Recent Researches in Electric Power and Energy Systems
The Load Unbalance Influence on the Power Factor Value in ThreePhase Distribution Networks
DENISA RUSINARU1
LEONARDO-GEO MANESCU1
MARIUS MERFU2
1
Energy Department - University of Craiova
107 Decebal Bld., 200440-RO, Craiova, ROMANIA
drusinaru@elth.ucv.ro; leman78@hotmail.fr
2
Company ENERGOTECH
Bucharest, ROMANIA
office@electropowersystems.ro
Abstract: - The paper approached issues regarding the 3-phase unbalanced loads, supplied by 3-phase 3-wires
distribution networks. The model of an unbalanced load was developed by associating of two basic
components: a 3-phase balanced load and a two-phase equivalent one, which is taking over the asymmetry of
the load currents as a transversal unbalance. Some particularities of the model were outlined for the cases of
supplying with sinusoidal and symmetrical or asymmetrical voltages. In each case, the dependence of the
power factor value of the load’s unbalance level was determined.
Key-Words: - unbalanced load, equivalent model, voltage asymmetry, power factor
buses with unbalanced loads are separated from the
rest of the network and replaced by equivalent
current injection. The advantage of this model is a
more simple structure of the equivalent network
configuration since its structural asymmetry is
substituted by a currents and voltages asymmetry
[2].
Concerns to reduce computation time led to the d
of integrated unbalanced load consumer – supplying
substation models [3, 4].
In generally, the unbalanced load models should
account their operation effect on the power flowing
into the system or their parameters variation
compared to those of equivalent balanced
consumers. No matter the equations that describe
the terminal behavior of these consumers, they must
meet the minimum complexity and restrictive scope
for the analysis of unbalanced operation. These
models are used not only in the detailed analysis of
the asymmetrical operation of power systems, but
they have direct applications in disturbances’
measurement or mitigation, as the model presented
below.
1 Introduction
Regardless the nature of the electromagnetic
disturbances in the actual power systems they affect
simultaneously more types of power quality
indicators. In generally the sources of power quality
problems are located at two sites: one at the actual
consumers load and equipments and the other at the
power system components.
There are already registered important works
regarding the model development for each of these
system components in generally, as well as for the
consumer load in particularly. Regarding the
consumer loads, the former experience proved the
efficiency of the models P=constant or Z=constant
for ideal normal operation regimes, as well as for
the disturbed ones (e.g. harmonic or unbalanced).
Undeniably, the consumers have characteristics
that may influence unbalanced system that includes
them. An interesting feature was highlighted by
Professor A. Tugulea, which characterized the
behavior of the unbalanced loads as similar to that
of symmetrical components converters [1].
According to the theory of Professor Tugulea if
these loads are supplied from a high power source
that generates positive sequence emf’s, they will
receive power only on the positive symmetric
component.
Another approach allows simplifying the
equivalent negative and zero sequence schemes. The
ISBN: 978-960-474-328-5
2 Equivalent model for unbalanced
load
Any unbalanced load is a typical case of transversal
asymmetry and can be modelled as asymmetrical
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Recent Researches in Electric Power and Energy Systems
[U ] = [U
three-phase admittances star or delta connected.
According to the principle of disturbance
representation as a deviation from an ideal situation,
this model has also a counterpart [4, 5] as
symmetrical three-phase admittances equal to Ye
with an additional admittance Ym connected between
two phases considered references for the load
unbalance, as in Fig.1.
sim
U − ]t – the column vector
U+
of the voltage symmetrical components supplying
the load;
[Ysim ] -the symmetrical bus admittance of
the equivalent unbalanced load model;
Y e
[Ysim ] =  0
 0
Ye
IA
0
0
Ye +Ym
−Y m
0 
− Y m 
Y e + Y m 
(7)
I'
Ye
IB
I'
2.1 Three-phase unbalanced loads supplied
with symmetrical voltages
Im
Ym
Ye
IC
For this system’s supplying conditions the load
currents symmetrical components can be written as
in (8) outlining the connectivity between the
equivalent sequence schemes as is given in Fig.2.
