CN108599167B - Linear power flow calculation method for radial power distribution network - Google Patents

Abstract

The invention relates to a linearized power flow calculation method for a radial distribution network, and belongs to the technical field of power system operation control. Based on the traditional linear equation that does not consider the network loss and interphase mutual impedance, the invention includes the linear expansion of the nonlinear network loss term into the equation, and establishes a linearized three-phase branch power flow equation, using the active power of all nodes. Load and reactive power load, calculate the approximate values of all branch active power, reactive power, and all node voltages, and obtain the power flow calculation results of the radial distribution network. Compared with the linearized branch power flow equation that does not consider the network loss, the present invention greatly improves the calculation accuracy, and at the same time, the approximate solution of the power flow can be directly obtained without iterative solution, the calculation is fast, and it is suitable for the real-time online analysis of the distribution network and other performances. in demanding scenarios.

Description

A Linearized Power Flow Calculation Method for Radial Distribution Networks

technical field

The invention relates to a linearized power flow calculation method for a radial distribution network, and belongs to the technical field of power system operation control.

Background technique

Compared with the transmission network, the distribution network has the following salient features:

1. It basically operates in a radial network, and there is no ring loop.

2. The three-phase parameters of the line are asymmetrical, the three-phase load of the node is unbalanced, and some branches operate in a single-phase or two-phase state.

3. The branch resistance and reactance are similar in size and do not meet the PQ decoupling conditions.

Therefore, it is necessary to consider the above characteristics and establish a power flow model that conforms to the operation characteristics of the distribution network.

At present, many scholars have proposed power flow algorithms suitable for distribution networks, such as Gaussian iteration method based on three-phase extended node admittance matrix, Newton-Raphson method and previous generation back-substitution method using the characteristics of radial operation of distribution network . Although the calculation efficiency of the previous generation back-substitution method to solve the power flow of the distribution network is quite high, it still needs to be solved iteratively, and in the optimal power flow and other problems, the power flow equation often appears in the problem as a non-convex equality constraint. In order to simplify the solution of the problem, the exact power flow equation needs to be transformed into an approximate simplified power flow equation. At present, existing literatures have proposed the single-phase branch power flow equation based on the radial operation characteristics of the distribution network, and omitting the network loss term as a linear equation to solve.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a linearized power flow calculation method for a radial distribution network in order to overcome the shortcomings of the prior art. This method selects a known power flow state of a power grid as a reference point, uses parameters such as power and voltage at the reference point to approximate the nonlinear terms in the power flow equation, and obtains a linear three-phase power flow equation, which can be quickly approximated by solving the linear equation. trend solution. The invention has the advantages of fast calculation speed and high result accuracy, and is suitable for use in scenarios with demanding performance requirements such as real-time online analysis of distribution networks.

A linearized power flow calculation method for a radial distribution network proposed by the present invention is characterized in that the method includes the following steps:

1) Suppose there are N+1 nodes in the radial distribution network to be calculated for power flow. The root node is numbered 0. Starting from the root node, the end nodes of each branch are numbered from 1 to N in turn. Then the number of branches of the distribution network is N;

2) arbitrarily select a node from the nodes numbered 1 to N and denote k, the branch formed by this node and its upstream node i is denoted as ik, and calculate the three-phase branch power flow equation of node k and branch ik; Concrete steps as follows:

2-1) Calculate the voltage drop on branch ik:

表示点除，v，I和s都是三维的列向量形式，分别代表三相的电压、电流和功率，下标i代表支路ik的首端节点编号，下标k代表支路ik的末端节点编号，*代表复数的共轭；v_k和v_i分别表示节点k和节点i的电压，I_ik表示支路ik的电流，s_ik表示支路ik的功率；z_ik是支路ik的三相阻抗矩阵，为3×3的对称复矩阵：in,

