**2. Data reliability criteria**

This section is intended to present the main data reliability criteria proposed to be employed in the power quality applications addressed in this chapter. The criteria are applied in a slightly different manner to fit the nature of each problem. As it will be explained in this chapter, in the context of the network impedance estimation, the concern is *ΔI(f)* and *ΔV(f)* (the variation of the voltage or current), whereas for the case of interharmonic measurement, the concern is the value of *I(f)* and *V(f)*. The reason for this will be explained in more detail in sections 3 and 4, and at this point it is just important to keep in mind that the introduced criteria is applied in both cases, but with this slight difference.

#### **2.1 The energy level index**

70 Power Quality Harmonics Analysis and Real Measurements Data

where *ΔV(h)* and *ΔI(h)* are the subtraction in frequency domain of one or more cycles previous to the transient occurrence from the corresponding cycles containing the transient disturbance. The objective of this chapter is not to promote the use of the transient-based approach for determining the network harmonic impedance, nor is it to explain the method in detail. The reader is encouraged to consult (Robert & Deflandre, 1997) for details. In this application, the level of accuracy of such estimation can be supported by a set of indices, which are (but not limited to) the quantization noise in the data acquisition, the frequency

resolution, the energy levels, and the scattering of the results obtained from the data.

which rely on the active power index (Kim et al, 2005), (Axelberg et al, 2008).

The objective of this research is to present a set of reliability criteria to evaluate recorded data used to assess power quality disturbances. The targets of the proposed methods are the data used in the determination of the network harmonic impedance and the identification of interharmonic sources. This chapter is structured as follows. Section 2 presents the data reliability criteria to be applied to both challenges. Section 3 presents the harmonic impedance determination problem and section 4 presents a network determination case study. Section 5 presents the interharmonic source determination problem and sections 6 and 7 present two case studies. Section 8 presents general conclusions and

This section is intended to present the main data reliability criteria proposed to be employed in the power quality applications addressed in this chapter. The criteria are applied in a slightly different manner to fit the nature of each problem. As it will be explained in this chapter, in the context of the network impedance estimation, the concern is *ΔI(f)* and *ΔV(f)* (the variation of the voltage or current), whereas for the case of interharmonic measurement,

Interharmonics are spectral components which frequencies are non-integer multiples of the supply fundamental frequency. This power quality event represents the target of the second application of the proposed reliability criteria. Diagnosing interharmonic problems is a difficult task for a number of reasons: (1) interharmonics do not manifest themselves in known and/or fixed frequencies, as they vary with the operating conditions of the interharmonic-producing load; (2) interharmonics can cause flicker in addition to distorting the waveforms, which makes them more harmful than harmonics; (3) they are hard to analyze, as they are related to the problem of waveform modulation (IEEE Task Force, 2007). The most common effects of interharmonics have been well documented in literature (IEEE Task Force, 2007), (Ghartemani & Iravani, 2005)-(IEEE Interhamonic Task Force, 1997), (Yacamini, 1996). Much of the published material on interharmonics has identified the importance of determining the interharmonic source (Nassif et al, 2009, 2010a, 2010b). Only after the interharmonic source is identified, it is possible to assess the rate of responsibility and take suitable measures to design mitigation schemes. Interharmonic current spectral bins, which are typically of very low magnitude, are prone to suffer from their inherently low energy level. Due to this difficulty, the motivation of the proposed reliability criteria is to strengthen existing methods for determining the source of interharmonics and flicker

**1.2 The Interharmonics measurement** 

**1.3 Objectives and outline** 

recommendations.

**2. Data reliability criteria** 

As shown in (1), the network impedance determination is heavily reliant on *ΔI(f)*, which is the denominator of the expression. Any inaccuracy on this parameter can result in great numerical deviance of the harmonic impedance accurate estimation. Therefore, the *ΔI(f)* energy level is of great concern. For this application, a threshold was suggested in (Xu et al, 2002) and is present in (2). If the calculated index is lower than the threshold level, the results obtained using these values are considered unreliable.

$$I\_{throldd} = \frac{\Delta I \left(f\right)}{\Delta I \left(60Hz\right)} > 1\,\%. \tag{2}$$

Fig. 1 shows an example on how this criterion can be used. The energy level for *ΔI(f)* is compared with the threshold. For this case, frequencies around the 25th harmonic order (1500Hz) are unreliable according to this criterion.

Fig. 1. Energy level of *ΔI(f)* seen in a three-dimensional plot

#### **2.2 Frequency-domain coherence index**

This index is used in the problem of the network impedance estimation, which relies on the transient portion of the recorded voltages and currents (section 3 presents the method in detail). The random nature of a transient makes it a suitable application for using the power density spectrum (Morched & Kundur, 1987). The autocorrelation function of a random process is the appropriate statistical average, and the Fourier transform of the autocorrelation function provides the transformation from time domain to frequency domain, resulting in the power density spectrum.

