A rectangular coordinate Newton method for current flow calculation suitable for research purposes

Abstract

The present invention discloses a rectangular coordinate Newton method for power flow calculation suitable for research purposes, and provides a Newton method for power flow calculation that is easy to modify and maintain for scientific researchers who conduct further research based on the rectangular coordinate Newton method for power flow calculation. For a power system with n nodes, the node number is independent of the node type and can be numbered arbitrarily. The Jacobian matrix in the power flow calculation is stored as (2n)×(2n), and the elements of the rows and columns related to the balance node are all 0, but the diagonal elements are not cleared to zero, and the original calculated values are retained. In this way, the linear equation solving function with excellent performance provided by the programming language can be used to solve the correction equation of the power flow calculation, which reduces the programming difficulty, improves the calculation speed, and ensures the reliability of the algorithm. The implementation example shows that the present invention can use the linear equation solving function provided by the programming language to solve the correction equation of the power flow calculation, improves the speed of the power flow calculation, and reduces the programming difficulty.

Description

Technical Field

[0001] The present invention relates to a Newton method for calculating power system power flow, in particular to a rectangular coordinate Newton method for calculating power flow suitable for research purposes. Background Art

[0002] Power system flow calculation is a fundamental calculation used to study the steady-state operation of power systems. It determines the operating status of the entire power system based on given operating conditions and network structure. It also forms the basis for other power system analyses, such as security analysis and transient stability analysis. Due to its reliable convergence, fast computational speed, and moderate memory requirements, the Newton method has become the mainstream approach for power flow calculations. The Newton method is available in two forms: polar coordinates and rectangular coordinates. Both forms of Newton's method for power flow calculations are widely used in power systems.

[0003] In the rectangular coordinate Newton method power flow calculation, the voltage at node i is expressed in rectangular coordinates as:

[0004]

[0005] like Figure 1 As shown in Figure 2, the conventional rectangular coordinate Newton method power flow calculation method mainly includes the following steps:

[0006] A. Input original data and initialization voltage.

[0007] According to the characteristics of power system nodes, power flow calculation divides power system nodes into three categories: nodes where the active power and reactive power injected by the node are known, but the node voltage amplitude and voltage phase angle are unknown are called PQ nodes; nodes where the active power and voltage amplitude injected by the node are known, but the node reactive power and voltage phase angle are unknown are called PV nodes; nodes where the voltage amplitude and voltage phase angle are known, but the node active power and reactive power injected by the node are unknown are called balance nodes.

[0008] The voltage is initialized using a flat start. This means that the real part of the voltage at the PV node and the balancing node is set to the given voltage amplitude, the real part of the voltage at the PQ node is set to 1.0, and the imaginary part of the voltage at all nodes is set to 0.0. Units are per unit.

[0009] B. Form a node admittance matrix.

[0010] C. Set the iteration count t=0.

[0011] D. Calculate the node power, node power imbalance, and node voltage imbalance, and find the maximum imbalance ΔW max .

[0012] The node power calculation formula is:

[0013]

[0014] Where, P i , Q i are the active power and reactive power of node i respectively; e i 、f i are the real and imaginary parts of the voltage at node i, respectively; e k and f k are the real and imaginary parts of the voltage at node k respectively; G ik 、B ik are the node admittance matrix elements Y ik The real and imaginary parts of ; n is the number of nodes.

[0015] Assume that nodes 1 to m are PQ nodes, nodes m+1 to n-1 are PV nodes, and node n is a balanced node. The calculation formulas for node power imbalance and node voltage imbalance are:

[0016]

[0017] Where, ΔP i , ΔQ i are the active power imbalance and reactive power imbalance of node i respectively; P is , Q is are the injected active power and injected reactive power given to node i respectively; ΔU i 2 is the voltage unbalance at node i; U is is the voltage amplitude given at node i; U i is the voltage amplitude of node i; m is the number of PQ nodes.

[0018] Balanced nodes do not participate in the iterative calculation, and there is no need to calculate the node power imbalance and node voltage imbalance.

[0019] The value with the largest absolute value among the power imbalance and voltage imbalance of each node is called the maximum imbalance.

