High-performance harmonic extraction algorithm fora 25 kV traction power quality conditioner

P.C. Tan, P.C. Loh and D.G. Holmes

Abstract: Many single-phase 25kV electrified railway systems carry a large number ofconventional thyristor-based locomotives, which draw current rich in harmonic content. Apossible solution to mitigate this harmonic complication is to connect a power quality conditionerbased on voltage detection at the end of the traction feeder. However, the performance of theinstalled power quality conditioner depends strongly on the accuracy and stability of the harmonicextraction algorithm used to generate the command harmonic current reference to be tracked bythe inverter inner current control loop. A high performance, computationally efficient harmonicextraction algorithm is presented, implemented using delta-operator-based digital resonant filtersfor traction applications. Unlike most other extraction algorithms, the implementation of thealgorithm presented does not require complex mathematical transformations, and therefore has asignificantly faster computation speed and is suitable for single-phase traction applications. Also,through delta-operator-based implementation, the proposed algorithm exhibits improved accuracyand robustness over a wider range of operating conditions. The accuracy and robustness of theproposed algorithm have been verified by both simulation and experiment.

1 Introduction

On many main line electrified railway systems that usesingle-phase 25kV industrial frequency supplies, a signifi-cant proportion of the locomotives still employ phase-controlled thyristor converters to feed the DC motor drives.These older thyristor-based locomotives draw current witha low displacement power factor and a rich harmoniccontent, and are expected to be in service for many years tocome. In particular, the lagging load current causes asignificant amount of reactive voltage drop along the feederline, and the flow of harmonic load current through thefeeder impedance results in a distorted pantograph voltagewaveform that has a reduced (rectified) average value. Theformer effect (low RMS voltage) can significantly limit thedistances between substations and also the number oflocomotives that the system can support. As a standard,IEC specification 349 has stipulated that the minimumpantograph voltage must be above 19kV continuously and17.5kV for short periods. The latter effect (low averagevoltage) is also important because, for phase-controlledlocomotives, the power output is proportional to theaverage voltage (rather than the RMS voltage) [1].

To enhance the supply quality, the authors haveproposed the installation of a hybrid compensation system,implemented using a multilevel voltage-source-inverterbased active power filter and a passive RLC filter, at theend of the traction feeder [2]. For easier reference, the

hybrid compensation system will be referred to as thetraction power quality conditioner (T-PQC) in the remain-der of this paper. The installed switching active filter iscapable of mitigating low-order harmonics (3rd, 5th and 7thharmonics) to bring the form factor (defined as the ratio ofthe RMS value to the rectified average value) closer to theideal value of 1.11 for a pure sinusoid, whilst simultaneouslyproviding a controllable amount of reactive power to boostthe RMS voltage. The passive filter serves to dampharmonic overvoltages and also to suppress switchingnoise, producing a much cleaner pantograph voltage. Asfor most power filtering systems, the level of compensationthat can be achieved by the T-PQC depends strongly on thetracking accuracy and stability of the harmonic extractionalgorithm implemented, whose task is to generate thecommand harmonic current reference to be tracked by theinverter inner current control loop. To date, most of thecommonly used harmonic extraction algorithms involvethree-phase synchronous frame conversion and are there-fore not directly applicable to single-phase traction systems.

This paper presents a high-performance, computationallyefficient harmonic extraction algorithm, based on voltagedetection and implemented using delta-operator-baseddigital resonant filters, for traction applications. Unlikemost other extraction algorithms, the implementation ofthis algorithm does not require complex mathematicaltransformations, and it therefore has a significantly fastercomputation speed and is suitable for single-phasetraction applications. Particular challenges in digitallyimplementing the continuous time concepts on a low-costfixed-point DSP, that is used to control the inverter, arealso highlighted. In particular, it has been shown that,with delta-operator-based implementation of the digitalresonant filters, the RMS voltage fluctuations can bebrought to below 1% even with the use of only a low-cost 16-bit DSP. The accuracy and robustness of theproposed algorithm have been confirmed through bothextensive simulation and experimental investigations using amodel traction system.

