CN104239280A - Method for quickly solving nodal impedance matrix of power system - Google Patents

Abstract

The invention provides a method for quickly solving a nodal impedance matrix of a power system, and relates to the field of analytical computation of the power system. The method mainly comprises the following steps of inputting data of a nodal admittance matrix Y; establishing an augmented matrix B by the nodal admittance matrix Y and an identity matrix E together; normalizing the augmented matrix B and carrying out a Gauss-Jordan elimination method on the augmented matrix B for n times; obtaining an inverse matrix Z. At present, traditional methods for solving the nodal impedance matrix comprise an LDU (Logic Data Unit) triangular decomposition method and a Gauss elimination method, and compared with the two traditional methods, the novel method for quickly solving the nodal impedance matrix by utilizing the Gauss-Jordan elimination method, provided by the invention, has the advantages that the principle is simple and easy to understand, the computation time is reduced, the programming is convenient, and the like; compared with the traditional LDU triangular decomposition method and the Gauss elimination method, by utilizing the method for verifying systems such as an IEEE-57 node, an IEEE-118 node and an IEEE-300 node, the computation speeds can be respectively increased by about 25%-50%.

Description

A Fast Method to Solve Power System Node Impedance Matrix

technical field

The invention relates to the field of power system analysis and calculation, and mainly relates to a calculation method for quickly obtaining a node impedance matrix.

Background technique

In power system analysis and calculation, the node impedance matrix is often used. In traditional methods, the LDU triangular decomposition method and the Gaussian elimination method are commonly used to solve the node impedance matrix.

Various triangular decomposition methods are similar to the factor table method, and are more suitable for the repeated solution of coefficient matrix invariant equations. Many literatures introduce the use of the LDU triangular decomposition method to solve the inverse matrix Z of the Y matrix, mainly to convert the solution of the Z matrix into the solution of the Z _k matrix. Since the calculation formulas of the triangular decomposition method are closely related to the Gaussian elimination calculation formula, if the repeated solution for multiple inverse matrices Z is not considered (only one inverse matrix is sought in most cases), the triangular decomposition method is not as good as The Gaussian elimination method is simple and direct, and the calculation speed is fast, and it is not more ideal than the Gauss-Jordan elimination method with normalization. It is only because the traditional Gaussian elimination method generally does not involve normalized calculations when it is used to solve the inverse matrix (see Figure 1 for the calculation flow chart), and the LDU triangular decomposition method includes normalized calculations (see Figure 2 for the calculation flow chart), so The LDU triangular decomposition method is faster than the Gaussian elimination method without normalization. Therefore, more literature uses the LDU triangular decomposition method instead of the Gaussian elimination method when introducing the inverse matrix. However, the traditional LDU triangular decomposition method and Gaussian elimination method have problems such as complex principles and long calculation time.

(1) Gaussian elimination method or LDU triangular decomposition method to obtain the inverse matrix

YZ=ZY=E (1)

According to the above formula, it can be obtained

YZ _k =E _k (k=1,2,...,n)(2)

The traditional Gaussian elimination method or LDU triangular decomposition method is used to obtain the inverse matrix in formula (2), that is, the Z matrix is obtained by obtaining the Z _k matrix in the 1st to n columns, instead of simultaneously obtaining the complete The entire Z matrix. Since the LDU triangular decomposition method includes normalized calculations, the LDU triangular decomposition method has an obvious advantage in computing speed over the Gaussian elimination method. The calculation flow charts are shown in Figure 1 and Figure 2, respectively.

Contents of the invention

The purpose of the invention is to provide a new method for quickly solving the node impedance matrix of the power system, which can improve the calculation speed in the analysis and calculation of the power system.

The present invention is achieved through the following technical solutions.

First, the node admittance matrix and the unit matrix are formed into a special augmented matrix, and then the augmented matrix is subjected to Gauss-Jordan elimination with normalization to obtain the final solution of the equation. The basic steps are as follows:

Step 1: Input node admittance matrix data;

Step 2: Construct the augmented matrix B=[Y E] by node admittance matrix Y matrix and unit square matrix E matrix;

Step 3: get B ⁽ⁿ⁾ ″=[E E ⁽ⁿ⁾ ″] to B matrix normalization and n times of Gaussian-Jordan elimination;

Step 4: get the inverse matrix Z=E ⁽ⁿ⁾ ";

Step 5: Output the result.

The new matrix E ⁽ⁿ⁾ ″ obtained after performing n times Gauss-Jordan normalization on the augmented matrix B is the node impedance matrix Z.

In step 2 of the present invention, it is different from the traditional LDU triangular decomposition method and the Gaussian elimination method for finding the inverse matrix. The augmented matrix formed in the process of traditional method elimination is B=[Y E _k ], and then the Z _k matrix is calculated column by column. The present invention utilizes the calculation principle of the Gaussian-Jordan elimination method to form a special augmented matrix B=[Y E] to the required inverse matrix and the unit matrix, and can directly obtain the complete inverse matrix Z after the elimination is completed, and No regression process.

In the step 3 and step 4 of the present invention, the special augmented matrix B that is formed is carried out after Gaussian-Jordan elimination element containing normalization and can obtain B ⁽ⁿ⁾ ″=[Y ⁽ⁿ⁾ ″ E ⁽ⁿ⁾ ″]. At this moment, the Y matrix has become the n-order unit square matrix Y ⁽ⁿ⁾ ″, and the n-order unit square matrix E has become the n-order square matrix E ⁽ⁿ⁾ ″ matrix, and the E ⁽ⁿ⁾ ″ matrix is exactly all The required nodal impedance matrix Z.