On the Reliability of Real Measurement Data for Assessing Power Quality Disturbances 73

and 0.0905 (ohms). Those values that are numerically distant from the rest of the data (shown inside the circles) may spoil the final result as those data are probably gross results. As per statistics theory, in the case of normally distributed data, 97 percent of the observations will differ by less than three times the standard deviation [14]. In the study presented in this chapter, the three standard deviation criterion is utilized to statistically

0 5 10 15 20 25 30

Snapshots

In the example presented in Fig. 2, once the resistance of the network is achieved by averaging the filtered data, the confidence on the obtained results might be questioned. Instead of estimating the parameter by a single value, an interval likely to include the parameter is evaluated. Confidence intervals are used to indicate the reliability of such an estimate (Harnett, 1982). How likely the interval is to contain the parameter is determined by the confidence level or confidence coefficient. Increasing the desired confidence level will widen the confidence interval. For example, a 90% confidence interval for the achieved resistance will result in a 0.0717 ± 0.0055 confidence interval. In the other words, the resistance of the network is likely to be between 0.662 and 0.772 (ohms) with a probability of

Fig. 3 shows the calculated harmonic impedance of the network. Error bars are used to show the confidence intervals of the results. Larger confidence intervals present less reliable values. In this regard, the estimated resistance at 420 Hz is more reliable than its counterpart

0 200 400 600 800

Frequency (Hz)

Fig. 3. Selected 5th harmonic resistance data showing confidence intervals.

filter the outlier data.

90%.

at 300 Hz.

0

0

0.02

0.04

R (Ohm)

0.06

0.08 0.1

Fig. 2. Calculated 5th harmonic resistance over a number of snapshots

0.05

R300 Hz

0.1

This relationship can be understood as a transfer function. The concept of transfer function using the power spectral method based on correlation functions can be treated as the result from dividing the cross-power spectrum by the auto-power spectrum. For electrical power systems, if the output is the voltage and the input is the current, the transfer function is the impedance response of the system (Morched & Kundur, 1987). The degree of accuracy of the transfer function estimation can be assessed by the coherence function, which gives a measure of the power in the system output due to the input. This index is used as a data selection/rejection criterion and is given by

$$\mathcal{Y}\_{\scriptscriptstyle{\rm V}}\left(f\right) = \frac{\left|P\_{\scriptscriptstyle{\rm V}}\left(f\right)\right|^{2}}{P\_{\scriptscriptstyle{\rm V}}\left(f\right)P\_{\scriptscriptstyle{\rm I}}\left(f\right)},\tag{3}$$

where *PVI(f)* is the cross-power spectrum of the voltage and current, which is obtained by the Fourier transform of the correlation between the two signals. Similarly, the auto-power spectrum *PVV(f)* and *PII(f)* are the Fourier transforms of the voltage and current autocorrelation, respectively. By using the coherence function, it is typically revealed that a great deal of data falls within the category where input and output do not constitute a cause-effect relationship, which is the primary requirement of a transfer function.

#### **2.3 Time-domain correlation between interharmonic current and voltage spectra**

This index is used for the interharmonic source detection analysis, and is the time-domain twofold of the coherence index used for the harmonic system impedance. The criterion is supported by the fact that, if genuine interharmonics do exist, voltage and current spectra should show a correlation (Li et al, 2001) because an interharmonic injection will result in a voltage across the system impedance, and therefore both the voltage and current should show similar trends at that frequency. As many measurement snapshots are taken, the variation over time of the interharmonic voltage and current trends are observed, and their correlation is analyzed. In order to quantify this similarity, the correlation coefficient is used (Harnett, 1982):

$$r\left(ih\right) = \frac{n\sum\_{i=1}^{n} I\_{\mathcal{H}}\left(i\right) V\_{\mathcal{H}}\left(i\right) - \sum\_{i=1}^{n} I\_{\mathcal{H}}\left(i\right) \sum\_{i=1}^{n} V\_{\mathcal{H}}\left(i\right)}{\sqrt{\left[n\sum\_{i=1}^{n} I\_{\mathcal{H}}\left(i\right) - \left(\sum\_{i=1}^{n} I\_{\mathcal{H}}\left(i\right)\right)^{2}\right] \left[n\sum\_{i=1}^{n} V\_{\mathcal{H}}^{2}\left(i\right) - \left(\sum\_{i=1}^{n} V\_{\mathcal{H}}\left(i\right)\right)^{2}\right]}},\tag{4}$$

where *IIH* and *VIH* are the interharmonic frequency current and the voltage magnitudes of the *n*-snapshot interharmonic data, respectively. Frequencies showing the calculated correlation coefficient lower than an established threshold should not be reliable, as they may not be genuine interharmonics (Li et al, 2001).

