1
An Instantaneous Phase Angle Detection Algorithm
under Unbalanced Line Voltage Condition
Hong-seok Song, Hyun-gyu Park, and Kwanghee Nam
Department of Electrical Engineering, POSTECH University
Hyoja San-31, Pohang, 790-784 Republic of Korea
Tel:(82)562-279-2218, Fax:(82)562-279-5699,
E-mail:kwnam@postech.ac.kr
Abstract {If a negative sequence is generated by voltage sag
and/or unbalance, it appears as an oscillating error in a synchronous reference frame(SRF). In the power conditioning
equipment, the exact value of positive sequence is needed for
achieving the desired goal of the system, e.g., unity power
factor and constant output voltage, whereas the exact value
of negative sequence is needed for compensation. To measure the positive sequence separately from the negative sequence, one uses normally a low pass lter having a narrow
bandwidth. But, such a lter causes a lot of phase delay or
measurement delay, thus the response time of the system
tends to be lengthened. We propose a method of estimating
the positive and the negative sequence voltages separately
without a signicant delay by utilizing the weighted leastsquares estimation (WLSE) method having the covariance
resetting technique. We demonstrate through simulation
and experiment the superior performance of the proposed
scheme in measuring the positive and the negative sequence
voltages at the time of abrupt transition. This method can
be applied to UPS, PWM AC/DC converter, active lter,
series voltage compensator, etc.
I. Introduction
The phase angle of a line voltage is used to calculate and
control the ow of active/reactive power, and transform
the feedback variables to a synchronous reference frame
(SRF) [1][2]. The phase angle, therefore, is critical piece of
information for the operation of most power conditioning
equipment such as: controlled AC/DC converter, UPS, series voltage compensator, static VAR compensator, active
harmonic lter, etc. If a voltage sag takes place in one or
two phases in a three-phase power system, it causes voltage unbalance by generating a negative sequence voltage.
The voltage unbalance induces an oscillating error in the
measurement of the phase angle. In the power conditioning
equipment, if there is an error in the phase angle estimation then it may directly aect the compensation voltage,
and thus deteriorate the performance. To avoid the performance deterioration, it is necessary to nd the positive
and the negative sequence voltage instantaneously. However, it is normally dicult to estimate the positive and the
negative sequences separately from their summed values.
Enjeti and Chaoudhury [3] measured the positive sequence voltage by using a lter and they used the phase
angle of the positive sequence under the voltage unbalance
condition. However, the use of a lter typically generates some measurement delay or phase delay so that an
AC voltage compensator looses half or several cycles in the
transient state. Haddad and Joos [4] obtained the phase
angle of each phase voltage in a three-phase voltage from
two sampled values in a short time interval. They used
the phase angles for three single-phase AC voltage compensators. But, in their scheme the relationships among
the phases in a three-phase system were neglected. Note
that the phase dierences among three phases under voltage sag and/or unbalance conditions are not generally even
[2]. Thus, we cannot compensate completely voltage unbalance or sag with such a phase detection scheme under
general voltage sag and/or unbalance conditions.
In this work, we propose an instantaneous phase angle detection algorithm under voltage sag and/or unbalance conditions using the weighted least-squares estimation
(WLSE) method having the covariance resetting technique.
With the proposed detection method, one can nd the positive and the negative sequence voltage almost without a
delay, and thus obtain the instantaneous phase angle.
