Affine power flow calculation method for power system uncertainty based on holomorphic embedding

Abstract

The present invention discloses a method for calculating an affine power flow of an electric power system based on full-morphic embedding. First, an embedded affine power flow model describing the uncertainty of load and new energy output is established. Then, a definite initial solution of the affine power flow is obtained, and the problem that the convergence of the affine power flow algorithm is affected by the initial value is solved. Thereafter, a mathematical relationship between the affine reactive power and the affine voltage of a PV node is established, and the affine reactive power of the PV node can be directly solved by using the affine node voltage obtained by each calculation. Finally, the problem of solving the affine power flow is converted into the problem of solving the Taylor power series coefficients, and the recursive relationship of the power series coefficients of the affine state quantity is derived. Only the linear algebra operation of the affine equation group is involved in the calculation process.

Description

Technical Field

[0001] The invention relates to the technical field of power systems, and in particular to a method for calculating power system uncertainty affine power flow based on holomorphic embedding. Background Art

[0002] With the rapid development of new power systems, large-scale grid connection of distributed power sources such as wind power and photovoltaics has increased the uncertainty of power system operation. Traditional deterministic power flow can only obtain the power grid operation status under specific working conditions, and it is difficult to quantify and analyze the impact of uncertain factors, and cannot adapt to the development of new power systems. Uncertain power flow can analyze the distribution characteristics of system state quantities under the influence of uncertain factors, and is an effective power flow analysis tool in new power systems.

[0003] Among the uncertainty power flow calculation methods, the interval power flow method has attracted widespread attention because it requires the least information and can solve the interval of the system state quantity based on the upper and lower bounds of the uncertainty. However, when the variables are interdependent, the traditional interval algorithm may experience interval explosion and it is difficult to reflect the true interval range of the state quantity. Therefore, the uncertainty power flow algorithm based on affine arithmetic has gradually become the main analysis method for interval power flow. At present, the convergence of the affine iterative power flow algorithm is easily affected by the initial value of the iteration, and the algorithm is difficult to apply to systems containing PV nodes, and the scope of application is narrow. In addition, complex nonlinear operations such as affine matrix inversion increase the algorithm complexity, conservatism and calculation time.

[0004] At present, when the variables of traditional interval algorithms are interdependent, interval explosion may occur, making it difficult to reflect the true interval range of the state quantity, and it is impossible to quantify and analyze the relative influence of various uncertain factors on the uncertain state quantity of the power system. The current affine power flow algorithm is difficult to achieve an organic unity of application scope and computational efficiency. The forward-backward affine power flow algorithm requires renumbering of nodes in different systems, and can only be applied to radial networks and is difficult to be directly applied to ring networks. The Ybus Gaussian affine power flow iteration algorithm is difficult to directly solve the system containing PV nodes due to the difficulty in handling PV nodes. The Newton affine power flow iteration method requires the inversion of the affine Jacobian matrix, which greatly increases the difficulty of calculation. The convergence of the above-mentioned iterative uncertainty power flow algorithm depends on the selection of initial values. Improper initial value setting may cause the algorithm to fail to converge. Summary of the invention

[0005] In view of the problems existing in the affine power flow calculation method, the present invention establishes an embedded affine power flow model that describes the uncertainty of load and renewable energy output, and proposes a holomorphic embedded affine recursive power flow analysis method, which can directly obtain the initial solution of the power flow calculation, avoids the complex nonlinear calculation of affine, and can quantitatively analyze the relative influence of various uncertain factors on the uncertain state quantity of the power system.

[0006] This method first establishes an embedded affine power flow model that describes the uncertainty of load and renewable energy output; then, the definite initial solution of the affine power flow is obtained, which solves the problem that the convergence of the affine power flow algorithm is affected by the initial value; thereafter, the mathematical relationship between the affine reactive power and the affine voltage of the PV node is established, and the affine node voltage obtained by each calculation is used to directly solve the affine reactive power of the PV node; finally, the affine power flow solution problem is converted into the solution problem of Taylor power series coefficients, and the recursive relationship of the power series coefficients of the affine state quantity is derived. The calculation process only involves linear algebra operations of the affine equation group.

