Area-Based Approach in Power Quality Assessment

Abstract

This paper presents an approach for assessment of power quality parameters using analysis of fundamental and harmonic voltage and current waveforms. Park transformation technique has been utilized for the analysis in three-phase system, which has reduced the computational effort to a great extent. Contributions of fundamental and harmonic components in power system voltage and current signals have been assessed separately. An algorithm has been developed to calculate the power quality parameters from online signals. This algorithm has been simulated for a radial system, and the results have been compared with that obtained from a standard FFT-based system. The results are seen to be in good agreement with that of the standard system.

1. Introduction

Quality of electric power refers to maintaining the near sinusoidal waveform of power distribution bus voltages and currents at rated magnitude and frequency. The causes of poor quality of power have nowadays been identified broadly as follows: (i) actual loads, equipment, and linear and nonlinear components and (ii) subsystems of transmission and distribution systems causing impulses, notches, voltage sag and swell, voltage and current unbalances, momentary interruption, and harmonic distortions. The other major contributors to poor power quality are harmonics and reactive power circulated by power circuit controllers in a power system.

Researches are going on for long to study the effects of unbalance and harmonics on power quality in a power system [1–3]. In a nonsinusoidal and unbalanced environment, some practical definitions have also been suggested [4–6]. A new approach for designing power acceptability curve has been proposed [7]. In such conditions, measurement and monitoring of power quality in respect of its unbalance and harmonics have been presented in different publications [8–13], and different modern mathematical tools have also been utilized to estimate the unbalance in power in a system [14, 15]. There are several existing methods for the detection of the current harmonics like fast Fourier transform (FFT), instantaneous 𝑝-𝑞 theory, synchronous 𝑑-𝑞 reference theory, analog or digital filters, and so forth [16]. In some cases, passive filters are also used, but they are having the drawbacks of bulky components, fixed compensated characteristics, sensitivity to line impedance, and series and parallel resonance with system.

In this paper, an area-based approach has been proposed to assess electric power quality in a three-phase power system. Harmonics-free system voltages 𝑣(𝑡) or currents 𝑖(𝑡) form a closed loop in voltage-voltage or current-current plane. In this loop, cleavages appear with the presence of harmonics, order of which is directly proportional to number of cleavages. An algorithm has been developed using area-based approach to calculate active power, reactive power, and total harmonic distortion factors. Park transformation technique has been used for three-phase power distortion factor measurement, which has reduced the computational effort to a great extent. The developed technique has been simulated showing good agreement to results of standard techniques.

2. Assessment of Predominant Harmonic

In a three-phase system, voltages and current waveform can be expressed as [𝑉_{𝑅,𝑌,𝐁}]=^{𝑛=1,2,3,…,𝑛𝑉}𝑉_𝑅𝑛sin(𝑛𝜔𝑡−𝜑_𝑅𝑛)_{𝑛=1,2,3,…,𝑛𝑉}𝑉_𝑌𝑛sin(𝑛𝜔𝑡−2𝜋/3−𝜑_𝑌𝑛)_{𝑛=1,2,3,…,𝑛𝑉}𝑉_𝐵𝑛sin(𝑛𝜔𝑡−4𝜋/3−𝜑_𝐵𝑛),[𝐼_{𝑅,𝑌,𝐁}]=^{𝑛=1,2,3,….}𝐼_𝑅𝑛sin(𝑛𝜔𝑡−𝜃_𝑅𝑛)_{𝑛=1,2,3,….}𝐼_𝑌𝑛sin(𝑛𝜔𝑡−2𝜋/3−𝜃_𝑌𝑛)_{𝑛=1,2,3,….}𝐼_𝐵𝑛sin(𝑛𝜔𝑡−4𝜋/3−𝜃_𝐵𝑛).(1) These phase voltages and line currents are transformed into 𝑑-𝑞 reference frame using well-known Park transformation matrix [16, 17] as follows: [𝑉_𝑑,𝑞,0𝑣]=𝑣^𝑑_𝑞𝑣₀=[ParkMatrix]×[𝑉_{𝑅,𝑌𝐵}],[𝐼_𝑑,𝑞,0𝐼]=𝐼^𝑑_𝑞𝐼₀=[ParkMatrix]×[𝐼_{𝑅,𝑌𝐵}].(2) In 𝑑-𝑞 plane, voltages and currents form closed loops in which the presence of harmonics brings cleavages (minima) as shown in Figure 1. The loop shown in this figure is created by the waves having 5th order of harmonics and has 4 cleavages. Table 1 gives the relation between the number of cleavages and highest order of predominant harmonic. The number of cleavages (minima) increases with the increase of order of highest harmonic [17]. In concludes that the highest order (𝑛_𝐻) is equal to the number of cleavages plus one, that is, 𝑛_𝐻=𝐶+1.(3)

Table 1 

Table for number of cleavage (𝐶) and order of highest harmonics (𝑛_𝐻).

