Proportional-resonant controllers and filters forgrid-connected voltage-source converters

R. Teodorescu, F. Blaabjerg, M. Liserre and P.C. Loh

Abstract: The recently introduced proportional-resonant (PR) controllers and filters, and theirsuitability for current/voltage control of grid-connected converters, are described. Using the PRcontrollers, the converter reference tracking performance can be enhanced and previously knownshortcomings associated with conventional PI controllers can be alleviated. These shortcomingsinclude steady-state errors in single-phase systems and the need for synchronous d–qtransformation in three-phase systems. Based on similar control theory, PR filters can also beused for generating the harmonic command reference precisely in an active power filter, especiallyfor single-phase systems, where d–q transformation theory is not directly applicable. Anotheradvantage associated with the PR controllers and filters is the possibility of implementing selectiveharmonic compensation without requiring excessive computational resources. Given theseadvantages and the belief that PR control will find wide-ranging applications in grid-interfacedconverters, PR control theory is revised in detail with a number of practical cases that have beenimplemented previously, described clearly to give a comprehensive reference on PR control andfiltering.

1 Introduction

Over the years, power converters of various topologies havefound wide application in numerous grid-interfacedsystems, including distributed power generation withrenewable energy sources (RES) like wind, hydro and solarenergy, microgrid power conditioners and active powerfilters. Most of these systems include a grid-connectedvoltage-source converter whose functionality is to synchro-nise and transfer the variable produced power over to thegrid. Another feature of the adopted converter is that it isusually pulse-width modulated (PWM) at a high switchingfrequency and is either current- or voltage-controlled usinga selected linear or nonlinear control algorithm. Thedeciding criterion when selecting the appropriate controlscheme usually involves an optimal tradeoff between cost,complexity and waveform quality needed for meeting ( forexample) new power quality standards for distributedgeneration in low-voltage grids, like IEEE-1547 in theUSA and IEC61727 in Europe at a commercially favour-able cost.

With the above-mentioned objective in view whileevaluating previously reported control schemes, the generalconclusion is that most controllers with precise referencetracking are either overburdened by complex computational

requirements or have high parametric sensitivity (sometimesboth). On the other hand, simple linear proportional–integral (PI) controllers are prone to known drawbacks,including the presence of steady-state error in the stationaryframe and the need to decouple phase dependency in three-phase systems although they are relatively easy to imple-ment [1]. Exploring the simplicity of PI controllers and toimprove their overall performance, many variations havebeen proposed in the literature including the addition of agrid voltage feedforward path, multiple-state feedback andincreasing the proportional gain. Generally, these variationscan expand the PI controller bandwidth but, unfortunately,they also push the systems towards their stability limits.Another disadvantage associated with the modified PIcontrollers is the possibility of distorting the line currentcaused by background harmonics introduced along thefeedforward path if the grid voltage is distorted. Thisdistortion can in turn trigger LC resonance especially whena LCL filter is used at the converter AC output for filteringswitching current ripple [2, 3].

Alternatively, for three-phase systems, synchronousframe PI control with voltage feedforward can be used,but it usually requires multiple frame transformations, andcan be difficult to implement using a low-cost fixed-pointdigital signal processor (DSP). Overcoming the computa-tional burden and still achieving virtually similar frequencyresponse characteristics as a synchronous frame PIcontroller, [4, 5], develops the P+resonant (PR) controllerfor reference tracking in the stationary frame. Interestingly,the same control structure can also be used for the precisecontrol of a single-phase converter [5]. In brief, the basicfunctionality of the PR controller is to introduce an infinitegain at a selected resonant frequency for eliminating steady-state error at that frequency, and is therefore conceptuallysimilar to an integrator whose infinite DC gain forces theDC steady-state error to zero. The resonant portion of thePR controller can therefore be viewed as a generalised ACintegrator (GI), as proven in [6]. With the introducedE-mail: fbl@iet.aau.dk

R. Teodorescu and F. Blaabjerg are with the Section of Power Electronics andDrives, Institute of Energy Technology, Aalborg University, Pontoppidan-straede 101, 9220 Aalborg East, Denmark

M. Liserre is with the Department of Electrotechnical and ElectronicEngineering, Polytechnic of Bari, 70125-Bari, Italy

P.C. Loh is with the School of Electrical and Electronic Engineering, NanyangTechnological University, Nanyang Avenue, S639798, Singapore

r The Institution of Engineering and Technology 2006

IEE Proceedings online no. 20060008

doi:10.1049/ip-epa:20060008

Paper first received 10th January and in final revised form 31st March 2006

750 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006

flexibility of tuning the resonant frequency, attempts atusing multiple PR controllers for selectively compensatinglow-order harmonics have also been reported in [6, 7] forthree-phase active power filters, in [8] for three-phaseuninterruptible power supplies (UPS) and in [9] for single-phase photovoltaic (PV) inverters. Based on similarconcept, various harmonic reference generators using PRfilters have also been proposed for single-phase tractionpower conditioners [10] and three-phase active powerfilters [11].

From the view point that electronic power converters willfind increasing grid-interfaced applications either as inver-ters processing DC energy from RES for grid injection or asrectifiers conditioning grid energy for different load usages,this paper aims to provide a comprehensive reference forreaders on the integration of PR controllers and filters togrid-connected converters for enhancing their trackingperformances. To begin, the paper reviews frequency-domain derivation of the ideal and non-ideal PR controllersand filters, and discusses their similarities as compared toclassical PI control. Generic control block diagrams forillustrating current or voltage tracking control are nextdescribed before a number of practical cases that theauthors have implemented previously are discussed toprovide readers with some implementation examples.Throughout the presentation, experimental results arepresented for validating the theoretical and implementationconcepts discussed.

2 PR control and filtering derivation

The transfer functions of single- and three-phase PRcontrollers and filters can be derived using internal modelcontrol, modified state transformation or frequency-domainapproach presented in [12, 13–15] and [4, 16], respectively.In this work, the latter approach is chosen for presentationas it clearly demonstrates similarities between PR con-trollers and filters in the stationary reference frame and theirequivalence in the synchronous frame, as shown in thefollowing Sections.