[0020] E. Determine the absolute value of the maximum imbalance |ΔW max |Is it less than the convergence precision ε? If so, go to step I; otherwise, go to step F.

[0021] F. Form the Jacobian matrix J.

[0022] The basic equations for power flow calculation are nonlinear equations, which are usually solved iteratively using the successive linearization method.

[0023] The correction equation (3) obtained by linearization is used to calculate the correction amount of the real and imaginary parts of the voltage.

[0024]

[0025] Where J is the Jacobian matrix of order (2n-2)×(2n-2), H, N, M, L, R, and K are the six block sub-matrices of the Jacobian matrix, with dimensions of (n-1)×(n-1), (n-1)×(n-1), m×(n-1), m×(n-1), (nm-1)×(n-1), and (nm-1)×(n-1), respectively; ΔP = [ΔP1,…,ΔP n-1 ] T is the column vector of node active power imbalance, the superscript T indicates transposition; ΔQ=[ΔQ1,…,ΔQ m ] T is the column vector of node reactive power unbalance; is the column vector of node voltage unbalance; Δe=[Δe1,…,Δe n-1 ] T is the column vector of the real part correction of the node voltage; Δf=[Δf1,…,Δf n-1 ] T is the column vector of the imaginary part correction of the node voltage.

[0026] The elements of the Jacobian matrix J (when i≠j) are calculated as follows:

[0027]

[0028]

[0029]

[0030]

[0031]

[0032]

[0033] Where H ij is the partial derivative of the active power imbalance at node i with respect to the real part of the voltage at node j; N ij is the partial derivative of the active power imbalance at node i with respect to the imaginary part of the voltage at node j; M ij is the partial derivative of the reactive power unbalance at node i with respect to the real part of the voltage at node j; L ij is the partial derivative of the reactive power unbalance at node i with respect to the imaginary part of the voltage at node j; R ij K is the partial derivative of the voltage unbalance at node i with respect to the real part of the voltage at node j; ij is the partial derivative of the voltage unbalance at node i with respect to the imaginary part of the voltage at node j.

[0034] The elements of the Jacobian matrix J (when i=j) are calculated as follows:

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041] Where H ii is the partial derivative of the active power imbalance at node i with respect to the real part of the voltage at node i; N ii is the partial derivative of the active power imbalance at node i with respect to the imaginary part of the voltage at node i; M ii is the partial derivative of the reactive power unbalance at node i with respect to the real part of the voltage at node i; L ii is the partial derivative of the reactive power unbalance at node i with respect to the imaginary part of the voltage at node i; R ii is the partial derivative of the voltage unbalance at node i with respect to the real part of the voltage at node i; K ii is the partial derivative of the voltage unbalance at node i with respect to the imaginary part of the voltage at node i.

[0042] G. Solve the correction equation and correct the real part e and imaginary part f of the node voltage.

[0043] Solve the corrected equation (3) to obtain the real and imaginary corrections of the node voltage.

[0044] The node voltage correction formula is:

[0045]

[0046] Where, the superscript (t) indicates the tth iteration; Δe i and Δf i are the real part correction value and imaginary part correction value of the voltage at node i respectively.

[0047] H. Set t = t + 1 and return to step D for the next iteration.

[0048] I. Output node and branch data.

[0049] In the linear equation shown in Equation (3), the PQ node has related ΔP and ΔQ equations, and the PV node has related ΔP and ΔU 2 equation, and the ΔP, ΔQ, and ΔU associated with the equilibrium node2 The equations do not exist. Thus, PQ nodes and PV nodes correspond to two equations, while equilibrium nodes have none. Furthermore, equilibrium nodes do not require solutions for Δe and Δf, while all other nodes do. This requires that the correspondence between nodes and equations be determined based on the node type. Changes in type require changes in the correspondence, significantly complicating programming and debugging. While it is possible to specify the first m nodes as PQ nodes, nodes m+1 to n-1 as PV nodes, and node n as the equilibrium node, this requirement may necessitate renumbering nodes when inputting data. Furthermore, changes in node type require readjustment of the correspondence between nodes and equations.

