observer pll
DESCRIPTION
Phase locking schemeTRANSCRIPT
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Phase Locked Loop Based on an Observer for Grid Synchronization
Yongsoon Park Student member, IEEE
Seoul National University, Seoul, Korea.
Seung-Ki Sul Fellow, IEEE
Seoul National University, [email protected]
Woo-Chull Kim LG Uplus Corp.,
Seoul, Korea.
Hyun-Young Lee PLASPO Co., Ltd.
Goyang City, Korea
Abstract— A grid synchronization technique is essential for
grid-connected power converters. Phase locked loop (PLL) has been widely exploited as an implementation technique, but additional efforts are required to deal with severely distorted grid voltages. In this paper, an observer is proposed to enhance the performance of PLL. A state equation is newly formulated to construct the observer, which extracts positive-sequence voltage from the distorted grid voltage. Additionally, guidelines are suggested to set the observer gains by considering its internal transfer functions. The PLL and proposed observer have been tested in simulations and experiments in contrast to dual second order generalized integrator frequency locked loop (DSOGI-FLL). The results have shown that the proposed method reveals better phase-tracking performance for grid synchronization.
Index terms – digital signal processing, grid-tied converter,
grid synchronization, observer, phase locked loop, positive-sequence voltage.
I. INTRODUCTION
Electricity is delivered from suppliers to consumers via an electrical grid. The grid voltage is expected to be sinusoidal, whose frequency is fundamentally 50 or 60 Hz. This normal voltage is called a positive-sequence voltage. In particular, electrical power is mainly transferred by modulating the AC current corresponding to the positive-sequence voltage, and it is useful to separate this from the rest of ‘apparent power’ [1].
The grid voltage phase angle should be detected to elaborately modulate active and reactive powers when a grid-connected converter participates in power delivery. For this purpose, the method based on phase locked loop in the synchronous reference frame (SRF-PLL) has been widely used [2], [3]. However, since the grid voltage is usually polluted by unexpected distortions such as harmonics, unbalances, and glitches, the detection of its phase angle may not be easy with the simple PLL method.
Therefore, many attempts have been made to enhance the detection performance under polluted grid conditions [4]-[15]. When the recent researches are considered, a better method for grid synchronization is to fulfill the following requirements simultaneously. First, the mitigation of unbalance and harmonics should be achieved under higher dynamics to track the phase angle variation. Second, the consistency of grid synchronization has to be guaranteed even if grid frequency varies.
In general, each required function can be implemented in several stages with filter-type blocks [4]-[9]. In some cases, the implementation can be based on parallel structure [12], [13]. Even if all these methods have already presented
effective performances in functional aspects, the computational efficiency is not high due to the distributed structures. Moreover, the integrated system design becomes more complicated when the number of separate parts increases. By using a specialized filter, the rejection at every harmonic has been attempted with the relatively simple structure [14], [15]. However, this method should save the previously sampled values, and its sampling frequency must be set carefully in conjunction with the grid frequency. Therefore, an appropriate compromise is required between the implementation effort and the grid synchronization performance.
In this paper, an observer is proposed to clearly extract the positive-sequence component of the grid voltage. The required functions for grid synchronization can be unified into a single observer by virtue of its state equation, which has been newly derived in this paper. As a result, the proposed method has a relatively simple structure while presenting the phase detection performance comparable to the dual second order generalized integrator frequency locked loop (DSOGI-FLL) [10].
In terms of design and analysis, the following points can be further noted: although the observer gains are generally set on the basis of the heuristic insight of the designer, this paper suggests explicit guidelines in setting the gains considering the internal transfer functions. In addition, the analyzing tools are newly introduced to discuss the unified effect of the proposed 4th order observer and to examine the distortions from digital implementation.
The organization of this paper is as follows. In section II, how the observer can extract positive-sequence voltage with less distortion is described. After the observer gains are set, the observer to PLL attachment is detailed in section III. The proposed method is then assessed via simulation and experimental results under grid fault situations in section IV. Finally, concluding remarks are noted in section V.
II. EXTRACTION OF POSITIVE-SEQUENCE VOLTAGES
A. Definitions of Symmetrical Components
In general, the three-phase voltage can be decomposed into symmetrical components: positive-, negative-, and zero-sequence voltages [18]. Among them, negative-sequence voltage arises mainly when the grid is under abnormal operating conditions such as phase-to-ground faults and unbalanced loading conditions. In SRF-PLL, it is hard to filter out the negative-sequence voltage in the synchronous reference frame even though the bandwidth of PLL is quite reduced. [3], [4]. To effectively remove the negative-
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sequence voltbe considered
Initially, th
a+ +
b+ +
c+ +
v V sin
v V sin(
v V sin(
= −
= − = −where both thsequence, anvoltage.
As per theonly exists reference framthe coherenceusing the samvoltage can th
a
b
c
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v V sin
v V sin
− −
− −
− −
= −
= − = −where the subφn is the phapositive-seque
For exampin terms of thV+ in (1) wasIn addition, φn
negative-sequin Fig. 1.