2.1 Derivation of single-phase PR transferfunctionsFor single-phase PI control, the popularly used synchro-nous d–q transformation cannot be applied directly, and theclosest equivalence developed to date is to multiply thefeedback error e(t), in turn, by sine and cosine functionsusually synchronised with the grid voltage using a phase-locked-loop (PLL), as shown in Fig. 1 [10, 17]. Thisachieves the same effect of transforming the component atthe chosen frequency to DC, leaving all other components

as AC quantities. Take for example an error signalconsisting of the fundamental and 3rd harmonic compo-nents, expressed as:

eðtÞ ¼ E1 cosðot þ y1Þ þ E3 cosð3ot þ y3Þ ð1Þwhere o, y1 and y3 represent the fundamental angularfrequency, fundamental and third harmonic phase shiftsrespectively. Multiplying this with cos(ot) and sin(ot) gives,respectively:

eCðtÞ ¼E1

2fcosðy1Þ þ cosð2ot þ y1Þg

þ E3

2fcosð2ot þ y3Þ þ cosð4ot þ y3Þg

eSðtÞ ¼E1

2fsinð�y1Þ þ sinð2ot þ y1Þg

þ E3

2fsinð�2ot � y3Þ þ sinð4ot þ y3Þg

ð2Þ

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, it also contributes toharmonic terms at a frequency of 2o (this is unlike three-phase synchronous d–q conversion where the chosenfrequency component contributes only towards the DCterm). Nevertheless, passing ec(t) and es(t) through integralblocks would still force the fundamental error amplitude E1

to zero, caused by the infinite gain of the integral blocks.Instead of transforming the feedback error to the

equivalent synchronous frame for processing, an alternativeapproach of transforming the controller GDC (s) fromthe synchronous to the stationary frame is also possible.This frequency-modulated process can be mathematicallyexpressed as:

GACðsÞ ¼ GDCðs� joÞ þ GDCðsþ joÞ ð3Þwhere GAC(s) represents the equivalent stationary frametransfer function [10]. Therefore, for the ideal and non-idealintegrators of GDCðsÞ¼Ki=s and GDCðsÞ¼Ki=ð1þ ðs=ocÞÞ(Ki and oc � o represent controller gain and cutofffrequency respectively), the derived generalised AC inte-grators GAC(s) are expressed as:

GACðsÞ ¼Y ðsÞEðsÞ ¼

2Kiss2 þ o2

ð4Þ

GACðsÞ ¼Y ðsÞEðsÞ ¼

2Kiðocsþ o2cÞ

s2 þ 2ocsþ ðo2c þ o2Þ

� 2Kiocss2 þ 2ocsþ o2

ð5Þ

Equation (4), when grouped with a proportional term Kp,gives the ideal PR controller with an infinite gain at the ACfrequency of o (see Fig. 2a), and no phase shift and gain atother frequencies. For Kp, it is tuned in the same way as fora PI controller, and it basically determines the dynamics ofthe system in terms of bandwidth, phase and gain margin.To avoid stability problems associated with an infinite gain,(5) can be used instead of (4) to give a non-ideal PRcontroller and, as illustrated in Fig. 2b, its gain is now finite,but still relatively high for enforcing small steady-state error.Another feature of (5) is that, unlike (4), its bandwidth canbe widened by setting oc appropriately, which can behelpful for reducing sensitivity towards ( for example) slightfrequency variation in a typical utility grid ( for (4), Ki canbe tuned for shifting the magnitude response vertically, but

Fig. 1 Single-phase equivalent representations of PR and synchro-nous PI controllers

IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006 751

this does not give rise to a significant variation inbandwidth). In passing, note that a third control structureof GACðsÞ ¼ 2Kio=ðs2 þ o2Þ, can similarly be used sinceaccording to the internal model principle, it introduces amathematical model that can generate the requiredsinusoidal reference along the open-loop control path, andtherefore can ensure overall zero steady-state error [12]. Thisthird form is, however, not preferred since the absence of azero at s¼ 0 causes its response to be relatively slower [12].

Besides single frequency compensation, selective harmo-nic compensation can also be achieved by cascading severalresonant blocks tuned to resonate at the desired low-orderharmonic frequencies to be compensated for. As anexample, the transfer functions of an ideal and a non-idealharmonic compensator (HC) designed to compensate forthe 3rd, 5th and 7th harmonics (as they are the mostprominent harmonics in a typical current spectrum) aregiven as:

GhðsÞ ¼X

h¼3;5;7

2Kihs

s2 þ ðhoÞ2ð6Þ

GhðsÞ ¼X

h¼3;5;7

2Kihocs

s2 þ 2ocsþ ðhoÞ2ð7Þ

where h is the harmonic order to be compensated for andKih represents the individual resonant gain, which must betuned relatively high (but within stability limit) forminimising the steady-state error. An interesting featureof the HC is that it does not affect the dynamics ofthe fundamental PR controller, as it compensates only forfrequencies that are very close to the selected resonantfrequencies.

Because of this selectiveness, (7) with Kih set to unity,implying that each resonant block now has a unity resonantpeak, can also be used for generating harmonic commandreference in an active filter. The generic block representationis given in Fig. 3a, where the distorted load current (orvoltage) is sensed and fed to the resonant filter Gh(s), whosefrequency response is shown in Fig. 3b for two differentvalues of oc, o¼ 2p� 50 rad/s and h¼ 3, 5, 7. Obviously,Fig. 3b shows the presence of unity (or 0dB) resonant peaksat only the selected filtering frequencies of 150, 250 and350Hz for extracting the selected harmonics as commandreference for the inner current loop. Also noted in theFigure is that as, oc gets smaller, Gh(s) becomes moreselective (narrower resonant peaks). However, using asmaller oc will make the filter more sensitive to frequencyvariations, lead to a slower transient response and make thefilter implementation on a low-cost 16-bit DSP moredifficult owing to coefficient quantisation and round-offerrors. In practice, oc values of 5–15 rad/s have been foundto provide a good compromise [10].