[0050] To this end, Chinese patent CN201010509556.5 proposes a Newton-based power flow calculation method suitable for research purposes, providing researchers conducting further research based on polar coordinate Newton-based power flow calculations with an easily modifiable and maintainable Newton-based power flow calculation algorithm. The Jacobian matrix for the power flow calculation in this patent is stored as (2n) × (2n). The elements in the rows corresponding to the PV node ΔQ and the columns corresponding to ΔU are all 0, and the elements in the rows and columns related to the equilibrium nodes are also all 0. Although this increases memory requirements, it simplifies the correspondence between the nodes and the rows and columns of the equation coefficient matrix, greatly reducing programming difficulty and not increasing the amount of calculation. When solving the modified equation for the power flow calculation, rows in the Jacobian matrix with a main diagonal element of 0 are skipped through judgment (a main diagonal element of 0 indicates that the elements in that row are all 0 and there is no corresponding equation) and are not processed.

[0051] Although the method in Chinese patent CN201010509556.5 simplifies the correspondence between nodes and rows and columns of the equation coefficient matrix, greatly reducing the programming difficulty, it requires special processing when solving the power flow calculation correction equation, and cannot use the high-performance linear equation solving function provided by the programming language. Summary of the Invention

[0052] In order to solve the above-mentioned problems existing in the prior art, the present invention proposes a rectangular coordinate Newton method for tidal current calculation suitable for research purposes, and provides a rectangular coordinate Newton method for tidal current calculation algorithm that is easy to modify and maintain for scientific researchers who conduct further research based on the rectangular coordinate Newton method for tidal current calculation.

[0053] To achieve the above objectives, the present invention improves the Jacobian matrix as follows: the Jacobian matrix is still stored as (2n) × (2n), with the rows and columns associated with the balancing nodes all zero, but the diagonal elements are not cleared to zero, retaining the original calculated values. This allows the use of the high-performance linear equation solver provided by the programming language to solve the corrected equations for power flow calculations.

[0054] The technical solution of the present invention is as follows: A rectangular coordinate Newton method power flow calculation method suitable for research purposes includes the following steps:

[0055] A. Input original data and initialization voltage.

[0056] B. Form a node admittance matrix.

[0057] C. Set the iteration count t=0.

[0058] D. Calculate the node power, calculate the unbalance value ΔW of each node, and find the maximum unbalance value ΔW max .

[0059] Calculate the node power according to formula (1);

[0060] The unbalanced quantity ΔW is a 2n-dimensional column vector, which is composed of the active power unbalanced quantity, the reactive power unbalanced quantity and the voltage unbalanced quantity.

[0061] If node i is a PQ node, the imbalance is calculated according to formula (17):

[0062]

[0063] Where, ΔP i , ΔQ i are the active power imbalance and reactive power imbalance of node i respectively; P is , Q is are the injected active power and injected reactive power given to node i respectively.

[0064] If node i is a PV node, the unbalanced value is calculated according to formula (18):

[0065]

[0066] Where ΔU i 2 is the voltage unbalance at node i; U is is the voltage amplitude given at node i.

[0067] If node i is a balanced node, the unbalanced value is calculated according to formula (19):

[0068]

[0069] Find the maximum absolute value of the active power imbalance, reactive power imbalance and voltage imbalance of each node as the maximum imbalance ΔW max .

[0070] E. Determine the absolute value of the maximum imbalance |ΔW max|Is it less than the convergence precision ε? If so, go to step I; otherwise, go to step F.

[0071] F. Form the Jacobian matrix J.

[0072] The steps to form the Jacobian matrix J are as follows:

[0073] F1. Regardless of the node type, calculate the Jacobian matrix elements according to equations (20) to (27) to form a (2n) × (2n) Jacobian matrix J;

[0074] The elements of the Jacobian matrix J (when i≠j) are calculated as follows:

[0075]

[0076]

[0077]

[0078]

[0079] Where, J i,j is the element in the i-th row and j-th column of the Jacobian matrix; is the partial derivative of the active power imbalance at node i with respect to the real part of the voltage at node j; is the partial derivative of the active power imbalance at node i with respect to the imaginary part of the voltage at node j; is the partial derivative of the reactive power unbalance at node i with respect to the real part of the voltage at node j; is the partial derivative of the reactive power unbalance at node i with respect to the imaginary part of the voltage at node j.

