CN109709378B - Frequency and Amplitude Adaptive Algorithm for Transient Electrical Signals - Google Patents
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Abstract
本发明公开了一种瞬变电信号的频率和幅值自适应算法，包括以下步骤：在时域上先对采样的电信号进行三次样条插值，获得插值序列后按照采样间隔进行数据截断，构建重构样本，最后采用Prony算法估算出各重构样本的频率幅值。实验结果验证，该算法快速准确地实现了电网频率幅值的同步自适应跟踪；本发明适应在电网信号在频率和幅值同时随时间变化情况下电参数的同步检测，算法准确度高，为电网三要素精准测量和分析提供一种可靠、有效的技术支持。
The invention discloses an adaptive algorithm for the frequency and amplitude of a transient electrical signal, comprising the following steps: firstly performing cubic spline interpolation on the sampled electrical signal in the time domain; The reconstructed samples are constructed, and the Prony algorithm is used to estimate the frequency amplitude of each reconstructed sample. The experimental results verify that the algorithm quickly and accurately realizes the synchronous adaptive tracking of the frequency and amplitude of the power grid; the invention is suitable for the synchronous detection of electrical parameters when the frequency and amplitude of the power grid signal change with time at the same time, and the algorithm has high accuracy and is The precise measurement and analysis of the three elements of the power grid provides a reliable and effective technical support.
Description
Technical Field
The invention relates to the field of time-varying electric signal analysis of an electric power system, in particular to a frequency and amplitude self-adaptive algorithm of a transient electric signal.
Background
The frequency and the amplitude of the power system are important parameter indexes for measuring the quality of the electric energy, not only are feedback quantity for realizing stability control of the power system, but also serve as a reference for accurate action of the relay protection device, and therefore the method has important significance for accurate measurement and analysis of the frequency and the amplitude of the power grid.
With the proposal of the smart grid concept, the access of a large-scale distributed power supply and a nonlinear load causes the voltage and current frequency amplitude of the power grid to be in a time-varying characteristic, and the power grid is an unstable time-varying power signal.
From the papers published at home and abroad at present, there are many research achievements about real-time tracking of power signal frequency under the unstable time-varying signal condition, and the main methods are as follows: wavelet analysis, least square, taylor's formula, kalman filtering, etc. However, the above algorithms are all established on the basis of a standard sinusoidal signal with constant amplitude, and when the amplitude changes greatly, the calculation accuracy will be obviously reduced, and effective tracking under the condition of synchronous time-varying frequency and amplitude cannot be realized.
Under the condition of unsteady time-varying signals generated by distributed energy grid connection, the research on synchronous real-time tracking of frequency and amplitude is still in a blank stage, and efficient exploration on the aspects of theory, algorithm and the like is urgently needed, and an effective solution is provided.
Disclosure of Invention
The invention aims to solve the technical problem of providing a frequency and amplitude self-adaptive algorithm of a transient electric signal, which can quickly and accurately realize synchronous self-adaptive tracking of the frequency amplitude of a power grid.
In order to solve the technical problems, the invention adopts a technical scheme that: there is provided a frequency and amplitude adaptive algorithm for a transient electrical signal, comprising the steps of:
s1: after being filtered by an anti-aliasing analog filter, the electric power system signal x (t) takes the sampling frequency as fsSampling duration of tsSampling to obtain N discrete electrical signals x (N), wherein N ═ fs*tsWith a sampling interval of Ts＝1/fs；
S2: establishing a cubic spline interpolation function to interpolate the discrete electric signal x (N), so as to obtain an interpolation sequence x' (k) containing interpolation points, wherein k is 1,2, ┄, (M-2) (N-1), and M is the number of interpolation points between two adjacent sampling points;
s3: truncating the interpolation sequence x' (k) according to sampling intervals to obtain (N-1) equal-interval reconstruction samples, and recording the samples as xi(M), wherein i ═ 1,2, ┄, N-1, M ═ 1,2, ┄, M;
s4: estimation of (N-1) reconstructed sample sequences x using the Prony algorithmiFrequency and amplitude of (m).
