Distortion and Power Factor of Nonlinear Loads

7/30/2019 Distortion and Power Factor of Nonlinear Loads

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DISTORTION AN D POWER FACTOROF NONLINEAR LOADS

Tristan A. Kneschke, P.E.LTK Engine ering ServicesTwo Valley Square, Suite 3005 12 Tow nship Line RoadBlue Bell, PA 19422

Telephone : 2 15-542-0700 Facsimile: 215-542-7676 E-mail: [email protected]

Ab strac t. In recent years some electrical power utilitycompanies have experienced a significant increase in thequantity and m agnitude of nonlinear loads being connected totheir system s. Load s such as fluore scen t lights, rectifiers,microprocessor-driven equipment, power supplies, andvariable speed drives are often relatively small and welldispersed to cause a major negative impact on the powersystem.However when large nonlinear loads, in the order of tens ofmegawatts, are connected to utility systems, significantharmonic voltage s and currents are produced. These loadsinclude static freq uenc y converte rs servin g 25 Hz and 16Y3 Hzintercity and commuter rail systems. The converterharmonics cause increased heating of the utility and othercustomer equipment and can lead to system resonance.Therefore, the harmonics need to be taken into account invarious system evaluations.The difficulties in dealing with nonline ar loads encountered bythe a uthor include the following:

Calculation of the harm onic distortion.Calculation of power factor.

The IEEE method of harmonic distortion calculation iscomp ared with an alternative method proposed in technicalliterature. Th e alternative me thod resolves the intuitivedifficulty of visualizing harmonic distortion of over 100%when o ne or more of the frequencies in the harmonic spectrumis higher than the fundamental frequ ency, but introduces otherdifficulties, suc h as unde rstating the m agnitude of distortion.The complexity of determinationof power factor for distortedvoltage an d current increases as m ore harm onics are includedin the calculation. Due to distorted waveforms, the true power

factor is always lower than the power factor calculated when,as often is the case, the harmo nics are ignored.In this paper the concepts of nonlinear load and harmon ic arereviewed first. Subseque ntly, calculation of the harmonicdistortion and power factor are discussed in mathematicalterms and supplemented with prac tical examp les.1. LINEAR AND NONLINEAR LOADSLinear loads have the following characteristics:0

0e

In

Linear loads, when con nected to a system with sinusoidalvoltage, draw sinusoidal currents.The supply voltage rem ains sinusoidal.Voltage and current waveforms are of the same shape a ndcontain only fundame ntal frequency.comparison, nonlinear loads have the following

characteristics:Nonlinear loads, when conne cted to a system withsinusoidal voltage, draw non sinusoidal cu rrents.

0 The supply voltage becom es nonsinusoidal.0 The voltage and current waveforms are not of the same

shape and contain fundamental frequency as well asnonfundamental frequencies, so-called harm onics.2. FUNDAMENTAL FREQUENCY AND HARMONICSPower system analysis, design procedures, and calculationmethods are developed for voltages, currents, and powerdem ands having purely sinusoidal, i .e. ,undistorted waveform s.To enable analyses and d esigns of systems with nonsinusoidalwaveforms, the nonsinusoidal (distorted) variables must berepresented by sinusoidal (undistorted) waveform s.

0-7803-55334/99/S10.00 1999 EEE 47
mailto:[email protected]:[email protected]


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Using Fourier analysis, each periodic distorted waveform canbe represented by a fundamental frequency and number ofharmonics. Any required system studies can then beperformed separately at each frequency and final resultsobtained by subsequent superposition of the results atindividual frequencies.The IEEE defies and the industry recognizes the followingtypes of harmonics:

Characteristic harmonics or integer harmonics whoseharmonic order is equal to an integer multiple of thefundamental frequency.Noncharacteristic harmonics or noninteger harmonicswhose harmo nic orde r is equal to a noninteger multiple ofthe fundam ental frequency. Two types of nonintegerharmonics are identified:Sub-harmonics - the fundamental f requencymultipliers are less than I , and therefore, theharmonic frequenc ies are lower than the fundame ntalfrequency.Inter-harmonics - the fundamental f requencymultipliers are larger than 1, and therefore, theharmonic freque ncies are higher than the fundamentalfrequency. The frequencies of inter-harmonics arebetween the frequencies of characteristic harmonics.

