A method and device for obtaining linear power flow based on optimal square approximation of function

Abstract

The present invention discloses a method and device for obtaining linear power flow based on the best square approximation of a function, which belongs to the field of electrical engineering technology. First, the structure and conventional parameters of the power grid are collected, and then the power system power flow equation is established based on the AC line model, and then the traditional best square approximation theory of a single variable function is generalized to the case of a three-variable function, and then the considered approximation area is defined, and finally the nonlinear power flow equation is linearized based on the generalized theory, and the variable linear transformation method is used to simplify the model building process. The overall accuracy of the present invention within a larger operating range can meet the optimization analysis requirements of a high proportion of renewable energy access to the power system; the variable linear transformation strategy adopted can effectively reduce the amount of algebraic calculations in the model building process and significantly improve the calculation efficiency; the linearized power flow equation can take into account the reactive power, voltage and loss in the power system, and can be widely used in various application scenarios of the power system.

Description

Technical Field

[0001] The present invention belongs to the field of electrical engineering technology, and more specifically, relates to a method and device for obtaining linear power flow based on optimal square approximation of a function. Background Art

[0002] The power flow model is the basis for the optimal operation of the power system. The AC power flow model has high calculation accuracy and can fully reflect the actual operation status of the power system, but its strong nonlinear characteristics make it impossible to guarantee the convergence of the calculation and it is easy to fall into the local optimal solution. In view of this, a variety of linearized power flow models have been proposed in existing studies, which are conducive to establishing the optimization problem in the form of linear programming and have great advantages in calculation efficiency and convergence.

[0003] At present, the most commonly used DC power flow model in the power industry is a classic linearized power flow model, but it ignores the influence of the active power loss of the system and the reactive power of the branch. In recent years, some scholars have constructed a linearized power flow model based on the Taylor expansion principle and combined with some assumptions in DCPF. Other scholars have proposed a general expression of the linearized power flow model based on Taylor expansion, revealing that the essence of its differences is to achieve different nonlinear transformations of the variable space by selecting different state variable function forms, and proposed a normalized expression method of the state variable function form and a method for selecting the optimal state space of the power flow equation.

[0004] However, as the penetration rate of renewable energy continues to increase, the power system will operate more frequently under high-power fluctuation conditions, which puts forward new requirements for linearized power flow models, that is, the linearized model needs to be able to adapt to system conditions with a wider range of power fluctuations. Due to the inherent local mathematical properties of Taylor expansion, the existing linearized power flow models usually have good accuracy near the expansion point, but when the system operating point is far away from the expansion point, the accuracy of the model is usually difficult to meet the accuracy requirements of practical applications. The access of a high proportion of renewable energy has seriously affected the accuracy of the existing linearized power flow models. Therefore, there is an urgent need for a linearized power flow model that can adapt to a larger operating range of the power system. Summary of the invention

[0005] In view of the above defects or improvement needs of the prior art, the present invention provides a method and device for obtaining linear power flow based on the best square approximation of function, the purpose of which is to extend the traditional best square approximation theory of univariate function to the case of ternary function, and apply it to the linearization of nonlinear power flow model, so as to establish a linearized power flow model with the minimum square error in the considered approximation area, so as to meet the accuracy requirements of power systems with large power fluctuation conditions, thereby solving the technical problem of low accuracy of existing linearized power flow models.

[0006] To achieve the above object, according to one aspect of the present invention, a linear power flow acquisition method based on optimal square approximation of a function is provided, comprising:

[0007] S1: Collecting conventional parameters of the power system, which at least include: the number of nodes N b , the resistance r of the line j formed by any two nodes j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j ;

[0008] S2: establishing a power system power flow equation based on the conventional parameters and the AC line model, wherein the power system power flow equation is used to describe the active power and reactive power transmitted from node i to node j on line (i, j);

[0009] S3: Extend the best square approximation theory of univariate function to the ternary case to obtain the ternary function approximation theory;

[0010] S4: construct an approximation region and use the ternary function approximation theory to linearize the power system power flow equation to obtain a linearized power flow model, where the linearized power flow model is used to describe linear power flow.

