JET  23
JET Volume 15 (2022) p.p. 23-50
Issue 1, June 2022
Type of article: 1.01 
www.fe.um.si/si/jet.html
THE IMPACT OF UNBALANCED POWER 
SUPPLY ON LOAD CURRENTS IN TRAN-
SIENT AND STEADY-STATE OPERATION
VPLIV NESIMETRIČNIH NAPAJALNIH 
NAPETOSTI NA TOKE BREMEN MED 
USTALJENIM OBRATOVANJEM IN MED 
PREHODNIMI POJAVI
Nina Štumberger
1
, Matej Pintarič
2
, Gorazd Štumberger
2 ℜ
Keywords: unbalanced power supply voltages, load currents, steady state, transients, Dommel’s 
method, increase in released heat Abstract
Abstract
The impact of unbalanced power supply voltages on power derating, increases in current values 
and the heat released from electrical machines and devices is typically analysed in quasi-steady-
state operation using sequence components. The ratios between the negative and positive se-
quence voltage and the homopolar (zero) and positive sequence voltage are used to evaluate 
the level of voltage unbalance. This paper proposes an extension of this approach that includes 
subtransient, transient and quasi-steady-state operation of wye (grounded and ungrounded) and 
delta connected loads, consisting of resistors and magnetically linear and non-linear iron core 
inductors, which is a novelty. Dommel’s method for dynamic simulation of electrical circuits, 
 
ℜ Corresponding author: Professor dr. Gorazd Štumberger, Tel.: +386 (0)2-220-7075, Mailing address: Koroška cesta 
46, 2000 Maribor, Slovenia. E-mail address: gorazd.stumberger@um.si, nina.stumberger@rwth-aachen.de
1
 RWTH Aachen University (Rheinisch Westfälische Technische Hochschule), Fakultät für Elektrotechnik und Infor-
mationstechnik, Schinkelstr. 2, 52062 Aachen, Germany
2
 University of Maribor, Faculty of Electrical Engineering and Computer Science, Koroška cesta 46, 2000 Maribor, 
Slovenia
Nina Štumberger, Matej Pintarič, Gorazd Štumberger
Method of the best available technology and low carbon future of a combined heat and power plant

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Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
based on the modified nodal potential method, time discretisation and implicit numerical inte-
gration, is applied to evaluate the impact of unbalanced power supply voltages on load currents. 
The magnetically non-linear behaviour of the iron core inductors is accounted for by introducing 
variable dynamic inductances inside Dommel’s method. The integrals of instantaneous values 
of squared load branch currents values in subtransient, transient and quasi-steady-state opera-
tion, determined for different levels of unbalance in supply voltages, are normalised with those 
calculated for the balanced power supply voltages in quasi-steady-state. The obtained ratios of 
the integrals are used to implicitly evaluate the impact of unbalance in supply voltages on the 
increase in released heat due to Joule losses. The proposed approach is generally applicable, 
while the results are shown for the case study of iron core inductors with magnetically linear and 
non-linear behaviour.
Povzetek
Vpliv nesimetrije v sinusnih napajalnih napetostih na zmanjšanje moči, povečanje tokov in Joul-
skih izgub električnih strojev in naprav običajno obravnavamo v kvazi stacionarnih (vnihanih) 
stanjih, pri tem pa uporabljamo simetrične komponente. Merilo nesimetrije sta najpogosteje 
razmerji med napetostjo negativnega in pozitivnega ter ničnega in pozitivnega zaporedja si-
metričnih komponent. To delo predstavlja razširitev opisanega pristopa saj vključuje subrtan-
zientno, tranzientno in kvazi stacionarno območje delovanja v zvezdo in v trikot vezanih bremen, 
sestavljenih iz zaporedne vezave upora in tuljave z železnim jedrom z magnetno linearnim in 
magnetni nelinearnim obnašanjem. To je novost, s katero ta prispevek presega obstoječe pris-
tope. Za ovrednotenje vpliva nesimetrije v sinusnih napajalnih napetostih na toke bremen je 
uporabljena Dommel-ova metoda dinamične simulacije električnih vezij, ki temelji na modifici-
rani metodi vozliščnih potencialov, implicitni metodi numerične integracije povezane s časovno 
diskretizacijo. Magnetno nelinearno obnašanje tuljave z železnim jedrom je upoštevano z upora-
bo spremenljive dinamične induktivnosti, ki je smiselno integrirana v Dommel-ovo metodo. Inte-
grali kvadratov trenutnih vrednosti tokov bremen, so izračunani za subtranzientno, tranzientno 
in kvazi stacionarno obratovanje in določeni za različne stopnje nesimetrije v sinusnih napajalnih 
napetostih. Za potrebe primerjave so podeljeni z vrednostmi integralov kvadratov tokov bremen 
pri napajanju s simetričnimi sinusnimi napetostmi pri kvazi stacionarnem obratovanju. S tem smo 
dobili normirano obliko, ki implicitno omogoča ovrednotenje vpliva nesimetrije v sinusnih napa-
jalnih napetostih na toploto sproščeno v obliki Joulskih izgub. Predstavljena metoda je splošno 
uporabna in jo je mogoče generalizirati, njeno delovanje pa je v tem delu predstavljeno na prim-
eru tuljav z železnim jedrom z magnetno linearnim in z magnetno nelinearnim obnašanjem.
1 INTRODUCTON
With a balanced three-phase power supply, the voltages are sinusoidal, of equal amplitude and 
phase-shifted to each other by 120°. An uneven distribution of loads over the three phases or 
errors in the electrical network, such as short circuits, other faults and different unexpected op-
erating conditions, can cause voltage unbalances [1]. The unbalances in voltage supply are usu-
ally treated in quasi-steady-state operation of electrical machines and devices using sequence 
components – the positive (1), the negative (2) and the homopolar (0) component. The level of 
supply voltage unbalances is evaluated through the ratios of the negative and the homopolar 
sequence component with respect to the positive one [1], [2].
The supply voltage unbalances can negatively affect the equipment connected to the power sup-
ply due to an increase in line currents. This increase causes an increase in Joule losses, which 

JET  25
Method of the best available technology and low carbon future of a combined heat and power plant
increases the operating temperature and can cause deterioration of the isolation and negatively 
affect the lifespan of the equipment [1]. Most of the authors of existing research have focused 
on the impact of supply voltage unbalances on the quasi-steady-state operation of induction 
machines. The impact of angle unbalance in supply voltages on operational properties of induc-
tion machines is investigated in [3]. The authors in [4, 5] deal with the performance analysis of 
induction machines supplied with unbalanced sinusoidal voltages in quasi-steady-state opera-
tion. The influence of unbalanced sinusoidal and balanced non-sinusoidal supply voltages on the 
performance of a squirrel-cage induction motor is analysed in [6]. The impact of supply voltage 
unbalances on the precise derating and energy performance of three-phase induction motors is 
discussed in [7] and [8] respectively. The modelling effect of supply voltage unbalances in sys-
tems with adjustable speed drives is described in [9]. The author in [10] analyses the influence 
of supply voltage unbalances, combined with over voltages and under voltages, on induction 
machine winding temperature and loss of insulation life. In [11], the authors deal with thermal 
effects in the induction machine stator caused by mechanical overloads and supply voltage un-
balances. Calorimetric and finite element simulations are applied in [12] to evaluate the impact 
of supply voltage unbalances on induction motor derating.
Transient operation and magnetically non-linear load behaviour are not considered in any of 
the aforementioned publications. Although the loads considered in this paper are iron core 
inductors, the proposed approach is new. It enables the treatment of magnetically linear and 
non-linear behaviour, including subtransient, transient and quasi-steady-state operation. Using a 
method similar to the one described in [13], the approach proposed in this paper can be further 
developed to be suitable for treating iron core inductors and electrical machines.
The aim of this paper is to propose a unified method for evaluating the impact of supply voltage 
unbalances on the increase in currents and Joule losses of wye and delta connected loads, con-
sidering the magnetically linear or non-linear behaviour of the iron core of the load, including 
subtransient, transient and quasi-steady-state operation.
This paper deals with the impact of supply voltage unbalances on the currents and Joule losses 
of a three-phase load consisting of iron core inductors connected into the delta, ungrounded 
and grounded wye. Voltage unbalances of up to 10% are considered. To observe the currents 
in the transient states as well as in quasi-steady-state operation, the load as well as the supply 
line are modelled using Dommel’s method and simulated in MATLAB. The load model represents 
iron core inductors with magnetically linear and non-linear behaviour. The iron core saturation is 
considered by introducing variable dynamic inductance, the value of which decreases when the 
current reaches a certain threshold. Since the aim of this paper is to explore the effect of supply 
voltage unbalances on a general load, no specific equipment is considered, although this is an 
area that could be explored in the future.
The theoretical background for evaluating the supply voltage unbalance and Dommel’s method 
are described in Section 2. Section 3 explains the proposed methodology. The results obtained 
considering different levels of unbalances in supply voltages in the cases of wye and delta con-
nected iron core inductors with magnetically linear and non-linear behaviour are illustrated in 
Section 4. In Section 5, the findings are summarised and conclusions are drawn.