P.C. Tan and D.G. Holmes are with the Department of Electrical andComputer Systems Engineering, Monash University, Wellington Road, ClaytonVIC 3800, Australia

P.C. Loh is with the Center for Advanced Power Electronics, School ofElectrical and Electronic Engineering, Nanyang Technological University,Nanyang Avenue, S639798, Singapore

r IEE, 2004

IEE Proceedings online no. 20040586

doi:10.1049/ip-epa:20040586

Paper first received 28th October 2003 and in revised form 2nd April 2004.Originally published online: 10th August 2004

IEE Proc.-Electr. Power Appl., Vol. 151, No. 5, September 2004 505

2 25 kV traction system with power qualityconditioning

2.1 25 kV traction systemThe traction system considered in this investigation consistsof a 30km single-phase contact feeder section withlongitudinal impedance of (0.169+j0.432)O/km at 50Hzand a shunt capacitance of 0.011mF/km, fed from asubstation stepdown transformer at 25kV, as shown inFig. 1. The 30km contact feeder is modelled as three 10kmpi-sections, and the substation transformer is represented byits equivalent series inductance and resistance. Up to fourlocomotive loads, each nominally rated at 2.5MW at 25kV,are considered. Each locomotive is modelled as a pair ofseries-connected half-controlled thyristor bridges, the detailsof which can be found in [3]. The system parameters usedare summarised in Table 1.

2.2 Traction power quality conditioner(T-PQC)The T-PQC is also shown in Fig. 1, and is shunt connectedto the end of the feeder section. This location of the T-PQCinstallation is chosen as it results in optimal compensationperformance and filter rating, as reported in [4]. Thecompensation system itself is made up of two distinctphysical subsystems: a voltage source inverter and a passivedamping filter. To cater for high-power applications, acascaded five-level voltage-source-inverter topology isadopted for implementing the T-PQC. This inverter isregulated by an inner hysteresis current regulation loop,which is chosen because, amongst all current regulationschemes, it has the best dynamic response and is well suited

for traction applications in which fast-changing nonlinearload conditions prevail.

The resulting hysteresis-regulated inverter is controlled tocompensate for the 3rd, 5th and 7th harmonics, and toinject fundamental reactive current, to restore the averagevoltage and boost the fundamental RMS voltage. Theoverall control structure of the T-PQC inverter is shown inFig. 2, where an additional PI controller is incorporated toregulate the inverter DC bus voltage. Further details of thecontrol algorithm can be found in a preceding paper [2]. Toimprove performance further, a second-order passive RLCfilter is also added as shown in Fig. 1, to damp out resonantovervoltages, usually 1–2kHz, on the railway system [2, 5],

traction power quality conditioner

R LRS

LS

iS

iLBiLA iLC iLD

vdc 1

vdc 2

iinvVS

vA vB vC

CP

RPLP

LCvDR L

C /2 C /2C C

R L

passivedamping

filtertrain load A train load B train load C train load D

30km contact feeder sectionfeeder substation

five-levelvoltage Source

inverter

Fig. 1 25 kV railway traction system model

Table 1: Traction system parameters

System voltage 25kV

Line frequency 50Hz

Substation transformer inductance LS 27.1mH

Substation transformer equivalentdamping resistance RS

1000O

PI-section inductance L 1.38mH

PI-section resistance R 1.69O

PI-section capacitance C 0.11mF

Inverter coupling inductance LC 47.3mH

Passive filter capacitance CP 0.84mF

Passive filter inductance LP 385mH

Passive filter resistance RP 344O

fundamental voltage control / reactive power compensation

150Hz resonantfilter G3(s)

250Hz resonantfilter G5(s)

350Hz resonantfilter G7(s)

proportional term P

vD (t )

vD (t )

vD (t )

vD

vD*

Iq*

id*

Id*

Vdc*

Vdc

iinv

iq*

ihar*

iinv*+

+++

gain K

single-phasefive-levelinverter

switchingsignals

currenttransducer

runningwindow

RMS calculator

P-I controller

zero-crossingdetectionalgorithm

50Hzsine wavegenerator

90° phase lead

P-I controller

low passfilter

phase angle of 50Hz component

of vD

sine wave in phase with vD

3rd, 5th and 7th harmonic compensation

DC bus voltage control

sine waveleading vD by 90°

+−

++

+

+−

+−

G(s) = P+G3(s) + G5(s) + G7(s)

five-level hysteresis regulator

Fig. 2 Controller block diagram of T-PQC inverter

506 IEE Proc.-Electr. Power Appl., Vol. 151, No. 5, September 2004

and to partially absorb the switching current ripple of theinverter.

