CN108599167B - Linear power flow calculation method for radial power distribution network - Google Patents
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Abstract
The invention relates to a linear power flow calculation method for a radial power distribution network, and belongs to the technical field of operation control of power systems. On the basis of a traditional linear equation without considering network loss and interphase mutual impedance, nonlinear network loss terms are linearly expanded and included in the equation, a linear three-phase branch power flow equation is established, and the active power load and the reactive power load of all nodes are utilized to calculate the approximate values of the active power, the reactive power and the voltage of all nodes of all branches, so that the power flow calculation result of the radial power distribution network is obtained. Compared with a linearized branch load flow equation without considering network loss, the method greatly improves the calculation precision, can directly obtain an approximate solution of the load flow without iterative solution, is rapid in calculation, and is suitable for being applied to scenes with strict performance requirements such as real-time online analysis of the power distribution network.
Description
Technical Field
The invention relates to a linear power flow calculation method for a radial power distribution network, and belongs to the technical field of operation control of power systems.
Background
Compared with a transmission network, the power distribution network has the following remarkable characteristics:
1. essentially operating as a radial network, there is no annular loop.
2. The three-phase parameters of the line are asymmetric, the three-phase load of the node is unbalanced, and part of branches operate in a single-phase or double-phase state.
3. The branch resistance and the reactance are close to each other, and the PQ decoupling condition is not met.
Therefore, the characteristics need to be considered, and a power flow model which accords with the operation characteristics of the power distribution network is established.
Currently, a power flow algorithm suitable for a power distribution network has been proposed by a few scholars, such as a gaussian iteration method based on a three-phase extended node admittance matrix, a newton-raphson method and a previous generation back-substitution method using the radial operation characteristics of the power distribution network. Although the calculation efficiency of solving the power flow of the power distribution network by the prior generation back substitution method is quite high, the prior generation back substitution method still needs iterative solution, and in the problems of optimal power flow and the like, a power flow equation often appears in the problems as a non-convex equality constraint condition. To simplify the solution of the problem, it is necessary to convert the exact power flow equations into approximate simplified power flow equations. The existing literature provides a single-phase branch power flow equation based on the radial running characteristic of a power distribution network, and omits a network loss item to be used as a linear equation to solve.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a linear power flow calculation method of a radial power distribution network. The method selects a known tidal current state of a power grid as a reference point, approximates nonlinear terms in a tidal current equation by using parameters such as power and voltage of the reference point to obtain a linear three-phase tidal current equation, and rapidly obtains an approximate tidal current solution by solving the linear equation. The method has the advantages of high calculation speed and high result precision, and is suitable for being applied to scenes with harsh performance requirements, such as real-time online analysis of the power distribution network.
The invention provides a linearization power flow calculation method of a radial distribution network, which is characterized by comprising the following steps:
1) supposing that a radial distribution network to be subjected to load flow calculation has N +1 nodes, wherein the number of a root node is 0, starting from the root node, numbering the tail end nodes of each branch from 1 to N in sequence, and then, the number of the branches of the distribution network is N;
2) randomly selecting one node from nodes numbered from 1 to N as k, recording a branch formed by the node and an upstream node i as ik, and calculating a three-phase branch load flow equation of the node k and the branch ik; the method comprises the following specific steps:
2-1) calculate the voltage drop over branch ik:
wherein,
the expression points are divided, v, I and s are all in a three-dimensional column vector form and respectively represent voltage, current and power of three phases, subscript I represents the number of a head node of a branch ik, subscript k represents the number of a tail node of the branch ik, and x represents the conjugate of a complex number; v. ofkAnd viRespectively representing the voltages of node k and node I, IikCurrent, s, representing branch ikikRepresents the power of branch ik; z is a radical ofikIs a three-phase impedance matrix of branch ik, which is a 3 × 3 symmetric complex matrix:
wherein z isikThe diagonal elements of (1) represent self-impedance of a, b and c three phases, and the non-diagonal elements represent interphase mutual impedance;
conjugate two sides of the equal sign of the formula (1) and multiply the conjugate with the formula (1) to obtain:
2-2) considering node power balance, obtaining:
where the left side of the equation is the incoming power of node k, the right side of the equation is the sum of the outgoing powers of node k,
is the load of node k;
substituting formula (1) for formula (4) to obtain:
the three-phase branch load flow equation of the node k and the branch ik is formed by the formulas (1) and (5);
3) carrying out linear approximation on the three-phase branch load flow equation obtained in the step 2); the method comprises the following specific steps:
3-1) selecting a reference state, wherein
The voltage representing the reference state of node i,
and
respectively representing the active power of the branch ik reference state and the reactive power of the reference state;
3-2) voltage of reference state with node i
Approximation
Approximation of equation (1):
for the first two terms on the right of the equal sign of formula (6), define
Obtaining:
wherein p isik，qikRespectively representing the active and reactive power of branch ik,
is a calculated auxiliary variable;
for the third term on the right of equation (6), define
diag denotes the conversion of a vector into a diagonal matrix,
are all calculated auxiliary variables, resulting in:
to the right of formula (8) is with respect to pikAnd q isikA quadratic function of
The first-order Taylor expansion is carried out and converted into a linear function:
further written in the form:
wherein R isik，XikIs a 3 x 3 matrix of the matrix,
is a 3 × 1 vector, and takes the following value:
3-3) voltage of reference state with node i
Approximation
Approximation of equation (5):
the second term on the left of equation (12) is written as:
order to
Are all calculated auxiliary variables, resulting in:
writing the real part and the imaginary part separately into the forms of active power and reactive power:
in equation (15), the first equation represents the active power balance of node k, the second equation represents the reactive power balance of node k, pik、pkmThe active power of the branches ik and km, qik、qkmThe reactive powers of the branches ik and km respectively, and the node m is a downstream node of the node k;
is the active load of the node k and,
is the reactive load of node k; f. ofp(pik,qik) And fq(pik,qik) Respectively as follows:
fp(pik,qik) And fq(pik,qik) Are all about pikAnd q isikOf a quadratic function of
And (3) performing first-order Taylor expansion, and converting into a linear function:
equation (15) is written as follows:
wherein B isik，Gik，Hik，KikIs a 3 x 3 matrix of the matrix,
is a 3 × 1 vector, and takes the following value:
after conversion, the equations (10) and (20) form a linearized three-phase branch load flow equation of the node k and the branch ik;
4) repeating the step 2) to the step 3) to obtain a linearized three-phase branch flow equation of all nodes of the power distribution network except the root node and corresponding branches taking the nodes as tail end nodes;
5) converting all the linearized three-phase branch load flow equations obtained in the step 4) into a matrix form to obtain a power distribution network load flow calculation result;
the expression of the linearized three-phase branch load flow equation in the form of a matrix is as follows:
solving a linear equation set composed of the formulas (22) to (24) to obtain the active power p of the branchbReactive power qbBranch voltage difference ubAnd node voltage unThe expression of (1) is as follows, wherein subscript b represents all branch sets, and subscript n represents all node sets;
wherein M is a three-phase extended node branch incidence matrix, and N ispRepresenting a network incidence matrix, Rb,Xb,Bb,Gb,Hb,KbRespectively representing R corresponding to each branchik,Xik,Bik,Gik,Hik,KikThe extension forms which are arranged along the main diagonal of the matrix according to the serial number sequence of the branch tail end nodes are all 3 Nx3N matrixes;
respectively indicating that each branch corresponds to
The extension forms arranged according to the serial number sequence of the branch ends are vectors of 3 Nx 1;
inputting the active power load p of all nodes by the equation (25)LAnd reactive power load qLCalculating the active power p of all branchesbReactive power qbAll node voltages unAnd obtaining the load flow calculation result of the radial distribution network.
The method has the characteristics and beneficial effects that:
in order to improve the accuracy of a linearized three-phase power flow equation, the invention provides a novel linearized approximation method, and a nonlinear network loss term is linearly expanded and included in the equation on the basis of a traditional linear equation without considering network loss and phase mutual impedance. As long as a proper reference load flow state is selected, the method can perform quite accurate estimation on the network loss, the accuracy of a calculation result is obviously higher than that of a simple linear equation, the advantages of a linear model are maintained, iterative solution is not needed, and the calculation speed is high.