2.2 Derivation of three-phase PR transferfunctionsFor three-phase systems, elimination of steady-state track-ing error is usually performed by first transforming thefeedback variable to the synchronous d–q reference framebefore applying PI control. Using this approach, doublecomputational effort must be devoted under unbalancedconditions, during which transformations to both thepositive- and negative-sequence reference frames are

101 102 1030

200

400

600

800

Mag

nitu

de (

dB)

Frequency (Hz)

101 102 103

Frequency (Hz)

101 102 103

Frequency (Hz)

101 102 103

Frequency (Hz)

-100

-50

0

50

100

Pha

se (

deg)

0

20

40

60

80

Mag

nitu

de (

dB)

-100

-50

0

50

100

Pha

se (

deg)

a

b

Fig. 2 Bode plots of ideal and non-ideal PR compensatorsKP¼ 1, Ki¼ 20, o¼ 314 rad/s and oc¼ 10 rad/sa Idealb Non-ideal

a

102 103

-80

-60

-40

-20

0

Frequency (Hz)

102 103

Frequency (Hz)

Mag

nitu

de (

dB)

-100

-50

0

50

100

Pha

se (

Deg

)

150Hz 350Hz 250Hz wc=1 rad/swc=10 rad/s

b

Fig. 3 Resonant filter for filtering 3rd, 5th and 7th harmonicsKih¼ 1, oc¼ 1 rad/s and 10 rad/sa Block representationb Bode plots

752 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006

required (see Fig. 4). An alternative simpler method ofimplementation is therefore desired and can be derived byinverse transformation of the synchronous controller backto the stationary a-b frame Gdq(s)-Gab(s). The inversetransformation can be performed by using the following2� 2 matrix:

GabðsÞ ¼1

2

Gdq1 þ Gdq2 jGdq1 � jGdq2

�jGdq1 þ jGdq2 Gdq1 þ Gdq2

24

35

Gdq1 ¼Gdqðsþ joÞ

Gdq2 ¼Gdqðs� joÞ

ð8Þ

Given that GdqðsÞ ¼ Ki=s and GdqðsÞ ¼ Ki=ð1þ ðs=ocÞÞ,the equivalent controllers in the stationary frame forcompensating for positive-sequence feedback error aretherefore expressed as:

Gþab sð Þ ¼ 1

2

2Kiss2 þ o2

2Kios2 þ o2

� 2Kios2 þ o2

2Kiss2 þ o2

2664

3775 ð9Þ

Gþab sð Þ’ 1

2

2Kiocss2 þ 2ocsþ o2

2Kiocos2 þ 2ocsþ o2

� 2Kiocos2 þ 2ocsþ o2

2Kiocss2 þ 2ocsþ o2

2664

3775 ð10Þ

Similarly, for compensating for negative sequence feedbackerror, the required transfer functions are expressed as:

G�abðsÞ ¼1

2

2Kiss2 þ o2

� 2Kios2 þ o2

2Kios2 þ o2

2Kiss2 þ o2

2664

3775 ð11Þ

G�abðsÞ’1

2

2Kiocss2 þ 2ocsþ o2

� 2Kiocos2 þ 2ocsþ o2

2Kiocos2 þ 2ocsþ o2

2Kiocss2 þ 2ocsþ o2

2664

3775 ð12Þ

Comparing (9) and (10) with (11) and (12), it is noted thatthe diagonal terms of GþabðsÞ and G�abðsÞ are identical, but

their non-diagonal terms are opposite in polarity. Thisinversion of polarity can be viewed as equivalent to thereversal of rotating direction between the positive- andnegative-sequence synchronous frames.

Combining the above equations, the resulting controllersfor compensating for both positive- and negative-sequencefeedback errors are expressed as:

GabðsÞ ¼1

2

2Kiss2 þ o2

0

02Kis

s2 þ o2

2664

3775 ð13Þ

GabðsÞ’1

2

2Kiocss2 þ 2ocsþ o2

0

02Kiocs

s2 þ 2ocsþ o2

2664

3775 ð14Þ

Bode plots representing (13) and (14) are shown in Fig. 5,where their error-eliminating ability is clearly reflected bythe presence of two resonant peaks at the positive frequencyo and negative frequency �o. Note that, if (9) or (10)((11) or (12)) is used instead, only the resonant peak at o(�o) is present since those equations represent PI controlonly in the positive-sequence (negative-sequence) synchro-nous frame. Another feature of (13) and (14) is that theyhave no cross-coupling non-diagonal terms, implying thateach of the a and b stationary axes can be treated as asingle-phase system. Therefore, the theoretical knowledgedescribed earlier for single-phase PR control is equallyapplicable to the three-phase functions expressed in(13) and (14).

Fig. 4 Three-phase equivalent representations of PR and synchro-nous PI controllers considering both positive- and negative-sequencecomponents

Fig. 5 Positive- and negative-sequence Bode diagrams of PR controller

IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006 753

3 Implementation of resonant controllers

The resonant transfer functions in (4) and (5) (similarly in(13) and (14)) can be implemented using analogueintegrated circuits (IC) or a digital signal processor (DSP),with the latter being more popular. Because of this, twomethods of digitising the controllers are presented in detailafter a general description of the analogue approach isgiven.

3.1 Analogue implementationThe rational function in (4) can be rewritten as [9]:

Y ðsÞEðsÞ¼

2Kiss2þo2

) Y sð Þ¼1s½2KiEðsÞ�V2ðsÞ�V2ðsÞ¼

1

so2Y ðsÞ

�

ð15Þ

Similarly, the function in (5) can be rewritten as:

Y ðsÞEðsÞ ¼

2Kiocss2 þ 2ocsþ o2

)

Y ðsÞ ¼ 1

s½2KiocEðsÞ � V1ðsÞ � V2ðsÞ�

V1ðsÞ ¼ 2ocY ðsÞ

V2ðsÞ ¼1

so2Y ðsÞ

8>>>><>>>>:

ð16Þ

Equations (15) and (16) can both be represented by thecontrol block representation shown in Fig. 6, where theupper feedback path is removed for representing (15). Fromthis Figure, it can be deduced that the resonant function canbe physically implemented using op-amp integrators andinverting/non-inverting gain amplifiers. Note also that,while implementing (15), parasitic resistance and othersecond-order imperfections would cause it to degenerateinto (16), but of course its bandwidth can only be tuned ifadditional components are added for implementing theupper feedback path.