II. Effects of a Voltage Unbalance
on Phase Measurement
An unbalanced three-phase line voltage fEa ; Eb ; Ec g
without zero sequence can be represented as the orthogonal
sum of a positive and a negative sequence such that
p
n (t );
Edqs (ti ) = ej!t Edqe
(ti ) + e,j!t Edqe
(1)
i
i
i
p
=
where Edqs = 23 [Ea + Eb ej2=3 + Ec e,j2=3 ], Edqe
p
p , E n = E n + jE n , and ! 120 (60Hz) is
Ede
+ jEqe
qe
dqe
de
the angular frequency. The subscripts s and e denote the
quantities in the stationary frame and in the SRF of !ti ,
respectively. The superscripts p and n denote the quantities of positive and negative sequences, respectively. Note
that one can ignore the eects of a zero sequence voltage
in a three-wired three-phase system, since a zero sequence
voltage does not appear in a three-wired three-phase system. Multiplying e,j!t to (1), we measure the phase angle
oset, '(ti ):
,
'(ti ) = ] e,j!t Edqs (ti ) ;
p
n (t ) ;
= ] Edqe
(ti ) + e,2j!t Edqe
i
= (ti ) + ~(ti );
(2)
p
where (ti ) ]Edqe
(ti ) denotes the phase angle of the
positive sequence voltage in the SRF of !ti and ~(ti ) i
i
i
2
(pos. seq. volt. axis)
ϕ(ti )
ω ti +ϕ (ti )
(unbalanced
volt. axis)
Ea
Stationary
Q-axis
ω ti + φ (ti )= θ (ti )
~
φ(ti)
Ea
Ec
Eb
Ec
arbitrary
SRF (60Hz)
(50V./div)
(50V./div)
φ (ti )
ω ti
Stationary
D-axis
ϕ
ϕ
Fig. 1. Phase relationship of (ti ), ~(ti ), and '(ti ).
'(ti ) ,
Eb
ωt+ϕ
p
]Edqe
(ti ) denotes the oscillatory phase error caused
by the negative sequence. Fig. 1 shows the relationship
among (ti ), ~(ti ), and '(ti ). It should be noted that
~(ti ) contains 120Hz AC ripple.
Under the normal balanced line voltage condition,
'(ti ) = (ti ) and ~(ti ) = 0, i.e., '(ti ) contains only
DC component. However, under voltage sag and/or unbalance conditions, '(ti ) contains AC component, since
~(ti ) 6= 0. Fig. 2 shows the eect of the unbalanced voltage on the phase angle measurement. Plots in column(a)
show the three-phase line voltage, fEa ; Eb ; Ec g, '(ti ) and
!ti + '(ti ), when a balanced three-phase voltage is applied. On the other hand, plots in column(b) show the
same things, when the three-phase voltage is unbalanced.
Plots in column(b) show clearly that the negative sequence
n resulting from the voltage unbalance causes
voltage Edqe
an 120Hz oscillating error ~(ti ) in the phase angle measurement.
For power conditioning equipment such as voltage compensator, UPS, active lter, etc., if one uses the distorted
phase angle without taking account of the eects of the
voltage sag and/or unbalance, then it results in the phase
angle error so that the output performance is degraded
correspondingly. For the complete compensation for voltage sag and/or unbalance, one may need to identify (ti ).
When one intends to measure (ti ) by using a conventional
PLL or a lter, there exists an inherent problem: to obtain
DC signal (ti ), one should employ a low pass lter (LPF)
having a narrow bandwidth enough to eliminate ~(ti ). But,
the LPF having the narrow bandwidth generates a lot of
phase delay or measurement delay, and causes sluggish response. Thus, these methods cannot be good solutions for
voltage compensator and UPS in which fast response is essential. In this work, we solve this problem by using the
WLSE method having the covariance resetting technique.
III. Instantaneous Phase Angle
Detection Algorithm using the WLSE
(60deg./div)
(60deg./div)
ωt+ϕ
(60deg./div)
(4msec./div)
(a)
(b)
(60deg./div)
(4msec./div)
Fig. 2. Phase angle measurement when a line voltage is (a) balanced,
(b) unbalanced.
