JP2006276006A - Harmonic analysis method in power system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide an accurate measuring method of an effective value including harmonics and phase angle of the voltage and current in a power system. <P>SOLUTION: In this method, a recursive DFT operation is applied to sampling data in a voltage and current waveform with two different setting frequencies in the power system, and weighted average according to the frequency deviation between the frequency of the power system and the set frequencies of the DFT operation is applied to each operation result. Thus, the operation error is corrected, and fundamental wave effective values and harmonic effective values of voltage and current is measured at high accuracy. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a highly accurate measurement method for effective values and phase angles including harmonics with respect to voltage and current in a power system.

  There is a Fourier transform as a method for analyzing frequency components contained in a periodic signal waveform. When this is obtained by digital signal processing, the signal waveform is sampled at equal intervals, and an operation is performed on the obtained data string. This operation is called discrete Fourier transform (hereinafter referred to as “DFT”). In the power system, it is used to analyze harmonic components contained in voltage and current waveforms.

Specifically, the calculation for obtaining the effective value V _e and the phase angle φ ₀ of the waveform represented by the following Equation 1 will be described as an example. Here, the fundamental wave component will be described, but the harmonic component can be obtained in the same way.

Here, f is the frequency of the measured voltage waveform of the power system (hereinafter referred to as “system frequency”). FIG. 1 shows a case where sampling is performed with one waveform period as the sampling number N = 12. T _S [s] is a sampling period. When sampling is performed in this way, the following data string is obtained.

v ₀ , v ₁ , ..., v ₁₀ , v ₁₁ (2)
When a DFT operation is performed on this data string according to the following Equation 2, a complex value V ₁₁ is obtained.

f ₀ represents a fundamental frequency of Fourier transform (hereinafter referred to as “set frequency”), and has a relationship of f ₀ = 1 / (N · T _S ). If the system frequency f is equal to the set frequency f ₀ , the calculation result of Equation 2 is V ₁₁ = V _e · exp (jφ ₀ ), and therefore the effective value V _e and the phase angle φ ₀ can be obtained by DFT calculation. .

Here, the function W _i defined by Equation 3 below
If we rewrite equation 2 using, it becomes like equation 4 below.

W _i is a point plotted on the circumference of radius 1 on the complex plane as shown in FIG. In this state, the data string and the value is that taken in the newly v ₁₂ the next sampling time,
v ₁ , ..., v ₁₁ , v ₁₂ (6)
Therefore, the DFT operation is as shown in Equation 5 below.

Subtracting Equation 4 from Equation 5 yields Equation 6 below.

As can be seen from FIG. 2, since W ₁₂ = W ₀ , the equation can be modified as in Equation 7 below.

This formula shows that the latest complex value V ₁₂ can be obtained recursively by adding the latest instantaneous value v ₁₂ to the previous complex value V ₁₁ and subtracting the instantaneous value v _{0 of the} previous cycle. Yes. This is called recursive discrete Fourier transform (hereinafter referred to as “RDFT”). Therefore, the calculation when the r-th data is taken in can be expressed as the following Equation 8 using the relationship shown in FIG.

Thus, if the voltage waveform v (t) is a repetitive waveform and f = f ₀ , the effective value V _e and the phase angle φ ₀ can be continuously obtained by the RDFT calculation.

However, the frequency of the power system fluctuates with a maximum width of ± 0.3 Hz with respect to the standard frequency (eastern Japan 50 Hz, western Japan 60 Hz). Therefore, since the frequency of the voltage waveform and the current waveform is also momentarily varies, deviation occurs between the set frequency f ₀ and the system frequency f. When this frequency deviation occurs, V _r = V _e · exp (jφ ₀ ) is not obtained, and there is a problem that the effective value V _e and the phase angle φ ₀ cannot be obtained accurately.

  As a specific example, FIG. 4 shows a result of performing RDFT calculation at a set frequency 60.0 [Hz] on an AC voltage having a system frequency 59.7 [Hz] and an effective value 100.0 [V]. For the RMS value waveform of 100.0 [V], the calculation result includes an error, and the RMS value cannot be obtained accurately.

In order to explain such a method for correcting an error due to a frequency deviation, the relationship between the frequency deviation and the error will be described. Consider a case where the system frequency f has a deviation of Δf (= f−f ₀ ) from the set frequency f ₀ . At this time, the r-th sampling data can be expressed by Equation 9 below.