I'
Fig.1 Equivalent model for unbalanced three-phase
loads in 3-phase 3-wire networks
 I + = (Y e + Y m )U +

 I − = −Y m U +
The correspondence between the two models is
given by the relation between the values in the buses
of the studied network:
[I n ] = [Yn,dez ]⋅ [U n ] = [Yn,echiv ]⋅ [U n ]
I+
(1)
U+
where [In], [Un] are column vectors of the currents,
respectively voltages in the terminal buses of the
load model;
[I n ]
not
=
[I
A
I B
I C
not
[U n ] = [U A
UB
UC
]t
]t
U-
0
YB
0
0
0 
Y C 
0
Ye +Y m
−Y m
0 
− Y m 
Y e + Y m 
(3)
sim
where
[I ] = [I
sim
0
I+
sim
sim
IYe
The dependency between the two unbalanced
load models is given in (9).
(
)
Y e = 2Y A − a 2 Y B − aY C / 3 = −a 2 Y B − aY C = Y A
(4)
(
)
Y m = − Y A + aY B + a 2 Y C / 3 =
a 2 (Y C − Y B )
a −1
(9)
2.2 Three-phase unbalanced loads supplied
with asymmetrical voltages
(5)
For the case of an unbalanced load supplied with
asymmetrical voltages in a 3-phase 3-wires network
the voltage sequence components can be written as:
The corresponding equation in symmetrical
components of (1) is given by (6):
[I ] = [Y ]⋅ [U ]
Ym
Fig.2 The connectivity between sequence schemes
of the equivalent unbalance load model in 3-phase
3-wires networks
[Yn,echiv] – the bus admittances matrix of the
equivalent unbalanced load model:
Y e
[Yn,echiv ] =  0
 0
Ye
(2)
[Yn,dez] – the bus admittances matrix of the initial
unbalanced load model:
Y A
[Yn ,dez ] =  0
 0
(8)
U 0 ≈ 0; U − = k U U +
(6)
where U + = U + ⋅ e j ⋅ϕ +
sequence component;
I − ]t is the column vector of
the load currents symmetrical components;
(10)
is the voltage positive
U − = U − ⋅ e j ⋅ϕ − - the voltage negative sequence
component
ISBN: 978-960-474-328-5
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Recent Researches in Electric Power and Energy Systems
kU – the complex voltage asymmetry factor:
kU =
U−
U+
= kU ⋅ e j ⋅(ϕ − −ϕ + )
with Ue, Ie – the equivalent voltage, respectively
current in 3-phase networks:
(11)
Ue =
The following relations between the bus values
at terminal of the load model result by replacing
(10) in the general equation (1):
nes
I A   Y A 
 I  = a 2 Y nes U
B 
+
 B 
 I C   aY Cnes 


(
not
(
; Ie =
I A2 + I B2 + I C2
3
(19)
Regarding the equivalent model of the
unbalanced load the powers’ repartition on the two
admittances components Ye and Ym, can give
additional useful information for the asymmetry
analysis of the power systems.
The power components and the corresponding
load unbalance factor are given for the two
supplying cases as following.
(12)
with:
not
2 +U 2 +U 2
UA
B
C
3
)
Y nes
A = 1+ kU Y A
)
2
Y nes
B = 1+ a kU Y B
3.1 Symmetrical supplying voltages
(13)
In the case of the 3-phase unbalanced load supplied
by symmetrical voltages, the two power components
associated to the equivalent model are:
not
nes = 1 + a k Y
YC
U C
(
)
)
(
S tot = 3 U + 2 (I +2 + I −2 )
The equivalent unbalanced load model keeps its
features for a frequent network operation case by
taking into consideration the influence of the
supplying voltages asymmetry.
The components of the equivalent unbalanced
load model can be written as:
Y enes = Y e + ∆Y e ; Y nes
m = Y m + ∆Y m
S m = j 3 ⋅ a ⋅ Y *B − Y *A U + 2
(21)
The load unbalanced factor results:
s = j 3 (s B − a s A )
(14)
(22)
with
with:
S A = Y *A ⋅ U + 2 ; S B = Y *B ⋅ a ⋅ U + 2 ; S C = Y *C ⋅ a 2 ⋅ U + 2
∆Y e = kU Y A
∆Y m = j
(20)
a 2 kU
3
(a2Y B − Y A )+ Y m
def S
def S
def S
A ;s
B ;s
=
= C
S tot B
S tot C
S tot
(15)
sA =
(23)
Sm/Stot consumator Y tens.sim.
Sm/Stot for Y load
the deviations of the equivalent unbalanced load
model related to the case of symmetrical voltages
supplying.
3. Power characteristics of the
equivalent unbalanced load model
Based on the equivalent model the load unbalance
factor s can be determined as [4]:
s=
Sm
S tot
SB/Stot
(16)
SA/Stot
where Sm is the equivalent power associated to the
biphasic admittance Ym of the equivalent load
model;
*
2
S m = Y m (U B − U C )
(17)
SC=ct.