B. An Obser
For the obmust first berelationship bthe grid voltavariation. In pthe synchrono
d
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= −
where the hat For each p
the right-sideconsidered, beThe d-q voltanegative-sequframe:
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v v
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where d pθ θ=
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p
p
p
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(θ 120 )
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he subscripts d θp is the p
definition of in the quad
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p
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n
φφφ
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bscript ‘−’ repase differenceence voltage. ple, each voltahe phase angles set to 179.6
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rver for Estim
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p p
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ˆ ˆθ cos(θ 12
ˆ ˆnθ sin(θ 12
−
− −
‘^’ hereafter rphase, the sume of (3). Thecause it is naages can also uence compon
d +
q +
v V s
v V co−
−
− + =
pθ̂− .
metrical comp
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‘+’ and ‘p’ rphase angle o
f (1), the positdrature axis onition has beenid and AC-mogle in (1), thlized into
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age in (1) ande as shown in V, and V− in
et to 30 °. Thes presents an
mating Positive
, the state equThis state eq
s and their dere discussed in grid voltage frame by
p
p
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° +
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nents in the sy
d
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ponent model
e can be defin
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egative-sequen-sequence vol
d (2) can be dFig. 1. In the(2) was 30 % sum of positiunbalance as
e-Sequence Vo
uation on grid quation explarivatives. In en terms of itcan be consid
a
b
c
v120 )
v120 ) v
° °
mated value .) can be insertence voltage d in the procesed into positiv
ynchronous re
p d
p d
ˆin (2θ θ
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ing can
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If dθd/dt andslow variation
e derivative oto
d d
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−
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Each negative other axis in
d
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here ω is anguThe novel de
per. In (7), thith respect to t
without thecause it has brive (7), the dn be regardedid voltage can
ps
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ˆ2ω 0V
0 0
0 0
t
−=
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= =
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e system is ob
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d
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ular frequencyerivation of (7he derivativesthemselves annegative-sequ
been assumedderivative of p
d as null. Thenn be derived as
p p
p
ˆ ˆω 0 2ω
ˆ0 2ω 0
0 0 0
0 0 0
−
s
0 0 0V
1 0 0
=
T
d+ q+v v
cripts ‘s’ and
er observer canservable with
nce voltages.
negligible undation error andapproximated
p d
p d
ˆV cos(2θ θ
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−
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+
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component in n (6) is then re
q q+
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. 7) has led to ths of vd and vq
nd the positiveuence voltag
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he proposal oq can be explae-sequence voge). Furthermis small enougence voltage iate equation o
state and ou
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mption ence,
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(6)
ed in
(7)
f this ained
oltage more, gh to in (4) of the
(8-a)
(8-b)
(8-c)
utput,
cause
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s m s m o m sˆ ˆ ˆV A V L [V C V ]
d
dt= + ⋅ − (9-a)
T
1 3 1 3m
2 4 2 4
p p q qL
p p q q
=
(9-b)
T
s d q d+ q+ˆ ˆ ˆ ˆ ˆV v v v v = (9-c)
where variables in Lm are observer gains. By virtue of the practical assumption on positive-sequence
voltage, the observer in (9) could be newly proposed, which can observe the positive-sequence voltage in its internal state.
C. Gain Settings for the Observer
The observer gains of Lm in (9) must be specified. In general, it is difficult to logically describe why a set of observer gains have to be selected since certain selections may have to rely on the insight of the designer. In this paper, the gain setting of the observer is described with the reason why they are selected. The internal transfer functions of the observer are used to explain the gain setting. These functions can be derived as follows. Initially, the state equation in (9) can be transformed into (10) when the derivative operator is replaced with the Laplace operator:
s m s m o m sˆ ˆ ˆs V A V L [V C V ]⋅ = + ⋅ − . (10)
