A variable-harmonic impedance estimation method based on Legendre polynomial fitting

Abstract

The application relates to a variable-harmonic impedance estimation method based on Legendre polynomial fitting, comprising the following steps: S1, acquiring harmonic voltage and current at a public connection point, establishing an equivalent relationship at the public connection point, and establishing a target function constraint by using harmonic impedance and system-side harmonic voltage variation characteristics; S2, fitting a harmonic impedance variation curve by using a Legendre polynomial, obtaining harmonic impedance estimation by solving through an optimization theory, and determining the optimal Legendre polynomial fitting order by using an order screening method; S3, determining an impedance mutation point by using a mutation point screening algorithm, segmenting data according to the impedance mutation point, performing harmonic impedance estimation on each segment by using steps S1 and S2, and finally integrating segmented estimation results to obtain a final harmonic impedance variation curve. Through the application, the estimation accuracy of the harmonic impedance can be improved.

Description

Technical Field

[0001] This application relates to the field of variable harmonic impedance estimation technology, specifically to a variable harmonic impedance estimation method based on Legendre polynomial fitting. Background Technology

[0002] With the transformation of power distribution networks into smart grids, a large number of power electronic devices (such as frequency converters, rectifiers, and electronic switches) and renewable energy sources are being integrated into the grid, leading to increasingly serious harmonic problems. Increased harmonic content not only affects the normal operation of electrical equipment but also can cause safety accidents and threaten the stable operation of the power system. Accurate estimation of harmonic impedance is crucial for subsequent harmonic emission level assessment, harmonic mitigation, and system stability prediction.

[0003] However, traditional harmonic estimation methods often assume that the system-side harmonic impedance is constant or fluctuates within a small range. But considering that the power system is a complex and time-varying system, this assumption no longer holds when there are changes in operating mode, capacitor switching, topology changes, etc. The integration of new energy sources such as photovoltaics further exacerbates this time-varying nature. In such cases, traditional estimation methods will have significant errors and will find it difficult to accurately track changes in harmonic impedance. At the same time, when the harmonic impedance ratio on both sides of the point of common coupling (PCC) and the background harmonics are large, the estimation error of traditional methods is often large. Summary of the Invention

[0004] The purpose of this application is to propose a variable harmonic impedance estimation method based on Legendre polynomial fitting, so as to improve the estimation accuracy of harmonic impedance.

[0005] To achieve the above objectives, embodiments of this application provide a method for estimating the impedance of variable harmonic waves based on Legendre polynomial fitting, the method comprising:

[0006] Step S1: Obtain the harmonic voltage and current at the point of common coupling, establish the equivalent relationship at the point of common coupling, and establish the objective function constraint using the harmonic impedance and the system-side harmonic voltage variation characteristics;

[0007] Step S2: Fit the harmonic impedance variation curve with Legendre polynomials, obtain the harmonic impedance estimate by solving through optimization theory, and then determine the optimal Legendre polynomial fitting order using the order screening method.

[0008] Step S3: The impedance abrupt change point is determined by the abrupt change point screening algorithm. The data is segmented according to the impedance abrupt change point. Harmonic impedance is estimated for each segment using steps S1 and S2. Finally, the segmented estimation results are integrated to obtain the final harmonic impedance change curve.

[0009] Preferably, the objective function constraint is:

[0010]

[0011]

[0012] α=[a1,a2,…,a K ] H

[0013] Among them, v pcc It is the voltage of the PCC at the point of common coupling, I pcc It is the current in the PCC at the point of common coupling. represent The changing second norm, minJ(α), represents finding α under specific basis functions such that... Minimum, For Legendre polynomials.