[0080] The elements of the Jacobian matrix J (when i=j) are calculated as follows:

[0081]

[0082]

[0083]

[0084]

[0085] Where, is the partial derivative of the active power imbalance at node i with respect to the real part of the voltage at node i; is the partial derivative of the active power unbalance at node i with respect to the imaginary part of the voltage at node i; is the partial derivative of the reactive power unbalance at node i with respect to the real part of the voltage at node i; is the partial derivative of the reactive power unbalance at node i with respect to the imaginary part of the voltage at node i.

[0086] F2. Set node count i = 1;

[0087] F3. Determine whether node i is a PV node. If not, go to step F8.

[0088] F4. Set count j = 1;

[0089] F5. Calculate the Jacobian matrix element J according to equations (28)-(31) n+i,j , J n+i,n+j ;

[0090] The elements of the Jacobian matrix J (when i≠j) are calculated as follows:

[0091]

[0092]

[0093] Where, is the partial derivative of the voltage unbalance at node i with respect to the real part of the voltage at node j; is the partial derivative of the voltage unbalance at node i with respect to the imaginary part of the voltage at node j.

[0094] The elements of the Jacobian matrix J (when i=j) are calculated as follows:

[0095]

[0096]

[0097] Where, is the partial derivative of the voltage unbalance at node i with respect to the real part of the voltage at node i; is the partial derivative of the voltage unbalance at node i with respect to the imaginary part of the voltage at node i.

[0098] F6, let j = j + 1;

[0099] F7. Determine whether j is greater than n. If j is greater than n, go to step F8; otherwise, return to step F5;

[0100] F8. Let i=i+1;

[0101] F9, determine whether i is greater than n, if i is greater than n, go to step F10; otherwise return to step F3;

[0102] F10, set node count i=1;

[0103] F11. Determine whether node i is a balanced node. If not, go to step F21.

[0104] F12, set count k = 1;

[0105] F13, determine whether k=i. If yes, go to step F16;

[0106] F14, let J i,k =0;

[0107] F15, let J k,i =0;

[0108] F16, determine whether k=n+i is true, if so, go to step F19;

[0109] F17, let J n+i,k =0;

[0110] F18, let J k,n+i =0;

[0111] F19, let k = k + 1;

[0112] F20, determine whether k is greater than 2n, if k is greater than 2n, go to step F21; otherwise return to step F13;

[0113] F21, let i = i + 1;

[0114] F22. Determine whether i is greater than n. If i is greater than n, go to step G; otherwise, return to step F11. G. Solve the modified equation (32) and modify the real part e and imaginary part f of the node voltage.

[0115]

[0116] Where J is the Jacobian matrix of order (2n)×(2n); Δe=[Δe1,…,Δe n ] T is the column vector of the real part correction of the node voltage, and the superscript T indicates the transposition; Δf=[Δf1,…,Δf n ] T is the column vector of the node voltage imaginary part correction; ΔW is the 2n-dimensional unbalance column vector.

[0117] The node voltage correction formula is:

[0118]

[0119] Where, the superscript (t) indicates the tth iteration; Δe i and Δf i are the real part correction value and imaginary part correction value of the voltage at node i respectively.

[0120] H. Set t = t + 1 and return to step D for the next iteration.

[0121] I. Output node and branch data.

[0122] Compared with the prior art, the present invention has the following beneficial effects:

[0123] 1. The Jacobian matrix of the power flow calculation of the present invention is stored as (2n)×(2n), which can simplify the correspondence between nodes and rows and columns of the equation coefficient matrix, greatly reducing the programming difficulty and not increasing the amount of calculation.

[0124] 2. For equations without equilibrium nodes, all elements in the corresponding rows of the Jacobian matrix, except for the diagonal elements, are zero, indicating the nonexistence of the corresponding equation. Since the diagonal elements are not zero, the linear equation solver provided by the programming language can be used to solve the corrected equations for the power flow calculation, significantly reducing programming complexity. These linear equation solvers have been optimized for high computation speed and high stability, significantly improving both the speed and stability of power flow calculations.

