2602 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005
Abstract— This paper investigates the inclusion of attenuation and diversity effects in harmonic distortion assessment for systems with distributed harmonic sources. The concept of device characteristic curves is introduced to represent the effects. An iterative method is proposed to include the effects in system wide harmonic assessment. The method is illustrated using a system with distributed personal computer loads. Test results have confirmed the effectiveness of the proposed method.
Index Terms— Distributed harmonic loads, harmonic analysis, harmonic attenuation and diversity effects, harmonic sources.
I. I
NTRODUCTION
A NOTICEABLE trend in power systems nowadays is the emergence of distributed harmonic-producing loads.
These loads typically have comparable sizes and are distributed all over an electric network. Traditional harmonic assessment techniques have difficulties in determining the collective distortion effects of these sources. There is a need to develop new techniques to assess harmonic distortions for systems with distributed harmonic sources.
The difficulties are due to two factors. One is that the supply voltage distortions will change the harmonic generation behavior of the distributed harmonic-producing loads. Both the amplitude and phase angle of the harmonic current injected by a load will vary with the degree of supply voltage distortion.
The former effect is called attenuation while the latter effect is
called diversity [1]. Both effects tend to reduce the harmonic
currents produced by the loads, resulting in reduced harmonic distortion levels in the system. The second factor is the random variations of the loads. It is necessary to determine the probabilistic harmonic distortion levels in such cases.
The objective of this paper is to introduce a method that can take into account the first factor in (deterministic) harmonic assessment. A typical case that can be analyzed by the proposed method is the commercial electric systems. Such systems have many distributed harmonic-producing office electronics. The
Fig. 1.
Main setup of the conducted experiments for the PC load harmonic measurements.
Fig. 2.
Investigating the effect of supply impedance variation.
main focus of this paper is a single-phase harmonic source uses a capacitor-filtered diode bridge rectifier circuit. Traditional harmonic assessment methods ignore the attenuation and diver-
sity effects [2]. They can result in significant overestimation of
the harmonic distortion levels when applied to such systems.
The concept of diversity and attenuation effects was intro-
duced in [1]. The attenuation factor “
” of the resultant th harmonic current with the operation of “ ” PCs sharing a common supply impedance is defined as follows:
(1) where eration and is the th harmonic current when PC’s are in opis the th harmonic current when 1 PC is in operation. The diversity factor “ ” is defined as follows:
Manuscript received May 11, 2004; revised December 31, 2004. This work was supported by the Alberta Energy Research Institute (AERI). Paper no.
TPWRD-00221-2004.
E. E. Ahmed is with the Department of Electrical Engineering, Cairo University-Fayoum Campus, Fayoum, Egypt (e-mail: emad@ualberta.net).
W. Xu is with the Department of Electrical and Computer Engineering,
University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: wxu@ece.
ualberta.ca).
G. Zhang with the University of Alberta, Edmonton, AB T6G 2V4, Canada
(e-mail: guibin@ece.ualberta.ca).
Digital Object Identifier 10.1109/TPWRD.2005.855441
(2) where is the th harmonic current injected by the th load.
The impact of the attenuation and diversity effects on the harmonic-generation characteristics of a PC-type load was investi-
gated in [3]. It revealed that a reduction of 30% of harmonic cur-
rent injection is possible due to these effects. However, no work
0885-8977/$20.00 © 2005 IEEE

AHMED et al.
: ANALYZING SYSTEMS WITH DISTRIBUTED HARMONIC SOURCES
Fig. 3.
PC current THD as a function of the supply voltage THD or CF (due to supply impedance variation).
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Fig. 4.
PC current harmonics as functions of supply voltage THD or CF (due to supply impedance variation).
has been published on including the effects for system wide harmonic assessment. Time-domain simulation that models all harmonic-producing loads in detail could be one method to solve the problem. But such a method is not practical when there are many distributed harmonic sources. In view of this situation, we propose a harmonic domain iterative method to address the need. Single-phase systems with distributed computer loads are used as an example to illustrate the method.
II. C HARACTERIZING THE
AND
A TTENUATION
D IVERSITY E FFECTS
As the proposed method deals with the attenuation and diversity effects, the first step is to establish a method to characterize the effects. For this purpose, extensive laboratory experiments were conducted to quantify the impact of supply voltage distortion on the harmonic currents generated by the switched mode power supplies. The power supply is commonly used in office appliances and consists of a capacitor-filtered diode bridge rectifier.
No relationship could be extracted between the input harmonic currents and the parameters of the capacitor-filtered rectifier circuit. Therefore, it is not easy to predict the current wave-
form from circuit parameters [4]. Rather, the distortion of the
terminal voltage waveform in this research work is adopted to predict the magnitude and the phase angle of the input harmonic currents to reflect the impact of the supply impedance or other system connected harmonic sources.
Two types of experiments were conducted. The first experiment is to change the supply impedance so a different level of harmonic distortion is created at the terminal of a test PC respecting a switched mode power supply. The second experiment is to change the number of other harmonic sources in the system, creating different amount of background voltage distortions.

