Proportional-resonant controllers and filters for grid-connected voltage-source converters
R.Teodorescu,F.Blaabjerg,M.Lirre and P.C.Loh
Abstract:The recently introduced proportional-resonant (PR)controllers and filters,and their suitability for current/voltage control of grid-connected converters,are described.Using the PR controllers,the converter reference tracking performance can be enhanced and previously known shortcomings associated with conventional PI controllers can be alleviated.The shortcomings include steady-state errors in single-pha systems and the need for synchronous d –q transformation in three-pha systems.Bad on similar control theory,PR filters can also be ud for generating the harmonic command reference precily in an active power filter,especially for single-pha systems,where d –q transformation theory is not directly applicable.Another advantage associated with the PR controllers and filters is the possibility of implementing lective harmonic compensation without requiring excessive computational resources.Given the advantages and the belief that PR control will find wide-ranging applications in grid-interfaced converters,PR control theory is revid in detail with a number of practical cas that have been implemented previously,described clearly to give a comprehensive reference on PR control and filtering.
1Introduction
Over the years,power converters of various topologies have found wide application in numerous grid-interfaced systems,including distributed power generation with renewable energy sources (RES)like wind,hydro and solar energy,microgrid power conditioners and active power filters.Most of the systems include a grid-connected voltage-source converter who functionality is to synchro-ni and transfer the variable produced power over to the grid.Another feature of the adopted converter is that it is usually pul-width modulated (PWM)at a high switching frequency and is either current-or voltage-controlled using a lected linear or nonlinear control algorithm.The deciding criterion when lecting the appropriate control scheme usually involves an optimal tradeoff between cost,complexity and waveform quality needed for meeting (for example)new power quality standards for distributed generation in low-voltage grids,like IEEE-1547in the USA and IEC61727in Europe at a commercially favour-able cost.
With the above-mentioned objective in view while evaluating previously reported control schemes,the general conclusion is that most controllers with preci reference tracking are either overburdened by complex computational
requirements or have high parametric nsitivity (sometimes both).On the other hand,simple linear proportional–
integral (PI)controllers are prone to known drawbacks,including the prence
of steady-state error in the stationary frame and the need to decouple pha dependency in three-pha systems although they are relatively easy to imple-ment [1].Exploring the simplicity of PI controllers and to
improve their overall performance,many variations have been propod in the literature including the addition of a grid voltage feedforward path,multiple-state feedback and increasing the proportional gain.Generally,the variations can expand the PI controller bandwidth but,unfortunately,they also push the systems towards their stability limits.Another disadvantage associated with the modified PI
controllers is the possibility of distorting the line current caud by background harmonics introduced along the feedforward path if the grid voltage is distorted.This distortion can in turn trigger LC resonance especially when a LCL filter is ud at the converter AC output for filtering switching current ripple [2,3].
Alternatively,for three-pha systems,synchronous frame PI control with voltage feedforward can be ud,but it usually requires multiple frame transformations,and can be difficult to implement using a low-cost fixed-point digital signal processor (DSP).Overcoming the computa-tional burden and still achieving virtually similar frequency respon characteristics as a synchronous frame PI [6].With the introduced
E-mail:fbl@iet.aau.dk
R.Teodorescu and F.Blaabjerg are with the Drives,Institute of Energy Technology,straede 101,9220Aalborg East,Denmark
M.Lirre is with the Department of Engineering,Polytechnic of Bari,70125-Bari,P.C.Loh is with the School of Electrical and Technological University,Nanyang Avenue,r The Institution of Engineering and IEE Proceedings online no.20060008doi:10.1049/ip-epa:20060008
Paper first received 10th January and in PI 缺点改进方法丗电压前馈丆多状态反馈丆
增大增益。
可能电流扭曲
concept,various harmonic reference generators using PR
filters have also been propod for single-pha traction
power conditioners[10]and three-pha active power
filters[11]
.
From the view point that electronic power converters will
find increasing grid-interfaced applications either as inver-
ters processing DC energy from RES
for grid injection or as
rectifiers conditioning grid energy for different load usages,
古风摄影this paper aims to provide a comprehensive reference for
readers on the integration of PR controllers andfilters to
grid-connected converters for enhancing their tracking
performances.To begin,the paper reviews frequency-
domain derivation of the ideal and non-ideal PR controllers
andfilters,and discuss their similarities as compared to
classical PI control.Generic control block diagrams for
illustrating current or voltage tracking control are next
described before a number of practical cas that the
authors have implemented previously are discusd to
provide readers with some implementation examples.