pressing (1) in the matrix form, we obtain
y(i) = H (i) x(ti );
(3)
where
!ti ) , sin(!ti ) cos(!ti ) sin(!ti ) ;
H (i) cos(
sin(!ti ) cos(!ti ) , sin(!ti ) cos(!ti )
p
p (t ) E n (t ) E n (t ) ]T ;
x(i) [ Ede
(ti ) Eqe
i
qe i
de i
y(i) Eds (ti ) :
Eqs (ti )
We choose the cost function such that:
J [x(i)] =
i
X
k=0
,k y(k) , H (k)x(k)
T
y(k) , H (k)x(k) + 0,1 x(k)T x(k); (4)
where 0 > 0 is the initial covariance constant and
2 (0; 1) is the forgetting factor. The solution x^(i) that
minimizes the cost function J [x(i)] is obtained by the following the least-squares algorithm:
x^(i) = x^(i , 1) + kp;i [y(i) , H (i)^x(i , 1)] ;
(5)
kp;i = P i,1 H (i)T re (i),1 ;
re (i) = I + H (i)P i,1 H (i)T ;
P i = ,1 P i,1 , ,1 P i,1 H (i)T re (i),1 H (i)P i,1 :
x^(,1) = 0, P ,1 = 0 I 2 IR44 . From (5), we obtain
In this section, we develop an estimation algorithm for where
the phase angle oset of the positive sequence voltage:
positive and negative sequence voltages by applying the
weighted least-squares estimation (WLSE) method. Ex-
p
^(ti ) = tan,1 [E^qep (ti )=E^de
(ti )]:
(6)
3
Ea [V]
80
no
yes
covariance
resetting
_
P i-1 = π 0
_
_
T
re(i) = I + H(i) P i-1 H(i)
_
_ -1
T
kp,i = P i-1 H(i) re(i)
_
_
_
_
-1
-1
T_
-1
Pi = λ Pi-1 - λ Pi-1 H(i) re(i) H(i) Pi-1
-80
80
Eb [V]
^
|| y(i) - H(i) x(i-1)
|| > ε
-80
80
Ec [V]
Measure Edqs ( t i ) ( = y(i) )
-80
^
^
-1 ^
φ (ti ) = tan [ Eqe(ti ) / Ede(ti )]
_
_
^
^
= x(i)
Pi-1 = Pi , x(i-1)
Fig. 3. Flowchart of the instantaneous phase detection algorithm.
Then, the phase angle of the the line voltage is obtained
such that ^(ti ) = !ti + ^(ti ).
To enhance the tracking speed of the phase angle jump
caused by sudden voltage sag and/or unbalance, we add
the covariance resetting technique. The existence of a
phase angle jump is discerned from the normal condition
by the estimation error magnitude. That is, if the estimation error is larger than a given boundary value ", i.e.,
ky(i) , H (i)^x(i , 1)k > ", then we reset P i,1 with the
initial covariance 0 I . To avoid frequent resetting caused
by noise under the normal condition, the boundary value
" needs not to be very small. A heuristic guide line is
to set " equal to 20 40% of the peak amplitude of the
nominal line voltage. Note also that with the WLSE, one
can increase noise immunity of the estimator by selecting a
larger . Fig. 3 shows the owchart of the proposed phase
detection algorithm.
IV. Simulation and Experimental Results
To verify the feasibility of the proposed phase angle
estimation scheme, computer simulation study was conducted. Fig. 4(a) shows an example line voltage used in
the simulation study and in experiments. Fig. 4(b) shows
its frequency spectrum. A balanced three-phase voltage
fEa ; Eb ; Ec g = f60 cos(!t)+6 cos(5!t); 60 cos(!t,120)+
6 cos(5!t , 126); 60 cos(!t +120)+6 cos(5!t +120)g was
applied for t = 250msec. Note that we added 5th harmonics (300Hz) intentionally in order to check the robustness of
FFT [pu]
^
^
^ = x(i-1)
x(i)
+ kp,i [ y(i) - H(i) x(i-1)
]
1
(a)
(50msec./div.)
Noise
(300Hz, 10% amplitude)
(100Hz/div.)
0
(b)
Fig. 4. Three-phase voltage used in the simulation study and the
experiments: (a) its waveforms and (b) the frequency spectrum.
the estimation algorithm against noise. After t = 250msec,
an unbalanced three-phase voltage f59 cos(!t + 31:3) +
6 cos(5!t); 33 cos(!t , 62:9)+6 cos(5!t , 126); 66 cos(!t +
181:1) + 6 cos(5!t + 120)g was applied in a step manner.
Note that the 5th harmonics are observed in the spectrum
shown in Fig. 4(b).
Fig. 5 shows the simulation results of the estimation with
" = 18V , fs = 3kHz. Plots in columns (a)-(d) show the
estimation results with f; 0 g = f0:95; 100g, f0:95; 10g,
f0:85; 100g and f0:85; 10g, respectively. Plots in the rst
and second row show ^ and ^ = !t + ^, respectively. Plots
p ^p
n; E
n g,
^de
in the third and forth row show fE^de
; Ede g and fE^de
respectively. As shown in the simulation results, we can
p
n; E
n g almost
^qe
separately measure fE^de
; E^qep g and fE^de
without a delay, and obtain the instantaneous phase angle of the positive sequence voltage even under voltage sag
and/or unbalance conditions.
