# Method for active checkup of voltage and current level for sinusoid distortions

FIELD: computer-aided checkup of electrical energy characteristics.

SUBSTANCE: proposed method involves evaluation of coefficients of current and voltage sinusoid distortions, checkup of these characteristics for compliance with their rated values, and generation of control signal by devices correcting sinusoid of voltage and current levels. This method is characterized in that subharmonic and higher fractional components of current and voltage are included in evaluation of coefficient of current and voltage sinusoid distortions due to determination of actual period of voltage and current variations with time.

EFFECT: enhanced precision of checking voltage and current for their sinusoid distortions.

1 cl, 4 dwg

The invention relates to electrical engineering and can be used in existing power systems containing nonlinear electrical load can be used as an integral part of an automated control of all parameters of quality of electrical energy.

The current interstate standard GOST 13109-97 "Rules of electric power quality in electrical systems General purpose" prescribes the level of nonsinusoidal voltage to estimate the distortion factor nonsinusoidal voltage and ratio ν-the harmonic component of the voltage.

For an objective and complete assessment of the quality of electric energy have the sense to define appropriate indicators and relatively current. Therefore, we are concerned about the level of nonsinusoidal current.

Assessment of the level of nonsinusoidal voltage and current traditionally involves the determination of the spectral composition of these characteristics of electric energy. This is commonly used Fourier transform. This process is repeatedly described in the literature. For example, in [1].

But the Fourier transform in the form in which it is used to control the level of nonsinusoidal voltage and current at the present time considers only integer and only in the ssie harmonic components of the specified characteristics of electrical energy, although it is known that the spectral composition of the voltages and currents of modern electric power systems are the highest fractional harmonic and sub-harmonic (frequency less than 50 Hz) components.

In addition, existing methods of monitoring levels of nonsinusoidal voltage and current are classified as passive control, i.e. involve only the receipt of the relevant information on the quality of electricity, without an attempt of active influence on the latter.

The closest technical solution to the proposed algorithm for active control of the level of nonsinusoidal voltage and current is a method of quantitative evaluation of subharmonic and fractional harmonic components periodically varying quantities [2].

Object of the invention is the formation of the structural scheme of the algorithm for active control of the level of nonsinusoidal voltage and current when taking into account not only the respective higher harmonic, and subharmonic components.

The technical result is achieved by using the Fourier transform with the preliminary determination of the actual period changes of voltage and current in time, the subsequent assessment of the level of sinusoidality the analyzed values and forming, if necessary, signal the Board for appropriate corrective devices.

As corrective devices you can use any of the technical solutions aimed at maintaining an adequate level of nonsinusoidal voltage and current. For example, filtercontrol device.

The classical representation of the Fourier transform is usually represented as:

where F(jω) is the spectral density of the original function f(t), ω cyclic frequency; ƒ(t) describe the time function.

In this case we are more interested in the inverse Fourier transform. Here it is better to present it in the form of a trigonometric series:

where A_{0}- constant component functions ƒ(t);,,, ...,and,,, ...,- sine and cosine coefficients at the frequency of the 1st, 2nd, and 3rd, ..., nth harmonic components.

Because here instead of ƒ(t) refers to the laws of the time variation of the voltage and current, the trigonometric series should be presented in a different form:

where U_{0}and I_{0}- constant component functions u(t) and i(t);,,, ...,,,,, ...,and,,, ...,,,,, ...,- sine and cosine coefficients of the functions u(t) and i(t) at the frequency of the 1st, 2nd, and 3rd, ..., nth harmonic components.

In the mathematical interpretation of this number is supposed to be infinite in duration summation of its sine and cosine parts. Of course in the real case, it is unattainable. Therefore, in the proposed wording summation is limited by the frequency of the highest noticeable in the spectra of voltage and current harmonics n. Its presence can be determined using widely used at the present time, analog or digital analyzers (meters) range. Of course, this definition is very tentative.

When determining the numerical value of n should be remembered that the greater will be its value is, the more will be the expected result.

From mathematics we know that

where T is the period of the function f(t), ν - the number of the current harmonic component.

