A transient voltage stability index analytical method based on gauss-legendre algorithm

Abstract

The application provides a transient voltage stability index analytical method based on a Gauss-Legendre algorithm, and belongs to the technical field of transient stability analysis of AC-DC systems. The technical scheme is as follows: the technical scheme is as follows: step 1, a polynomial approximation method of the transient voltage stability index is proposed; step 2, a suitable base function is selected to construct a polynomial approximation model; step 3, the coefficients of the polynomial are solved through a Gauss-Legendre quadrature method; and step 4, based on a piecewise approximation strategy, the order of each polynomial is determined, and the order of the approximation model is dynamically adjusted according to the change rate of the voltage response of the power grid. The application quantifies the stability performance of the power grid under different operating conditions in real time by performing polynomial approximation on the transient voltage stability margin.

Description

Technical Field

[0001] This invention belongs to the field of transient stability analysis technology of AC / DC systems, and particularly relates to an analytical method for transient voltage stability indices based on the Gauss-Legendal algorithm. Background Technology

[0002] With the continuous development of power systems, the power grid structure has become increasingly complex. In particular, emerging characteristics such as large-scale renewable energy integration, long-distance power transmission, and multi-regional grid interconnection present new challenges to power system stability analysis. Under these complex operating conditions, the transient voltage stability of the power grid has gradually become a crucial factor affecting the safe operation of the system. Especially when the power system encounters faults or sudden events (such as short circuits, load fluctuations, generator disconnection, etc.), voltage fluctuations can rapidly trigger transient instability in the power grid, and even lead to system collapse.

[0003] Traditional power grid stability assessment methods typically focus on steady-state analysis, primarily evaluating system safety under steady-state conditions using indicators such as static voltage stability margin and load margin. However, these methods have limitations because they neglect the impact of transient voltage stability on system operation. Especially when the power grid is subjected to disturbances, the dynamic voltage response often changes very complexly, and steady-state methods cannot effectively capture rapid voltage fluctuations and the system's transient stability.

[0004] To more comprehensively assess the power grid's receiving capacity and stability, a dynamic evaluation method that considers transient voltage stability must be introduced. Currently, with the development of high-precision time-domain simulation technology and advanced measurement equipment such as phasor measurement units (PMUs), the dynamic response of the power grid can be captured and analyzed more accurately, providing new technical means for assessing transient voltage stability.

[0005] The challenge in assessing transient voltage stability lies in the accurate quantification and analytical expression of the indices. In quantifying transient voltage stability indices, the binary table method uses the duration of voltage below a certain threshold as the criterion for transient voltage stability, simplifying complex calculations and enabling rapid quantification, thus gaining widespread application in transient voltage stability assessment. The key to using a binary table as a transient voltage stability index for assessing the maximum DC feed-in power of the receiving-end grid lies in its analytical expression. To address this, a polynomial approximation method is introduced, using polynomials to represent the system's dynamic trajectory to analyze transient stability. However, in polynomial approximation, global approximation methods are often employed, and the order of the polynomial basis functions cannot be flexibly selected based on the rate of change of the fitted curve, still presenting a trade-off between approximation accuracy and computational complexity.

[0006] How to solve the above-mentioned technical problems is the challenge facing this invention. Summary of the Invention

[0007] The purpose of this invention is to provide an analytical method for transient voltage stability indices based on the Gauss-Legendal algorithm. Employing an analytical method based on Gauss-Legendal approximation theory, this method constructs a parameterized approximation model to transform the nonlinear relationship between DC power and transient stability margin into a polynomial analytical form. By appropriately selecting basis functions and coefficients, high accuracy of the approximation results is ensured while reducing computational complexity. The main advantage of this method is its ability to quantify the stability performance of the power grid under different operating conditions in real time through polynomial approximation of the transient voltage stability margin. This method not only compensates for the shortcomings of traditional analytical methods in dynamic evaluation but also provides a more reliable theoretical basis for power system planning and operation, thereby improving the power system's ability to cope with complex disturbances and its security.