Since the entire Z matrix of the present invention is obtained simultaneously, it is not necessary to use the LDU triangular decomposition method mainly for multiple solutions of constant coefficient equations and the Gaussian elimination method without normalization with low calculation efficiency. Therefore, the invention can quickly solve the node impedance matrix, and has simple principle and convenient programming. Use the present invention to calculate IEEE-57, -118, -300 nodes and other systems, compared with the traditional LDU triangular decomposition method and Gaussian element elimination method to find the inverse matrix, the calculation speed can be increased by about 25% to 50% ( See Example 1).

The invention proposes a new method for quickly solving the node impedance matrix of the power system, which utilizes the advantages of the Gauss-Jordan elimination method in calculation speed, forms a special augmented matrix for the inverse matrix and the identity matrix, and performs elimination Yuan. Different from the traditional LDU triangular decomposition method and the Gaussian elimination method for finding the inverse matrix, instead of seeking the Z _k matrix column by column, the whole Z matrix is solved at the same time, that is, the present invention directly uses formula (1) instead of Use formula (2) to complete the calculation of the inverse matrix, and there is no back-substitution process. These characteristics determine the speed advantage of this method. The calculation flow chart is shown in Figure 3.

When calculating an inverse matrix Z, the calculation speed of the Gauss-Jordan elimination method is not only better than the Gaussian elimination method, but also better than the LDU triangular decomposition method.

Description of drawings

Figure 1 Gaussian elimination method to obtain the inverse matrix Z calculation flow of node admittance matrix Y

Figure 2 Calculation flow chart of calculating the inverse matrix Z of node admittance matrix Y by LDU triangular decomposition method

Fig. 3 present invention calculates the inverse matrix Z calculation flowchart of node admittance matrix Y

Detailed ways

The invention will be further illustrated by the following examples.

The invention relates to a new method for quickly solving the node impedance matrix of the power system, which can greatly improve the calculation speed compared with the traditional Gaussian elimination method and the LDU triangular decomposition method for solving the node impedance matrix. The invention can also be applied to quickly obtain the inverse matrix of the coefficient matrix of the linear equation system in the analysis and calculation of the power system.

The present invention forms a special augmented matrix with the node admittance matrix Y and the entire unit matrix E, and then carries out normalized Gauss-Jordan elimination on the augmented matrix, and the final solution of the equation can be obtained directly after the elimination.

Form Y array and E array into a special augmented matrix B=[Y E], B array is expanded as follows.

The constant item matrix F in the traditional B array has only 1 column of elements, while the constant item matrix E in the B array here has n columns of elements.

After carrying out Gauss-Jordan elimination with normalization on formula (3), matrix B becomes B ⁽ⁿ⁾ "matrix.

In formula (4), there is Y ⁽ⁿ⁾ ″=E, that is, after n times of Gauss-Jordan elimination with normalization, the Y matrix changes from an n-order square matrix to an n-order unit square matrix E, and the E matrix From the n-order unit matrix to the n-order square matrix E ⁽ⁿ⁾ ″. At this time, the equation corresponding to B ⁽ⁿ⁾ "matrix is Y ⁽ⁿ⁾ "Z=E ⁽ⁿ⁾ ", which has the same solution as formula (1). And because Y ⁽ⁿ⁾ "=E, Z=E can be obtained ⁽ⁿ⁾ ".

Therefore, it can be concluded that the augmented matrix B=[Y E] formed by the Y matrix and the E matrix is carried out after the Gauss-Jordan elimination of normalization is obtained, and the E ⁽ⁿ in the new augmented matrix B ⁽ⁿ⁾ " ⁾ "array is exactly required Z matrix.

Example 1.

For systems such as IEEE-57, 118, and -300 nodes, the programming calculation is performed without considering the element sparsity, and the traditional Gaussian elimination method, LDU triangular decomposition method and the present invention are compared to obtain the inverse matrix of the node admittance matrix Y The calculation time of Z. The calculation results are shown in Table 1.

Table 1 Comparing the calculation time of various calculation methods to obtain the inverse matrix Z of Y matrix

As can be seen from Table 1, taking the IEEE-300 node system as an example, although the calculation speed of the LDU triangular decomposition method containing normalization is about 25% faster than the Gaussian elimination method without normalization, the present invention is respectively faster than the traditional The calculation speed of LDU triangular decomposition method and Gaussian elimination method is about 30-50% faster respectively. The above table is enough to prove the advantages of the present invention.

The present invention can adopt any programming language and programming environment to realize, is to adopt C++ programming language in the present embodiment, and development environment is Visual C++.

Claims (1)

1. A method for quickly solving the power system node impedance matrix is characterized in that the steps are as follows: Step 1: Input node admittance matrix data; Step 2: Construct the augmented matrix B=[Y E] by node admittance matrix Y matrix and unit square matrix E matrix; Step 3: get B ⁽ⁿ⁾ ″=[E E ⁽ⁿ⁾ ″] to B matrix normalization and n times of Gaussian-Jordan elimination; Step 4: get the inverse matrix Z=E ⁽ⁿ⁾ "; Step 5: Output the result.