Probabilistic Modeling of Negative-Sequence Current in Three-Phase
Power Systems with Statistically Unbalanced Phases
DIEGO BELLAN
Department of Electronics, Information and Bioengineering,
Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milan,
ITALY
Abstract: - In this work the parameters of an unbalanced three-phase component, such as a load or a
transmission line, are described as random variables with known statistical properties. It is well-known that
phase unbalancing leads to injected current in the negative-sequence circuit producing propagation of voltage
unbalance and malfunctioning of power system components. In this paper, the injected current is described in
probabilistic terms as a function of the statistical properties of the component unbalancing. In particular,
approximate expressions for the probability density function, the mean value, and the variance of the injected
current are derived in closed form and numerically validated.
Key-Words: - Asymmetrical three-phase systems, negative sequence current, probabilistic models, statistical
analysis, unbalanced phases, voltage unbalance.
Received: September 19, 2021. Revised: September 14, 2022. Accepted: October 18, 2022. Published: November 11, 2022.
1 Introduction
Three-phase power systems under steady-state
conditions are commonly analyzed by resorting to
the well-known Symmetrical Component
Transformation (SCT), [1], [2], [3], [4]. The main
advantage of the SCT consists in the derivation of
three uncoupled circuits named
positive/negative/zero-sequence circuits, with
transformed voltages/currents named
positive/negative/zero-sequence voltages/currents.
This result can be achieved because the SCT is
able to decouple the originally coupled equations
describing the given three-phase system, [4]. The
main assumption underlying the effectiveness of
the SCT is that the three phases are symmetrical,
i.e., the passive part of the three phases is balanced
(e.g., equal self-impedances, and equal mutual
impedances). This is an ideal condition that can be
only approximated in real-world applications by 1)
using transposition of transmission lines, 2)
distributing the loads uniformly between the three
phases.
In modern power systems, however, balancing
the three phases is becoming more and more
complicated, especially because of the increasing
number of large single-phase loads. The main
consequence of unbalanced phases is that the SCT
leads to coupled sequence circuits. Thus, the
positive-sequence circuit injects current in both the
negative and the zero-sequence circuits, [5]. This is
a critical point because negative and zero-sequence
currents can easily cause malfunctioning of many
electrical apparatus connected to the power system,
[6]. For this reason, many research contributions can
be found in the literature about modeling and
calculation of voltages and currents due to
unbalanced phases (i.e., asymmetrical lines and
loads), [7], [8], [9], [10], [11], [12], [13], [14], [15],
[16], [17], [18], [19]. Such contributions, however,
assume a deterministic knowledge of system
asymmetries. In many practical conditions it would
be more useful to consider the statistical uncertainty
of line/load parameters, [20].
In this paper, it is assumed that unbalancing of
the three phases of a given component is described
in statistical terms. More specifically, it is assumed
that the parameters of a three-phase component (i.e.,
line or load) can be treated as random variables with
a given statistical distribution. Such an unbalanced
component is responsible for injection of current in
the negative-sequence circuit (the zero-sequence
circuit is not involved since the analysis is limited to
three-wire systems). The injected current, as a
function of the unbalanced component parameters,
is a random variable as well. The main objective of
the paper is providing the probabilistic description
of the injected negative-sequence current as a
function of the statistical properties of the
unbalanced component parameters. In particular,
approximate analytical expressions for the
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2022.17.36
Diego Bellan
E-ISSN: 2224-350X
364
Volume 17, 2022
probability density function, the mean value, and the
variance of the injected current are derived in closed
form.
The paper is organized as follows. In Section 2 an
approximate expression for the negative-sequence
current is derived as a function of the parameters of
an unbalanced three-phase component. In Section 3
the probabilistic modeling of the injected current is
derived in closed form by assuming Gaussian
distribution of the unbalanced component
parameters. In Section 4 the analytical results are
validated by means of numerical simulations.