The only complication associated with the T-PQC is thespreading of its generated switching harmonics across awide frequency range due to the use of asynchronoushysteresis switching. This does increase the potential for theswitching harmonics to coincide with the resonant modesof the overhead system. However, with the use of aharmonically less distorted five-level inverter and a passiveRLC resonant-damping filter, there is generally insufficientenergy at any one harmonic frequency to excite a significantovervoltage (as confirmed in Section 5). It is alsocommented that traction loads are quite tolerant ofa distorted power supply, and therefore the high-frequency switching ripple in the supply voltage is usuallynot an issue [3].

3 High-performance harmonic extraction algo-rithm for railway applications

The harmonic extraction algorithm presented in this paperuses a frequency-modulation process in the Laplace domain[6, 7] to convert well-established synchronous frame filteringprinciples in one part of the frequency spectrum tostationary frame forms suitable for single-phase tractionapplications in another part of the frequency spectrum. Tobetter illustrate the concept, this Section begins with adescription of the criteria used to select the appropriatesystem variable detection method before the frequencymodulation process is applied to derive the proposedharmonic extraction algorithm.

3.1 System variable detection methodsThe first consideration when implementing a harmonicextraction algorithm is to select an appropriate systemvariable detection method. Referring to the literature, for ashunt active power filter, three detection methods arecommonly used to determine the harmonic currentreference i�har:

� by directly measuring the load harmonic current to becompensated and using the current signal as a referencecommand [8, 9]

� by controlling the filter to minimise the source harmoniccurrent [10, 11]

� by measuring and controlling the harmonic voltage at thepower filter point of coupling to minimise particularharmonics [4, 11].

For a traction system, since the shunt active power filter isto be added to the far end of a feeder while the locomotiveloads are physically moving, it is not feasible to measure theload and source currents. Therefore, the harmonic voltagedetection approach seems to be the only practical possibilityfor the hybrid power filter control. Consequently, the nodevoltage vD at the far end of the feeder section is measuredand fed into the harmonic extraction algorithm (presentednext) to provide a harmonic current reference i�har, given by:

i�har ¼ KG sð Þ � vD ð1Þ

where K is the controller gain and G(s) is the equivalenttransfer function of the harmonic extraction algorithm, asindicated in Fig. 2.

3.2 High-performance harmonic extractionalgorithmMost of the commonly used harmonic filtering approachesrequire a synchronous frame conversion of the measuredsignals, which obviously cannot be applied to single-phasetraction systems. To date, the closest equivalent (synchro-nous) method developed for a single-phase system is tomultiply the measured signal, in turn, by sine and cosinefunctions at a chosen frequency [12], as shown in Fig. 3.This achieves the same effect of transforming the compo-nent at the chosen frequency to DC, leaving all othercomponents as AC quantities. Take for example a voltagesignal consisting of the fundamental and 3rd harmoniccomponents:

V ¼ V1 cosðot þ y1Þ þ V3 cosð3ot þ y3Þ ð2Þ

Multiplying this by cos (ot) and sin (ot) gives, respectively:

VC ¼V1

2cosðy1Þ þ cosð2ot þ y1Þf g

þ V3

2cosð2ot þ y3Þ þ cosð4ot þ y3Þf g ð3Þ

VS ¼V1

2sinð�y1Þ þ sinð2ot þ y1Þf g

þ V3

2sinð�2ot � y3Þ þ sinð4ot þ y3Þf g ð4Þ

It is observed that the fundamental term now appears asDC quantities cos(y1) and sin(�y1). The only complicationwith this equivalent single-phase conversion is that thechosen frequency component not only appears as a DCquantity in the synchronous frame, but also it contributes toharmonic terms at a higher frequency (cos(2ot+y1) andsin(2ot+y1) for the example above). This is unlike thethree-phase synchronous d-q conversion, where the chosenfrequency component contributes only towards the DCterm. Therefore, using the single-phase equivalent conver-sion, the harmonics cannot be extracted from the measuredsignal by simply using a highpass filter.

