CN107390022B - Electric energy metering method based on discrete spectrum correction - Google Patents

Electric energy metering method based on discrete spectrum correction Download PDF

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CN107390022B
CN107390022B CN201710729394.8A CN201710729394A CN107390022B CN 107390022 B CN107390022 B CN 107390022B CN 201710729394 A CN201710729394 A CN 201710729394A CN 107390022 B CN107390022 B CN 107390022B
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CN107390022A (en
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周杰文
任智仁
汪龙峰
黄杰
杨辉
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Wasion Group Co Ltd
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    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R22/00—Arrangements for measuring time integral of electric power or current, e.g. electricity meters
    • G01R22/06—Arrangements for measuring time integral of electric power or current, e.g. electricity meters by electronic methods
    • G01R22/10—Arrangements for measuring time integral of electric power or current, e.g. electricity meters by electronic methods using digital techniques

Abstract

The invention discloses an electric energy metering method based on discrete spectrum correction, which comprises the steps of carrying out time domain truncation on voltage and current signals converted into digital quantity by adopting a Nuttall four-term third-order window with the window length of N; calculating the spectrum centroid and the fundamental frequency of the fundamental by adopting a spectrum centroid formula; estimating the position of a harmonic peak value, and calculating the centroid and the fundamental wave of a harmonic spectrum and the maximum spectral peak position and the offset of each harmonic; determining a fundamental wave harmonic amplitude correction coefficient, and calculating fundamental waves, harmonic amplitudes and phases of the voltage and the current to be measured; and calculating the electrical parameters of fundamental waves and harmonic waves to finish electric energy metering. Therefore, the method does not need to carry out peak value search, does not need to calculate the offset according to the type of the added window function, omits zero crossing point detection, directly adopts an FFT analysis method to carry out calculation, omits complex window spectrum operation, greatly improves the frequency precision obtained by calculation and analysis, and has simple program implementation and good real-time property.

Description

Electric energy metering method based on discrete spectrum correction
Technical Field
The invention particularly relates to an electric energy metering method based on discrete spectrum correction.
Background
With the development of economic technology and the improvement of living standard of people, electric energy becomes essential secondary energy in daily production and life of people, and brings endless convenience to production and life of people.
The harmonic waves of the power system have great influence on the accuracy of electric energy metering, the mode of respectively metering fundamental wave electric energy and harmonic wave electric energy is reasonable, and the requirement of the market on the metering precision of the fundamental wave harmonic wave is higher and higher along with the development of the technology. According to the current development in the field of electric energy metering, the design schemes of harmonic wave metering mainly include two types: one is a special metering chip, and the other is DSP + special AD + MCU. The former scheme has lower cost and is mainly applied to the design of common single-phase and three-phase meters; the latter has relatively high cost and is mainly applied to a gateway meter platform, the gateway meter is mainly used for metering points with frequently changed loads, such as regional power grids, provincial power grids, photovoltaic power, wind power, hydropower stations, electrified railways and the like, and in the face of the occasions with large harmonic content and variable loads, compared with the former, the scheme has more flexible functions and can more easily meet complex requirements.
The customs meter is a mature product, and not only requires higher electric energy metering precision, but also requires lower operation power consumption. At present, a plurality of algorithms have been proposed for high-precision measurement of the amplitude, phase and frequency of fundamental waves and harmonic waves, and a considerable part of the algorithms are difficult to meet the real-time requirement in an embedded system due to complex implementation, and the more complex the implementation is, the more the system operation power consumption can be correspondingly improved, so that the algorithm which is high in measurement precision and easy to implement and meets the requirements of the existing hardware platform on complex time and space is especially important.
Disclosure of Invention
The invention aims to provide an electric energy metering method based on discrete spectrum correction, which has relatively high precision, good real-time performance and simple algorithm.