[0007] Its main design points include:

[0008] (1) Based on the holomorphic embedding method, a holomorphic embedding affine power flow model is established to describe the uncertainty of renewable energy output and load.

[0009] (2) According to the actual physical meaning corresponding to the initial state of the affine current, the initial solution of the affine current is obtained.

[0010] (3) The solution of the uncertain affine power flow is converted into the solution of the affine Taylor power series coefficients to obtain the recursive relationship of the affine state quantity power series coefficients.

[0011] (4) Establish the mathematical relationship between the affine reactive power and affine voltage of the PV node and directly obtain the affine reactive power of the PV node.

[0012] To achieve the above object, the present invention adopts the following technical solutions:

[0013] A method for calculating uncertain affine power flow of power system based on holomorphic embedding is characterized by: firstly, an embedded affine power flow model describing the uncertainty of load and new energy output is established; then, a definite initial solution of the affine power flow is obtained to solve the problem that the convergence of the affine power flow algorithm is affected by the initial value; thereafter, a mathematical relationship between the affine reactive power and the affine voltage of the PV node is established, and the affine node voltage obtained by each calculation is used to directly solve the affine reactive power of the PV node; and the affine power flow solution problem is converted into the solution problem of Taylor power series coefficients, and the recursive relationship of the power series coefficients of the affine state quantity is derived, so that the calculation process only involves linear algebra operations of the affine equation group.

[0014] Furthermore, the process of establishing the embedded affine power flow model describing the uncertainty of load and new energy output specifically includes:

[0015] Step S11: Establish a PQ node holomorphic embedding affine model:

[0016] The full embedded power balance equation of the PQ node is shown in formula (1):

[0017]

[0018] Where: ^ represents the affine number, * represents conjugation, and s is the holomorphic embedding factor; represents the holomorphic embedded affine voltage of the kth node; represents the holomorphic embedded affine voltage of the i-th node; represents the conjugate of the affine apparent power of the i-th node; Y ik,tr represents the line and transformer impedance between lines i and k, where the subscript tr represents the transformer; Y i,sh The line and transformer admittance of the node and the ground capacitance of the node, where the subscript sh indicates parallel connection; N is the total number of nodes;

[0019] In the formula, The Taylor expansion of is an affine infinite series with respect to s, and is analytic for s, as shown in formula (2):

[0020]

[0021] Where: n is the nth order Taylor expansion; is the affine number of the n-th order Taylor expansion of the voltage at node i; V i，0 [n] is the central value of the n-th order Taylor expansion affine coefficient; V i,j [n] is the noise element coefficient corresponding to the jth noise element in the nth-order Taylor expansion affine coefficient; ε j is the jth noise element; M i is the total number of noise elements;

[0022] Step S12: Establish a PV node holomorphic embedding affine model:

[0023] The fully embedded power balance equation of the PV node is shown in equation (3):

[0024]

[0025] Where: is the affine active power of node i; is the holomorphic embedded affine reactive power of node i;

[0026] Since the voltage of the PV node is given, the voltage equation shown in formula (4) must be satisfied:

[0027]

[0028] Where: V i sp represents the set value of node i, and sp represents the set value;

[0029] Step S13: Establish a balanced node holomorphic embedding affine model:

[0030] The balanced node satisfies the node voltage equation as shown in the formula:

[0031]

[0032] Furthermore, the solution process of the embedded affine power flow model describing the uncertainty of load and new energy output specifically includes:

[0033] Step S21: Calculate the initial solution of affine power flow recursion:

[0034] For the holomorphic embedded affine power flow equation, when the initial state s = 0, the solutions of equations (1) to (5) are:

[0035]