An alternative approach presented in this paper toovercome the above-mentioned complication is to useseveral single-phase harmonic synchronous frame filters inparallel, each extracting a harmonic frequency of interestfrom the measured vD signal. This approach is illustrated inFig. 4 for extracting the 3rd, 5th and 7th harmonics, andindeed is more suitable for railway systems in which theprimary objective is to compensate only for the first fewlow-order harmonics, which have the most impact on theaverage voltage. To compensate for anything more thanthose selected low-order harmonics would firstly be a waste

v (t )

low-passfilter

low-passfilter

++ × 2 y (t )

sin(ωot)

cos(ωot) cos(ωot)

sin(ωot)

GAC(s )

GDC(s )

Fig. 3 Single-phase equivalent synchronous reference frame filter

IEE Proc.-Electr. Power Appl., Vol. 151, No. 5, September 2004 507

of inverter capacity and, more importantly, risk thepossibility of severe switching distortion as the T-PQCattempts to compensate for the 41kHz resonance. This isunlike compensation for public power systems in which thesuppression of all harmonics is preferred so as to lower theTHD as much as possible [4].

In the harmonic synchronous frame, the lowpass filteredDC output is the harmonic voltage magnitude. The lowerthe filter cut-off frequency, the more accurately theharmonic voltage component will be extracted in the steadystate, but the transient response time will increasecorrespondingly as a consequence. For three-phase syn-chronous transformation, the lowest ripple component is atsix times the fundamental frequency. This contrasts withonly twice the fundamental frequency for the equivalentsingle-phase transformation [12]. In other words, for thesame extraction accuracy, the lowpass cut-off frequency hasto be lower in a single-phase system, resulting in thetransient performance being significantly poorer. All theseconsiderations, plus the difficulty of performing lineartransfer analysis on Fig. 4, point to the more challengingnature of power filtering design for traction systems.

To assist in the design process and to derive the proposedharmonic extraction algorithm, the filtering transfer func-tion GDC(s) in the synchronous reference frame is frequencymodulated to an equivalent stationary filtering function

GAC(s) (see Fig. 3) using [6, 7]:

GACðsÞ ¼ GDCðs� josÞ þ GDCðsþ josÞ ð5Þwhere os is the selected angular rotating frequency of thesynchronous frame. If a simple first-order lowpass filterwith cut-off frequency oc is used in the synchronous frame,i.e.:

GDCðsÞ ¼1

1þ ðs=ocÞð6Þ

then using (5) it can be shown that:

GACðsÞ ¼2ocsþ 2o2

c

s2 þ 2ocsþ ðo2c þ o2

s Þð7Þ

which is a second-order resonant type network with a peakmagnitude at the resonant frequency of

or ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffios

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

s þ 4o2c

q� o2

c

rð8Þ

The parameter oc determines the width of the resonantpeak and if oc{os, (7) can be simplified to:

GACðsÞ ¼2ocs

s2 þ 2ocsþ o2s

ð9Þ

which has a resonant peak of 1 centred exactly at s¼ jos.Using (7) and (9) as the mathematical basis, linear control

+

+

+

+

+

+

++

+v (t )

low-passfilter

low-passfilter

low-passfilter

low-passfilter

low-passfilter

low-passfilter

× 2

× 2

× 2

sin(3ωot)

cos(3ωot) cos(3ωot)

sin(3ωot)

sin(5ωot)

cos(5ωot) cos(5ωot)

sin(5ωot)

sin(7ωot)

cos(7ωot) cos(7ωot)

sin(7ωot)

vD 3

vD 5

vD 7

Fig. 4 3rd, 5th and 7th synchronous frame selective harmonic filter for traction applications(oo¼ system line frequency)

508 IEE Proc.-Electr. Power Appl., Vol. 151, No. 5, September 2004

analysis can now be easily carried out on the entire system,hence simplifying the design process. Another advantage ofusing (9) is that it requires less signal processing andtherefore can be implemented using a low-cost 16-bit DSP.