[0008] The inventive concept of this invention is as follows: Based on the smoothness and continuity characteristics of the dynamic response of power systems, this invention proposes a polynomial approximation method for transient voltage stability indices. This method maps the nonlinear relationship between DC power and transient stability margin indices into polynomial form by constructing a parameterized function model. This significantly improves the computational efficiency of the indices while ensuring approximation accuracy. For the transient voltage stability margin index M... i Subject to DC power P dl (l represents the l-th DC line) directly affects, when P dl When the voltage changes, the dynamic response U i (t;P dl The DC power P changes accordingly. dl A continuous function in parameter space. Due to the voltage response U i (t;P dl For parameter P dl The change is smooth, therefore the transient voltage stability margin index M i Regarding parameter P dl It is continuous and approximate. According to the Weierstrass approximation theorem, a continuous function on a closed interval can be uniformly approximated by a polynomial.

[0009] To achieve the aforementioned objectives, the present invention employs the following technical solution: a method for analyzing transient voltage stability indices based on the Gauss-Legendal algorithm, comprising the following steps:

[0010] Step 1: Propose a polynomial approximation method for transient voltage stability index. By constructing a parameterized function model, the nonlinear relationship between DC power and transient stability margin index is mapped into a polynomial form.

[0011] Step 2: Select appropriate basis functions to construct a polynomial approximation model. The basis functions are Legendre polynomials. By analyzing the rate of change of the grid voltage response, select an appropriate order of basis functions.

[0012] Step 3: Solve for the coefficients of the polynomial using the Gauss-Legend quadrature method to obtain the polynomial relationship between the transient voltage stability margin index and the DC power.

[0013] Step 4: Based on the piecewise approximation strategy, determine the order of the polynomial in each segment, and dynamically adjust the order of the approximation model according to the rate of change of the grid voltage response.

[0014] Furthermore, the polynomial approximation method is employed, which uses a linear combination of polynomial basis functions and their coefficients to approximate the transient voltage stability margin index M. i Expanded into parameters DC power P dl The polynomial representation of M. Ignoring the subscripts bus i and DC line l, the transient voltage stability index M is related to the DC power P. d The functional relationship is denoted as M(P) d The general expression for polynomial approximation is as follows:

[0015]

[0016] In the formula, n represents the order of the polynomial basis functions, and c j Let Φ be the undetermined basis function coefficients. j (P d ) represents the selected basis functions.

[0017] Let the approximation error be...

[0018]

[0019] In the formula, P d,max P d,min DC power P d The upper and lower limits, ω(P) d ) is the probability density function. Since the parameter DC transmission power P d It varies arbitrarily within a given bounded closed interval, therefore the parameter P can be... d Assuming a uniform distribution, i.e., ω(P) d ) = 1.

[0020] The principle of the polynomial approximation method is to minimize the error between the fitted curve and the original function. Let the error equation (2) be a multivariate function I(c0,c1,…,c n Then the original approximation problem is transformed into a multivariate function I(c0,c1,…,c n The problem of finding the extrema of a multivariable function. From the necessary condition for finding the extrema of a multivariable function, we obtain:

[0021]

[0022] Simplifying, we get:

[0023]

[0024] Introducing the concept of inner product, let's denote it as:

[0025]

[0026] In the formula, d k Let be the calculated constant. Then equation (3) can be simplified to the following equation:

[0027] Gc=d (7)

[0028] In the formula:

[0029]

[0030] To obtain the coefficients c of the undetermined basis functions k This requires solving the system of equations shown in equation (7). However, if the basis functions are not chosen appropriately, resulting in ill-conditioned matrix G, then equation (7) cannot be inverted, and the coefficients c will be difficult to solve. k This becomes difficult, so it is necessary to choose the basis functions appropriately.