The invention provides an electric energy metering method based on discrete spectrum correction, which comprises the following steps:
s1, converting voltage and current signals into digital quantities through sampling, and performing time domain truncation on the converted digital quantities by adopting a Nuttall four-term third-order window with the window length of N to obtain a spectrum signal Y (k) of a truncated signal; the spectrum signal Y (k) comprises a voltage spectrum value Yu(k) Sum current spectral value Yi(k)；
S2, obtaining the voltage frequency spectrum value Y according to the step S1u(k) Calculating to obtain the spectrum centroid k of the fundamental wave by adopting a spectrum centroid formulasc1And fundamental frequency f0；
S3, adopting the fundamental frequency f obtained in the step S20Estimating harmonic peak locations
Figure BDA0001386768130000023
And calculating by adopting a spectrum centroid formula to obtain harmonic spectrum centroid kschAnd further calculating to obtain the maximum spectral peak position k of fundamental wave and each subharmonichAnd offset lambdah；
S4, determining a fundamental harmonic amplitude correction coefficient g (lambda) according to the offset obtained in the step S3h) Calculating to obtain fundamental wave, harmonic amplitude and phase of the voltage and current to be measured;
s5, calculating the electrical parameters of the fundamental wave and the harmonic wave according to the measured voltage, the fundamental wave of the current, the harmonic amplitude and the phase obtained in the step S4, and thus completing the measurement of the electrical energy.
The step S1 of obtaining the spectrum signal y (k) of the truncated signal is specifically to obtain the spectrum signal y (k) of the truncated signal by FFT analysis.
The nuttally quadrinomial third-order window with the window length of N in step S1, specifically, the main lobe width of the nuttally quadrinomial third-order window is
Figure BDA0001386768130000021
There are 8 spectral lines in the main lobe.
The Nuttall four-term third-order window with the window length of N meets the following formula:
Figure BDA0001386768130000022
wherein N is 1,2, …, N-1; bmIs a window function coefficient, satisfies
Figure BDA0001386768130000031
Calculating to obtain the centroid k of the fundamental spectrum in step S2sc1And fundamental frequency f0The method specifically comprises the following steps:
A. calculating to obtain a fundamental spectrum centroid k according to a spectrum centroid formulasc1：
Figure BDA0001386768130000032
In the formula
Figure BDA0001386768130000033
And is
Figure BDA0001386768130000034
For the rounding function, fmin is the minimum of the fundamental frequency of the signal, fmax is the maximum of the fundamental frequency of the signal, and Δ f ═ fsN, fs is the sampling frequency of analog-to-digital conversion, and N is the number of analysis points of the FFT algorithm;
B. according to the fundamental spectrum centroid k obtained in the step Asc1The fundamental frequency f is calculated and measured0：
f0＝ksc1*Δf。
Calculating to obtain harmonic spectrum centroid k in step S3schMaximum spectral peak position k of fundamental wave and each subharmonichAnd offset lambdahThe method specifically comprises the following steps:
a. the harmonic spectrum peak position is estimated by the following formula
Figure BDA0001386768130000035
Figure BDA0001386768130000036
Wherein h is 2,3, …, M; m is the maximum harmonic number of the signal;
b. obtained according to step a
Figure BDA0001386768130000037
Calculating to obtain harmonic spectrum centroid k by adopting the following formulasch：
Figure BDA0001386768130000038
c. C, obtaining harmonic spectrum centroid k according to step bschCalculating the maximum spectral peak position k of fundamental wave and each subharmonic by the following formulah：
kh＝round(ksch)
Where round () is a rounding function;
d. according to the maximum spectrum peak position k obtained in the step chCalculating to obtain offset lambdah：
λh＝kh-ksch。
Determining the fundamental wave amplitude correction factor g (λ) as set forth in step S4h) Specifically, the following formula is adopted for calculation:
Figure BDA0001386768130000041
wherein p is1～p5Are all constants.
Step S4, calculating to obtain the fundamental wave, harmonic amplitude and phase of the measured voltage and current, specifically, correcting the amplitude and phase by using the following correction formulas:
A＝|Y(kh)|g(λh)
Figure BDA0001386768130000042
where A is the corrected amplitude, theta is the corrected phase, angle (Y (k)h) Is Y (k)h) The phase value of (a).
Step S5, calculating the electrical parameters of the fundamental wave and the harmonic wave, specifically, calculating the electrical parameters of the fundamental wave and the harmonic wave by using the following equations:
defining fundamental harmonic voltage amplitude UhVoltage phase θ uhAmplitude of current IhCurrent phase θ ih，h＝1,2,…,63；
Phase difference: thetah＝θuh-θih
Fundamental harmonic active power: ph＝UhIhcos(θuh-θih)
Fundamental harmonic reactive power: qh＝UhIhsin(θuh-θih)
Full-wave voltage:
Figure BDA0001386768130000051
full wave current:
Figure BDA0001386768130000052
full-wave active power:
Figure BDA0001386768130000053
full-wave reactive:
Figure BDA0001386768130000054
fundamental wave active electric energy: eP＝P1T, t is time;
fundamental wave reactive electric energy: eQ＝Q1T, t is time;
harmonic active electric energy:
Figure BDA0001386768130000055
t is time;
harmonic reactive electric energy:
Figure BDA0001386768130000056
t is time;
full-wave active electric energy: ePP · t, t is time;
full-wave reactive electric energy: eQQ · t, t is time.