[0011] In one embodiment, the power system flow equation in S2 is:

[0012]

[0013]

[0014] Among them, P ij (v i ,v j ,θ ij ) represents the active power transmitted from node i to node j on line (i, j), Q ij (v i ,v j ,θ ij ) represents the reactive power transmitted from node i to node j on line (i, j); v i represents the voltage amplitude at node i, v j represents the voltage amplitude at node j; θ ij Indicates the phase angle difference between the first and the last two ends of line (i, j); g ii represents the conductance of node i, b ii represents the susceptance of node i, g ij represents the conductance of line (i, j), b ij Represents the susceptance of line (i, j).

[0015] In one embodiment, the S3 includes:

[0016] Utilizing functions Make the best square approximation for f(x, y, z) in the approximation area D, that is, find a set of coefficients Make the function satisfy:

[0017]

[0018] in, H n for The generated function space; represents the basis function, whose linear combination For any (x, y, z)∈D if and only if a0=a1=...=a n =0 is established;

[0019] According to the necessary conditions for extreme values, the equation derived from the above optimization problem can be written as: Solving this equation yields the best square approximation function generated by the basis functions:

[0020]

[0021] In one embodiment, the S4 includes:

[0022] S41: Constructing the approximation region Among them, v i,ub and v i,lb are the maximum voltage amplitude and minimum voltage amplitude of node i allowed for normal system operation; v j,ub and v j,lb are the maximum voltage amplitude and minimum voltage amplitude of node j allowed for normal system operation; θ ij,max θ is the value allowed for normal operation of the system ij The maximum value of V i,ub and V i,lb are the upper and lower limits of the voltage amplitude of the set node i, V j,ub and V j,lb are the upper and lower limits of the voltage amplitude of the node j, respectively. ij,ub and θ ij,lb are respectively set to ij The upper and lower limits of

[0023] S42: Substitute the power system flow equation and the approximation region D into the formula The linearized power flow model is obtained; the linearized power flow model includes: an active power flow linear optimal square approximation function and a reactive power flow linear optimal square approximation function.

[0024] In one embodiment, the approximation interval D corresponds to θ ij,lb =-θ ij,ub .

[0025] In one embodiment, the S42 includes:

[0026] Taking active power flow as an example, the power system power flow Translates to:

[0027]

[0028] The approximation area D is converted into D′, which is expressed as: in,

[0029] The basis function The transformed active power flow equation f(v i ′,v j ′,θ ij ′) and the approximate region D′ are substituted into The best square approximation function is s(v i ′,v j ′,θ ij ′)=a3θ ij ′+a2v j ′+a1v i ′+a0; for s(v i ′,v j ′,θ ij ′) performs an inverse transformation to obtain the linearized power flow model; the linearized power flow model includes: an active power flow linear optimal square approximation function and a reactive power flow linear optimal square approximation function; wherein,

[0030]

[0031]

[0032] In one embodiment, the active power flow linear optimal square approximation function is:

[0033] f PL (v i ,v j ,θ ij )=p3θ ij +p2v j +p1v i +p0;

[0034] The reactive power flow linear optimal square approximation function expression is:

[0035] f QL (v i ,v j ,θ ij )=q3θ ij +q2v j +q1v i +q0; where

[0036]

[0037]

[0038] According to another aspect of the present invention, there is provided a linear power flow acquisition device based on optimal square approximation of a function, comprising:

[0039] The acquisition module is used to collect the conventional parameters of the power system, and the conventional parameters include at least: the number of nodes N b , the resistance r of the line j formed by any two nodes j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j ;

[0040] An establishment module is used to establish a power system power flow equation based on the conventional parameters and the AC line model, wherein the power system power flow equation is used to describe the active power and reactive power transmitted from node i to node j on line (i, j);

[0041] A generalization module is used to generalize the best square approximation theory of univariate function to the ternary case to obtain the ternary function approximation theory;

[0042] The processing module is used to construct an approximation region and use the ternary function approximation theory to linearize the power system power flow equation to obtain a linearized power flow model, wherein the linearized power flow model is used to describe linear power flow.