The characteristic harmonics are the conventional harmonicsproduced by sem iconductor conve rter equipment in the courseof normal operation. The noncharacteristic harmonics are aresult of abnormal operation, and are caused by beatfrequencies, demodulation, and unbalance of the electricalpower supply network. The noncharacteristic harmonics areare also produced by cycloconverters in the course of theirnormal operation.Regardless of the harmonic type, it can be uniformly statedthat all harmonics, characteristic and noncharacteristic, arepotentially harmful to electrical equipment and should belimited to the lowest practical level. Harm onics add to thefundamental current already present in the equipment andincrease the apparatus heating, could be a source ofelectromagnetic interference, and, under particularly onerousconditions may cause system resonance.3. HARMONIC DISTORTIONIEEE DefinitionsIn order to quantify the level of harmonic distortion, the IE EEStandard 5 19 [11 defines ha rmon ic distortion with respect tothe fundamental frequency.The Individual Harmonic Distortion (IHD) at a particularharmonic freque ncy is the ratio of the root-mean-square v alue

(FWS) f the harm onic under con sideration to the RMS valueof the fundamental as shown in the following expression:Harmonic Frequency

Fundamental FrequencyIH D = ' 1 00 (1)

The Total Harmonic Distortion (THD) is defined as the ratioof the RMS sum of all harmonic frequencies to the RMSvalueof the fundamental frequency as shown below:

(2)RMS Su m of al l HarmonicsFundamental FrequencyTH D = .loo

These definitions are accepted throughout the industry and on eof their advantages is their linear relationship between themagnitude of the harmonic components and the IHD o r THDvalues. For exam ple, it is possible to say that the value of aharmonic in a waveform with IHD =4% i s mice as h igh theharmonic value in another waveform with IHD =2%.For most nonlinear loads the magnitude of the fundamentalfrequency is much larger than the magnitude o f an y individualharmonic frequency and also much larger than the RMS su mof all harmonics. In such cases the IHD and TH D are wellbelow 100%.Alternative DefinitionsLoads such as cycloconverters ha ve a very high content o f lowharmonic. For example, a 60 H z to 25 Hz cycloconverter,such as the one used by SEPTA to supply a part of itscommuter rail lines [2] produces the highest harmonics at 10Hz, 40Hz, 110 Hz, 160 Hz. The magnitude of the 10 Hzfrequency ma y, especially during light load cond itions, exceedthe magnitude ofthe fundamental frequency. In such an event,the individual and total harmonic distortion, if calculated bythe IEEE method, would be higher than 100%. Whenexpressing harmonic distortion with values over loo%, anintuitive feeling for how distorted a particular variable is ma ybe lost. For example, when THD =130%, does it make senseto say a waveform is 130% distorted?To avoid this disadvantage, the following alternative methodfor calculation of Total Harmonic Distortion was proposed[3]. The method expresses the THD relative to the RMSmagnitude of the entire waveform, not relative to themagnitude of the fundamental frequency, as shown below:

(3)RMS Sum of AN HarmonicsRM S Sum of AN FrequenciesTH D = '100~ ~ ~ ~

This method could also be extended for calculation of

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Individual Harmonic Distortion as shown in the followingequation:

P = v m , lm , c o s4 =V l c o s 4 (8 )2

Harmonic FrequencyRMS Su m of A ll FrequenciesIH D = . O0 (4)

Using the alternative definitions, the IH D a nd the THD wouldnever be above 100%. However, the linear relationshipbetween the m agnitude of the harmonic components and theresulting IHD and T HD would be lost as shown in Figure 1 .

