1 Abstract--This paper provides a concise review on the main

developments and conclusions in the area of power systemharmonic analysis. Commonly accepted methods for conductingharmonic studies have been summarized. The area of harmonicanalysis is still a very fertile ground for exploration. With thehelp of four example research topics, the paper alsodemonstrates possible future directions of power systemharmonic analysis research.

Index Terms--Harmonics, Harmonic Analysis

I. INTRODUCTION

ower system harmonic analysis is to determine the impactof harmonic producing loads on a power system.

Harmonic analysis has been widely used for system planning,operation criteria development, equipment design,troubleshooting, verification of standard compliance, and soon. Over the past two decades, significant efforts andprogresses have been made in the area of power systemharmonic analysis. Well-accepted component models,simulation methods and analysis procedures for conductingharmonic studies have been established. Harmonic studies arebecoming an important component of power system analysisand design.

With the widespread use of digital computers, computersimulation has become the preferred method to conductharmonic analysis. This has transformed the subject ofharmonic analysis into two main areas: computer modeling ofpower systems for harmonic analysis and computersimulation of harmonic propagation in power systems.Research work in the above two areas has resulted in anumber of software packages for harmonic analysis andsimulation. It is important to note, however, that harmonicanalysis still requires a good knowledge on the subject areaand the skill of analytical thinking.

One of the purposes of this paper is to review the status ofcomputer-aided harmonic analysis methods. The mostcommonly accepted techniques for harmonics modeling andsimulation are discussed. The readers will find from thisreview that the research and application of harmonic analysishave entered a relatively mature stage. This naturally leads tothe following question: what are the future directions ofharmonic analysis? In the second part of this paper, somethoughts to this question are presented using examples ofresearch work conducted at the University of Alberta and

W. Xu is with the Department of Electrical and Computer Engineering,University of Alberta, Canada. He is also an adjunct professor of ShandongUniversity, China. (e-mail: wxu@ee.ualberta.ca).

other places. It is hoped that these examples or thoughts willserve as a step stone for continued research efforts in thisarea. For people who are interested in applying harmonicanalysis techniques, this paper could serve as a channel toobtain feedback from industry.

II. MODELING OF POWER SYSTEM COMPONENTS FOR HARMONICANALYSIS

Harmonics are a high frequency phenomenon involvingfrequencies from 50/60Hz to about 3000Hz. For harmonicstudies, it is important to take into account the componentresponse characteristics at the harmonic frequency range.Since most harmonic analysis tools are based on frequencydomain algorithms, component models for harmonic analysisnormally takes the frequency domain form as well. A detailedsummary on this topic can be found from reference [1] andother papers presented in this panel session. In this section,we will focus on the development status of componentmodeling research.Overhead Lines and Underground Cables: There is aconsensus that lines and cables can be modeled with amultiphase coupled equivalent pi-circuit. For balancedharmonic analysis, the model can be further simplified into asingle-phase pi-circuit determined from the positivesequence impedance data of the component. It is important toinclude the shunt element and its associated long-line effectin the model. The shunt admittance of the component, thoughsmall at the fundamental frequency, can become quitesignificant at higher frequencies. This effect can be easilyrepresented using the exact or equivalent pi-circuit model [1].Transformers: A transformer affects harmonic flowsthrough its series impedance, winding connection, andmagnetizing branch. Many research results in this topic haveconcluded that a transformer can be modeled using its short-circuit impedance for most harmonic studies. It is alsoimportant to include the transformer phase-shift effect. Thisphase-shifting effect could lead to significant harmoniccancellations in a system [2]. The magnetizing branch of atransformer is a harmonic source. Inclusion of the saturationcharacteristic of the branch is important only when theharmonics generated by a transformer are of primaryconcern.Rotating Machines: This component includes synchronousand induction machines. The consensus is that a machine canbe represented using its short-circuit impedance for mostharmonic analysis tasks, although more complex models areavailable for advanced applications.

Wilsun Xu, Senior Member, IEEE

Status and Future Directions ofPower System Harmonic Analysis

P

2Aggregate Loads: Aggregate loads refer to a group of loadbuses that are treated as one component in harmonic analysis.Typical aggregate loads are distribution feeders seen from asubstation bus or a customer plant seen at the point ofcommon coupling. Although such loads typically containharmonic sources, the main concern is the frequencyresponse characteristics of its equivalent impedance. If theharmonic sources are of concern, the load should not betreated as an aggregate one.