Figure 1 

Loop formed by $I_d$ and $I_q$ of three-phase system containing fifth harmonic.

3. Principle of Area-Based Approach

3.1. Real and Imaginary Parts

If voltage and current waveform are plotted in 𝑣-𝑖 plane, then the area covered in one cycle is given by 𝐴^𝑣-𝑖_TOTAL=𝑣𝑑𝑖=𝑘_1𝑛𝑛𝑉_𝑛𝐼_𝑛sin𝜙_𝑛,(4) where, 𝜙_𝑛=𝜑_𝑛−𝜃_𝑛, and 𝑘₁=0.1𝜔.

Similarly, in 𝑣𝑖-𝑡 plane, the area covered by voltages and currents in one cycle is given by 𝐴^{𝑣𝑖-𝑡}_TOTAL=𝑣𝑖𝑑𝑡=𝑘_2𝑛𝑉_𝑛𝐼_𝑛cos𝜙_𝑛,(5) where, 𝜙_𝑛=𝜑_𝑛−𝜃_𝑛, and 𝑘₂ is a constant.

Equations (4) and (5) indicate that 𝐴^𝑣-𝑖_TOTAL contains the information of real part of power, and 𝐴^{𝑣𝑖-𝑡}_TOTAL contains the information of imaginary or reactive part of power. Also, it is noted that only harmonic components of the same order have contributions in the areas.

3.2. Introduction of Reference Signal

If any one of the voltage or current signal from a power system is available, then, it’s real and imaginary parts can be separated by plotting the signal with a known reference signal. General form of such a signal is defined as 𝑅(𝑡)=sin𝑛𝜔₁𝑡=sin𝑛2𝜋𝑓₁𝑡.(6) Here, 𝑓₁ is fundamental frequency of the reference signal. 𝑓₁ should be equal to the fundamental frequency 𝑓 of the voltage and current signals collected from the power system, which is initially an unknown quantity. 𝑓₁ is taken equal to 50 Hz, the standard power frequency. Then, the reference signal is plotted with any one of voltage and current signals. If the loop formed in this plotting is closed one then the fundamental frequency is confirmed as 50 Hz. Otherwise, the loop will be open, that is, there will be a finite gap between the starting and ending point as shown in Figure 2. Then, 𝑓₁ is set as (50±Δ𝑓), where, Δ𝑓 is a small incremental value. Δ𝑓 is to be adjusted so that starting and ending point meet each other. The frequency corresponding to this loop is taken as the fundamental frequency of the voltage and current waveform collected from the power system. Then, the reference signal is modified as 𝑅(𝑡)=sin𝑛2𝜋𝑓𝑡.(7)

Figure 2 

Curves formed by $R$ and $I_d$ having unequal fundamental frequency.

3.3. Assessment of Harmonic Component

The above reference signal is plotted with voltage and current signals, respectively, to find real and imaginary parts as discussed in Section 3.1. Contribution of fundamental component is assessed taking 𝑛=1 in the reference signal. For harmonic assessment, maximum limit of 𝑛 is taken as 𝑛_𝐻 as observed from (4).