[0014] Preferably, step S2 includes:

[0015] Step S21: The basis functions are Legendre polynomials, which are usually represented by P... n (x) represents the recursive formula as follows:

[0016] (n+1)P n+1 = (2n+1)xP n (x)-nP n-1 (x)

[0017] The specific expression is as follows

[0018]

[0019] Substituting the Legendre polynomial into φ(t), then,

[0020]

[0021] Step S22, will Substitution middle, Differentiating with respect to α, we get:

[0022]

[0023] Representing I pcc Taking the conjugate transpose of the given expression and setting it to zero, we obtain:

[0024]

[0025] According to optimization theory, the optimal estimate of α that minimizes J(α)J is obtained at this point, and thus the harmonic impedance of the system can be obtained. Optimal estimation of harmonic voltage

[0026] Step S23: Determine the optimal order K by traversing the sequence. best Starting from 1, the order is continuously increased, and the corresponding harmonic impedance estimate is obtained based on α. After reaching a certain point, the harmonic impedance estimation error tends to level off or even slightly increases; this is the optimal order K. best K can be determined based on this feature. best The order is determined as follows:

[0027]

[0028] η(p) represents the average difference between the two estimates. and These represent the estimated harmonic impedance values ​​when K = p and K = p+1, respectively. It is set that K is reached when η(p) first becomes less than σ1. best Point σ1 is a preset value.

[0029] Preferably, step S3 includes:

[0030] Step S31: Identify mutation points and solve the problem in segments, based on... Calculate the C(i) function, divide the entire time interval into N-2*win segments, and the sequence of each time interval is (t) i-win ,…,t i ,…,t i+win (i∈(win+1,N-win)), where win is the window length, calculate each segment separately. The variance, C(i), is calculated using the following function:

[0031]

[0032] Where var represents the variance of the data segment, the mutation point is determined by finding the maximum value of C(i), and win is the preset value;

[0033] Step S32, modify C(i) as follows:

[0034]

[0035] Where, medium(C(i)) represents the median of the C(i) function, σ2 is the preset screening threshold, the mutation segment is obtained by the screening criterion above, and the location estimate of the mutation point is obtained by selecting the maximum value of each screening segment;

[0036] Step S33: Remove the located abrupt change point and several surrounding points, and then re-evaluate the harmonic impedance for each data segment. The connection point is replaced by the average value of the two endpoints to obtain the final harmonic impedance estimate.

[0037] Preferably, the value of σ1 is 0.5%-1%.

[0038] Preferably, win is 10-20; σ2 is 10-15.

[0039] Compared with the prior art, the embodiments of this application have the following advantages and beneficial effects:

[0040] (1) The embodiment of this application is based on the Legendre polynomial fitting method for estimating variable harmonic impedance. It assumes that the harmonic impedance is a curve that changes with time, which is more in line with the actual situation. The objective function constraint is established according to the law of harmonic voltage change, which can estimate the changing harmonic impedance of the system side. Compared with the traditional method, the embodiment of this application estimates the variable system harmonic impedance more accurately.

[0041] (2) The variable harmonic impedance estimation method based on Legendre polynomial fitting in the embodiments of this application is less affected by the background harmonic level and the magnitude of the harmonic impedance on both sides.

[0042] (3) The variable harmonic impedance estimation method based on Legendre polynomial fitting in this application embodiment can identify the system-side harmonic impedance mutation points in the power system. By fitting the impedance mutation points in segments, the application scenarios of this application embodiment are effectively expanded.

[0043] Other features and advantages of the embodiments of this application will be set forth in the following description. Attached Figure Description

[0044] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the accompanying drawings required in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0045] Figure 1 This is a flowchart illustrating a method for estimating the impedance of a variable harmonic wave based on Legendre polynomial fitting, as described in an embodiment of this application.

[0046] Figure 2 This is the equivalent circuit diagram at PCC in the embodiments of this application.

[0047] Figure 3 This is a flowchart illustrating a variable harmonic impedance estimation method based on Legendre polynomial fitting in an embodiment of this application.

[0048] Figure 4 This is a schematic diagram of the simulation circuit in the embodiments of this application.