In a preferred embodiment of the present invention, in step S2, the step of establishing the cubic spline interpolation function is:
given interval [ a, b]Function f above and a set of nodes a ═ x0<x1<┄<xqB, the cubic spline interpolation S of the function f is a function satisfying the following condition:
condition 1: for the subinterval [ x ]j,xj+1](j is 0,1,2, ┄, q-1), and S (x) is a cubic polynomial in the subinterval denoted as Sj(x)；
Condition 2: s (x)j)＝f(xj)，(j＝0,1,2,┄,q-1)；
Condition 3: sj+1(xj+1)＝Sj+1(xj+1)，(j＝0,1,2,┄,q-2)；
Condition 4: s'j+1(xj+1)＝S′j(xj+1)，(j＝0,1,2,┄,q-2)；
Condition 5: s ″)j+1(xj+1)＝S″j(xj+1)，(j＝0,1,2,┄,q-2)；
One of the boundary conditions is satisfied, S ″ (x)0)＝S″(xq)＝0；②S′(x0)＝f′(x0) And S' (x)q)＝f′(xq)。
In a preferred embodiment of the present invention, the step S2 includes the following steps:
s2.1: the interpolated function f corresponds to the signal x (t) of the power system, the interpolated function point f (x)j) Discrete electrical signals x (n) obtained by sampling corresponding to the power system;
s2.2: difference h between arbitrary cellsj＝xj+1-xj(j ═ 1,2, ┄, q-1), corresponding to sampling interval Ts;
s2.3: determining the number M of interpolation points between two adjacent sampling points;
s2.4: after the cubic spline interpolation function is obtained, the independent variable x is sequentially set to 0, Ts,2Ts,3Ts, ┄, (N-1) MTs, and a new interpolation sequence x' (k) is obtained.
In a preferred embodiment of the present invention, the step S4 includes the following steps:
s4.1: setting the transient power system signal as
Reconstructed sample x after sampling interpolation reconstruction and truncation processingi(m) constructing an extended Prony detection model for a group of harmonic signals with p random amplitudes, phases and frequencies, wherein the discrete time function form of the extended Prony detection model is as follows:
where M is 0,1,2, …, M-1, p is the rank of the model matrix, ajIs the amplitude, fjIs the frequency, thetajIs a phase，αjAn attenuation factor;
s4.2: the Prony algorithm utilizes the principle of error square sum minimum to realize model parameter estimation and constructs a cost function, namely:
s4.3: calculating a Prony detection model, and solving the amplitude and the frequency of the transient electric signal at a certain moment;
s4.4: and repeating the steps S4.1 to S4.3, and solving the amplitude and the frequency of the transient electric signal at each moment.
Further, the calculation process of step S4.3 includes:
s4.3.1: calculating a reconstructed sample function R (i, j) and constructing an expansion matrix RiDetermining RiAn effective rank p;
s4.3.2: establishing a linear matrix equation and solving the parameter aj：Ri[1,a1,…,ap]T＝[ξi,0,…,0]TIn which epsilonpiFor minimum error energy:
s4.3.3: solving the characteristic root z of a polynomialj：1+a1z-1+…+apz-p＝0；
S4.3.4: solving for the amplitude and frequency of the transient signal: a. thej＝2|aj|，fj＝arctan[Im(zj)/Re(zj)]/(2πTs)。
Further, in step (ii)S4.4, the amplitude and frequency of the transient electrical signal at each time instant is a ═ a1 Ai … AN-1]，f＝[f1 fi … fN-1]Wherein A isi＝[2|a1| 2|aj| … 2|ap|]，
The invention has the beneficial effects that:
(1) in the invention, cubic spline interpolation is firstly carried out on a sampled electric signal in a time domain, an interpolation sequence is obtained, then data truncation is carried out according to a sampling interval, a reconstructed sample is constructed, and finally a Prony algorithm is adopted to estimate the frequency amplitude of each reconstructed sample. The experimental result verifies that the algorithm quickly and accurately realizes the synchronous self-adaptive tracking of the frequency amplitude of the power grid;
(2) the method is suitable for synchronous detection of the electrical parameters of the power grid signals under the condition that the frequency and the amplitude of the power grid signals simultaneously change along with time, has high algorithm accuracy, and provides a reliable and effective technical support for accurate measurement and analysis of three elements of the power grid.