[0031] Furthermore, for ease of approximation, the basis functions of the polynomial are usually orthogonal polynomials, since the DC power P d If the possible values ​​of ω(P) are considered to be uniformly distributed, then we choose ω(P) as the value of ω(P). d The orthogonal Legendre polynomials with weights of 1 are used as basis functions. The recurrence formula for the Legendre polynomials is:

[0032] (n+1)Φ n+1 (P d )=(2n+1)P d Φ n (P d )-nΦ n-1 (P d ), n=1,2,... (9)

[0033] In the formula, Φ0(P d )=1,Φ1(P d ) = P d n is the order of the basis functions. The higher the order, the higher the power of the approximating polynomial, and the higher the accuracy of the approximation. However, polynomials are relatively more complex, so it is necessary to choose the appropriate order n.

[0034] Given that the domain of the Legendre polynomial is [-1, 1], it is necessary to define the DC transmission power P. dThe actual values ​​are linearly mapped to the standard interval [-1, 1], and the actual parameter values ​​are obtained by inverse mapping after polynomial approximation. The corresponding mapping expression is:

[0035]

[0036] In the formula These are the DC power parameter values ​​after normalization to the standard range.

[0037] Considering the orthogonality of orthogonal polynomials, i.e., the inner product of two distinct elements in a space is zero, we have:

[0038]

[0039] In the formula, Φ m (P d ), Φ n (P d () represents Legendre polynomials of order m and n. The value is a constant; the Legendre polynomial used in this embodiment has the value of...

[0040] Furthermore, after determining the basis functions, matrix G is determined according to the orthogonality property of equation (11). At this point, matrix G becomes a sparse matrix:

[0041]

[0042] Further, the coefficients c of the basis functions can be obtained by solving equation (7). k Its expression is:

[0043]

[0044] The integrand in the numerator of the above equation still contains the transient voltage binary index value M(P) that cannot be directly calculated. d This makes it difficult to analytically solve for the coefficient c. k This section uses the Gaussian integral method to approximate the integral value of the integrand, thereby realizing the coefficient c. k Analytical calculation. Considering the DC power parameter P d The weight function ω(P) d ) = 1, and by using the Gauss-Legend quadrature formula to approximate the integral term in equation (13), we have:

[0045]

[0046] Substituting into equation (13), we can obtain the polynomial coefficients c. k for:

[0047]

[0048] In the formula, A m To find the product coefficient, For the integration nodes, N represents the integration order. N+1 groups of DC transmission power are obtained through simulation or historical data. The values ​​and corresponding voltage trajectories, and by calculating each transmission power The corresponding transient voltage stability margin index This forms N+1 sets of quadrature nodes. Substituting these into equation (15) calculates the coefficient c. k The value of is then obtained. The polynomial approximation expression. Among them... The inverse mapping is to the actual power magnitude P d This will give us the final approximation polynomial. The higher the product order N, the more simulation or actual data points are obtained, and the higher the accuracy of the approximation. However, the amount of data required is also larger, which will increase the time complexity of the calculation.

[0049] Furthermore, as can be seen from the above analysis, the proposed polynomial approximation method fits the analytical relationship M(P) between DC power and transient voltage stability margin index. d The accuracy and computational complexity of the approximation are directly affected by the order of the basis functions (n) and the order of the quadrature (N). The order of the basis functions (n) represents the power of the approximation polynomial. When the rate of change of the fitted curve is large, the error of fitting with low-order basis functions is large, so the order of the selected basis functions (n) should be increased. However, when the rate of change of the fitted curve is small, the computational complexity of fitting with high-order basis functions will increase significantly. The order of the quadrature (N) represents the amount of simulation or historical data required for approximation. The larger the order of N, the smaller the approximation error and the more accurate the calculation, but the computational complexity will also increase. Therefore, it is necessary to reasonably determine the order of the basis functions (n) and the order of the quadrature (N). Therefore, this section proposes the idea of ​​piecewise approximation, determining the required order of the basis functions (n) according to different rates of change of the fitted curve, and determining the quadrature (N) for each stage based on the approximation error and computational complexity requirements.