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2 THEORETICAL BACKGROUND
Subsection 2.1 describes the relations among the phasors of phase voltages and sequence com-
ponent voltages. The latter are used to introduce two factors for evaluating unbalances in sinusoi-
dal supply voltages. Subsection 2.2 provides a comprehensive description of Dommel’s method.
2.1 Voltage unbalances
Whilst the degree of voltage unbalance can be described in multiple ways, this paper uses 
the definition of the International Electrotechnical Commission (IEC), which can be found 
in [2]. It describes how much negative (2) and zero-sequence (0) voltage is superposed on 
the positive-sequence (1) voltage. The connection between positive-sequence (V
1
), zero-se-
quence (V
0
), and negative-sequence (V
2
) voltages and the phase voltages (V
a
, V
b
, V
c
) is de-
scribed by equation (2.1):
The impact of unbalanced power supply on load currents in transient and
steady state operation
5
----------
[
V
 a
V
 b
V
 c
]
= 
[
1 1 1
1 a
2
a
1 a a
2 ]
.
[
V
 0
V
 1
V
 2
]
,      (2.1)
where a=exp
(
j2
π
3
)
 [4].
The voltage unbalance factor (VUF) is defined by the ratio between the negative-sequence and
positive sequence voltage, as given by (2.2):
(2.1)
where
The impact of unbalanced power supply on load currents in transient and
steady state operation
5
----------
[
V
 a
V
 b
V
 c
]
= 
[
1 1 1
1 a
2
a
1 a a
2 ]
.
[
V
 0
V
 1
V
 2
]
,      (2.1)
where a=exp
(
j2
π
3
)
 [4].
The voltage unbalance factor (VUF) is defined by the ratio between the negative-sequence and
positive sequence voltage, as given by (2.2):
[4].
The voltage unbalance factor (VUF) is defined by the ratio between the negative-sequence and 
positive sequence voltage, as given by (2.2):
6 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
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 VUF=
V
2
V
1
.   (2.2)
Along with the delta and ungrounded wye connection of the load, where zero-sequence voltage
unbalance has no effect on the load currents, this paper also deals with the grounded wye
connection of the load. Therefore, the zero-sequence voltage unbalance factor (ZSVUF), defined
by (2.3): 
(2.2)
Along with the delta and ungrounded wye connection of the load, where zero-sequence 
voltage unbalance has no effect on the load currents, this paper also deals with the ground-
ed wye connection of the load. Therefore, the zero-sequence voltage unbalance factor 
(ZSVUF), defined by (2.3):
The impact of unbalanced power supply on load currents in transient and
steady state operation
7
----------
ZSVUF= 
V
0
V
1
   (2.3)
is used.
2.2 Dommel’s method
In order to analyse the transient-state currents, as well as the quasi-steady-state currents, the
inductors representing the loads and the connection lines were modelled using Dommel's
method, as described in [14]. The basis for this model is the equation for the inductor voltage
u
L
given by (2.4): 
(2.3)
is used.
2.2 Dommel’s method
In order to analyse the transient-state currents, as well as the quasi-steady-state currents, the 
inductors representing the loads and the connection lines were modelled using Dommel’s meth-
od, as described in [14]. The basis for this model is the equation for the inductor voltage U
 L
 given 
by (2.4):
8 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
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u
L
(t)= v
k
(t)- v
m
(t)=L
d i
km
dt
,                                                                                           (2.4)
from which the equation for the inductor current i
km
, given in equation (2.5)
(2.4)
from which the equation for the inductor current i
km
, given in equation (2.5)

JET  27
Method of the best available technology and low carbon future of a combined heat and power plant
The impact of unbalanced power supply on load currents in transient and
steady state operation
9
----------
i
km
(t )= i
km
(t-∆t )+
1
L
∫
t-∆t
t
( v
k
( )- v
m
( ) ) d                                                                   (2.5)
is derived by integration over a time period ∆t . t is the current point in time,   is an
auxiliary variable, L is the inductance of the inductor while v
k
(t) and v
m
(t) are
the potentials on either end of the inductor, as shown in Figure 1.
The integral is approximated using the trapezoidal rule, given by (2.6):
Figure 1: Inductor with defined potentials and inductor current
(2.5)
is derived by integration over a time period Δt. t is the current point in time, τ is an auxiliary vari-
able, L is the inductance of the inductor while v
k
(t) and v
m
(t) are the potentials on either end of 
the inductor, as shown in Figure 1.
The impact of unbalanced power supply on load currents in transient and
steady state operation
5
----------
2.2 Dommel’s method
In order to analyse the transient-state currents, as well as the quasi-steady-state currents, the
inductors representing the loads and the connection lines were modelled using Dommel's
method, as described in [14]. The basis for this model is the equation for the inductor voltage
u
L
given by (2.4): 
u
L
(t)= v
k
(t)- v
m
(t)=L
d i
km
dt
,                                                                                           (2.4)
from which the equation for the inductor current i
km
, given in equation (2.5)
i
km
(t )= i
km
(t-∆t )+
1
L
∫
t-∆t
t
(
v
k
( )- v
m
( )
)
d                                                                   (2.5)
is derived by integration over a time period ∆t . t is the current point in time,   is an
auxiliary variable, L is the inductance of the inductor while v
k
(t) and v
m
(t) are
the potentials on either end of the inductor, as shown in Figure 1.
The integral is approximated using the trapezoidal rule, given by (2.6):
x (t+∆t )=x (t )+ ∆t 
1
2
 
(
f (x (t+∆t ),t+∆t )+f (x (t ),t ) )
,                                                                (2.6)
where f is the integrated function and x is the value of the integral.
When the trapezoidal rule (2.6) is applied to (2.5) and the potentials are grouped according to
the points in time ( t, t-∆t ), equation (2.7) is obtained.
i
km
(t )= i
km
(t-∆t )+
∆t
2L
(
v
k
(t-∆t )- v
m
(t-∆t )
)
+
∆t
2L
(
v
k
(t )- v
m
(t )
)
                                           (2.7)
It is evident that the inductor current at a given time depends on the inductor current and
potential difference over the inductor in the previous time step, and the potential difference
over the inductor in the current time step. 
2L
∆t
 is a constant and represents the effective
resistance of the inductor. The terms describing the conditions in the previous time step can be
Figure 1: Inductor with defined potentials and inductor current
Figure 1: Inductor with defined potentials and inductor current
The integral is approximated using the trapezoidal rule, given by (2.6):
10 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
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x (t+∆t )=x (t )+ ∆t 
1
2
 (f (x (t+∆t ),t+∆t )+f (x (t ),t ) ),                                                                (2.6)
where f is the integrated function and x is the value of the integral.
When the trapezoidal rule (2.6) is applied to (2.5) and the potentials are grouped according to
the points in time ( t, t-∆t ), equation (2.7) is obtained.
(2.6)
where f is the integrated function and x is the value of the integral.
When the trapezoidal rule (2.6) is applied to (2.5) and the potentials are grouped according to 
the points in time (t, t-Δt), equation (2.7) is obtained.
The impact of unbalanced power supply on load currents in transient and
steady state operation
11
----------
i
km
(t )= i
km
(t-∆t )+
∆t
2L
( v
k
(t-∆t )- v
m
(t-∆t ) )+
∆t
2L
(v
k
(t )- v
m
(t ) )                                           (2.7)
It is evident that the inductor current at a given time depends on the inductor current and
potential difference over the inductor in the previous time step, and the potential difference
over the inductor in the current time step. 
2L
∆t
 is a constant and represents the effective
resistance of the inductor. The terms describing the conditions in the previous time step can be
summarised into the current I
km
(t-∆t) , which describes the history of the inductor current.
Thus, equation (2.7) can be rewritten as equation (2.8)
(2.7)
It is evident that the inductor current at a given time depends on the inductor current and poten-
tial difference over the inductor in the previous time step, and the potential difference over the 
inductor in the current time step. 
2L
Δt
 is a constant and represents the effective resistance of the 
inductor. The terms describing the conditions in the previous time step can be summarised into 
the current, I
km
(t-Δt) which describes the history of the inductor current. Thus, equation (2.7) can 
be rewritten as equation (2.8)
The impact of unbalanced power supply on load currents in transient and
steady state operation
13
----------
i
km
(t )= I
km
(t-∆t )+
∆t
2L
( v
k
(t )- v
m
(t ) ) ,                                                                             (2.8) 
and the inductor can be modelled as a current source connected in parallel with the effective
resistance, as can be seen in the circuit shown in Figure 2.
3
PROPOSED METHOD
I n t h i s s e c t i o n , t h e c i r c u i t m o d e l s f o r t h e d i f f e r e n t t y p e s o f l o a d c o n n e c t i o n s a r e s h o w n .
Furthermore, the types and degrees of supply voltage unbalance used in the simulation are
described, together with the approach used to consider the magnetically non-linear behaviour
of the iron core inductor.  
3.1 Circuit models
The circuit model consists of the power supply and connection line, which are the same for all
types of load connection, and the load consisting of iron core inductors, which was connected in
either delta, ungrounded or grounded wye. The power supply is modelled as a real current
source in each of the phases, as can be seen in Figure 3, where the current of a phase i
j
(t )
is defined by equation (3.1) 
Figure 2: Model of an inductor according to Dommel's method
(2.8)
and the inductor can be modelled as a current source connected in parallel with the effective 
resistance, as can be seen in the circuit shown in Figure 2.
6 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
----------
summarised into the current I
km
(t-∆t) , which describes the history of the inductor current.
Thus, equation (2.7) can be rewritten as equation (2.8)
i
km
(t )= I
km
(t-∆t )+
∆t
2L
(
v
k
(t )- v
m
(t )
)
,                                                                             (2.8) 
and the inductor can be modelled as a current source connected in parallel with the effective
resistance, as can be seen in the circuit shown in Figure 2.
3
PROPOSED METHOD
I n t h i s s e c t i o n , t h e c i r c u i t m o d e l s f o r t h e d i f f e r e n t t y p e s o f l o a d c o n n e c t i o n s a r e s h o w n .
Furthermore, the types and degrees of supply voltage unbalance used in the simulation are
described, together with the approach used to consider the magnetically non-linear behaviour
of the iron core inductor.  
3.1 Circuit models
The circuit model consists of the power supply and connection line, which are the same for all
types of load connection, and the load consisting of iron core inductors, which was connected in
either delta, ungrounded or grounded wye. The power supply is modelled as a real current
source in each of the phases, as can be seen in Figure 3, where the current of a phase i
j
(t )
is defined by equation (3.1) 
i
n
(t )= 
1
R
u
n
(t ),                                                                                                                (3.1)
where R is the inner resistance of the current source, which is chosen to be 10m Ω.
u
j
(t ) is the supply voltage of that phase and n denotes one of the phases, namely ‘a’,
‘b’, or ‘ c’, n  {a,b,c } . The supply voltage is sinusoidal with a frequency of 50Hz and is
defined by (3.2):
u
n
(t )=Re
{ √2 V
 n
 exp (j 2 π 50 t )
}
,                                                                         (3.2)
Figure 2: Model of an inductor according to Dommel's method
Figure 2: Model of an inductor according to Dommel’s method