Represents point division, v, I and s are all three-dimensional column vectors, representing the three-phase voltage, current and power respectively, the subscript i represents the head node number of the branch ik, and the subscript k represents the end of the branch ik Node number, * represents the conjugate of complex number; v _k and vi represent the voltage of node k and node _i respectively, I _ik represents the current of branch ik, s _ik represents the power of branch ik; z _ik is the power of branch ik The three-phase impedance matrix is a 3×3 symmetric complex matrix:

Among them, the diagonal elements of _zik represent the self-impedance of the three phases a, b, and c, and the off-diagonal elements represent the mutual impedance between phases;

Taking the conjugate of both sides of the equation (1) and multiplying it with the equation (1), we get:

2-2) Considering the node power balance, we get:

是节点k的负荷；where the left side of the equation is the inflow power of node k, and the right side of the equation is the sum of the outgoing power of node k,

is the load of node k;

Substituting equation (1) into equation (4), we get:

Equations (1) and (5) form the three-phase branch power flow equation of node k and branch ik;

3) Linearize and approximate the three-phase branch power flow equation obtained in step 2); the specific steps are as follows:

代表节点i参考状态的电压，

和

分别代表支路ik参考状态的有功功率和参考状态的无功功率；3-1) Select a reference state, where

voltage representing the reference state of node i,

and

respectively represent the active power of the reference state of the branch ik and the reactive power of the reference state;

近似

对式(1)进行近似：3-2) Use node i to refer to the voltage of the state

approximate

Approximate equation (1):

得到：For the first two terms on the right side of the equal sign of Equation (6), define

get:

是计算得到的辅助变量；Among them, p _ik , q _ik represent the active power and reactive power of branch ik, respectively,

is the calculated auxiliary variable;

diag表示将向量转化为对角矩阵，

都是计算得到的辅助变量，得到：For the third term on the right-hand side of equation (6), define

diag means to convert the vector into a diagonal matrix,

are calculated auxiliary variables, and get:

处进行一阶Taylor泰勒展开，转化为线性函数：The right side of equation (8) is the quadratic function about p _ik and q _ik , in

Perform a first-order Taylor expansion at , and convert it to a linear function:

It is further written in the following form:

是3×1的向量，取值如下式所示：where R _ik , Xi _ik are 3×3 matrices,

is a 3×1 vector with the following values:

近似

对式(5)进行近似：3-3) Use node i to refer to the voltage of the state

approximate

Approximate equation (5):

The second term on the left side of equation (12) is written as:

都是计算得到的辅助变量，得到：make

are calculated auxiliary variables, and get:

Separate the real and imaginary parts in the form of active power and reactive power:

是节点k的有功负荷，

是节点k的无功负荷；f_p(p_ik,q_ik)和f_q(p_ik,q_ik)的计算表达式分别如下：In equation (15), the first equation represents the balance of active power at node k, the second equation represents the balance of reactive power at node k, p _ik , p _km are the active powers of branch ik and km, respectively, q _ik and q _km are the reactive powers of branches ik and km respectively, and node m is the downstream node of node k;

is the active load of node k,

is the reactive load of node k; the calculation expressions of f _p (p _ik , q _ik ) and f _q (p _ik , q _ik ) are as follows:

处进行一阶Taylor展开，转化为线性函数：Both f _p (p _ik , q _ik ) and f _q (p _ik , q _ik ) are quadratic functions of p _ik and q _ik , which can be expressed in

Perform a first-order Taylor expansion at and convert to a linear function:

Write equation (15) in the following form:

是3×1的向量，取值如下式所示：where _Bik , _Gik , _Hik , _Kik are 3×3 matrices,

is a 3×1 vector with the following values:

After transformation, equations (10) and (20) form the linearized three-phase branch power flow equation of node k and branch ik;

4) Repeat step 2) to step 3) to obtain the linearized three-phase branch power flow equation of all nodes of the distribution network except the root node and the corresponding branch with this node as the end node;

5) Convert all the linearized three-phase branch power flow equations obtained in step 4) into matrix form to obtain the power flow calculation result of the distribution network;