In this occasion:

The source data for the analysis of the level of nonsinusoidal main characteristics are formed in the analog-to-digital Converter in the form of discrete values of voltages and currents. The definition of permanent part time functions, as well as the sine and cosine coefficients in the above formulas in practice difficult to implement. So here you should use the discrete Fourier transform. Otherwise, this operation is called by a geographical decomposition functions ƒ(t).

The drawings show the following: figure 1 - illustration of the considered decay function ƒ(t)figure 2 - illustration of the process of determining the actual period changes of the analyzed characteristics of the electrical energy figure 3 - block diagram of the algorithm for active control of ur is una nonsinusoidal voltage, figure 4 - block diagram of the algorithm for active control of the level of nonsinusoidal current.

Figure 1 illustrates the process of converting analog signals into discrete in analog-to-digital Converter.

Here analog function splits on the x-axis into equal parts Δωt. Otherwise, these areas are called quantizations. When determining the number of quantizations for achieving objectives makes sense to use the recommendations of the sampling theorem. From corollaries of this theorem is that the number of quantizations for the period of the function should be defined by the formula N=2^{k}, where k is any natural number (k=1, 2, 3, 4, 5, ...). It is from this condition, in addition to performance and multi-channel, you should choose analog-to-digital Converter.

The summation function ƒ(t)according to figure 1, should be based on the argument ωt. Therefore, the DC component and the sine and cosine coefficients for this function in this case will appear in a slightly different form:

In relation to the voltage and current these formulas are transferred so:

Considering that

the last formulation, taking into account the discreteness of the functions u(t) and i(t) can be converted and presented in a different form.

Thus, the constant components of the analyzed functions with respect to this equality is converted as follows:

where u_{p}(t) and i_{p}(t) - discrete p-tide values of the functions u(t) and i(t); p is the current number of quantization.

Sine coefficients on frequency ν-the harmonic component will be defined like this:

where (ωt)_{p}- quantitative argument value functions u_{p}(t) and i_{p}(t) in radians, which is defined by the formula

Similarly for the cosine coefficients on frequency ν-the harmonic component:

Next, based on the properties of mathematical formulations, it is possible to determine the amplitude, ν's harmonic components of voltage and current as the geometric sum of the respective sine and cosine ratios:

The initial phase of voltage and current at the frequency of this harmonic component of the phase angle shift of the zero harmonic component relative to the origin) will be determined by the formula

As a result of the transformations, it turns out that

It is now possible to determine the effective values of voltages and currents at the frequency of each harmonic component:

The above method of determining the spectral content of the voltage and current has a fairly serious limitation, which can be attributed to the discharge of the shortcomings of the Fourier transform, restricting its use to solve technical problems: it allows you to identify only integer harmonics. This is particularly challenging for modern electric power systems, saturated power electronic devices and other nonlinear elements, providing electric energy processes, characterized by a nonlinear volt-ampere and Weber-ampere characteristics, equipped with a long line and transmission of high power. In modern power systems have a significant share of the fractional harmonic components, which when using the above methods is not possible.

Extremely noticeable drawback of this technique, which consists in ignoring the subharmonic components, whose frequencies are less than the frequency of the main harmonic (in Russia, 50 Hz). And it is subharmonic components have an extremely negative impact on biological objects, including the human body.

The solution of this problem based on the analysis of [2] is relatively simple: quite simply the most reliable way to determine the period changes of time-varying values. But sometimes it is very difficult to do. And this operation is performed on the assumption that the voltage and current is periodically varying in value, due to a catastrophic deterioration in the quality of electrical energy is becoming more questionable. In this case it is necessary to set the period of the lowest harmonic component, and only then to use the above mathematical formulation to determine the spectral content of the voltage and current.

In the real case, still often the voltage and current of periodically varying in value. In this case, the su is basically a very real possibility to determine the actual period changes of these quantities. This procedure is illustrated in figure 2.