To show the characteristics of the estimation with respect to the forgetting factor , the initial covariance constant 0 and the sampling frequency fs , we performed the
simulation study with varying , 0 and fs . Fig. 6 shows
the convergence time t with respect to , 0 , and fs . The
convergence time t is dened by the required time for the
estimated phase oset ^ to reach the real phase oset after
a sudden phase angle change. Note that t is signicantly
aected by 0 . Fig. 7 shows the oscillation error of the estimated phase oset, caused by the 5th harmonic noise,
when we add 10% 5th harmonic disturbance. Note that
is signicantly aected by . In other words, the use
4
[deg]
360
^
φ
^
φ
^
φ
^
φ
^
θ
^
θ
^
θ
^
θ
0
[deg]
360
0
80
[V]
^p
Ede
^p
Eqe
^p
Ede
0
^p
Eqe
^p
Eqe
^p
Ede
^p
Ede
^p
Eqe
-80
80
[V]
^n
Eqe
^n
Eqe
^n
Eqe
^n
Eqe
^n
Ede
^n
Ede
^n
Ede
^n
Ede
0
-80
(a)
(10msec./div.)
(b)
(10msec./div.)
(c)
(10msec./div.)
(d)
(10msec./div.)
Fig. 5. Simulation results of phase angle detection under the voltage sag and/or unbalance condition with f; 0 g set at (a) f0:95; 100g,
(b) f0:95; 10g, (c) f0:85; 100g and (d) f0:85; 10g.
Fig. 6. Simulation results of the convergence time t with respect Fig. 7. Simulation results of the oscillation error caused by the
5th harmonic disturbance with respect to and fs .
to , 0 and fs .
of the larger 0 presents the faster convergence, i.e., the
smaller t, and the use of the larger presents the smaller
. For example, t = 780sec: and = 0:7, when we
adopt = 0:95, 0 = 100 and fs = 3kHz.
For the experiments, the control board was made with
the TMS320C31 DSP processor. The unbalanced voltage
sag was made by a programmable AC power supply. The
voltage limit of the AC power supply prevented us from
increasing the magnitude of the line voltage more than 80V.
Fig. 8 shows the experimental results of the estimation with
fs = 3kHz and " = 18V . Plots in row (a) show ^, ^,
fE^dep ; E^qep g, and fE^dep ; E^qep g with 0 = 100, = 0:95.
Plots in row (b) show the same things with 0 = 10, =
0:95. The overall shapes are quite similar to the simulation
results. One can clearly see that the proposed method
can estimate the instantaneous phase angle of the positive
sequence under voltage sag and/or unbalance conditions.
V. Concluding Remarks
We proposed a phase angle detection algorithm that
acted promptly when a sudden voltage sag and/or unbalance took place. The proposed algorithm was derived from
5
80
780usec
80
0
^n
Ede [V]
φ^ [deg]
^p
Ede [V]
360
-80
80
(a)
-80
80
0
(10msec./div.)
-80
^ n [V]
Eqe
θ^ [deg]
^ p [V]
Eqe
360
(10msec./div.)
80
1.6msec
-80
(10msec./div.)
80
0
^n
Ede [V]
φ^ [deg]
^p
Ede [V]
360
-80
80
(b)
-80
80
0
(10msec./div.)
-80
^ n [V]
Eqe
θ^ [deg]
^ p [V]
Eqe
360
(10msec./div.)
-80
(10msec./div.)
Fig. 8. Experimental results of phase angle detection under voltage sag/unbalance conditions with f; 0 g set at: (a) f0:95; 100g, (b)
f0:95; 10g.
the weighted least-squares estimation method having the
covariance resetting technique. It enables us to measure instantaneously both the positive and the negative sequences
separately. In many applications, the estimated phase angle of the positive sequence will be used for setting reference frame, and the estimated negative sequence component will be used for unbalance compensation. The proposed method can be distinguished from the conventional
lter technique by the fast transient response. Hence, it
is thought to be suitable to UPS, active lter, or voltage
compensator in which the fast response is essential. Technically, one can increase the robustness to noise by choosing a larger and obtain the faster convergence speed by
choosing the larger 0 . According to simulation study, recommended parameters are 2 [0:9; 1) and 0 2 [10; 1000].
We have demonstrated the superior performance through
computer simulations as well as through experiments.
References
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