#### **2.4 Statistical data filtering and confidence intervals**

In many power quality applications, the measured data are used in calculations to obtain parameters that are subsequently used in further analyses. For example, in the network impedance estimation problem, the calculated resistance of the network may vary from 0.0060 to 0.0905 (ohms) in different snapshots (see Fig. 2). The resistance of the associated network is the average of these results. Most of the calculated resistances are between 0.0654

This relationship can be understood as a transfer function. The concept of transfer function using the power spectral method based on correlation functions can be treated as the result from dividing the cross-power spectrum by the auto-power spectrum. For electrical power systems, if the output is the voltage and the input is the current, the transfer function is the impedance response of the system (Morched & Kundur, 1987). The degree of accuracy of the transfer function estimation can be assessed by the coherence function, which gives a measure of the power in the system output due to the input. This index is used as a data

where *PVI(f)* is the cross-power spectrum of the voltage and current, which is obtained by the Fourier transform of the correlation between the two signals. Similarly, the auto-power spectrum *PVV(f)* and *PII(f)* are the Fourier transforms of the voltage and current autocorrelation, respectively. By using the coherence function, it is typically revealed that a great deal of data falls within the category where input and output do not constitute a cause-effect

**2.3 Time-domain correlation between interharmonic current and voltage spectra**  This index is used for the interharmonic source detection analysis, and is the time-domain twofold of the coherence index used for the harmonic system impedance. The criterion is supported by the fact that, if genuine interharmonics do exist, voltage and current spectra should show a correlation (Li et al, 2001) because an interharmonic injection will result in a voltage across the system impedance, and therefore both the voltage and current should show similar trends at that frequency. As many measurement snapshots are taken, the variation over time of the interharmonic voltage and current trends are observed, and their correlation is analyzed. In order to quantify this similarity, the correlation coefficient is used

*VI*

relationship, which is the primary requirement of a transfer function.

1 1 1

where *IIH* and *VIH* are the interharmonic frequency current and the voltage magnitudes of the *n*-snapshot interharmonic data, respectively. Frequencies showing the calculated correlation coefficient lower than an established threshold should not be reliable, as they

In many power quality applications, the measured data are used in calculations to obtain parameters that are subsequently used in further analyses. For example, in the network impedance estimation problem, the calculated resistance of the network may vary from 0.0060 to 0.0905 (ohms) in different snapshots (see Fig. 2). The resistance of the associated network is the average of these results. Most of the calculated resistances are between 0.0654

*n I iV i I i V i*

*nIi Ii nVi Vi*

*n n n IH IH IH IH i i i nn n n IH IH IH IH ii i i*

2 2 11 1 1

2 2

,

(4)

*VV II P f <sup>f</sup> <sup>P</sup> <sup>f</sup> <sup>P</sup> <sup>f</sup>*

2 , *VI*

(3)

selection/rejection criterion and is given by

(Harnett, 1982):

may not be genuine interharmonics (Li et al, 2001).

**2.4 Statistical data filtering and confidence intervals** 

*r ih*

and 0.0905 (ohms). Those values that are numerically distant from the rest of the data (shown inside the circles) may spoil the final result as those data are probably gross results. As per statistics theory, in the case of normally distributed data, 97 percent of the observations will differ by less than three times the standard deviation [14]. In the study presented in this chapter, the three standard deviation criterion is utilized to statistically filter the outlier data.

Fig. 2. Calculated 5th harmonic resistance over a number of snapshots

In the example presented in Fig. 2, once the resistance of the network is achieved by averaging the filtered data, the confidence on the obtained results might be questioned. Instead of estimating the parameter by a single value, an interval likely to include the parameter is evaluated. Confidence intervals are used to indicate the reliability of such an estimate (Harnett, 1982). How likely the interval is to contain the parameter is determined by the confidence level or confidence coefficient. Increasing the desired confidence level will widen the confidence interval. For example, a 90% confidence interval for the achieved resistance will result in a 0.0717 ± 0.0055 confidence interval. In the other words, the resistance of the network is likely to be between 0.662 and 0.772 (ohms) with a probability of 90%.

Fig. 3 shows the calculated harmonic impedance of the network. Error bars are used to show the confidence intervals of the results. Larger confidence intervals present less reliable values. In this regard, the estimated resistance at 420 Hz is more reliable than its counterpart at 300 Hz.

Fig. 3. Selected 5th harmonic resistance data showing confidence intervals.

On the Reliability of Real Measurement Data for Assessing Power Quality Disturbances 75

Many methods that deal with measuring the harmonic impedance have been proposed and published (Xu et al, 2002), (Morched & Kundur, 1987), (Oliveira et al, 1991). They can be classified as either invasive or non-invasive methods. Invasive methods are intended to produce a disturbance with energy high enough to change the state of the system to a different post-disturbance state. Such change in the system is necessary in order to obtain data records to satisfy (9) and (10), but low enough not to affect the operation of network equipment. The applied disturbance in the system generally causes an obvious transient in the voltage and current waveforms. The transient voltage and current data are used to obtain the impedance at harmonic frequencies. For the case presented in this chapter, the source of disturbance is a low voltage capacitor bank, but other devices can also be used, as