3.2 Shift-operator digital implementationThe most commonly used digitisation technique is the pre-warped bilinear (Tustin) transform [18], given by:

s ¼ o1

tanðo1T =2Þz� 1

zþ 1¼ KT

z� 1

zþ 1ð17Þ

where o1 is the pre-warped frequency, T is the samplingperiod and z is the forward shift operator. Equation (17)can then be substituted into (5) ((4) is not considered hereowing to possible stability problems associated with itsinfinite resonant gain [4, 5]) for obtaining the z-domaindiscrete transfer function given in (18), from whichthe difference equation needed for DSP implementation is

derived and expressed in (19) (where n represents the pointof sampling):

Y ðzÞEðzÞ ¼

a1z�1 � a2z�2

b0 � b1z�1 þ b2z�2

a1 ¼ a2 ¼ 2KiKToc

b0 ¼K2T þ 2KToc þ o2

b1 ¼ 2K2T � 2o2

b2 ¼K2

T þ 2KToc þ o2

K2T þ 2KToc þ ðhoÞ2 for h ¼ 3; 5; 7

8<:

ð18Þ

yðnÞ ¼ 1

b0fa1½eðn� 1Þ � eðn� 2Þ� þ b1yðn� 1Þ

� b2yðn� 2Þg ð19Þ

Equations (18) and (19) can similarly be used forimplementing the HC compensator after the desiredharmonic order h is substituted. The resulting differenceequation can conveniently be programmed into a floating-point DSP, but when a fixed-point DSP is used instead,coefficients of (19) have to be normalised by multiplyingthem with the maximum integer value of the chosen wordlength [10, 19]. This multiplication is needed for minimisingthe extent of coefficient quantisation error, and the choiceof word length is solely dictated by the size of error that canbe tolerated (large coefficient quantisation error should beavoided since it can change the frequency characteristics ofa resonant peak, and even render it ‘open-loop’ unstable).Unfortunately, no standard method of choosing this wordlength is available and, as discussed in [10, 19], theappropriate word length is usually determined experimen-tally with the aim of achieving the best tradeoff betweenexecution speed and accuracy.

3.3 d-operator digital implementationGenerally, when the shift-operator resonant implementationgiven in (18) and (19) is programmed into a fixed-pointDSP, some performance degradations can usually beobserved and are caused mainly by round-off errorsassociated with the use of integer variables on the fixed-point DSP (so-called finite word length effect). 16-bit fixed-point implementation always has finite word length effects,but the problem is particularly pronounced at a fastsampling rate and for sharply tuned filters such as theresonant function used for PR control. Specifically, theroundoff errors cause the voltage or current wave shape tochange slightly from cycle to cycle, resulting in significantfluctuations in its RMS value, as proven in [10].

To improve the resonant precision, the use of deltaoperator d in place of the conventional shift operator hasbeen investigated. The delta operator has recently gainedimportance in fast digital control owing to its superior finiteword length performance [19–22], and it can be defined interms of the shift operator z as:

d�1 ¼ Dz�1

1� z�1ð20Þ

Essentially, delta-operator resonant implementation in-volves converting a second-order section in z into a

Fig. 6 Decomposition of resonant block into two interlinkedintegrators

754 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006

corresponding second-order Section in d, as follows:

HðzÞ ¼ b0 þ b1z�1 þ b2z�2

1þ a1z�1 þ a2z�2

)HðdÞ ¼ b0 þ b1d�1 þ b2d

�2

1þ a1d�1 þ a2d

�2 ð21Þ

where b0¼ b0, b1¼ ð2b0þb1Þ=D, b2¼ ðb0þb1þb2Þ=D2,

a0 ¼ 1, a1 ¼ ð2þ a1Þ=D, a2 ¼ ð1þ a1 þ a2Þ=D2, and D isa positive constant less than unity, which is carefully chosento select the appropriate ranges for the a and b coefficients,and to minimise other internal variable truncation noise[22]. Equation (21) is then implemented using thetransposed direct form II (DFIIt) structure shown inFig. 7. The DFIIt structure is chosen out of the many filterstructures available because it has the best roundoff noiseperformance for delta-operator-based filters [22]. FromFig. 7, the difference equations to be coded for the DSP canbe written, in processing order, as:

w4ðnÞ ¼ Dw3ðn� 1Þ þ w4ðn� 1Þ

w2ðnÞ ¼ Dw1ðn� 1Þ þ w2ðn� 1Þ

yðnÞ ¼ b0xðnÞ þ w4ðnÞ

w3ðnÞ ¼ b1xðnÞ � a1yðnÞ þ w2ðnÞ

w5ðnÞ ¼ b2xðnÞ � a2yðnÞ

ð22Þ

Note that the first two equations in (22) for w4(n) and w2(n)are obtained from the definition of the delta operator givenin (20). In addition, similar to (19), the coefficients in (22)will initially be floating-point numbers and must benormalised by multiplying them with the maximum integer

value of the chosen word length for faster and accurateexecution in a fixed-point DSP. This required word lengthand the constant D together represent two degrees of designfreedom that can be used for optimising the round-offperformance against coefficient quantisation and potentialoverflows, often through experimental testing.

4 Example cases using PR controllers or filters

Given the advantages of PR controllers and filters, anumber of applications have since been proposed in theliterature with most focusing on the control of convertersinterfaced directly to the utility grid. In this Section, twoexample cases are presented for demonstrating the effec-tiveness of using PR controllers in a single-phase PVconverter [9], and a three-phase microgrid power qualitycompensator [14].

4.1 Single-phase PV grid-connectedinverterSingle-phase grid inverters are commonly used in applica-tions like residential RES (typically PV or fuel cell systems)and UPS. Figure 8 shows a typical RES where the DC-linkvoltage, active P and reactive Q power are controlled in theouter control loops (labelled as voltage controller andreference generator in the Figure). The reference currentoutputs of the outer loops (i�dd and i�) are next tracked by aninner current loop whose output is eventually fed to a PWMmodulator for switching the inverter.

Typically, the inner current loop is implemented using astationary PI current controller with voltage feedforward, asshown in Fig. 9a. Using PI control, however, leads tosteady-state current error (both in phase and magnitude)when tracking sinusoidal input, and hence a poor harmoniccompensation performance is expected [9]. Synchronous PIcontrol described in Section 2.1 can mitigate the trackingerror, but is generally difficult to apply. Instead, theequivalent stationary PR controller can be used as theinner current controller, as shown in Fig. 9b. Compared toa stationary PI controller, the only computational require-ment imposed by the PR controller is an extra integrator forimplementing a second-order system, but with a modernlow-cost 16-bit fixed-point DSP, this increase in computa-tion can generally be ignored [9]. Besides that, using a PRcontroller would allow the removal of the grid voltagefeedforward path, as proven in [9], and the simple cascadingof a HC compensator for eliminating selected low-orderharmonics.