When an RDFT operation is performed on this data using Equation 8, an operation result represented by Equation 10 below is obtained.

Since the effective value is obtained as the magnitude of the complex value V _r , the following Expression 11 is obtained from the calculation result of Expression 10.

Therefore, when compared with the effective value V _e of the original waveform, the result of Equation 12 below is an error caused by the frequency deviation.

Therefore, Err expressed by the following formula 13 is the ratio of the error to the effective value.

As shown in Equation 10, since A, B, α _r , and β _r are values including the frequency deviation Δf, the relationship between the frequency deviation Δf and the error rate Err is calculated as shown in FIG. The calculation conditions in this calculation example are N = 128, T _S = 1 / (60 Hz × 128 samples) = 130.2 [μs]. As can be seen from the figure, within the range of ± 0.3 Hz, the relationship between the frequency deviation Δf and the error rate Err can be regarded as a proportional relationship. FIG. 5 shows the case where r = 0, but only the slope of the straight line changes depending on the value of r, and the proportionality does not change.

Next, FIG. 6 shows the result of calculating the time change of the error ratio Err. As shown in the figure, the time variation of Err changes in a vibration manner, so the vibration frequency will be described. In Equation 13, A and B are values that do not change with time. Therefore, (α _r −β _r ) is expressed as θ, and this is calculated as follows.

θ = α _r −β _r
= 4π (f ₀ + Δf) r · T _S + 2φ ₀ −2π (f ₀ + Δf) · (N−1) · T _S (16)
In this equation, r · T _S in the first item represents the time when the r-th data is sampled, and is expressed as _tr . That is, the first item is a term that changes with time, and the second item and the third item are terms that do not change with time. From this, if θ = ωt _r + φ, ω and φ can be expressed by the following equations, respectively.

ω = 4π (f ₀ + Δf) = 2π · 2 (f ₀ + Δf)
φ = 2φ ₀ −2π (f ₀ + Δf) · (N−1) · T _S (17)
Therefore, the frequency of vibration of Err is 2 (f ₀ + Δf) = 2f, which is twice the system frequency.

  As described above, the error caused by the frequency deviation has a property that the magnitude of the error is proportional to the frequency deviation, and the vibration of the error changes at twice the system frequency.

  The correction method is explained using the nature of these errors.

For one voltage waveform, RDFT calculation is simultaneously performed at two different set frequencies. These set frequencies are defined as a first set frequency f _0a and a second set frequency f _0b , respectively. If the number of samplings is N _a and N _b , the first set frequency has a relationship of f _0a = 1 / (N _a · T _S ), and the second set frequency has a relationship of f _0b = 1 / (N _b · T _S ). It becomes. The first set frequency f _0a and the second set frequency f _0b can be selected arbitrarily, but considering that the frequency deviation of the power system is ± 0.3 Hz at the maximum, it is ± 0.4 Hz of the standard frequency of the power system Select so as not to exceed. For example, in the case of a 60 Hz system, the first setting frequency f _0a = 59.6 [Hz] and the second setting frequency f _0b = 60.4 [Hz] are selected. Then, when the RDFT operation is performed according to the following equation 14, complex values V _ra and V _rb are obtained as the respective operation results, as shown in FIG.

The error ratio included in V _ra and V _rb respectively Err _a, When Err _b, is proportional to the Err _a and Err _b each frequency deviation Delta] f _a and Delta] f _b, is shown in Figure 8. Therefore, the ratio of error magnitudes (= a: b) is approximately equal to the ratio of frequency deviations of the two RDFT algorithms.

a ： b ≒ Δf _a ： Δf _b (18)
The frequency deviations Δf _a and Δf _b can be obtained by the method described in the document [1].
[1] Nakano, et al .: “Real-time power system frequency detection based on synchronous phasor measurement”,
IEEJ Transaction C, 122,12, pp2076-2082,2002

Since the frequency of the error vibration is twice the system frequency regardless of the set frequency, the frequency of the error vibration included in V _ra and V _rb is equal. For example, for AC voltage with system frequency f = 60.2 [Hz] and effective value 100.0 [V], RDFT calculation is performed with the first set frequency f _0a = 59.6 [Hz] and the second set frequency f _0b = 60.4 [Hz]. The results of performing are shown in FIG. It can be seen that the frequencies of vibrations of V _ra and V _rb are equal to each other. In this case,
a: b≈Δf _a : Δf _b = (60.2−59.6) :( 60.2−60.4) = 3−1 (19)
It has become. Therefore, as shown in Equation 15 below, by calculating the weighted average of the respective effective values using a: b as weighting factors, it is possible to obtain a complex value V _{rp in} which the error is canceled and the error is reduced.