Therefore, the transient signal is extracted by subtracting one or more intact pre-disturbance

*transient disturbance pre disturbance transient disturbance pre disturbance*

*VV V V II I I*

Traditionally, transients are characterized by their magnitude and duration. For the application of network impedance estimation, the harmonic content of a transient is a very useful piece of information. A transient due to the switching of a capacitor has the following

Magnitude: up to 2 times the pre-existing voltage (assuming a previously discharged

The energization of the capacitor bank (isolated switching) typically results in a mediumfrequency oscillatory voltage transient with a primary frequency between 300 and 900 Hz and magnitude of 1.3-1.5 p.u., and not longer than two 60Hz cycles. Fig. 5 shows typical transient waveforms and frequency contents due to a capacitor switching. For this case, the higher frequency components (except the fundamental component) are around 5th to 10th

\_ \_

, .

(9)

. *Z h Vh Ih eq* (10)

explained in (Xu et al, 2002), (Morched & Kundur, 1987), (Oliveira et al, 1991)

Fig. 4. Equivalent circuit for system impedance measurement.

cycles from the cycles containing the transient, as

Finally, the network impedance is calculated by using

characteristics (IEEE Std. 1159-1995):

Duration: From 0.3ms to 50ms.

harmonic (300-600Hz).

Main frequency component: 300Hz to 5 kHz.

capacitor).

**3.1 Characterization of the capacitor switching transient** 

#### **2.5 Quantization error**

Quantization refers to the digitalization step of the data acquisition equipment. This value dictates the magnitude threshold that a measurement must have to be free of measurement quantization noise (Oppenheim & Shafer, 1999). The A/D conversion introduces quantization error. The data collected are in the form of digital values while the actual data are in analog form. So the data are digitalized with an A/D converter. The error associated with this conversion is the quantization step. As the energy of current signals drops to a level comparable to that of quantization noises, the signal may be corrupted, and the data will, therefore, be unreliable. For this reason, if the harmonic currents are of magnitude lower than that of the quantization error, they should not be trusted. This criterion was developed as follows:

1. The step size of the quantizer is

$$
\Delta = V\_{in} \not\!\!\!\!\!\!\!\!\!\!\!\!\!\!\/\!\!\!\/\!\!\/\!\!\/\!\/\!\/\,\tag{5}
$$

where *n* is the number of bits and *Vin* is the input range.


$$
\Delta\_l = \Delta \not{k}\_{pro\text{\textquotedblleft}}\tag{6}
$$

4. Finally, the maximum quantization error will be half of the step size.

The input range, number of bits and current probe ratio will depend on the data acquisition equipment and measurement set up. The measurements presented later in this chapter are acquired by high-resolution equipment (NI-6020E - 100kbps, 12-bit, 8 channels). For the case of the system impedance estimation, equation (7) should hold true in order to generate reliable results for single-phase systems. This criterion is also used for *ΔV(f)*:

$$\left|\Delta I(f)\right| > I\_{err}.\tag{7}$$

For the interharmonic case, the interharmonic current level *I(ih)* is monitored rather than the *ΔI(f)*:

$$\left| I(ilh) \right| > I\_{err} \,\, . \tag{8}$$

#### **3. Network harmonic impedance estimation by using measured data**

The problem of the network harmonic impedance estimation by using measured data is explained in this section. Fig. 4 presents a typical scenario where measurements are taken to estimate the system harmonic impedance. Voltage and current probes are installed at the interface point between the network and the customer, called the point of common coupling (PCC). These probes are connected to the national instrument NI-6020E 12-bit data acquisition system with a 100 kHz sampling rate controlled by a laptop computer. Using this data-acquisition system, 256 samples per cycle were obtained for each waveform. In Fig. 4, the impedance *Zeq* is the equivalent impedance of the transmission and distribution lines, and of the step-down and step-up transformers.

Fig. 4. Equivalent circuit for system impedance measurement.

Quantization refers to the digitalization step of the data acquisition equipment. This value dictates the magnitude threshold that a measurement must have to be free of measurement quantization noise (Oppenheim & Shafer, 1999). The A/D conversion introduces quantization error. The data collected are in the form of digital values while the actual data are in analog form. So the data are digitalized with an A/D converter. The error associated with this conversion is the quantization step. As the energy of current signals drops to a level comparable to that of quantization noises, the signal may be corrupted, and the data will, therefore, be unreliable. For this reason, if the harmonic currents are of magnitude lower than that of the quantization error, they should not be trusted. This criterion was

2 , *<sup>n</sup> Vin* (5)

. *I probe k* (6)

() . *error If I* (7)

() . *error I ih I* (8)

**2.5 Quantization error** 

developed as follows:

*ΔI(f)*:

1. The step size of the quantizer is

3. Therefore, the step size in amperes is

and of the step-down and step-up transformers.

where *n* is the number of bits and *Vin* is the input range. 2. The current probe ratio is *kprobe*, which is the ratio V/A.