The model for aggregate loads, therefore, has the form ofa frequency-dependent impedance. Research has shown thatthe impedance is not only a function of the individual loadscontained in the component but also dependent on the lines orcables connecting the loads. For example, the distributionfeeder conductors and shunt capacitors can have a largerimpact on the frequency-dependent impedance seen at thefeeder terminal than those of the loads connected to thefeeder. As a result, it is almost impossible to use a set ofgeneral formulas to construct an adequate impedance modelfor aggregate loads. Although a few models for aggregateloads have been proposed in the past, the validity of thesemodels has not been fully verified. Modeling of aggregateloads is one of the weakest areas in the harmonics modelingfield. A lot of research work is still needed. In author'sopinion, solutions to the following topics would be valuable: Verified techniques to construct aggregate load models. The impact of model inaccuracy on harmonic analysis

results. Measured harmonic impedance data for aggregate loads

and analysis on the correlation of the measured data withthe load composition.

External System: External system for power quality analysistypically refers to either the utility supply system seen at thepoint of common coupling from a customer's perspective orthe neighboring networks of a utility system under study.Reference [1] has more information on this subject.Power Electronic Devices: Power electronic devices can bea load or a compensator. In terms of harmonic analysis, acommon characteristic of them is that they are harmonicsources. As a result, a number of different models have beenproposed to represent such devices. Among these models,the most commonly accepted one is the current sourcemodel. The model simply treats a power electronic device asa harmonic current source. The magnitude and phase of thesource can be calculated, for example, from the typicalharmonic current spectrum of the device. A more accurateprocedure to establish the current source model is asfollows:1) The power electronic load device is treated as a PQ load

at the fundamental frequency, and the fundamentalfrequency power flow of the system is determined;

2) The current injected from the load to the system is thencalculated and is denoted as I1/?1.

3) The magnitude of the harmonic current sourcerepresenting the load can be determined as follows:

( )1 1

1spectrum

spectrumhh I

III

-

-=

4) The phase angle of the harmonic current source can becalculated using the following formula:

( )2 ) qqqq spectrum-11spectrum-hh -( h+=

where subscript spectrum standards for the typicalharmonic current spectrum of the load. This data can bemeasured, obtained from manufacturers, or calculatedaccording to formulas. The current source model is the mostcommon one used in commercial power system harmonicanalysis programs. Its main disadvantage is the use of typicalharmonic spectrum and, as such, assessment of casesinvolving non-typical operating conditions becomes difficult.This has prompted the development of other modelingtechniques such as detailed time-simulation based models anda hybrid of current source model and the detailed model.

It is reasonable to say that power electronic devicemodeling techniques for harmonic calculations have becomea mature subject. What is needed is to improve ourunderstanding on the harmonic characteristics of the devicesfor other applications. For example, harmonic sources arecommonly treated as current sources for harmonic sourcedetection applications. There is an urgent need to determinethe degree of validity of this assumption. Another caseinvolves the generation of interharmonics from variablefrequency drives. As will be demonstrated later, thetraditional models are insufficient to analyze some aspects ofthe interharmonic problems.

III. NETWORK SOLUTION TECHNIQUES FOR HARMONIC ANALYSIS

Over the past two decades, considerable progress has beenmade in the area of computing harmonic power flows for apower system. Mature techniques are now available forassessing harmonic distortions in a network containingsignificant harmonic sources. In this section, two mostcommon and useful harmonic analysis techniques arepresented and discussed.

A. Frequency Scan AnalysisFrequency scan is the simplest and most commonly used

technique for harmonic analysis [3]. The input datarequirement is minimal. It calculates the frequency responseof a network seen at a particular bus or node. Typically, anone per unit sinusoidal current is injected into the bus ofinterest and the voltage response is calculated. Thiscalculation is repeated using discrete frequency stepsthroughout the range of interest. Mathematically, the processis to solve the following network equation at frequency f:

( )3 ][]][[ IVY fff =

where [If] is the known current vector and [Vf] is the nodalvoltage vector to be solved. In a typical frequency scananalysis, only one entry of [If] is nonzero. In other analysis, aset of positive or zero sequence currents may be injected intothree phases of a bus respectively. The results are the positive

3or zero sequence driving-point impedance of the network.Frequency scan analysis is the most effective tool to detectharmonic resonance conditions in a system. It has also beenwidely used for filter design.