2 Negative-Sequence Current due to
Unbalanced Phases
Let us consider a basic three-phase radial network
consisting of a voltage source, transmission line,
and load (see Fig. 1). Under sinusoidal steady-state
conditions, the system can be readily analyzed by
means of the well-known SCT. In fact, under the
crucial assumption of balanced (i.e., symmetrical)
phases, the SCT leads to three uncoupled circuits in
the transformed variables, i.e., the so-called
positive/negative/zero-sequence circuits with
corresponding sequence voltages and currents. Each
sequence circuit can be solved independently from
the other circuits through standard techniques.
Finally, the inverse SCT can be applied to the
sequence-domain solution to obtain the solution in
the original a,b,c, domain.
The procedure outlined above is effective under
the crucial assumption of balanced phases, i.e., a
passive three-phase component characterized by a
symmetrical matrix with equal elements on the main
diagonal, and equal off-diagonal elements. For
example, by considering a three-phase component
characterized by an impedance matrix, the three
self-impedances must be equal to each other, and the
three mutual impedances must be equal to each
other. In case this crucial assumption is not met, the
SCT leads to coupled sequence circuits instead of
uncoupled. By assuming that the system depicted in
Fig. 1 is a three-wire three-phase system, only the
positive and the negative-sequence circuits are
involved in the analysis since the zero-sequence
circuit is open. Therefore, an unbalanced three-
phase component in Fig. 1 results in coupled
positive and the negative-sequence circuits (Fig. 2).
Fig. 1: Three-phase radial network.
Fig. 2: Positive and negative-sequence circuits.
Balanced three-phase components result in the total
sequence impedance Zs. Coupling is due to the
unbalanced three-phase component with impedance
matrix Zu.
In general terms, the solution of the circuit in
Fig. 2 requires the evaluation of coupled equations.
Therefore, in this case the SCT does not lead to a
simpler solution with respect to the original problem
in the a,b,c variables. However, in [19] it was
shown that, under proper and general assumptions,
the circuit coupling in Fig. 2 can be regarded as a
weak coupling. It means that an approximate
approach can be adopted where the feedback from
the negative to the positive-sequence circuit can be
neglected. Thus, the positive-sequence current can
be evaluated by neglecting circuit coupling, whereas
the negative-sequence current can be evaluated by
considering a simple controlled source taking into
account the circuit coupling. This approximate
approach can be easily explained by observing that,
under ideal conditions, the negative-sequence
current is zero. Thus, a component unbalance can be
regarded as a perturbation of an ideal system where
only the positive-sequence current is flowing. The
unbalanced three-phase component injects current in
the negative-sequence circuit. As a result, the
negative-sequence current is normally a small
fraction of the positive-sequence current. This point
explains the above mentioned weak-coupling
assumption.
More specifically, let us consider an unbalanced
three-phase component characterized by the
following impedance matrix:
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(1)
By denoting with Z the balanced value of the
impedances, we can write:
(2)
By using the SCT in (2), the balanced matrix Z
remains unchanged, whereas the deviation in the
sequence domain is given by:
(3)
where:
(4)
(5)
(6)
and the SCT matrix S in rational form is defined as:
(7)
where
.
Since we assume the system in Fig. 2 as a three-wire
system, only the positive and negative-sequence
circuits must be considered. Thus, circuit coupling
between those sequence circuits is described by the
first two rows and columns of the matrix in (3):
(8)
Notice that by assuming small impedance
deviations, i.e.,
(9)
the terms and in (9) can be neglected.
Moreover, the above-mentioned approximate
approach based on the assumption of weak coupling
between the two sequence circuits allows neglecting
the term representing the feedback from the
negative to the positive-sequence circuit. Thus, from
(8) we obtain a voltage perturbation in the negative-
sequence circuit given by:
Fig. 3: Approximate equivalent circuit for the
unbalanced component Zu. Unbalancing is
represented by a current-controlled voltage source in
the negative-sequence circuit.