[0050] First, select relatively small initial order n0 of the basis functions and initial order N0 of the quadrature, and then apply this to the DC power P. d With transient voltage stability margin M(P) d By performing polynomial approximation on the analytic relation of ), the initial polynomial expression M0(P) is obtained. d Secondly, utilizing curvature

[23] The rate of change of the fitted curve is characterized, and different approximation intervals are then defined based on the curvature corresponding to different DC powers, thereby improving the accuracy of the approximation and meeting computational requirements. Curvature represents the rate of rotation of the tangent direction angle at a point on the fitted curve with respect to the arc length, indicating the degree to which the curve deviates from a straight line. Its expression is:

[0051]

[0052] Based on the curvature expression obtained above, its effect on DC power P is further calculated. d first derivative Setting its value to 0 yields the extreme points of the derivative function. Using these extreme points as the basis for interval segmentation, the power interval can be divided into several sub-intervals, with the degree of variation of the fitted curve remaining relatively consistent within each sub-interval. Furthermore, the order of the basis functions and the order of the quadrature are determined based on the curvature within different sub-intervals, thereby significantly improving the overall approximation accuracy and computational efficiency.

[0053] When performing polynomial approximation on interval i, the initial approximation polynomial for the interval is first obtained using the order of the basis functions n0 and the order of the quadrature N0. The maximum curvature is obtained from the curvature expression of this polynomial. and average Defining the ratio of the maximum curvature to the average curvature to represent the smoothness of the curve, the smoothness value r0 of the initial approximation polynomial curve can be expressed as:

[0054]

[0055] The curve smoothing value r0 and the set threshold r max Compare, if r0 > r max This indicates that the order of the basis functions of the curve is relatively small, resulting in a larger error compared to the approximated curve. It is necessary to further increase the order of the basis functions n0 before performing polynomial approximation, until it is increased to n0. f This ensures that the smooth value of the fitted curve is less than the threshold, at which point n... f The value of is the order of the basis functions. Subsequently, the order of the basis functions n is used... f The approximate polynomial for the first iteration of the interval is obtained by summing the product order N0. Let the error between the actual value and the actual value be ε1. If ε1 is greater than the set error requirement ε0, then increase the order of the quadrature N0 and perform polynomial approximation again until the error meets the set requirement. Then the final polynomial approximation expression M for this interval can be obtained. i (P d ).

[0056] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0057] 1. This invention utilizes the Gauss-Legendal approximation theory to approximate the transient voltage stability margin index of the power grid using a polynomial function. By selecting appropriate polynomial basis functions and combining voltage response data of the power grid under different DC power conditions, the nonlinear relationship between transient voltage stability margin and DC power can be transformed into an analytical form, thereby achieving rapid and efficient stability assessment.

[0058] 2. This invention proposes a polynomial order determination method based on piecewise approximation, which can intelligently select the polynomial order according to the rate of change of the grid voltage response, thereby reducing computational complexity while ensuring computational accuracy. By adjusting the order of the basis functions and the number of quadrature nodes, the method can achieve an optimal balance between computational efficiency and accuracy under different accuracy requirements. Attached Figure Description

[0059] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0060] Figure 1 This is the improved IEEE 39-node network topology diagram in this invention.

[0061] Figure 2 This is the initial polynomial fitting curve diagram in this invention.

[0062] Figure 3 This is a graph showing the fitting effect of different quadrature orders N for interval 1 in this invention.

[0063] Figure 4 This is a fitting effect diagram for different quadrature orders N in the interval two of this invention.

[0064] Figure 5 This is a comparison chart of the approximation effects of different approximation methods in this invention. Detailed Implementation

[0065] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0066] Example 1

[0067] See Figure 1 and Figure 5 This embodiment provides a technical solution: a transient voltage stability index analysis method based on the Gauss-Legendal algorithm, comprising the following steps:

[0068] Step 1: Propose a polynomial approximation method for transient voltage stability index. By constructing a parameterized function model, the nonlinear relationship between DC power and transient stability margin index is mapped into a polynomial form.