Equation (10) can be rearranged in terms of Vo: 1
s m m m m m oV̂ [s I A L C ] L V−= ⋅ − + ⋅ (11)
where Im represents the identity matrix. Hence, the estimated states in (9-c) can be expressed with
respect to the measurable states in (8-b):
d t t d t t q
q t t d t t q
d+ t t d t t q
q+ t t d t t q
v̂ A /P v E /P v
v̂ B /P v F /P v
v̂ C /P v G /P v
v̂ D /P v H /P v
= ⋅ + ⋅ = ⋅ + ⋅ = ⋅ + ⋅ = ⋅ + ⋅
(12)
where all transfer functions in (12) are detailed with 3 2
t 1 p 3 3 1 4 2 3
2p 1 2 2 1 3 4 4 3 p 1 p 1 4 2 3
ˆA p s {2ω (p q ) p p p p }s
ˆ ˆ ˆ2ω (p q p q p q p q 2ω q )s 4ω (q q q q )
= + − + −
+ − + − + + −(13-a)
3 2 2t 3 p 1 1 p 3ˆ ˆB p s 2ω (q p )s 4ω q s= + − + (13-b)
3 2t 1 4 1 3 2
2p 1 2 2 1 p 1 p 1 4 2 3
C q s (p q p q )s
ˆ ˆ ˆ+2ω (p q p q 2ω q )s 4ω (q q q q )
= + −
− + + − (13-c)
3 2t 3 4 3 3 4
p 3 2 2 4 1 1 p 3
D q s (p q p q )s
ˆ ˆ2ω {q (q p ) q (p q ) 2ω q }s
= + −+ − + − +
(13-d)
3 2 2t 2 p 4 4 p 2ˆ ˆE p s 2ω (p q )s 4ω q s= + − + (13-e)
3 2t 4 p 2 2 1 4 2 3
2p 1 2 2 1 3 4 4 3 p 4 p 1 4 2 3
ˆF p s {2ω (q p ) p p p p }s
ˆ ˆ ˆ2ω (p q p q p q p q 2ω q )s 4ω (q q q q )
= + − + −
+ − + − + + −(13-f)
3 2t 2 1 2 2 1
p 2 3 3 1 4 4 p 2
G q s (p q p q )s
ˆ ˆ2ω {q (p q ) q (q p ) 2ω q }s
= + −+ − + − +
(13-g)
3 2t 4 1 4 2 3
2p 3 4 4 3 p 4 p 1 4 2 3
H q s (p q p q )s
ˆ ˆ ˆ+2ω (p q p q 2ω q )s 4ω (q q q q )
= + −
− + + − (13-h)
t m m m m
4 3 21 4 p 1 4 2 3
2 2p p 3 2 2 3 1 4 2 3
p 1 2 2 1 3 4 4 3 p 1 4
P det[s I A L C ]
ˆs (p p )s 4ω (q q q q )
ˆ ˆ{4ω 2ω (p p q q ) p p p p }s
ˆ ˆ2ω {p q p q p q p q 2ω (q q )}s
= ⋅ − +
= + + + −
+ + − + − + −
+ − + − + +
. (14)
Among the transfer functions in (12), the most important factors are Ct/Pt and Ht/Pt, which explain the influence of each axis voltage to its estimated positive-sequence voltage. Therefore, these transfer functions have been crucially considered in setting the observer gains. Initially, the roll-off rate of Ct/Pt and Ht/Pt can simply be increased by setting q1 and q4, the highest order coefficients in the numerators, to zero:
1 4q q 0= = . (15)
The settings of (15) are intended to enhance the filtering performance to high-frequency distortions. By (15), Ct and Ht are simplified into
2 2t 3 2 p 1 2 p 2 3ˆ ˆC p q s 2ω p q s 4ω q q= − + − (16)
2 2t 2 3 p 4 3 p 2 3ˆ ˆH p q s 2ω p q s 4ω q q= − − − . (17)
To prevent non-minimum phase responses, all coefficients must have the same sign in each of (16) and (17). This condition causes the roots of each equation to be placed in the left half plane (LHP) and is presented as
1 3 3 3 4 2 2 2p / p 0, q / p 0, p / p 0, q / p 0< > > > . (18)
Meanwhile, the observer poles can be determined by using the concept of the general second order system as follows:
2 2 2 2set 1 n1 n1 2 n2 n2P (s 2ζ ω s ω )(s 2ζ ω s ω )= + + + + (19)
where ζx is damping ratio and ωnx is natural frequency. After the damping ratios and natural frequencies are
determined, the observer gains are set by comparing the coefficients between (14) and (19). Considering that all the coefficients are positive in (19), the condition in (20) must be satisfied in the coefficient comparison after (15) is applied:
1 4 2 3p p 0, q q 0+ > < . (20)
The next design point is to set the observer gains so that the observer structure is symmetric in the synchronous d-q reference frame. This can contribute to the simple implementation of the observer, because similar gains can be repeated in the d-q reference frame, as shown in Fig. 6. The symmetric structure can be achieved with (21) when the conditions of (15), (18), and (20) are assumed:
1 4 2 3 2 3p p , p p , q q= = − = − . (21)
By the settings of (15) and (21), the equations of (13) and (14) are simplified into (22) and (23), respectively. That is, the internal transfer functions in the d-axis become similar to those in the q-axis.
3 2 2 2t t 1 p 2 2 1 2
2 2p 1 2 p 2
ˆA F p s {2ω (q p ) p p }s
ˆ ˆ4ω p q s 4ω q
= = + − + +
+ + (22-a)
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t tB E (p= − = −
t t 2 2C H p q= =
t tD G {q= − = −4 3
t 1
p 1 2
P s 2p s
ˆ4ω p q s
= +
+ +
The obserrespect to pfrequencies inharmonic disfrequency, it wto be proporti
n1 1 p nˆω k ω , ω=
where pω̂ is e
used for the dFor (21) an
and (23) resul
1 4 1 1p p (ζ k= =2p p 2ˆ ˆ4ω 4ω (q+ −
p 1 2 1ˆ4ω p q 2k k=2 2 2 2p 2 1 2ˆ ˆ4ω q k k ω=
If p1 in (25This resultantinto (29) by th
2
1 2 2 1
1 1 2 2
ζ k ζ k
ζ k ζ k
+ +
Because epositive, (29)
2 1 1(ζ ζ )(k k− −
When the gain settings oOne of two fwork. When (31) must be sk1 is equal to
22 pˆ(p /ω 2)− =
22 pˆ(p /ω 2)− =
It is imponumber in (31different undsame. Howevequal to k2. increase the d
21(1 ζ ) 0 − ≥ →
In a generwould occur Therefore, in the observer g
3 22 p 1ˆp s 2ω p s+ +2
2 p 1 2ˆs +2ω p q s +3 2
2 1 2q s p q s 2+ +2p p
2 2p 2
ˆ ˆ{4ω 4ω (q
ˆ4ω q
+ +
rver gains in predeterminedn (19) by thetortions commwould be convional to the gri
n2 2 pˆk ω=
estimated grid
design since (7nd (24), the colts in
2 2 pˆζ k )ω+ 2 2
2 1 2p ) p p− + + =
2 1 2 2 1 ˆk (ζ k ζ k )ω+4pω̂ .