2604 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005
Fig. 5.
PC harmonic current phase angles as functions of the supply voltage THD or CF (due to supply impedance variation).
The measurements are conducted using a Data Acquisition
Nicolet Transient Recorder BE256-LE. This instrument has eight channels for simultaneous recording with an adjustable sampling rate. The voltage and current waveforms are measured using voltage and current probes. The waveform data can be acquired by connecting the recorder to a laptop via
IEEE-488 interface unit. The recorder has special software called “TEAM256” (Transient Evaluation and Analysis Manager) which is installed on the laptop to control the recording process. The main set up of the conducted experiments and the instruments used in the measurements are shown in Fig. 1.
Adjustable speed drive loads, the most common three-phase harmonic sources, use the same circuit topology used in the PC power supply, but this load is out of the scope of this paper.
Different harmonic source types can be considered by other researchers and following the way presented in this paper is an option while finding a new way is encouraged to further explore the inclusion of harmonics attenuation and diversity in harmonic analysis for additional load types.
A. Supply Impedance Variation
The circuit configuration for the first experiment is shown in
Fig. 2 where the PC load is supplied through variable impedance components and . The supply impedance is changed in 10 steps for a given ratio, and in each step the feeding voltage
“ ” and the supply current “ ” are measured with a sampling rate of 7.5 kHz. This process is repeated for different ratios of supply impedance with magnitudes causing a voltage drop up to 3.15%.
Fig. 3 shows the variation of the supply current total harmonic distortion (THD) with the feeding voltage THD or crest factor
Fig. 6.
Investigating the effect of background voltage distortion.
(CF) indices. The results show that there is a consistent negative or positive slope relationship between the between the PC current THD and supply voltage THD or CF.
Fig. 4 depicts the variation of the individual harmonic current magnitudes in percent of the fundamental component and
Fig. 5 presents the variation of the phase angles of the harmonic currents.
The figures confirm that the harmonic current spectrum can no longer be assumed as constant. The supply voltage distortion can significantly change the harmonic current output of the load. The magnitude of each harmonic current is almost linearly related to the THD or CF of the supply voltage. The variation of the individual harmonic current phase angles, though scattered slightly, has a specific trend. The phase angles become more delayed with the increase in voltage THD and the decrease in voltage CF.
B. Variation of Background Voltage Distortion
The set up for this experiment is shown in Fig. 6. Seven computer and monitor loads were connected in sequence to the

AHMED et al.
: ANALYZING SYSTEMS WITH DISTRIBUTED HARMONIC SOURCES
Fig. 7.
Variation of PC current THD with voltage THD or CF due to background voltage distortion variation.
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Fig. 8.
Variation of PC individual harmonic current magnitudes with voltage THD or CF due to background voltage distortion variation.
common point of the test PC to create different background voltage distortions. The feeding voltage and supply current of the test PC were measured. The first measurement point is taken when none of the external loads is connected so that we got a total of eight measurement points for the same supply impedance.
The measurement was repeated for different ratios.
Fig. 7 shows the variation of the PC current THD with the feeding voltage indices THD or CF. There is also a good correlation between the PC current THD and either voltage indices. Fig. 8 depicts the variation of the harmonic current magnitudes while
Fig. 9 illustrates the variation of the current phase angles. Both figures show similar trends to those observed in the impedance variation case. Note that there is a plateau in the curves. This is caused by the connection of computer monitors. The monitor loads have different harmonic current characteristics from the PC loads. The monitors are connected just to provide wider range of voltage distortion. The figures show that the voltage distortion can uniquely characterize the harmonic currents except in the range where the monitors are connected. Therefore, with only PC loads, i.e., with distributed harmonic sources of comparable sizes and same circuit topology dominating the system, the harmonic currents can be identified by the voltage distortion.
C. Characterizing the Attenuation and Diversity Effects
Figs. 10 and 11 compare the harmonic current magnitudes and phase angles obtained under both test conditions. An important conclusion is that the harmonic magnitude results are comparable. As for the harmonic phase angles, even they are scattered, they show a specific trend and averaging the two effects is acceptable to consider the interaction with the supply voltage distortion which is overlooked by the traditional harmonic analysis methods. It will be shown later that the slight spread of the phase angles will not result in a pronounced effect on the results. Therefore, it can be inferred that the cause of voltage distortion has no significant effect on the harmonic current generated by the loads. The diversity and attenuation effect can be represented approximately using either the supply voltage THD or CF as the sole independent variable.
The main concern of this paper is a single-phase harmonic source employs a capacitor-filtered diode bridge rectifier cir-