[0043] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0044] 1. The best square approximation technology used can minimize the square error of the linearized power flow model in the defined area and improve the overall accuracy of the power flow model within a larger system operation range; the principle of this method is clear and easy to implement;

[0045] 2. The proposed model is based on the AC power flow model and can be iterated using continuous linearization technology to improve the quality of the optimization solution;

[0046] 3. The linearized power flow equation can take into account the reactive power, voltage and loss in the power system, and can be widely used in various application scenarios of the power system, such as reactive power optimization, loss electricity price analysis, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 A flowchart of a linear power flow acquisition method based on optimal square approximation of a function provided by the present invention;

[0048] Figure 2 A schematic diagram of the structure of an AC line model provided by the present invention;

[0049] Figure 3 This is the topological diagram of the IEEE39 node example provided by the present invention. DETAILED DESCRIPTION

[0050] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0051] like Figure 1 As shown, the present invention provides a linear power flow acquisition method based on optimal square approximation of a function, comprising the following steps:

[0052] S1: Collecting conventional parameters of the power system, which at least include: the number of nodes N b , the resistance r of the line j formed by any two nodes j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j ;

[0053] S2: establishing a power system power flow equation based on the conventional parameters and the AC line model, wherein the power system power flow equation is used to describe the active power and reactive power transmitted from node i to node j on line (i, j);

[0054] S3: Extend the best square approximation theory of univariate function to the ternary case to obtain the ternary function approximation theory;

[0055] S4: construct an approximation region and use the ternary function approximation theory to linearize the power system power flow equation to obtain a linearized power flow model, where the linearized power flow model is used to describe linear power flow.

[0056] The following is a detailed description of the application method:

[0057] Step 1: Collect general parameters of the power system.

[0058] Conventional technical parameters of power systems include: number of nodes N b , the active load P at the i-th node Di and reactive load Q Di , the upper limit of voltage V imax and the lower limit V imin 、The upper and lower limits of the active output of the generator P Gimin , P Gimax And the upper and lower limits of reactive power output Q Gimin , Q Gimax ; Total number of lines N l , the resistance r of the jth line j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j , Rated capacity f Lj .

[0059] Step 2: Based on the AC line model, establish the power system flow equation.

[0060] AC line model as attached Figure 2 As shown, the variable in the figure is the voltage v at the node at the beginning of the line i and the voltage at the end node v j , the phase angle difference between the first and the last two ends of the line is θ ij , the active power P transmitted at the head end of the line ij and reactive power Q ij The subscript i is the node number at the beginning of the line, and the subscript j is the node number at the end of the line. Figure 3 This is the topological diagram of the IEEE39 node example provided by the present invention.

[0061] According to the AC line model, the power system flow equation is as follows:

[0062]

[0063]

[0064] Step 3: Generalize the theory of best square approximation for one-variable functions to the case of three-variable functions.

[0065] Assume that f(x, y, z), g(x, y, z) are continuous in region D. The inner product of the functions f(x, y, z), g(x, y, z) in region D is defined as the integral shown below:

[0066]

[0067] If the function family Satisfies the condition: any linear combination of n functions For any (x, y, z)∈D if and only if a0=a1=...=a n = 0, then these n functions are called linearly independent in region D, and this function family is called a linearly independent function family on region D. Given n linearly independent functions on region D, a function space H can be generated from these n linearly independent functions, and any function in H can be represented by a linear combination of these n functions, so these n functions are also called a linearly independent basis.

[0068] Let function in It is linearly independent in region D. If s(x, y, z) is used to make the best square approximation to f(x, y, z) in region D, it is to find a set of coefficients Make the function satisfy:

[0069]

[0070] in H n for The function space generated. This problem is equivalent to finding the function space about a k (k=0,1,...,n) n ) minimum value problem. Among them:

[0071]

[0072] According to the necessary conditions for the extreme value of multivariate functions, we have

[0073]

[0074] The equation derived from the necessary conditions for extremum can finally be written as:

[0075]

[0076] The above equation is a linear equation system. The coefficient matrix on the left is called the Gram matrix, whose elements are the inner products of the two corresponding basis functions. The elements of the constant vector on the right are the inner products of the corresponding basis functions and the quasi-approximation functions. Solving the equation, we can get the best square approximation function generated by the basis function.

[0077] Step 4.1: Construct the approximation region.

[0078] Considering the requirements for safe operation of power systems, there are usually the following constraints:

[0079]

[0080] The first two inequalities in the above formula represent the voltage fluctuation range under normal system operation. The third inequality indicates that the phase angle difference between the two ends of the line should be limited to a certain range. In order to weaken the impact of conservative problems, the approximate area D of interest is considered as a sub-area of ​​the area, as shown below:

[0081]

[0082]

[0083]

[0084]

[0085] Step 4.2: Linear transformation.