5) is inserted int q2 must thenhe substitution
1= .
each value inis further sim
2k ) 0= .
observer poleof (15) and (2factors in (30)(26) is considsatisfied if ζ1 k2:
21 2(k k ) (1 ζ− −
2 21 1 2k (ζ ζ )− ⋅ −
ortant to reme1) and (32). Aer (33)’s con
ver, ζ1 must bTherefore, th
degree of freed
1 1 ζ 1→ − ≤ ≤ .
ral second orif the damp
(33), the damgains in (21) a
2p 2ˆ4ω q s)+
2 2p 2ˆ4ω q
p 2 2 2ˆ2ω q (q p− +2
2 2 1q p ) p p− + +
(23) can thed damping e coefficient cmonly occur venient to set tid frequency:
d frequency. ω
7). oefficient com
2 21 2 1(k k 4ζ= + +
3pω̂
nto (27), q2 is n satisfy (28),n of the resulta
n (29) has bmplified into
es are placed a1) are used on) must be zerdered in additis equal to ζ2
21ζ )
.
ember that evAccording to (nstraint even be identical the setting fordom in the obs
rder system, oping ratio is
mping ratio couare finally spec
pˆ2ω )s}+ 2 22p }s
.
en be specifieratios and
comparison. Bat multiples the natural fre
pω̂ has alread
mparison betwe
2
2 1 2 pˆζ k k )ω
uniquely dete which is rearant q2:
been assumed
according to (nly if (30) is saro at least forion to (30), ewhile (32) do
very value is31), k1 and k2
if ζ1 and ζ2
o ζ2 in (32) r (31) is seleserver poles.
oscillatory ressmaller than
uld be set to ucified as
(22-b)
(22-c)
(22-d)
(23)
ed with natural
Because of grid
equency
(24)
dy been
een (19)
(25)
(26)
(27)
(28)
ermined. rranged
(29)
d to be
(30)
19), the atisfied. r this to equation oes so if
(31)
(32)
s a real 2 can be are the if k1 is
ected to
(33)
sponses n unity. unity so
Fig
p
p
q
D.
seq
v
v
sim
1k
(38
t
t
C
P
t
D
P
whspenefirfreaccfir
4th2nveoriHopro
inta llin
g. 2. Pole placeme
1 4 1
2 3 p
2 3 1 2
p p (k k
ˆp p 2ω
q q k k ω
= = +
= − =
= − =
Pole Placem
Considering quence voltag
d+ t t
q+ t t
v̂ C /P v
v̂ D /P v
= ⋅
= ⋅
The substitumilar to the ge
2k, k kρ= = .
The transfer 8) after reduci
t2
(
s (1 )k
ρρ
=+ +
t2
t
1
2 s (1= −
+ +
The meaninghere the locatecific valuesgative real ax
rst, two polesequency. Thecording to set
rst placement.By the propo
h order obsernd order transfry helpful in iginal system owever, it is oposed observTo analyze
troduced baselinear system
near systems,
ents of the propos
2 p
p
ˆk )ω
ω̂ / 2
.
ment of the Ob
(12) and (ge is expressed
d t t q
d t t q
v D /P v
v C /P v
− ⋅
+ ⋅.
ution of (36)eneral second o
functions in (ing common f
2p
2p p
ˆ(kω )
ˆ ˆkω s (kω )ρ+2
p
p
ˆk ω s
ˆ)kω s (kω
ρρ ρ+ +
g of k and ρ tions of obser
of k and ρ.xis pertaining s are placed aen, the rematting ρ, which
osed setting inrver shown infer functions i
adjusting them can be und
not clear wver is with onlthe unified ed on the fact t
m. Since the sthe response
sed observer acco
bserver
(22), the esd by
.
makes the order system:
35) are then dfactors:
2pω̂ )
.
can be undersrver poles are All poles ato (31) and (3at a distance aining two ph is the relati
n this paper, thn (9) can be din (37) and (3e observer reerstood with
what the unify (35), (37), a
effect, the folthat the derive
superposition to three-phas
ording to k and ρ
stimated posi
transfer func
derived as (37
stood from Fidepicted with
are located on33). When k i
k−times thepoles are plive distance to
he operation odescribed with38). This featuesponses since
simple modefied effect ofand (38). llowing methoed system in (principle holdse voltage at
.
(34)
itive-
(35)
ctions
(36)
) and
(37)
(38)
ig. 2, h the n the is set grid
laced o the
of the h the ure is e the eling. f the
od is (8) is ds in each
Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
frequency can[14], the three
ah h
bh h
ch h
v (t) V s
v (t) V s
v (t) V s
= −
= − = −where ‘h’ can
Through Pvoltage can be
hdh
qh h
Vv (t)
v (t) V
=
If the Laplestablished be
tdh qh
ωv (s) v
s=
Based on three-phase vcan be derived
d+ t
q+ t
v̂ (s) (C /
v̂ (s) (D /
= =
The Bode replacing s widesign of the of Tq. It becoin frequency d
As shown serves as lowframe while oIn particular, pole placemeestimated freqcommon propresults from negative-sequ
Although estimation errband is more the grid freqnegative-sequthe estimatedwithin the freestimated freqthe magnitudeto be mitigate
II
The observPLL. By virtube readily acPLL to increa
n be separatele-phase voltag
p
p
p
sin(hω t)
sin(hω t 120 )
sin(hω t 120 )
− °
+ °
n be any real vPark’s transfore derived as
p
p
sin((h 1)ω t)
cos((h 1)ω t)
− −
ace transformetween the d- a
(s) .
