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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 dqtransformation 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 dq 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 and

filtering.

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-connected

voltage-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 proportionalintegral ( 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, synchronous

frame 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 the

DC 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: [email protected]

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

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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 to

provide 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, 1315] 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 dq 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:

et E1 cosot y1 E3 cos3ot y3 1

where o, y1 and y3 represent the fundamental angularfrequency, fundamental and third harmonic phase shiftsrespectively. Multiplying this with cos(ot) and sin(ot) gives,respectively:

eCt E1

2fcosy1 cos2ot y1g

E3

2fcos2ot y3 cos4ot y3g

eSt E1

2fsiny1 sin2ot y1g

E3

2fsin2ot y3 sin4ot y3g

2

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 to

harmonic terms at a frequency of 2o (this is unlike three-phase synchronous dq 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 E1to zero, caused by the infinite gain of the integral blocks.

Instead of transforming the feedback error to theequivalent 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:

GACs GDCs jo GDCs jo 3

where GAC(s) represents the equivalent stationary frametransfer function [10]. Therefore, for the ideal and non-idealintegrators of GDCsKi=s and GDCsKi=1 s=oc(Ki and oc ( o represent controller gain and cutofffrequency respectively), the derived generalised AC inte-grators GAC(s) are expressed as:

GACs Ys

Es

2Kis

s2 o24

GACs Ys

Es

2Kiocs o2c

s2 2ocs o2c o2

%

2Kiocs

s2 2ocs o2 5

Equation (4), when grouped with a proportional term Kp,gives the ideal PR controller with an infinite gain at the ACfrequency ofo (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 be

helpful 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

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this does not give rise to a significant variation inbandwidth). In passing, note that a third control structure

of GACs 2Kio=s2 o2, can similarly be used since

according 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 a

zero at s 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:

Ghs

Xh3;5;72Kihs

s2 ho26

Ghs X

h3;5;7

2Kihocs

s2 2ocs ho2

7

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 ofoc, o 2p 50 rad/s and h 3, 5, 7. Obviously,Fig. 3b shows the presence of unity (or 0 dB) resonant peaksat only the selected filtering frequencies of 150, 250 and350 Hz 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 the

filter implementation on a low-cost 16-bit DSP moredifficult owing to coefficient quantisation and round-offerrors. In practice, oc values of 515rad/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 dq reference framebefore applying PI control. Using this approach, doublecomputational effort must be devoted under unbalancedconditions, during which transformations to both the

positive- and negative-sequence reference frames are

101 102 1030

200

400

600

800

Magnitude(dB)

Frequency (Hz)

101 102 103

Frequency (Hz)

101 102 103

Frequency (Hz)

101 102 103

Frequency (Hz)

-100

-50

0

50

100

Phase(de

g)

0

20

40

60

80

Magnitude(dB)

-100

-50

0

50

100

Phase(deg)

a

b

Fig. 2 Bode plots of ideal and non-ideal PR compensatorsKP 1, Ki 20, o 314 rad/s and oc 10 rad/sa Ideal

b Non-ideal

a

102

103

-80

-60

-40

-20

0

Frequency (Hz)

102

103

Frequency (Hz)

Magnitude(dB)

-100

-50

0

50

100

Phase(Deg)

150Hz 350Hz250Hzwc=1 rad/swc=10 rad/s

b

Fig. 3 Resonant filter for filtering 3rd, 5th and 7th harmonicsKih 1, oc 1 rad/s and 10rad/sa Block representation

b Bode plots

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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:

Gabs 1

2

Gdq1 Gdq2 jGdq1 jGdq2

jGdq1 jGdq2 Gdq1 Gdq2

24

35

Gdq1 Gdqs jo

Gdq2 Gdqs jo

8

Given that Gdqs Ki=s and Gdqs Ki=1 s=oc,the equivalent controllers in the stationary frame forcompensating for positive-sequence feedback error aretherefore expressed as:

Gab s 1

2

2Kis

s2 o22Kio

s2 o2

2Kio

s2 o22Kis

s2 o2

2664

3775 9

Gab s 1

2

2Kiocs

s2 2ocs o22Kioco

s2 2ocs o2

2Kioco

s2 2ocs o22Kiocs

s2 2ocs o2

2664

3775 10

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

Gabs 1

2

2Kis

s2 o2

2Kio

s2 o2

2Kio

s2 o22Kis

s2 o2

2664

3775 11

Gabs 12

2Kiocs

s2

2ocs o2

2Kioco

s2

2ocs o2

2Kioco

s2 2ocs o22Kiocs

s2 2ocs o2

2664 3775 12

Comparing (9) and (10) with (11) and (12), it is noted that

the diagonal terms of Gabs and Gabs 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:

Gabs 1

2

2Kis

s2 o20

02Kis

s2 o2

2664

3775 13

Gabs 1

2

2Kiocs

s2 2ocs o20

02Kiocs

s2 2ocs o2

2664

3775 14

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

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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]:

Ys

Es

2Kis

s2o2) Y s

1

s2KiEsV2sV2s

1

so2Ys

&15

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

Ys

Es

2Kiocs

s2 2ocs o2

)

Ys 1

s2KiocEs V1s V2s

V1s 2ocYs

V2s 1

so2Ys

8>>>>>>>:

16

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 other

second-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 o1

tano1T=2

z 1

z 1 KT

z 1

z 117

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):

Yz

Ez

a1z1 a2z

2

b0 b1z1 b2z2

a1 a2 2KiKToc

b0 K2T 2KToc o

2

b1 2K2

T 2o2

b2 K2T 2KToc o

2

K2T 2KToc ho2

for h 3; 5; 7

8