4. Finally, the maximum quantization error will be half of the step size.

reliable results for single-phase systems. This criterion is also used for *ΔV(f)*:

**3. Network harmonic impedance estimation by using measured data** 

The input range, number of bits and current probe ratio will depend on the data acquisition equipment and measurement set up. The measurements presented later in this chapter are acquired by high-resolution equipment (NI-6020E - 100kbps, 12-bit, 8 channels). For the case of the system impedance estimation, equation (7) should hold true in order to generate

For the interharmonic case, the interharmonic current level *I(ih)* is monitored rather than the

The problem of the network harmonic impedance estimation by using measured data is explained in this section. Fig. 4 presents a typical scenario where measurements are taken to estimate the system harmonic impedance. Voltage and current probes are installed at the interface point between the network and the customer, called the point of common coupling (PCC). These probes are connected to the national instrument NI-6020E 12-bit data acquisition system with a 100 kHz sampling rate controlled by a laptop computer. Using this data-acquisition system, 256 samples per cycle were obtained for each waveform. In Fig. 4, the impedance *Zeq* is the equivalent impedance of the transmission and distribution lines, Many methods that deal with measuring the harmonic impedance have been proposed and published (Xu et al, 2002), (Morched & Kundur, 1987), (Oliveira et al, 1991). They can be classified as either invasive or non-invasive methods. Invasive methods are intended to produce a disturbance with energy high enough to change the state of the system to a different post-disturbance state. Such change in the system is necessary in order to obtain data records to satisfy (9) and (10), but low enough not to affect the operation of network equipment. The applied disturbance in the system generally causes an obvious transient in the voltage and current waveforms. The transient voltage and current data are used to obtain the impedance at harmonic frequencies. For the case presented in this chapter, the source of disturbance is a low voltage capacitor bank, but other devices can also be used, as explained in (Xu et al, 2002), (Morched & Kundur, 1987), (Oliveira et al, 1991)

Therefore, the transient signal is extracted by subtracting one or more intact pre-disturbance cycles from the cycles containing the transient, as

$$\begin{aligned} V\_{t\_{\text{trainint}}} &= V\_{t\_{\text{disturbnar}}} - V\_{pv\_{-\text{disturbnar}}} = \Delta V\_{\prime} \\ I\_{t\_{\text{twoister}}} &= I\_{disturbnar} - I\_{pv\_{-\text{disturbnar}}} = \Delta I. \end{aligned} \tag{9}$$

Finally, the network impedance is calculated by using

$$Z\_{\
u} \begin{pmatrix} h \end{pmatrix} = -\Delta V \begin{pmatrix} h \end{pmatrix} \Big/ \Delta I \begin{pmatrix} h \end{pmatrix}. \tag{10}$$

#### **3.1 Characterization of the capacitor switching transient**

Traditionally, transients are characterized by their magnitude and duration. For the application of network impedance estimation, the harmonic content of a transient is a very useful piece of information. A transient due to the switching of a capacitor has the following characteristics (IEEE Std. 1159-1995):

Magnitude: up to 2 times the pre-existing voltage (assuming a previously discharged capacitor).


The energization of the capacitor bank (isolated switching) typically results in a mediumfrequency oscillatory voltage transient with a primary frequency between 300 and 900 Hz and magnitude of 1.3-1.5 p.u., and not longer than two 60Hz cycles. Fig. 5 shows typical transient waveforms and frequency contents due to a capacitor switching. For this case, the higher frequency components (except the fundamental component) are around 5th to 10th harmonic (300-600Hz).

On the Reliability of Real Measurement Data for Assessing Power Quality Disturbances 77

Resistance (R) Reactance (X)

0 500 1000 1500 2000 2500 3000

*Ratio Successfull cases Total cases* 100%, (11)

Frequency [Hz]

As the reliability criteria are applied, it is useful to define the following ratio of success:

Fig. 7 shows the success rate of cases for each index in function of frequency. Fig. 7a. also shows that the application of this index will affect *ΔI* much more than for *ΔV*, since the latter is acquired using voltage probes, which are inherently much more reliable. Since the impedance measurement is calculated from the ratio -*ΔV*/*ΔI*, the voltage threshold is applied to the denominator and is therefore less sensitive, as shown in (Xu et al, 2002). Fig. 7b. shows that the coherence index does not reveal much information about reliability of the measurements; however it provides an indicator of the principal frequency of the transient signals, highlighted in the dotted circle. The standard deviation results presented in Fig. 7c. show that the impedances measured at frequencies between 1260 and 2000 Hz are very spread out and are, therefore, unreliable. The same situation occurs for frequencies above 2610 Hz. These results agree with those presented in Fig. 7a. for the threshold used for *ΔI*. Fig. 7d. shows that the quantization is not a critical issue and the measurements taken in the field are accurate enough to overcome quantization noises. However the low quantization values, especially for current, are of lower values for the unreliable ranges presented in Fig.