(35) and (41oltage at the ad as
t t tt
tt t t
s/P D /P )
ω
ω/P C /P )
s
− ⋅
⋅ +
plot of Td anith jωt as showobserver, the
omes evident hdomain. in Fig. 3, the
w-pass filters offering the bthe bandwidth
ent, and the stquency. The lperty of obserthe proposed
uence voltage. both filtering
ror of the gridcritical. Then
quency error uence voltage d frequency inequency errorquency deviatee of negative-
ed by the obser
II. PLL BASE
ver proposed ue of this conchieved with ase its bandwid
y considered. ge can be gene
)
)
value. rmation in (3)
h t
h t
V sin(ω t)
V cos(ω t)
=
m is applied to and q-axis vol
1), the frequearbitrary frequ
dh d
qh q
v (s) T (s)
v (s) T (s)
⋅ =
⋅ =
nd Tq in (42)wn in Fig. 3. DBode plot of Thow the prop
e proposed obsin the synchr
band-stop funch of low-pass top band is lolow-pass functrver while thed modeling o
g functions ad frequency, thn, it has been d
as shown inwas considere
n (37) and (38r of 20%. As es from the gr-sequence volrver by up to o
ED ON THE OBS
in section II nstruction, filtless effort indth and contrib
Considering eralized into
), the generali
)
.
(40), equationltages:
ency responseuency, denoted
dh
qh
v (s)
v (s)
⋅
⋅.
can be obtaiDue to the symTd is the same
posed observer
server fundamronous d-q rection simultanfiltering is setocated at doution originatee band-stop fof (7) to rej
are affected he movement discussed in ten Fig. 4. Whed in (42) (ωt
8) has been ca result, even
rid frequency bltage is still exone tenth.
SERVER
is attached ttering harmon
n PLL, whichbutes to the hi
(1) and
(39)
zed d-q
(40)
n (41) is
(41)
e to the d by ωt,
(42)
ined by mmetric e as that r works
mentally eference neously. t by the
uble the s in the
function ect the
by the of stop
erms of hen the = 2ωp),
changed n if the by 20%, xpected
to SRF-nics can
allows igher
Fig
Fig
Fig
phshoAi
thehabe dirdrafredisass
dθ̂
pθ̂
g. 3. Frequency re
g. 4. Deterioration
g. 5. Entire PLL s
hase-angle tracown in Fig. 5ided PreprocesThe estimate
e observer gas been used asensitive to d
rectly deliverawback, ωe1
equency. The scussed with sumed, (44) ca
d p pˆθ θ θ≈ = −
pp ip
2pp i
k s k
s k s k
+=
+ +
esponse of the ob
n of band-stop fun
structure.
cking dynami and is referressing phase loed frequency iins accordings the estimatedistortions becrs scaled ang
in Fig. 5 ieffectivenesthe followi
an be derived
pip
θ
server to three-ph
nction by grid fre
ics. The entireed to as Synchcked loop (SOin PLL must
g to (34). Usued frequency. Hcause kpp, the
gle errors to s used as ths of this moing equationsfrom Fig. 5:
hase voltage.
equency error.
e PLL structuhronous Obse
OAP-PLL). be fed back t
ually, ωe2 in FHowever, ωe2
proportional it. To refine
he estimated odification cas. When (43
ure is erver-
to set Fig. 5
2 may gain,
e this grid
an be 3) is
(43)
(44)
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A
where kpp andPLL, respectiv
From Fig.
ipe1 d
k kˆω θs
= ≈
If pθ̂ in (44
ipe1 2
pp
k sω
s k s≈
+
ωe1 is a lousing this freq
less sensitive function in (4system, the Pof the second
pp pll npllk 2ζ ω ,=
where ζpll anfrequency of P
The gain stogether, so ais larger than is set to four pole of the dynamics inte
Meanwhileboth sides ospecifically imof obtaining according to design seemimplementatiodiscussed earand band-stop
IV. SIM
A. Evaluatio
The perforaspects: the mitigating abbeen examinecontaminated frequency wa
In particulas compared because its imgain settings a
d kip are propovely. 5 with (43), ω
ipp p
k ˆ(θ θ )s
− .
4) is inserted i
p 2ip
θk s
=+ +
ow-pass filterquency as pω̂
to distortions46) takes the I gains can beorder system:
2ip npllk ω=
nd ωnpll are PLL, respectivsettings of theall the gains ar
that of (44). Etimes greater observer is n
ended in (44) we, based on thf (10) by ‘s,mplemented, a
Fig. 6, the(15), (34), an
ms to be raon of the prlier, this singl
p functions sim
MULATION AND
on of SOAP-P
rmance of SOtracking err
bility against ded under grid f
by harmonics changed. ar, the effectivto DSOGI-FL
mplementationare known [10
ortional and in
ωe1 is derived a
into (45), it ca
ipp
pp ip
kω
k s k+.
ed value of a, the observer
s. In addition,form of the g
e set accordin:
the dampingvely. e observer andre set where thEmpirically, ththan that of (
not larger thawould be distu
he equation ob,’ the proposas shown in F
e observer gand (36). Even ather complicroposed obsele observer camultaneously.