The invention provides an electric energy metering method based on discrete spectrum correction, which comprises the steps of firstly, accurately finding out a signal spectrum peak value by adopting a spectrum centroid method, and eliminating spectrum leakage and fence response generated under asynchronous sampling; further correcting the frequency spectrum values of the voltage and the current by adopting an amplitude correction formula and a phase correction formula, and finally obtaining full wave, fundamental wave, harmonic wave electric parameters and electric energy; therefore, the method does not need to carry out peak value search, does not need to calculate the offset according to the type of the added window function, omits zero crossing point detection, directly adopts an FFT analysis method to carry out calculation, omits complex window spectrum operation, greatly improves the frequency precision obtained by calculation and analysis, and has simple program implementation and good real-time property.
Drawings
FIG. 1 is a process flow diagram of the process of the present invention.
FIG. 2 is a schematic diagram of the spectral centroid calculation of the method of the present invention.
FIG. 3 is a calibration coefficient fit curve for the method of the present invention.
FIG. 4 is a graph of harmonic frequency error accuracy of the method of the present invention.
FIG. 5 is a graph of harmonic amplitude error accuracy of the method of the present invention.
FIG. 6 is a graph of harmonic phase error accuracy of the method of the present invention.
Detailed Description
FIG. 1 shows a flow chart of the method of the present invention: the invention provides an electric energy metering method based on discrete spectrum correction, which comprises the following steps:
s1, converting voltage and current signals into digital quantities through sampling, performing time domain truncation on the converted digital quantities by adopting a Nuttall four-term third-order window with the window length of N, and obtaining a spectrum signal Y (k) of the truncated signals by adopting FFT analysis; the spectrum signal Y (k) comprises a voltage spectrum value Yu(k) Sum current spectral value Yi(k)；
In specific implementation, the main lobe width of the Nuttall four-term third-order window is preferably as follows
Figure BDA0001386768130000061
8 spectral lines are arranged in the main lobe;
the Nuttall four-term third-order window with the window length N meets the following formula:
Figure BDA0001386768130000062
wherein N is 1,2, …, N-1; bmIs a window function coefficient, satisfies
Figure BDA0001386768130000063
b0＝0.338946，b1＝0.481973，b2＝0.161054，b3＝0.018027；
The applied window function is to process the discrete voltage u (n) and the current signal i (n) as follows: u '(n) ═ u (n) × w (n), i' (n) ═ i (n) × w (n); u '(n) and i' (n) are subjected to FFT analysis to obtain a discrete spectrum Yu(k) And Yi(k) For the purpose of analysis, it is uniformly represented by Y (k);
s2, obtaining the voltage frequency spectrum value Y according to the step S1u(k) Calculating by adopting a spectrum centroid formula (figure 2 is a spectrum centroid calculation schematic diagram of the method of the invention) to obtain a spectrum centroid k of the fundamental wavesc1And fundamental frequency f0(ii) a The method specifically comprises the following steps:
A. calculating to obtain a fundamental spectrum centroid k according to a spectrum centroid formulasc1：
Figure BDA0001386768130000071
In the formula
Figure BDA0001386768130000072
And is
Figure BDA0001386768130000073
For a downward integer function, fmin is the minimum value of the fundamental frequency of the signal, specifically 45 Hz; fmax is the maximum value of the fundamental frequency of the signal, specifically 65 Hz; f ═ fsN, fs is the sampling frequency of analog-to-digital conversion, and N is the number of analysis points of the FFT algorithm;
B. according to the fundamental spectrum centroid k obtained in the step Asc1The fundamental frequency f is calculated and measured0：
f0＝ksc1*Δf
S3, adopting the fundamental frequency f obtained in the step S20Estimating harmonic peak locations
Figure BDA0001386768130000074
And calculating by adopting a spectrum centroid formula to obtain harmonic spectrum centroid kschAnd further calculating to obtain the maximum spectral peak of fundamental wave and each subharmonicPosition khAnd offset lambdah(ii) a The method specifically comprises the following steps:
a. the harmonic spectrum peak position is estimated by the following formula