Throughout the prentation,experimental results are
prented for validating the theoretical and implementation
concepts discusd.
as it clearly demonstrates similarities between PR con-
trollers andfilters in the stationary reference frame and their
equivalence in the synchronous frame,as shown in the
following Sections.
csgo怎么下载2.1Derivation of single-pha PR transfer
functions
For single-pha PI control,the popularly ud synchro-
nous d–q transformation cannot be applied directly,and the
as AC quantities.Take for example an error signal
consisting of the fundamental and3rd harmonic compo-
nents,expresd as:
eðtÞ¼E1cosðo tþy1ÞþE3cosð3o tþy3Þð1Þ
where o,y1and y3reprent the fundamental angular
frequency,fundamental and third harmonic pha shifts
respectively.Multiplying this with cos(o t)and sin(o t)gives,
respectively:
e CðtÞ¼
E1
2
f cosðy1Þþcosð2o tþy1Þg
þ
E3情人之间的情话
2
f cosð2o tþy3Þþcosð4o tþy3Þg
ð2Þ
frequency component contributes only towards the DC
term).Nevertheless,passing e c(t)and e s(t)through integral
blocks would still force the fundamental error amplitude E1
to zero,caud by the infinite gain of the integral blocks.
Instead of transforming the feedback error to the
DC i DC i c
(K i and o c(o reprent controller gain and cutoff
frequency respectively),the derived generalid AC inte-
grators G AC(s)are expresd as:
YðsÞ2K i s
Fig.1Single-pha equivalent reprentations of PR and synchro-
nous PI controllers
通过积分模块
会强制基础
误差E1为0.
因为积分的
增益无限大。
公众号如何赚钱
harmonic compensator (HC)designed to compensate for the 3rd,5th and 7th harmonics (as they are the most prominent harmonics in a typical current spectrum)are given as:
ð6Þ
G h ðs Þ¼
X
h ¼3;5;72K ih o c s
s 2
þ2o c s þðh o Þ
2ð7Þ
where h is the harmonic order to be compensated for and
K ih reprents the individual resonant gain,which must be tuned relatively high (but within stability limit)for minimising the steady-state error.An interesting feature
of the HC is that it does not affect the dynamics of the fundamental PR controller,as it compensates only for frequencies that are very clo to the lected resonant frequencies.
frequency respon is shown in Fig.3b for two different values of o c ,o ¼2p Â50rad/s and h ¼3,5,7.Obviously,Fig.3b shows the prence of unity (or 0dB)resonant peaks at only the lected filtering frequencies of 150,250and 350Hz for extracting the lected harmonics as command reference for the inner current loop.Also noted in the Figure is that as,o c gets smaller,G h (s )becomes more lective (narrower resonant peaks).However,using a smaller o c will make the filter more nsitive to frequency variations,lead to a slower transient respon and make the filter implementation on a low-cost 16-bit DSP more functions
For three-pha systems,elimination of steady-state track-ing error is usually performed by first transforming the feedback variable to the synchronous d –q reference frame before applying PI control.Using this approach,double computational effort must be devoted under unbalanced conditions,during which transformations to both the positive-and negative-quence reference frames are
101102
103
0200400600800M a g n i t u d e (d B )Frequency (Hz)
101
102
103
Frequency (Hz)
101102
103
Frequency (Hz)
101
102
103
Frequency (Hz)
-
100
-50050100P h a s e (d e g )
020406080M a g n i t u d e (d B )
-100
-50050100P h a s e (d e g )
a
b
Fig.2Bode plots of ideal and non-ideal PR compensators
K P ¼1,K i ¼20,o ¼314rad/s and o c ¼10rad/s a Ideal
a
10
10
3
Frequency (Hz)
赏什么悦什么10
10
3
Frequency (Hz)
M a g n i t u d e (d B )
-100
大气压力-50050100P h a s e (D e g )
b
Fig.3
Resonant filter for filtering 3rd,5th and 7th harmonics
K ih ¼1,o c ¼1rad/s and 10rad/s a Block reprentation b Bode plots
低阶谐波谐波补偿HC
Ki 足够大丆for 减小稳态
误差。
并且谐波补偿HC 不影响
基本的PR
控制器。
required (e Fig.4).An alternative simpler method of
implementation is therefore desired and can be derived by inver transformation of the synchronous controller back
to the stationary a -b frame G dq (s )-G ab (s ).The inver transformation can be performed by using the following
Given that G dq ðs Þ¼K i =s and G dq ðs Þ¼K i =ð1þðs =o c ÞÞ,
the equivalent controllers in the stationary frame for compensating for positive-quence feedback error are therefore expresd as:
G þab s ðÞ¼122K i s s þo 2K i o s þo À2K i o s þo 2K i s s þo 26643
775G þab s ðÞ’122K i o c s s þ2o c s þo 2K i o c o
s þ2o c s þo À2K i o c o s 2
þ2o c s þo 2
2K i o c
s
s 2þ2o c s þo 2
2
66437
75Similarly,for compensating for negative quence feedback
error,the required transfer functions are expresd as:
G Àab ðs Þ¼122K i s s þo À2K i o
s þo 2K i o s þo 2K i s s þo 2664
3775ð11ÞG Àab ðs Þ’122K i o c s s 2þ2o c s þo 2À2K i o c o
s 2
奥硝唑
þ2o c s þo 22K i o c
o
s þ2o c s þo 2K i o c
噫吁嚱危乎高哉
s
s þ2o c s þo 2
66437
75ð12ÞComparing (9)and (10)with (11)and (12),it is noted that
the diagonal terms of G þab ðs Þand G À
ab ðs Þare identical,but their non-diagonal terms are opposite in polarity.This inversion of polarity can be viewed as equivalent to the reversal of rotating direction between the positive-and 0
i s 2þo 2
G ab ðs Þ’12
2K i o c s s þ2o c s þo 00
2K i o c s s 2þ2o c s þo 2
2
6643775
ð14Þ
Bode plots reprenting (13)and (14)are shown in Fig.5,
where their error-eliminating ability is clearly reflected by the prence of two resonant peaks at the positive frequency o and negative frequency Ào .Note that,if (9)or (10)((11)or (12))is ud instead,only the resonant peak at o (Ào )is prent since tho equations reprent PI control only in the positive-quence (negative-quence)synchro-nous frame.Another feature of (13)and (14)is that they
have no cross-coupling non-diagonal terms,implying that each of the a and b stationary axes can be treated as a single-pha system.Therefore,the theoretical knowledge described earlier for single-pha PR control is equally applicable to the three-pha functions expresd in (13)and (14).