D EXPERIMENT
PLL
OAP-PLL can or on grid distortions. Eafaults where thcs and unbal
veness of SOALL. This methn is relatively 0].
ntegral (PI) ga
as
an be rearrang
actual frequenr performance
because the tgeneral secon
ng to the settin
g ratio and
d PLL have the bandwidth he bandwidth (44). If the doan that of PLurbed. btained from dsed observer Fig. 6. In the ains have bethough the o
cated, the perver is simpan offer the lo
TAL RESULTS
be discussedparameters aach performanhe grid voltaglances, and th
AP-PLL is dihod has been ssimple, and e
ains for
(45)
ed into
(46)
ncy. By e can be
transfer nd order ng rules
(47)
natural
to come of (37) of (37)
ominant LL, the
dividing can be process een set
observer practical ple. As ow-pass
d in two and the nce has es were he grid
scussed selected essential
Fig
Fig
B.
froidenothe
1V
5V
fauf
whord‘no
allthe
eargasim(48ρ
of PLspetim
phaftoscsyn
g. 6. Block diagra
g. 7. Grid fault un
Simulation
The fault conom the literatueal grid is sel
ormal voltage e fault conditio
0.5 30 ,+ = ∠ − °
5 70.2 0 ,V− += ∠ °
ult norf 5Hz− = −
here the subscder, and the nor’. When the nor
l the conditione fault wavefoInitially, the
rlier work [10ins of SOAP-P
milar to those8). The specifi
pll1, k 1.7, ζ= =
The gains of Fig. 6, and t
LL of Fig. 5 ecific values,
me under the cDue to the st
hase angle andter the fault, cillations disnchronization
am of the propose
nder conditions of
Results
nditions for simure [12]. Whlected as the is denoted as
ons are given
1V 0.25 11− = ∠
110.2 0 ,V+ −= ∠ °
z
cript number onormal conditi
rmal grid is asns in (48), (49)orms are show
gains of DSO0]. For a fair cPLL had to beof DSOGI-FL
fic values of th
l npll1, ω 2π= =
ρ and k are dithe gains of ζ
through the SOAP-PLL h
condition spectep variations d frequency pas captured
appear, both s as if there
ed observer.
f (48), (49), and (
mulation testshen the voltag
base voltage, s 1∠0°. Basedas follows:
10°
0.2 0− = ∠ °
of a voltage reion is indicate
ssumed to be 2), and (50) aren in Fig. 7.
OGI-FLL wercomparison ofe set such thatLL under the hose gains wer
20⋅ .
irectly inserteζpll and ωnpll a
PI gains in has presentedified in (48), ain (48), the e
present oscillain Fig. 8. Hmethods sh
are no unba
(50).
s have been quge amplitude
the phasor od on this con
epresents harmed by the subs
220Vrms-60Hze applied after
re set accordinf performancest its dynamics unbalance fau
re
d into the obsare reflected in
(47). With d the same setas shown in Fistimation erroations for a w
However, afterhows perfect alance distort
uoted at an
of the ncept,
(48)
(49)
(50)
monic script
z, and fault,
ng to s, the were
ult of
(51)
erver n the these ttling ig. 8. ors of while r the grid
tions,
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
which confirmcomparable w
The perforDSOGI-FLL discrete time in fault condFLL was baimplemented implementatiotested under tsimulation res
As shown DSOGI-FLL PLL consists Even though timproved by [12], this extcode lines forDSOGI-FLL due to harmon
The other the steady-staTo discuss thiin Fig. 10. following tran
2
ˆkωD(s)
ˆs kω=
+
2
ˆkωQ(s)
ˆs kω=
+
where the subreference framsignal of inpuharmonics.
When a sydomain, its orcomputationacircles in Figspecify thesedistortion (ZT
H
HZTD (ω
H) =
where ‘H’ symoriginal systetransformed sfrequency of cimplementatio
The meandomain systemZTD correspoof each system
10 H20log | ZTD
ms that the unwith that of DSrmance of SO
when each domain, or th
ditions. The dased on [21]
with backwons under 10 the fault condsult is presente
in Fig. 9, Sin harmonic of two filteri
the filtering peextending to m
tension can onr implementatpresents the
nics. conspicuous f
ate error of 2.is phenomenoTo estimate
nsfer functions
p α2
p p α
ω s v̂ˆs ω v
=+
2p α
2p p α
ω̂ v̂ˆω s ω v
q=+
bscript ‘α’ refme. Q(s) in put while both
ystem is transfriginal proper
al delay of z-
gs. 6 and 10, me types of TD) is introduc
sωΤz
s
(e )
H ( ω)
j
j
mbolizes the em in the ssystem in theconcern and Ton.
ning of ZTD m from the s-onds to the dim:
H 10(ω) | 20 log=
nbalance rejectSOGI-FLL. OAP-PLL was
method washe harmonics idigital implem
while SOAward Euler m
kHz samplingditions of (48)ed in Fig. 9. OAP-PLL obfiltering. Thi
ing stages: therformance ofmultiple SOGnly be done btion. In additiaverage error
feature is that7 ° in estimat
on, one of the e positive-seqs are importan
fers to the d-aparticular geneh D(s) and Q
formed from thrty can be dist-1, which is imay be includdistortions, tced:
system of inte-domain, and
e z-domain. ITs is the sampl
is the distort-domain systeifferences betw
sωΤz| H (e ) | 20j −
tion of SOAP
s different to s implementein (49) were in
mentation of DAP-PLL was method. Withgs, each meth), (49), and (5
bviously outpeis is because e observer anf DSOGI-FLL
GI-FLL (MSOGby increasing ion, the frequer of 0.79 Hz
t DSOGI-FLLting the phasedual SOGI is
quence voltagnt in Fig. 10:
axis in the staerates the qua(s) serve as f
he s-domain ttorted. Moreovindicated by ded for feedbathe z-transfor
erest, Hs referd Hz represen addition, ωling period for
ted degree ofem. The Bode ween the Bod
10 s0 log | H ( ω) |j
-PLL is
that of d in a ncluded DSOGI-
simply h these hod was 50). The
erforms SOAP-
nd PLL. L can be GI-FLL) source ency of mainly
L shows e angle. s shown ge, the
(52-a)
(52-b)
ationary adrature filtering
o the z-ver, the dashed
ack. To rmation
( 5 3 )
rs to the nts the
ω is the r digital
f the z-plot of
de plots
(54)
Figimp
Figdig
Fig
Z∠
DSfremawhAsthefre
DSPLPLvo
g. 8. Simulation replementations.
