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International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX
Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com
Review and Analysis of Different Methods of
Synchronization in Power System
Ali Asghar Ghadimi1, Hamid Reza Mahdian2, Mohammad Reza Hashemi3
1
Arak University, Faculty of Electrical Engineering, Arak, Iran
Arak Science and Research branch, Islamic Azad University, Arak, Iran
2,3
Abstract: Conventional techniques of phasor estimation which based on constant frequency with dynamic changes
in the frequency of a power system are severely affected. In addition, the frequency changes from its nominal
value had led to change in line and utility reactances, that causes a change in relay operation. Consequently,
frequency in protection, monitoring, control of power quality and performance of system is highly sensitive and is
a key parameter. Current techniques that estimate the frequency are sampled voltage and current. If we consider
pure sinusoidal voltage system, by measuring the time between zero crossings of the signal, the frequency can be
easily calculated. Furthermore, the wide spread use of power electronics devices in the power generation,
transmission, distribution and utilization of electrical energy, has led to change the actual shape of the voltage
wave form. Consequently, in practice given signal from power system is associated with noise and distortion, so
we need to be able to use more advanced techniques that frequency can be calculated with reasonable speed and
accuracy. With acceptance of this difficulty. the fact that the behaviour of the power system is a flexible system
that cannot be predicted accurately, is even more visible. So we need a way to calculate frequency in minimum
possible time with maximum accuracy. So we can eliminate distortion caused by the frequency. Furthermore, for
connecting with AC system and AC power injection, power electronic converters require accurate knowledge of
voltage phase, so that the injected voltage must be in phase with grid voltage.
Keywords: Phase-Locked Loop (PLL), voltage-sourced converters (VSC), synchronization, current control,
unbalanced voltage
I.
Introduction
T In this paper we introduce a method for the synchronization which is a kind of enhanced PLL. In comparison with other
commonly synchronization methods used by PLL, this method gives a high degree of resistance to noise and harmonics.
The very important property of EPLL is an ability to adjust the input frequency. In other words, if the frequency of the
system is changing, EPLL can adjust itself. Before discussing EPLL, we will took a glance at some simple methods and
also methods based on PLL.
II.
Existing methods of synchronization
In this section, we examine different methods of synchronization. In general, these methods can be divided in to two
categories: Open loop and closed loop.
Open loop methods directly acquire the phase angle of the voltage vector by using the αβ space vector, while in the
method, for closed loop, voltages are processed in αβ space vector and the phase estimating adaptively performed by a
control loop. At last, this loop must be locked in the phase which is consistent with the actual value.
A. Open loop synchronization methods
Ideally if there is no distortion and imbalance, we reach the following equation for the three phase voltages:
[Va , Vb , Vc ] = [Vcosθ, Vcos(θ − 120), Vcos(θ + 120)]
(1)
And they would be transformed in to αβ frame like:
[Vα , Vβ ] = [V cos θ , −V sin θ]
(2)
This conversion is due to the fact that Vα and Vβ are respectively in phase and orthogonal components of the fundamental
voltage that amplitude of the vectors are equal. We need proper filtering to reduce distortion. Filtering methods are
described in this section:
B. Methods based on low pass filter (LPF)
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In this method (Figure 1. ) voltages [𝑉𝑎 , 𝑉𝑏 , 𝑉𝑐 ] are converted to voltages [𝑉𝛼 , 𝑉𝛽 ] and then passed on to typical low pass
filters to remove distortion. Filtered signals are normalized and passed through the rotation matrix (R) so the phase
difference is compensated by the filter. In design of a low pass filter, a trade-off between strength and speed should be
selected. Lower cut off frequency reduces the distortion, meanwhile the convergence rate decreased. A major drawback of
the LPF is dependency to system frequency. If the system frequency changed, the phase difference caused by the filter
would not fully compensate. Another problem of the LPF is that it shows weakness in front of the grid voltage imbalance.
Figure 1.
