Post on 15-Dec-2015
description
Comparison and evaluation of the PLL techniques
for the design of the grid-connected inverter systems
A.Nicastri, A. Nagliero Dept of Electrotechnical and Electronic Eng.
Polytechnic of Bari
Via E.Orabona 4
70125 – Bari - Italy
antonionicastri@aol.com, nagliero@deemail.poliba.it
Abstract- The knowledge of the phase, amplitude and
frequency of the utility voltage is a fundamental aspect
for the design of the grid-connected inverter systems. In
this paper are presented the basic features of the PLL
technique. A particular attention is dedicated to the
Synchronous Reference Frame-PLL scheme. About the
generation of the orthogonal voltages system an
evaluation of the most employed techniques is shown:
Transport Delay, Inverse Park Transformation, Hilbert
Transformation, Second Order Generalized Integrators
(SOGI). Moreover some problems in filtering are treated.
The internal filtering – due to the PD structure – and the
external filtering – due to the harmonics presence in the
grid voltage. In both the cases the most employed
solutions are shown. For the internal filtering: Low Pass
Filter, Resonant Filter, Moving Average Filter, Repetitive
Controller. For the external filtering two alternative
schemes are presented: the Dual SOGI-PLL and the
Enanched-PLL (EPLL).
I. INTRODUCTION
Phase, amplitude and frequency of the utility voltage are
critical information for the operation of the grid-connected
systems. The grid voltage monitoring is used to ensure that
the performance of a grid-connected system comply with the
standard requirements for operation under comon utility
distortions In such applications, an accurate and fast detection
of the phase angle of the utility voltage is essential to assure
the correct generation of the reference signals. Thus, phase-
locked loop topologies must handle distorted utility voltages
if they are intented to applications that required the tracking
of the utility voltage vector [1]. The Phase-Locked Loop
(PLL) structure is a feedback control system that
automatically adjusts the phase of a locally generated signal
to match the phase of an input signal [2].
Fig. 1. Conceptual scheme of the PLL
Fig. 2. Block diagram of the PLL
Basically it consists of three blocks: a Phase Detector (PD), a
Loop Filter (LP) and a Voltage Control Oscillator (VCO)
(Fig.1). The VCO generates the output oscillation while the
PD block values the phase difference between the input and
output signals namely the phase error between the two
signals. The Loop Filter, typically a PI controller, is
employed to minimize the phase error and to provide an
opportune driving signal to the VCO [3].
A first classification of the different PLL-based techniques
can be made considering the PD structure. The simplest PD
implementation consists in a signal multiplier. Under the
assumption of a small-signal analysis, the phase difference
can be approximate by its sinus value according to the
following trigonometric formula
ˆ ˆ ˆ ˆsin( ) sin cos cos sinε θ θ θ θ θ θ θ θ= − = − = − (1)
in which θ and θ̂ are respectively the phase of reference and
the output signal [4].
II. SYNCHRONOUS REFERENCE FRAME PLL
At present, one of the most employed PLL topology is the
Synchronous Reference Frame PLL (SRF-PLL) (Fig.3).
Fig. 3. Block diagram of the single-phase SRF-PLL
If vα is the single-phase voltage input, vβ is an internally
generated signal that is a 90 degrees shifted version of vα [5].
The Park transformation block changes the reference frame,
bringing the voltages system from an α β− stationary
978-1-4244-6392-3/10/$26.00 ©2010 IEEE 3865
reference frame to a d-q rotating synchronous reference
frame.
ˆ ˆcos sin
ˆ ˆsin cos
d
q
v v
v v
α
β
θ θ
θ θ
� �−� � � �= � �� � � �� � � �� � � �
(2)
The feedback loop controls the angular position of this d-q
reference frame. In particular the utility voltage vector is
totally lined up to the d-axis. In this way it coincides with all
its d-component; consequently the q-component is made
equal to zero. The d-component describes the voltage vector
amplitude course [6].
Fig. 4. Reference transform αβ -dq
III. QUADRATURE SIGNAL GENERATION
About the quadrature signal generation, there are different
implementation techniques; the most simple way is using a
Transport Delay of T/4 that introduces a phase shift of 90
degrees. In this case, all the harmonics of the input signal are
characterized by the same time delay.
Other approaches are based on Inverse Park Transformation,
Inverse Hilbert Transformation or through the use of a
Second Order Generalized Integrator (SOGI)
A. Inverse Park Transformation
The Inverse Park Transformation is given by
cos sin
sin cos
d
q
vv
vv
α
β
δ δ
δ δ
� �� � � �= � �� � � �� �� � � �
(3)
The resulting scheme is shown in Fig.5. In this case two
interdependent nonlinear loops are formed. In fact the outputs
of the Direct Park Transformation block are used as inputs of
the dq/αβ block and vice versa. A pair of first-order low-pass
filters, one for each d-q voltage signal, is employed between
the two blocks as energy storage elements to avoid algebric
loops [7].