To summarise, the transfer function of the proposedharmonic extraction algorithm for filtering the 3rd, 5th and7th harmonics is given by:

GðsÞ ¼ 2ocs

s2 þ 2ocsþ ð3ooÞ2þ 2ocs

s2 þ 2ocsþ ð5ooÞ2

þ 2ocs

s2 þ 2ocsþ ð7ooÞ2ð10Þ

where oo represents the angular fundamental frequency of2p(50) rad/s, and its frequency response is plotted in Fig. 5for two values of oc, which obviously shows the presence ofresonant peaks at only the selected compensation frequen-cies of 150Hz, 250Hz and 350Hz. Also noted is that, as oc

gets smaller, G(s) becomes more selective (narrowerresonant peaks). However, using a smaller oc will makethe filter more sensitive to frequency variations, lead to aslower transient response and make the filter implementa-tion on a low-cost 16-bit DSP more difficult due tocoefficient quantisation and round-off errors. In practice, oc

values of 5–15 rad/s have been found to provide a goodcompromise.

4 Implementation of harmonic extractionalgorithm

To implement the proposed harmonic extraction algorithmdigitally using a low-cost fixed-point Texas InstrumentsTMS320F240 DSP, the filtering transfer function (10) mustfirst be digitised using an appropriate digitisation method.This Section presents and compares two methods ofdigitally implementing the discrete equivalence of (10),identifying in particular the optimal technique for use withthe proposed algorithm.

4.1 Z-operator implementationThe most commonly used digitisation technique is the pre-warped bilinear (Tustin) transform [13], given by:

s ¼ o1

tanðo1T=2Þz� 1

zþ 1ð11Þ

where o1 is the pre-warped frequency, T is the samplingperiod and z is the forward shift operator. Equation (11)can then be substituted into (10) to obtain the z-domain(discrete) transfer function, from which the differenceequations for DSP implementation can be easily derived.

Note that, when expanded, (10) represents a 6th ordertransfer function. In analogue implementation or withfloating-point data representation, whether (10) is calculatedas a single 6th order network or a summation of three 2ndorder networks has no effect on the output value. Withfixed-point digital representation, the common practice is torealise a high-order filter as a cascade or parallel combina-tion of 1st or 2nd order stages to minimise the effects ofcoefficient quantisation errors and round-off errors. Equa-tion (10) is already conveniently expressed in the parallelform and can be implemented as such. Thus, for the 3rdharmonic resonant branch, the first term in (10) digitises to

G3ðzÞ ¼0:00099752417� 0:00099752417z�2

1� 1:9891377z�1 þ 0:99800495z�2ð12Þ

where oc¼ 10 rad/s and oo¼ 2p(50) rad/s. The correspond-ing difference equation can then be written as:

yðnÞ ¼ 0:00099752417 xðnÞ � xðn� 2Þ½ �þ 1:9891377yðn� 1Þ � 0:99800495yðn� 2Þ ð13Þ

For fixed-point implementation, the coefficients in (13) haveto be normalised with multiplying them by the maximuminteger value of the chosen word length. The choice of wordlength here is dictated by the size of coefficient quantisationerrors that can be tolerated. Large coefficient quantisationerrors can change the frequency characteristics of a filter,and even render it ‘open-loop’ unstable. Furthermore, themore selective the filter is, the more significant is the effectof coefficient quantisation, and thus a larger word length isrequired for representation.

For the 16-bit DSP used, a word length of 16 bits resultsin the fastest execution, since (13) would then involve three16-bit� 16-bit multiplications and each of these can bedone by the DSP’s 16-bit� 16-bit hardware multiplier in asingle machine cycle. The C compiler also supports a 32-bit� 32-bit multiplication, which takes about five timeslonger to execute. However, the 32-bit multiplication givesmeaningful results only when the product can be repre-sented with 32 bits. Through experiments, it was found that,although computationally fast, 16-bit representation re-sulted in significant performance deterioration, and hence a20-bit representation of the coefficients in (13) was used.This is possible because the input to the filter, i.e. themeasured system voltage vD, comes from the analogue-to-digital-converter (ADC), which has only a 10-bit resolution.A larger than 20-bit representation for the coefficients is notused here since it would result in overflows when the termsin (13) are added under certain conditions.