450.00400.00

II- 350.002 300.00

250.00200.00

2 150.00

sv.-90.-5FTk 100.00

50.000.00

I IEEE Distortiona

' Alternative D istortion0 50 100 150 200 250 300 350 400

Magnitude of Harmonic Content

Figure 1 - Relationship Between the Magnitude of theHarmonic Components and the Resulting Distortion

IEEE LimitsIt is acknowledged that the IEEE Std. 519 limits are severe forcertain types of loads and particularly for cycloconverters.However, the approaches discussed in this paper appear to bedesigned to understate the distortion levels, a nd therefore arenot recommend ed for general use. Instead, the followingcauses of action are proposed:

The IEEE Standard limits could be modified to reflectcharacteristicsof various types of equipm ent. This can beaccomplished by participation at the appropriatecomm ittees, task force groups, or w orking groups.Parties associated with a particular project, including thepower utility, owner, and manufacturer, could agree thatthe installation will not comply w ith the IEE E S tandardlimits which, after all, are only recomm endations.A harm onic distortion s tudy could be performe d todemonstrate that any adverse effects due to harmonicdistortion will be acceptable.

Expressions for calculating he Individual Harm onic Distortionand Total H armonic Distortion for voltages and currents usingthe IEEE and the alternative methods are provided in theAppendix.4. POWER FACTORPrior to developing a method of pow er factor calculation fornonlinear circuits with harmonic voltages and currents, lineartheory w ith sinusoidal variables is reviewed.Linear Circuits - Single Frequency AnalysisInstantaneous values of voltage and current are given by:Calculation of Harmonic Distortion in Practice

The alternative methods of distortion calculation have onefurther disadvantage. Since the denominator of the alternativedefinition (fundamental a nd harmonics) is always higher thanthe denominator in the IEEE definition (fundamental only),calculations using the alternative definition would alwaysunderstate the magn itude of distortion.Another exam ple of ha rmonic distortion calculations used inpractice is the m ethod proposed in [4]. Here, the magnitude ofthe fundamental frequency of the waveform in thedenominator is replaced with a value corresponding to therating of a line or a supply transformer. The magnitude ofharmonic distortion depends on the c haracteristics of the loadand not on rating of the supply equipment. According toIEEE, the harmon ic distortion isa measure of distortion of thefundamental waveform, and not of the equipment rating.Clearly, distortion values would no t change in the event thatthe rating of the supply line or transformer is changed.

v(t)=V m , s in ( w t +a ) ( 5 )

The instantaneous power demand is a product of theinstantaneous voltage and current:

(7)(t) =v(i). i(t) =I =Vm,sin(mt +a ) . ,,sin(cd +p )

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The apparent power is defined as:

In any circuit; h e a r or nonlinear, regardless of the voltage andcurrent waveforms, the powe r factor is de fined as a factor bywhich the apparent power nee ds to be multiplied in order toobtain the real power [5] as sh own in the following equation:I P =Power Factor *SFrom the foregoing equation, the pow er factor is equal to aratio of real and apparent powe r. In linear circuits withsinusoidal voltage and current waveform s, the power factor iscalled displacement power factor (dPF) and is also equal to thecosine of angle between the voltage and current.

Nonlinear Circuits - Harmonic Frequency AnalysisWhen a nonsinusoidal voltage containing fundamentalfrequency and harm onics up to the m order, as shown below

v(5)=p,,Isin(wt +a ] )+Vmm2sin(2wt+a2)++Vm,3sin(3wt +a3)+ ..+V,,,sin(mwt +a,)

h=I

is applied to a nonlinea r circuit, the n the resulting curren t willcontain fundamental frequency and harmonics up to the nthorder:

The instantaneous power is ag ain calculated as a product of theinstantaneous voltage and c urrent:

p( t )=v(t). (t)==/VmmIsin(wtil)+Vm,2sin(20t +a2)++Vm,3sin(3wt +a3)+ ..+V,,,,,sin(mwt +am)J..[I,,Isin(wt +P I )+Im,2sin(2wt +p2 )++Im,3sin(3wt +p3)+.. .+l,,sin(not +p,)J (15)m n

h=l k = I

Using trigonometrical expansions, assuming that a,,pk =hhk,and rew riting the above equation for Rh4S values yields thefollowing expressions for real and reactive power. Since theseexpressions include harmonic terms, nomenclature for the realand the reactive power ha s been changed from P to SRea,nd Qto SReact:

m n

~

Defming Rh4S values of vo ltage and current as:

enables to write equation for apparent power. The de signationof app arent power for linear circuits S is changed to SAppornonlinear circuits:

The P and Q are real and reactive values of power atfundamental frequency and are corresponding directly to theP an d Q values in the linear system equations (8 ) an d (9).The power factor is again defined as a ratio of real andappa rent power. In nonlinear circuits with distorted voltageand current waveforms, the power factor is called true power

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factor (tPF) and is no longer equal to the cosine of anglebetween the voltage and current. distortion power is defined using the following equation:

Comparing he equation for displacement power factor and theequation for true power factor it can be concluded that latterpower factor will always be lower.

The difference between the true and displacement power factorincrease with co ntent of harm onics and can be substantial forvariables with large harmonic content.Distortion PowerWhen a nonsinusoidal voltage contains fundamental frequencyand harmonics up to the mth order, two sets of voltagefrequencies can be considered. One set, x =1 ..p, is due to thenonsinusoidal voltage, and the other set y = l...q is due to thenonlinear load. Similarly, a current, containing harmonics upto n" order, will include two sets of frequen cies, x= l...p dueto the nonsinusoidal voltage, and the other set z=1... due tothe nonlinear load.With this definition of harmonic content, power com ponentsdue nonsinusoidal voltage and due to to nonlinear load can beidentified and sepa rated.Expressions for the real and reactive power due tononsinusoidal voltage can be w ritten in the form of equations(1 6) and ( 1 7). Since the variables have a different harmoniccontent than SRea,nd SReact,he nomenclature for the powerexpressions has be en changed from SRealo Sp and from S,,,,,to s,:

~I I

I x = I x = I I

Frequencies due to nonlinear load, y =1 ..q for voltages and z= I...r for cu rrents fo rm so-called distortion power SD. The

y= I x = la r

y = l r =

The apparent power is defined as:

~ ~

and the true pow er factor is:

The true power factor component are defined as follows:P Ps$= v ; ' c I, cos2 4xx (28)

x = l x = l

x = l r = l

y = l := I IField Test ExampleA traction pow er substation for a light rail transit system withone transfonnerlrectifier unit rated at 1,000 kW and inputvoltage rating of 4.16 kV was tested using a BM I 303013060Power Profiler measuring instrument. A short circuit wasapplied at the rectifier output terminals a nd the input voltagewas increased only to a level permitting flow of full loadcurrent. Due to the fact that the load in the test circuit was

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formed by the highly reactive transformer windings, lowpower factor results were expected . The magnitude of thephase voltages and currents, true and displacement powerfactors, as well as the total harmonic distortion data are shownin Figure 2. POWERPROFILER SITE Sep 17 1998 (Ihu)BMI 3838 METERS

U c l t . age :Phase A - E :Phase B - C :Phase C - A :I mba lance :

Phase A :Phase 8 :Phase C :I m b a l a n c e :

C u r r e n t :

Power :UOlt-AlbpS:UA React . iwe:P ow er Fac t o r :D i s p l a c e m e n t F a c t o r :

Voltage THD:A-Nm V o l t :E-Nm Volt:c - t h V o l t :

A C u r r e n t :B C u r r e n t :C C u r r e n t :

Current THD :

1 1 : 2 9 : 3 2 AM

292 Urrtis38 9 Urms284 Urms284 Urnis5 , 6 %2 4 1 , 3 A t l i \ S141. 4 A rm s1 4 4 , 8 I? rm s132. 5 A rm s4 , 9 %

8 . 8 3 kW7 8 . 6 1 kUA69 I88 kUAR

8. 12 P F8, 15 dP F

4. 4% THD4 , 4 % THD3.2% THD5.5% THD8.1%THD8 . 5 % TH D~ 7 %HD

8 . 9 % TH D

Figure 2 -Harmonic Distortion andPower Factor MeasurementIt is interesting to note that for overall voltage THD of 4.4%and current THD of 0.7% the true power factor, whenharmonics are considered, is 0.12 in comparison with thedisplacement power factor of 0.15 at fundamental frequency.This represents a change of 25%.Compensation of Reactive PowerExamining the above equations for various voltage and loadconditions, the following four circumstances can arise:

Sinusoidal Voltages - Linear Loads, =VIcosq5=PS, =VIsinq5 =QS, can be completely compensated with capacitorsand in that case tPF =dPF=1.S,=O

Sinusoidal Voltages - Nonlinear Load

S,#Osp=VIcosq5=PS, =VIsinq5=QS, cannot be fully compensated by capacitors due tocross products at different frequencies. Therefore,tPF< 1 even if S, is completely com pensated.

0 Nonsinusoidal Voltages - Linear Loadsp=CVCICOSq5# PS, =C V C I s i n 4 * QS, cannot be fully compensated by capacitors due tocross products at different freque ncies. ThereforetPF< 1.

S,=O

Nonsinusoidal Voltages - Nonlinear LoadS, =CVCIsinq5 Qs, =p c 1 c o s 4 +PS D # oS, and So cannot be fully compensated by capacitorsdue to cross products at different frequencies.Therefore tPF < 1.

It should be noted that for linear and nonlinear circuits withsinusoidal voltage S, = P and S, =Q, SI, exists only fornonlinear loads, for linear loads So is zero.The above evaluation presents an interesting proposition thatin systems with distorted voltages a nd or nonlinear load, thepower factor cannot be compen sated to unity with capacitors.5. APPENDICESList of Symbolsa Voltage phase shiftP Current phase shift40 Angular velocity1 , 2 , 3, h, k, n, x, y, and z are harmonic frequency indexes

Difference between voltage and current phase shifts

Displacement power factorInstantaneous value of currentRMS value of current for single frequencyMaximum value o f currentMaximum value of current at fundamental frequencyMaximum value o f current at frequency 2Maximum value of current at frequency hMaximum value o f current at frequency nRMS value of current at fundamental frequencyRMS value of current at frequency 2RMS value of current at frequency hRMS value of current at frequency nIndividual Harmonic Distortion of current atfrequency hIndividual Harmonic Distortion of voltage atfrequency h

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Instantaneous value of powerRMS value of real pow er for single frequencyRMS value of real power for harmonic frequenciesRMSvalue of reactive power for single frequencyRMS value of reactive power for harm. frequenciesRMS value of apparent power for single frequencyDistortion powerRMS value of apparen t power for all frequenciesRMS value of real power for fundamental frequencyand all frequencies due to nonsinusoidal voltageRMS value of reactive power for all frequenciesRMS value of real powe r for all frequenciesRMS value of reactive power for fundamentalfrequency and all frequencies due to nonsinusoidalvoltageTimeTotal Harmonic Distortion of currentTotal Harmonic Distortion of voltageTrue power factorInstantaneous value of voltageRMSvalue of voltage for single frequencyMaximum value of voltageMaximum value of voltage at fundamental frequencyMaximum value of voltage at frequency 2Maximum value of voltage at frequency hMaximum value of voltage at frequency nRMS value of voltage at fundamental frequencyRMS value of voltage at frequency 2RMS value of voltage at frequency hRMS value of voltage at frequency n

IEEE Std. 519 Definitions of Harmonic DistortionIndividual Harmonic Distortion of Voltage

I H h =&-I00 (31) VI

Individual Harmonic Distortion of Current

IHDI,, =k.00 (32)I I

Total Harmonic Distortion of Current

Alternative Definitions of Harmonic DistortionIndividual Harmonic Distortion of Voltage

(34)

Individual Harmonic D istortion of Current

(33)

IC= I

Total Harmonic D istortion of Voltage

JJv:+v22+vj2+ ...+,2THDv =

Total Harmonic Distortion of Voltage

(37)

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Total Harmonic Distortion of Current

6.1.

2.

3.

4.

5.

7.