Synchronization method based on LPF
C. Methods based on space vector filters
In fact space vector filters or SVF is a low pass filter, based on the fact that the αβ components of the grid voltage are
independent from each other. SVF equations are written as follows:
𝑋[𝑛 + 1] = 𝛾𝑅(𝜔0 𝑇𝑠 )𝑋[𝑛] + 𝐵𝑈[𝑛]
(3)
𝑌[𝑛] = 𝛾𝑅(𝜔0 𝑇𝑠 )𝑋[𝑛] + 𝐷𝑈[𝑛]
(4)
T
Where V[n] = [Vα [n] , Vβ [n]] , B = D = ( 1 − γ )I2 that 𝐼2 is a 2 × 2 matrix and 𝛾𝜖(0,1) is known as the forgotten
factor. Whatever the amount of γ is closer to 1, better filtering is done. And as a result there can be less distortion in the
estimated frequency. A method based on SVF can be adjusted so that the carried out estimation is without distortion in very
high levels. The main problem with this method is dependency to the input frequency and grid voltage imbalance. It may
be noted that there is a problem in high sensitivity to noise and harmonics and also the problem of fine tuning of
parameters.
D. Closed loop synchronization methods
A closed loop approach acts such that with a closed structure, the error signal would bring to zero. One of the known
methods is using PLL. Block diagram of a PLL is shown in Figure 2. .
Figure 2.
Block diagram of a typical single phase PLL
The basic structure of a Phase-locked loop (PLL) is shown in Figure 2. . It consists of three fundamental blocks:

The phase detector (PD): This block generates an output signal proportional to the phased difference between the
input signal, 𝑣, and the signal generated by the internal oscillator of the PLL, 𝑣́ . Depending on the type of PD,
high frequency AC components appear together with the DC phase angle difference signal.

The loop filter (LF): This block presents a low pass filtering characteristics to attenuate the high frequency AC
components from the PD output. Typically, this block is constituted by a first-order low pass filter or a PI
controller.

The voltage controlled oscillator (VCO): This block generates at its output an AC signal whose frequency is
shifted with respect to a given central frequency, 𝜔𝑐 , as a function of the input voltage provided by the LF.
Briefly, phase difference between the input signal and the output is measured using a PD and then passed through a low
pass filter. Error signal stimulates VCO to produce an output signal. Output low pass filter calculate the total of phase
difference between two input PD signals. Ideally the error signal is proportional to the total phase difference between two
input signals of the PD.
E. Synchronization using three phase PLL
Single phase Phase-Locked Loop shown in Figure 2. can be extended to be used in power applications. Suppose that
voltages are correspond to equation (1), so that the αβ conversion is like equation (2) and dq conversion for this voltages
will be:
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[𝑉𝑞 , 𝑉𝑑 ] = [𝑉 cos( 𝜃 − 𝜃̂ ), −𝑉 sin( 𝜃 − 𝜃̂ )]
(5)
A closed loop system that helps regulate 𝑉𝑑∗ = 0 , is able to adapt 𝜃̂ to the actual value of θ . Three phase PLL block
diagram is shown in Figure 3. The fact that we use for the design of K f (s) is that the term sin ( θ − θ̂ ) in small signal
analysis mode is almost equal to ( θ − θ̂ ) and there are different methods for the design of K f (s) that one of these
methods is the use of wiener method.
Block diagram of a three phase PLL
Figure 3.
A well designed closed loop system must be fast and do well in filtering operation. However these requirements do not
fulfill simultaneously and always should be considered a trade-off or an intermediate state for them.