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3 PROPOSED METHOD
In this section, the circuit models for the different types of load connections are shown. Further-
more, the types and degrees of supply voltage unbalance used in the simulation are described, 
together with the approach used to consider the magnetically non-linear behaviour of the iron 
core inductor.
3.1 Circuit models
The circuit model consists of the power supply and connection line, which are the same for all 
types of load connection, and the load consisting of iron core inductors, which was connected 
in either delta, ungrounded or grounded wye. The power supply is modelled as a real current 
source in each of the phases, as can be seen in Figure 3, where the current of a phase i
j
(t) is 
defined by equation (3.1)
14 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
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i
n
(t )= 
1
R
u
n
(t ),                                                                                                                (3.1)
where R is the inner resistance of the current source, which is chosen to be 10m Ω.
u
j
(t ) is the supply voltage of that phase and n denotes one of the phases, namely ‘a’,
‘b’, or ‘ c’, n  {a,b,c } . The supply voltage is sinusoidal with a frequency of 50Hz and is
defined by (3.2):
(3.1)
where R is the inner resistance of the current source, which is chosen to be 10mΩ. u
j
(t) is the 
supply voltage of that phase and n denotes one of the phases, namely ‘a’, ‘b’, or ‘c’, n {a,b,c}. The 
supply voltage is sinusoidal with a frequency of 50Hz and is defined by (3.2):
a)
The impact of unbalanced power supply on load currents in transient and
steady state operation
15
----------
u
n
(t )=Re { √2 V
 n
 exp (j 2 π 50 t ) },                                                                         (3.2)
where V
 n
 denotes the phasor of a phase voltage defined by (2.1). The voltage source is
transformed into a current source to enable nodal potential analysis, which is used to calculate
the line currents in transient and quasi-steady-state operation.
The connection line supplying the load in each phase is modelled as an inductor, in accordance
with Dommel's method, in series with a resistor, as shown in Figure 3. The chosen inductance L
of the connection line is 0.2µH. The resistance of the connection line R is chosen to equal the
internal resistance of the current source, namely 10mΩ.
Similarly to the connection line, the load in the form of iron core inductors is also modelled as
an RL-combination connected in series for each phase of the load in the case of a wye
connection, and each load branch in the case of a delta connection. The iron core inductors are
modelled according to Dommel's method as either iron core inductors with magnetically linear
behaviour, where the inductance is constant, or iron core inductor with magnetically non-linear
behaviour, where iron core saturation is considered by variable dynamic inductance. The
discussed circuit models are shown in Figures 3 a) to 3 c). The time interval ∆t equals 1µs.
The chosen inductances of the iron core inductors L
n
, where n stands for one of the load
phases or branches, n  {a,b,c } , are 0.233H. The distinction between the load inductances
enables the modelling of the iron core inductors with magnetically non-linear behaviour, which
is described in subsection 3.3. The load resistance R
L
is equal in all branches of the load
with a value of 2.08Ω. 
c)
b)
Figure 3: Circuits models with the three-phase loads a) ungrounded wye connected, b) grounded wye
connected and c) delta connected
(3.2)
where V
n
 denotes the phasor of a phase voltage defined by (2.1). The voltage source is trans-
formed into a current source to enable nodal potential analysis, which is used to calculate the 
line currents in transient and quasi-steady-state operation.
The connection line supplying the load in each phase is modelled as an inductor, in accordance 
with Dommel’s method, in series with a resistor, as shown in Figure 3. The chosen inductance 
L of the connection line is 0.2µH. The resistance of the connection line R is chosen to equal the 
internal resistance of the current source, namely 10mΩ.
Similarly to the connection line, the load in the form of iron core inductors is also modelled as an 
RL-combination connected in series for each phase of the load in the case of a wye connection, 
and each load branch in the case of a delta connection. The iron core inductors are modelled 
according to Dommel’s method as either iron core inductors with magnetically linear behaviour, 
where the inductance is constant, or iron core inductor with magnetically non-linear behaviour, 
where iron core saturation is considered by variable dynamic inductance. The discussed circuit 
models are shown in Figures 3 a) to 3 c). The time Δt interval equals 1µs. The chosen inductances 
of the iron core inductors L
n
, where n stands for one of the load phases or branches, n {a,b,c}, 
are 0.233H. The distinction between the load inductances enables the modelling of the iron core 
inductors with magnetically non-linear behaviour, which is described in subsection 3.3. The load 
resistance R
L
 is equal in all branches of the load with a value of 2.08Ω.

JET  29
Method of the best available technology and low carbon future of a combined heat and power plant
a)
The impact of unbalanced power supply on load currents in transient and
steady state operation
7
----------
where V
 n
 denotes the phasor of a phase voltage defined by (2.1). The voltage source is
transformed into a current source to enable nodal potential analysis, which is used to calculate
the line currents in transient and quasi-steady-state operation.
The connection line supplying the load in each phase is modelled as an inductor, in accordance
with Dommel's method, in series with a resistor, as shown in Figure 3. The chosen inductance L
of the connection line is 0.2µH. The resistance of the connection line R is chosen to equal the
internal resistance of the current source, namely 10mΩ.
Similarly to the connection line, the load in the form of iron core inductors is also modelled as
an RL-combination connected in series for each phase of the load in the case of a wye
behaviour, where the inductance is constant, or iron core inductor with magnetically non-linear 
b)
a) 
a)
The impact of unbalanced power supply on load currents in transient and
steady state operation
7
----------
where V
 n
 denotes the phasor of a phase voltage defined by (2.1). The voltage source is
transformed into a current source to enable nodal potential analysis, which is used to calculate
the line currents in transient and quasi-steady-state operation.
The connection line supplying the load in each phase is modelled as an inductor, in accordance
with Dommel's method, in series with a resistor, as shown in Figure 3. The chosen inductance L
of the connection line is 0.2µH. The resistance of the connection line R is chosen to equal the
internal resistance of the current source, namely 10mΩ.
Similarly to the connection line, the load in the form of iron core inductors is also modelled as
an RL-combination connected in series for each phase of the load in the case of a wye
behaviour, where the inductance is constant, or iron core inductor with magnetically non-linear 
b)
b) 
8 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
----------
To perform the nodal potential analysis, the current vector I and the admittance matrix G are
determined based on the circuit models. The matrix G’, which is inverse of G, is then calculated.
Using (3.3)
V=G' I,                                                                                                                         (3.3)
the node potential vector V is determined. Based on this vector, the load currents i
L
n
(t) are
determined using equation (2.8). To evaluate the Joule losses, the calculated instantaneous
current values are recorded for 0.5s in 1 µs time intervals ∆t . This time interval is divided
into three regions: the first is the subtransient region with a duration of 0.1s, the next is the
transient region with a duration of 0.3s, and the last is the quasi-steady-state region with a
duration of 0.1s. Within each of these regions, the sum of the squared instantaneous values of
the load current is calculated using (3.4)
i
n,sum
2
=
∑
k=0
t
start
- t
stop
∆t
i
L
n
2
(
t
start
+k∆t
)
.                                                                                  (3.4)
To better compare these values in different regions, a sum of squared current values normalised
over one period i
n,sum,rel
2
 is calculated for all the discussed regions using (3.4) and (3.5): 
i
n,sum,rel
2
=
i
n,sum
2
N*i
n,sum,0
2
, (3.5)
c)
Figure 3: Circuits models with the three-phase loads a) ungrounded wye connected, b) grounded wye
connected and c) delta connected
c)
Figure 3: Circuits models with the three-phase loads a) ungrounded wye connected, b) ground-
ed wye connected and c) delta connected
To perform the nodal potential analysis, the current vector I and the admittance matrix G are 
determined based on the circuit models. The matrix G’, which is inverse of G, is then calculated. 
Using (3.3)
The impact of unbalanced power supply on load currents in transient and
steady state operation
17
----------
V=G' I,                                                                                                                         (3.3)
the node potential vector V is determined. Based on this vector, the load currents i
L
n
(t) are
determined using equation (2.8). To evaluate the Joule losses, the calculated instantaneous
current values are recorded for 0.5s in 1 µs time intervals ∆t . This time interval is divided
into three regions: the first is the subtransient region with a duration of 0.1s, the next is the
transient region with a duration of 0.3s, and the last is the quasi-steady-state region with a
duration of 0.1s. Within each of these regions, the sum of the squared instantaneous values of
the load current is calculated using (3.4)
(3.3)