[0036] Therefore, the affine center value of the constant term of the voltage and reactive power series in the initial state is V i，0 [0]=1,Q i，0 [0] = 0, the noise element coefficients are all 0; the above initial state corresponds to the grid being unloaded, all generators not working, and the PV node reactive output being zero;

[0037] Step S22: Construct the recursive relationship of the affine power flow power series coefficients

[0038] Define the inverse of the affine voltage function because According to the principle of numerical equality of coefficients of each power series, we can deduce The calculation of is shown in formula (7);

[0039]

[0040] For the PQ node, let the coefficients of the different power terms of s on both sides of the equal sign in equation (1) be equal, and obtain the recursive relationship of equation (8):

[0041]

[0042] For the PV node, let the coefficients of different power terms of s on both sides of the equal sign in equation (3) and equation (4) be equal, and obtain the recursive relationship of equation (9) and equation (10):

[0043]

[0044]

[0045] Where: n1 For the pulse function, it takes 1 when n=1 and 0 in other cases;

[0046] In formula (9), and As the unknown quantity is moved to the left, we get:

[0047]

[0048] At this time, the PV node reactive power is directly solved using the desired affine node voltage;

[0049] For the equilibrium node, let the coefficients of different power terms of s on both sides of the equal sign of formula (5) be equal, and obtain the recursive relationship shown in formula (12):

[0050]

[0051] Step S23: Construct a matrix representation of the recursive relationship of the power series coefficients

[0052] Decompose the recursive relationship into two parts, the real part and the imaginary part, and write it in the form of a matrix equation; for the node voltage and admittance, both are decomposed into the real part and the imaginary part, that is:

[0053]

[0054] Y ik,tr =G ik,tr +jB ik,tr

[0055] Where: for The real part of , the subscript re represents the real part; for The imaginary part of G, the subscript im represents the imaginary part; ik,tr is conductivity; B ik,tr is the electrical admittance;

[0056] Convert the PQ node recursive relation (8) into (14):

[0057]

[0058]

[0059] Convert the PV node recursive relations (9) and (10) into (15) and (16):

[0060]

[0061]

[0062] Convert the equilibrium node recursive relation (12) into (17):

[0063]

[0064]

[0065] The recursive relationship from equation (14) to equation (17) is further organized into the form of algebraic linear equations:

[0066]

[0067] Where: A is the recursive matrix; is the coefficient vector of the n-th order affine power series to be determined; is a known vector not greater than the coefficient of the n-1th order affine power series;

[0068] Step S24: Calculate the holomorphic embedded uncertainty power flow affine solution

[0069] Since s=1 is the analytical solution of the target state, after the recursion converges, the affine solution of the calculated state quantity is:

[0070]

[0071] In the formula, k is the number of recursions to converge;

[0072] The recursive convergence condition is shown in formula (20):

[0073]

[0074] Where: represents the k-order affine power series coefficient of the i-th state quantity; sup and inf represent the maximum and minimum values ​​of the affine interval respectively; {} represents a set; ξ is the convergence accuracy.

[0075] The present invention and its preferred solution introduce a holomorphic embedding factor into an uncertain affine power flow model, and establish an embedded affine power flow model that describes the uncertainty of load and new energy output; then, according to the actual physical meaning corresponding to the initial state, a definite initial solution of the affine power flow is obtained, and the problem that the convergence of the affine power flow algorithm is affected by the initial value is solved; thereafter, a mathematical relationship between the affine reactive power and the affine voltage of the PV node is established, and the affine reactive power of the PV node can be directly solved by using the affine node voltage obtained by each calculation; finally, the affine power flow solution problem is converted into a Taylor power series coefficient solution problem, and the recursive relationship of the power series coefficients of the affine state quantity is derived. The calculation process only involves linear algebra operations of the affine equation group, thereby simplifying the calculation complexity and improving the calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 The figure is a schematic diagram of the overall solution flow of an embodiment of the present invention. DETAILED DESCRIPTION

[0077] In order to make the features and advantages of this patent more obvious and easy to understand, the following embodiments are specifically described in detail as follows:

[0078] It should be noted that the following detailed descriptions are illustrative and are intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meanings as those commonly understood by those skilled in the art to which the present application belongs.