Fig.4Three-pha equivalent reprentations of PR and synchro-nous PI controllers considering both positive-and negative-quence components
Fig.5Positive-and negative-quence Bode diagrams of PR controller
只对正负W 有谐振作用且没有交叉
耦合。
3Implementation of resonant controllers
The resonant transfer functions in (4)and (5)(similarly in (13)and (14))can be implemented using anal
ogue integrated circuits (IC)or a digital signal processor (DSP),with the latter being more popular.Becau of this,two methods of digitising the controllers are prented in detail after a general description of the analogue approach is given.
3.1Analogue implementation
The rational function in (4)can be rewritten as [9]:Y ðs ÞE ðs Þ¼2K i s s þo )Y s ðÞ¼1s ½2K i E ðs ÞÀV 2ðs Þ V 2ðs Þ¼1s
o 2
Y ðs Þ&
ð15Þ
Similarly,the function in (5)can be rewritten as:
Y ðs ÞE ðs Þ¼2K i o c s
s 2
þ2o c s þo 2
)Y ðs Þ¼1s ½2K i o c E ðs ÞÀV 1ðs ÞÀV 2ðs Þ V 1ðs Þ¼2o c Y ðs ÞV 2ðs Þ¼1s
o 2Y ðs Þ
8
>>>><
>>>>:ð16Þ
Equations (15)and (16)can both be reprented by the control block reprentation shown in Fig.6,where the upper feedback path is removed for reprenting (15).From this Figure,it can be deduced that the resonant function can be physically implemented using op-amp integrators and
The most commonly ud digitisation technique is the pre-warped bilinear (Tustin)transform [18],given by:
s ¼
o
1tan ðo 1T =2Þz À1z þ1
¼K T
z À1
z þ1ð17Þ
where o 1is the pre-warped frequency,T is the sampling
period and z is the forward shift operator.Equation (17)can then be substituted into (5)((4)is not considered here owing to possible stability problems associated with its infinite resonant gain [4,5])for obtaining the z -domain discrete transfer function given in (18),from which the difference equation needed for DSP implementation is
derived and expresd in (19)(where n reprents the point of sampling):
Y ðz ÞE ðz Þ¼
a 1z À1Àa 2z À2
b 0Àb 1z À1þb 2z À2a 1¼a 2¼2K i K T o c
b 0¼K 2
T þ2K T o c þo 2b 1¼2K 2T À2o 2
b 2¼
K 2T þ2K T o c þo 2
K 2
T þ2K T o c þðh o Þ2
for h ¼3;5;7
8
<:ð18Þ
y ðn Þ¼
1
b 0
f a 1½e ðn À1ÞÀe ðn À2Þ þb 1y ðn À1ÞÀb 2y ðn À2Þg
ð19Þ
Equations (18)and (19)can similarly be ud for implementing the HC compensator after the desired harmonic order h is substituted.The resulting difference equation can conveniently be programmed into a floating-point DSP,but when a fixed-point DSP is ud instead,coefficients of (19)have to be normalid by multiplying them with the maximum integer value of the chon word length [10,19].This multiplication is needed for minimising the extent of coefficient quantisation error,and the choice of word length is solely dictated by the size of error that can be tolerated (large coefficient quantisation error should be avoided since it can change the frequency characteristics of 3.3d -operator digital implementation
Generally,when the shift-operator resonant implementation given in (18)and (19)is programmed into a fixed-point DSP,some performance degradations can usually be
obrved and are caud mainly by round-off errors associated with the u of integer variables on t
he 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 fast sampling rate and for sharply tuned filters such as the
Esntially,delta-operator resonant implementation in-volves converting a cond-order ction in z into a
Fig.6Decomposition of resonant block into two interlinked integrators
见文献9
从这里分为两式丆
应为其中V2=xx
离散化
定点浮点数处理时需要
乘上选择数
据位数的最大整数值丆减小量化误差。舍入误差舍入误差