Drawings
FIG. 1 is a flow chart of a frequency and amplitude adaptive algorithm of a transient electrical signal of the present invention;
FIG. 2 is a schematic illustration of the reconstructed sample;
FIG. 3 is a graph of amplitude tracking for the first embodiment of the present invention;
FIG. 4 is a graph of frequency tracking for the first embodiment of the present invention;
FIG. 5 is a graph of amplitude tracking error in accordance with a first embodiment of the present invention;
FIG. 6 is a graph of frequency tracking error according to a first embodiment of the present invention;
FIG. 7 is a graph of amplitude tracking for a second embodiment of the present invention;
FIG. 8 is a graph of frequency tracking for a second embodiment of the present invention;
FIG. 9 is a graph of amplitude tracking error for a second embodiment of the present invention;
fig. 10 is a frequency tracking error graph according to the second embodiment of the present invention.
Detailed Description
The following detailed description of the preferred embodiments of the present invention, taken in conjunction with the accompanying drawings, will make the advantages and features of the invention easier to understand by those skilled in the art, and thus will clearly and clearly define the scope of the invention.
Referring to fig. 1, an embodiment of the present invention includes:
a frequency and amplitude adaptive algorithm for transient electrical signals comprising the steps of:
s1: before an AD sampling module, an anti-aliasing analog filter is additionally arranged on a power system signal x (t), so that the frequency spectrum leakage of a sampling signal is reduced to the maximum extent, and the sampling frequency is fs(unit: Hz), sampling duration tsSampling a power system signal x (t) (unit: V or A) to obtain an N-point discrete electric signal x (N), wherein N ═ fs*tsWith a sampling interval of Ts＝1/fs；
S2: establishing a cubic spline interpolation function to interpolate the discrete electric signal x (N), so as to obtain an interpolation sequence x' (k) containing interpolation points, wherein k is 1,2, ┄, (M-2) (N-1), and M is the number of interpolation points between two adjacent sampling points;
the method comprises the following steps of:
given interval [ a, b]Function f above and a set of nodes a ═ x0<x1<┄<xqB, the cubic spline interpolation S of the function f is a function satisfying the following condition:
condition 1: for the subinterval [ x ]j,xj+1](j is 0,1,2, ┄, q-1), and S (x) is a cubic polynomial in the subinterval denoted as Sj(x)；
Condition 2: s (x)j)＝f(xj)，(j＝0,1,2,┄,q-1)；
Condition 3: sj+1(xj+1)＝Sj+1(xj+1)，(j＝0,1,2,┄,q-2)；
Condition 4: s'j+1(xj+1)＝S′j(xj+1)，(j＝0,1,2,┄,q-2)；
Condition 5: s ″)j+1(xj+1)＝S″j(xj+1)，(j＝0,1,2,┄,q-2)；
One of the boundary conditions is satisfied, S ″ (x)0)＝S″(xq)＝0；②S′(x0)＝f′(x0) And S' (x)q)＝f′(xq)。
The specific steps of applying the cubic spline interpolation in the power signal sample reconstruction include:
s2.1: the interpolated function f corresponds to the signal x (t) of the power system, the interpolated function point f (x)j) Discrete electrical signals x (n) obtained by sampling corresponding to the power system;
s2.2: difference h between arbitrary cellsj＝xj+1-xj(j ═ 1,2, ┄, q-1), corresponding to sampling interval Ts;
because the signals of the power system adopt real-time sampling and equivalent time sampling, and the sampling interval Ts is fixed, hjThe value of (A) is also constant;
s2.3: determining the number M of interpolation points between two adjacent sampling points;
s2.4: after the cubic spline interpolation function is obtained, the independent variable x is sequentially set to 0, Ts,2Ts,3Ts, ┄, (N-1) MTs, and a new interpolation sequence x' (k) is obtained.
S3: truncating the interpolation sequence x' (k) according to sampling intervals to obtain (N-1) equal-interval reconstruction samples, and recording the samples as xi(M), wherein i ═ 1,2, ┄, N-1, M ═ 1,2, ┄, M; the method comprises the following specific steps:
sequentially combining two adjacent sampling points (shown as triangles in figure 2) in the new interpolation sequence x '(k) and (M-2) interpolation points (shown as pentagons in figure 2) between the two adjacent sampling points into a reconstruction sample (except for the initial sampling point and the tail sampling point, the rest sampling points are used twice), and finally, substantially truncating the interpolation sequence x' (k) according to sampling intervals to obtain (N-1) equidistant reconstruction samples which are recorded as xi(M), wherein i is 1,2, ┄, N-1, M is 1,2, ┄, M.