[0069] Step 2: Select appropriate basis functions to construct a polynomial approximation model. The basis functions are Legendre polynomials. By analyzing the rate of change of the grid voltage response, select an appropriate order of basis functions.

[0070] Step 3: Solve for the coefficients of the polynomial using the Gauss-Legend quadrature method to obtain the polynomial relationship between the transient voltage stability margin index and the DC power.

[0071] Step 4: Based on the piecewise approximation strategy, determine the order of the polynomial in each segment, and dynamically adjust the order of the approximation model according to the rate of change of the grid voltage response.

[0072] This embodiment proposes a polynomial approximation method for transient voltage stability indices based on the smoothness and continuity of the dynamic response of power systems. This method constructs a parameterized function model to map the nonlinear relationship between DC power and the transient stability margin indices into a polynomial form, significantly improving the computational efficiency of the indices while ensuring approximation accuracy.

[0073] For the transient voltage stability margin index M i Subject to DC power P dl (l represents the l-th DC line) directly affects, when P dl When the voltage changes, the dynamic response U i (t;P dl The DC power P changes accordingly. dl A continuous function in parameter space. Due to the voltage response U i (t;P dl For parameter P dl The change is smooth, therefore the transient voltage stability margin index M i Regarding parameter P dl It is continuous and approximate. According to the Weierstrass approximation theorem, a continuous function on a closed interval can be uniformly approximated by a polynomial. Based on this, this embodiment adopts a polynomial approximation method, which uses a linear combination of polynomial basis functions and their coefficients to approximate the transient voltage stability margin index M. i Expanded into parameters DC power P dl The polynomial representation of M. Ignoring the subscripts bus i and DC line l, the transient voltage stability index M is related to the DC power P. d The functional relationship is denoted as M(P) d The general expression for polynomial approximation is as follows:

[0074]

[0075] In the formula, n represents the order of the polynomial basis functions, and c j Let Φ be the undetermined basis function coefficients. j (P d ) represents the selected basis functions.

[0076] Let the approximation error be...

[0077]

[0078] In the formula, P d,max P d,min DC power P d The upper and lower limits, ω(P) d ) is the probability density function. Since the parameter DC transmission power P d It varies arbitrarily within a given bounded closed interval, therefore the parameter P can be... d Assuming a uniform distribution, i.e., ω(P) d ) = 1.

[0079] The principle of the polynomial approximation method is to minimize the error between the fitted curve and the original function. Let the error formula (19) be a multivariate function I(c0,c1,…,c n Then the original approximation problem is transformed into a multivariate function I(c0,c1,…,c n The problem of finding the extrema of a multivariable function. From the necessary condition for finding the extrema of a multivariable function, we obtain:

[0080]

[0081] Simplifying, we get:

[0082]

[0083] Introducing the concept of inner product, let's denote it as:

[0084]

[0085] In the formula, d k Let be the calculated constant. Then equation (20) can be simplified to the following equation:

[0086] Gc=d (24)

[0087] In the formula:

[0088]

[0089] To obtain the coefficients c of the undetermined basis functions k This requires solving the system of equations shown in equation (24). However, if the basis functions are not chosen appropriately, resulting in ill-conditioned matrix G, then equation (24) cannot be inverted, and the coefficients c will be difficult to solve. k This becomes difficult, so it is necessary to choose the basis functions appropriately.

[0090] To facilitate approximation, the basis functions of the polynomial are usually orthogonal polynomials, since the DC power P d If the possible values ​​of ω(P) are considered to be uniformly distributed, then we choose ω(P) as the value of ω(P). d The orthogonal Legendre polynomials with weights of 1 are used as basis functions. The recurrence formula for the Legendre polynomials is:

[0091] (n+1)Φ n+1 (P d )=(2n+1)P d Φ n (P d )-nΦ n-1 (P d ), n=1,2,... (26)

[0092] In the formula, Φ0(P d )=1,Φ1(P d ) = P d n is the order of the basis function. The higher the order, the higher the power of the approximating polynomial and the higher the approximation accuracy. However, polynomials are relatively more complex, so it is necessary to choose the appropriate order n.

