CN104714927B - Method for solving node impedance matrix of power system based on symmetric CU triangular decomposition - Google Patents

Abstract

A method for solving a node impedance matrix Z of a power system based on symmetric CU triangular decomposition belongs to the field of analysis and calculation of the power system. The method comprises the following steps: forming a node admittance matrix Y, and carrying out CU triangular decomposition on the Y matrix according to symmetry; for CW_k＝E_kEquation solving for w only_kkAn element; to UZ_k＝W_kFinding Z_kArray diagonal element Z_kkAnd off-diagonal elements above; finding the diagonal Z from symmetry_kkOff-diagonal elements to the left; and writing the Z-array data to a data file. The method utilizes the symmetry of Y, C, U, Z array elements and E in an E array of an identity matrix_kThe structural characteristics of the array are that CU triangular decomposition is carried out according to symmetry, and only the diagonal element C of the U array element and the C array is calculated_iiObtaining the off-diagonal element C of the C array according to the symmetrical relation_ij(ii) a Pair equation CW_k＝E_kIs simplified to find w_kk＝1/c_kk(ii) a To Z_kArray only computing Z_kkAnd the above elements. Compared with the traditional CU triangular decomposition method, the method for checking the IEEE-30, -57, -118 node system has the advantage that the calculation speed can be improved by about 62 percent.

Description

Method for solving node impedance matrix of power system based on symmetric CU triangular decomposition

Technical Field

The invention belongs to the field of analysis and calculation of an electric power system, and relates to a method for solving a node impedance matrix of the electric power system.

Background

The node impedance matrix Z is widely applied and plays an important role in the power system. The traditional method for solving the Z matrix comprises a branch addition method, an admittance matrix Y elimination inversion method, an LDU triangular decomposition method and the like. In the traditional method, the LDU trigonometric decomposition method has the fastest calculation speed relatively, so the method is used most, and the method is characterized in that the trigonometric decomposition method suitable for solving constant coefficient linear equation is utilized, after the Y array is subjected to LDU trigonometric decomposition, the solution of the Z array elements of n multiplied by n orders can be divided into n array matrixes Z_kAnd (5) solving the elements.

In general, when solving equations of the same kind, the calculation speed of the LDU triangulation method is slower than that of the LR or CU triangulation method. Therefore, the method of LDU trigonometric decomposition for obtaining Z array elements is not three to threeOptimal choice of angular decomposition application. On the other hand, the CU trigonometric solution has a calculation speed about 3% faster than that of the LR trigonometric solution when solving the equation of the same kind. Therefore, when the trigonometric decomposition method is used to obtain the Z-array elements, the CU trigonometric decomposition method should be the best choice. However, the traditional CU trigonometric decomposition method does not utilize the symmetry of Y, C, U array elements in the decomposition process, and does not consider utilizing Z array elements in the process of solving the Z array elements_kCalculation order of array elements, symmetry of Z array elements and E in unit matrix E_kThe structural characteristics of the array are such that the actual computation time is not ideal.

The conventional triangulation methods establish their respective factor arrays independently, and thus it is not easy to find and utilize the interrelations between the factor array elements. When the elements of each factor array are calculated, all the elements are calculated from the upper left corner to the lower right corner in a 'L' (inverse L) mode of figure 3 or from the top to the next one in a 'row' mode of figure 4, and the calculation modes of the 2 elements cannot utilize the symmetrical relation of the Y array and the elements of the factor array. The calculation of these 2 elements is completed by one calculation using a calculation formula for all the elements, which is herein referred to as a formula method. Through analysis, the formula method is not beneficial to improving the element calculation process. Therefore, the efficiency of computing the factor array formed by various traditional triangulation methods is low, and the computing time is not ideal.

The calculation sequence of the Z array elements by the traditional LDU triangular decomposition method is as follows: z₁,…,Z_k,…,Z_nThe process is shown in fig. 5.

Disclosure of Invention

The invention aims to overcome the defects of the existing method and provides a method for rapidly solving an impedance matrix based on symmetric CU triangular decomposition.

The invention is realized by the following technical scheme.

The invention comprises the following steps:

a method for solving a node impedance matrix of a power system based on symmetric CU triangular decomposition is characterized by comprising the following steps:

step 1: reading each branch data file;

step 2: forming a node admittance matrix Y;

and step 3: rapidly forming a synthetic array of a C, U factor array by utilizing the symmetrical relation of the Y array and the C, U array elements;

the specific implementation process of the step 3 is as follows:

according to YZ_k＝E_kLet Y equal CU to obtain CUZ_k＝E_k. CUZ will be replaced_k＝E_kFurther decomposition into CW_k＝E_k，UZ_k＝W_kTwo equations.