30 JET 30 JET
Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
the node potential vector V is determined. Based on this vector, the load currents i
L n
(t) are de-
termined using equation (2.8). To evaluate the Joule losses, the calculated instantaneous current 
values are recorded for 0.5s in 1µs time intervals Δt. This time interval is divided into three re-
gions: the first is the subtransient region with a duration of 0.1s, the next is the transient region 
with a duration of 0.3s, and the last is the quasi-steady-state region with a duration of 0.1s. 
Within each of these regions, the sum of the squared instantaneous values of the load current is 
calculated using (3.4)
18 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
----------
i
n,sum
2
=
∑
k=0
t
start
- t
stop
∆t
i
L
n
2
(
t
start
+k∆t
)
.                                                                                  (3.4)
To better compare these values in different regions, a sum of squared current values normalised
over one period i
n,sum,rel
2
 is calculated for all the discussed regions using (3.4) and (3.5): 
(3.4)
To better compare these values in different regions, a sum of squared current values normalised 
over one period i
2
n,sum,rel
 is calculated for all the discussed regions using (3.4) and (3.5):
The impact of unbalanced power supply on load currents in transient and
steady state operation
19
----------
i
n,sum,rel
2
=
i
n,sum
2
N* i
n,sum,0
2
, (3.5)
Where N is the number of periods in the region of observation, while i
n,sum,0
2
 is the sum of
the squared instantaneous current values over one period (20ms) in the quasi-steady-state
operation determined for balanced power supply.
Since the Joule losses of the load are proportional to the sum of the squared instantaneous
values of the load currents, the normalised sum (3.5) represents the increase of Joule losses
o v e r o n e p e r i o d c o m p a r e d t o t h o s e o v e r o n e p e r i o d ( 2 0 m s ) i n n o r m a l q u a s i - s t e a d y - s t a t e
operation of the load supplied with balanced sinusoidal voltages. 
3.2 Unbalanced power supply 
The negative-sequence and zero-sequence voltage unbalance factors are used to evaluate the
level of voltage unbalance. The negative-sequence voltage unbalance factor VUF (2.2) is applied
to all three types of load connection, with the VUF ranging from 0 to 10% in 2% intervals. The
zero-sequence voltage unbalance factor ZSVUF (2.3) is only applied to the circuit with grounded
wye connected load, since it has no effect on the currents of ungrounded wye and delta
c o n n e c t e d l o a d s . T h e Z S V U F , d e f i n e d b y ( 2 . 3 ) , i s i n c r e a s e d f r o m 0 t o 1 0 % i n 2 % s t e p s .
Additionally, the effect of equal negative-sequence and zero-sequence voltage unbalance
simultaneously is investigated for the grounded wye connected load. Figure 4 shows the three-
phase supply voltages: the balanced (Figure 4 a)), those with a 10% VUF (Figure 4 b)), those with
a 10% ZSVUF (Figure 4 c)), and those with a 10% VUF and a 10% ZSVUF (Figure 4 d)).
a) b)
c)
d)
Figure 4: Supply voltages u
a 
, u
b
, u
c
, and u
a 
+ u
b 
+ u
c 
 over time for a) balanced power
(3.5)
Where N is the number of periods in the region of observation, while i
2
n,sum,0
 is the sum of the 
squared instantaneous current values over one period (20ms) in the quasi-steady-state opera-
tion determined for balanced power supply.
Since the Joule losses of the load are proportional to the sum of the squared instantaneous val-
ues of the load currents, the normalised sum (3.5) represents the increase of Joule losses over 
one period compared to those over one period (20ms) in normal quasi-steady-state operation of 
the load supplied with balanced sinusoidal voltages.
3.2 Unbalanced power supply
The negative-sequence and zero-sequence voltage unbalance factors are used to evaluate the 
level of voltage unbalance. The negative-sequence voltage unbalance factor VUF (2.2) is applied 
to all three types of load connection, with the VUF ranging from 0 to 10% in 2% intervals. The 
zero-sequence voltage unbalance factor ZSVUF (2.3) is only applied to the circuit with grounded 
wye connected load, since it has no effect on the currents of ungrounded wye and delta connect-
ed loads. The ZSVUF, defined by (2.3), is increased from 0 to 10% in 2% steps. Additionally, the 
effect of equal negative-sequence and zero-sequence voltage unbalance simultaneously is inves-
tigated for the grounded wye connected load. Figure 4 shows the three-phase supply voltages: 
the balanced (Figure 4 a)), those with a 10% VUF (Figure 4 b)), those with a 10% ZSVUF (Figure 4 
c)), and those with a 10% VUF and a 10% ZSVUF (Figure 4 d)).

JET  31
Method of the best available technology and low carbon future of a combined heat and power plant
Figure 4: Supply voltages u
a
, u
b
, u
c
 and u
a
+ u
b
+ u
c
 over time for a) balanced power supply, b) 10% 
VUF, c) 10% ZSVUF, d) 10% VUF and ZSVUF
3.3 Iron core saturation
To model a load consisting of iron core inductors with magnetically non-linear behaviour, the 
current dependent dynamic inductance L
d
 [15] is introduced (3.6):
The impact of unbalanced power supply on load currents in transient and
steady state operation
21
----------
u=
d Ψ(i)
d t
=
∂ Ψ(i)
∂ i
d i
d t
= L
d
( i)
d i
d t
,                                                                           (3.6)
where Ψ is the flux linkage, u is the voltage over the iron core inductor, and i the
current through the iron core inductor. 
In the case of an iron core inductor with magnetically linear behaviour, where iron core
saturation does not occur, the load inductance is constant for all currents, with the value
L
n
= 0.233H . In the case of an iron core inductor with magnetically non-linear behaviour,
the dynamic inductance decreases when the absolute value of the current through the inductor
exceeds the threshold. It is set to 4.5A, which is the amplitude of the load currents in quasi-
steady-state operation when supplied with balanced sinusoidal supply voltages. The saturated
dynamic inductance is 10 times smaller than the unsaturated one, and has the value
L
dn
= 0.0233H . In each step of the simulation, the dynamic inductances of all iron core
inductors in the load are adjusted according to the absolute values of the current flowing
through them. The inductors used to model the connection lines behave magnetically linearly
and have constant values of inductances. 
The iron core inductors with magnetically non-linear behaviour, representing the load, are only
used for the cases of delta and grounded wye connected loads. In the case of an ungrounded
wye connected load, convergency issues might appear, which would require further
investigation.
4 RESULTS
This section illustrates the results of the simulation. Section 4.1 contains the results for a load
containing iron core inductors with magnetically linear behaviour, meaning no saturation effects
are considered. Section 4.2 shows the results for a load containing iron core inductors with
magnetically non-linear behaviour, meaning saturation effects are considered. All the results are
shown in two ways. The first is shown using a graph illustrating the time behaviour of the
current in the case of a balanced power supply voltage, and the worst case, meaning the case
with the highest voltage unbalance for the discussed case. The second is shown using tables that
illustrate the values of the sums of the squared instantaneous current values normalised over
one period (3.5) as described in subsection 3.1. They represent an increase in per period Joule
losses due to the voltage supply unbalances relative to the per period Joule losses in quasi-
steady-state operation in the case of symmetrical supply voltages. 
4.1 Effects of supply voltage unbalances on iron core inductor loads
with magnetically linear behaviour 
The simulation was initially conducted for the loads consisting of iron core inductors with
magnetically linear behaviour, meaning the inductance in each branch of the load was constant. 
(3.6)
where Ψ is the flux linkage, u is the voltage over the iron core inductor, and i the current through 
the iron core inductor.
In the case of an iron core inductor with magnetically linear behaviour, where iron core satura-
tion does not occur, the load inductance is constant for all currents, with the value L
n
= 0.233H. In 
the case of an iron core inductor with magnetically non-linear behaviour, the dynamic inductance 
decreases when the absolute value of the current through the inductor exceeds the threshold. It 
is set to 4.5A, which is the amplitude of the load currents in quasi-steady-state operation when 
supplied with balanced sinusoidal supply voltages. The saturated dynamic inductance is 10 times 
smaller than the unsaturated one, and has the value L
dn
= 0.0233H. In each step of the simulation, 
the dynamic inductances of all iron core inductors in the load are adjusted according to the ab-
solute values of the current flowing through them. The inductors used to model the connection 
lines behave magnetically linearly and have constant values of inductances.