Satisfactory performance has been achieved in simula-tions and experiments with the conventional shift-operatorimplementation of resonant filters given in (12) and (13).However, when compared to PSCAD simulations basedon a continuous time s-domain transfer function, someperformance degradation can be observed. With the shift-operator implementation, the shape of the compensatedsystem voltage vD changes slightly from cycle to cycle,causing significant fluctuations in its RMS value (as verifiedin Section 5). This is due to round-off errors associated withthe use of integer variables on a fixed-point DSP (so-calledfinite-word-length effect). 16-bit fixed-point implementationalways has finite-word-length effects, but the problem isparticularly pronounced at a fast sampling rate and for

−80

−60

−40

−100

−50

−20

0

102 103

frequency, Hz

mag

nitu

de, d

B

wc=1 rad/swc=10 rad/s

0

50

100

phas

e, d

eg

150Hz 250Hz 350Hz

Fig. 5 Frequency response of (10) for oc¼ 1 rad/s and 10 rad/s

IEE Proc.-Electr. Power Appl., Vol. 151, No. 5, September 2004 509

sharply tuned filters such as the resonant filters used in theharmonic extraction algorithm.

4.2 d-operator implementationTo improve performance, the use of delta operator d inplace of the conventional shift operator has been investi-gated. The delta operator has recently gained interest in fastdigital control due to its superior finite word lengthperformance [14–16]. The delta operator can be defined interms of the shift operator z as:

d�1 ¼ Dz�1

1� z�1ð14Þ

Essentially, delta-operator filter implementation involvesconverting a 2nd order section in z:

HðzÞ ¼ b0 þ b1z�1 þ b2z�2

1þ a1z�1 þ a2z�2ð15Þ

into a corresponding 2nd order section in d:

HðdÞ ¼ b0 þ b1d�1 þ b2d

�2

1þ a1d�1 þ a2d

�2 ð16Þ

where b0¼ b0, b1 ¼ 2b0þb1D , b2 ¼ b0þb1þb2

D2 , a0¼ 1, a1 ¼ 2þa1D ,

a2 ¼ 1þa1þa2D2 , and D is a positive constant less than unity,

which is carefully chosen to select the appropriate ranges forthe a and b coefficients, and to minimise other internalvariable truncation noise [15]. Equation (16) is thenimplemented using the transposed direct form II (DFIIt)structure shown in Fig. 6. The DFIIt structure is chosen outof the many filter structures available because it has the bestround-off noise performance for delta-operator-based filters[15]. From Fig. 6, the difference equations to be coded forthe DSP can be written, in processing order, as:

w4ðnÞ ¼ Dw3ðn� 1Þ þ w4ðn� 1Þw2ðnÞ ¼ Dw1ðn� 1Þ þ w2ðn� 1ÞyðnÞ ¼ b0xðnÞ þ w4ðnÞw3ðnÞ ¼ b1xðnÞ � a1yðnÞ þ w2ðnÞw5ðnÞ ¼ b2xðnÞ � a2yðnÞ

ð17Þ

Note that the first two equations in (17) for w4(n) and w2(n)are obtained from the definition of the delta-operator givenin (14). Similar to (13), the coefficients in (17) will initially befloating-point numbers and must be normalised by multi-plying them by the maximum integer value of the chosenword length. Here, the word length and the constant Drepresent two degrees of design freedom that are used tooptimise the round-off performance against coefficientquantisation and potential overflows, often by trial anderror. In this paper, PSCAD simulations and experimentalconfirmation have shown that a word length of 21 bits and

D¼ 1/16 produce excellent results for the harmonicextraction algorithm presented.

5 Simulation and experimental results

The accuracy and practicality of the proposed harmonicextraction algorithm have been confirmed by simulationand experimentally under one number of different tractionloading conditions, but only results for one case arepresented here due to space limitation. In simulation, afive-level inverter having a fundamental RMS current ratingof 210A was simulated using the PSCAD/EMTDC soft-ware package and, for the experimental work, a 150V scalemodel of the traction system consisting of lumped LCelements was constructed with the T-PQC implementedusing a 2kVA cascaded five-level inverter. The inverter inturn was controlled by a TMS320F240 DSP.

The function of the harmonic extraction algorithm is toextract and pass only harmonics of interest from vD, andstrongly attenuate other frequency components. Ideally,G(s) in (10) should have a value of 1 at the harmonicfrequencies of interest and a value of 0 at all otherfrequencies. In other words, the T-PQC should behave as alow resistor of �1/K O according to (1) (a negative sign isincluded here as the inverter current iinv is indicated asoutflowing in Fig. 1) at the frequencies of interest and aninfinite resistor at all other frequencies. This is verified inFig. 7, which shows the harmonic impedance seen by alocomotive load at various locations along a 30km feedersection for oc¼ 5 rad/s and K¼�0.2 O�1 (implying animpedance termination of 1/7K7¼ 5O). Also observed inFig. 7 is the flattening of the traction system resonant peakat around 1.3kHz. This is expected due to the presence ofthe passive filter.