Fig. 5. Block diagram of Inverse Park Transformation based PLL
B. Hilbert Transformation
The Hilbert Transform of a generic signal x(t) is defined as
following
( )( )
P xH x d
t
ττ
π τ
∞
−∞
=−� (4)
in which P is the Cauchy principal value.
The ideal Hilbert transformer violates the systems causality
property therefore it is not practically realizable [6].
However, it is possible to approximate the transformation
through the use of a Finite Impulse Response (FIR) filter,
with coefficients defined as
1 cos[( 0.5 ) ] for 0.5
( 0.5 )[ ]
0 for 0.5
n Nn N
n Nh n
n N
π
π
− −�≠
−= =�
(5)
where N is the filter order, n is the coefficient index
( 0 n N< < ) and [ ]h n are the coefficents of the filter.
The last method considered uses a structure based on the
Second Order Generalized Integrator (SOGI)
C. Second Order Generalized Integrator
The transfer function of adaptive filter based on the Second
Order Generalized Integrator is defined as
2 2( )
sGI s
s
ω
ω=
+ (6)
where ω represents the resonance frequency of the SOGI [8].
The close-loop scheme in Fig. 6 is essentially composed by a
pair of integrators. The first one needs to the generation of
the signal 'v , which has the same phase of the input signal
v , while the second one is employed in the generation of the
quadrature signal 'qv that is a 90 degrees shifted version of
v . The closed-loop transfer functions are
'
2 2( ) ( )d
v k sH s s
v s k s
ω
ω ω= =
+ + (7)
' 2
2 2( ) ( )q
qv kH s s
v s k s
ω
ω ω= =
+ + (8)
where the parameter k adjusts the system filtering capability.
All the presented methods, except the last one, have same
shortcomings: frequency dependency, high complexity,
nonlinearity, problems in filtering. The SOGI-based
technique, on the contrary, provides a couple of orthogonal
and already filtered signals only through the use of the simple
scheme in Fig.6.
Fig. 6. Adaptive filter based on the SOGI
IV. FILTERING OPTIONS FOR PLL
The main problem in the phase detection is the rejection of
the harmonics. It is possible distinguishing between an
internal and an external filtering. The internal filtering is
exclusively due to the PD structure while the external
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filtering is not linked to any intrinsic features of the system
but it regards the grid voltage conditions in terms of
waveforms purity.
The operation implemented in the PD block is the following
n d
2
2
2 h a rm o n ic t e r m
ˆ( ) ( ) c o s ( ) s in ( )
ˆ ˆs in ( ) s in ( 2 )2
in o u tv t v t A k t t
A kt
ω θ ω θ
θ θ ω θ θ
= + + =
� �� �= − + + +� �� �
� � �� � ��
(9)
in which A is the amplitude factor and k is the PD gain [9].
Under the assumption of a small-signal analysis, it is possible
rewriting the term between brackets as
nd nd2 harmonic term 2 harmonic term
ˆ ˆ ˆ( ) sin(2 ) sin(2 )t tθ θ ω θ θ ε ω θ θ− + + + = + + +������� �������
(10)
The PD output is given by two terms: the first one is actually
the phase error namely the phase difference between the input
and output signals; the other one is a 2nd
harmonic term
superimposed to the useful signal that it needs filtering.
There are various approaches to improve the PD scheme. All
the techniques are based on the addition of a further filter
above the PI controller.
A. Low Pass Filter
The first technique uses a Low Pass Filter (LPF) (Fig.7). The
transfer function of the LPF with the unitary gain is
1( )
(1/ 2 ) 1c
H sf sπ
=+
(11)
where cf is the cut-off frequency. In this way the tracking
precision is improved but the dynamic response is slowed and
the filtering performance results frequency sensitive because
the LPF introduces phase shifting in signals [10].
Fig. 7. LPF technique for harmonics rejection
B. Resonant Filter
Alternatively the LPF can be replaced with a Second Order
Resonant Filter (RF) as shown in Fig. 8
The RF transfer function is
2 2
2( )
fk sH s
s ω=
+ (12)
where fk determines the bandwidth and ω is the resonant
frequency. This filter is more stable that the first one. It
guarantees a superior capability in the harmonics rejection
but it doesn’t introduce any phase shifting at the resonant
frequency. At the others frequencies the shifting introduced is
90± ° [11].
