# Area-Based Approach in Power Quality Assessment

Authors: Surajit Chattopadhyay; Samarjit Sengupta; Madhuchhanda Mitra
Journal: International Journal of Power Management Electronics
 (2008)

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## 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.


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## Body

## 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 voltagesv(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(1)[VR,Y,B]=[∑n=1,2,3,…,nVVRnsin(nωt−φRn)∑n=1,2,3,…,nVVYnsin(nωt−2π/3−φYn)∑n=1,2,3,…,nVVBnsin(nωt−4π/3−φBn)],[IR,Y,B]=[∑n=1,2,3,….IRnsin(nωt−θRn)∑n=1,2,3,….IYnsin(nωt−2π/3−θYn)∑n=1,2,3,….IBnsin(nωt−4π/3−θBn)]. These phase voltages and line
currents are transformed into d-q reference frame using well-known Park transformation
matrix [16, 17] as follows: (2)[Vd,q,0]=[vdvqv0]=[ParkMatrix]×[VR,YB],[Id,q,0]=[IdIqI0]=[ParkMatrix]×[IR,YB].
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 (nH) is equal to the number of
cleavages plus one, that is, (3)nH=C+1.Table 1
Table for number
of cleavage (C) and order of highest harmonics (nH).
nH1235713C0124612Figure 1
Loop formed byId and Iq 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 inv-i plane, then the area covered in one cycle is given by (4)ATOTALv-i=∫vdi=k1∑nnVnInsinϕn, where, ϕn=φn−θn, and k1=0.1ω.Similarly, invi-t plane, the area
covered by voltages and currents in one cycle is given by (5)ATOTALvi-t=∫vidt=k2∑nVnIncosϕn, where, ϕn=φn−θn, and k2 is a constant.Equations (4) and (5) indicate  that ATOTALv-i contains the information of real part of power,
and ATOTALvi-t 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(6)R(t)=sinnω1t=sinn2πf1t.
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 (7)R(t)=sinn2πft.Figure 2
Curves formed byR
and Id 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
Section3.1. Contribution of fundamental component
is assessed taking n=1 in the reference signal. For harmonic assessment,
maximum limit of n is taken as nH as observed from (4).Alongd-axis, amplitudes and
phase angles for nth order of voltage and current harmonic component are
derived as (8)Vdn=(K1Adnvd−R)2+(K2AdnvdR−t)2,φdm=tan−1(K1Admvd−RK2AdmvdR−t),Idn=(K1Adnid−R)2+(K2AdnidR−t)2,θdn=tan−1(K1Adnid−RK2AdnidR−t).
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
(9)Qdn=12VdnIdnsin(φdn−θdn),Pdn=12VdnIdncos(φdn−θdn).
Similarly, along q-axis,
amplitudes and phase angles for nth of voltage and current component are
derived as
(10)Vqm=(K1Aqmvq−R)2+(K2AqmvqR−t)2,φqm=tan−1(K1Aqmvq−RK2AqmvqR−t),iqn=(K1Aqniq−Rn)2+(K2AqniqR−t)2,θqn=tan−1(K1Aqniq−RK2AqniqR−t).
Using (10),  reactive and active
power contributed by nth order of voltage and current harmonics are written as
(11)Qqn=12VqnIqnsin(φqn−θqn),Pqn=12VqnIqncos(φqn−θqn).