32 JET 32 JET
Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
The iron core inductors with magnetically non-linear behaviour, representing the load, are only 
used for the cases of delta and grounded wye connected loads. In the case of an ungrounded wye 
connected load, convergency issues might appear, which would require further investigation.
4 RESULTS
This section illustrates the results of the simulation. Section 4.1 contains the results for a load 
containing iron core inductors with magnetically linear behaviour, meaning no saturation ef-
fects are considered. Section 4.2 shows the results for a load containing iron core inductors with 
magnetically non-linear behaviour, meaning saturation effects are considered. All the results are 
shown in two ways. The first is shown using a graph illustrating the time behaviour of the current 
in the case of a balanced power supply voltage, and the worst case, meaning the case with the 
highest voltage unbalance for the discussed case. The second is shown using tables that illustrate 
the values of the sums of the squared instantaneous current values normalised over one period 
(3.5) as described in subsection 3.1. They represent an increase in per period Joule losses due to 
the voltage supply unbalances relative to the per period Joule losses in quasi-steady-state oper-
ation in the case of symmetrical supply voltages.
4.1 Effects of supply voltage unbalances on iron core inductor loads with 
magnetically linear behaviour
The simulation was initially conducted for the loads consisting of iron core inductors with mag-
netically linear behaviour, meaning the inductance in each branch of the load was constant.
Figures 5 shows the current over time in each of the load branches for a balanced voltage supply 
and a supply with a 10% VUF, and the load connected in ungrounded wye. Figures 6 and 7 show 
the load currents for a grounded wye and delta connected load.
As can be seen in Figures 5 and 6, in the cases of a grounded or ungrounded wye connected 
load, the negative-sequence voltage unbalance causes an increase in amplitude of the current in 
phase ‘a’, compared to operation with a balanced power supply, while the currents in phases ‘b’ 
and ‘c’ decrease. In the cases of a delta connected load, as seen in Figure 7, the amplitude of the 
current increases in phases ‘a’ and ‘c’, and decreases in phase ‘b’.
To better illustrate the change in the current time behaviour and its influence on the normalised 
per period increase in Joule losses, caused by supply voltage unbalances, the level of which is 
defined by the value of VUF (2.2), the normalised sums of squared instantaneous current val-
ues i
2
n,sum,rel
 (3.5) are used. The values of i
2
n,sum,rel
 in load branches are given in Tables 1 to 3 for all 
three types of load connection, considering different values of the VUF in the subtransient, tran-
sient and quasi-steady-state region of operation. The results shown are completed by the sum of 
i
2
n,sum,rel
 in all three load branches, which represents the total Joule losses in the load.

JET  33
Method of the best available technology and low carbon future of a combined heat and power plant
a)
b)
c)
Figure 5: Load currents over time for an ungrounded wye connected load with balanced power 
supply and a 10% VUF in a) branch ‘a’, b) branch ‘b’, and c) branch ‘c’ of the load
a)
b)
c)
Figure 6: Load currents over time for a grounded wye connected load with balanced power 
supply and a 10% VUF in a) branch ‘a’, b) branch ‘b’, and c) branch ‘c’ of the load

34 JET 34 JET
Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
a)
b)
c)
Figure 7: Load currents over time for a delta connected load with balanced power supply and a 
10% VUF in a) branch ‘a’, b) branch ‘b’, and c) branch ‘c’ of the load
Table 1: Relative sum i
2
n,sum,rel
 for values of VUF between 0 to 10% for an ungrounded wye con-
nected load
VUF [%] 
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 1.221 1.271 1.321 1.372 1.425 1.478
Branch ‘b’ 1.925 1.886 1.849 1.812 1.777 1.742
Branch ‘c’ 1.239 1.201 1.164 1.129 1.096 1.064
Total 4.385 4.358 4.334 4.314 4.297 4.284
Transient
Branch ‘a’ 1.010 1.051 1.093 1.135 1.178 1.222
Branch ‘b’ 1.060 1.039 1.019 0.999 0.981 0.963
Branch ‘c’ 1.019 0.998 0.978 0.959 0.940 0.923
Total 3.089 3.088 3.089 3.093 3.099 3.108
Quasi-steady-state
Branch ‘a’ 1.000 1.040 1.082 1.124 1.166 1.210
Branch ‘b’ 1.000 0.980 0.962 0.943 0.926 0.910
Branch ‘c’ 1.000 0.980 0.962 0.944 0.926 0.910
Total 3.000 3.001 3.005 3.011 3.019 3.030

JET  35
Method of the best available technology and low carbon future of a combined heat and power plant
Table 2: Relative sum i
2
n,sum,rel
 for values of VUF between 0 to 10% for a grounded wye connected 
load
VUF [%] 
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 1.209 1.258 1.308 1.359 1.411 1.463
Branch ‘b’ 1.924 1.885 1.847 1.811 1.775 1.740
Branch ‘c’ 1.252 1.213 1.176 1.141 1.107 1.074
Total 4.385 4.357 4.332 4.310 4.292 4.277
Transient
Branch ‘a’ 1.009 1.050 1.092 1.134 1.177 1.221
Branch ‘b’ 1.060 1.039 1.019 0.999 0.981 0.963
Branch ‘c’ 1.020 0.999 0.979 0.960 0.941 0.924
Total 3.089 3.088 3.089 3.093 3.099 3.108
Quasi-steady-state
Branch ‘a’ 1.000 1.040 1.082 1.124 1.166 1.210
Branch ‘b’ 1.000 0.980 0.962 0.943 0.926 0.910
Branch ‘c’ 1.000 0.980 0.962 0.944 0.926 0.910
Total 3.000 3.001 3.005 3.011 3.019 3.030
Table 3: Relative sum i
2
n,sum,rel
 for values of VUF between 0 to 10% for a delta connected load
VUF [%] 
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 1.667 1.686 1.707 1.728 1.750 1.772
Branch ‘b’ 1.709 1.641 1.575 1.510 1.446 1.384
Branch ‘c’ 0.999 1.019 1.040 1.062 1.086 1.111
Total 4.375 4.346 4.321 4.300 4.282 4.268
Transient
Branch ‘a’ 1.038 1.058 1.079 1.101 1.123 1.147
Branch ‘b’ 1.049 1.007 0.966 0.927 0.888 0.849
Branch ‘c’ 1.000 1.020 1.041 1.063 1.085 1.109
Total 3.087 3.085 3.087 3.090 3.097 3.105
Quasi-steady-state
Branch ‘a’ 1.000 1.020 1.042 1.064 1.086 1.110
Branch ‘b’ 1.000 0.960 0.922 0.884 0.846 0.810
Branch ‘c’ 1.000 1.020 1.042 1.063 1.086 1.110
Total 3.000 3.001 3.005 3.011 3.019 3.030

36 JET 36 JET
Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
As can be seen in Tables 1-3, the relative sum, and with it the Joule losses in the load, increase lin-
early with the VUF in the quasi-steady-state region of operation. In the cases of a grounded and 
ungrounded wye connected load, an increase only occurs in branch ‘a’, with a factor of about 2, 
meaning a 10% VUF results in a 20% increase in Joule losses in branch ‘a’. Whilst the Joule losses 
in the other two branches decrease, there is still a minor net increase in Joule losses in the load. 
Similarly, in a delta connected load, the Joule losses increase in branches ‘a’ and ‘c’ in accordance 
with the VUF, meaning a 10% VUF causes a 10% increase in Joule losses in each branch. The Joule 
losses in branch ‘b’ decrease, however, there is still a minor net increase in Joule losses, as is the 
case for the wye connected load.
The results are similar in the transient and subtransient region of operation, namely with the wye 
connected loads there is an increase in Joule losses in branch ‘a’ amounting to about twice the 
VUF, while the losses in the other two phases decrease. Similarly, there is an increase in losses in 
branches ‘a’ and ‘c’ in the delta connected load, and a decrease in branch ‘b’. There is minor net 
increase in Joule losses in all three load connections in the transient region of operation, and a 
minor decrease in the subtransient region.
These results show that, whilst the total Joule losses in the load increase slightly or even de-
crease with a higher VUF, the losses in a particular branch increase by a factor of the VUF or twice 
that, depending on the connection of the load.
The effect of a zero-sequence unbalance on the Joule losses was only observed in the case of a 
grounded wye connected load. Figure 8 shows the currents in each branch of the load over time 
for a 10% ZSVUF and balanced power supply. The zero-sequence unbalance in supply voltages 
does not affect the currents of ungrounded and delta connected loads since there is no path 
where the zero-sequence current could close.
a)
b)
c)
Figure 8: Load currents over time for a grounded wye connected load with balanced power 
supply and a 10% ZSVUF in a) branch ‘a’, b) branch ‘b’, and c) branch ‘c’ of the load

JET  37
Method of the best available technology and low carbon future of a combined heat and power plant
As can be seen in Figure 8, similarly to the negative-sequence voltage unbalance, the zero-se-
quence voltage unbalance causes an increase in the amplitude of the current in branch ‘a’, while 
the amplitude in branches ‘b’ and ‘c’ decreases.
The values of i
2
n,sum,rel
 in the subtransient, transient and quasi-steady-region of operation are show 
in Table 4 for different values of the ZSVUF.
Table 4: Relative sum i
2
n,sum,rel
 for values of ZSVUF between 0 to 10% for a grounded wye connect-
ed load
ZSVUF [%] 
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 1.209 1.258 1.308 1.359 1.411 1.463
Branch ‘b’ 1.924 1.887 1.851 1.816 1.782 1.748
Branch ‘c’ 1.252 1.242 1.232 1.224 1.217 1.210
Total 4.385 4.387 4.391 4.399 4.409 4.422
Transient
Branch ‘a’ 1.009 1.050 1.092 1.134 1.177 1.221
Branch ‘b’ 1.060 1.039 1.019 1.001 0.982 0.965
Branch ‘c’ 1.020 1.001 0.983 0.966 0.949 0.933
Total 3.089 3.091 3.094 3.100 3.109 3.120
Quasi-steady-state
Branch ‘a’ 1.000 1.040 1.082 1.124 1.166 1.210
Branch ‘b’ 1.000 0.980 0.962 0.944 0.927 0.910
Branch ‘c’ 1.000 0.980 0.962 0.944 0.926 0.910
Total 3.000 3.001 3.005 3.011 3.019 3.030
As can be seen in Table 4, the ZSVUF has the same effect as the VUF, as shown in Table 2, with 
an increase in Joule losses of twice the ZSVUF in branch ‘a’, a decrease in the other two branches 
and a minor net increase of i
2
n,sum,rel
 in the transient and quasi-steady-state region of operation. 
However, there is also a minor net increase in losses in the subtransient region.
Furthermore, the effects of a simultaneous negative-sequence and zero-sequence supply voltage 
unbalance on the currents of a grounded wye connected load were investigated. Figure 9 shows 
the currents in each branch for the cases of a 10% VUF and ZSVUF and balanced power supply 
voltages.
Figure 9 shows an increase in the amplitude of the current in branch ‘a’ in the case of a voltage 
unbalance, compared to the case of the balanced voltage supply, and a decrease in the ampli-
tude of the current in the other two branches.
The values of i
2
n,sum,rel
 for different values of simultaneous VUF and ZSVUF are given in Table 5.