F. Synchronization using developed 3 phase PLL
The main problem of a three phase PLL is sensitivity to grid voltage imbalance. Some solutions to solve the problem of
voltage imbalance is presented, using the theory of symmetrical components. Although the theory of symmetrical
components is stated for phasors but it can be expanded as a function of time. This idea is obtained by replacing complex
°
phasor β = ej120 with a phase shift operator in 120 degrees in the time domain. In practice, the 90 degrees phase shift is
°
1
√3
°
easier. Using equation e±j120 = ±
ej90 positive sequence component can be found in the time domain. Positive
2
2
sequence component of a system is defined as follows:
𝑉𝑎+ (𝑡)
1
1
( 𝑉𝑏+ (𝑡)) = ( 𝛽 2
3
𝑉𝑐+ (𝑡)
𝛽
𝛽
1
𝛽2
𝑓
𝑉𝑎 (𝑡)
𝛽2
𝛽 ) ( 𝑉𝑏𝑓 (𝑡))
𝑓
1
𝑉 (𝑡)
(6)
𝑐
In the above equation, f is the fundamental component. By rewriting equation (6) using the 90 degrees phase shift we reach
the following equation:
𝑣𝑎+ (𝑡) =
1 𝑓
1
1
𝑓
𝑓
𝑓
𝑓
𝑣 (𝑡) − ( 𝑣𝑏 (𝑡) + 𝑣𝑐 (𝑡)) −
𝑆90 ( 𝑣𝑏 (𝑡) − 𝑣𝑐 (𝑡))
3 𝑎
6
2√3
𝑣𝑏+ = − 𝑣𝑎+ (𝑡) − 𝑣𝑐+ (𝑡)
𝑣𝑐+ (𝑡) =
(7)
(8)
1 𝑓
1
1
𝑓
𝑓
𝑓
𝑓
𝑣𝑐 (𝑡) − ( 𝑣𝑎 (𝑡) + 𝑣𝑏 (𝑡)) −
𝑆90 ( 𝑣𝑎 (𝑡) − 𝑣𝑏 (𝑡))
3
6
2√3
(9)
In (7) to (9), S90 is a function of phase shift as 90 degrees. A block diagram for separating the positive sequence
components using equations (7) to (9) is shown in Figure 4. . Filters shown in Figure 4. must be all pass filter and create a
S
ω0
S
1+
ω0
1−
90 degrees phase shift in their center frequency. A first order filter like
can be used for this purpose. The idea of using
positive sequence component is very efficient to eliminate distortion. But it should be noted that the use of all pass filter has
two basic problems:
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1) This filter does not have capability to frequency adaptability therefore cannot accurately create 90 degrees phase shift,
when the frequency is changed from the nominal value.
2) All pass filters cannot prevent harmonics and distortion and therefore performance of PD may be subject to some degree
of error. A low pass filter can be used in output of d component so to eliminate the harmonic effects. But we should
eliminate some of the mentioned problems caused by the filter.
Figure 4.
Positive sequence extractor with all pass filter and 90 degrees phase shift around central frequency
G. Synchronization using EPLL
A block diagram of EPLL is shown in Figure 5. . Positive sequence component is extracted by the first block of EPLL and
then output signal is given to the second block of EPLL to estimate the phase angle. In other words this structure has two
stages:
1) the positive sequence component signal extraction.
2) phase and frequency signal extraction.
This structure includes the three phase voltages and also insensitive to the conditions of imbalances. Block diagram for use
of EPLL in estimating the parameters is shown in Figure 5. .
Figure 5.
Use of EPLL for parameter estimation
The structure is composed of three EPLL units and a computational unit. EPLLs adaptively extract the fundamental
components of the system voltages and shift them for 90 degrees. Computational blocks received the fundamental
component and also 90 degrees shifted component and find the positive sequence of grid voltage components with regard
to equation (7). Methods of calculation such as the calculation unit is shown in Figure 4. . The advantage of this method
according to the developed three phase PLL is extended to the following: All pass filters have been replaced with EPLL
with the structure shown. Actually EPLL is an ANF that its frequency varies to the system frequency. This adaptation
causes the EPLL to be non-sensitive to frequency. This change in frequency is of the most basic problems of the developed
three phase PLL. Since EPLL is also a low pass filter in relation to all pass filter has the property that it is free from
distortion. But the input signal is filtered in two stages. In first stage, positive sequence component is extracted and in the
second stage, phase and frequency of signal is obtained.