The method starts from an accurate branch power flow equation, the voltage in the equation is replaced by the voltage in a reference state, the secondary power term is subjected to first-order Taylor expansion linearization at the power of a reference point, and no other approximation or some term is removed. Therefore, the accuracy of the power flow equation only depends on the selection of the datum point and is not influenced by the three-phase imbalance degree. The application of the method in the practical power distribution network calculation example shows that the result provided by the method is high in precision and the precision is related to the selected reference state, when the precise power flow solution is selected as the reference state, the precise solution is obtained, and even if the non-precise power flow solution is selected as the reference state, the error is far smaller than the linearized branch power flow equation without considering the network loss. Meanwhile, the method does not need iterative solution, is rapid in calculation, and is suitable for being applied to scenes with harsh performance requirements, such as real-time online analysis of the power distribution network.
Drawings
FIG. 1 is a block diagram of the overall flow of the method of the present invention.
Fig. 2 is a schematic power flow diagram of a radial distribution network in accordance with an embodiment of the present invention.
Detailed Description
The invention provides a linearized power flow calculation method for a radial distribution network, which is further described in detail below with reference to the accompanying drawings and specific embodiments.
The invention provides a linearization power flow calculation method of a radial distribution network, aiming at the network characteristics of radial operation and three-phase unbalance of the distribution network, the overall flow is shown in figure 1, and the linearization power flow calculation method comprises the following steps:
1) supposing that a radial distribution network to be subjected to load flow calculation has N +1 nodes, wherein the number of a root node is 0, starting from the root node, numbering the tail end nodes of each branch from 1 to N in sequence, and then, the number of the branches of the distribution network is N;
2) randomly selecting one node from nodes numbered from 1 to N as k, recording a branch formed by the node and an upstream node i as ik, and calculating a three-phase branch load flow equation of the node k and the branch ik;
taking the trend of node k and branch ik as shown in fig. 2 as an example, node i is an upstream node of node k (i.e., node i is closer to the root node than node k, and there is only one upstream node in each node in the radial grid, (if the selected node is numbered 1. then the upstream node is the root node, the same method is also adopted for calculation), and node m is a downstream node of node k (there may be more than one downstream node of each node, and any one downstream node m is taken as an illustration here).
The method comprises the following specific steps:
2-1) calculate the voltage drop over branch ik:
wherein,
the notation dot division, v, I and s are all in the form of three-dimensional column vectors representing the voltage, current and power of the three phases, respectively, with subscript I representing the number of the leading node of branch ik, subscript k representing the number of the trailing node of branch ik, and x representing the conjugate of the complex number. v. ofkAnd viRespectively representing the voltages of node k and node I, IikCurrent, s, representing branch ikikRepresenting the power of branch ik. z is a radical ofikIs a three-phase impedance matrix of branch ik, which is a 3 × 3 symmetric complex matrix:
wherein z isikThe diagonal elements of (a) represent the self-impedance of the three phases a, b and c, and the non-diagonal elements represent the interphase mutual impedance.
Conjugate two sides of the equal sign of the formula (1) and multiply the conjugate with the formula (1) to obtain:
2-2) considering node power balance, obtaining:
where the left side of the equation is the incoming power of node k, equationThe right side is the sum of the outgoing powers of node k,
is the load of node k.
Substituting formula (1) for formula (4) to obtain:
the three-phase branch power flow equation of the node k and the branch ik is formed by the formulas (1) and (5). Compared with the single-phase branch load flow equation, the mathematical form of the three-phase branch load flow equation is more complicated due to the interphase coupling.
3) Carrying out linear approximation on the three-phase branch load flow equation obtained in the step 2); the method comprises the following specific steps:
3-1) for a linearized approximation, a reference state is selected, which includes values of branch power and node voltage, and the result of the state estimation can be generally selected as the reference state, whose value is denoted by the superscript 0 in the following derivation, including:
the voltage representing the reference state of node i,
and
respectively representing the active power of the branch ik reference state and the reactive power of the reference state;
3-2) approximating the equation (1), the voltage per unit of the actual power grid is about 1, so the voltage of the state is referred to by the node i
Approximation
For the first two terms on the right of the equal sign of formula (6), define
Obtaining:
wherein p isik，qikRespectively representing the active and reactive power of branch ik,
are all calculated auxiliary variables.
For the third term on the right of equation (6), define
diag denotes the conversion of a vector into a diagonal matrix,
are all calculated auxiliary variables, resulting in:
wherein
Respectively, the active power and the reactive power of the branch ik reference state.