B. Harmonic Power Flow AnalysisIf one needs to find the harmonic distortion levels for

certain operating conditions, a network harmonic power flowanalysis should be conducted. Several harmonic power flowtechniques have been proposed in the past [4], [5]. Many yearsof application experiences have shown that a non-iterativetechnique that represents harmonic sources as currentsources is sufficient for many common harmonic analysistasks. This technique is summarized as follows:Step 1: Compute the fundamental frequency power flow of

the network. The results define the operating scenario forharmonic analysis. In this step, the harmonic-producingloads are modeled as constant power loads

Step 2: Determine the harmonic current source models forthe harmonic-producing loads. The model consists of themagnitudes and phase angles of current source at variousharmonic frequencies. Equations needed to establish themodel are described in Section II, Equations (1) and (2).

Step 3: Calculate the network harmonic voltages and currentsby solving the following network nodal equation forharmonic number h of interest:

( )4 ][=]][[ hhh IVYwhere [Ih] is a known vector that has all harmonic currentsources included, and [Vh] is the nodal voltage vector tobe solved. Equation (4) is solved for all harmonicsinterested.

Step 4: The results of Steps 1 and 3 jointly define theharmonic power flow conditions of the study system.Harmonic indices such as total harmonic distortions andtransformer k factors can be calculated from the results.The main disadvantage of the above method is the use of

typical harmonic spectra to represent harmonic-producingdevices. It prevents an adequate assessment of cases involvingnon-typical operating conditions. Such conditions include,for example, partial loading of harmonic-producing devices,excessive harmonic voltage distortions and unbalancednetwork conditions. The applications of harmonic power flowanalysis include 1) network harmonic distortion assessment,2) harmonic limit compliance verification, 3) filterperformance evaluation, and 4) equipment de-ratingcalculation.

C. Comments on Network Harmonic Analysis TechniquesThe development of harmonic analysis techniques

followed mainly two directions over the past many years. Onedirection is to improve solution algorithms for harmonicpower flow calculation. Examples of this direction are theiterative harmonic method [4] and the Newton-Raphson basedsolution method [5]. The second direction is to include morecomplete network models in the solution. A good example ofthis direction is the three-phase or multiphase harmonicanalysis [6]. This direction is needed since certain harmonic

problems such as harmonic-telephone interference, must beanalyzed in three phase.

For both directions, the focus has been on the analysis ofnetworks with dominant harmonic sources. This is a refectionof the industry situation in the past, where harmonic-producing loads are large and are concentrated in a fewlimited locations. With the proliferation of power electronicloads, a power system may contain many harmonic sourceswith comparable sizes. How to analysis such systems hasbecome one of the challenges in harmonic analysis research.

IV. FUTURE DIRECTIONS OF HARMONIC ANALYSIS

It is always dangerous to predict research directions forthe future. Taking into account the fact that there is only asingle author for this paper and the author's knowledge on thesubject is limited, possibilities of error are high. Instead ofpredicting the future, what we attempt to do in this section isto illustrate some interesting research topics in the area ofharmonic analysis. It is hoped that the examples or thoughtspresented will serve as a step stone for continued researchefforts in the area.

A. Distributed Harmonic SourcesA noticeable trend in power systems nowadays is the

emergence of distributed harmonic-producing loads. Theseloads typically have comparable sizes and are distributed allover an electric network. Traditional techniques for harmonicpower flow analysis generally have difficulties in determiningthe collective impact of these sources. Determination ofharmonic distortions for systems with distributed sources is anew and challenging research topic in the area of harmonicanalysis. In order to make the problem manageable, someresearchers have sub-divided the problems into two. Oneassumes that the harmonic-producing loads are deterministicand the other assumes the loads vary randomly.