EPLL is a conventional tool for power applications, because not only the produced signal is in phase to the input signal but
the amplitude and frequency of the output signal is the same as input signals. In other words EPLL is able to give us an
online estimate of a signal that have variable parameters. These parameters include the frequency, the phase and the
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amplitude of signal. Another important feature of EPLL is that it extracted a 90 degrees phase shift from the fundamental
signal component. A block diagram of EPLL is shown if Figure 6. . The figure illustrated is based on the conventional PLL
structures, including PD, LF and VCO that are shown. Input signal u(t) compared with an estimated signal y(t) and an error
signal e(t) is produced. So with using LF, a trigger signal for VCO be obtained.
EPLL structure
Figure 6.
Simple structure shown in Figure 6. contains three independent parameters. Parameter k is determine the convergence
speed of amplitude of the signal (A) and parameter K P K V , K i K V , determine the convergence speed of phase and
frequency. A low pass filter after integrator unit VCO can be used in order to achieve a more accurate estimation of phase
wherein the input signal is distorted. In fact EPLL is for use in single phase systems. But because to have a high resistance
to noise and harmonic, the structure in Figure 6. is used.
H. Digital implementation of EPLL
In the presented method several EPLL has been used. Basic continuous time equations that govern EPLL, calculated by
using block diagram in Figure 6. as follows. In these equations dot at the top of parameters represent the derivative with
respect to time.
̇ = 𝐾 𝑒(𝑡) sin( 𝜑(𝑡))
𝐴 (𝑡)
(8)
𝜔̇ (𝑡) = 𝐾𝑖 𝑒 (𝑡) cos( 𝜑(𝑡))
(9)
𝜑̇ (𝑡) = 𝜔(𝑡) +
𝐾𝑃 𝐾𝑉
𝜔̇ ( 𝑡)
𝐾𝑖
(10)
e(t) = u(t) − y(t) represents the error signal and u(t) is the input signal. y (t) = A(t)sin ( φ(t)) is the fundamental
component. A(t), ω(t), φ(t) indicate the amplitude, frequency and phase of the signal. For digital implementation we need
to write the continuous time equations to discrete time equations. So we use the recursive first order derivation estimation
for this equations. Assuming 𝑇𝑠 for a sampling period, to reach the following equations:
𝐴 [𝑛 + 1] = 𝐴 [𝑛] + 𝜇1 𝑒[𝑛] sin( 𝜑 [𝑛])
(11)
𝜔 [𝑛 + 1] = 𝜔[𝑛] + 𝜇2 𝑒[𝑛] sin( 𝜑 [𝑛] )
(12)
𝜑 [𝑛 + 1] = 𝜑[𝑛] + 𝑇𝑠 𝜔[𝑛] + 𝜇3 𝑒 [𝑛] sin( 𝜑[𝑛] )
(13)
Where:
𝜇1 = 𝐾 𝑇𝑠
(14)
𝜇2 = 𝐾𝑖 𝑇𝑠
(15)
𝜇3 = 𝐾𝑃 𝐾𝑉 𝑇𝑠
(16)
Are known as step size parameters that is very similar to LMS algorithm.
III.
Quadrature Phase locked loop (QPLL)
Another method for extracting the frequency of a signal is known as QPLL. We expect that like the same
way as EPLL, QPLL is also having high resistance to unwanted input signal distortions.
In the next section, we introduce the basic principle of QPLL.
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Introduction to QPLL
QPLL obtain the fundamental signal components and also phase and frequency changes in input
signals using two orthogonal signals.
A.
A block diagram of QPLL is shown in Figure 7. . This block is based on the conventional PLL. QPLL
structure is based on the steepest descent method.
Figure 7.
Schematic diagram of the QPLL
QPLL output is considered in to two orthogonal components as follows:
𝑦(𝑡) = 𝐾𝑠 (𝑡) sin(𝜑(𝑡)) + 𝐾𝑐 (𝑡) cos(𝜑(𝑡))
(17)
Output signal is the estimated of the input signal and so we define the error as follows:
𝑒(𝑡) = 𝑢(𝑡) − 𝑦(𝑡)
(18)
Obtained error is used for estimation of amplitude of (K s (t), K c (t)) and phase φ(t). In QPLL, first the
variables being derivated, and then being integrated until the desired result is obtained. So, the frequency
of the input signal can be estimated and this frequency is equal to the derivative of phase.