[0093] Given that the domain of the Legendre polynomial is [-1, 1], it is necessary to define the DC transmission power P. d The actual values ​​are linearly mapped to the standard interval [-1, 1], and the actual parameter values ​​are obtained by inverse mapping after polynomial approximation. The corresponding mapping expression is:

[0094]

[0095] In the formula These are the DC power parameter values ​​after normalization to the standard range.

[0096] Considering the orthogonality of orthogonal polynomials, i.e., the inner product of two distinct elements in a space is zero, we have:

[0097]

[0098] In the formula, Φ m (P d ), Φ n (P d () represents Legendre polynomials of order m and n. The value is a constant; the Legendre polynomial used in this embodiment has the value of...

[0099] After determining the basis functions, matrix G is determined according to the orthogonality property of equation (28). At this point, matrix G becomes a sparse matrix:

[0100]

[0101] Further, the coefficients c of the basis functions can be obtained by solving equation (24). k Its expression is:

[0102]

[0103] The integrand in the numerator of the above equation still contains transient voltage binary index values ​​that cannot be directly calculated. This makes it difficult to analytically solve for the coefficient c. k This section uses the Gaussian integral method to approximate the integral value of the integrand, thereby realizing the coefficient c. k Analytical calculation. Considering the DC power parameter P d The weight function ω(P) d ) = 1, and by using the Gauss-Legend quadrature formula to approximate the integral term in equation (30), we have:

[0104]

[0105] Substituting into equation (30), we can obtain the polynomial coefficients c. k for:

[0106]

[0107] In the formula, A m To find the product coefficient, For the integration nodes, N represents the integration order. N+1 groups of DC transmission power are obtained through simulation or historical data. The values ​​and corresponding voltage trajectories, and by calculating each transmission power The corresponding transient voltage stability margin index This forms N+1 sets of quadrature nodes. Substituting these into equation (32) calculates the coefficient c. k The value of is then obtained. The polynomial approximation expression. Among them... The inverse mapping is to the actual power magnitude P d This will give us the final approximation polynomial. The higher the product order N, the more simulation or actual data points are obtained, and the higher the accuracy of the approximation. However, the amount of data required is also larger, which will increase the time complexity of the calculation.

[0108] As can be seen from the above analysis, the proposed polynomial approximation method fits the analytical relationship M(P) between DC power and transient voltage stability margin index. dThe accuracy and computational complexity of the approximation are directly affected by the order of the basis functions (n) and the quadrature order (N). The order of the basis functions (n) represents the power of the approximation polynomial. When the rate of change of the fitted curve is large, the error of fitting with low-order basis functions is large, so the selected basis function order (n) should be increased. However, when the rate of change of the fitted curve is small, the computational complexity of fitting with high-order basis functions will increase significantly. The quadrature order (N) represents the amount of simulation or historical data required for approximation. The larger the order (N), the smaller the approximation error and the more accurate the calculation, but the computational complexity will also increase. Therefore, it is necessary to reasonably determine the order of the basis functions (n) and the quadrature order (N). Therefore, this section proposes the idea of ​​piecewise approximation, determining the required basis function order (n) according to different rates of change of the fitted curve, and determining the quadrature order (N) for each stage based on the approximation error and computational complexity requirements.

[0109] First, select relatively small initial orders n0 for the basis functions and N0 for the quadrature, and then apply this to the DC power P. d With transient voltage stability margin M(P) d By performing polynomial approximation on the analytic relation of ), the initial polynomial expression M0(P) is obtained. d Secondly, utilizing curvature

[23] The rate of change of the fitted curve is characterized, and different approximation intervals are then defined based on the curvature corresponding to different DC powers, thereby improving the accuracy of the approximation and meeting computational requirements. Curvature represents the rate of rotation of the tangent direction angle at a point on the fitted curve with respect to the arc length, indicating the degree to which the curve deviates from a straight line. Its expression is:

[0110]

[0111] Based on the curvature expression obtained above, its effect on DC power P is further calculated. d first derivative Setting its value to 0 yields the extreme points of the derivative function. Using these extreme points as the basis for interval segmentation, the power interval can be divided into several sub-intervals, with the degree of variation of the fitted curve remaining relatively consistent within each sub-interval. Furthermore, the order of the basis functions and the order of the quadrature are determined based on the curvature within different sub-intervals, thereby significantly improving the overall approximation accuracy and computational efficiency.