2606 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005
Fig. 9.
Variation of PC individual harmonic current phase angles with voltage THD or CF due to background voltage distortion variation.
cuit as the most common single-phase harmonic source used in residential and commercial systems. However, in order to confirm the drawn conclusions for other harmonic sources use the same circuit topology, the same procedure was followed using simulation studies for adjustable speed drive load (ASD).
Fig. 12 depicts the obtained results for the fifth harmonic current. It can be asserted that the harmonics generated from this circuit topology for either single-phase or three-phase harmonic sources can be reasonably characterized by the distortion of the supply voltage. This adds a significant feature that can be utilized in harmonic analysis to include the harmonics attenuation and diversity which are commonly ignored by traditional harmonic analysis methods.
From the aforementioned, we propose to model the diversity and attenuation effects using the following functions:
(3)
(4) where (VTHD) denotes the magnitude of the th harmonic current as a function of the supply voltage THD.
(VTHD) denotes the phase angle of the th harmonic current as a function of the supply voltage THD. The above functions are determined using curve fitting techniques. Sample results are as follows:
Similar functions can also be determined with the voltage CF as the variable.
In summary, it is shown that attenuation and diversity effects of a harmonic-producing load can be characterized using the harmonic currents versus voltage THD or voltage CF curves.
Some of the significant implications of these curves are as follows.
• It is interesting to note that when the results of (5) and (6) are the harmonic current spectrum of the load used for traditional harmonic power flow calculations
[5]. Since the traditional methods can only use one point
on the curves, it is clear that they are unable to include the diversity and attenuation effects.
• The curves are of general application values. They represent the input-output or terminal characteristics of a harmonic-producing load. The complexity involved in modeling the internal working of the load and the associated impact on the diversity/attenuation effects is thus eliminated. Similar curves can be determined for different types of loads to form the associated harmonic current source model for the loads.
• The curves or current source models are represented using either voltage THD or CF as independent variables. It is possible that other indices could also be used. As will be shown later, the THD is a much better index than CF for modeling the diversity and attenuation effects.
(5)
(6)
III. A N I TERATIVE M ETHOD FOR H ARMONIC A SSESSMENT
The next problem to be solved is how to include the diversity and attenuation curves for system wide harmonic assessment.
For this purpose, we propose an iterative frequency domain

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Fig. 10.
Variation of PC harmonic current magnitudes due to the change of supply impedance or background voltage distortion.
Fig. 11.
Variation of PC harmonic current phase angles due to the change of supply impedance or background distortion.
harmonic analysis method. The basic idea of the proposed method is to revise the current spectrum of the harmonic sources according to the device characteristic curves in an iterative process.
The proposed method starts with the traditional harmonic analysis method. The method produces results on the harmonic voltage distortions at various buses of the system, without taking into account the diversity and attenuation effects. With the cal-

2608 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005
Fig. 12.
Variation of ASD fifth harmonic current magnitude and phase angle with voltage THD due to the change of supply impedance or background voltage distortion.
culated voltage THD results, the harmonic current injected by the harmonic source at bus is adjusted, with the help of the device characteristics developed earlier, as follows:
(7)
(8)
Note that the above two equations are somewhat similar to those used in traditional harmonic analysis, which are shown below
[5]
(9)
(10)
The difference is that the magnitude and phase angle of the updated harmonic current are no longer scaled and shifted according to a simple linear relationship. The scaling of magnitude and shifting of phase angle are now determined from two nonlinear equations that are the functions of bus voltage THD. This adjustment has therefore considered the attenuation and diversity effects. The adjusted harmonic currents are then re-injected into the system to get the new and improved system harmonic voltages using the nodal voltage equations.
The above process will yield a new set of bus voltage THD results. In turn, a new set of adjusted harmonic currents is obtained according to (7) and (8). This iterative process can be repeated until a convergence of the voltage THDs at all buses is reached. A graphical illustration of the iterative process is shown in Fig. 13.
As discussed earlier, either THD or CF can be used to characterize the attenuation and diversity effects of a device. There is a need to determine which index is more suitable for the proposed iterative method. This subject is investigated by analyzing the interaction between the device curve and the system characteristic curve. Fig. 14 shows both the system characteristic curves and the device curve together for a simple system of Fig. 15. In this case, the supply system impedance is varied so that a family of system curves is obtained.
In Fig. 14, the characteristic curves are represented using either voltage THD or CF index. It can be seen that there is usually no intersection between the system and load characteristic curves when the CF index is used. As a result, one can conclude that a device characteristic curve that is based on the CF index
Fig. 13.
Iterative process for harmonic distortion assessment.
Fig. 14.
System voltage CF and THD response to the PC current distortion.
is not suitable for the proposed iterative method. On the other hand, the voltage THD-based characteristic curves usually have intersection points. This implies that the voltage THD index is