The linearized three-phase branch power flow equation in matrix form is expressed as follows:

Solving the linear equations composed of equations (22) to (24), the expressions of branch active power p _b , reactive power q _b , branch voltage difference _ub and node voltage u _n are obtained as follows, where the subscript b represents the set of all branches, and the subscript n represents the set of all nodes;

分别表示将每条支路对应的

按照支路末端编号顺序排布的扩展形式，都是3N×1的向量；Among them, M is the three-phase expansion node branch correlation matrix, N _p represents the network correlation matrix, R _b , X _b , B _b , G _b , H _b , K _b represent the R _ik , X corresponding to each branch, respectively _ik ,B _ik ,G _ik ,H _ik ,K _ik are the extended forms arranged along the main diagonal of the matrix according to the number sequence of the branch end nodes, which are all 3N×3N matrices;

Respectively represent the corresponding

The extended forms arranged in the order of branch end numbers are all 3N×1 vectors;

Using equation (25), input the active power load p _L and reactive power load q _L of all nodes, calculate the approximate values of all branches active power p _b , reactive power q _b , and all node voltage u _n , and obtain the radial pattern The result of the power flow calculation of the distribution network.

The characteristics and beneficial effects of the method of the present invention are:

In order to improve the accuracy of the linearized three-phase power flow equation, the present invention proposes a new linearization approximation method, which linearly expands the nonlinear network loss term on the basis of the traditional linear equation that does not consider the network loss and interphase mutual impedance. included in the equation. As long as an appropriate reference power flow state is selected, this method can make a fairly accurate estimation of the network loss. The accuracy of the calculation result is obviously higher than that of the simple linear equation, and the advantages of the linear model are maintained. Iterative solution is not required, and the calculation speed is very fast. .

This method starts from the accurate branch power flow equation, replaces the voltage in the equation with the voltage of the reference state, and linearizes the secondary power term at the reference point power by first-order Taylor expansion, and there is no other approximation or a certain term is removed. Therefore, the accuracy of the power flow equation only depends on the selection of the reference point, and is not affected by the degree of three-phase unbalance. The application in the actual distribution network example shows that the accuracy of the results given by this method is very high, and the accuracy is related to the selected reference state. When the precise power flow solution is selected as the reference state, the exact solution is obtained. The inaccurate power flow solution is used as the reference state, and the error is also much smaller than the linearized branch power flow equation without considering the network loss. At the same time, the method does not need iterative solution, and the calculation is fast, which is suitable for real-time online analysis of distribution network and other scenarios with demanding performance requirements.

Description of drawings

Fig. 1 is an overall flow chart of the method of the present invention.

FIG. 2 is a schematic diagram of the power flow of a radial distribution network according to an embodiment of the present invention.

Detailed ways

A linearized power flow calculation method for a radial distribution network proposed by the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

A linearized power flow calculation method for a radial distribution network proposed by the present invention, aiming at the radial operation of the distribution network and the network characteristics of three-phase unbalance, the overall process is shown in Figure 1, including the following steps:

1) Suppose there are N+1 nodes in the radial distribution network to be calculated for power flow. The root node is numbered 0. Starting from the root node, the end nodes of each branch are numbered from 1 to N in turn. Then the number of branches of the distribution network is N;

2) arbitrarily select a node from the nodes numbered 1 to N and denote k, the branch formed by this node and its upstream node i is denoted as ik, and calculate the three-phase branch power flow equation of node k and branch ik;

Taking the power flow of node k and branch ik as shown in Figure 2 as an example, node i is the upstream node of node k (that is, node i is closer to the root node than node k, and each node in the radial grid has one and only one Upstream node, (if the selected node number is 1. Then the upstream node is the root node, and the same method is used to calculate), node m is a downstream node of node k (each node may have more than one downstream node, take it out here Any downstream node m is used as an illustration).