S4: estimation of (N-1) reconstructed sample sequences x using the Prony algorithmiFrequency and amplitude of (m). This step includes the construction and calculation of a test model,the method comprises the following specific steps:
s4.1: setting the transient power system signal as
Reconstructed sample x after sampling interpolation reconstruction and truncation processingi(m) constructing an extended Prony detection model for a group of harmonic signals with p random amplitudes, phases and frequencies, wherein the discrete time function form of the extended Prony detection model is as follows:
where M is 0,1,2, …, M-1, p is the rank of the model matrix, ajIs the amplitude, fjIs the frequency, thetajIs a phase, αjAn attenuation factor;
s4.2: the Prony algorithm utilizes the principle of error square sum minimum to realize model parameter estimation and constructs a cost function, namely:
s4.3: calculating a Prony detection model, and solving the amplitude and the frequency of the transient electric signal at a certain moment;
further, the calculation process of step S4.3 includes:
s4.3.1: calculating a reconstructed sample function R (i, j) and constructing an expansion matrix RiDetermining RiAn effective rank p;
s4.3.2: establishing a linear matrix equation and solving the parameter aj：Ri[1,a1,…,ap]T＝[ξi,0,…,0]TIn which epsilonpiFor minimum error energy:
s4.3.3: solving the characteristic root z of a polynomialj：1+a1z-1+…+apz-p＝0；
S4.3.4: solving for the amplitude and frequency of the transient signal: a. thej＝2|aj|，fj＝arctan[Im(zj)/Re(zj)]/(2πTs)。
S4.4: and repeating the steps S4.1 to S4.3, solving the amplitude and the frequency of the transient electric signal at each moment: a ═ A1 Ai… AN-1]，f＝[f1 fi … fN-1]Wherein A isi＝[2|a1| 2|aj| … 2|ap|]，
In the field of power system control, the transfer function is generally in the form of
Wherein m and n are positive integers and n>m, the time domain form of the transfer function without the heavy root is:
therefore, the frequency and the amplitude of the constructed signal are transient according to the e-t and te-t rules, and the tracking performance of the algorithm under an MATLAB simulation platform is given.
The invention describes the tracking performance of the algorithm on the frequency and amplitude of the transient electric signal by two embodiments:
signal 1 is: x (t) ═ A1(t)cos(2πf1(t)t+45°)+A2(t)cos(2πf2(t) t +45 °), amplitudes a1, a2 are:
the frequencies f1 and f2 are:
the sampling interval is 0.001s, the sampling length N is 100, and the reconstructed sample length M is 1000. Algorithm frequency and amplitude tracking curves and error curves are plotted as shown in fig. 3-6.
As can be seen from the frequency and amplitude tracking curves and the error curves of the two experiments, the algorithm well realizes the frequency and amplitude synchronous tracking of the transient signal. Comparing the first and second embodiments, the tracking effect is different for different signal algorithms: (1) the overall tracking effect of the second embodiment is better than that of the first embodiment because the signal variation amplitude of the second embodiment is larger than that of the first embodiment; (2) the frequency tracking effect of the first and second embodiments is better than the amplitude tracking effect because the Prony algorithm is more sensitive to frequency.
After cubic spline interpolation reconstruction is carried out on a sampling signal of the electric power system in a time domain, data truncation is carried out according to a sampling interval, and an obtained reconstructed sample has three advantages: firstly, the truncated equidistant data sequence is not only strictly cycle truncated, but also contains an integer number of reconstruction points in each cycle; secondly, the cubic spline interpolation value has small signal reconstruction error, and the frequency spectrum and the amplitude hardly leak, so that a foundation is provided for realizing accurate detection of Prony parameters; thirdly, the length of the reconstructed sample can be reduced by reducing the length of the truncated reconstructed sample, so that the sampling interval of the software is reduced, the time interval is ensured to be small enough, the accuracy of the algorithm is improved, and the cost of sampling equipment is reduced. And finally, estimating the frequency amplitude of each reconstructed sample by adopting a Prony algorithm. Experimental results prove that the algorithm quickly and accurately realizes the synchronous self-adaptive tracking of the frequency amplitude of the power grid.
The above description is only an embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes performed by the present specification and drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.