𝑑𝜑(𝑡) (19)
𝑑𝑡
Three parameters μf , μc , μs control the behaviour of QPLL. These parameters are respectively,
responsible for controlling the speed of convergence of , k c , k s . If μs = μc = μf , we get the simple
structure of Figure 8. . QPLL phase estimation process is substantially different from what is usual in
PLL. The complex structure of PD in QPLL allows which can be directly obtain time derivation of phase
and also the desired parameters of the input signal. The main problem of the conventional
PLL is that the error signal is related to the non-linear phase shift. This problem is overcome in QPLL
and VCO is triggered with signal that is equal to phase changes. So it is expected that the speed of
convergence and the interval of convergence in QPLL performance is better than conventional PLL. PD
shown in Figure 7. is composed of six multipliers, two integrators, four different gain and three adders.
Output direct and in-quadrature components of VCO are multiplied by k c , k s , the result of these two
signals are added together so the output signal y(t) is obtained, the output signal is then subtracted from
the input to obtain the error signal e(t).
𝜔(𝑡) =
Error signal is multiplied to the VCO output, so after they were added the proper gain, two integrators
would work. If μs = μc we will use only two different gains and result in Figure 7. .can be simplified and
come to Figure 8. . μa , μf are the same and equal to 2μs = 2μc .
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Figure 8.
Schematic diagram of the simplified QPLL
Obtain the differential equation
Suppose u(t) is a continuous periodic function of time. A sinusoidal component of this function is as
follows:
B.
y(t) = A sin(ωt + δ)
(20)
It is assumed that A is the amplitude of signal, ω is signal frequency, and δ is the constant phase
signal and φ = ωt + δ is the total phase of signal. In general u(t) has the following form:
𝑢(𝑡) = 𝑢0 (𝑡) + ∑𝑖≠1 𝐴𝑖 sin(𝑖𝜔0 𝑡 + 𝛿𝑖 ) + 𝑛(𝑡)
(21)
In the above equation, components in sigma show harmonic. In practice, all parameters that are
expressed in u equation may be time varying.
Suppose M is a vector space that may take all the signals which varies sinusoidally with time:
M = {y(t, θ), t ∈ R, y: sinusoid with time}
(22)
Where θ(t) = [θ1 (t), θ2 (t), … , θn (t)]T is the vector of parameters that belong to the following
parameter space:
θ = {[θ1 (t), θ2 (t), … , θn (t)]T |θi ∈ [θimin , θimax ], i = 1,2, … , n}
(23)
And T is a matrix indicating transpose. We are looking for a member of M that have a shortest
distance with sinusoidal desired components of u(t). For this we must look for a suitable θ that minimize
the distance function between y and u. The form is shown below:
𝜃𝑜𝑝𝑡 = arg min 𝑑[𝑦(𝑡, 𝜃), 𝑢(𝑡)]
(24)
The function of d is defined as follows:
𝑑(𝑡, 𝜃(𝑡)) = [𝑢(𝑡) − 𝑦(𝑡, 𝜃)] = 𝑒(𝑡)
(25)
So the cost function is defined as: j(t, θ(t)) = d2 (t, θ(t)). Parameter vector θ using SD method is
estimated as follows:
𝑑𝜃(𝑡)
𝜕[𝑗(𝑡, 𝜃(𝑡)]
= −𝜇
𝑑𝑡
𝜕𝜃(𝑡)
(26)
In which the positive element μ is known as the regulatory parameter of this algorithm. Value of M is
for determining the speed of convergence and stability of algorithm. From SD method, we know that this
solution will lead us toward the minimum point. In QPLL, assume that u(t) is the general form of
t
equation (17), where (t) = ∫0 (ω0 + ∆ω(ξ))dξ . In other words, the vector of parameters are defined as
θ = [k s , k c , ∆ω]. Estimated K s , K c , ∆ω must satisfy the following conditions:
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̇
𝐾𝑠 (𝑡)
−𝜇𝑠
̇
[ 𝐾𝑐 (𝑡) ] = 2 𝑒(𝑡) [ 0
0
̇
∆𝜔(𝑡)
0
−𝜇𝑐
0
0
− sin 𝜑(𝑡)
0 ][
− cos 𝜑(𝑡)
]
−𝜇𝑓 𝑡 ( − 𝐾𝑠 cos 𝜑(𝑡) + 𝐾𝑐 sin 𝜑(𝑡))
(27)
In the above equation, the time derivative is indicated by dot. The problem we have here is we have a
set of time varying equations that are not desirable. But since the input signal u(t) is periodic, we expect
that alternative solutions exist to solve the above derivative relations. There are two ways to overcome
this problem:
2π
1) We consider the time varying component to ω .