In this case it is necessary to consider several visually certain periods of the periodically varying function. These several periods must be carried out through an analog-to-digital Converter, converting them into many discrete values. Procedure determine the actual period of the periodically varying function is to find groups with the same discrete values of the analyzed function. The same discrete values it can be determined with the only other set in advance, the degree of reliability because of the absolute requirement N=2^{k}it is impossible without a preliminary determination period. After that, if you ［ analog-to-digital Converter, there is a sense in accordance with this equality to specify the number of quantizations for a period of time approximately equal to a certain period, and to clarify in the stipulated herein, the quantitative value of the last. If this is not achieved absolute equality of groups of discrete values, the number of quantizations attributable to the period, again to clarify... And so on up until either not achieved absolute equality of groups of discrete values, then the actual period will be op is adalats time interval one of the mentioned groups of discrete values,
or the difference between the corresponding values of the considered groups will obviously increase.

In the latter case, the increase in the difference between the corresponding values of groups of discrete values indicates either an incorrect definition of these groups and then after a preliminary reduction of the allowable error in the determination of equality between the respective discrete values, increasing the considered time interval, it is necessary to repeat the definition stated earlier groups, followed by a more reliable determination of the actual period of the periodically varying function or of insufficient resolution analog-to-digital Converter. In this case, if it is not possible to use an analog-to-digital Converter with higher resolution, for the actual value desired value should be the period with the least difference in the corresponding values of the groups under consideration. Naturally, the reliability of the actual period of the periodically varying function will be greater, the greater the resolution of the analog-to-digital Converter, the less time range of one quantization, the greater the number of quantizations have one period.

Substituting therefore a certain FA is the critical period changes of interest to the researcher functions (voltage or current) in the above mathematical formulation, you can determine the spectral content of the voltage and current.

The minimum frequency of the harmonic component is defined as the reciprocal of the actual period of the investigated functions:

The frequency of the other harmonic components are determined by the formula

where v is the current number of harmonics having only positive integer values (ν=1, 2, 3, 4, ...).

The minimum frequency of the harmonic component may be much less accepted in one country or another, the frequency of the main harmonic component (in Russia 50 Hz):

In this case we are talking about subharmonic components.

The actual frequency of the main harmonic component is determined by the maximum value of its amplitude.

If the number of the primary harmonic component is equal to the unit, and non other harmonics identified by the symbol νwhen observing the last inequality will find numbers mentioned harmonic components, smaller units (sub-harmonic components), and fractional numbers:

Now, finally, have the opportunity to determine the degree of conformity of the actual spectra of voltage and current your the regulatory requirements. These requirements can be, if necessary, differentiated for each node in the power system. In other cases it is limited to the applicable interstate standard on electric power quality in electrical systems General purpose" GOST 13109-97.

Mentioned here the standard defines two indicators of the quality of electric energy, assessing the level of nonsinusoidal voltage: distortion factor nonsinusoidal voltage curve k_{U}and the factor ν-the harmonic component of the voltage k_{νU}. According to the recommendations of this standard, these indicators are defined as:

where U_{ν}- the effective value of the voltage to frequency ν-the harmonic component; U_{1}- the RMS voltage at the frequency of the main harmonic component; n is the largest number of registered harmonic components (in the quoted text of the interstate standard greatest number of harmonic components is limited by the numeral 40).

Unfortunately, this equality does not take into account the presence in the spectrum of the analyzed voltage fractional harmonic and subharmonic sostavlyajushie is. In this case, to assess the level of nonsinusoidal voltage distortion factor nonsinusoidal voltage curve is proposed to determine a little differently:

where ν_{min}- the number of the harmonic component, characterized by the lowest frequency

For effective control of quality of electrical energy by means of ASK the SCE need to assess the level of nonsinusoidal not only voltage, but current. In this case, the corresponding indicators of the quality of electric energy will appear as follows:

the distortion factor of the nonsinusoidal current curve

factor ν-the harmonic current component

where I_{ν}- the effective value of current at the frequency ν-the harmonic component; I_{1}- the effective value of current at the frequency of the main harmonic component.

The current standard for specified values of THD and factor individual harmonic components of the voltage. These standards are found quite widely used in power supply systems for General use. Normative values for the coefficient travesty what s sinusoidally and factor individual harmonic components of the currents can be evaluated in the same percentage, what is the recommended standard to estimate these coefficients for voltage. However, in some special cases the normative values for these coefficients have to tighten.