38 JET 38 JET
Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
a)
b)
c)
Figure 9: Load currents over time for a grounded wye connected load with balanced power 
supply and a simultaneous 10% ZSVUF and VUF in a) branch ‘a’, b) branch ‘b’, and c) branch ‘c’ 
of the load
Table 5: Relative sum i
2
n,sum,rel
 for values of simultaneous ZSVUF and VUF between 0 to 10% for a 
grounded wye connected load
VUF and ZSVUF [%] 
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 1.209 1.308 1.411 1.517 1.627 1.741
Branch ‘b’ 1.924 1.848 1.773 1.700 1.629 1.559
Branch ‘c’ 1.252 1.202 1.154 1.106 1.060 1.014
Total 4.385 4.358 4.338 4.323 4.315 4.314
Transient
Branch ‘a’ 1.009 1.092 1.177 1.266 1.358 1.454
Branch ‘b’ 1.060 1.018 0.977 0.936 0.897 0.858
Branch ‘c’ 1.020 0.980 0.940 0.901 0.863 0.826
Total 3.089 3.089 3.094 3.104 3.119 3.138
Quasi-steady-state
Branch ‘a’ 1.000 1.082 1.166 1.254 1.346 1.440
Branch ‘b’ 1.000 0.960 0.922 0.884 0.846 0.810
Branch ‘c’ 1.000 0.960 0.922 0.884 0.846 0.810
Total 3.000 3.002 3.010 3.022 3.038 3.060

JET  39
Method of the best available technology and low carbon future of a combined heat and power plant
The results shown in Table 5 show that the Joule losses in branch ‘a’ increase by four times 
the value of the simultaneous VUF and ZSVUF. As is the case for the VUF and ZSVUF changing 
separately, there is a decrease in Joule losses in the other two branches of the load, and like the 
increase in branch ‘a’, it is about double the decrease in the cases with just a VUF or ZSVUF. The 
minor net increase of losses in the transient and quasi-steady-state region of operation also dou-
bles, and the minor net decrease in the subtransient region is lower than the case of just a VUF, 
as shown in Table 2. These results suggest the effects of a negative-sequence and zero-sequence 
supply voltage unbalance are additive.
The results shown in Tables 1 to 5 represent the relative Joule losses over one period. In order to 
calculate the total relative losses over the 0.5s period shown in Figures 5 to 9, a sum of the i
2
n,sum,rel
 
values should be formed, with the subtransient value multiplied by 5, the transient i
2
n,sum,rel
 value 
multiplied by 15, and the quasi-steady-state i
2
n,sum,rel
 value multiplied by 5, since that is the number 
of periods in the three discussed regions.
Due to supply voltage unbalances to an iron core inductor load with linear magnetic behaviour, 
the Joule losses in one branch can increase by the VUF if the load is connected into delta, twice 
the VUF if the load is connected into ungrounded wye, and twice the sum of the VUF and ZSVUF 
when the load is connected into grounded wye.
4.2 Effects of supply voltage unbalances on iron core inductor loads with 
magnetically non-linear behaviour
The simulation was conducted for the loads consisting of iron core inductors with magnetically 
non-linear behaviour, where variable dynamic inductances were applied to consider the iron 
core saturation as described in section 3.3. Due to convergence issues, the results for unground-
ed wye connected loads are inconclusive and therefore not included in this paper.
Figure 10 shows the load current over time in each branch of the delta connected load for the 
cases of a balanced power supply, and a 10% VUF, while Figure 11 shows the same results for a 
grounded wye connected load.
As can be seen in Figures 10 and 11, the currents generally behave similarly to the case with a 
load with magnetically linear behaviour. Namely, due to the supply voltage unbalance the cur-
rent amplitude increases in branch ‘a’ and decreases in the other two branches for the grounded 
wye connected load, whilst it increases in branches ‘a’ and ‘c’, and decreases in branch ‘b’ for the 
delta connected load. However, the decrease in the dynamic inductance due to the saturation, 
after reaching the saturation value of the current, causes a rapid increase in the current ampli-
tude. This can be observed in the subtransient and partially the transient region of operation for 
both load connections. Additionally, in the case of the grounded wye connected load, the effects 
of saturation are also visible in the quasi-steady-state region of operation in branch ‘a’, when the 
VUF is 10%, since the current increase due to the supply voltage unbalance causes the current 
to reach values above the saturation threshold. This is not the case for the delta connected load, 
due to the current increase being in both branches ‘a’ and ‘c’ with a lower value.
To better present the impact of these effects on current responses and thus Joule losses, the rela-
tive sums i
2
n,sum,rel
 are given in Table 6 for the delta connected load and in Table 7 for the grounded 
wye connected load.

40 JET 40 JET
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a)
b)
c)
Figure 10: Load currents over time for a delta connected iron core inductor with magnetically 
non-linear behaviour with a balanced voltage supply and a 10% VUF in a) branch ‘a’, b) branch 
‘b’, and c) branch ‘c’ of the load
a)
b)
c)
Figure 11: Load currents over time for a grounded wye connected iron core inductor with mag-
netically non-linear behaviour with a balanced voltage supply and a 10% VUF in a) branch ‘a’, b) 
branch ‘b’, and c) branch ‘c’ of the load

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Method of the best available technology and low carbon future of a combined heat and power plant
Table 6: Relative sum i
2
n,sum,rel
 for values of VUF between 0 to 10% for a delta connected iron core 
inductor load
VUF [%]
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 5.446 5.611 5.777 5.968 6.143 6.344
Branch ‘b’ 5.468 4.957 4.481 4.026 3.613 3.222
Branch ‘c’ 0.999 1.019 1.040 1.063 1.087 1.113
Total 11.913 11.587 11.298 11.057 10.843 10.679
Transient
Branch ‘a’ 1.010 1.030 1.051 1.073 1.096 1.120
Branch ‘b’ 1.015 0.977 0.939 0.903 0.866 0.831
Branch ‘c’ 1.000 1.020 1.041 1.063 1.085 1.109
Total 3.025 3.027 3.031 3.038 3.048 3.060
Quasi-steady-state
Branch ‘a’ 1.000 1.020 1.042 1.064 1.086 1.110
Branch ‘b’ 1.000 0.960 0.922 0.884 0.847 0.810
Branch ‘c’ 1.000 1.020 1.042 1.063 1.086 1.110
Total 3.000 3.001 3.005 3.011 3.019 3.030
Table 7: Relative sum i
2
n,sum,rel
 for values of VUF between 0 to 10% for a grounded wye connected 
iron core inductor load
VUF [%]
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 3.504 3.860 4.249 4.667 5.068 5.535
Branch ‘b’ 9.951 9.579 9.199 8.855 8.507 8.182
Branch ‘c’ 3.387 3.095 2.813 2.553 2.311 2.087
Total 16.843 16.534 16.261 16.075 15.885 15.804
Transient
Branch ‘a’ 1.023 1.087 1.206 1.391 1.636 1.942
Branch ‘b’ 1.052 1.024 0.998 0.975 0.954 0.934
Branch ‘c’ 1.027 1.001 0.978 0.956 0.936 0.918
Total 3.102 3.112 3.182 3.322 3.526 3.794

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VUF [%]
i
2
n,sum,rel
0 2 4 6 8 10
Quasi-steady-state
Branch ‘a’ 1.000 1.054 1.188 1.385 1.637 1.943
Branch ‘b’ 1.000 0.980 0.962 0.944 0.926 0.910
Branch ‘c’ 1.000 0.980 0.962 0.944 0.927 0.910
Total 3.000 3.014 3.112 3.273 3.490 3.763
The comparison of the results shown in Tables 3 and 6 shows that in the quasi-steady-state re-
gion of operation there is no difference between the relative sum i
2
n,sum,rel
 of the delta connected 
iron core inductors with magnetically linear and non-linear behaviour, due to the currents re-
maining below the saturation threshold even in the case of unbalanced supply voltages. This is 
not the case with the grounded wye connected load, as can be seen from Tables 2 and 7. Whilst 
the decrease in Joule losses (i
2
n,sum,rel
) in branches ‘b’ and ‘c’ remains equal for the iron core in-
ductors with magnetically linear and non-linear behaviour, the increase in Joule losses in branch 
‘a’ is not linear but instead approximately proportional to the square of the VUF. This causes the 
Joule losses to nearly double in the case of a 10% VUF compared to the case of a balanced power 
supply. This subsequently leads to larger increase in net Joule losses in the load.
The results are similar in the transient region of operation, with the losses in the case of the delta 
connected load (Table 6) remaining approximately equal to those in Table 3. In the case of a wye 
connected load, the Joule losses increase proportional to the square of the VUF (Table 7). In the 
a)
b)
c)
Figure 12: Load currents over time for a grounded wye connected iron core inductor with mag-
netically non-linear behaviour with a balanced voltage supply and a 10% ZSVUF in a) branch ‘a’, 
b) branch ‘b’, and c) branch ‘c’ of the load