0
2π
2) We consider t to the average value of t av ϵ [0 , ω ] .
0
We choose the second method, which is simpler. Using the definition of y(t), e(t), ∆ω(t) we get:
𝐾𝑠̇ (𝑡) = 2 𝜇𝑠 𝑒(𝑡) sin( 𝜑(𝑡))
(28)
𝐾𝑐̇ (𝑡) = 2 𝜇𝑐 𝑒(𝑡) cos( 𝜑(𝑡))
(29)
∆̇𝜔(𝑡) = 2 𝜇𝑓 𝑒(𝑡)[𝐾𝑠 cos( 𝜑(𝑡)) − 𝐾𝑐 sin(𝜑(𝑡))]
(30)
𝜑̇(𝑡) = 𝜔0 + ∆𝜔(𝑡)
(31)
𝑦(𝑡) = 𝐾𝑠 sin( 𝜑(𝑡)) + 𝐾𝑐 cos( 𝜑(𝑡))
(32)
𝑒(𝑡) = 𝑢(𝑡) − 𝑦(𝑡)
(33)
Equation (27) shows the equations based on QPLL. To make this equation, capable to be implemented
on digital methods, the first order derivative approximation can be used and operate like an EPLL. If Ts is
the sampling period, we reached the following equations:
𝐾𝑠 [𝑛 + 1] = 𝐾𝑠 [𝑛] + 2 𝑇𝑠 𝜇𝑠 𝑒[𝑛] sin( 𝜑[𝑛])
(34)
𝐾𝑐 [𝑛 + 1] = 𝐾𝑐 [𝑛] + 2 𝑇𝑠 𝜇𝑐 𝑒 [𝑛] cos( 𝜑[𝑛])
(35)
∆𝜔[𝑛 + 1] = ∆𝜔 [𝑛] + 2 𝑇𝑠 𝜇𝑓 𝑒[𝑛] [𝐾𝑠 [𝑛] cos( 𝜑[𝑛]) − 𝐾𝑐 [𝑛] sin( 𝜑[𝑛])]
(36)
𝜑 [𝑛 + 1] = 𝜑[𝑛] + 𝑇𝑠 ( 𝜔0 + ∆𝜔[𝑛])
(37)
𝑦 [𝑛 + 1] = 𝑦 [𝑛] + 𝑇𝑠 ( 𝐾𝑠 [𝑛] sin( 𝜑[𝑛]) + 𝐾𝑐 [𝑛] cos( 𝜑[𝑛] ) )
(38)
𝑒 [𝑛 + 1] = 𝑒[𝑛] + 𝑇𝑠 ( 𝑢[𝑛] − 𝑦[𝑛] )
(39)
References
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Wiley/IEEE Press, 2010
[2]. Remus Teodorescu, Marco Liserre and Pedro Rodriguez, “Grid Converters For Photovoltaic And Wind Power Systems”,
Wiley/IEEE Press, 2011
[3]. Frede Blaabjerg , Remus Teodorescu , Marco Liserre , Adrian V. Timbus , " Overview of Control and Grid Synchronization for
Distributed Power Generation Systems " , IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 5,
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