(10)
Such perturbation can be represented as a current-
controlled voltage source (see Fig. 3). The negative-
sequence current is given by:
(11)
where is the total balanced impedance in the
positive and negative-sequence circuits. Thus, the
ratio between negative and positive-sequence
currents is given by:
(12)
The current ratio given in (12) will be analyzed as
a random variable in the next Section.
3 Probabilistic Modeling of Negative-
Sequence Current
The negative-sequence current (normalized by the
positive-sequence current) given by (12) can be
regarded as a random variable when the impedance
deviations of the unbalanced three-phase
component are given in statistical terms. For the
sake of simplicity, in this paper we assume real
impedance deviations (i.e., we assume that
only the resistive parts are deviating from the
balanced value). Therefore, from (5) we obtain that
(12) can be rewritten as:
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(13)
Therefore, can be regarded as a function of the
random variables . In this Section the
statistical properties of the random variable will
be derived as functions of the statistical properties
of the uncorrelated random variables . To
this aim, we study the following transformation of
random variables:
(14)
where:
(15)
Assuming with zero mean values, the
random variables x and y are also unbiased, i.e.,
. The variances of x and y are given by:
(16)
Notice that even in the case of with
Gaussian distribution, the random variable w given
by (14) is not a Rayleigh variable for two reasons.
First, the two variances in (16), in general, are not
equal. Second, since and are both involved in
the definition of x and y in (15), the random
variables x and y, in general, are correlated. The
correlation coefficient can be readily evaluated,
[21], [22], [23]:
(17)
From (17) it can be observed that in the special case
the random variables x and y are
uncorrelated. In this case, the random variable w in
(14) can be approximated by a Rayleigh random
variable with parameter given by the average
value of the variances and , i.e.,
(18)
where is the probability density function (PDF)
of w, and
(19)
In the general case, however, the random variables x
and y are correlated, and the approximate PDF (18)
cannot be used. In this case an approximate
approach based on the Taylor expansion can be used
to estimate the mean value and the variance of w,
[24]. To this aim, in order to obtain simpler and
consistent derivations, a further change of variables
is used. By letting and , from (14) we
obtain:
(20)
By assuming Gaussian distribution for the random
variables , the mean values and the
variances of u and v are given by:
(21)
(22)
Notice again that, in general, the random variables u
and v are correlated since and are both in the
definition of u and v. Therefore, the evaluation of
the correlation coefficient is required. After
simple algebra we obtain:
(23)
The correlation coefficient is the square of
because of the definition of u and v as the square of
x and y, respectively. We can observe that keeps
the information about the sign of the correlation,
whereas is always positive. Also in this case we
obtain when .
The mean value and the variance of the random
variable w can be estimated by resorting to the
Taylor series expansion, [21], [22], [23], [24]:
(24)
(25)
where the derivatives are evaluated at and
. Notice that if the same approach was used
for w as a function of x and y, instead of u and v, all
the derivatives in (24)-(25) were zero. This explains
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the reason for the use of the variables u and v.
From (24)-(25) after simple algebra we obtain:
(26)
(27)
where , , and are given by (16) and (23) as
functions of , .
Finally, it can be noticed that the above analytical
results can be readily used for the original random
variable given in (13). In particular, the
approximate PDF (18) becomes:
(28)
whereas the mean value and the variance (26)-(27)
provide:
(29)
(30)
4 Numerical Validation
Analytical results derived in Section 3 have been
validated through numerical simulations. In
particular, the formulas (17) and (23) for the
correlation coefficient, (18) for the PDF, and (26)-
(27) for the mean value and the variance have been
assessed against numerical simulations. Notice that
the random variable w has been tested instead of .
Indeed, since (see (13)), analytical
results for and for w are related in a
straightforward way as shown by (28)-(30).