[0125] 3. The node numbering of the power flow calculation of the present invention is flexible and has nothing to do with the node type. The PQ node number is not required to be in the front, and the balance node is the last node. There is no need to change the original numbering of the system nodes according to the program design requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0126] The present invention has the following Figure 2 Zhang. Among them:

[0127] Figure 1 This is the flow chart of the conventional rectangular coordinate Newton method power flow calculation.

[0128] Figure 2 It is a flowchart of the Jacobian matrix formation of the present invention. DETAILED DESCRIPTION

[0129] The present invention will be further described below with reference to the accompanying drawings. Figure 1 The flowchart of the rectangular coordinate Newton method power flow calculation is shown in Figure 2 The flowchart for the Jacobian matrix formation process of the present invention is shown, and a power flow calculation is performed on a real large-scale power grid. This large-scale power grid has 445 nodes and contains a large number of low-impedance branches. To enable conventional power flow calculation methods, these low-impedance branches are converted to normal branches. The convergence accuracy of the power flow calculation is 0.00001.

[0130] For comparison, the following three methods were used to calculate the power flow of the actual large-scale power grid:

[0131] Conventional method: conventional rectangular coordinate Newton method for current flow calculation;

[0132] Patented method: Patent CN201010509556.5;

[0133] Comparison method: The method of patent CN201010509556.5 is adopted, but the correction equation for the power flow calculation is solved using the linear equation solving function of Matlab.

[0134] Various power flow calculation methods are programmed using Matlab M files. The calculation time of different power flow calculation methods is shown in Table 1.

[0135] Table 1 Calculation time (s) of different power flow calculation methods

[0136] method Conventional methods Patented method Comparison Method The present invention Calculation time 23.103303 21.780145 The coefficient matrix is singular and cannot be solved 0.316549

[0137] As shown in Table 1, for the modified 445-node real-world power system example, the computation time for the conventional Cartesian Newton method for power flow calculation and the patented method is similar. However, the proposed method is much faster because it uses the linear equation solver provided by the Matlab programming language to solve the equations. However, some diagonal elements in the Jacobian matrix of the comparison method are zero, resulting in a singular coefficient matrix in the modified equations, making them unsolvable using the linear equation solver provided by the programming language.

[0138] The present invention can be implemented using any programming language and programming environment, such as C language, C++, FORTRAN, Delphi, MATLAB M-file programming language, etc. The development environment can be Visual C++, Borland C++Builder, Visual FORTRAN, MATLAB, etc.

[0139] The present invention is not limited to this embodiment, and any equivalent concepts or modifications within the technical scope disclosed by the present invention are included in the protection scope of the present invention.