β2

βο 1

δ-1

δ-1

β1 -α1

-α2

x(n) y(n)

w1(n)

w2(n)

w3(n)

w4(n)

Fig. 7 Direct form II transpose (DFIIt) structure for second-orderdigital filter

RES(PV, FC)

VoltageController

ReferenceGenerator

CurrentController

DC

DC

Filter ⊃

i

u

+-

GridFilterGrid ConverterDC-DCBoost Module

RES1 x 240 V

PWMidd

*

i*P

Q

Ud*

Ud

1-ph VSI

Fig. 8 Block diagram of typical single-phase RES system

IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006 755

The designed control scheme in Fig. 9b has been testedusing an experimental 3kW PV full-bridge inverter with anoutput LCL filter, as shown in Fig. 10. The inverter ispowered from a regulated DC power supply (set toUD¼ 350V) for simulating a PV string, and is interfacedto the utility grid with a voltage of Ug¼ 230VRMS and abackground THD of 1.46%. The resulting system iscontrolled digitally using a 16-bit fixed-point TMS320F24xxDSP platform with an execution time of 40ms (includingHC compensation) and the controller gains set as Kp¼ 2,Ki¼ 300 and Kih¼ 300 for h¼ 3, 5 and 7. With thesesettings, the grid current and voltage at 50% load using PRand PR+HC controllers are shown in Figs. 11 and 12respectively. As seen in Fig. 11, there is no phase error notedbetween the grid current and voltage, confirming the properfunctioning of the PR controller. The harmonic distortionin Fig. 11 can be further reduced by cascading an HCcompensator, as demonstrated by the smoother currentwaveform in Fig. 12.

The improved performance achieved here with a single-phase inverter can obviously be extended to a three-phaseRES (e.g. small wind or water turbines and high-power PVplant) since as explained in Section 2.2, three-phase controlin the stationary a–b frame can be viewed as twoindependent control paths along the a and b axes,respectively. For illustration, Fig. 13b shows the innercurrent control scheme of a three-phase RES inverter,where a second PR controller is added, as compared to thatin Fig. 9b. Also shown in Fig. 13a is the conventionalsynchronous PI method of implementation, where multipleframe transformations and control decoupling are needed.These complications are obviously removed from Fig. 13bwhen PR controllers are used instead.

4.2 Three-phase microgrid power qualitycompensatorIn Section 4.1, the precise current tracking and selectiveharmonic compensation functionalities of the PR control-lers in a single-phase inverter have been demonstrated. ThisSection now presents a second example on a microgridpower quality compensator for demonstrating that the PRcontrollers can equally be used in a voltage control loop andcan simultaneously compensate for both positive- andnegative-sequence components.

In general, microgrids can be viewed as ‘local areanetworks’ where clusters of micro-generators are installedfor distributed power generation. For interfacing thesemicrosystems to the utility grid, while simultaneouslyrefining the waveform quality at the point of coupling(PCC), a microgrid power quality compensator, consistingof a shunt and a series inverter (labelled as inverters A andB, respectively), can be used [14], as shown in Fig. 14. Inprinciple, shunt inverter A is controlled to maintain abalanced set of three-phase voltages in the microgrid underall grid and load operating conditions. Besides voltageregulation, inverter A is also tasked to perform otherfunctions such as the proper dispatch of active and reactivepower, and the synchronisation of the micro- and utilitygrids during the transition from islanding to grid-connectedmode [14], but these are not described here since the focusof this paper is mainly on the application of PR controllersfor voltage or current tracking. On the other hand, seriesinverter B is controlled to inject appropriate voltagecomponents along the distribution feeder for blocking large

i*Ki

s

Kp

+

+

+

+

-

idd*

i u u*

i*Ki s

s ω⋅

+

Kp

+

+

+

+

-

idd*

i u*

( )

Ki s

s hω⋅

+ ⋅

+

b

a

Fig. 9 Single-phase grid inverter controla Stationary PI controlb Stationary PR inner current control

CURRENT CONTROL

Li

H-VSI

Cfug ui

ii

us

ig LgZs

PWM

Ud

Fig. 10 Schematic representation of experimental single-phase PVinverter

Fig. 11 Waveforms captured using PR controller at 50% loadGrid voltage Ch2 [100V/div.], grid current Ch1 [5A/div.] and DCvoltage Ch4 [250V/div.]

Fig. 12 Waveforms captured using PR+HC controller at 50%loadGrid voltage Ch2 [100V/div.], grid current Ch1 [5A/div.] and DCvoltage Ch4 [250V/div.]

756 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006

negative-sequence currents that might flow along the low-impedance line if the PCC voltages are unbalanced.

With the assigned control tasks in view, Fig. 15 shows thecontrol block representation of shunt inverter A, where themeasured inverter voltage phasor VabðsÞ is forced to trackits reference V �abðsÞ precisely using the PR control block

GabðsÞ. The generated output is then fed to an innerproportional current regulator for providing a fasterdynamic response. (In passing, it is commented that thesame control structure can be used for controlling a UPSand a dynamic voltage restorer (DVR), as presented in [23]and [24], respectively.) Similarly, the control block diagram

3

2 -ωL

ωLe jθ

SVM

3

2

PLL

3

2

u

id*

iq*(=0)

idd*

i*

iiy

ix

θ

id

iq

uq

ud

ux*

uy*

θ

e-jθ e-jθ

ux

uy iy*

ix* id

*

iq*

Ki

s

Ki

s

Kp

Kp

ud

uq

-

+

+

++

+

+

+

++

+

+

+

+

-

6e-jθ

3

2

SVM

PLL

3

2

u

ix*

i y*

idd*

i*

ii y

ix

θ

ux*

uy*

θ

iy*

ix*

Kp

Kp

-

+

+

+

+

+

+

+

-

6

e jθ

Ki s

s ω⋅

+

Ki s

s ω⋅

+

i xd*

a

b

Fig. 13 Three-phase grid inverter current controla Using Synchronous PI controllerb Using PR controllers

Fig. 14 Schematic of microgrid interfaced to utility grid using power quality compensator

IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006 757

of series inverter B is shown in Fig. 16, where the linecurrent phasor ILine(s) is measured and used for generatingthe required negative-sequence voltage command V �CabðsÞneeded for forcing the negative-sequence line current tozero. This command reference is then closely tracked by themeasured inverter voltage phasor VCabðsÞ, again using a PRcontrol block GabðsÞ. Note that an inner current loop forenhancing the dynamic response of inverter B is deemedunnecessary since the line current response is primarilylimited by the feeder line impedance.