[0079] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0080] like Figure 1 As shown, in the present embodiment, firstly, an embedded affine power flow model describing the uncertainty of load and new energy output is established; then, a definite initial solution of the affine power flow is obtained to solve the problem that the convergence of the affine power flow algorithm is affected by the initial value; thereafter, a mathematical relationship between the affine reactive power and the affine voltage of the PV node is established, and the affine node voltage obtained by each calculation is used to directly solve the affine reactive power of the PV node; and the affine power flow solution problem is converted into a Taylor power series coefficient solution problem, and by deriving the recursive relationship of the power series coefficients of the affine state quantity, the calculation process involves only linear algebra operations of the affine equation group.

[0081] The following is a detailed introduction to this embodiment.

[0082] 1. Power System Holomorphic Embedded Affine Power Flow Model

[0083] (1) PQ node holomorphic embedding affine model:

[0084] The full embedded power balance equation of the PQ node is shown in formula (1):

[0085]

[0086] Where: ^ represents the affine number, * represents conjugation, and s is the holomorphic embedding factor; represents the holomorphic embedded affine voltage of the kth node; represents the holomorphic embedded affine voltage of the i-th node; represents the conjugate of the affine apparent power of the i-th node; Y ik,tr represents the line and transformer impedance between lines i and k, where the subscript tr represents the transformer; Y i,shThe line and transformer admittance of the node and the ground capacitance of the node, where the subscript sh indicates parallel connection; N is the total number of nodes;

[0087] In the formula, The Taylor expansion of is an affine infinite series with respect to s, and is analytic for s, as shown in formula (2):

[0088]

[0089] Where: n is the nth order Taylor expansion; is the affine number of the n-th order Taylor expansion of the voltage at node i; V i，0 [n] is the central value of the n-th order Taylor expansion affine coefficient; V i,j [n] is the noise element coefficient corresponding to the jth noise element in the nth-order Taylor expansion affine coefficient; ε j is the jth noise element; M i is the total number of noise elements;

[0090] (2) PV node holomorphic embedding affine model

[0091] The fully embedded power balance equation of the PV node is shown in equation (3):

[0092]

[0093] Where: is the affine active power of node i; is the holomorphic embedded affine reactive power of node i;

[0094] Since the voltage of the PV node is given, the voltage equation shown in formula (4) must be satisfied:

[0095]

[0096] Where: V i sp represents the set value of node i, and sp represents the set value;

[0097] (3) Balanced Node Holomorphic Embedding Affine Model

[0098] The balanced node satisfies the node voltage equation as shown in the formula:

[0099]

[0100] 2 Affine power flow solution based on the holomorphic embedding method

[0101] (1) Affine power flow recursion initial solution

[0102] For the holomorphic embedded affine power flow equation, when the initial state s = 0, the solutions of equations (1) to (5) are:

[0103]

[0104] Therefore, the affine center value of the constant term of the voltage and reactive power series in the initial state is V i，0 [0]=1,Q i，0 [0] = 0, the noise element coefficients are all 0; the above initial state corresponds to the grid being unloaded, all generators not working, and the PV node reactive output being zero, which has actual physical meaning.

[0105] (2) Recursive relation of affine power series coefficients

[0106] Define the inverse of the affine voltage function because According to the principle of numerical equality of coefficients of each power series, we can deduce The calculation of is shown in formula (7);

[0107]

[0108] For the PQ node, let the coefficients of the different power terms of s on both sides of the equal sign in equation (1) be equal, and obtain the recursive relationship of equation (8):

[0109]

[0110] For the PV node, let the coefficients of different power terms of s on both sides of the equal sign in equation (3) and equation (4) be equal, and obtain the recursive relationship of equation (9) and equation (10):

[0111]

[0112]

[0113] Where: n1 For the pulse function, it takes 1 when n=1 and 0 in other cases;

