1The equivalent node admittance matrix after the cascade connection;
then, admittance matrix Y is applied to two different order nodesaAnd YbRespectively establishing a node voltage equation as follows:
Figure FDA0002658126860000044
Figure FDA0002658126860000045
wherein: y isaAnd YbNode admittance matrices, Y, for cable segment a and cable segment b, respectivelybOrder ratio Y ofa2 more; i issaAnd IsbCurrent vectors of the head ends of the cable section a and the cable section b respectively flow from the head end to the tail end of the cable; i ismaAnd ImbThe current vectors of the tail ends of the cable section a and the cable section b respectively flow from the tail end to the head end of the cable; u shapesaAnd UsbVoltage vectors to ground of the head ends of the cable segment a and the cable segment b are respectively; u shapemaAnd UmbVoltage vectors to earth at the tail ends of the cable segment a and the cable segment b respectively; a. the2、B2、C2And D2Is YaThe quarter-sub-matrix of (1); m2、N2、O2And P2Is YbThe quarter-sub-matrix of (1);
subsequently, a current transformation matrix T is introduced1Voltage transformation matrix T2Sum node admittance transformation matrix TAAnd TBThe following expression is obtained:
Figure FDA0002658126860000051
Figure FDA0002658126860000052
Figure FDA0002658126860000053
wherein:
Figure FDA0002658126860000054
Figure FDA0002658126860000055
thus obtaining:
Figure FDA0002658126860000056
wherein: y iseq2Is YbTransformed equivalent node admittance matrix, order and YaThe same; i'mbAnd U'mbRespectively converting the tail end current and voltage vectors of the cable section b;
finally, according to the equivalent matrix Y after transformationeq2And the cascade connection of matrixes of different orders can be completed by combining the formula of the cascade connection of matrixes of the same order.
5. The method for calculating the sheath induced voltage and the circulating current of the cable system laid along the bridge according to claim 1, wherein the method comprises the following steps: the specific process of calculating the induced voltage and the circulation distribution of the cable sheath and the return conductor along with the bridge section by section according to the boundary conditions and the node admittance matrix in the step 4 is as follows:
if the number of the nodes of the cable section is not changed, determining the distribution of the first end current and the tail end current according to the boundary condition, determining the vector of the tail end voltage and current according to the first end voltage and current and determining the vector of the head end voltage and current according to the tail end voltage and current respectively adopt the following expressions:
Figure FDA0002658126860000061
Figure FDA0002658126860000062
Figure FDA0002658126860000063
wherein: i'sIs head end current vector, I'mIs terminal current vector, U'sIs head end to ground voltage vector, U'mA terminal voltage vector to earth is adopted, Y 'is a node admittance matrix of the cable segment, and A', B ', C' and D 'are quarter submatrices of Y';
if the number of the nodes of the cable section changes, determining the voltage and current distribution by adopting the following steps:
firstly, introducing a voltage inverse transformation matrix T4Having the following node voltagesThe equation:
Figure FDA0002658126860000064
Figure FDA0002658126860000065
wherein: u'sAnd U'mThe voltage vectors, U', of the head end and the tail end of the low-order cable segment, which are obtained according to the boundary condition and the formula (22) or the formula (23), are relative to the groundsAnd U ″)mFor the voltage vector of head end and tail end of high-order cable segment to be solvedsAnd I ″)mCurrent vectors of the head end and the tail end of the high-order cable section to be solved are obtained;
finally, the induced voltage and circulating current distributions with the bridge cable jacket and return conductor are obtained by combining formula (22), formula (23) and formula (24).
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