Fig.3 Ratio Sm/Stot dependency on the phase powers
of the unbalanced load supplied with symmetrical
voltages
Stot – the total power absorbed by the load;
S tot = 3 ⋅ U e I e
ISBN: 978-960-474-328-5
The load unbalanced factor s meets a minimum
in the medium area of the values of SA/Stot and
(18)
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Recent Researches in Electric Power and Energy Systems
SB/Stot and defines the case of an almost equal
repartition of load on the three–phases of the
system, as it can be seen in Fig. 3.
SA/Stot and SB/Stot (high and almost equal loads on the
specified phases).
In the case of star connection if the power ratio
of the reference phase SA/SB is higher than 1 the
dependency of s=f(kU) is a direct one, and inverse
for a subunit ratio.
3.2 Asymmetrical supplying voltages
For the case of supplying with symmetrical
voltages, the two power components associated to
the equivalent unbalance load model are:
)
(U +2 + U −2 )(I +2 + I −2 )
(
*
nes*
2
S m = j 3 ⋅ a ⋅ Y nes
B −Y A U+
(24)
S tot = 3
(25)
4 The influence of power components
of the equivalent unbalanced load
model on the power factor
Additionally the dependency of the power factor
value on the power component Sm of the unbalanced
load model can be outlined.
Since the power factor is simultaneously
influenced by the load unbalanced and voltages
asymmetry, this information can be useful for
optimizing the measures and equipments for power
factor correction in distribution networks with
asymmetrical voltages.
The power factor characteristics are studied also
for the two cases of supplying voltages.
The load unbalanced factor becomes:
* (s − s )] |=
s =| j ⋅ 3 ⋅ [(s B − a s A ) + a k U
B
A
= S m + ∆S m
(26)
where ∆Sm is the deviation of the power on the load
component Y m equivalent related to the case of
supplying with symmetrical voltages:
* ⋅ (s − s )
∆ S m = j ⋅ 3 ⋅ a ⋅ kU
B
A
(27)
The load unbalanced factor s as a function of the
voltage asymmetry factor (Fig.4) has a minimum
depending on SB/Stot.
1
4.1 Symmetrical supplying voltages
The power factor is given as:
λP =
Sm/Stot pt.consumator Y cu tens.nesim
1
s1( a)
t1 ( a) 0.8
s(nes)
0.6
t3 ( a)
λp =
sB=0.3
u3( a)
U +2 (I +2 + I −2 )
sB=0.1
0.2
0.4
0
kU=0.12
cos ϕ UI+
1 + k I2
(29)
)
+ - the natural power factor of the supplying
cos ϕUI
0
0
=
+ = ϕ + − ϕ + is the phase shift between
where
ϕUI
U
I
positive sequence components of the voltage and
current;
t5 ( a)
0
U + I + cos ϕ UI+
(
0.4
s5( a)
u5( a) 0.2
(28)
By expressing the two powers as functions on the
symmetrical components of the currents and
voltages, the power factor results as:
sB=0.5
u1( a)
s3( a)
P
S tot
0.6
a
SA/Stot
kU=0.22
0.8
network;
1
1
kI =
kU=1.22
Further by writing the power components of the
equivalent unbalanced load model as a function of
the current sequence components, the power factor
can be written as in (30) and plotted in Fig. 5.
Fig.4 Ratio s=Sm/Stot dependency on the phase
powers of the unbalanced load supplied with
asymmetrical voltages
+ 1 − s( s , s ) 2
λP = cosϕUI
A B
In the area of the low values of SA/Stot and SB/Stot
(low and almost equal loads on the specified phases)
s is inverse dependent on kU . On the other hand, s
increases with kU in the area the high values of
ISBN: 978-960-474-328-5
I−
- the factor of negative current asymmetry.
I+
with:
317
(30)
Recent Researches in Electric Power and Energy Systems
[
not
]
the area with values greater than the critical ratio
and increases for the field of inferior values.