Figure BDA0001386768130000075
Figure BDA0001386768130000076
Wherein h is 1,2,3, …, M; m is the maximum harmonic number of the signal;
b. obtained according to step a
Figure BDA0001386768130000077
Calculating to obtain harmonic spectrum centroid k by adopting the following formulasch：
Figure BDA0001386768130000078
c. C, obtaining harmonic spectrum centroid k according to step bschCalculating the maximum spectral peak position k of fundamental wave and each subharmonic by the following formulah：
kh＝round(ksch)
Where round () is a rounding function;
d. calculating to obtain an offset lambda according to the maximum spectral peak position kh obtained in the step ch：
λh＝kh-ksch
S4, determining a fundamental harmonic amplitude correction coefficient g (lambda) according to the offset obtained in the step S3h) Calculating to obtain fundamental wave, harmonic amplitude and phase of the voltage and current to be measured;
the following formula is used to calculate g (lambda)h)：
Figure BDA0001386768130000081
Wherein p is1～p5Are all constants; preferred is p1＝0.2073，p2＝-0.01546，p3＝0.9941，p4＝-0.000251，p5＝2.9503；
The amplitude and phase are corrected using the following correction formula:
A＝|Y(kh)|g(λh)
Figure BDA0001386768130000082
where A is the corrected amplitude, theta is the corrected phase, angle (Y (k)h) Is Y (k)h) The phase value of (a);
the derivation processes of the above-mentioned amplitude correction coefficient, amplitude correction formula and phase correction formula are briefly described as follows:
first, consider a single frequency signal y (n) at a sampling frequency fsThe uniformly sampled discrete-time signal is:
Figure BDA0001386768130000083
wherein A is amplitude, f0In order to be the frequency of the radio,
Figure BDA0001386768130000084
is an initial phase;
adding Nuttall fourth-term third-order window to y (n) to obtain yw(n)＝y(n)w(n)，yw(n) continuous Fourier transform
Figure BDA0001386768130000091
The above equation is discretely sampled and the negative frequency points-f are ignored0And (3) obtaining the side lobe influence of the frequency peak, wherein the expression of the discrete Fourier transform of the windowed signal is as follows:
Figure BDA0001386768130000092
let f0＝kscΔ f, then kscIs the spectral centroid of the signal; as shown in FIG. 2, k*To be close to the spectrum centroid kscThe maximum spectral peak position of (a), thus:
Figure BDA0001386768130000093
let λ be k*-kscλ is the offset, λ ∈ [ -0.5,0.5]；
It is further deduced that the discrete spectrum function of the nuttally quadrinomial third-order window is approximated as follows:
Figure BDA0001386768130000094
in summary, the obtained amplitude correction formula is:
Figure BDA0001386768130000095
in the formula
Figure BDA0001386768130000096
The magnitude of the window function at offset;
Figure BDA0001386768130000097
is shown in fig. 3; it can be seen that the curve can be fitted by a polynomial, and in order to reduce the amount of operation, a fourth-order polynomial fitting formula is firstly adopted for approximation
Figure BDA0001386768130000098
The curve, the fitting equation is as follows:
g(λ)＝p1λ4+p2|λ|3+p3λ2+p4|λ|+p5
wherein p is1＝0.2073，p2＝-0.01546，p3＝0.9941，p4＝-0.000251，p5＝2.9503；
The formula for the amplitude can thus be found as:
A＝|Y(k*)|g(λ)
the phase correction formula is as follows:
Figure BDA0001386768130000101
s5, calculating electric parameters of fundamental waves and harmonic waves according to the measured voltage, fundamental waves of current, harmonic amplitude and phases obtained in the step S4, and accordingly completing electric energy metering; specifically, the electrical parameters of fundamental waves and harmonic waves are calculated by adopting the following formula:
defining fundamental harmonic voltage amplitude UhVoltage phase θ uhAmplitude of current IhCurrent phase θ ih，h＝1,2,…,63；
Phase difference: thetah＝θuh-θih
Fundamental harmonic active power: ph＝UhIhcos(θuh-θih)
Fundamental harmonic reactive power: qh＝UhIhsin(θuh-θih)
Full-wave voltage:
Figure BDA0001386768130000102
full wave current:
Figure BDA0001386768130000103
full-wave active power:
Figure BDA0001386768130000104
full-wave reactive:
Figure BDA0001386768130000105
fundamental wave active electric energy: eP＝P1T, t is time;
fundamental wave reactive electric energy: eQ＝Q1T, t is time;
harmonic active electric energy:
Figure BDA0001386768130000111
t is time;
harmonic reactive electric energy:
Figure BDA0001386768130000112
t is time;
full-wave active electric energy: ePP · t, t is time;
full-wave reactive electric energy: eQQ · t, t is time.