The first sub-problem can be described as the following: adistribution system has a number of harmonic-producingloads of comparable sizes. The loads operate at a constantload level. An office building with a lot of PCs is a typicalexample of such a case. It is required to determine theharmonic distortion level for this type of systems. Traditionalmethods to solve this problem cannot take into account thediversity and attenuation effects of the harmonic sources [7].Consequently, the results can be quite conservative, sinceboth effects help to reduce the over all harmonic distortionlevel in the system. Although several papers have beenpublished on this subject [7], [8], almost no method has beenproposed to estimate the harmonic distortion levels for suchsystems.

At the University of Alberta, we attempt to solve thisproblem by enabling the traditional methods to include thediversity and attenuation effects. One attractive idea is basedon the observation that each type of harmonic-producingloads has unique characteristic curves representing itsharmonic diversity and attenuation effects. Fig. 1 shows anexample curve for PC power supplies. As a result, one coulduse the following generic iterative method to analyze systemswith distributed harmonic sources:

41) Determine attenuation and diversity characteristic curvesby using, for example, measurements for the loads to bestudied. The relationship can be represented using thefollowing equations:

( )5 )( ,)( VTHDVTHDI hhhh bqa ==The above equations state that the voltage THD at the

terminal of the harmonic-producing load will affect thespectrum of the harmonic current produced by the load.2) The harmonic power flow results are first obtained using

the traditional methods. The bus voltage THDs aredetermined from the results.

3) Using the THD results, the harmonic current spectrum ofthe sources are adjusted according to the characteristicscurves defined in (5): This adjustment takes into accountthe attenuation and diversity effects.

4) Using the adjusted harmonic current spectrum data, thesystem harmonic power flow is re-solved. This providesimproved harmonic distortion results. The bus voltageTHDs are determined.

5) With the updated THD values, the calculation isredirected to Step 2). This iterative procedure stops whenthere is no significant change on the bus voltage THDlevel.

1 2 3 4 5 6 7 88 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

V o l t a g e T H D ( % )

Cur

rent

TH

D (

%)

Z c h a n g e V b a c k g r o u n d

Fig. 1. Characterization of the attenuation effects for single-phase powersupply.

Sample results for a system multiple personal computersare shown in Fig. 2. It compares the results obtained by theproposed method, the traditional method, and the measuredresults. The x-axis represents different sizes of the supplysystem impedance. The results show that the traditionalmethods are very conservative and the proposed method issignificant more accurate than the traditional methods.

Voltage at node 2

0

5

10

15

20

25

30

35

1 2 3 4 5 6 7Study cases

Vo

ltag

e T

HD

(%

)

Measurements

Iterative approach

Traditional method

Current through branch 1

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7Study cases

Cur

rent

TH

D (

%)

Measurements Iterative approach Traditional method

Fig. 2. Verification of the iterative method for distributed harmonic sources.

B. Distributed Random Harmonic SourcesA more difficult subject is to consider the random

variation of the loads in addition to its distributed nature. Atypical case can be described as the following: a distributionsystem has a number of harmonic-producing loads of

comparable sizes. The loads vary randomly from Pi-min to Pi-max, where subscript i is the bus number of a particular load.How can one determine the mean value and the 95%probability value of the harmonic distortion levels in thesystem? The 95% probability level is defined as the THD orIHD values that could be exceeded only for 5% probability. Arepresentative case for this problem is a distribution systemwith DC drive powered ski lifts. For the sake of simplicity,let's omit the diversity and attenuation effects at present stageof analysis.

A lot of work has been done to establish methods that cancharacterize the total impact of randomly operating harmonicsources [9]. A consensus on the subject has been reachedrecently [10]. It is generally agreed that if there are manyharmonic sources in a system the net impact follows thenormal distribution. Since a normal distribution can becharacterized by two parameters, mean value and standarddeviation, the problem becomes how to find the twoparameters for a given set of load variation patterns. Severalmethods, such as Monte Carlo simulation, have beenproposed to find the parameters [9-14]. The University ofAlberta has also investigated this problem. Our approach isnot to develop a brand new method, as its practical applicationcould be limited due to the need for the users to learn anothercomputer program. We try to use existing deterministicpower flow methods to estimate answers with reasonableaccuracy.