[0049] Figure 5This is a schematic diagram illustrating the simulation experiment effect in an embodiment of this application;

[0050] Figure 6 This is a schematic diagram illustrating another simulation experiment effect in an embodiment of this application. Detailed Implementation

[0051] The detailed description of the accompanying drawings is intended to illustrate the present preferred embodiments of this application and is not intended to represent only the forms in which this application can be implemented. It should be understood that the same or equivalent functions can be achieved by different embodiments intended to be included within the spirit and scope of this application.

[0052] like Figure 1 As shown, the method for estimating the impedance of variable harmonic waves based on Legendre polynomial fitting includes the following steps:

[0053] Step S1: Obtain the harmonic voltage and current at the point of common coupling, establish the equivalent relationship at the point of common coupling, and establish the objective function constraint using the harmonic impedance and the system-side harmonic voltage variation characteristics;

[0054] Step S2: Fit the harmonic impedance variation curve with Legendre polynomials, obtain the harmonic impedance estimate by solving through optimization theory, and then determine the optimal Legendre polynomial fitting order using the order screening method.

[0055] Step S3: The impedance abrupt change point is determined by the abrupt change point screening algorithm. The data is segmented according to the impedance abrupt change point. Harmonic impedance is estimated for each segment using steps S1 and S2. Finally, the segmented estimation results are integrated to obtain the final harmonic impedance change curve.

[0056] Furthermore, in the variable harmonic impedance estimation method based on Legendre polynomial fitting, the objective function constraint establishment in step S1 includes the following steps:

[0057] Step S11, according to Figure 2 The equivalent circuit diagram at PCC is shown. The relationship between the harmonic voltage and harmonic current at PCC is as follows:

[0058] V s =V pcc +I pcc Z s (1)

[0059] Among them, Z s and Z c These are the equivalent harmonic impedances on the system side and the user side, respectively; V s The system measures the equivalent harmonic voltage, I c This is the user-side equivalent harmonic current. V pcc and I pcc These represent the harmonic voltage and harmonic current at point PCC, respectively. When Zs When the variable is a variable, for ease of calculation, equation (1) is changed to:

[0060] v s =v pcc +I pcc z s (2)

[0061] In the formula, the matrix representation of each variable is as follows:

[0062]

[0063] N represents the total number of sample points, t i (i = 1, 2, ..., N) represent the sampling time points. T represents the matrix transpose.

[0064] Step S12, Z s An approximation is made using a combination of a series of smooth functions, as shown in equation (4).

[0065]

[0066] For continuous real-valued functions, this invention uses Legendre polynomials, a j Let V be its coefficient, and K be the number of fitted functions. s The fluctuations are constrained using the objective function of equation (5). ||Bv s ||2 represents the L2 norm of harmonic voltage variation, which should be minimized.

[0067] minJ=||Bv s ||2 (5)

[0068] V s In the short term, the mean can be assumed to remain constant, and V can be calculated. s The difference between each point and the mean of its interval is used as the fluctuation evaluation for that point. However, the total sampling period is relatively long, and it is obviously unreasonable to use the average value of the entire period for analysis. Therefore, a sliding window is used for calculation, with the window length set to p and B set according to equation (6).

[0069]

[0070] Find the v that minimizes J. s This is the system harmonic voltage to be determined. For ease of calculation, equation (5) is expressed in matrix form.

[0071]

[0072] in, α=[a1,a2,…,a K ] HSubstituting equations (7) and (2) into equation (4), we get...

[0073]

[0074] At this point, the problem is transformed into finding α under specific basis functions that minimizes equation (8). For Legendre polynomials.

[0075] Furthermore, step S2 includes the following sub-steps:

[0076] Step S21: The basis functions are Legendre polynomials, which are usually represented by P... n (x) represents the recursive formula as follows:

[0077] (n+1)P n+1 = (2n+1)xP n (x)-nP n-1 (x) (9)

[0078] The specific expression is as follows

[0079]

[0080] Substituting the Legendre polynomial into φ(t), then,

[0081]

[0082] Step S22, substitute equation (11) into equation (8), and differentiate equation (8) with respect to α to obtain:

[0083]

[0084] Representing I pcc Taking the conjugate transpose of the given expression and setting it to zero, we obtain...