g. 9. Simulation regital implementati
g. 10. Second ord
HZTD (ω H) = ∠
The Bode pSOGI-FLL arequency is seagnitude and hich correspons shown in Fige frequency equency.
From Fig. 1SOGI-FLL anLL and DSOGLL can sepaoltages via Pa
esults under unba
esults under faultions.
er generalized int
sωΤz sH (e ) Hj − ∠
plots of critire depicted inet at 10 kHzphase must b
nd to unity, whg. 11, the ZTD
of concern
11, the reasond not in SOAP
GI-FLL are diffarate the poark’s transform
alance fault of (48
t conditions of (48
tegrator.
( ω)j .
cal ZTDs inn Fig. 11 whz. In the Bobe 0 dB and hen the z-tranDs deviate fro
approaches
n for the steP-PLL can be
fferent theoretisitive- and mation. By th
8) with analog
8), (49), and (50)
n SOAP-PLL here the samode plot of Z
0 °, respectinsformation is om the unity w
to the Ny
eady-state erroexplained. SO
ically since SOnegative-sequ
his separation
) with
(55)
and mpling
ZTD, ively, ideal. when
yquist
or in OAP-OAP-uence n, the
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
negative-sequfrequency whband. In contnegative-sequwhere both vo
As shownexhibit the phfrequency, wDSOGI-FLL.parallel displadistortion of pstate error in sampling fre@5kHz). In cunity at the Dappears in SOvoltage wouldobserver funcfrequency as stop functionresult of Fig.are required f[22].
In additionthe fault test property of SOare not observrejects negatvariation.
C. Experime
To demonwas emulatedalgorithms w(DSP), TMSthrough voltaPLL were set
The most f[17], which isby grid-connephase-to-phasconsidered:
a
b
c
V 1
V 1 / 2
V 1 / 2
= = − −
= − +
where Vsag isvoltage sag. In
The magnphase of Vsag 40° . The morders were variation in (5
uence voltagehile the posititrast, DSOGI-uence voltagesoltages appearn in Fig. 11, thhase distortion
where the posi For AC signacements in tpositive-sequeDSOGI-FLL,quencies as
contrast, the ZDC band, wh
OAP-PLL. In od not be distoction is slighshown in Fig.
n in SOAP-PL. 9. Unlike Sfor digital imp
n, the conditionof Fig. 9 to
OAP-PLL. Beved often in Ftive-sequence
ental Results
strate the feasd using an AC were implemen
320F28335, aage sensors pe
according to (frequent fault s commonly rected convertese fault betw
sag
sag
V 3 / 2
V 3 / 2
j
j
−
+
s a complex n addition, all
nitude of Vsag
[19]. As an exmagnitudes of
set to about 50) was includ
e appears ave-sequence v-FLL deals ws in the stationr at the grid frehe ZTDs of D
ns of 2.2 and 3itive-sequencenals, these nothe time-domaence voltage c, and can be ishown in FTDs of SOAP
here the positiother words, torted in SOAPhtly distorted . 11, the effecLL can be co
SOAP-PLL, caplementation o
n of (50) has ao examine theecause the secFig. 9, the SOA
voltage ev
sibility of SOAprogrammabl
nted in a digand grid voltr 100 μs. All (51). is the single p
recognized as ers due to tra
ween b- and
number indic magnitudes a
g is determinxperiment, Vsa
harmonics at0.08 per unit
ded as well. Th
at double thvoltage is at
with the positivnary referenceequency. D(s) and Q(s) .2 degrees at t
e voltage apponzero phaseain. Then, thecan cause the intensified at ig. 11 (deno
P-PLL presentive-sequence the positive-seP-PLL. Althou
at double thctiveness of thorroborated wareful consideof DSOGI-FL
also been incle frequency-acond order distAP-PLL succe
ven after fre
AP-PLL, a grle source, MX
gital signal prtages were sthe gains for
phase-to-grounphase-to-pha
ansformers [1c-phases ha
cating the deare in per unit.
ed according ag was set to 0t 5th, 7th, ant, and the frehe grid voltage
he grid the DC ve- and e frame,
in (52) the grid
pears in s mean e phase steady-smaller
oted by t almost voltage
equence ugh the he grid
he band-with the erations
LL [20]-
luded in adaptive tortions essfully equency
rid fault X30. All rocessor sampled SOAP-
nd fault ase fault 9]. The
as been
( 5 6 )
gree of
to the 0.38∠nd 11th equency es
Figsam
Fig
arothe
stato oscgenset
haeffcobeneSOsituSOFLthePL
err
g. 11. Bode plots mplings.
g. 12. Grid fault fo
ound the faulte voltages recoInitially, the c
arting point ofcapture sever
cilloscope. Itnerate almostttings. Under the f
rmonics in thefect of the promparison withen eliminatedgative-sequen
OAP-PLL wheuations, the
OAP-PLL, 1.5LL. Furthermoe average devLL.