[0114] In formula (9), and As the unknown quantity is moved to the left, we get:

[0115]

[0116] At this time, the affine node voltage can be used to directly solve the PV node reactive power;

[0117] For the equilibrium node, let the coefficients of different power terms of s on both sides of the equal sign of formula (5) be equal, and obtain the recursive relationship of formula (12):

[0118]

[0119] (3) Matrix representation of the recursive relationship of power series coefficients

[0120] In order to facilitate calculation, the recursive relationship is decomposed into two parts, the real part and the imaginary part, and written in the form of a matrix equation; for the node voltage and admittance, they are decomposed into the real part and the imaginary part, that is:

[0121]

[0122] Y ik,tr =G ik,tr +jB ik,tr

[0123] Where: for The real part of , the subscript re represents the real part; for The imaginary part of G, the subscript im represents the imaginary part; ik,tr is conductivity; B ik,tr is the electrical admittance;

[0124] Convert the PQ node recursive relation (8) into (14):

[0125]

[0126]

[0127] Convert the PV node recursive relations (9) and (10) into (15) and (16):

[0128]

[0129]

[0130] Convert the equilibrium node recursive relation (12) into (17):

[0131]

[0132]

[0133] The recursive relationship from equation (14) to equation (17) is further organized into the form of algebraic linear equations:

[0134]

[0135] Where: A is the recursive matrix; is the coefficient vector of the n-th order affine power series to be determined; is a known vector not greater than the coefficient of the n-1th order affine power series;

[0136] (4) Affine solution and convergence conditions of holomorphic embedded uncertainty power flow

[0137] Since s=1 is the analytical solution of the target state, after the recursion converges, the affine solution of the calculated state quantity is:

[0138]

[0139] In the formula, k is the number of recursions to converge;

[0140] The recursive convergence condition is shown in formula (20):

[0141]

[0142] Where: represents the k-order affine power series coefficient of the i-th state quantity; sup and inf represent the maximum and minimum values ​​of the affine interval respectively; {} represents a set; ξ is the convergence accuracy.

[0143] The above is only a preferred embodiment of the present invention, and does not limit the present invention in other forms. Any technician familiar with the profession may use the above disclosed technical content to change or modify it into an equivalent embodiment with equivalent changes. However, any simple modification, equivalent change and modification made to the above embodiment according to the technical essence of the present invention without departing from the technical solution of the present invention still belongs to the protection scope of the technical solution of the present invention.

[0144] This patent is not limited to the above-mentioned optimal implementation mode. Anyone can derive other forms of uncertainty affine power flow calculation methods for power systems based on holomorphic embedding under the inspiration of this patent. All equal changes and modifications made according to the scope of the patent application of the present invention should be covered by this patent.