Fig. 8. RF technique for harmonics rejection
C. Moving Average Filter
Another method to improving the PD rejection capability is
based on the Moving Average Filter (MAF) as shown in
Fig.9 [12]. The MAF operator of a generic signal x(t) is
defined as shown in (13)
1( ) ( )
t
t Tx t x d
T ωω
τ τ−
= � (13)
Fig. 9. MAF technique for harmonics rejection
where Tω is the window width and 1/f Tω ω= the equivalent
frequency of the MAF. When the input signal contains
sinusoidal components with a multiple frequency of fω , the
MAF output is a constant value. Choosing an opportune
value of the window width, the behaviour of this filter
approximates a Low Pass Filter. The transfer function of the
MAF is
[ ]1 cos( ) sin( )( )( )
( )MAF
j k T k TX jkH jk
X jk k T
ω ω
ω
ω ωωω
ω ω
− −= = (14)
D. Repetitive Controller
The last option considered is the employment of a Repetitive
Controller (RC) (Fig.10). This kind of filter improves the
rejection capability of the PI controller amplifying the second
harmonic [13]. The Repetitive Controller is essentially a
bandpass filter in which the odd harmonics are filtered while
the even harmonics no. Indirectly the proportional gain of the
PI controller is increased and so the rejection capability too.
Fig. 10. Repetitive Controller technique for harmonics rejection
The model of the Repetitive Controller is based on a DFT
algorithm (Fig. 11). The discrete transfer function of the
controller is given by
( ) ( )1
0
2 2cos
h
N i
DFT ai h NF z h i N z
N N
π− −
= ∈
� � �= + ⋅� �� �
� �� �� � (15)
that represents practically the equation of a N order FIR
filter with unity gain on all selected harmonics h.
Fig. 11. Repetitive Controller
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V. THREE PHASE PLL
The three phase SRF-PLL scheme is shown in Fig.12. The
three-phase topology is not too different from the analogous
single-phase. The system doesn’t change neither in its
constitutive elements nor in the technique to obtain the
synchronization between the inputs and outputs.
The Park transformation block brings the voltages system
from the α β− stationary reference frame to the d-q rotating
synchronous reference frame. The q component is made
equal to zero and it is controlled by the feedback loop, while
the d component depicts the voltage vector amplitude. The
utility voltage can be represented as
, ,
c o s
2c o s
3
2c o s
3
a b c mv V
θ
θ π
θ π
� � �� �� ��
= −� �� �� �� �
� �� +� �� �� �
� �� �
(16)
A further transformation block (abc/ αβ ) is put above the
Park Transformation Block to obtain a diphase voltage
system from the three phase inputs. This is the unic
considerable difference of this scheme in comparison with
the single-phase SRF-PLL. The transformation block
implements the following vectorial equation
1 1
12 2 2
03 3 3
2 2
a
b
c
vv
vv
v
α
β
� � � �− −� �� � � �� �=� � � �� �� � � �− � �� �� �
(17)
The performances of all these schemes are evaluated
considering two basic aspects. First of all the tracking
precision: the phase error between the output and input
signals must converge to zero. Moreover the system must
exhibit a fast dynamic response that corresponds to
considering a short transitory. Generally a system
characterized by a rapid dynamic response, presents a greater
tracking error and vice versa. For this reason, nowadays, the
internal parameters of the schemes are derived as trade-off
between those two aspects making use of the settling time
[14]. The SRF-PLL behaviour is not very satisfactory in
presence of harmonics or notches in the grid voltage. [15] In
these cases the filtering capability of the system makes worse
considerably. The outputs are yet synchronized with the input
voltage system but the waveforms are not filtered: inputs and
outputs have the same THD. An external filtering is
necessary in order to obtain synchronized but also cleaned
output signals. All the employed techniques are based on the
extraction of the positive sequence.
A pair of SOGIs are employed in the Quadrature-Signals
Generator (QSG) to obtain two couples of orthogonal and
cleaned signals (Fig.13). These four signals enters in the
Positive-Sequence Calculator. (PSC). This technique exhibits
a fast, precise, and frequency-adaptive response under faulty
grid conditions. Practically the grid disturbances are filtered
before entering in the PLL scheme. The global performances
results improved because the PLL is connected to a filtered
version of qv+ [16].
The Enhanched PLL (EPLL) uses a more advanced scheme
in phase detection. This scheme is obtained by the
combination of an Adaptative Notch Filter (ANF) with a
conventional PLL (Fig.14). This kind of PD exhibits superior
performances about the rejection capability. The main feature
of the EPLL is its possibility of estimating the input
fundamental component. This last signal is employed to
derivate the error value.