When C, U two factor array synthesis array is formed, the element calculation mode has great influence on the speed of forming factor array, and the CU trigonometric decomposition method adopted by the invention fully utilizes the relationship among the elements to quickly form C, U factor array synthesis array, which is mainly characterized as follows:

(1) and for the lower left 4-order Y array, the lower right 4-order composite array can be obtained according to the characteristics of the C, U array structure.

The relationship between the C, U matrix elements and the Y matrix elements in the composite matrix is shown below.

(2) The symmetric relation of the elements of the Y array and the C, U array and the relation among the elements in the synthetic array are fully shown when the array is synthesized, which is beneficial to understanding and applying the process method provided by the method of the invention according to ' reverse L ' and ' Gamma

C, U array elements are computed in steps.

(3) The diagonal elements and the values of the right elements are determined according to a 'Gamma' mode, and then the values of the elements below the diagonal elements are determined according to the symmetry.

(4) The upper triangle in the whole rectangular frame contained in 'Gamma' is calculated step by step in a similar elimination way

All elements of (1), and a lower triangle

All elements of (a) are obtained according to symmetry. The step-by-step calculation mode is called a process method, and is completely different from the calculation of a formula method in the traditional trigonometric decomposition method.

(5) The above analysis shows that when C, U factor arrays are formed according to the process of the present invention, each c is_ij、u_ijThe elements being calculated not once, but several times, in steps, in particular u_ijAnd (4) calculating elements.

(6) The procedural calculation procedure is shown in fig. 6. If similar elimination calculation is performed on a certain element at the left of each row of diagonal elements, only the diagonal elements at the same row and the elements at the right of the diagonal elements at the same row are calculated, and all the elements (shaded parts) at the left of the diagonal elements at the same row are not calculated, all the elements are obtained through symmetry. E.g. for c respectively_k1And c_k2When class elimination calculation is carried out, all the calculation is carried out by only calculating the diagonal elements and c above the diagonal elements step by step_kk～u′_knAll elements, and diagonal element c_kkAll elements to the left are not calculated. C is calculated_kk～u′_knAfter the elements, all u 'can be directly determined by symmetry'_kjThe element gets all elements c under the diagonal element_jkU 'to the right of the diagonal element'_kjElement divided by the row diagonal element c_kkAll u are obtained_kjAnd (4) elements.

Then, the diagonal elements and the upper triangles to the right in the whole rectangular frame contained in the 'Gamma' are calculated step by step according to the elimination mode

All elements in (1), and the lower triangle

All elements in (a) are triangulated from top according to symmetry

Is obtained from the elements (1). And circulating in sequence.

And 4, step 4: according to equation CUZ_k＝E_kPair equation CW_k＝E_kFinding W_kArraying; by the equation UZ_k＝W_kFinding Z_kArray diagonal element Z_kkAnd off-diagonal elements above;

the specific implementation process of the step 4 is as follows:

(1) pair equation CW_k＝E_kSolving for W_kProcedure for array

Opposite direction CW in traditional CU trigonometric decomposition method_k＝E_kTo solve for the whole W_kAnd (5) arraying.

The invention is due to the fact that_kThe array element only needs to solve the diagonal element Z_kkAnd the way of calculation of the above elements, hence the equation CW_k＝E_kCan be simplified to find only W_kDiagonal element w in the matrix_kkAnd above elements thereof without solving the whole W_kArray, corresponding to using only E_kThe k-th row of the array has diagonal elements and more. Due to E_kThe array elements are characterized in that: the element in the k-th row is 1 and the remaining elements are all zeros. Thus, the equation CW_k＝E_kW in_kSimplifying the array solving into only solving W_kDiagonal element w in the matrix_kk＝1/c_kkIs calculated, and w_kkThe above elements are still all 0, w_kkThe following elements are not considered.

Thus in the method of the invention the equation CW is_k＝E_kOnly need to solve for E_k Diagonal element 1 of the matrix divided by diagonal element C of the corresponding C-matrix position_kkAnd (4) finishing.