Claims

1. A rectangular coordinate Newton method for calculating current flow suitable for research purposes, characterized by: The following steps are involved: A. Input original data and initialization voltage; According to the characteristics of power system nodes, power flow calculation divides power system nodes into three categories: nodes with known active power and reactive power injection but unknown voltage amplitude and voltage phase angle are called PQ nodes; nodes with known active power and voltage amplitude but unknown reactive power injection and voltage phase angle are called PV nodes; nodes with known voltage amplitude and voltage phase angle but unknown active power and reactive power injection are called balancing nodes; The voltage initialization adopts flat start, that is, the real part of the voltage of the PV node and the balance node takes the given value of the voltage amplitude, the real part of the voltage of the PQ node takes 1.0; the imaginary part of the voltage of all nodes takes 0.0; the unit here adopts the per unit value; B. Forming a node admittance matrix; C. Set the iteration count t = 0; D. Calculate the node power and the imbalance of each node Δ W , find the maximum unbalance Δ W max ; The node power calculation formula is: (1) Where, P i 、 Q i Node i Active power and reactive power; e i 、 f i Node i The real and imaginary parts of the voltage; e k and f k Node k The real and imaginary parts of the voltage; G ik 、 B ik are the node admittance matrix elements Y ik The real and imaginary parts of n is the number of nodes; Unbalance Δ W is 2 n Dimensional column vector, consisting of active power imbalance, reactive power imbalance, and voltage imbalance; If the node i is a PQ node, and the unbalance amount is calculated according to formula (2): (2) Where, Δ P i , Δ Q i Node i Active power imbalance and reactive power imbalance; P is 、 Q is Node i Given injected active power and injected reactive power; If the node i is the PV node, and the unbalanced quantity is calculated according to formula (3): (3) Where, Δ U i 2 For nodes i The voltage unbalance amount; U is For nodes i Given voltage amplitude; If the node i is a balanced node, and the unbalanced amount is calculated according to formula (4): (4) Find the maximum absolute value of the active power imbalance, reactive power imbalance, and voltage imbalance at each node, which is called the maximum imbalance. E. Determine the absolute value of the maximum imbalance |Δ W max Is it less than the convergence accuracy? ε ; If it is less than the convergence precision ε , execute step I; Otherwise, go to step F; F. Forming the Jacobian Matrix J ; F1, regardless of the node type, calculate the Jacobian matrix elements according to formula (5)-formula (12), forming (2 n )×(2 n )'s Jacobian matrix J ; when i ≠ j When the Jacobian matrix J The element calculation formula is as follows: (5) (6) (7) (8) Where, J i,j The Jacobian matrix i Rank j Elements of the column; For nodes i The active power imbalance of the node j The partial derivative of the real part of the voltage; For nodes i The active power imbalance of the node j The partial derivative of the imaginary part of the voltage; For nodes i The reactive power unbalance to the node j The partial derivative of the real part of the voltage; For nodes i The reactive power unbalance to the node j The partial derivative of the imaginary part of the voltage; when i = j When the Jacobian matrix J The element calculation formula is as follows: (9) (10) (11) (12) Where, For nodes i The active power imbalance of the node i The partial derivative of the real part of the voltage; For nodes i The active power imbalance of the node i The partial derivative of the imaginary part of the voltage; For nodes i The reactive power unbalance to the node i The partial derivative of the real part of the voltage; For nodes i The reactive power unbalance to the node i The partial derivative of the imaginary part of the voltage; F2, set node count i = 1; F3, judgment node i Is it a PV node? If not, go to step F8; F4, set count j = 1; F5. Recalculate the Jacobian matrix elements of the PV node according to equations (13) to (16) J n+i , j , J n+i , n+j ; when i ≠ j When the Jacobian matrix J The element calculation formula is as follows: (13) (14) Where, For nodes i The voltage unbalance amount to the node j The partial derivative of the real part of the voltage; For nodes i The voltage unbalance amount to the node j The partial derivative of the imaginary part of the voltage; when i = j When the Jacobian matrix J The element calculation formula is as follows: (15) (16) Where, For nodes i The voltage unbalance amount to the node i The partial derivative of the real part of the voltage; For nodes i The voltage unbalance amount to the node i The partial derivative of the imaginary part of the voltage; F6, Command j = j +1; F7. Judgment j Is it greater than n ,if j Greater than n , go to step F8; Otherwise return to step F5; F8, Command i = i +1; F9, Judgement i Is it greater than n ,if i Greater than n , go to step F10; otherwise return to step F3; F10, set node count i = 1; F11, judgment node i Is it a balanced node? If not, go to step F21; F12, set count k = 1; F13, Judgment k = i Is it established? If so, go to step F16; F14, command J i,k = 0; F15, Command J k,i = 0; F16, Judgment k = n + i Is it established? If so, go to step F19; F17, Order J n+i,k = 0; F18, Command J k,n+i = 0; F19, Order k = k +1; F20, Judgment k Is it greater than 2? n ,if k Greater than 2 n , go to step F21; otherwise return to step F13; F21, Order i = i +1; F22, Judgment i Is it greater than n ,if i Greater than n , go to step G; otherwise return to step F11; G. Solve the correction equation and correct the real part of the node voltage e and the imaginary part f ; The corrected equation is: (17) Where, J For (2 n )×(2 n )-order Jacobian matrix; is the column vector of the real part correction of the node voltage, and the superscript T indicates transposition; is the column vector of the node voltage imaginary part correction; Δ W is 2 n dimensional imbalance column vector; The node voltage correction formula is: (18) In the formula, the superscript ( t ) indicates the t Iterations; Δ e i and Δ f i Node i The real part correction value of voltage and the imaginary part correction value of voltage; H. Order t = t + 1, return to step D for the next iteration; I. Output node and branch data.

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