The approach adopted at the University of Alberta can bestated as the follows: Suppose yk is the distortion index of thekth bus or branch of interest (such as THD or IHD of a busvoltage) and pi are the load levels of the ith harmonic source(i=1,2 .. N, where N is the total number of harmonic sources),it is required to determine the mean value and standarddeviation of yk using existing deterministic harmonic analysistools. Our proposed method estimates the mean value of ykby running a traditional harmonic power flow program 3times. The standard deviation of yk is estimated by running atraditional harmonic power flow program N time. Due tospace limitation, details are omitted here.

Verification of the proposed methods has been conductedusing test systems. The results are compared with thoseobtained by Monte Carlo simulation. Three sets of resultswere obtained. The results labeled as Monte Carlo areobtained from Monte Carlo simulation without anyapproximations. They can therefore be considered as thecorrect results. The results labeled as NormalDistribution are obtained by assuming they follow a Normaldistribution. Parameters of the distribution (mean value andstandard deviation) are extracted from the Monte Carlosimulation results. A good agreement of these results withthe "Monte Carlo" results confirms the validity of the normaldistribution. The results labeled as Proposed Method arethose obtained from the proposed method. Sample studyresults are shown in Fig. 3 for the PCC current distortionindex of a test system. It can be seen that the proposedmethod has an acceptable performance and it is simpleenough for engineers to use.

5Branch 1_2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 7 11 13 1 7 19 2 3 2 5 29 31 35 37Harmonic Order

Har

mo

nic

Cu

rren

t M

agn

itu

de

(%)

95% - Monte Carlo

95% - Normal Distribution

95% - Proposed Method

Branch 1_2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 7 11 13 17 19 23 25 29 31 35 37Harmonic Order

Har

mo

nic

Cu

rren

t Mag

nit

ud

e (%

)

50% - Monte Carlo

50% - Normal Distribution

50% - Proposed Method

Fig. 3. Sample results of the proposed method.

C. Assessment of InterharmonicsInter-harmonics are those spectra components whose

frequencies are not integer multiple of the fundamentalfrequency. In recent years, interharmonics have gained someattention because they can cause voltage flicker. A voltageflicker can be viewed as the modulation of the peak (or RMS)value of a voltage waveform. The inter-harmonics can beatwith the fundamental frequency component to causemodulated peak voltage. Fig. 4 shows the waveformcomposed by a 175 Hz component and a 60Hz component. Itcan be seen that the waveform magnitude beats at about 5 Hz.

Fig. 4. Voltage flicker waveform caused by a 175Hz inter-harmonic.

Variable frequency drives, particularly the current sourceinverter based and the voltage source inverter based drives,can be a source of interharmonics. The drive can beconsidered as a frequency converter. It converts a 60Hz inputinto an output with a different frequency. Technicallyspeaking, the system can be considered as bi-directional,meaning the motor side voltages and currents could also beconverted to the supply system side (Fig. 5). For example, ifthe motor is running at 50Hz, this frequency will be seen atthe DC link as 50Hz ripples. As a result, the current of theDC link actually contains both 60Hz ripples (due to supplyvoltage) and 50Hz ripples (due to motor voltage). Since thesupply side current is related to the DC link current throughthe converter, the 50Hz ripples will penetrate into the supplyside and present themselves as inter-harmonics (since 50Hzis not an integer multiplier of 60Hz). Reference [15] providesan excellent description on this phenomenon. The inter-harmonic components are normally very small and they varywith the drive output frequency. If their frequencies coincidewith the system resonance frequency, the inter-harmonics canbe amplified. Problems could also arise if the drive producesexcessive inter-harmonics due to poor design or aggressivecost cutting.

InverterConverter MotorSource

60Hz 50Hz

DC link reactor

60Hz ripple 50Hz ripple

Fig. 5. Generation of inter-harmonics.

Although several papers have been published on the subjectof predicting interharmonics for such drive systems, a lotmore work is still needed. One topic of good interest toindustry and research communities is to develop an adequateinter-harmonic model for such a drive system. This is adifficult task due to the following reasons:1) The inter-harmonics cannot be modeled as a pure current

source. This is because the inverter produced DC-linkripple is voltage ripple (assuming the inverter side has astrong AC source to simplify this analysis). The currentripple is therefore a function of the impedances of theDC link and the supply system. If the inverter side is aweak system, the impedance of the inverter side couldalso play a role.