In harmonic analysis, many polluters are usually present in a power distribution system for each harmonic order because power system harmonics always occur in fixed frequencies, i.e., integer multiples of the fundamental frequency. All harmonic loads usually generate all harmonic orders, and therefore, it is common to try to determine the harmonic contribution of each load rather than the harmonic sources. As opposed to harmonics, interharmonics are almost always generated by a single polluter. This property of interharmonics can be

Fig. 6. Harmonic impedance for a sample field test used in the case study.

where each case is one data snapshot taken at each site.

**5. Interharmonic source determination** 


0

0.5

Impedance [ohms]

7a. and Fig. 7c.

explained as follows.

1

1.5

Fig. 5. Characterization of the transients resulting from a capacitor switching: (a) voltage and current waveforms during a disturbance, (b) Transient waveforms and frequency contents.

#### **3.2 Transient Identification**

The perfect extraction of the transient is needed for the present application. Several classification methods were proposed to address this problem, such as neural networks and wavelet transforms (Anis & Morcos, 2002). Some other methods use criteria detection based on absolute peak magnitude, the principal frequency and the event duration less than 1 cycle (Sabin et al, 1999).

In this chapter, a simple approach is proposed to perform this task. It calculates the numerical derivative of the time-domain signals, and assumes that if a transient occurred, this derivative should be higher than 10. As a result, the numerical algorithm monitors the recorded waveforms and calculates the derivatives at each data sample; when this derivative is higher than 10, it can be concluded that a capacitor switching occurred.

### **4. Impedance measurement case study**

More than 120 field tests have been carried out in most of the major utilities in Canada (in the provinces of British Columbia, Ontario, Alberta, Quebec, Nova Scotia and Manitoba), and a representative case is presented in this section. Over 70 snapshots (capacitor switching events) were taken at this site. Using the techniques described in section II, the impedance results were obtained and are presented in Fig. 6. This figure shows that in the range of 1200-1750Hz there is an unexpected behavior in both components of the impedance. A resonant condition may be the reason of this sudden change. However, it might be caused by unreliable data instead. Further investigation is needed in order to provide a conclusion for this case.

Based on extensive experience acquired by dealing with the collected data, the following thresholds were proposed for each index:


0

20

Transient Current [A]

Fig. 5. Characterization of the transients resulting from a capacitor switching: (a) voltage and current waveforms during a disturbance, (b) Transient waveforms and frequency

The perfect extraction of the transient is needed for the present application. Several classification methods were proposed to address this problem, such as neural networks and wavelet transforms (Anis & Morcos, 2002). Some other methods use criteria detection based on absolute peak magnitude, the principal frequency and the event duration less than 1

In this chapter, a simple approach is proposed to perform this task. It calculates the numerical derivative of the time-domain signals, and assumes that if a transient occurred, this derivative should be higher than 10. As a result, the numerical algorithm monitors the recorded waveforms and calculates the derivatives at each data sample; when this

More than 120 field tests have been carried out in most of the major utilities in Canada (in the provinces of British Columbia, Ontario, Alberta, Quebec, Nova Scotia and Manitoba), and a representative case is presented in this section. Over 70 snapshots (capacitor switching events) were taken at this site. Using the techniques described in section II, the impedance results were obtained and are presented in Fig. 6. This figure shows that in the range of 1200-1750Hz there is an unexpected behavior in both components of the impedance. A resonant condition may be the reason of this sudden change. However, it might be caused by unreliable data instead. Further investigation is needed in order to provide a conclusion

Based on extensive experience acquired by dealing with the collected data, the following

derivative is higher than 10, it can be concluded that a capacitor switching occurred.

40

Transient Voltage [V]

0 10 20 30 40 50

0 10 20 30 40 50

Frequency (p.u.)

<sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> -200

<sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> -400

**4. Impedance measurement case study** 

thresholds were proposed for each index: Energy level: *ΔI(f)* > 1% and *ΔV(f)* > 1%.

Quantization error: *ΔI(f)* > 0.0244A.

.

 Coherence: *γ(f)* ≥ 0.95. Standard deviation: 0.5

Time [samples]

0 200 400


contents.

**3.2 Transient Identification** 

cycle (Sabin et al, 1999).

for this case.

Current [A]

Voltage [V]

As the reliability criteria are applied, it is useful to define the following ratio of success:

$$\text{Ratio} = \text{Successfull cases} \{ \text{Total cases} \times 100\%,\tag{11}$$

where each case is one data snapshot taken at each site.