One of the final stages of analysis of nonsinusoidal voltage and current is to verify the conformity of the mentioned factors to their normative values. This operation involves the review of inequalities:

where ε_{U}that ε_{νU}that ε_{I}that ε_{νI}- the normative values of the coefficients, respectively, waveform distortion and ν-the harmonic component of voltage and current.

Figure 3 presents the recommended structural diagram of the algorithm of analysis of the level of nonsinusoidal voltage at the node power system. Here R_{1}and R_{2}- current values (numbers) quantization of the analyzed curve (figure 2); Δt - time sample rate; m is the number of quantizations specific usually from the resolution of the analog-to-digital Converter.

Here in the first five blocks of the above method is determined by the actual period of time variation of the analyzed voltage is Then determined by the formula (1) DC component of this voltage and the minimum number of harmonic components. Then, for each harmonic component of the measured spectrum of the voltage by the formulas(3), (5), (7) and (9) are the sine and cosine coefficients, its peak value and the initial phase angle shift of the zero harmonic component relative to the origin), by the formulas (12) and (11) determines the distortion factor nonsinusoidal curve and the factor ν-the harmonic component of the voltage. And in the last two blocks of the inequalities (15) and (16) checks whether the found indicators of quality of electric energy to its normative values. If you violate this match makes sense to start forming the control signal for the corresponding corrective devices.

In parallel with the analysis of the level of nonsinusoidal voltage curve is quite possible the implementation of the analysis of the level of nonsinusoidal current curve. Structural diagram of such analysis is shown in figure 4.

Here, as with the analysis of the level of nonsinusoidal voltage, the first five units is determined by the actual period of time variation of the analyzed current. Then by the formula (2) is determined by the constant component of current and the number of its minimal harmonic component. Next, for each harmonic component of the measured spectrum of the current in the form of the s (4), (6), (8) and (10) are the sine and cosine coefficients, its peak value and the initial phase angle shift of the zero harmonic component relative to the origin), by the formulas (13) and (14) determines the distortion factor nonsinusoidal curve and the factor ν-the harmonic current component. Then in the last two blocks of the inequalities (17) and (18) checks whether the found indicators of quality of electric energy to its normative values. If you violate this match here, as in the previous case, it makes sense to start forming the control signal for the corresponding corrective devices.

So look the structural scheme of the algorithm for active control of the level of nonsinusoidal voltage and current in modern power systems.

Sources of information

1. Bunyak A.F. Use of the discrete Fourier transform to determine the power quality parameters of the computing devices. Izvestiya vuzov. Energy (Minsk), 1982, No. 6. - P.7-12.

2. The method of quantitative evaluation of subharmonic and fractional harmonic components periodically varying quantities. Gaebelein, Englobal, Swiderian, Eaaea, Avinadav, Caimari, Mailfrom. - The patent of the Russian Federation 2122186, MCI G 01 J 3/18. - Bratsk industrial Institute, No. 96112228/25; Appl. 14.06.96; Publ. 20.11.98. Bull. No. 32.

The method of active control of the level of nonsinusoidal voltage and current, providing a geographical decomposition of the spectral composition of the periodically varying voltage and current, at which the actual period changes of these values, the number of the harmonic component, characterized by the lowest frequency, and the effective value of each harmonic component, wherein the determined coefficient k_{U}waveform distortion voltage according to the formula

where ν_{min}- the number of the harmonic component, characterized by the lowest frequency;

U_{ν}- the RMS voltage at the frequency v of the harmonic component;

U_{1}- the RMS voltage at the frequency of the main harmonic component;

n is the largest number of registered harmonic components,

determine the THD of the current by the formula

where I_{ν}- the effective value of current at the frequency ν-the harmonic component;

I_{1}- the effective value of current is and the frequency of the main harmonic component

determine the coefficient of ν-the harmonic current component by the formula

check the conformity of these indicators with their normative values and the violation of this compliance form the control signal devices, adjustment level nonsinusoidal voltage and current.

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