## 3.1. Real and Imaginary Parts

If voltage and current waveform are
plotted inv-i plane, then the area covered in one cycle is given by (4)ATOTALv-i=∫vdi=k1∑nnVnInsinϕn, where, ϕn=φn−θn, and k1=0.1ω.Similarly, invi-t plane, the area
covered by voltages and currents in one cycle is given by (5)ATOTALvi-t=∫vidt=k2∑nVnIncosϕn, where, ϕn=φn−θn, and k2 is a constant.Equations (4) and (5) indicate  that ATOTALv-i contains the information of real part of power,
and ATOTALvi-t 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(6)R(t)=sinnω1t=sinn2πf1t.
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 (7)R(t)=sinn2πft.Figure 2
Curves formed byR
and Id 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
Section3.1. Contribution of fundamental component
is assessed taking n=1 in the reference signal. For harmonic assessment,
maximum limit of n is taken as nH as observed from (4).Alongd-axis, amplitudes and
phase angles for nth order of voltage and current harmonic component are
derived as (8)Vdn=(K1Adnvd−R)2+(K2AdnvdR−t)2,φdm=tan−1(K1Admvd−RK2AdmvdR−t),Idn=(K1Adnid−R)2+(K2AdnidR−t)2,θdn=tan−1(K1Adnid−RK2AdnidR−t).
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
(9)Qdn=12VdnIdnsin(φdn−θdn),Pdn=12VdnIdncos(φdn−θdn).
Similarly, along q-axis,
amplitudes and phase angles for nth of voltage and current component are
derived as
(10)Vqm=(K1Aqmvq−R)2+(K2AqmvqR−t)2,φqm=tan−1(K1Aqmvq−RK2AqmvqR−t),iqn=(K1Aqniq−Rn)2+(K2AqniqR−t)2,θqn=tan−1(K1Aqniq−RK2AqniqR−t).
Using (10),  reactive and active
power contributed by nth order of voltage and current harmonics are written as
(11)Qqn=12VqnIqnsin(φqn−θqn),Pqn=12VqnIqncos(φqn−θqn).


## 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(12)PDF=PHP1=∑n(Pdn+Pqn)P1.


### 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(13)QDF=QHQ1=∑n(Qdn+Qqn)Qd1+Qq1.


### 4.3. Total Power

Total power is given by(14)S=P+jQ=∑n(Pdn+Pqn)+j∑n(Qdn+Qqn).


### 4.4. Total Harmonic Distortion Factors

Total harmonic distortion of voltage alongd-axis is
given by
(15)THDdV=∑nVdn2Vd1. Total harmonic distortion of
voltage along q-axis is given by
(16)THDqV=∑nVqn2Vq1. Total harmonic distortion of current along d-axis is
given by
(17)THDdI=∑nIdn2Id1. Total harmonic  distortion of current along q-axis is
given by
(18)THDqI=∑nIqn2Iq1.


## 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(12)PDF=PHP1=∑n(Pdn+Pqn)P1.


## 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(13)QDF=QHQ1=∑n(Qdn+Qqn)Qd1+Qq1.


## 4.3. Total Power

Total power is given by(14)S=P+jQ=∑n(Pdn+Pqn)+j∑n(Qdn+Qqn).


## 4.4. Total Harmonic Distortion Factors

Total harmonic distortion of voltage alongd-axis is
given by
(15)THDdV=∑nVdn2Vd1. Total harmonic distortion of
voltage along q-axis is given by
(16)THDqV=∑nVqn2Vq1. Total harmonic distortion of current along d-axis is
given by
(17)THDdI=∑nIdn2Id1. Total harmonic  distortion of current along q-axis is
given by
(18)THDqI=∑nIqn2Iq1.


## 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 Figure3. 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 Sections3 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
Tables2, 3, and 4. Error decreases
with the increase of the sampling rate of the sampler.Table 2PDFTruePDFCalculatedERROR0.12460.12470.00010.12840.12840.00000.14220.14230.00010.23500.23500.00000.35300.35320.0002Table 3QDFTrueQDFCalculatedERROR1.16621.16640.00021.20421.20410.00011.42001.42000.00001.56671.56680.00011.16001.16000.0000Table 4THDd-TrueTHDd-CalculatedERROR0.12420.12420.00000.20240.20250.00010.30320.30280.00020.19980.19990.00010.36000.36020.0002


## 5.1. Simulated Network

A three-phase radial power system having motor and
static load has been simulated in MATLAAB as shown in Figure3. 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 Sections3 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
Tables2, 3, and 4. Error decreases
with the increase of the sampling rate of the sampler.Table 2PDFTruePDFCalculatedERROR0.12460.12470.00010.12840.12840.00000.14220.14230.00010.23500.23500.00000.35300.35320.0002Table 3QDFTrueQDFCalculatedERROR1.16621.16640.00021.20421.20410.00011.42001.42000.00001.56671.56680.00011.16001.16000.0000Table 4THDd-TrueTHDd-CalculatedERROR0.12420.12420.00000.20240.20250.00010.30320.30280.00020.19980.19990.00010.36000.36020.0002


## 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 ofd- and q-axis
currents and voltages reduces the number of variable to four (Vd, Vq,
Id, and Iq) from six (VR, VY, VB,
IR, IY, and IB). 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.



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*Source: 147359-2008-07-08.xml*