R E F E R E N C E SIEEE Std . 519, IEEE Recommended Pract ices andRequi rements for Harmonic Cont rol in Elect r icalPower Systems, Institute of Electrical and ElectronicsEngineers, 1992.T. Kneschke, Traction Power Augmentation ofSEPTAS Wayne Junct ion Conver ter Stat ion, IEEEPaper No. CH2020-6/84/0000-0007, IEEE PublicationNo. 84CH 2020-6, 1984 Joint ASME/IEEE Conference,Chicago, 11.T. M. Gruzs, Uncertainties in Compliance withHarmonic Current Distor t ion in Elect r ic PowerSyste ms, IEEE Trans. on Industry Applications, Vol. 27,No.4, July/August 1991.R. F. Chu, J. J. Bums, Impact of CycloconverterHa rm on ics , IEEE Trans. On Industry Application, Vol.25, No.3, MayIJune 1989.W. Shepherd, P. Zakikhani, Suggested Definition ofRea ctive pow er for Nonsinusoidal Systems, Proc. IEE,Vol. 119, No. 9, Septemb er 1972.A B O U T T H E A U T H O R

Tris tan A. K nes chk e (SM8 2) graduated from the Universityof Sussex, UK , where he also received his Ph.D. Both degreeswere in electrical engineering. Earlier in his career he wasresponsible fo r conducting a variety of studies and designs ofpower utility system s and industrial plants for clients in manycountries. Since his arrival to the U.S. in 197 7,. heconcentrated on analyses and design activities for mainlinerailroads, commuter railroads, and urban transit systems.Since joining LTK, he participated in more than 80 projectassignments of varying lengths and complexities. Currently,he is serving as a Project Manager on the Ohio-Kentucky-Indiana (OKI) Cincinnati 1-71 Corridor light rail transitproject, the New Jersey Transit (NJT) Newark City Subway

(NCS) upgrade, the NJT Newark to Elizabeth Rail Link(NERL ) light rail transit project, and on se veral system studiesfor Am trak, the National Railroad Passenger C orporation. Hisresponsibilities include studies and engineering of tractionpower, catenary, trolley, corrosion control, signaling,comm unication, and fare collection systems, a s well as of lightrail vehicles, yards, and shops.His assignments for mainline railroads include work on theNortheast Corridor Improvement Project for the FederalRailroad Administration (FRA) and a major investigation onbehalf of the U.S. Department o f Transportation (DO T) intothe feasibility of electrification of 10,000 miles 0fU .S. freightrailroads. Also, he worked on traction power supp ly systemdesigns for Michigan DOT, M issouri-Kansas-Texas (MKT)freight railroad, and Quebec Cartier mining railroad.Projects for commuter railroads includ e implementation of thefirst static frequency converter for traction power applicationon behalf of the southeastern Pennsylvania TransportationAuthority (SEPTA), NJTs North Jersey Coast Lineelectrification, and upgrade of Metro-North CommuterRailroad Harlem & Hudson lines.Relevant transit projects include the Hennepin County TwinCities Corridor LRT, the Walt Disney World monorail, thePortland Banfield, Westside, and Vintage Trolley, the SEPTABroad Street Subway, and the Market-Frankford Subway/Elevated line systems. He also performed studie s and designson the Los Angeles, Buffalo, San Diego, and Dallas AreaRapid Transit (DART) light rail transit system projects.Mr. Kneschke has numerous technical publications to hiscredit with topics including power supply and distributionsystem design, substation design, catenary system design,alternative system analyses, power utility interfaces, harmonicsand power quality issues, corrosion control issues, andelectrification system design optimization and costeffectiveness.He is the Chairman of Subcommittee No. 6 - Power Supplyand Distribution of the American Railway Engineering andMaintenance of Way Association (AR EMA ), the Chairman ofthe American Public Transportation Association (APTA)Power, Signals and Commu nications Committee, and he alsoserved as the Chairman of the IEEE Vehicular TechnologySocietysLand Transportation Division Executive Committee.He is a registered Professional Engineer in eight states and isa Recipient ofthree IEEE Industry Applications SocietysLandTransportation Committee and Vehicular TechnologyCommittee Awards.

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Distortion and Power Factor of Nonlinear Loads

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