Fig. 7 shows the success rate of cases for each index in function of frequency. Fig. 7a. also shows that the application of this index will affect *ΔI* much more than for *ΔV*, since the latter is acquired using voltage probes, which are inherently much more reliable. Since the impedance measurement is calculated from the ratio -*ΔV*/*ΔI*, the voltage threshold is applied to the denominator and is therefore less sensitive, as shown in (Xu et al, 2002). Fig. 7b. shows that the coherence index does not reveal much information about reliability of the measurements; however it provides an indicator of the principal frequency of the transient signals, highlighted in the dotted circle. The standard deviation results presented in Fig. 7c. show that the impedances measured at frequencies between 1260 and 2000 Hz are very spread out and are, therefore, unreliable. The same situation occurs for frequencies above 2610 Hz. These results agree with those presented in Fig. 7a. for the threshold used for *ΔI*. Fig. 7d. shows that the quantization is not a critical issue and the measurements taken in the field are accurate enough to overcome quantization noises. However the low quantization values, especially for current, are of lower values for the unreliable ranges presented in Fig. 7a. and Fig. 7c.

#### **5. Interharmonic source determination**

In harmonic analysis, many polluters are usually present in a power distribution system for each harmonic order because power system harmonics always occur in fixed frequencies, i.e., integer multiples of the fundamental frequency. All harmonic loads usually generate all harmonic orders, and therefore, it is common to try to determine the harmonic contribution of each load rather than the harmonic sources. As opposed to harmonics, interharmonics are almost always generated by a single polluter. This property of interharmonics can be explained as follows.

On the Reliability of Real Measurement Data for Assessing Power Quality Disturbances 79

The interharmonic active power can be obtained from the voltage and current

\* Re cos , *P V I VI IH IH IH IH IH*

where |*VIH*| and |*IIH*| are the interharmonic voltage and current magnitudes, respectively,

The conclusion of the power direction method, therefore, is the following (Kim et al, 2005),

If this criterion is extended to a multi-feeding system like that shown in Fig. 9, the interharmonic source for each interharmonic can be identified. In such a case, monitoring equipment should be placed at each feeder suspected of injecting interharmonics into the system. For the system side measurements (point A), if the measured *PIH > 0*, the interharmonic component comes from the system. For the customer side measurements (points B and C), if the measured *PIH < 0*, the interharmonic component comes from the

As a given interharmonic frequency has only one source, the power direction method (in theory) could always reveal the interharmonic source correctly. In reality, as the active power of the interharmonics is typically very small, the measured data may not be reliable, so this index may not provide reliable conclusions. On the other hand, the angle *φIH* can be very close to either *π*/2 or –*π*/2, oscillating around these angles because of measurement errors, and resulting in the measured active power index swinging its sign and potentially causing misjudgment. The drifting nature of interharmonics in frequency and the supply

and *φIH* is the angle displacement between the interharmonic voltage and current.

 If *PIH > 0*, the interharmonic component comes from the upstream side. If *PIH < 0*, the interharmonic component comes from the downstream side.

Fig. 9. System diagram for locating the interharmonic source.

fundamental frequency variation also influence this inaccuracy.

*IH* (14)

Fig. 8. Determination of interharmonic source – two different scenarios.

measurements at a metering point as

(Axelberg, 2008):

measured customer.

Fig. 7. Indices in function of frequency: (a) energy level, (b) coherence, (c) standard deviation, (d) quantization error.

The main interharmonic sources are adjustable speed drives (ASDs) with a *p1*-pulse rectifier and a *p2*-pulse inverter and periodically varying loads such as arc furnaces. Their interharmonic generation characteristics can be expressed as in (12) for ASDs (Yacamini, 1996) and (13) for periodically varying loads (IEEE Task Force, 2007), respectively:

$$f\_{\mid \mathbb{H}} = \left| \left( p\_1 m \pm 1 \right) f \pm p\_2 n f\_z \right|\_{\prime} \quad m = 0, 1, 2...; \quad n = 1, 2, 3... \tag{12}$$

where *f* and *f*z are the fundamental and drive-operating frequency.

$$f\_{\mu\mu} = \left| f \pm \mu f\_v \right|, \quad \mathfrak{n} = \mathbf{1}, \mathbf{2}, \mathfrak{E} ... \tag{13}$$

where *fv* is the load-varying frequency. According to equations (12) and (13), the interharmonic frequency depends on many factors such as the number of pulses of the converter and inverter, the drive-operating frequency, or the load-varying frequency. Therefore, the same frequency of interharmonics is rarely generated by more than one customer.

Based on the above analysis, for interharmonic source determination, the analysis can be limited to the case of a single source for each of the interharmonic components. The most popular method currently being used to identify the interharmonic sources is based on the active power index. Fig. 8 helps to explain the power direction method. For this problem, the polluter side is usually assumed to be represented by its respective Norton equivalent circuit. Fig. 8 shows two different scenarios at the metering point between the upstream (system) and the downstream (customer) sides. Fig. 8a and Fig. 8b show the case where the interharmonic components come from the upstream side and the downstream side, respectively. The circuits presented in Fig. 8 are used for each frequency.

**Coherence successful rate [%]**

0%

0% 20% 40% 60% 80% 100% 120%

**S. D. successful rate [%]**

60

360

660

deviation, (d) quantization error.

960

1260

1560

**Frequency [Hz]**

1860

2160

where *f* and *f*z are the fundamental and drive-operating frequency.

frequency of interharmonics is rarely generated by more than one customer.

respectively. The circuits presented in Fig. 8 are used for each frequency.