  In measuring the voltage and current of a power system whose frequency is fluctuating, it is possible to obtain an accurate effective value by correction by simple four arithmetic operations without losing the continuity of the RDFT operation.

  Figure 10 shows a power system harmonic analyzer that implements the proposed method.

  The voltage input box 1a includes a voltage converter (PT) 1 and steps down to a value that does not exceed the input upper limit voltage of the analog-digital converter (ADC) 2 in the input section of the DSP board 7. In this example, 100 [V] AC voltage is converted to 1 [V] AC voltage. The ADC 2 converts analog data into digital data at a constant sampling interval. The sampling frequency of ADC2 used in this example is 8 [kHz]. The converted digital data is taken into the DSP 3 via the memory (RAM) 4 and the above-described RDFT operation and correction operation are performed in the DSP 3. The calculation result stored in the RAM 4 is transferred to the personal computer (PC) 8 through the LAN port 6 and the calculation result is displayed on the PC 8.

Using this system, a waveform generator with a voltage effective value of 100 [V] and a frequency of f = 59.9 [Hz] was created and input to the PT1 input in the voltage input box 1a. FIG. 11 shows the result of the RDFT calculation performed at the set frequency f ₀ = 60.15 [Hz] (= 8 [kHz] / 133 [sample]). Since there is a difference between the frequency of the voltage waveform and the set frequency, an error occurs in the calculation result. On the other hand, the first set frequency f _0a = 59.70 [Hz] (= 8 [kHz] / 134 [sample]) and the second set frequency f _0b = 60.15 [Hz] (= 8 [kHz] / 133 [sample] FIG. 12 shows the result of performing the RDFT operation in step) and performing correction by the method described above. It can be seen that the error can be greatly reduced as compared with the conventional method.

  The error correction method described here is for the fundamental component of the voltage waveform, but the harmonic component can also be corrected by the same method, and the effective value can be obtained with high accuracy. If the current waveform can be taken into the ADC using the current converter (CT), only the magnitude of the effective value to be obtained can be changed, and the error can be corrected by the same method.

  In this embodiment, if the sin function data and the cos function data corresponding to the two set frequencies are stored in the ROM in advance, high-accuracy RMS value calculation can be performed without performing trigonometric functions in the DSP. It becomes possible. For this reason, the DSP program can be simplified, the sampling cycle can be shortened, and high frequency components can be calculated.

Sample points when N = 12 samples Plot on the complex plane of equation (4) Sample points and recursive DFT operation timing Calculation error caused by frequency deviation Relationship between frequency deviation and error rate Time variation of error rate Sample points and recursive DFT operation timing when performing RDFT operation at two different design frequencies Relationship between frequency and error rate when RDFT calculation is performed at two different design frequencies RMS value when RDFT operation is performed at two different design frequencies Configuration diagram of power system harmonic analyzer RMS value calculation result when error correction is not performed in the device of FIG. RMS value calculation result when error correction is performed in the device of FIG.

Explanation of symbols

1 Voltage converter (PT)
1a Voltage input box
2 Analog-to-digital converter (ADC)
3 Digital signal processor (DSP)
4 Memory (RAM)
5 Memory (ROM)
6 LAN port
7 DSP board
8 PC (PC)

Claims (3)

  In the power system, recursive DFT operation is performed on the sampling data of voltage / current waveform at two different set frequencies, and each operation result depends on the frequency deviation between the power system frequency and the set frequency of DFT operation. A method of correcting the calculation error by applying a weighted average and measuring the effective value of voltage and current with high accuracy.   In the power system, recursive DFT operation is performed on the sampling data of voltage / current waveform at two different set frequencies, and each operation result depends on the frequency deviation between the power system frequency and the set frequency of DFT operation. A method of correcting the calculation error by applying a weighted average and measuring the harmonics of voltage and current with high accuracy.   A power system harmonic analysis apparatus comprising the measurement method according to claim 1 or 2.