The electric energy metering method provided by the invention is verified through tests as follows:
the method is applied to a gateway table (the specification model of an electric meter is DTSD341-MA2,3 multiplied by 57.7V, 3 multiplied by 1.5(6) A, 20000imp/kWh, 50Hz), and a program is realized on a Blackfin BF533 DSP processing chip of the electric meter platform.
The fundamental wave active pulse basic error precision of the method is shown in figure 1:
TABLE 1 fundamental active pulse basic error accuracy
Figure BDA0001386768130000121
Harmonic pulse experimental setup:
fundamental wave voltage Un is 57.7V; the fundamental current In is 0.5 × Imax, i.e. In ═ 2 × Ib ═ 3A; the fundamental power factor is 1.0; the fundamental frequency is 50 Hz; single harmonic waves are superposed on the basis of fundamental waves, and the harmonic frequencies are superposed to 48 times at most; harmonic voltage amplitude 0.05 × Un; harmonic current amplitude 0.4 In; the harmonic power factor is 0.5L; harmonic pulse constant 1000000 imp/kWh;
the following is that under the condition that the gateway table is superposed with single harmonic, the error of the harmonic pulse is shown in table 2:
TABLE 2 error schematic of harmonic pulse
Figure BDA0001386768130000131
From the above table, it can be found that the measured data of the gateway table is far from the theoretical value, because the experimental conditions are limited, the experimental data cannot reach the best, even if the error data of the above table meets the highest requirement of GB/T17215.302-2013 on the harmonic electric energy metering precision, and each strict test is performed, and the standard requirement is completely met. The harmonic pulse error is mainly influenced by the specific difference and phase difference of a hardware sampling circuit and a mutual inductor, and the harmonic error data in the meter is measured after the amplitude and the phase of the electric meter are corrected. After the experimental conditions allow, the data precision can be improved qualitatively.
MATLAB simulation results are shown in FIGS. 3-6, which are respectively a correction coefficient fitting curve, harmonic frequency error accuracy, harmonic amplitude error accuracy and harmonic phase error accuracy; as can be seen from the figure, the algorithm provided by the invention can realize high-precision measurement of harmonic signals.

Claims (9)

1. An electric energy metering method based on discrete spectrum correction comprises the following steps:
s1, converting voltage and current signals into digital quantities through sampling, and performing time domain truncation on the converted digital quantities by adopting a Nuttall four-term third-order window with the window length of N to obtain a spectrum signal Y (k) of a truncated signal; the spectrum signal Y (k) comprises a voltage spectrum value Yu(k) Sum current spectral value Yi(k)；
S2, obtaining the voltage frequency spectrum value Y according to the step S1u(k) Calculating to obtain the spectrum centroid k of the fundamental wave by adopting a spectrum centroid formulasc1And fundamental frequency f0；
S3, adopting the fundamental frequency f obtained in the step S20Estimating harmonic peak locations
Figure FDA0001386768120000011
And calculating by adopting a spectrum centroid formula to obtain harmonic spectrum centroid kschAnd further calculating to obtain the maximum spectral peak position k of fundamental wave and each subharmonichAnd offset lambdah；
S4, determining a fundamental harmonic amplitude correction coefficient g according to the offset obtained in the step S3(λh) Calculating to obtain fundamental wave, harmonic amplitude and phase of the voltage and current to be measured;
s5, calculating the electrical parameters of the fundamental wave and the harmonic wave according to the measured voltage, the fundamental wave of the current, the harmonic amplitude and the phase obtained in the step S4, and thus completing the measurement of the electrical energy.
2. The discrete spectrum correction-based electric energy metering method according to claim 1, wherein the step S1 is to obtain the spectrum signal y (k) of the truncated signal, specifically, the spectrum signal y (k) of the truncated signal is obtained by FFT analysis.