2460

R X

2760

Fig. 7. Indices in function of frequency: (a) energy level, (b) coherence, (c) standard

1996) and (13) for periodically varying loads (IEEE Task Force, 2007), respectively:

The main interharmonic sources are adjustable speed drives (ASDs) with a *p1*-pulse rectifier and a *p2*-pulse inverter and periodically varying loads such as arc furnaces. Their interharmonic generation characteristics can be expressed as in (12) for ASDs (Yacamini,

where *fv* is the load-varying frequency. According to equations (12) and (13), the interharmonic frequency depends on many factors such as the number of pulses of the converter and inverter, the drive-operating frequency, or the load-varying frequency. Therefore, the same

Based on the above analysis, for interharmonic source determination, the analysis can be limited to the case of a single source for each of the interharmonic components. The most popular method currently being used to identify the interharmonic sources is based on the active power index. Fig. 8 helps to explain the power direction method. For this problem, the polluter side is usually assumed to be represented by its respective Norton equivalent circuit. Fig. 8 shows two different scenarios at the metering point between the upstream (system) and the downstream (customer) sides. Fig. 8a and Fig. 8b show the case where the interharmonic components come from the upstream side and the downstream side,

60

360

660

960

1260

1560

**Frequency Hz**

1860

2160

2460

2760

DV DI

20% 40% 60% 80% 100% 120%

**E.L. successful rate [%]**

0% 20% 40% 60% 80% 100% 120%

0% 20% 40% 60% 80% 100% 120%

**Q. E. successful rate [%]**

60

360

*IH* 1 2 1 , 0,1,2...; 1,2,3..., *<sup>z</sup> f p m f p nf m n* (12)

, 1,2,3..., *IH <sup>v</sup> f f nf n* (13)

660

960

1260

1560

**Frequency Hz**

1860

2160

2460

Voltage Current

2760

60

360

660

960

1260

1560

**Frequency (Hz)**

1860

2160

2460

2760

Fig. 8. Determination of interharmonic source – two different scenarios.

The interharmonic active power can be obtained from the voltage and current measurements at a metering point as

$$P\_{\rm II^\circ} = \text{Re}\left\{ V\_{\rm II^\circ} \times I\_{\rm II^\circ}^\circ \right\} = \left| V\_{\rm II^\circ} \right| \left| I\_{\rm II^\circ} \right| \cos(\phi\_{\rm II^\circ}) \,\tag{14}$$

where |*VIH*| and |*IIH*| are the interharmonic voltage and current magnitudes, respectively, and *φIH* is the angle displacement between the interharmonic voltage and current.

The conclusion of the power direction method, therefore, is the following (Kim et al, 2005), (Axelberg, 2008):


If this criterion is extended to a multi-feeding system like that shown in Fig. 9, the interharmonic source for each interharmonic can be identified. In such a case, monitoring equipment should be placed at each feeder suspected of injecting interharmonics into the system. For the system side measurements (point A), if the measured *PIH > 0*, the interharmonic component comes from the system. For the customer side measurements (points B and C), if the measured *PIH < 0*, the interharmonic component comes from the measured customer.

Fig. 9. System diagram for locating the interharmonic source.

As a given interharmonic frequency has only one source, the power direction method (in theory) could always reveal the interharmonic source correctly. In reality, as the active power of the interharmonics is typically very small, the measured data may not be reliable, so this index may not provide reliable conclusions. On the other hand, the angle *φIH* can be very close to either *π*/2 or –*π*/2, oscillating around these angles because of measurement errors, and resulting in the measured active power index swinging its sign and potentially causing misjudgment. The drifting nature of interharmonics in frequency and the supply fundamental frequency variation also influence this inaccuracy.

On the Reliability of Real Measurement Data for Assessing Power Quality Disturbances 81

**Time [hours]**

16:16

17:02

17:50

18:37

19:24

20:11

**Frequency [Hz]**

The active power index was monitored at the three loads. This is shown in Fig. 12-Fig. 15. The system was observed to be fairly balanced, and therefore only the power in phase A is shown. By looking into these figures, one would conclude that Customer 2 is the source of interharmonics 228Hz, 348Hz and 384Hz, whereas Customer 3 is the source of interharmonics 264Hz and 348Hz. As explained in equation (2), it is almost impossible that an interharmonic component is generated by two sources at the same time. Furthermore, after deeper investigation, it is shown that this apparent identification of the interharmonic polluters is incorrect, and the reliability criteria proposed in this chapter is useful in aiding

**228 Hz**

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67

**Time Step**

0.1

Customer 3 Customer 2 Customer 1 0.15

0.2

0.25

0.3

100 200 300 400 500 600

Fig. 11. Contour plot of the interharmonic data recorded at the feeder

the researcher to drawing correct conclusions.

Fig. 12. Active power at the loads for *fIH = 228Hz* 


**Pa (W)**