Along 𝑑-axis, amplitudes and phase angles for 𝑛th order of voltage and current harmonic component are derived as 𝑉_𝑑𝑛=(𝐾₁𝐴^{𝑣𝑑−𝑅}_𝑑𝑛)²+(𝐾₂𝐴^{𝑣𝑑𝑅−𝑡}_𝑑𝑛)²,𝜑_𝑑𝑚=tan⁻¹(𝐾₁𝐴^{𝑣𝑑−𝑅}_𝑑𝑚𝐾₂𝐴^{𝑣𝑑𝑅−𝑡}_𝑑𝑚𝐼),_𝑑𝑛=(𝐾₁𝐴^{𝑖𝑑−𝑅}_𝑑𝑛)²+(𝐾₂𝐴^{𝑖𝑑𝑅−𝑡}_𝑑𝑛)²,𝜃_𝑑𝑛=tan⁻¹(𝐾₁𝐴^{𝑖𝑑−𝑅}_𝑑𝑛𝐾₂𝐴^{𝑖𝑑𝑅−𝑡}_𝑑𝑛).(8) As in active and reactive power only, harmonics components of same order contribute as discussed in Section 3.1, using (8), reactive and active power contributed by 𝑛th order of voltage and current harmonics are written as 𝑄_𝑑𝑛=12𝑉_𝑑𝑛𝐼_𝑑𝑛sin(𝜑_𝑑𝑛−𝜃_𝑑𝑛𝑃),_𝑑𝑛=12𝑉_𝑑𝑛𝐼_𝑑𝑛cos(𝜑_𝑑𝑛−𝜃_𝑑𝑛).(9) Similarly, along 𝑞-axis, amplitudes and phase angles for 𝑛th of voltage and current component are derived as 𝑉_𝑞𝑚=(𝐾₁𝐴^{𝑣𝑞−𝑅}_𝑞𝑚)²+(𝐾₂𝐴^{𝑣𝑞𝑅−𝑡}_𝑞𝑚)²,𝜑_𝑞𝑚=tan⁻¹(𝐾₁𝐴^{𝑣𝑞−𝑅}_𝑞𝑚𝐾₂𝐴^{𝑣𝑞𝑅−𝑡}_𝑞𝑚𝑖),_𝑞𝑛=(𝐾₁𝐴^{𝑖𝑞−𝑅𝑛}_𝑞𝑛)²+(𝐾₂𝐴^{𝑖𝑞𝑅−𝑡}_𝑞𝑛)²,𝜃_𝑞𝑛=tan⁻¹(𝐾₁𝐴^{𝑖𝑞−𝑅}_𝑞𝑛𝐾₂𝐴^{𝑖𝑞𝑅−𝑡}_𝑞𝑛).(10) Using (10), reactive and active power contributed by 𝑛th order of voltage and current harmonics are written as 𝑄_𝑞𝑛=12𝑉_𝑞𝑛𝐼_𝑞𝑛sin(𝜑_𝑞𝑛−𝜃_𝑞𝑛𝑃),_𝑞𝑛=12𝑉_𝑞𝑛𝐼_𝑞𝑛cos(𝜑_𝑞𝑛−𝜃_𝑞𝑛).(11)

4. Power Quality Parameters

4.1. Active Power Distortion Factor

Active power distortion factor is defined as the ratio of active power contributed by harmonic components to that contributed by fundamental component and is given by 𝑃PDF=^𝐻𝑃₁=∑_𝑛(𝑃_𝑑𝑛+𝑃_𝑞𝑛)𝑃₁.(12)

4.2. Reactive Power Distortion Factor

Reactive power distortion factor is defined as the ratio of reactive power contributed by harmonic components to that contributed by fundamental component and is given by 𝑄QDF=^𝐻𝑄₁=∑_𝑛(𝑄_𝑑𝑛+𝑄_𝑞𝑛)𝑄_𝑑1+𝑄_𝑞1.(13)

4.3. Total Power

Total power is given by 𝑆=𝑃+𝑗𝑄=_𝑛(𝑃_𝑑𝑛+𝑃_𝑞𝑛)+𝑗_𝑛(𝑄_𝑑𝑛+𝑄_𝑞𝑛).(14)

4.4. Total Harmonic Distortion Factors

Total harmonic distortion of voltage along 𝑑-axis is given by THD_𝑑𝑉=∑_𝑛𝑉²_𝑑𝑛𝑉_𝑑1.(15) Total harmonic distortion of voltage along 𝑞-axis is given by THD_𝑞𝑉=∑_𝑛𝑉²_𝑞𝑛𝑉_𝑞1.(16) Total harmonic distortion of current along 𝑑-axis is given by THD_𝑑𝐼=∑_𝑛𝐼²_𝑑𝑛𝐼_𝑑1.(17) Total harmonic distortion of current along 𝑞-axis is given by THD_𝑞𝐼=∑_𝑛𝐼²_𝑞𝑛𝐼_𝑞1.(18)

5. Computer Simulation

5.1. Simulated Network

A three-phase radial power system having motor and static load has been simulated in MATLAAB as shown in Figure 3. Voltage at bus 1 and current drawn from the alternator are observed in the alternator bus. Voltage signal is stepped down using potential transformer, and current signal is stepped down using current transformer. These signals are then sampled using a sampling unit. The sampled data are digitized and taken into consideration to form loops.

Figure 3 

Schematic diagram of simulated network.

5.2. Algorithm

The analyzer works obeying an algorithm developed based on the principles discussed in Sections 3 and 4. The algorithm has been explained with the help of a flow diagram shown in Figure 4. First fundamental frequency is determined. Then, reference signal is set, and different loops are to be formed. Areas are calculated to assess fundamental as well as harmonic components. After this active power, reactive power and apparent power are calculated.