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Method of the best available technology and low carbon future of a combined heat and power plant
subtransient region, the saturation effects are visible in both loads – the grounded wye and delta 
connected iron core inductors with magnetically non-linear behaviour (Tables 6 and 7) – due to 
the currents surpassing the saturation threshold. This causes a larger increase in the Joule losses 
in branches ‘a’ and ‘c’ of the delta connected load and in branch ‘a’ of the wye connected load. 
However, these increases are not proportional to the square of the VUF as in the case of the tran-
sient or quasi-steady-state region with the wye connected load. Additionally, the relative Joule 
losses are generally much higher compared to the load with magnetically linear behaviour, even 
in the balanced case. Even though the total losses increase in general, they decrease with the 
VUF in a similar way as in the case with a load with magnetically linear behaviour (Tables 2 and 3).
The effects of zero-sequence voltage unbalance can be seen in Figure 12, which shows the load 
currents over time in a grounded wye connected iron core inductor load with magnetically 
non-linear behaviour for a 10% ZSVUF and balanced supply voltages.
It is evident from Figure 12 that a zero-sequence voltage unbalance has a similar effect on the 
current responses as the negative-sequence voltage unbalance seen in Figure 11. The effects of 
the magnetically non-linear behaviour of the iron core inductor on the relative sum i
2
n,sum,rel
 can be 
seen in the subtransient and partially transient region of operation in all three branches in the 
case of the unbalanced and balanced power supply, whilst in quasi-steady-state region it is only 
present in branch ‘a’ in the case of a supply voltage unbalance.
The relative sum i
2
n,sum,rel
 for values of the ZSVUF between 0 and 10% are shown in Table 8.
Table 8: Relative sum i
2
n,sum,rel
 for values of ZSVUF between 0 to 10% for a grounded wye connect-
ed iron core inductor load
ZSVUF [%]
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 3.504 3.860 4.249 4.667 5.068 5.535
Branch ‘b’ 9.940 9.572 9.207 8.850 8.527 8.192
Branch ‘c’ 3.387 3.357 3.328 3.299 3.281 3.272
Total 16.832 16.789 16.784 16.816 16.875 16.998
Transient
Branch ‘a’ 1.023 1.087 1.206 1.391 1.636 1.942
Branch ‘b’ 1.052 1.024 0.999 0.976 0.955 0.935
Branch ‘c’ 1.027 1.002 0.980 0.960 0.941 0.923
Total 3.102 3.113 3.185 3.326 3.532 3.800
Quasi-steady-state
Branch ‘a’ 1.000 1.054 1.188 1.385 1.637 1.943
Branch ‘b’ 1.000 0.980 0.962 0.944 0.926 0.910
Branch ‘c’ 1.000 0.980 0.962 0.944 0.927 0.910
Total 3.000 3.014 3.112 3.273 3.491 3.764

44 JET 44 JET
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Similarly to the effects of the VUF, the ZSVUF causes an increase of i
2
n,sum,rel
, and thus the Joule 
losses in branch ‘a’, which is proportional to the square of the ZSVUF in the transient and quasi-
steady-state region of operation. However, the decrease in Joule losses in branches ‘b’ and ‘c’ in 
Table 8 remains the same as in the case with the load with magnetically linear behaviour, causing 
a higher increase in the total Joule losses in these regions than in Table 4. In the subtransient 
region, a larger increase in total Joule losses is also noticeable, compared to Table 4, however, 
unlike the values in Table 7, the total losses increase with the ZSVUF, as was in the case with a 
grounded wye connected load with magnetically linear behaviour (Table 4).
The effects of a simultaneous negative-sequence and positive sequence voltage unbalance are 
shown in Figure 13, which shows the load currents over time in a grounded wye connected iron 
core inductor load with magnetically non-linear properties for a simultaneous 10% ZSVUF and 
VUF and balanced power supply.
a)
b)
c)
Figure 13: Load currents over time for a grounded wye connected iron core inductor with mag-
netically non-linear behaviour with balanced power supply and a simultaneous 10% ZSVUF and 
VUF in a) branch ‘a’, b) branch ‘b’, and c) branch ‘c’ of the load
As seen in Figure 13, the amplitude of the current in branch ‘a’ increased compared to the cases 
seen in Figures 11 and 12, however, the overall behaviour remains the same.
Table 9 shows the relative sums i
2
n,sum,rel
 representing the Joule losses.

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Method of the best available technology and low carbon future of a combined heat and power plant
Table 9: Relative sum i
2
n,sum,rel
 for values of simultaneous ZSVUF and VUF between 0 to 10% for a 
grounded wye connected iron core inductor load
VUF and ZSVUF [%]
i
2
n,sum,rel
0 2 4 6 8 10
Subtransient
Branch ‘a’ 3.504 4.249 5.068 6.032 7.191 8.533
Branch ‘b’ 9.951 9.186 8.462 7.770 7.102 6.483
Branch ‘c’ 3.387 3.051 2.736 2.444 2.176 1.920
Total 16.843 16.486 16.265 16.246 16.469 16.936
Transient
Branch ‘a’ 1.023 1.206 1.636 2.306 3.215 4.353
Branch ‘b’ 1.052 0.996 0.949 0.904 0.862 0.823
Branch ‘c’ 1.027 0.978 0.933 0.892 0.854 0.818
Total 3.102 3.180 3.518 4.102 4.932 5.994
Quasi-steady-state
Branch ‘a’ 1.000 1.188 1.637 2.311 3.226 4.361
Branch ‘b’ 1.000 0.960 0.922 0.884 0.846 0.810
Branch ‘c’ 1.000 0.961 0.922 0.884 0.847 0.810
Total 3.000 3.109 3.481 4.078 4.919 5.982
The results shown in Table 9 indicate that the simultaneous negative-sequence and zero-se-
quence voltage unbalances result in an increase in Joule losses in branch ‘a’ of the load, which 
is proportional to approximately double the square value of the VUF and ZSVUF in the transient 
and quasi-steady-state region. In the case of a 10% simultaneous VUF and ZSVUF this means 
more than four times the Joule losses in that branch than in the case of a balanced power supply. 
Furthermore, due to the losses not increasing in other branches compared to the case with a 
load with magnetically linear behaviour (Table 5), compared to the case with a balanced power 
supply, the total losses almost doubled as well. In the subtransient region, the Joule losses in 
branch ‘a’ also increased compared to Table 5, although not proportional to the square value of 
the VUF and ZSVUF. At a 10% zero-sequence and negative-sequence voltage unbalance, the loss-
es in branch ‘a’ are more than double those in the case of a balanced power supply. Additionally, 
due to the larger increase in losses caused by the zero-sequence voltage unbalance, the total 
Joule losses slightly increased with the VUF and ZSVUF. These results suggest that the effects 
of the VUF and ZSVUF are additive in the case of an iron core inductor load with magnetically 
non-linear behaviour.
The results show that due to supply voltage unbalances of an iron core inductor load with mag-
netically linear behaviour, the Joule losses in one branch, compared to the case of a balanced 
power supply, can increase by the VUF if the load is connected into delta, and twice the sum of 
the square values of the VUF and ZSVUF when the load is connected into grounded wye. This 

46 JET 46 JET
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means that in a grounded wye connected load with magnetically non-linear behaviour, even 
small supply voltage unbalances can lead to a larger loss of efficiency and overheating.
5 CONCLUSION
This work proposes a unified method for evaluating the impact of supply voltage unbalances 
on the increase in currents and Joule losses of wye and delta connected loads. It considers the 
magnetically linear or non-linear behaviour of iron core conductors in subtransient, transient and 
quasi-steady-state operation. The supply voltage unbalances are characterised by the VUF and 
ZSVUF. The proposed method is based on Dommel’s method, which utilises a modified nodal 
potential method, implicit numerical integration and time discretisation for dynamic and quasi-
steady-state simulations of electric circuits. The magnetically non-linear behaviour of iron core 
indictors is considered in the form of variable current dependent dynamic inductances.
Although the proposed method is demonstrated in the case studies of wye and delta connected 
loads consisting of iron core inductors with magnetically linear and non-linear behaviour, it can 
be extended towards applicability to electrical machines. The results presented in the paper 
clearly show to what extent the supply voltage unbalances influence the currents in the branch-
es and corresponding Joule losses of the discussed loads in subtransient, transient and quasi-
steady-state operation.
References
[1] A. von Jouanne, B. Banerjee: Assessment of voltage unbalance, IEEE Transactions on 
Power Delivery, vol.16, no. 4 pp.782-790, 2001
[2] IEEE STANDARDS ASSOCIATION: IEEE Recommended Practice for Monitoring Electric 
Power Quality, IEEE, p.p.25-28, 2019, DOI: 10.1109/IEEESTD.2019.8796486
[3] Y . Wang: Analysis of effects of three-phase voltage unbalance on induction motors with 
emphasis on the angle of the complex voltage unbalance factor, IEEE Transactions on 
Energy Conversion, vol.16, no.3, p.p.270-275, 2001
[4] Y. Wang: An analytical study on steady-state performance of an induction motor con-
nected to unbalanced three-phase voltage, 2000 IEEE Power Engineering Society Win-
ter Meeting. Conference Proceedings, p.p.159 – 164, 2000
[5] J. Faiz, H. Ebrahimpour, P. Pillay: Influence of unbalanced voltage on the steady-sta-
te performance of a three-phase squirrel-cage induction motor, IEEE Transactions on 
Energy Conversion, vol.19, no.4, pp.657-662, 2004
[6] C. T. Raj, P. Agarwal, S. P. Srivastava: Performance Analysis of a Three-Phase Squir-
rel-Cage Induction Motor under Unbalanced Sinusoidal and Balanced Non-Sinusoidal 
Supply Voltages, 2006 International Conference on Power Electronic, Drives and Energy 
Systems, p.p.1-4, 2006
[7] J. Faiz, H. Ebrahimpour: Precise derating of three-phase induction motors with unbal-
anced voltages, Fourtieth IAS Annual Meeting. Conference Record of the 2005 Industry 
Applications Conference, p.p.485-491, 2005