To physically confirm the performance of the T-PQC,Figs. 8 and 9 show the experimental waveforms of thesystem before and after compensation (using the delta-operator-based harmonic extraction algorithm). Note the

β0 1

δ−1

δ−1β1

β2

−α1

−α2

x (n) y (n)

w4(n)

w3(n)

w2(n)

w1(n)

Fig. 6 Direct form II transpose (DFIIt) structure for 2nd-orderdigital filter

020406080

uncompensatedpassive damping filter onlyT-PQC with K = −0.2

020406080

020406080

104101 102 1030

20406080

a

b

c

d

impe

danc

e, d

Bim

peda

nce,

dB

impe

danc

e, d

Bim

peda

nce,

dB

Fig. 7 Impedance seen by locomotive load at variance distancefrom feeder substation of a 30km feeder sectiona 10km,b 10km,c 20kmd 30km

510 IEE Proc.-Electr. Power Appl., Vol. 151, No. 5, September 2004

improvements of voltage quality at the end of the feedersection (vD). To better illustrate the selective compensationof low order 3rd, 5th and 7th harmonics, Figs. 10 and 11show the corresponding experimental harmonic spectra ofvD. These figures clearly show the reduction of the selectedlow-order harmonics to less than 2% of the fundamentalafter compensation. The figures also confirm that the spreadspectral characteristics of the hysteresis-regulated five-levelT-PQC do not introduce resonant overvoltage complica-tions to the system.

To demonstrate the improved performance of thed-operator based implementation over the z-operatorimplementation, the experimental waveforms for thez-operator implementation are also given in Fig. 12.Comparing the figures, it is observed that the deltacompensated vD in Fig. 9 looks very much sinusoidal andits shape does not change appreciably from one cycle toanother (except for the high-frequency switching compo-nents, since hysteresis switching is asynchronous). On theother hand, the shift compensated vD waveform in Fig. 12varies from a sine wave with sharp peak to one with a flatcrest from one cycle to another as the low-frequencycompensation struggles to settle to a steady level, giving riseto fluctuations in the instantaneous RMS values

Vrms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

PNk¼1ðvkÞ2

s. These fluctuations in RMS values

were dynamically captured in this work by configuring theDSP to record the instantaneous values of vD at a samplingrate of 10kHz over 22 fundamental cycles (in total, 4400data points recorded). The recorded data were then post-processed in Microsoft Excel to calculate the instantaneousRMS values by moving across one sample point at a time.The results obtained are presented in Figs. 13 and 14 for thed-operator and z-operator implemented algorithms respec-

vD

vA

iLD

iS

Fig. 8 Experimental waveforms before compensationvD: voltage at end of feeder sectionvA: voltage at node AiLD: traction load current at node DiS: feeder substation transformer current

vD

vinv

iLD

iinv

Fig. 9 Experimental waveforms after compensation with delta-operator resonant filtersvD: voltage at end of feeder sectionvinv: filter inverter switched voltageiLD: traction load current at node Diinv: filter inverter current

10−4

10−3

10−2

10−1

100

0 1 2 3 4 5 6

frequency, kHz

norm

alis

ed m

agni

tude

THD (up to 6.4kHz) = 26.7%3rd harmonic

5th harmonic

7th harmonic

systemresonance

Fig. 10 Experimental vD harmonic spectrum before compensation

10−4

10−3

10−2

10−1

100

0 1 2 3 4 5 6

frequency, kHz

norm

alis

ed m

agni

tude

THD (up to 6.4kHz) = 10.0%

3rd, 5th and 7thharmonics suppressed

Fig. 11 Experimental vD harmonic spectrum after compensation

vD

vinv

iLD

iinv

Fig. 12 Experimental waveforms after compensation with shift-operator resonant filtersvD: voltage at end of feeder sectionvinv: filter inverter switched voltageiLD: traction load current at node Diinv: filter inverter current

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tively. (Note that the instantaneous RMS values have beennormalised against their mean value (set to 100%) tohighlight the fluctuation levels.) Fig. 13 shows that the RMSfluctuations are less than 0.6%, indicating a more stable andsatisfactory filter operation with the d-operator basedalgorithm. On the other hand, Fig. 14 shows fluctuationsin excess of 2%, indicating the poorer round-off andcoefficient quantisation properties of the z-operator basedimplementation at a high sampling rate.