Figure 4 

Flow diagram for analyzer.

5.3. Results

Different distortion factors are calculated using above algorithm. The result is designated as calculated value. The same data are assessed using Fourier analysis, and the result is designated as true value. The true value is compared with the calculated value. The error has been calculated, and satisfactory results containing low error have been achieved. Sample results in measurement of active power distortion, reactive power distortion, and total harmonic distortion are presented in Tables 2, 3, and 4. Error decreases with the increase of the sampling rate of the sampler.

Table 2 

Table 3 

Table 4 

6. Conclusions

Power quality parameters have been assessed using analysis of fundamental and harmonic voltage and current waveforms. Instead of using three line currents and three-phase voltages, use of 𝑑- and 𝑞-axis currents and voltages reduces the number of variable to four (𝑉_𝑑, 𝑉_𝑞, 𝐼_𝑑, and 𝐼_𝑞) from six (𝑉_𝑅, 𝑉_𝑌, 𝑉_𝐵, 𝐼_𝑅, 𝐼_𝑌, and 𝐼_𝐵). Hence, it decreases the executing time and memory space required for the analysis. Thus, Park transformation technique, in three-phase system, has reduced the computational effort to a great extent. The same approach can be applied for quality assessment of each individual phase. In such cases, Park transformation will not be required and will again reduce the computational effort. An algorithm has been developed to calculate the power quality parameters from the online signals to assess contributions of fundamental and harmonic components in power system voltage and current signals separately. The results obtained from simulated network using the algorithm have been compared with that obtained using a standard FFT-based technique. The results are seen to be in good agreement with that of the standard way. Small error speaks of the acceptability of the above area-based approach.

List of Symbols and Abbreviations

𝑁:	Order of current harmonics
𝑛𝐻:	Order of highest harmonics
𝜑𝑚:	Phase angle of voltage of harmonics of order 𝑚
𝜃𝑛:	Phase angle of current of harmonics of order 𝑛
𝐴𝑣𝑑-𝑅𝑑𝑚:	Area covered by curve 𝑣𝑑 and 𝑅𝑉𝑚 waveforms in 𝑣𝑑-𝑅
𝐴𝑣𝑑𝑅-𝑡𝑑𝑚:	Area formed by curve in 𝑣𝑑 𝑅-𝑡 plane
𝐴𝑖𝑑-𝑅𝑑𝑛:	Area covered by curve in 𝑖𝑑-𝑅 plane
𝐴𝑖𝑑𝑅-𝑡𝑑𝑛:	Area formed by curve in 𝑖𝑑𝑅-𝑡 plane
𝐴𝑣𝑞-𝑅𝑞𝑚:	Area formed by curve in 𝑣𝑞-𝑅 plane
𝐴𝑣𝑚𝑅-𝑡𝑞𝑚:	Area formed by curve in 𝑣𝑞𝑅-𝑡 plane
𝐴𝑖𝑞-𝑅𝑞𝑛:	Area formed by curve in 𝑖𝑞-𝑅 plane
𝐴𝑖𝑞𝑅-𝑡𝑞𝑛:	Area formed by curve in 𝑖𝑞𝑅-𝑡 plane
𝑄𝑑𝑛:	Reactive power contributed by harmonic component with frequency of order 𝑛 along 𝑑 axis
𝑄𝑞𝑛:	Reactive power contributed by harmonic component with frequency of order 𝑛 along 𝑞 axis
𝑄𝐻:	Total reactive power contributed by harmonic components
𝑄1:	Total reactive power contributed by fundamental component
QDF:	Reactive power distortion factor
𝑃𝑑𝑛:	Active power contributed by harmonic component with frequency of order 𝑛 along 𝑑 axis
𝑃𝑞𝑛:	Active power contributed by harmonic component with frequency of order 𝑛 along 𝑞 axis
PDF:	Active power distortion factor
THD𝑑𝑉:	Along 𝑑-axis total harmonic distortion of voltage
THD𝑞𝑉:	Along 𝑞-axis total harmonic distortion of voltage
THD𝑑𝐼:	Along 𝑑-axis total harmonic distortion of current
THD𝑞𝐼:	Along 𝑞-axis total harmonic distortion of current
THD𝑌𝑉:	Total harmonic distortion factor for voltage in phase 𝑌
THD𝑌𝐼:	Total harmonic distortion factor for current in phase 𝑌.