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Method of the best available technology and low carbon future of a combined heat and power plant
[8] E. C. Quispe, I. D. Lopez: Effects of unbalanced voltages on the energy performance of 
three-phase induction motors, 2015 IEEE Workshop on Power Electronics and Power 
Quality Applications (PEPQA), p.p.1-6, 2015
[9] K. Lee, G. Venkataramanan, T. M. Jahns: Modeling Effects of Voltage Unbalances in 
Industrial Distribution Systems With Adjustable-Speed Drives, IEEE Transactions on In-
dustry Applications, vol.44, no.5, p.p.1,322-1,332, 2008
[10] P. Gnacinski: Windings Temperature and Loss of Life of an Induction Machine Un-
der Voltage Unbalance Combined With Over- or Undervoltages, IEEE Transactions on 
Energy Conversion, vol.23, no.2, p.p.363-371, 2008
[11] J. L. Gonzalez-Cordoba, R. A. Osornio-Rios, D. Granados-Lieberman, R. D. J. Rome-
ro-Troncoso, M. Valtierra-Rodriguez: Correlation Model Between Voltage Unbalance 
and Mechanical Overload Based on Thermal Effect at the Induction Motor Stator, IEEE 
Transactions on Energy Conversion, vol.32, no.4, pp.1,602-1,610, 2017
[12] P. Sudasinghe, U. Jayatunga, P. Commins, J. Moscrop, S. Perera: Dependancy of Three 
Phase Induction Motor Derating Aspects on Complex Voltage Unbalance Factor: A Calo-
rimetric and Finite Element Simulation Study, IEEE Access, vol.9, p.p.147,063-147,071, 
2021
[13] G. Štumberger, B. Štumberger, T. Maričič: Magnetically nonlinear dynamic models 
of synchronous machines and experimental methods for determining their parame-
ters, Energies, vol.12, no.18, p.p.1-22, 2019
[14] J. Arrillaga, Computer modelling of electrical power systems, Wiley, 2001
[15] G. Štumberger, Ž. Plantić, B. Štumberger, T. Maričič: Impact of static and dynamic 
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p.p.190-193, 2011
36 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
----------
N om e n c l a t ur e
(Symbols) (Symbol meaning)
V
 0
Phasor of the zero-sequence voltage
V
 1
Phasor of the positive-sequence voltage
V
 2
Phasor of the negative-sequence voltage
V
 a
Phasor of the voltage phase voltage in phase ‘a’
V
 b
Phasor of the voltage phase voltage in phase ‘b’
V
 c
Phasor of the voltage phase voltage in phase ‘c’
exp Exponential function
a exp
(
j2
π
3
)
j Imaginary unit
VUF Voltage unbalance factor
ZSVUF Zero sequence voltage unbalance factor
u
L
(t) Inductor voltage
v
k
(t) Potential in point k
v
m
(t) Potential in point m
i
km
(t) Inductor current at time t
L Inductance
t Time
∆t Time period
Auxiliary variable
f (x (t ),t )
Integrated function
x (t ) Value of the integral of function f
I
km
(t-∆t ) Current describing the history of the inductor current
i
n
(t ) Current of a phase
u
n
(t ) Supply voltage of a phase
n Index denoting one of the phases, n  {a,b,c }
R Resistance of the current source and cable

48 JET 48 JET
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36 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
----------
N om e n c l a t ur e
(Symbols) (Symbol meaning)
V
 0
Phasor of the zero-sequence voltage
V
 1
Phasor of the positive-sequence voltage
V
 2
Phasor of the negative-sequence voltage
V
 a
Phasor of the voltage phase voltage in phase ‘a’
V
 b
Phasor of the voltage phase voltage in phase ‘b’
V
 c
Phasor of the voltage phase voltage in phase ‘c’
exp Exponential function
a exp
(
j2
π
3
)
j Imaginary unit
VUF Voltage unbalance factor
ZSVUF Zero sequence voltage unbalance factor
u
L
(t) Inductor voltage
v
k
(t) Potential in point k
v
m
(t) Potential in point m
i
km
(t) Inductor current at time t
L Inductance
t Time
∆t Time period
Auxiliary variable
f (x (t ),t )
Integrated function
x (t ) Value of the integral of function f
I
km
(t-∆t ) Current describing the history of the inductor current
i
n
(t ) Current of a phase
u
n
(t ) Supply voltage of a phase
n Index denoting one of the phases, n  {a,b,c }
R Resistance of the current source and cable
The impact of unbalanced power supply on load currents in transient and
steady state operation
37
----------
V
 n
Phasor of the supply voltage of phase n
L
n
Inductance of the iron core inductor in branch n
R
L
Load resistance
I Current vector
G Impedance matrix
G’ The inverse of the impedance matrix
V Node potential vector
i
L
n
(t) Load current in branch n
i
n,sum
2
The sum of the squared instantaneous value of the load current in branch n
t
start
Start time of a region of operation
t
stop
Stop time of a region of operation
i
n,sum,rel
2 Sum of the squared instantaneous values of the load current in branch n 
normalised over one period
N Number of periods
i
n,sum,0
2 The sum of the squared instantaneous value of the load current in branch n
during a balanced power supply and quasi-steady-state region of operation
Ψ Flux linkage
L
dn
Dynamic inductance of the load in branch n
36 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
----------
N om e n c l a t ur e
(Symbols) (Symbol meaning)
V
 0
Phasor of the zero-sequence voltage
V
 1
Phasor of the positive-sequence voltage
V
 2
Phasor of the negative-sequence voltage
V
 a
Phasor of the voltage phase voltage in phase ‘a’
V
 b
Phasor of the voltage phase voltage in phase ‘b’
V
 c
Phasor of the voltage phase voltage in phase ‘c’
exp Exponential function
a exp
(
j2
π
3
)
j Imaginary unit
VUF Voltage unbalance factor
ZSVUF Zero sequence voltage unbalance factor
u
L
(t) Inductor voltage
v
k
(t) Potential in point k
v
m
(t) Potential in point m
i
km
(t) Inductor current at time t
L Inductance
t Time
∆t Time period
Auxiliary variable
f (x (t ),t )
Integrated function
x (t ) Value of the integral of function f
I
km
(t-∆t ) Current describing the history of the inductor current
i
n
(t ) Current of a phase
u
n
(t ) Supply voltage of a phase
n Index denoting one of the phases, n  {a,b,c }
R Resistance of the current source and cable

JET  49
Method of the best available technology and low carbon future of a combined heat and power plant
The impact of unbalanced power supply on load currents in transient and
steady state operation
37
----------
V
 n
Phasor of the supply voltage of phase n
L
n
Inductance of the iron core inductor in branch n
R
L
Load resistance
I Current vector
G Impedance matrix
G’ The inverse of the impedance matrix
V Node potential vector
i
L
n
(t) Load current in branch n
i
n,sum
2
The sum of the squared instantaneous value of the load current in branch n
t
start
Start time of a region of operation
t
stop
Stop time of a region of operation
i
n,sum,rel
2 Sum of the squared instantaneous values of the load current in branch n 
normalised over one period
N Number of periods
i
n,sum,0
2 The sum of the squared instantaneous value of the load current in branch n
during a balanced power supply and quasi-steady-state region of operation
Ψ Flux linkage
L
dn
Dynamic inductance of the load in branch n
36 Nina Štumberger, Matej Pintarič, Gorazd Štumberger JET Vol. 15 (2022)
Issue 1
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N om e n c l a t ur e
(Symbols) (Symbol meaning)
V
 0
Phasor of the zero-sequence voltage
V
 1
Phasor of the positive-sequence voltage
V
 2
Phasor of the negative-sequence voltage
V
 a
Phasor of the voltage phase voltage in phase ‘a’
V
 b
Phasor of the voltage phase voltage in phase ‘b’
V
 c
Phasor of the voltage phase voltage in phase ‘c’
exp Exponential function
a exp
(
j2
π
3
)
j Imaginary unit
VUF Voltage unbalance factor
ZSVUF Zero sequence voltage unbalance factor
u
L
(t) Inductor voltage
v
k
(t) Potential in point k
v
m
(t) Potential in point m
i
km
(t) Inductor current at time t
L Inductance
t Time
∆t Time period
Auxiliary variable
f (x (t ),t )
Integrated function
x (t ) Value of the integral of function f
I
km
(t-∆t ) Current describing the history of the inductor current
i
n
(t ) Current of a phase
u
n
(t ) Supply voltage of a phase
n Index denoting one of the phases, n  {a,b,c }
R Resistance of the current source and cable

50 JET