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: ANALYZING SYSTEMS WITH DISTRIBUTED HARMONIC SOURCES 2609
Fig. 15.
Investigation of system voltage response to the PC current distortion.
suitable for characterizing the attenuation and diversity effects.
Based on the results, the voltage THD index is adopted for the proposed iterative method.
The reason that the CF index is not suitable for characterizing the diversity and attenuation effects has been investigated. The
CF index represents the sharpness of a waveform peak. Fig. 16 shows that when more distorted load current is injected to the system, the voltage waveform actually becomes sharper instead of flatter as commonly believed. This response implies a positive slope for system characteristic curve. The increased sharpness of the waveform is due to the voltage drop on the inductive component of the system impedance.
IV. C ASE S TUDIES FOR D ISTRIBUTED PC L OADS
The proposed iterative method has been verified experimentally. Harmonic measurements were conducted on two PC systems shown in Fig. 17. The magnitude of the supply impedance
“ ” was increased in steps to vary the system distortion level.
The harmonic distortion levels were determined using the traditional and the iterative methods. The results are compared with the measurements.
Fig. 16.
Supply voltage waveform as affected by the load current.
A. Measurement and Calculation Results
The supply impedance of PC system 1 is increased in seven steps with the magnitude of 0.5, 1.25, 1.75, 2.5, 3, 3.75, and 4.25
ohm in turn and with a constant X/R ratio of 0.754. The supply impedance of PC system 2 is varied in three steps with the same values as that in the first three steps for PC system 1. For both systems, the voltage at node 2 and the current in branch 1 are measured. The convergence criterion of the iterative method is set at 0.05% of the maximum absolute difference of the voltage
THD at all nodes between two successive iterations. Sample measurement and calculation results are compared in Figs. 18 and 19.
Fig. 18 compares the voltage and current THD results obtained by the measurements, the proposed iterative method and the traditional method. The results are presented for different study cases in ascending order of the supply impedance magnitude.
Fig. 19 compares the individual harmonic magnitudes of supply voltage at node 2 and the individual harmonic current magnitudes of the PC at node 2 for the test system 1. The figure reveals that the results obtained using the proposed iterative method are in a good agreement with the measurement results while the traditional method leads to considerable overestimation.
Fig. 17.
Single-line diagram of the two PC systems under study.
B. Sensitivity Study
According to Fig. 11, the phase angle has a large variation range as a function of the voltage THD. So it is difficult to model the diversity effect accurately using the function. In this section, the impact due to the inaccuracy of the function is investigated. In the investigation, the iterative method is applied to both PC systems without considering the diversity effect. The results are seen in Fig. 20 and are compared with those obtained with both effects being considered.
It can be seen that there is no noticeable difference between the results for both of the two PC systems. Therefore, one may conclude that diversity effect is not pronounced and there is no need to determine the function accurately. This observation

2610 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005
Fig. 18.
Comparison of voltage and current THD results.
Fig. 19.
Comparison of individual harmonic magnitudes (system 1).
can be explained according to the following analysis. According to (8), the phase angle difference between the th harmonic current components for two loads connected at buses and is obtained by
Comparing (11) with (12), it can be seen that an external term is introduced. If the two buses have similar voltage distortion level, this term can be very small. As a result, the effect of phase angle diversity becomes insignificant.
(11)
On the other hand, if the traditional method is used where the two loads have the same typical harmonic current spectrum, the phase angle difference, using (10), can be obtained by
(12)
C. Convergence Performance of the Proposed Method
Fig. 21 presents the convergence process of the voltage THD at different nodes for the test system 1. The variation of the number of iterations required for convergence for different study cases is depicted in Fig. 22. One can see that the number of iterations increases with the voltage distortion level and the number of harmonic sources. It was also found that the convergence is faster when the diversity effect is ignored. So for systems with a

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Fig. 20.
Impact of diversity effect on harmonic distortion levels.
Fig. 21.
Convergence of the voltage THD for different study cases (system 1).
Fig. 22.
Number of iterations required for convergence.