3. The discrete spectrum correction-based electric energy metering method according to claim 2, wherein the window length of step S1 is N, and specifically the main lobe width of the nuttal quadrinomial third-order window is N
Figure FDA0001386768120000012
There are 8 spectral lines in the main lobe.
4. The discrete spectrum correction-based electric energy metering method of claim 3, wherein the Nuttall quadrinomial third-order window with the window length N satisfies the following formula:
Figure FDA0001386768120000021
wherein N is 1,2, …, N-1; bmIs a window function coefficient, satisfies
Figure FDA0001386768120000022
5. The discrete spectrum correction-based electric energy metering method according to claim 4, wherein the calculation in step S2 is to obtain the centroid k of the fundamental spectrumsc1And fundamental frequency f0The method specifically comprises the following steps:
A. according toCalculating to obtain a basic spectrum centroid k by a spectrum centroid formulasc1：
Figure FDA0001386768120000023
In the formula
Figure FDA0001386768120000024
And is
Figure FDA0001386768120000025
For the rounding function, fmin is the minimum of the fundamental frequency of the signal, fmax is the maximum of the fundamental frequency of the signal, and Δ f ═ fsN, fs is the sampling frequency of analog-to-digital conversion, and N is the number of analysis points of the FFT algorithm;
B. according to the fundamental spectrum centroid k obtained in the step Asc1The fundamental frequency f is calculated and measured0：
f0＝ksc1*Δf。
6. The discrete spectrum correction-based electric energy metering method of claim 5, wherein the calculation of step S3 yields harmonic spectrum centroid kschMaximum spectral peak position k of fundamental wave and each subharmonichAnd offset lambdahThe method specifically comprises the following steps:
a. the harmonic spectrum peak position is estimated by the following formula
Figure FDA0001386768120000026
Figure FDA0001386768120000027
Wherein h is 2,3, …, M; m is the maximum harmonic number of the signal;
b. obtained according to step a
Figure FDA0001386768120000028
Using the following formulaCalculating to obtain harmonic spectrum centroid ksch：
Figure FDA0001386768120000031
c. C, obtaining harmonic spectrum centroid k according to step bschCalculating the maximum spectral peak position k of fundamental wave and each subharmonic by the following formulah：
kh＝round(ksch)
Where round () is a rounding function;
d. according to the maximum spectrum peak position k obtained in the step chCalculating to obtain offset lambdah：
λh＝kh-ksch。
7. The discrete spectrum correction-based electric energy metering method according to claim 6, wherein the step S4 is implemented by determining the fundamental wave amplitude modification factor g (λ)h) Specifically, the following formula is adopted for calculation:
Figure FDA0001386768120000032
wherein p is1～p5Are all constants.
8. The discrete spectrum correction-based electric energy metering method according to claim 7, wherein the fundamental wave, harmonic amplitude and phase of the measured voltage and current are obtained through calculation in step S4, and the amplitude and phase are corrected by using the following correction formulas:
A＝|Y(kh)|g(λh)
Figure FDA0001386768120000033
where A is the corrected amplitude, theta is the corrected phase, angle (Y (k)h) Is Y (k)h) The phase value of (a).
9. The discrete spectrum correction-based electric energy metering method of claim 8, wherein the step S5 is to calculate the electrical parameters of the fundamental wave and the harmonic wave by using the following formula:
defining fundamental harmonic voltage amplitude UhVoltage phase θ uhAmplitude of current IhCurrent phase θ ih，h＝1,2,…,63；
Phase difference: thetah＝θuh-θih
Fundamental harmonic active power: ph＝UhIhcos(θuh-θih)
Fundamental harmonic reactive power: qh＝UhIhsin(θuh-θih)
Full-wave voltage:
Figure FDA0001386768120000041
full wave current:
Figure FDA0001386768120000042
full-wave active power:
Figure FDA0001386768120000043
full-wave reactive:
Figure FDA0001386768120000044
fundamental wave active electric energy: eP＝P1T, t is time;
fundamental wave reactive electric energy: eQ＝Q1T, t is time;
harmonic active electric energy:
Figure FDA0001386768120000045
t is timeA (c) is added;
harmonic reactive electric energy:
Figure FDA0001386768120000046
t is time;
full-wave active electric energy: ePP · t, t is time;
full-wave reactive electric energy: eQQ · t, t is time.
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