For verifying the tracking performance of the PRcontrollers in the inverter control schemes, a hardwareprototype has been built in the laboratory using the systemparameters listed in Table 1. For the experimental system, aprogrammable AC source is used to represent the utilitygrid and is connected to an emulated microgrid. Themicrogrid consists of shunt inverter A, series inverter B with

an injection transformer and a connected RL load. Bothinverters are controlled using a single dSPACE DS1103processor card with the slave TMS320F240 processor onthe card configured to perform carrier-based PWM.

Under grid-connected mode of operation, Figs. 17a and bshow the utility voltages and microgrid load voltages,respectively, where the utility voltages become unbalancedwith 0.1p.u. negative-sequence and 0.1p.u. zero-sequencevoltage components added at t¼ 3.8 s. Despite thisunbalance in utility voltages, the load voltages in themicrogrid are kept balanced by controlling shunt inverterA. Similarly, by controlling series inverter B, the currentsflowing between the microgrid and utility grid canbe balanced. This is demonstrated in Fig. 18, wherethe captured line current waveforms are converted to thenegative-sequence synchronous frame (post-processing inMatlab) for a better illustration of how the DC negative-sequence current components vary. As anticipated, thenegative-sequence d–q components gradually decrease tozero, implying the proper functioning of series inverter B.

5 Other recent areas of development

Besides being used as PR controllers and filters, thefrequency-domain resonant concept has also been used ina number of related control developments. These develop-ments are summarised herein to give an insight into someperspective applications of the resonant concept.

5.1 Highpass equivalent stationary framefilterIn a three-phase active power filter, it is a common practiceto transform the measured load current to the (positive-sequence) synchronous reference frame before extractingthe harmonic components using a highpass filter [11, 25].The extracted harmonics are then used as commandreference for the active filter inner current loop, as shownin Fig. 19. Using a similar concept as in Section 2.2, thehighpass filter block, expressed as GþdqðsÞ ¼ s=ðsþ ocÞ, can

Fig. 15 Voltage control scheme of shunt inverter A

Fig. 16 Control scheme of series inverter B for compensating negative-sequence current

Table 1: Parameters of implemented microgrid powercompensator

Parameter Value

Nominal line-to-line grid voltage 120V

Frequency 50Hz

DC supply voltage 250V

Switching frequency for both inverters 10kHz

Series inverter filter capacitance 10mF

Series inverter filter inductance 3.9mH

Series transformer turns ratio 1 :1

Shunt inverter filter capacitance 30mF

Shunt inverter filter inductance 5mH

Line resistance RLine 3O

Line inductance LLine 10mH

Grid dispatch power 300W, 160var

Sensitive load in the microgrid 120W, 90var

758 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006

be inverse-transformed to the stationary a–b frame, and isexpressed as [11]:

GþabðsÞ ¼

s2 þ ocsþ o2

s2 þ 2ocsþ o2c þ o2

�ocos2 þ 2ocsþ o2

c þ o2

ocos2 þ 2ocsþ o2

c þ o2

s2 þ ocsþ o2

s2 þ 2ocsþ o2c þ o2

26664

37775

ð23ÞSince (23) is directly derived from the highpass filter in thepositive-sequence synchronous frame, it is expected to filterout all positive- and negative-sequence harmonics from the

distorted load current for compensation. The source currentsupplied by the grid would therefore consist only of apositive-sequence fundamental component assuming thatthe inner current loop of the active filter is implementedwith high tracking precision.

5.2 Hybrid repetitive controlIn [26–30], two alternative repetitive control schemes arepresented, whose control block representations are shown inFigs. 20a and b. Empirically, the control schemes can beviewed as the cascaded connection of a delayed feedbackpath and a feedforward path that resemble classicalrepetitive [31, 32] and Posicast control [33–35], respectively.With the cascading of these two classical control theories, itis interesting that it is shown in [26–28] that the controlscheme in Fig. 20a can be expressed as (24), while the

Fig. 17 Experimental utility grid voltages and sensitive load voltages in microgrida Utility grid voltagesb Sensitive load voltages

Fig. 18 Experimental line currents in the negative-sequencesynchronous d–q frame

Fig. 19 Block representation of typical active filter control scheme

Fig. 20 Block representations of hybrid repetitive and Posicastcontrola Positive feedback and feedforwardb Negative feedback and feedforward

IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006 759

scheme in Fig. 20b can be expressed as (25):

Y ðsÞEðsÞ ¼

1þ e�sTd

1� e�sTd¼ 2

Td

1

sþX1h¼1

2s

s2 þ ðhoÞ2

( )ð24Þ

Y ðsÞEðsÞ ¼

1� e�sTd=2

1þ e�sTd=2¼ 4

Td

X1k¼1

2s

s2 þ ½ð2k � 1Þo�2

( )ð25Þ

Obviously, (24) and (25) feature multiple harmonic resonantcompensators for eliminating all harmonics in (24) and oddharmonics in (25). This extent of harmonic compensationwould be computationally intensive if multiple resonantcompensators in (4) or (5) are used, but with the schemespresented in Fig. 20, only a single time delay block isneeded. These hybrid repetitive schemes are thereforeattractive alternatives with promising application in gridconverters.

Besides the schemes described above, another hybridrepetitive scheme with a degree of control freedom forselecting the desired harmonics to be compensated for isproposed in [36]. The proposed controller is recommendedfor discrete-time implementation using a DSP, and is shownschematically in Fig. 21. Compared with a traditionalpositive feedback repetitive controller, the controllerdescribed in [36] has an additional ‘discrete-Fourier-trans-form (DFT)’ filter block FDFT(z) inserted along the forwardpath, which is mathematically expressed as:

FDFT ðzÞ ¼Xh2Nh

FdhðzÞ ¼Xh2Nh

2

N

XN�1i¼0

cos2pN

hðiþ NaÞ� �

z�i

!