[0112] When performing polynomial approximation on interval i, the initial approximation polynomial for the interval is first obtained using the order of the basis functions n0 and the order of the quadrature N0. The maximum curvature is obtained from the curvature expression of this polynomial. and average Defining the ratio of the maximum curvature to the average curvature to represent the smoothness of the curve, the smoothness value r0 of the initial approximation polynomial curve can be expressed as:

[0113]

[0114] The curve smoothing value r0 and the set threshold r max Compare, if r0 > r max This indicates that the order of the basis functions of the curve is relatively small, resulting in a larger error compared to the approximated curve. It is necessary to further increase the order of the basis functions n0 before performing polynomial approximation, until it is increased to n0. f This ensures that the smooth value of the fitted curve is less than the threshold, at which point n... f The value of is the order of the basis functions. Subsequently, the order of the basis functions n is used... f The approximating polynomial M for the first iteration of the interval is obtained by summing the product order N0. i (1) (P d Let the error between the actual value and the given value be ε1. If ε1 is greater than the set error requirement ε0, then increase the quadrature order N0 and perform polynomial approximation again until the error meets the set requirement. Then the final polynomial approximation expression M for this interval can be obtained. i (P d ).

[0115] Example 2

[0116] To verify the effectiveness of the proposed evaluation method, an improved IEEE 39-node model was built on the PSCAD / EMTDC platform. For example... Figure 1 As shown, an LCC-HVDC (Line-Commutated Converter based High Voltage Direct Current) bipolar DC system is connected at bus node 8. The rated DC transmission power is 1200MW and the rated voltage is 500kV. The rectifier side adopts constant current control, and the inverter side adopts constant voltage, low voltage current limiting and other control links.

[0117] Figure 2 The power P is given d Regarding the initial polynomial curve of the transient voltage stability margin index M and its key point curvature values, the approximation interval is divided into interval 1: [0, 0.59]pu and interval 2: [0.59, 1]pu.

[0118] Figure 3 Table 1 shows the fitting results and computational cost for interval 1 at different orders. It can be seen that the fitting curve becomes more accurate with increasing quadrature order N. When the order is increased to 6, the error has decreased to 1.23%. Further increasing the quadrature order N to 8 reduces the error to 1.08%, but also significantly increases the number of simulations to 39. Therefore, considering both approximation accuracy and computational cost, N=6 is chosen as the quadrature order for interval 1. The corresponding polynomial approximation analytical expression is:

[0119]

[0120] Table 1 shows the number of simulations and errors for different quadrature orders N in interval 1.

[0121]

[0122] For interval 2, the ratio of the maximum to the average curvature of the interval, r = 5.72 > 1, indicates that the order of the basis function is set too low, resulting in a large error compared to the approximated curve. The order of the basis function needs to be further increased until the curvature meets the threshold requirement. As shown in Table 2, as the order of the basis function n increases, the curvature ratio r continuously decreases, and the fitted curve more closely matches the actual value. Therefore, based on the threshold requirement r < 1, the order of the basis function n is chosen to be 5.

[0123] Table 2 shows the relationship between the curvature ratio r and the order of the basis functions n.

[0124]

[0125] Subsequently, the order of the quadrature N is increased, such as... Figure 3 As shown in Table 3, it can be seen that the higher the quadrature order N, the closer the curve is to the actual data points. When N increases to order 6, the approximation polynomial error has decreased to 1.56%. Further increasing the quadrature order gradually increases the number of simulations, but the error remains almost unchanged. Therefore, N = 6 is chosen as the quadrature order for interval 2, and the polynomial expression at this time is:

[0126]

[0127] Table 3 shows the computational complexity and error for different quadrature orders N in interval two.