2) The frequency and magnitude of the inter-harmonics varywith the drive operating condition. As a result, theanalysis of the impact of interharmonics have to cover awide range of operating conditions.

These research results can be very valuable to the analysis ofdrive-caused voltage flicker problems and the design ofmitigation measures to such problems.

D. Advanced analysis of harmonic resonanceAlthough the phenomenon of harmonic resonance is well

known, tools available to analyze the phenomenon are verylimited. Frequency scan analysis can normally identify theexistence of resonance. It is more important to know whichbus can excite the resonance more easily and which powersystem components play more significant role for theresonance. One approach that has the potential to answerthese questions is to analyze the eigen-structure of thesystem admittance matrix [Yf] shown in (3).

Imagine a system experiences a sharp parallel resonance atfrequency f according to the frequency scan analysis. Itimplies that some elements of the voltage vector of (3) havelarge values at f. This in turns implies that the inverse of the[Yf] matrix has large elements. This phenomenon, in turn, isprimarily caused by the fact that one of the eigen-values ofthe Y matrix is close to zero. In fact, if the system had nodamping, the Y matrix would become singular due to one ofits eigenvalues becomes zero. The above reasoning leads usto believe that the characteristics of the smallest eigenvalueof the [Yf] matrix could contain useful information about thecause of the harmonic resonance. The above analysis can beformally stated as follows:

Let ? TT? ? TY 1-==

6be the eigen-decomposition of the Y matrix at frequency f. Yand T are the left and right eigenvector matrices of Y (thesubscript f is omitted here to simplify notations) and L is thecorresponding eigenvalue matrix. Y=T-1 is due to the factthat Y is 'block' symmetric. Define

( )6 TVVTII

==

m

m

as the modal current and voltage vectors respectively. It canbe seen that the original frequency scan equation has beentransformed into the following form:

( )7...

000

0...00000000

...

3

2

1

1

12

11

3

2

1

=

-

-

-

m

m

m

nm

m

m

I

I

I

V

V

V

l

ll

With the above equation, one can easily see that if l1=0 or isvery small, a small injection of mode 1 current will lead to alarge mode 1 voltage. On the other hand, the other modalvoltages will not be affected since they have no 'coupling'with mode 1 current. In other words, one can easily identifythe 'locations' of resonance in the modal domain. Afteridentifying the critical mode of resonance, it is easy to findthe 'participation' of each bus in the resonance. This can bedone using the well-known participation factor theory [16].Fig. 6 shows a sample analysis on an IEEE 14 bus harmonictest system. The size of the circle shows the degree ofparticipation of each bus to the resonance occurred ath=10.3. It can be seen from the figure that this particularresonance condition mainly involves buses 1, 2 and 5. At theUniversity of Alberta, we have studied harmonic resonanceusing modal analysis for sometime and a lot of veryinteresting results have been obtained. In this paper, we canonly present very limited information due to space limitation.

Weighted bus partifation factors to critical mode at 10.7 pu

2

3

4

5

6

7

8

0 2 4 6 8 10 12

G

G

C

13

1214

11 10

9

4

8

7

6

5

23

1

converter

SVC

Fig. 6. Relative significance of each bus in the 10.7th harmonic resonance.

V. CONCLUSIONS

In this paper, we have reviewed some significantprogresses in the area of power system harmonic analysis.Commonly accepted methods for conducting harmonicstudies have been summarized. The area of harmonic analysisis still a very fertile ground for exploration. With the help of

four example research topics, we attempt to demonstratepossible future directions of harmonic analysis research. Theconclusion is that we need to approach the subject from awider perspective. It is important to remember that there arestill many problems remaining to be solved. The mostdifficult part facing us is how to extract the problems into aform that can be researched with clear directions.

VI. ACKNOWLEDGMENT

The sample research works presented in this paper aresupported by the Natural Science and Engineering ResearchCouncil of Canada and other granting agencies. Some figuresare provided by my graduate students.

VII. REFERENCES[1] W. Xu, Component Modeling Issues for Power Quality Assessment

(Invited Paper for a Power Quality Tutorial Series), IEEE PowerEngineering Review, Vol. 21, No. 11, November 2001, pp.12-15+17.

[2] Derick Paice, Power Electronic Converter Harmonics: MultipulseMethods for Clean Power, IEEE Publication, New York, 1995, ISBN 0-7803-1137-X.