Claims

1. A method for calculating power system uncertainty affine power flow based on holomorphic embedding, characterized by: First, an embedded affine power flow model is established to describe the uncertainty of load and renewable energy output. Then, a definite initial solution of the affine power flow is obtained to solve the problem that the convergence of the affine power flow algorithm is affected by the initial value. After that, a mathematical relationship between the affine reactive power and the affine voltage of the PV node is established, and the affine node voltage obtained each time is used to directly solve the affine reactive power of the PV node. The affine power flow solution problem is converted into the solution problem of Taylor power series coefficients. By deriving the recursive relationship of the power series coefficients of the affine state quantity, the calculation process only involves linear algebra operations of the affine equation group. The process of establishing the embedded affine power flow model describing the uncertainty of load and new energy output specifically includes: Step S11: Establish a PQ node holomorphic embedding affine model: The full embedded power balance equation of the PQ node is shown in formula (1): Where: ^ represents the affine number, * represents conjugation, and s is the holomorphic embedding factor; represents the holomorphic embedded affine voltage of the kth node; represents the holomorphic embedded affine voltage of the i-th node; represents the conjugate of the affine apparent power of the i-th node; Y ik,tr represents the line and transformer impedance between lines i and k, where the subscript tr represents the transformer; Y i,sh represents the line and transformer admittance of the node and the ground capacitance of the node, where the subscript sh represents parallel connection; N is the total number of nodes; In formula (1), The Taylor expansion of is an affine infinite series with respect to s, and is analytic for s, as shown in formula (2): Where: n is the nth order Taylor expansion; is the affine number of the n-th order Taylor expansion of the voltage at node i; V i，0 [n] is the central value of the n-th order Taylor expansion affine coefficient; V i,j [n] is the noise element coefficient corresponding to the jth noise element in the nth-order Taylor expansion affine coefficient; ε j is the jth noise element; M i is the total number of noise elements; Step S12: Establish a PV node holomorphic embedding affine model: The fully embedded power balance equation of the PV node is shown in equation (3): Where: is the affine active power of node i; is the holomorphic embedded affine reactive power of node i; Since the voltage of the PV node is given, the voltage equation shown in formula (4) must be satisfied: Where: represents the set value of node i, and sp represents the set value; Step S13: Establish a balanced node holomorphic embedding affine model: The equilibrium node satisfies the node voltage equation shown in equation (5): The solution process of the embedded affine power flow model describing the uncertainty of load and new energy output specifically includes: Step S21: Calculate the initial solution of affine power flow recursion: For the holomorphic embedded affine power flow equation, when the initial state s = 0, the solutions of equations (1) to (5) are: Therefore, the affine center value of the constant term of the voltage and reactive power series in the initial state is V i，0 [0]=1,Q i，0 [0] = 0, the noise element coefficients are all 0; the above initial state corresponds to the grid being unloaded, all generators not working, and the PV node reactive output being zero; Step S22: Construct the recursive relationship of the affine power flow power series coefficients Define the inverse of the affine voltage function because According to the principle of numerical equality of coefficients of each power series, we can deduce The calculation of is shown in formula (7); For the PQ node, let the coefficients of the different power terms of s on both sides of the equal sign in equation (1) be equal, and obtain the recursive relationship of equation (8): For the PV node, let the coefficients of different power terms of s on both sides of the equal sign in equation (3) and equation (4) be equal, and obtain the recursive relationship of equation (9) and equation (10): Where: n1 It is an impulse function and takes 1 when n=1 and takes 0 otherwise; In formula (9), and As the unknown quantity is moved to the left, we get: At this time, the PV node reactive power is directly solved using the desired affine node voltage; For the equilibrium node, let the coefficients of different power terms of s on both sides of the equal sign of formula (5) be equal, and obtain the recursive relationship of formula (12): Step S23: Construct a matrix representation of the recursive relationship of the power series coefficients Decompose the recursive relationship into two parts, the real part and the imaginary part, and write it in the form of a matrix equation; for the node voltage and admittance, both are decomposed into the real part and the imaginary part, that is: Y ik,tr =G ik,tr +jB ik,tr Where: for The real part of , the subscript re represents the real part; for The imaginary part of G, the subscript im represents the imaginary part; ik,tr is conductivity; B ik,tr is the electrical admittance; Convert the PQ node recursive relation (8) into (14): Convert the PV node recursive relations (9) and (10) into (15) and (16): Convert the equilibrium node recursive relation (12) into (17): The recursive relationship from equation (14) to equation (17) is further organized into the form of algebraic linear equations: Where: A is the recursive matrix; is the coefficient vector of the n-th order affine power series to be determined; is a known vector not greater than the coefficient of the n-1th order affine power series; Step S24: Calculate the holomorphic embedded uncertainty power flow affine solution Since s=1 is the analytical solution of the target state, after the recursion converges, the affine solution of the calculated state quantity is: In the formula, k is the number of recursions to converge; The recursive convergence condition is shown in formula (20): Where: represents the k-order affine power series coefficient of the i-th state quantity; sup and inf represent the maximum and minimum values ​​of the affine interval respectively; {} represents a set; ξ is the convergence accuracy.

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