[0086] The active power flow formula of the power system is reformulated as:

[0087]

[0088] Since the constructed model is a linear approximation model, the basis function is selected as {1,v i ,v j ,θ ij}. Because 4 basis functions are selected, it is noted that the solution requires the calculation of 4*4 triple integrals to obtain the elements of the Gram matrix, and the calculation of 4 triple integrals to obtain the elements of the constant term on the right side of the equation system. A total of 20 triple integrals need to be calculated, and algebraic elimination is required to obtain an analytical solution. However, if the Gram matrix is ​​a diagonal matrix, only 8 triple integrals and 4 division operations need to be calculated to obtain an analytical solution. Therefore, consider linear transformation:

[0089]

[0090] To simplify the representation, let

[0091]

[0092] Then there is

[0093]

[0094] Furthermore, we have:

[0095] f′(v i ′,v j ′,θ ij ′)=(g ii +g ij )(a i v i ′+bi ) 2 -

[0096] g ij (a i v i ′+b i )(a j v j ′+b j )cos(θ ij,off θ ij ′+θ ij,cen )-

[0097] b ij (a i v i ′+b i )(a j v j ′+b j )sin(θ ij,off θ ij ′+θ ij,cen )

[0098] At this time, the approximation area D becomes D′:

[0099]

[0100] Now select the basis function {1,v i ′,v j ′,θ ij ′} to approximate the new nonlinear function. Although the formula is more complicated than that, since the integral region D′ is symmetric about the origin, the selected basis function {1,v i ′,v j ′,θ ij ′} are orthogonal in region D′, so the Gram matrix is ​​a diagonal matrix. In addition, after the integration region becomes a symmetric region about the origin, the calculation of the elements of the Gram matrix and the constant vector on the right side of the equation system becomes very simple by using the property that the integral of an odd function on an interval symmetric about the origin is 0.

[0101] Step 4.3: Based on the generalized theory of best square approximation of ternary functions, construct a linearized power flow model.

[0102] The style and Substituting into the formula, we get

[0103]

[0104] in:

[0105]

[0106] Therefore, the best square linear approximation function of the formula is

[0107] s(v i ′,v j ′,θ ij ′)=a3θ ij ′+a2v j ′+a1v i ′+a0

[0108] Substitute the inverse transformation of the equation into the equation:

[0109] f PL (v i ,v j ,θ ij )=p3θ ij +p2v j +p1v i +p0

[0110] The formula is the best square approximation function of the formula. Among them:

[0111]

[0112] Since the formulas for the reactive power flow and the active power flow of the line are the same, it is only necessary to replace g in the linear best square approximation expression of the active power flow. ii Replace with -b ii , g ij Replace with -b ij , b ij Replace with g ij , b ii Replace with g ii . Then the linear optimal square approximation expression of reactive power flow can be obtained:

[0113] f QL (v i ,v j ,θ ij )=q3θ ij +q2v j +q1v i +q0

[0114] in:

[0115]

[0116]

[0117] According to another aspect of the present invention, there is provided a linear power flow acquisition device based on optimal square approximation of a function, comprising:

[0118] The acquisition module is used to collect the conventional parameters of the power system, and the conventional parameters include at least: the number of nodes N b , the resistance r of the line j formed by any two nodes j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j ;

[0119] An establishment module is used to establish a power system power flow equation based on the conventional parameters and the AC line model, wherein the power system power flow equation is used to describe the active power and reactive power transmitted from node i to node j on line (i, j);

[0120] A generalization module is used to generalize the best square approximation theory of univariate function to the ternary case to obtain the ternary function approximation theory;

[0121] The processing module is used to construct an approximation region and use the ternary function approximation theory to linearize the power system power flow equation to obtain a linearized power flow model, wherein the linearized power flow model is used to describe linear power flow.