[0085]

[0086] According to the optimization theory, the optimal estimate of α that minimizes J is obtained at this point. Then, the harmonic impedance of the system can be obtained according to equations (7) and (2). Optimal estimation of harmonic voltage

[0087] Step S23: Determine the optimal order K by traversing the sequence. best Starting from 1 and continuously increasing the order, the first harmonic impedance estimate is obtained using step B2. After reaching a certain point, the estimation error tends to level off or even slightly increases; this is the optimal order K. best K can be determined based on this feature. best The order is determined as follows:

[0088]

[0089] η(p) represents the average difference between the two estimates. and These represent the harmonic impedance values ​​when K = p and K = p+1, respectively. It is set that when η(p) first becomes less than σ1, K is reached. best Point. The value of σ1 is generally set to 0.5%-1%.

[0090] Furthermore, step S3 includes the following sub-steps:

[0091] Step S31: Determine the harmonic impedance abrupt change point, and obtain the first system-side equivalent harmonic voltage estimate through step B.

[0092] according to Calculate the C(i) function, divide the entire time interval into N-2*win segments, and the sequence of each time interval is (t) i-win ,…,t i ,…,t i+win (i∈(win+1,N-win)), where win is the window length, calculate each segment separately. The variance, C(i), is calculated using the following function:

[0093]

[0094] var represents the variance of the data segment. The mutation point can be determined by finding the maximum value of C(i).

[0095] Step S32: The existence of outliers will affect the final judgment result, making it difficult to determine the judgment threshold. C(i) is corrected as follows:

[0096]

[0097] Where, medium(C(i)) represents the median of the C(i) function, and σ2 is the screening threshold, which is generally set to 10-15. The mutation segments are obtained through the screening criterion (16). Then, the location estimate of the mutation point is obtained by selecting the maximum value of each screening segment.

[0098] Step S33: Considering the influence of error, the estimation of the mutation point may not be completely accurate. The located mutation point and several surrounding points are removed. Then, the harmonic impedance is re-estimated for each data segment. The connection point is replaced by the average value of the two endpoints to obtain the final harmonic impedance estimate.

[0099] Based on the principles of the above embodiments, this embodiment discloses a specific implementation method:

[0100] Set up Figure 4 The simulation circuit model shown is... Figure 4 Essentially with Figure 2 Equivalent, only the system-side modeling method is different; the system-side harmonic current is represented as I. s =V s / Z s The specific parameter settings are shown in Table 1:

[0101] Table 1Z s Value settings

[0102]

[0103]

[0104] In Z s Add 5% random fluctuation to Z c Set to 30+j40, add 15% random fluctuation to the real and imaginary parts. User-side harmonic I c The amplitude is set to 100A, with a superimposed 30% sinusoidal fluctuation and 20% random fluctuation. For easy comparison of background harmonic magnitude, the system-side harmonic current I is set to... s Amplitude is set to I c By multiplying the amplitude by k, and simultaneously superimposing 20% ​​sinusoidal fluctuation and 10% random fluctuation, different levels of background harmonics can be generated by changing k. c and I s The phase angles were set to 30% and -40%, respectively, and random fluctuations of 20% and 15% were superimposed on them. The number of sampling points was set to 1000 in the simulation.

[0105] The formula for calculating the harmonic impedance estimation error is as follows:

[0106]

[0107] After obtaining the initial harmonic voltage estimate Subsequently, the mutation points were determined to be 200 and 700 using the mutation point discrimination method. The data was then segmented accordingly, and the optimal fitting order for each segment was recalculated. The optimal fitting order for each segment was 1, 10, and 1, respectively. The estimation results for each segment were then obtained, and the results were integrated to obtain the final harmonic impedance estimation result.