6 Conclusions

A high performance, computationally efficient harmonicextraction algorithm for traction applications has been

presented. Implementation of the algorithm does notrequire complex co-ordinate frame transformations, andtherefore the algorithm is computationally efficient andsuitable for single-phase traction applications. Throughdelta-operator based implementation, the algorithm canalso be made less sensitive to coefficient quantisation errorsand round-off errors, hence improving the accuracy androbustness of the overall filtering system. The accuracy androbustness of the proposed algorithm have been verifiedboth in simulation and experimentally.

7 References

1 Hu, L.: ‘The application of reactive power compensation to ACelectrified railway systems’. PhD Thesis, Staffordshire University,1995

2 Tan, P.C., Loh, P.C., and Holmes, D.G.: ‘A robust multilevel hybridcompensation system for 25kV electrified railway applications’. Proc.IEEE-PESC’03, Acapulco, Mexico, 2003, pp. 1020–1025

3 Tan, P.C., Morrison, R.E., and Holmes, D.G.: ‘Voltage form factorcontrol and reactive power compensation in a 25kV electrified railwaysystem using a shunt active filter based on voltage detection’, IEEETrans. Ind. Appl., 2003, 39, pp. 575–581

4 Akagi, H.: ‘Control strategy and site selection of a shunt active filterfor damping of harmonic propagation in power distribution systems’,IEEE Trans. Power Deliv., 1996, 12, pp. 354–363

5 Morrison, R.E., and Corcoran, J.C.W.: ‘Specification of an over-voltage damping filter for the national railways of Zimbabwe’, IEEProc. B, Electr. Power. Appl., 1989, 136, pp. 249–256

6 Zmood, D.N., Holmes, D.G., and Bode, G.H.: ‘Frequency-domainanalysis of three-phase linear current regulators’, IEEE Trans. Ind.Appl., 2001, 37, pp. 601–610

7 D’Azzo, J.J., and Houpis, C.H.: ‘Feedback control system analysisand synthesis’, 2nd Edn. (McGraw-Hill, New York, USA, 1966)

8 Moran, L.A., Dixon, J.W., and Wallace, R.R.: ‘A three-phase activepower filter operating with fixed switching frequency for reactivepower and current harmonic compensation’, IEEE Trans. Ind.Electron., 1995, 42, pp. 402–408

9 Bhattacharya, S., Frank, T.M., Divan, D.M., and Banerjee,B.: ‘Active filter system implementation’, IEEE Ind. Appl. Mag.,1998, pp. 47–63

10 Mattavelli, P.: ‘A closed-loop selective harmonic compensation foractive filters’, IEEE Trans. Ind. Appl., 2001, 37, pp. 81–89

11 Akagi, H.: ‘New trends in active filters for power conditioning’, IEEETrans. Ind. Appl, 1996, 32, pp. 1312–1322

12 Tnani, S., Mazaudier, M., Berthon, A., and Diop, S.: ‘Comparisonbetween differential real-time harmonic analysis methods forcontrol of electrical machines’. Proc. PEVSD’94, London, UK,1994, pp. 342–345

13 Franklin, G.F., Powell, J.D., and Workman, M.: ‘Digital control ofdynamic systems’, 3rd Edn. (Addison Wesley Longman Inc., 1998)

14 Middleton, R.H., and Goodwin, G.C.: ‘Improved finite word lengthcharacteristics in digital control using delta operators’, IEEE Trans.Autom. Control, 1986, AC-31, pp. 1015–1021

15 Kauraniemi, J., Laakso, T.I., Hartimo, I., and Ovaska, S.J.: ‘Deltaoperator realizations of direct-form IIR filters’, IEEE Trans. Circuits &Systems II-Analog Digit. Signal Process., 1998, 45, pp. 41–52

16 Newman, M.J., and Holmes, D.G.: ‘Delta operator digital filters forhigh performance inverter applications’, IEEE Trans. Power Electron.,2003, 18, pp. 447–454

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Fig. 14 Experimental vD RMS values over 22 fundamental cycleswith conventional shift-operator resonant filter

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