2 − 2 s s cos(ϕ − ϕ + 2π / 3)
3 s 2A + sB
A B
A
B
s ( s A , sB ) =
(31)
where ϕ A ,ϕ B are phases of the power vectors SA ,
respectively SB.
de putere voltages
tens.nesim.
asymmetrical
λP forFactor
Factor
de putere
Power
factor
in case of
λP tens.sim
symmetrical voltages
kU
cosϕ+, cosϕ- = ct.
s
cosϕ+
λ
g1p, g1
Fig.6 The power factor dependency on the ratio
Sm/Stot in case of asymmetrical supplying voltages
λp
kp
Fig.5 The power factor dependency on the ratio
Sm/Stot in case of symmetrical supplying voltages
5 Conclusion
It can be observed that the power factor
decreases while the load unbalance factor increases
and is as lower as the natural power factor of the
network (cosϕUI+) is lower.
This paper deals with the modeling aspects of the
power network components to determine the
propagation of voltage asymmetry in electrical
distribution networks.
The presented model of the unbalanced threephase load is based on representation in terms of
phase admittance and is further developed by using
an equivalent single-phase admittance load
superimposed on three symmetrical admittances that
perfectly describe the real load asymmetry. The
model is developed both in phase coordinates and
symmetrical components, by considering the supply
voltages as sinusoidal.
One of the applications of this unbalanced load
model is the sizing of the equipments for load
currents symmetrization [4, 7, 8]. By using these
compensators the control of the power components
is achieved in the connection bus of the load.
The unbalanced load model can adjusted so that
it can be applied for analysis of asymmetrical fault
operations or to study the behavior of transformers
with special connections (connection open-delta or
directly between two phases) used for supplying of
unbalanced large industrial consumers such as
electric railway traction.
4.2 Asymmetrical supplying voltages
In this case the dependency of the power factor on
the voltage asymmetry (32) is additionally
influenced by the load distribution on the phases as
its can be seen in (33).
U + I + cos ϕ UI+ + U − I − cos ϕ UI−
λp =
(U
2
+
+ U −2 )(I +2 + I −2 )
=
cos ϕ UI+ + k U k I cos ϕ UI−
(1 + k )(1 + k )
2
U
2
I
(32)
The resulting equivalent equation in per units
(33) was graphically drawn as is Fig. 5.
+ 1 − s 2 (s , s , k ) +
λ P = cos ϕUI
A B U
− ⋅ k ⋅ s (s , s , k )
+ cos ϕUI
U
A B U
(33)
with:
not
∆s m =
∆S m
= ∆s m e jϕ ∆sm
S tot
(34)
References:
[1] A. Tugulea, Consderations regarding the
energy effects in asymmetrical nonsinusoidal
regimes of the power systems (In Romanian),
The power factor value meets an inflection point
in the medium area of the ratio SB/Stot. Therefore the
power factor decreases for a higher asymmetry in
ISBN: 978-960-474-328-5
318
Recent Researches in Electric Power and Energy Systems
[2]
[3]
[4]
[5]
ENERG, vol.3, Technica Publ., Bucharest,
1987.
A.A. Arie, N. Golovanov, C. Dumitriu,
Calculus of the asmmetric and nonsinusoidal
regime produced by the electrical railways in
power system, Energetica 34, 1986, no.11,
pp.510-517.
T.H. Chen, Y.L. Chang, Integrated Models of
Distribution Transformers and Their Loads for
Three-Phase Power Flow Analysis, IEEE
Trans. on Power Delivery, vol.11, no.1,
Jan.1996, pp.507-513.
D. Ruşinaru D., Unbalanced permanent
operation of the power grids (in Romanian),
Universitaria Publ., Craiova, 2005.
D.
Ruşinaru,
C.
Bratu,
Optimized
Compensator’ Configurations for Load Current
Symmetrization, Iasi Politechnic Bulletin, LIV
(LVIII), fasc. 3, 2008, pp.131-138.
ISBN: 978-960-474-328-5
[6] IEEE Working Group on Non-sinusoidal
Situation, Practical Definitions on Power in
Systems with Nonsinusoidal Waveforms and
Unbalanced Loads: Effects on Meter
Performances, IEEE Trans. on Power Delivery,
vol.11, no1, Jan. 1996, pp.79-101.
[7] L.S. Czarnecki, Supply and Loading Quality
Improvement in Sinusoidal Power Systems
with Unbalanced Loads Supplied with
Asymmetrical
Voltage,
Archiv
fűr
Electrotechnik 77, Springer-Verlag 1994,
pp.169-177.
[8] J.C. Wang a.o., Capacitor Placement and Real
Time Control in Large-Scale Unbalanced
Distribution
Systems:
Loss
Reduction
Formula, Problem Formulation, Solution
Methodology and Mathematical Justification –
IEEE Trans. on Power Delivery, vol.12, no.2,
April 1997, pp.953-958.
319