Because the rors were calcu
of ZTDs in SOAP
for experiment.
t occurrence aognized by thec-phase voltag
f the fault, wasral waveformt was assumet time-invarian
fault situatione frequency eoposed observh SRF-PLL, wd from SOAPnce distortionen compared RMS of freq
52 Hz for SRFore, the frequiation of 0.10
actual phase ulated with re
P-PLL and DSOG
are shown in Fe DSP board. ge, whose negs selected as th
ms from the lied that the nt faults acco
n, SOAP-PLLstimation show
ver can be mowhere only theP-PLL. As s
ns were evidwith SRF-PL
quency rippleF-PLL, and 0.4ency of DSO2 Hz when co
angle was noespect to the ph
GI-FLL under 10
Fig. 12, whic
gative peak wahe reference simited channeAC source c
ording to the
L has the fewn in Fig. 13
ore obvious vie observer parhown in Fig
dently rejectedLL. After trane was 0.1 Hz45 Hz for DSO
OGI-FLL preseompared to SO
ot accessed, ahase angle of
0 kHz
h are
as the signal els of could same
ewest . The ia the rt has . 13, d by nsient z for OGI-ented OAP-
angle
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
Fig. 13. Frequenc
Fig. 14. Angle es
SOAP-PLL, θAfter transienθDSOGI, deviatin Fig. 14.
In this papextract positivoltage. This of PLL by uequation has bthe observer, suggested whand their com
In a functiospecifically srejecting unbparticular, theimplemented implementatiomethod has bresults. The distortions owithout steady
[1] A. E. EmanMeasuremeNonsinusoiInd. Appl.,
cy estimations in
stimations in expe
θSOAP, when nt situations, ted from that
V. C
per, an obserive-sequence extraction is sing the obsebeen newly deand the gain
hen considerinmbination with
onal aspect, thcrutinized in balances unde performancestructures un
on. As a resultbeen confirme
proposed mf the estimay-state errors.
RE
nuel, “Summary oent of Electric idal, Balanced, ovol. 40, no. 3, pp
experiment: time
eriment: time [10
considering ththe phase anof SOAP-PLL
CONCLUSION
rver has been voltage fromintended to re
erver as a preerived on gridsettings of theg the mitigatioPLL.
he proposed Pterms of filte
der polluted es have been dnder the consit, the effective
ed by simulatimethod exce
ated frequenc
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[21] F. J. Rodrigand M. C. generalizedIECON200
[22] A. G. YepePerformancIntegrators,
University, wherthe Vice Dean o2008 to 2011, hScience Researcpower electronicship drives, and p
guez, E. Bueno, MCavalcanti, “Disc
d integrators for08, Nov. 10-13, 20
es, F. D. FreiJedoce Digital Reson,” IEEE Trans. Po
re he is currentlyof Electrical Engihe was the Presch Institute (KEc control of electrpower-converter
M. Aredes, L. G.crete-time implemr grid converter008, pp. 176-181
o, O. Lopez and nant Controllers ower Electron., v
Yongsoon Parkand M.S. degreefrom Seoul NaKorea in 2008 where he is currdegree. His currsensorless driveand power conve
Seung-Ki Sul received the B.Sin electrical eNational Unive1980, 1983, and 1986 to 1988, Researcher in theand Computer EWisconsin, Madhe was a Princwith Gold-StaCompany. Sincemember of the Electrical Engin
y a Professor. Froineering, Seoul Nident of Korea
ESRI). His currerical machines, ecircuits.
Woo-Chull Kim1976. He receiInformation engineering froSuwon, Korea, ia research enginSeoul, Korea. interests includeof renewable ene
B. Rolim, F. A. mentation of secors,” in Conf. Re.
J. Doval-GandoyImplemented W
vol. 26, no. 2, Feb
k (S’12) receiveds in electrical eng
ational Universityand 2010, resp
rently pursuing trent research intes of electrical mersion circuits.
(S’78-M’80-SM’S, M.S, and Ph.Dengineering fromersity, Seoul, K
1986, respectivehe was an A
e Department of EEngineering, Univdison. From 1988cipal Research ar Industrial e 1991, he hasfaculty of the Sneering, Seoul om 2005 to 2007National UniversiElectrical Engin
ent research inteelectric/hybrid ve
m was born in ived the B.S. d
and commom Suwon Unn 2003. Since 20
neer with LG UpHis current
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S. Neves ond order ec. IEEE
y, “High-With Two b. 2011.
d the B.S. gineering y, Seoul, pectively, the Ph.D. erests are machines,
’98-F’00) D. degrees m Seoul
Korea, in ely. From Associate Electrical versity of to 1990, Engineer Systems
s been a School of
National 7, he was ity. From eering &
erests are ehicle and
Korea in degree in
munication niversity, 011, he is plus, Inc.,
research ic control
HydeKoElKoposwco
yun-Young Lee egree from Ulsan orea, in 2004. Helectrical Engineerorea. His researchower electronics, witching, photovoommunication tec
received the M.SUniversity, Ulsa
e is currently a Ser at PALSPO Co.h interests includmotor controls, s
oltaic system and chniques.
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