[0122] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A linear power flow acquisition method based on optimal square approximation of function, characterized in that: include: S1: Collecting conventional parameters of the power system, which at least include: the number of nodes N b , the resistance r of the line j formed by any two nodes j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j ; S2: establishing a power system power flow equation based on the conventional parameters and the AC line model, wherein the power system power flow equation is used to describe the active power and reactive power transmitted from node i to node j on line (i, j); S3: Extend the best square approximation theory of univariate function to the ternary case to obtain the ternary function approximation theory; S4: constructing an approximation region and linearizing the power system power flow equation using the ternary function approximation theory to obtain a linearized power flow model, wherein the linearized power flow model is used to describe linear power flow; The power system flow equation in S2 is: P ij (v i ,v j ,θ ij ) represents the active power transmitted from node i to node j on line (i, j), Q ij (v i ,v j ,θ ij ) represents the reactive power transmitted from node i to node j on line (i, j); v i represents the voltage amplitude at node i, v j represents the voltage amplitude at node j; θ ij Indicates the phase angle difference between the first and the last two ends of line (i, j); g ii represents the conductance of node i, b ii represents the susceptance of node i, g ij represents the conductance of line (i, j), b ij represents the susceptance of line (i, j); The S3 includes: utilizing function Make the best square approximation for f(x,y,z) in the approximation area D, that is, find a set of coefficients Make the function satisfy: H n for The generated function space; represents the basis function, whose linear combination For any (x, y, z)∈D if and only if a0=a1=…=a n =0 is established; According to the necessary conditions for extreme values, the equation derived from the above optimization problem can be written as: Solving this equation yields the best square approximation function generated by the basis functions: r∈[0,n],r∈Z,h∈[0,n],h∈Z; The S4 includes: S41: Constructing the approximation region Among them, v i,ub and v i,lb are the maximum voltage amplitude and minimum voltage amplitude of node i allowed for normal system operation; v j,ub and v j,lb are the maximum voltage amplitude and minimum voltage amplitude of node j allowed for normal system operation; θ ij,max θ is the value allowed for normal operation of the system ij The maximum value of V i,ub and V i,lb are the upper and lower limits of the voltage amplitude of the set node i, V j,ub and V j,lb are the upper and lower limits of the voltage amplitude of the node j, respectively. ij,ub and θ ij,lb are the set θ ij The upper and lower limits of S42: Substitute the power system flow equation and the approximation region D into the formula The linearized power flow model is obtained; the linearized power flow model includes: an active power flow linear optimal square approximation function and a reactive power flow linear optimal square approximation function.

2. The linear power flow acquisition method based on the best square approximation of function according to claim 1, characterized in that: The approximation region D corresponds to θ ij,lb =-θ ij,ub .

3. The linear power flow acquisition method based on the best square approximation of function according to claim 2, characterized in that: The S42 includes: Taking active power flow as an example, the power system power flow Transformed into f′(v i ′,v j ′,θ ij ′)=(g ii +g ij )(a i v i ′+b i ) 2 -g ij (a i v i ′+b i (a) j v j ′+b j )cos(θ ij,off i ij ′+θ ij,cen ); -b ij (a i v i ′+b i (a) j v j ′+b j )sin(θ ij,off i ij ′+θ ij,cen ) The approximation area D is converted into D′, which is expressed as: in, The basis function The transformed active power flow equation f(v i ′,v j ′,θ ij ′) and the approximate region D′ are substituted into The best square approximation function is s(v i ′,v j ′,θ ij ′)=a3θ ij ′+a2v j ′+a1v i ′+a0; for s(v i ′,v j ′,θ ij ′) to perform inverse transformation to obtain the linearized power flow model; the linearized power flow model includes: an active power flow linear optimal square approximation function and a reactive power flow linear optimal square approximation function; wherein, 4. The linear power flow acquisition method based on the best square approximation of function according to claim 3 is characterized in that: The linear optimal square approximation function of the active power flow is: f PL (v i ,v j ,θ ij )=p3θ ij +p2v j +p1v i +p0; The reactive power flow linear optimal square approximation function expression is: f QL (v i ,v j ,θ ij )=q3θ ij +q2v j +q1v i +q0; in, 5. A linear power flow acquisition device based on optimal square approximation of function, characterized in that: The method for obtaining linear power flow based on optimal square approximation of function according to claim 1 comprises: The acquisition module is used to collect the conventional parameters of the power system, and the conventional parameters include at least: the number of nodes N b , the resistance r of the line j formed by any two nodes j , Reactance x j 、Earth Susceptance b j , conductivity to ground g j ; An establishment module is used to establish a power system power flow equation based on the conventional parameters and the AC line model, wherein the power system power flow equation is used to describe the active power and reactive power transmitted from node i to node j on line (i, j); A generalization module is used to generalize the best square approximation theory of univariate function to the ternary case to obtain the ternary function approximation theory; The processing module is used to construct an approximation region and use the ternary function approximation theory to linearize the power system power flow equation to obtain a linearized power flow model, wherein the linearized power flow model is used to describe linear power flow.

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