Fig. 12. Three phase SRF-PLL scheme
Fig. 13. Dual SOGI-PLL
Moreover this mechanism provides additional informations
such as the amplitude and the phase angle of the input signal.
[17]. It does not need any subsystem for quadrature signal
generation because the EPLL provides also a 90° shifted
version of the input signal.
The Quadrature PLL (QPLL) is an interesting variant of the
EPLL. It is based on the estimating in-phase and in-
quadrature phase amplitudes of the fundamental component
of the input signal [18].
The output signal is defined as a linear combination between
these signals with coefficients sk and qk . About the EPLL
and QPLL three-phase configurations the topology is equal in
both the cases (Fig.15). The three-phase scheme employs six
single-phase EPLL. The first group (above the detection
block) is used to extract, phase by phase, the fundamental
components of the input voltages system. The second group
(below the detection block) is used for the evaluation of the
angle-phase. In the Positive Sequence Detection block is
implemented the following vectorial equation
2
2
2
( ) 1 ( )1
( ) 1 ( )3
( ) 1 ( )
f
a a
f
b b
f
c c
v t v t
v t v t
v t v t
β β
β β
β β
+
+
+
� � � � � � �� �
=� � � �� �� � � �� �� � � �� �
(18)
in which ( )( ) ( ) ( )a b cv t v t v t+ + + is the instantaneous positive-
sequence system in input to the second group of EPLLs,
( )( ) ( ) ( )f f f
a b cv t v t v t are the fundamental components of the
input voltages system and β is a 120° phase shift in the time
domain.
Fig. 14. Sigle phase EPLL scheme 3868
Fig. 15. Three-phase EPLL scheme
VI. EXPERIMENTAL RESULTS
The various PLL schemes have been implemented on DSP
TMS320F2812. This DSP uses a 16 bits fixed-point
representation of signals and parameters thus an internal
conversion is necessary. The position of the fixed point sets
univocally the number of bits employed for the integer part
and the number of bits useful to the conversion of the
fractional part. In the case of negative number, the most
significant bit (MSB) is used as sign bit. For the signals, the
point is fixed considering the greater and the smaller assumed
value and making sure that this two extremes result correctly
codified without underflow or overflow. In the case of a
parameter is sufficient codifying the exact value without
underflow or overflow. The fixed-point range for signed
number is given by the following equation 12 2 1I Iα−− ≤ ≤ − (16)
in which I is the number of bits ordained for the integer
part.
The hardware setup (Fig.16) is composed by:
• measurement boards for grid voltages (VR, VS, VT)
and the DC voltageVdc;
• the eZdsp TMS320F2812 board equipped with the
TMS320F2812 DSP;
• interface board;
• grid simulator PACIFIC 345-AMX for the
generation of the input voltages system with or
without harmonics presence.
The Code Composer Studio has been employed for the
waveforms visualization.
In the following, an evaluation of the external filtering
capability of the various techniques is made.
The Matlab simulations have been made considering a
symmetric harmonic contribution of 5th
and 7th
(THD=12%)
(Fig.17-18-19-20).
For the real test the harmonic contribution has been
implemented as a transient disturbance on the waveforms. In
the following figures are shown the experimental result
(Fig.21-22-23-24).
Real and simulated results are perfectly adherent. All the
proposed schemes, except the SRF-PLL, filter the harmonics
in input. In the following table (Tab.I) is shown quantitatively
the filtering that the various methods allow in term of THD.
TABLE I
THD OF THE INPUT AND OUTPUT SIGNALS
Device Input THD [%] Output THD [%]
SRF-PLL 12,09 12,09
EPLL 12,09 0,28
QPLL 12,09 3,60
DSOGI-PLL 12,09 4,04
Fig. 16: Experimental setup.
Fig. 17. SRF-PLL filtering capability
Fig. 18. EPLL filtering capability
Fig. 19. QPLL filtering capability
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Fig. 20. DSOGI-PLL filtering capability
Fig. 21. SRF-PLL filtering capability
Fig. 22. EPLL filtering capability
Fig. 23. QPLL filtering capability
Fig. 24. DSOGI-PLL filtering capability
VII. CONCLUSIONS
In this paper the grounds of the PLL technique has been
showed with particular reference to the SRF-PLL scheme. A
focus on the principal matters and an evaluation of the actual
solutions have been presented.
The filtering capability is a crucial aspect in these
applications. It is possible optimize the resultant
performances essentially in two ways:
- improving the basic SRF-PLL scheme with the addition of -
futher controllers and/or filters.
- making use of more complex schemes in phase detection in
order to obtain cleaned signals above the PLL scheme.
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