¼ 2

N

XN�1i¼0

Xh2Nh

cos2pN

hðiþNaÞ� � !

z�i¼ 2

N

XN�1i¼0

Wiz�i

ð26Þ

where i, N, h and Nh represent the ith sample point, numberof samples within a fundamental period, harmonic orderand set of harmonics selected for compensation respectively.Equation (26) also includes an additional term Na forintroducing a defined number of leading steps (equivalent toa leading phase shift), which, when used with the feedbackz-Na block, stabilises the system against phase delays,rounding and quantisation errors introduced by the digitalsampling process.

For showing that the DFT scheme approximates the HCcompensator in (6), the transfer functions of both schemesshould be re-expressed as (assuming Na¼ 0 for the DFTscheme):

DFT repetitive

GDFT ðzÞ ¼Y ðzÞEðzÞ ¼ KF

Ph2Nh

FdhðzÞ

1�P

h2Nh

FdhðzÞð27Þ

Resonant HC

GhðsÞ ¼Y ðsÞEðsÞ ¼ KHC

Xh2Nh

FhðsÞ1� FhðsÞ

’KHC

Ph2Nh

FhðsÞ

1�P

h2Nh

FhðsÞ

FhðsÞ ¼2xhhos

s2 þ 2xhhosþ ðhoÞ2; KHC ¼

Kih

xhhoð28Þ

where KF and KHC are gain constants, and xh is thearbitrary damping factor of bandpass filter Fh(s) (note thatthe approximation in (28) is always valid when multiple(very selective) bandpass filters are cascaded together [36]).Comparing (27) and (28), and noting that both Fdh(z) andFh(s) have bandpass characteristics, the DFT repetitivescheme reported in [36] is virtually equivalent to theresonant HC compensator. However, observing (26), anidentified feature of the DFT scheme is that its computa-tional complexity does not worsen as the number ofharmonics to be compensated for increases. Instead, theincrease in harmonic number can simply be adapted bychanging coefficient Wi in (26). The DFT scheme is thereforea recommended choice for digital implementation, espe-cially when a fixed-point DSP is used.

5.3 Synchronous frame selective harmoniccompensationIn [37], a synchronous frame HC scheme is proposed forthree-phase systems, where multiple resonant compensatorsare again used for eliminating selected harmonics. The soledifference here is that compensation is performed in thepositive-sequence synchronous frame rotated at the funda-mental frequency, where all (6k71)o harmonics in thestationary frame are transformed to 76ko positive-and negative-sequence components in the rotated frame.The number of resonant compensators needed in thesynchronous frame is therefore one-half those in itsstationary frame counterpart since, as noted in Section2.2, the resonant functions in (13) and (14) can simulta-neously compensate for opposite rotating sequence compo-nents. This method of implementation is thus highlysuitable for use when the number of harmonics to becompensated for is high. A further development of the ideaof using a harmonic controller in a synchronous frame hasbeen proposed in [38], where a frame rotating at a genericspeed is considered, and the advantages and limits of theapproach are discussed.

5.4 Resonant phase-locked-loopIn [37], the application of a resonant filter in a standardPLL is also explored. As shown in Fig. 22a, the measuredgrid voltage Vg is assumed to be distorted and anorthogonal system generation block is used to extract thefundamental voltages Va and Vb. The orthogonal genera-tion block consists of a resonant filter whose feedforwardpath produces an undistorted sinusoidal signal Va, while itsinner feedback path produces a second sinusoidal signal Vbphase-shifted by 901 (see Fig. 22b). The filtered voltages Vaand Vb are then fed to a standard PLL, whose input blockfirst converts Va and Vb to the synchronous d–q frame. Inthe synchronous frame, the d-axis component Vd is forcedto zero by a PI controller, whose output is added to anominal frequency value off to give the commandedangular frequency of o�. o� is next integrated to give an

angle y synchronised with the utility grid (this angle isused in [37] for implementing the synchronous frame

Fig. 21 Block representation of repetitive control with selectiveharmonic compensation

760 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006

compensator presented in Section 5.3). Besides y, the gridfrequency and voltage amplitude can also be estimated byperforming simple mathematical manipulations, which arealso shown in Fig. 22a.

6 Conclusions

In this paper, single- and three-phase PR control schemeshave been reviewed and their implementation options andsuitability for current/voltage control of grid-interfacedconverters evaluated. Advantages of the PR controllersinclude the possibility of tuning their individual resonantpeaks to the grid frequency for precise fundamentalreference tracking and to some low-order harmonicfrequencies for selective harmonic compensation, and thepossibility of implementing harmonic reference generatorin the stationary frame needed for active filters. Implemen-tation wise, the PR technique requires lesser computationaloverhead and does not require an explicit grid voltagefeedforward control path, while still achieving thesame performance as a synchronous PI controller. Forthree-phase systems, the PR technique also has the uniquefeature of compensating for both positive- and negative-sequence components simultaneously, unlike synchronousPI where separate frame transformations are needed. Giventhese advantages, it is in the view of the authors that PRcontrollers can certainly replace their PI counterparts. Thisview has been supported by some recent developmentssummarised in the paper for a comprehensive review on PRcontrol.

7 References

1 Kazmierkowski, M., Krishnan, R., and Blaabjerg, F.: ‘Control inpower electronics. Selected problems’ (Academic Press, 2002)

2 Teodorescu, R., Blaabjerg, F., Liserre, M., and Dell’Aquila, A.: ‘Astable three-phase LCL-filter based active rectifier without damping’.Proc. IEEE-IAS, Salt Lake City, UT, USA, 2003, pp. 1552–1557

3 Liserre, M., Blaabjerg, F., and Hansen, S.: ‘Design and control of anLCL-filter based active rectifier’, IEEE Trans. Ind. Appl., 2001, 38,pp. 299–307

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

5 Zmood, D.N., and Holmes, D.G.: ‘Stationary frame currentregulation of PWM inverters with zero steady-state error’, IEEETrans. Power Electron., 2003, 18, pp. 814–822

6 Yuan, X., Merk, W., Stemmler, H., and Allmeling, J.: ‘Stationary-frame generalized integrators for current control of active power filterswith zero steady-state error for current harmonics of concern underunbalanced and distorted operating conditions’, IEEE Trans. Ind.Appl., 2002, 38, pp. 523–532

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

8 Mattavelli, P.: ‘Synchronous-frame harmonic control for high-performance AC power supplies’, IEEE Trans. Ind. Appl., 2001, 37,pp. 864–872