(2) Pair equation UZ_k＝W_kSolving for Z_kProcedure for array

The traditional CU trigonometric decomposition method is to solve each column Z_kAll elements of the matrix。

The method of the invention is to equation UZ_k＝W_kSolving for Z_kThe calculation order of the array elements is: z_n,…,Z_k,…,Z₁And in calculating each Z_kIn the matrix process, only Z is calculated_kArray diagonal element Z_kkAnd off-diagonal elements above it, i.e. Z_kk,Z_k,k-1,…,Z_k1。

And 5: finding the diagonal Z from symmetry_kkOff-diagonal elements to the left;

since in step 4 each Z is calculated_kComputing only diagonal elements Z in time matrix_kkAnd above elements, if necessary, the diagonal element Z can be obtained according to the symmetry of the Z array elements_kkThe left element. This way of calculation can reduce the calculation of 50% off-diagonal elements.

The process of finding the Z-array elements in the method of the present invention is shown in fig. 7, where the elements with prime symbol indicate the off-diagonal elements obtained from symmetry.

Step 6: and writing the Z matrix into a data file for a subsequent program.

In view of the structuring of the program, the forming of the Z-matrix program ends here, and the calling of the formed Z-matrix data file is executed by the next program.

Compared with the traditional CU triangular decomposition method, the method disclosed by the invention has the following advantages:

(1) the use of the synthetic array not only saves memory cells, but also facilitates understanding and application of the computational process of the present invention.

(2) The symmetric relation of Y, C, U array elements is utilized to quickly form C, U factor array, and only the diagonal elements C of the U array element and the C array are calculated_iiObtaining the off-diagonal element C of the C array according to the symmetrical relation_ijThe calculation of about 50% of the elements in the formation of the factor array can be reduced.

(3) The back substitution process utilizes E_kThe structural characteristics of the array. Will be paired with equation CW_k＝E_kIs simplified to directly pair the diagonal elements w_kk＝1/c_kkThe calculation of the equation is greatly simplified.

(4) The back substitution process utilizes the symmetry of Z array elements to calculate only Z_kArray diagonal element Z_kkAnd the non-diagonal elements above the diagonal elements obtain the diagonal elements Z according to the symmetry_kkBy the off-diagonal elements on the left, the Z is reduced by 50 percent_kAnd calculating off-diagonal elements of the array. If necessary, the off-diagonal elements of the triangle under the Z matrix are not required, so as to further accelerate the calculation speed.

Therefore, when the Y array is used for solving the Z array element, the method of the invention is superior to the traditional CU trigonometric decomposition method.

Drawings

FIG. 1 is a flowchart of a conventional calculation for solving Z-matrix elements by CU trigonometric decomposition.

FIG. 2 is a flowchart of the calculation of the method of the present invention to obtain Z-array elements.

FIG. 3 shows a process of forming factor array elements in a "" trans-L "" manner in a conventional trigonometric decomposition method.

FIG. 4 is a process of forming factor array elements in a "row" manner in a conventional triangulation method.

FIG. 5 is a calculation sequence of Z-array elements in the conventional LDU triangulation method.

FIG. 6 is a process of forming factor array elements in a "symmetric elimination" manner by the process of the present invention.

FIG. 7 is a sequence of the Z-array elements calculated in the method of the present invention.

Detailed Description

The invention will be further illustrated by the following examples.

Example 1.

Taking an n × n-order node system as an example, the processes of solving the Z-array elements by the traditional CU trigonometric decomposition method and the method of the invention are respectively compared. The comparison results are shown in table 1.

TABLE 1 comparison of the conventional CU trigonometric decomposition method and the method of the present invention for solving the Z-array element

As can be seen from table 1:

(1) the traditional CU triangular decomposition method solves all elements of the C, U matrix according to a formula method, and the symmetry of the elements cannot be utilized; the method only solves the U array element U according to the process method_ijAnd the diagonal element C of the C matrix_iiObtaining the off-diagonal element C of the C array by using the symmetric relation_jiThe calculated amount of elements in the triangular decomposition process is greatly reduced.

(2) The traditional CU trigonometric decomposition method is used for solving the Z-array elements: one row Z_kAll the array elements are solved. Therefore, 2 equation sets each having n equations are solved, each equation is solved for n variables, and the total number of variable elements is 2n²One, the intermediate matrix variable is 1.

(3) The method for solving the Z array elements only needs to solve Z_kArray diagonal element Z_kkAnd the elements above. Solving equation CW_k＝E_kOnly n simple intermediate variables need to be calculated, and the solution of the equation set can be omitted; solving equation UZ_k＝W_kThe number of variables to be calculated is n (n +1)/2 ≈ n ²2, even if the lower triangular element of the Z matrix is obtained according to the symmetry, the total number of the calculated variables is n²。

Therefore, the comparison between the triangular decomposition process and the back substitution process can show that the element calculation amount and the calculation process of the method are greatly simplified.