Specific steps are as follows:

2-1) Calculate the voltage drop on branch ik:

表示点除，v，I和s都是三维的列向量形式，分别代表三相的电压、电流和功率，下标i代表支路ik的首端节点编号，下标k代表支路ik的末端节点编号，*代表复数的共轭。v_k和v_i分别表示节点k和节点i的电压，I_ik表示支路ik的电流，s_ik表示支路ik的功率。z_ik是支路ik的三相阻抗矩阵，为3×3的对称复矩阵：in,

Represents point division, v, I and s are all three-dimensional column vectors, representing the three-phase voltage, current and power respectively, the subscript i represents the head node number of the branch ik, and the subscript k represents the end of the branch ik Node number, * represents the conjugate of a complex number. v _k and vi represent the voltage of node k and node _i respectively, I _ik represents the current of branch ik, and s _ik represents the power of branch ik. z _ik is the three-phase impedance matrix of branch ik, which is a 3×3 symmetric complex matrix:

Among them, the diagonal elements of _zik represent the self-impedance of the three phases a, b, and c, and the off-diagonal elements represent the mutual impedance between phases.

Taking the conjugate of both sides of the equation (1) and multiplying it with the equation (1), we get:

2-2) Considering the node power balance, we get:

是节点k的负荷。where the left side of the equation is the inflow power of node k, and the right side of the equation is the sum of the outgoing power of node k,

is the load of node k.

Substituting equation (1) into equation (4), we get:

Equations (1) and (5) form the three-phase branch power flow equation for node k and branch ik. Compared with the single-phase branch power flow equation, the mathematical form of the three-phase branch power flow equation is more complicated due to the existence of phase-to-phase coupling.

3) Linearize and approximate the three-phase branch power flow equation obtained in step 2); the specific steps are as follows:

代表节点i参考状态的电压，

和

分别代表支路ik参考状态的有功功率和参考状态的无功功率；3-1) In order to perform the linearization approximation, it is necessary to select a reference state, which includes the values of branch power and node voltage. Usually, the result of state estimation can be selected as the reference state, and the superscript 0 is used in the following derivation Values representing reference states, including:

voltage representing the reference state of node i,

and

respectively represent the active power of the reference state of the branch ik and the reactive power of the reference state;

近似

3-2) Approximate Equation (1), the voltage per unit value of the actual power grid is all around 1, so the voltage of node i is used as a reference state

approximate

得到：For the first two terms on the right side of the equal sign of Equation (6), define

get:

都是计算得到的辅助变量。Among them, p _ik , q _ik represent the active power and reactive power of branch ik, respectively,

are calculated auxiliary variables.

diag表示将向量转化为对角矩阵，

都是计算得到的辅助变量，得到：For the third term on the right-hand side of equation (6), define

diag means to convert the vector into a diagonal matrix,

are calculated auxiliary variables, and get:

分别代表支路ik参考状态的有功功率和无功功率。in

represent the active power and reactive power of the branch ik reference state, respectively.

处进行一阶Taylor泰勒展开，转化为线性函数：The right side of equation (8) is the quadratic function about p _ik and q _ik , in

Perform a first-order Taylor expansion at , and convert it to a linear function:

It is further written in the following form:

是3×1的向量，取值如下式所示：where R _ik , Xi _ik are 3×3 matrices,

is a 3×1 vector with the following values:

近似

对式(5)进行近似：3-3) Use node i to refer to the voltage of the state

approximate

Approximate equation (5):

The second term on the left side of equation (12) is written as:

都是计算得到的辅助变量，得到：make

are calculated auxiliary variables, and get:

Separate the real and imaginary parts in the form of active power and reactive power:

是节点k的有功负荷，

是节点k的无功负荷。f_p(p_ik,q_ik)和f_q(p_ik,q_ik)的计算表达式分别如下：In equation (15), the first equation represents the balance of active power at node k, the second equation represents the balance of reactive power at node k, p _ik , p _km are the active powers of branch ik and km, respectively, q _ik and q _km are the reactive powers of branches ik and km respectively, and node m is the downstream node of node k;

is the active load of node k,

is the reactive load of node k. The calculation expressions of f _p (p _ik , q _ik ) and f _q (p _ik , q _ik ) are respectively as follows:

处进行一阶Taylor展开，转化为线性函数：Both f _p (p _ik , q _ik ) and f _q (p _ik , q _ik ) are quadratic functions of p _ik and q _ik , which can be expressed in

Perform a first-order Taylor expansion at and convert to a linear function:

After finishing, formula (15) can be written in the following form:

是3×1的向量，取值如下式所示：where _Bik , _Gik , _Hik , _Kik are 3×3 matrices,

is a 3×1 vector with the following values:

After transformation, equations (10) and (20) form a linearized three-phase branch power flow equation for node k and branch ik.

4) Repeat steps 2) to 3) to obtain the linearized three-phase branch power flow equations of all nodes of the distribution network except the root node and the corresponding branch with this node as the end node.

5) Convert all the linearized three-phase branch power flow equations obtained in step 4) into matrix form to obtain the power flow calculation result of the distribution network;

The linearized three-phase branch power flow equation in matrix form is expressed as follows:

To solve the linear equation system composed of equations (22) to (24), it can be solved by linear algebra software such as MATLAB, and it can be obtained that the branch active power p _b (subscript b represents the set of all branches), reactive power q _b , The explicit expressions of the branch voltage difference _ub and the node voltage un (the subscript _n represents the set of all nodes) are as follows, and it can be seen that it is the linear form of the node loads p _L and q _L.

分别表示将每条支路对应的

按照支路末端编号顺序排布的扩展形式，都是3N×1的向量。Among them, M is the three-phase expansion node branch correlation matrix, N _p represents the network correlation matrix (these two matrices can be directly written according to the grid topology), R _b , X _b , B _b , G _b , H _b , K _b Respectively represent the extended form of arranging _Rik , _Xik , _Bik , _Gik , _Hik , _Kik corresponding to each branch along the main diagonal of the matrix according to the number sequence of the end nodes of the branch, all of which are 3N×3N matrix.

Respectively represent the corresponding

The extended forms arranged in the order of branch end numbers are all 3N×1 vectors.

Using formula (25), input the active power load p _L and reactive power load q _L of all nodes, and perform matrix operation, the active power p _b of all branches, reactive power q _b , and voltage u of all nodes can be directly obtained. The approximate value of _n can be used to obtain the calculation result of the power flow 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. of_kAnd v_iRespectively representing the voltages of node k and node I, I_ikCurrent, s, representing branch ik_ikRepresents the power of branch ik; z is a radical of_ikIs a three-phase impedance matrix of branch ik, which is a 3 × 3 symmetric complex matrix:

wherein z is_ikThe 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 is_ik，q_ikRespectively 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 p_ikAnd q is_ikA quadratic function of

The first-order Taylor expansion is carried out and converted into a linear function:

further written in the form:

wherein R is_ik，X_ikIs 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, p_ik、p_kmThe active power of the branches ik and km, q_ik、q_kmThe 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. of_p(p_ik,q_ik) And f_q(p_ik,q_ik) Respectively as follows:

f_p(p_ik,q_ik) And f_q(p_ik,q_ik) Are all about p_ikAnd q is_ikOf 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 is_ik，G_ik，H_ik，K_ikIs 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 branch_bReactive power q_bBranch voltage difference u_bAnd node voltage u_nWherein 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 is_pRepresenting a network incidence matrix, R_b,X_b,B_b,G_b,H_b,K_bRespectively representing R corresponding to each branch_ik,X_ik,B_ik,G_ik,H_ik,K_ikThe 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 q_LCalculating the active power p of all branches_bReactive power q_bAll node voltages u_nAnd obtaining the load flow calculation result of the radial distribution network.

Images