[0108] To analyze the effectiveness of the proposed method compared with traditional harmonic impedance estimation methods, four methods were used to estimate the system harmonic impedance: Method 1 is the bivariate regression method, Method 2 is the covariance method, Method 3 is the dominant fluctuation method, and Method 4 is the method proposed in this paper. The estimation errors of these methods were compared. 100 simulations were performed, and the estimation results are as follows: Figure 5 As shown. To compare the influence of the impedance ratio on the two sides on the estimation results, Z is... cSet the value to 100+j200, keep other conditions unchanged, and recalculate the estimation error of the above method as k changes, such as... Figure 6 As shown.

[0109] from Figure 5 and Figure 6 As can be clearly seen, the method in this embodiment has lower error compared to the traditional method, and is less affected by the background harmonic level and the magnitude of the harmonic impedance on both sides. It can better track the changes in harmonic impedance, and the error level is low under various conditions, making it suitable for a wide range of applications.

[0110] The various embodiments of this application have been described above. These descriptions are exemplary and not exhaustive, nor are they limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terminology used herein is chosen to best explain the principles, practical applications, or technological improvements to the embodiments in the market, or to enable others skilled in the art to understand the embodiments disclosed herein.

Claims

1. A method for estimating the impedance of variable harmonic waves based on Legendre polynomial fitting, characterized in that, The method includes: Step S1: Obtain the harmonic voltage and harmonic current at the point of common coupling, establish the equivalent relationship at the point of common coupling, and establish the objective function constraint using the harmonic impedance and the system-side harmonic voltage variation characteristics. The objective function constraint is: in, It is the voltage of the PCC at the point of common coupling. It is the current in the PCC at the point of common coupling. represent The changing second norm, This indicates finding the answer under specific basis functions. ,make Minimum, For Legendre polynomials, It is a sliding window; Step S2: Fit the harmonic impedance variation curve with Legendre polynomials, obtain the harmonic impedance estimate by solving through optimization theory, and then determine the optimal Legendre polynomial fitting order using the order screening method. Step S2 includes: Step S21, the basis functions are Legendre polynomials, which are used in... The recursive formula is as follows: The specific expression is: Substituting the Legendre polynomial ,at this time, Step S22, will Substitution middle, right Taking the derivative, we get: represent Taking the conjugate transpose of the given expression and setting it to zero, we obtain: According to optimization theory, this yields the result that... smallest The optimal estimate is obtained, and thus the system-side harmonic impedance is obtained. Optimal estimation of harmonic voltage ; Step S23: Determine the optimal order by traversing the sequence. Starting from 1, the order is continuously increased, and according to... Obtain the corresponding harmonic impedance estimate Once a certain point is reached, the harmonic impedance estimation error tends to level off or even slightly increase; this is the optimal order. Based on this feature, determine The order is determined as follows: This represents the average difference between the two estimates. and Represent and Time harmonic impedance estimate, set when The first time less than When, reach point, This is the default value; Step S3: The impedance abrupt change point is determined by the abrupt change point screening algorithm. The data is segmented according to the impedance abrupt change point. Harmonic impedance is estimated for each segment using steps S1 and S2. Finally, the segmented estimation results are integrated to obtain the final harmonic impedance change curve. Step S3 includes: Step S31: Identify mutation points and solve the problem in segments, based on... calculate The function divides the entire time period into... Segment, each time interval sequence is , Given the window length, calculate the value for each segment separately. variance The function is calculated as follows: Where var represents the variance of the data segment, which is determined by finding... The maximum value determines the mutation point, and win is the preset value; Step S32, for The correction method is as follows: in, represent The median of the function, Using the preset screening threshold, the mutation segments are obtained through the screening criteria in the above formula. The location of the mutation point is estimated by selecting the maximum value of each screening segment. Step S33: Remove the located abrupt change point and several surrounding points, and then re-evaluate the harmonic impedance for each data segment. The connection point is replaced by the average value of the two endpoints to obtain the final harmonic impedance estimate.

2. The method for estimating the impedance of variable harmonic waves based on Legendre polynomial fitting according to claim 1, characterized in that, The value is 0.5%-1%.

3. The method for estimating the impedance of variable harmonic waves based on Legendre polynomial fitting according to claim 2, characterized in that, It is 10-20; It is 10-15.

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