Example 2.

Respectively using the traditional CU trigonometric decomposition method (figure 1) and the method of the invention (figure 2) to obtain Z array elements for the Y array of the IEEE-30, -57 and-118 node system, and comparing the average calculation time in the processes of decomposition, retrogradation and decomposition and retrogradation. The calculation results are shown in table 2.

TABLE 2 comparison of the computation times of the conventional method and the method of the present invention in the decomposition, the back substitution and the decomposition and back substitution processes

(1) The "decomposition" process average iteration time:

T₁: the traditional method does not utilize the relation characteristics of C array elements and U array elements to calculate all the elements;

T₂: the method only calculates the U array element U by utilizing the relation characteristic of the C array and the U array element_ijAnd the diagonal element C of the C matrix_iiObtaining the off-diagonal element C of the C array according to the symmetrical relation_ji；

(2) The "back substitution" process average iteration time:

T′₁: the traditional method is replaced by the whole column;

T′₂: the method of the invention is based on the symmetrical back substitution (Z is solved)_kArray diagonal element Z_kkAnd the above elements, and calculating the diagonal element Z according to the symmetry_kkLeft element), consider E-matrix specificity;

(3) average iteration time of the "decomposition + back substitution" process:

T”₁: the traditional method does not use the relation characteristics of C array elements and U array elements in the decomposition process to calculate all the elements; the process of 'back substitution' is back substitution according to the whole column;

T”₂: the method utilizes the relation characteristic of elements of the C array and the U array in the decomposition process; the process of 'back substitution' is based on symmetric back substitution and considers the particularity of the E matrix.

Taking 118 nodes as an example, as can be seen from table 2, the method of the present invention has the following differences compared with the conventional CU triangulation method:

(1) the calculation time of the 'decomposition' process of the method is reduced by about 47 percent;

(2) the calculation time of the 'back substitution' process of the method is reduced by about 71 percent;

(3) the calculation time of the 'decomposition + back substitution' process of the method is reduced by about 62 percent;

(4) the advantages of the method of the invention increase slightly as the number of system nodes increases.

The points are enough to show that compared with the traditional CU triangular decomposition method, the method disclosed by the invention can greatly accelerate the calculation speed in the triangular decomposition process and the backward substitution process when the Z array element is solved, so that the speed of solving the Z array element of the power system is greatly accelerated.

The method can be realized by adopting any programming language and programming environment, wherein the C + + programming language is adopted, and the development environment is Visual C + +.

Claims (1)

1. A method for solving a node impedance matrix of a power system based on symmetric CU triangular decomposition is characterized by comprising the following steps:

step 1: reading each branch data file;

step 2: forming a node admittance matrix Y;

and step 3: rapidly forming a synthetic array of a C, U factor array by utilizing the symmetrical relation of the Y array and the C, U array elements;

(1) establishing a CU synthetic array, and determining the relationship between elements c and u and elements of a Y array;

(2) according to the synthesis array, pair c_kjThe element is subjected to quasi-elimination element calculation, and only the diagonal element c is calculated step by step_kkAnd all u's to the right'_kjAn element;

(3) in calculating u_kj＝u′_kj/c_kkBefore elements, u 'is firstly prepared according to symmetry'_kjDirect assignment of elements to corresponding c_jkElement, recalculate u_kj＝u′_kj/c_kk；

And 4, step 4: according to equation CUZ_k＝E_kPair equation CW_k＝E_kFinding W_kArraying; by the equation UZ_k＝W_kFinding Z_kArray diagonal element Z_kkAnd off-diagonal elements above;

(1) pair equation UZ_k＝W_kSolving for Z_kThe calculation order of the array elements is: z_n,…,Z_k,…,Z₁And in calculating each Z_kIn the matrix process, only Z is calculated_kArray diagonal element Z_kkAnd off-diagonal elements above;

(2) pair equation CW_k＝E_kW in_kThe array solution only needs to obtain W_kDiagonal element w in the matrix_kk＝1/c_kk；

(3) Pair equation UZ_k＝W_kOnly computing Z_kArray pairCorner element Z_kkAnd off-diagonal elements above;

said E_kThe matrix is a matrix with the k-th row having 1 element and all the other elements being zero;

and 5: finding the diagonal Z from symmetry_kkOff-diagonal elements to the left;

the step 3 to the step 5 accelerate the speed of obtaining the Z array element of the power system;

step 6: and writing the Z matrix into a data file.

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