[3] IEEE Harmonics Modelling and Simulation Task Force, Modelling andSimulation of the Propagation of Harmonics in Electric Power Networks:Part I, IEEE Trans. on Power Delivery, vol. 11, n 1, Jan 1996, pp. 466-474.

[4] D. Xia and G. T. Heydt, Harmonic Power Flow Studies, Part I-Formulation and Solution, Part II-Implementation and PracticalApplication, IEEE Trans. on Power Apparatus and Systems, vol. PAS-101, no. 6, June 1982, pp. 1257-1270.

[5] Vinay Sharma, R. J. Fleming, and Leo Niekamp, An Iterative Approachfor Analysis of Harmonic Penetration in the Power TransmissionNetworks, IEEE Trans. on Power Delivery, vol. 6, no. 4, October 1991,pp. 1698-1706.

[6] W. Xu, J.R. Jose and H.W. Dommel, A Multiphase Harmonic Load FlowSolution Technique, IEEE Trans. on Power Systems, vol. PS-6, Feb.1991, pp. 174-182.

[7] A. Mansoor, W. M. Grady, A. H. Chowdhury, and M. J. Samotyj, AnInvestigation of Harmonics Attenuation and Diversity AmongDistributed Single-Phase Power Electronic Loads, IEEE Trans. onPower Delivery, vol. 10, no. 1, Jan. 1995, pp. 467-473.

[8] T. Hiyama, M. S. A. A. Hammam, and T. H. Ortmeyer, DistributionSystem Modeling with Distributed Harmonic Sources, IEEE Trans. onPower Delivery, vol. 4, no. 2, April 1989, pp. 1297-1304.

[9] M. Lehtonen, A General Solution to the Harmonics SummationProblem, European Transaction on Electrical Power Engineering, vol. 3,no. 4, July/Aug. 1993, pp. 293-297.

[10] Probabilistic Aspects Task Force of the Harmonics Working GroupSubcommittee of the Transmission and Distribution Committee, Time-Varying Harmonics: Part II Harmonic Summation and Propagation,IEEE Trans. on Power Delivery, vol. 17, no. 1, January 2002, pp. 279-285.

[11] S. R. Kaprielian, A. E. Emanuel, R. V. Dwyer and H. Mehta, PredictionVoltage Distortion in a System with Multiple Random HarmonicSources, IEEE Trans. on Power Delivery, vol. 9, no. 3, July 1994, pp.1632-1638.

[12] Y. J. Wang, L. Pierrat, and L. Wang, Summation of Harmonic CurrentsProduced by AC/DC Static Power Converters with Randomly FluctuatingLoads, IEEE Trans. on Power Delivery, vol. 9, no. 2, April 1994, pp.1129-1135.

[13] E. Duggan and R. E. Morrison, Prediction of Harmonic VoltageDistortion when a Nonlinear Load is Connected to an Already DistortedSupply, IEE Proceedings, Part C , vol. 140, no. 3, May 1993, pp. 161-166.

[14] A Cavallini and G. C. Montanari, A Deterministic/Stochastic Frameworkfor Power System Harmonics Modeling, IEEE Trans. on Power Systems,Vol. 12, No. 1, February, 1997, pp. 407-415.

[15] R. Yacamini, Power System Harmonics: Part 4 Interharmonics, IEEPublications (Tutorial series deals with harmonics), Power EngineeringJournal, August 1996, pp. 185-193

7[16] I. J. Perez-Arriaga, G. C. Verghese, and F. C. Schweppe, Selective ModalAnalysis with Applications to Electric Power Systems, Part I: HeuristicIntroduction, IEEE Trans. on Power Apparatus and Systems, vol. PAS-101, no. 9, Sept. 1982, pp. 3117-3125.

VIII. BIOGRAPHIES

Wilsun Xu (M'90, SM'95) received Ph.D. from the University of BritishColumbia, Vancouver, Canada in 1989. He worked in BC Hydro from 1990 to1996 as an electrical engineer. Dr. Xu is presently a professor at the Universityof Alberta and an adjunct professor of the Shandong University of China. Hismain research interests are harmonics and power quality. He can be reached atwxu@ee.ualberta.ca.