2612 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 4, OCTOBER 2005 large number of distributed harmonic sources and high voltage distortion levels, one may need to use some techniques to accelerate the convergence. As an example, damping factor has been found useful to improve the convergence by attenuating the oscillations in the iterative process.
V. C ONCLUSION
An iterative harmonic analysis method has been proposed to assess harmonic distortions for systems with distributed singlephase harmonic sources. It has been focus on harmonic sources employing capacitor-filtered diode bridge rectifier circuit. The method can take into account the harmonics attenuation and diversity effects and has the potential to be investigated for further harmonic source types. It overcomes the limitations of the traditional harmonic analysis method and provides more accurate results. Main contributions of this paper are summarized as follows:
• It introduced a general method to characterize the diversity and attenuation effects of harmonic-producing loads.
The result, device specific harmonic current curves as functions of supply voltage THD, is essentially a current source model for harmonic sources.
• It proposed an iterative method to include the device characteristics for system wide harmonic analysis. Performance of the proposed method has been verified on two test systems with good results. The impact of several factors on the method and its results has also been analyzed.
The concepts introduced in this paper are a promising way to take into account the attenuation and diversity effects. As shown in the case study results, such effects cannot be ignored for system with a large number of harmonic sources. Otherwise, one may overestimate the cost to mitigate the harmonic problem. As with any new methods, there is a lot of room for improvement. For example, more sophisticated iterative method could be used to improve the convergence speed.
[2] Task Force on Harmonics Modeling and Simulation, “Characteristics and modeling of harmonic sources—Power electronic devices,” IEEE
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, vol. 16, no. 4, pp. 791–800, Oct. 2001. IEEE PES
Harmonic Working Group.
[3] A. Mansoor, W. M. Grady, P. T. Staats, R. S. Thallam, M. T. Doyle, and M. J. Samotyj, “Predicting the net harmonic currents produced by large numbers of distributed single-phase computer loads,” IEEE Trans.
Power Del.
, vol. 10, no. 4, pp. 2001–2006, Oct. 1995.
[4] D.-G. Kim, T. Nakajima, and E. Masada, “Harmonic analysis of a capacitor-filtered rectifier with line impedance,” Electron. and Commun.
in Japan , pt. Part 1, vol. 72, no. 4, 1989.
[5] Task Force on Harmonics Modeling and Simulation, “Modeling and simulation of the propagation of harmonics in electric power networks, part I: Concepts, models, and simulation techniques,” in IEEE Trans.
Power Del.
, S. Ranade, W. Xu, and J. Mahseredjian, Eds., Jan. 1996, vol. 11, pp. 452–465.
Emad Ezzat Ahmed was born in Egypt in 1971. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Electrical Power and Machines
Department, Cairo University, Cairo, Egypt, in 1993 and 1998, respectively, and the Ph.D. degree from the University of Alberta, Edmonton, AB, Canada in
June 2003.
He then became an Assistant Professor in the
Electrical Engineering Department, Cairo University-Fayoum Campus. His research interests include power quality, harmonic impedance measurement, harmonic filters, optimization, and distributed generation.
Wilsun Xu (M’90–SM’95–F’05) received the Ph.D. degree from the University of British Columbia, Vancouver, BC, Canada, in 1989.
He was with BC Hydro from 1990 to 1996 as an electrical engineer. He is presently a Professor at the University of Alberta, Edmonton, AB, Canada, and an Adjunct Professor at Shandong University of China. His main research interests are harmonics and power quality.
Dr. Xu became an IEEE Fellow for his contributions to the analysis, simulation, and measurement of power system harmonics.
R EFERENCES
[1] A. Mansoor, W. M. Grady, A. H. Chowdhury, and M. J. Samotyj, “An investigation of harmonics attenuation and diversity among distributed single-phase power electronic loads,” IEEE Trans. Power Del.
, vol. 10, no. 1, pp. 467–473, Jan. 1995.
Guibin Zhang received the B.Sc. and M.Sc. degrees in electrical engineering from Shandong University, China in 1995 and 1998, respectively, and the Ph.D.
degree from Zhejiang University, China, in 2001.
He is currently a Postdoctoral Fellow at the University of Alberta, Edmonton,
AB, Canada. His main research interests include power quality, HVDC and
FACTS.