[0128]

[0129]

[0130] In summary, the polynomial approximation expression of the transient voltage stability margin index is as follows:

[0131]

[0132] The approximation polynomial is as follows Figure 4 As shown. To compare the superiority of the method proposed in this embodiment, Figure 4Meanwhile, the fitting results of the traditional method are given for comparison. The gray data points in the figure are the true values ​​of the transient voltage stability margin index obtained by simulation. The green curve and the purple curve are the polynomial approximation results of the method in reference

[20] and the method proposed in this embodiment, respectively. It can be seen that when the DC power changes in interval 1, the corresponding transient voltage stability margin index changes relatively smoothly and has strong linear characteristics. The approximation results obtained by different methods can all have good fitting effects. In interval 2, due to the relatively large DC power, the relationship between the transient voltage stability margin index and the power shows strong nonlinear change characteristics. The curve fitted by the traditional method has an error of 4.29%, which is larger than the error (1.38%) of the method proposed in this embodiment. Moreover, the number of simulations is much higher than that of the piecewise approximation method. Therefore, when using the method proposed in this embodiment, it is possible to ensure a low approximation error while greatly reducing the number of simulations and time complexity required for calculation.

[0133] Table 4 compares the computational cost and error of different approximations under the same approximation accuracy.

[0134]

[0135] 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, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An analytical method for transient voltage stability indices based on the Gauss-Legendal algorithm, characterized in that, Includes the following steps: Step 1: Propose a polynomial approximation method for transient voltage stability index. By constructing a parameterized function model, the nonlinear relationship between DC power and transient stability margin index is mapped into a polynomial form. Step 2: Select basis functions to construct a polynomial approximation model. The basis functions are Legendre polynomials. The order of the basis functions is selected by analyzing the rate of change of the grid voltage response. Step 3: Solve for the coefficients of the polynomial using the Gauss-Legend quadrature method to obtain the polynomial relationship between the transient voltage stability margin index and the DC power. Step 4: Based on the piecewise approximation strategy, determine the order of the polynomial in each segment, and dynamically adjust the order of the approximation model according to the rate of change of the grid voltage response. In step 4, the piecewise approximation method includes the following steps: 41) Constructing a curvature formula for determining segmented intervals ； In the formula: The initial polynomial curvature; The first derivative of the initial polynomial; The second derivative of the initial polynomial; 42) Construct curvature formulas that determine the order of basis functions. ； In the formula: DC power This is the curve smoothing value; The maximum curvature of the initial approximation polynomial in interval i; The average curvature of the initial approximation polynomial in interval i.

2. The analytical method for transient voltage stability index based on the Gauss-Legendal algorithm according to claim 1, characterized in that, In step 1, by constructing a parameterized function model, the nonlinear relationship between DC power and transient stability margin index is transformed into a polynomial expression as follows: ； In the formula, The order of the basis functions of the polynomial is represented. The basis function coefficients are to be determined. These are the selected basis functions.

3. The analytical method for transient voltage stability index based on the Gauss-Legendal algorithm according to claim 1, characterized in that, In step 2, the selection of basis functions includes choosing the Legendre polynomial order based on the dynamic characteristics of the grid's voltage response. The Legendre polynomial is as follows: ； In the formula, , , Let be the order of the basis functions.

4. The analytical method for transient voltage stability index based on the Gauss-Legendal algorithm according to claim 1, characterized in that, In step 3, the coefficients are solved using the Gauss-Legend quadrature formula, and the formula for solving the coefficients is as follows: ； In the formula, To find the product coefficient, For the integration nodes, N represents the integration order. N+1 groups of DC transmission power are obtained through simulation or historical data. The values ​​and corresponding voltage trajectories, m=0,1,…,N, are obtained by calculating the transmission power. The corresponding transient voltage stability margin index This forms N+1 sets of quadrature nodes.

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