To the right of formula (8) is with respect to pikAnd q isikA quadratic function of
The first-order Taylor expansion is carried out and converted into a linear function:
further written in the form:
wherein R isik，XikIs a 3 x 3 matrix of the matrix,
is a 3 × 1 vector, and takes the following value:
3-3) voltage of reference state with node i
Approximation
Approximation of equation (5):
the second term on the left of equation (12) is written as:
order to
Are all calculated auxiliary variables, resulting in:
writing the real part and the imaginary part separately into the forms of active power and reactive power:
in equation (15), the first equation represents the active power balance of node k, the second equation represents the reactive power balance of node k, pik、pkmThe active power of the branches ik and km, qik、qkmThe reactive powers of the branches ik and km respectively, and the node m is a downstream node of the node k;
is the active load of the node k and,
is the reactive load of node k. f. ofp(pik,qik) And fq(pik,qik) Respectively as follows:
fp(pik,qik) And fq(pik,qik) Are all about pikAnd q isikOf a quadratic function of
And (3) performing first-order Taylor expansion, and converting into a linear function:
after work-up, formula (15) is written as follows:
wherein B isik，Gik，Hik，KikIs a 3 x 3 matrix of the matrix,
is a 3 × 1 vector, and takes the following value:
after transformation, equations (10) and (20) constitute a linearized three-phase branch power flow equation for node k and branch ik.
4) And (4) repeating the steps 2) to 3) to obtain the linearized three-phase branch flow equation of all the nodes of the power distribution network except the root node and the corresponding branches taking the nodes as the tail end nodes.
5) Converting all the linearized three-phase branch load flow equations obtained in the step 4) into a matrix form to obtain a power distribution network load flow calculation result;
the expression of the linearized three-phase branch load flow equation in the form of a matrix is as follows:
solving the linear equation set composed of the equations (22) to (24) can be achieved by using linear algebra software such as MATLAB and the like to obtain the active power p of the branchb(subscript b represents the set of all branches), reactive power qbAm, amDifference u of the path voltagebNode voltage un(subscript n represents the set of all nodes) the explicit expression is as follows, it can be seen that the node load p isLAnd q isLLinear form of (a).
Wherein M is a three-phase extended node branch incidence matrix, and N ispRepresenting the network incidence matrix (these two matrices can be written directly according to the power grid topology), Rb,Xb,Bb,Gb,Hb,KbRespectively representing R corresponding to each branchik,Xik,Bik,Gik,Hik,KikThe expansion forms arranged along the main diagonal of the matrix according to the numbering sequence of the branch end nodes are all 3 Nx3N matrixes.
Respectively indicating that each branch corresponds to
The extension forms arranged according to the numbering sequence of the branch ends are vectors of 3 Nx 1.
Inputting the active power load p of all nodes by the equation (25)LAnd reactive power load qLBy performing matrix operation, all branch active power p can be directly obtainedbReactive power qbAll node voltages unAnd obtaining the load flow calculation result of the radial distribution network.
Claims (1)
1. A linearization power flow calculation method of a radial distribution network is characterized by comprising the following steps:
1) supposing that a radial distribution network to be subjected to load flow calculation has N +1 nodes, wherein the number of a root node is 0, starting from the root node, numbering the tail end nodes of each branch from 1 to N in sequence, and then, the number of the branches of the distribution network is N;
2) randomly selecting one node from nodes numbered from 1 to N as k, recording a branch formed by the node and an upstream node i as ik, and calculating a three-phase branch load flow equation of the node k and the branch ik; the method comprises the following specific steps:
2-1) calculate the voltage drop over branch ik:
wherein,
the expression points are divided, v, I and s are all in a three-dimensional column vector form and respectively represent voltage, current and power of three phases, subscript I represents the number of a head node of a branch ik, subscript k represents the number of a tail node of the branch ik, and x represents the conjugate of a complex number; v. ofkAnd viRespectively representing the voltages of node k and node I, IikCurrent, s, representing branch ikikRepresents the power of branch ik; z is a radical ofikIs a three-phase impedance matrix of branch ik, which is a 3 × 3 symmetric complex matrix:
wherein z isikThe diagonal elements of (1) represent self-impedance of a, b and c three phases, and the non-diagonal elements represent interphase mutual impedance;
conjugate two sides of the equal sign of the formula (1) and multiply the conjugate with the formula (1) to obtain:
2-2) considering node power balance, obtaining:
where the left side of the equation is the incoming power of node k, the right side of the equation is the sum of the outgoing powers of node k,
is the load of node k;
substituting formula (1) for formula (4) to obtain:
the three-phase branch load flow equation of the node k and the branch ik is formed by the formulas (1) and (5);
3) carrying out linear approximation on the three-phase branch load flow equation obtained in the step 2); the method comprises the following specific steps:
3-1) selecting a reference state, wherein
The voltage representing the reference state of node i,
and
respectively representing the active power of the branch ik reference state and the reactive power of the reference state;
3-2) voltage of reference state with node i
Approximation
Approximation of equation (1):
for the first two terms on the right of the equal sign of formula (6), define
Obtaining:
wherein p isik，qikRespectively representing the active and reactive power of branch ik,
is a calculated auxiliary variable;
for the third term on the right of equation (6), define
diag denotes the conversion of a vector into a diagonal matrix,
are all calculated auxiliary variables, resulting in:
to the right of formula (8) is with respect to pikAnd q isikA quadratic function of
The first-order Taylor expansion is carried out and converted into a linear function:
further written in the form:
wherein R isik，XikIs a 3 x 3 matrix of the matrix,
is a 3 × 1 vector, and takes the following value:
3-3) voltage of reference state with node i
Approximation
Approximation of equation (5):
the second term on the left of equation (12) is written as:
order to
Are all calculated auxiliary variables, resulting in:
writing the real part and the imaginary part separately into the forms of active power and reactive power:
in equation (15), the first equation represents the active power balance of node k, the second equation represents the reactive power balance of node k, pik、pkmThe active power of the branches ik and km, qik、qkmThe reactive powers of the branches ik and km respectively, and the node m is a downstream node of the node k;
is the active load of the node k and,
is the reactive load of node k; f. ofp(pik,qik) And fq(pik,qik) Respectively as follows:
fp(pik,qik) And fq(pik,qik) Are all about pikAnd q isikOf a quadratic function of
And (3) performing first-order Taylor expansion, and converting into a linear function:
equation (15) is written as follows:
wherein B isik，Gik，Hik，KikIs a 3 x 3 matrix of the matrix,
is a 3 × 1 vector, and takes the following value:
after conversion, the equations (10) and (20) form a linearized three-phase branch load flow equation of the node k and the branch ik;
4) repeating the step 2) to the step 3) to obtain a linearized three-phase branch flow equation of all nodes of the power distribution network except the root node and corresponding branches taking the nodes as tail end nodes;
5) converting all the linearized three-phase branch load flow equations obtained in the step 4) into a matrix form to obtain a power distribution network load flow calculation result;
the expression of the linearized three-phase branch load flow equation in the form of a matrix is as follows:
solving a linear equation set composed of the formulas (22) to (24) to obtain the active power p of the branchbReactive power qbBranch voltage difference ubAnd node voltage unWherein subscript b represents all branch sets and subscript n representsThere is a set of nodes;
wherein M is a three-phase extended node branch incidence matrix, and N ispRepresenting a network incidence matrix, Rb,Xb,Bb,Gb,Hb,KbRespectively representing R corresponding to each branchik,Xik,Bik,Gik,Hik,KikThe extension forms which are arranged along the main diagonal of the matrix according to the serial number sequence of the branch tail end nodes are all 3 Nx3N matrixes;
respectively indicating that each branch corresponds to
The extension forms arranged according to the serial number sequence of the branch ends are vectors of 3 Nx 1;
inputting the active power load p of all nodes by the equation (25)LAnd reactive power load qLCalculating the active power p of all branchesbReactive power qbAll node voltages unAnd obtaining the load flow calculation result of the radial distribution network.
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| CN110568265A (en) * | 2019-10-17 | 2019-12-13 | 江苏远致能源科技有限公司 | Impedance identification method and system based on low-voltage distribution network |
| CN112542838B (en) * | 2020-11-26 | 2023-01-03 | 云南电网有限责任公司 | Three-phase power flow linearization method for power distribution network based on support vector regression |
| CN114221344A (en) * | 2021-11-05 | 2022-03-22 | 三峡大学 | Improved affine algorithm-based forward-backward substitution load flow calculation method |
| CN114566971B (en) * | 2022-03-01 | 2024-09-24 | 东北大学秦皇岛分校 | Real-time optimal power flow calculation method based on near-end strategy optimization algorithm |
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