9 Teodorescu, R., Blaabjerg, F., Liserre, M., and Borup, U.: ‘A newcontrol structure for grid-connected PV inverters with zero steady-state error and selective harmonic compensation’. Proc. IEEE,APEC4, Anaheim, CA, USA, 2004, pp. 580–586

10 Tan, P.C., Loh, P.C., and Holmes, D.G.: ‘High performanceharmonic extraction algorithm for a 25kV traction power condi-tioner’, IEE Proc., Electr. Power Appl., 2004, 151, pp. 505–512

11 Newman, M.J., Zmood, D.N., and Holmes, D.G.: ‘Stationary frameharmonic reference generation for active power filter systems’, IEEETrans. Ind. Appl., 2002, 38, pp. 1591–1599

12 Fukuda, S., and Yoda, T.: ‘A novel current-tracking method for activefilters based on a sinusoidal internal model’, IEEE Trans. Ind. Appl.,2001, 37, pp. 888–895

13 Jacobina, C.B., Correa, M.B. DER., Oliveiro, T.M., Lima, A.M.N.,and da Silva, E.R.C.: ‘Current control of unbalanced electricalsystems’, IEEE Trans. Ind. Electron., 2001, 48, pp. 517–525

14 Li, Y.W., Vilathgamuwa, D.M., and Loh, P.C.: ‘A grid-interfacingpower quality compensator for three-phase three-wire microgridapplications’, IEEE Trans. Power Electron., 2006, (to be published)

15 Li, Y.W., Vilathgamuwa, D.M., and Loh, P.C.: ‘Micro-gridpower quality enhancement using a three-phase four-wire grid-interfacing compensator’, IEEE Trans. Ind. Appl., 2005, 41,pp. 1707–1719

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

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

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

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

20 Sera, D., Kerekes, T., Lungeanu, M., Nakhost, P., Teodorescu, R.,Andersen, G.K., and Liserre, M.: ‘Low-cost implementation ofproportional-resonant current controllers for PV inverter applications

Fig. 22 Block representation of PLL using a resonant filtera General structureb Orthogonal system generation structure

IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006 761

using delta operator’. Proc. IEEE IECON, Raleigh, NC, USA, 2005,pp. 2517–2522

21 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

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

23 Loh, P.C., Newman, M.J., Zmood, D.N., and Holmes, D.G.: ‘Acomparative analysis of multi-loop voltage regulation strategies forsingle and three-phase UPS systems’, IEEE Trans. Power Electron.,2003, 18, pp. 1176–1185

24 Loh, P.C., Vilathgamuwa, D.M., Tang, S.K., and Long, H.L.:‘Multilevel dynamic voltage restorer’, IEEE Power Electron. Lett.,2004, 2, pp. 125–130

25 Fujita, H., and Akagi, H.: ‘Unified power quality conditioner: theintegration of series and shunt-active filters’, IEEE Trans. PowerElectron., 1998, 13, pp. 315–322

26 Escobar, G., Valdez, A.A., Leyva-Ramos, J., and Mattavelli, P.:‘A repetitive-based controller for UPS using a combinedcapacitor/load current sensing’. Proc. IEEE-PESC, Recife, Brazil,2005, pp. 955–961

27 Escobar, G., Martinez, P.R., Leyva-Ramos, J., Valdez, A.A., andHernandez-Gomez, M.: ‘A repetitive-based controller for a powerfactor precompensator with harmonic compensation’. Proc. IEEE-PESC, Recife, Brazil, 2005, pp. 2363–2369

28 Escobar, G., Leyva-Ramos, J., Martinez, P.R., and Valdez, A.A.: ‘Arepetitive-based controller for the boost converter to compensate theharmonic distortion of the output voltage’, IEEE Trans. Control Syst.Tech., 2005, 13, pp. 500–508

29 Leyva-Ramos, J., Escobar, G., Martinez, P.R., and Mattavelli, P.:‘Analog circuits to implement repetitive controllers for tracking and

disturbance rejection of periodic signals’, IEEE Trans. Circuits Syst. II,2005, 52, pp. 466–470

30 Escobar, G., Ortega, R., Astolfi, A., Martinez, M.F., and Leyva-Ramos, J.: ‘A repetitive based controller for a shunt active filter tocompensate for reactive power and harmonic distortion’. Proc. 44thIEEE Conf. on Decision and Control and European Control Conf.(ECC), Seville, Spain, 2005, pp. 6480–6485

31 Tzou, Y.Y., Ou, R.S., Jung, S.L., and Chang, M.Y.: ‘High-performance programmable AC power source with low harmonicdistortion using DSP-based repetitive control technique’, IEEE Trans.Power Electron., 1997, 12, pp. 715–725

32 Costa-Castello, R., Grino, R., and Fossas, E.: ‘Odd-harmonic digitalrepetitive control of a single-phase current active filter’, IEEE Trans.Power Electron., 2004, 19, pp. 1060–1068

33 Smith, O.J.M.: ‘Posicast control of damped oscillatory systems’, Proc.IRE, 1957, 45, pp. 1249–1255

34 Hung, J.Y.: ‘Feedback control with posicast’, IEEE Trans. Ind.Electron., 2003, 50, pp. 94–99

35 Loh, P.C., Gajanayake, C.J., Vilathgamuwa, D.M., and Blaabjerg, F.:‘Evaluation of resonant damping techniques for Z-source current-typeinverter’. Proc. IEEE, APEC, Dallas, TX, USA, 2006

36 Mattavelli, P., and Marafao, F.P.: ‘Repetitive-based control forselective harmonic compensation in active power filters’, IEEE Trans.Ind. Electron., 2004, 51, pp. 1018–1024

37 Bojoi, R.I., Griva, G., Bostan, V., Guerriero, M., Farina, F., andProfumo, F.: ‘Current control strategy for power conditioners usingsinusoidal signal integrators in synchronous reference frame’, IEEETrans. Power Electron., 2005, 20, pp. 1402–1412

38 Liserre, M., Teodorescu, R., and Blaabjerg, F.: ‘Double harmoniccontrol for three-phase systems with the use of resonant currentcontrollers in a rotating frame’, IEE Trans. Power Electron., 2006, 21,pp. 836–841

762 IEE Proc.-Electr. Power Appl., Vol. 153, No. 5, September 2006