Claims (5)
1. A frequency and amplitude adaptive algorithm for transient electrical signals comprising the steps of:
s1: after being filtered by an anti-aliasing analog filter, the electric power system signal x (t) takes the sampling frequency as fsSampling duration of tsSampling to obtain N discrete electrical signals x (N), wherein N ═ fs*tsWith a sampling interval of Ts＝1/fs；
S2: establishing a cubic spline interpolation function to interpolate the discrete electric signal x (N), so as to obtain an interpolation sequence x' (k) containing interpolation points, wherein k is 1,2, - - -, (M-2) (N-1), and M is the number of the interpolation points between two adjacent sampling points;
s3: truncating the interpolation sequence x' (k) according to a sampling interval to obtain (N-1) equal-interval reconstruction samples, which are denoted as xi (M), wherein i is 1,2, N-1, M is 1,2, M;
s4: estimation of (N-1) reconstructed sample sequences x using the Prony algorithmi(m) frequency and amplitude; the method comprises the following specific steps:
s4.1: setting the transient power system signal as
Reconstructed sample x after sampling interpolation reconstruction and truncation processingi(m) constructing an extended Prony detection model for a group of harmonic signals with p random amplitudes, phases and frequencies, wherein the discrete time function form of the extended Prony detection model is as follows:
where M is 0,1,2, …, M-1, p is the rank of the model matrix, ajIs the amplitude, fjIs the frequency, thetajIs a phase, αjAn attenuation factor;
s4.2: the Prony algorithm utilizes the principle of error square sum minimum to realize model parameter estimation and constructs a cost function, namely:
s4.3: calculating a Prony detection model, and solving the amplitude and the frequency of the transient electric signal at a certain moment;
s4.4: and repeating the steps S4.1 to S4.3, and solving the amplitude and the frequency of the transient electric signal at each moment.
2. The adaptive algorithm for frequency and amplitude of transient electric signal according to claim 1, wherein in step S2, the step of establishing said cubic spline interpolation function is:
given interval [ a, b]Function f above and a set of nodes a ═ x0＜x1＜---＜xqB, the cubic spline interpolation S of the function f is a function satisfying the following condition:
condition 1: for the subinterval [ x ]j，xj+1](j is 0,1, 2' -, q-1), S (x) is a cubic polynomial of the subinterval, denoted as Sj(x)；
Condition 2: s (x)j)＝f(xj)，(j＝0，1，2，---，q-1)；
Condition 3: sj+1(xj+1)＝Sj+1(xj+1)，(j＝0，1，2，---，q-2)；
Condition 4: s'j+1(xj+1)＝S′j(xj+1)，(j＝0，1，2，---，q-2)；
Condition 5:S″j+1(xj+1)＝S″j(xj+1)，(j＝0，1，2，---，q-2)；
one of the boundary conditions is satisfied, S ″ (x)0)＝S″(xq)＝0；②S′(x0)＝f′(x0) And S' (x)q)＝f′(xq)。
3. The adaptive algorithm for frequency and amplitude of transient electric signal according to claim 1, wherein the step S2 comprises the following steps:
s2.1: the interpolated function f corresponds to the signal x (t) of the power system, the interpolated function point f (x)j) Discrete electrical signals x (n) obtained by sampling corresponding to the power system;
s2.2: difference h between arbitrary cellsj＝xj+1-xj(j-1, 2, -q-1), corresponding to the sampling interval Ts;
s2.3: determining the number M of interpolation points between two adjacent sampling points;
s2.4: after the cubic spline interpolation function is solved, the independent variable x is sequentially set to 0, Ts,2Ts,3Ts, - - -, (N-1) MTs, and a new interpolation sequence x' (k) is obtained.
4. Frequency and amplitude adaptive algorithm for transient electrical signals according to claim 1, characterized in that the calculation procedure of step S4.3 comprises:
s4.3.1: calculating a reconstructed sample function R (i, j) and constructing an expansion matrix RiDetermining RiAn effective rank p;
s4.3.2: establishing a linear matrix equation and solving the parameter aj：Ri[1，a1，…，ap]T＝[ξi，0，…，0]TIn which epsilonpiFor minimum error energy:
s4.3.3: solving the characteristic root z of a polynomialj：1+a1z-1+…+apz-p＝0；
S4.3.4: solving for the amplitude and frequency of the transient signal: a. thej＝2|aj|，fj＝arctan[Im(zj)/Re(zj)]/(2πTs)。
5. Frequency and amplitude adaptive algorithm for transient electrical signals according to claim 1, characterized in that in step S4.4, the amplitude and frequency of the transient electrical signal at each moment is a ═ a1 Ai…AN-1]，f＝[f1 fi…fN-1]Wherein
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