Numerical simulations were performed by
selecting as uncorrelated zero-mean
Gaussian random variables with standard deviations
. The standard deviation was assumed
as reference, with value . Thus, the two
ratios
and
were considered as
parameters or variables. Repeated runs were
performed (106 runs) to obtain the numerical
estimate of each quantity to be validated (i.e.,
correlation coefficient, PDF, mean value and
variance). In the following figures, numerical results
from simulations are represented with dotted lines,
whereas analytical results derived in Section 3 are
represented with solid lines.
Fig. 4: Correlation coefficient (17) between
variables x and y as defined in (15), as a function of
, for four different values of
.
Numerical results (not visible in this figure) are
perfectly overlapped by the solid lines
corresponding to analytical results.
Fig. 5: Correlation coefficient (23) between
variables u and v as defined before (20), as a
function of
, for four different values of
. Numerical results (not visible in this figure)
are perfectly overlapped by the solid lines
corresponding to analytical results.
Fig. 4 shows the behavior of the correlation
coefficient , given by (17), as a function of
by assuming
as a parameter. The solid lines
corresponding to (17) overlap the dotted lines
corresponding to numerical results. Therefore,
dotted lines are not visible in Fig. 4. Notice that all
the curves provide for as it was
clearly shown by (17). Moreover, according to (15),
correlation becomes negative as increases, and
vice versa. From (15), it is also clear that by
increasing correlation decreases because the
random variable is involved only in the definition
of x.
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Fig. 5 shows the behavior of the correlation
coefficient , given by (23), as a function of
, by assuming
as a parameter. As
already mentioned
, thus Fig. 5 is strongly
related to Fig. 4.
Fig. 6. Probability density function of the random
variable w given by (20) for and three
different values of
.
Fig. 7: Probability density function of the random
variable w given by (20) for and three
different values of
.
Indeed, the information about the sign of the
correlation is lost, but from Fig. 5 the magnitude of
the correlation coefficient allows a quantitative
evaluation about the assumption of weakly
correlated random variables. In fact, in case of weak
correlation the random variable w given by (20) can
be approximated by a Rayleigh random variable.
This point is investigated in Fig. 6 where the PDF of
w is reported in the case . In Fig. 5 the curve
for shows a wide excursion of the
correlation coefficient which equals zero for
, whereas it takes increasing values as
moves away from 1. In Fig. 6 three values
were selected for
, i.e., 0.5, 1, 2. It can be
noticed that for
the Rayleigh PDF (18) is
a good approximation of the numerical PDF because
the random variables u and v are uncorrelated. On
the contrary, for
and
the
strong correlation between u and v leads to a worse
approximation of the Rayleigh PDFs to the
numerical results. This is what we expected on the
basis of the correlation coefficient in Fig. 5.
Fig. 8: Mean value of w, given by (26), as a function
of
, for three different values of
,
compared with numerical simulations.
Fig. 9: Standard deviation of w, given by the square
root of (27), as a function of
, for three
different values of
, compared with numerical
simulations.
Fig. 7 is similar to Fig. 6, but in this case
was assumed. From Fig. 5 we see that for
the correlation coefficient has a lower excursion.
Therefore, we expect that the Rayleigh PDF is a
better approximation to the numerical PDF with
respect to Fig. 6. Moreover, we expect that for
we obtain the best approximation
because in this case the correlation coefficient is
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Diego Bellan
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Volume 17, 2022
zero. This is confirmed by the three curves depicted
in Fig. 7.
Fig. 8 shows the behavior of the mean value of
w, given by (26), as a function of
, for three
different values of
, compared with numerical
simulations. The approximation provided by (26) is
satisfactory, even if it always provides a slight
underestimation with respect to numerical values.
Fig. 9 shows the behavior of the standard deviation
of w, given by the square root of (27), as a function
of
, for three different values of
,
compared with numerical simulations. In this case,
the approximation provided by (27) is less
satisfactory with respect to the mean value
approximation. The reason is likely due to the fact
that the Taylor approximation (25) for the variance
takes into account only the first derivatives, whereas
the approximate mean value (24) takes into account
the second derivatives. Nevertheless, the maximum
error in Fig. 9 is lower than 20%.
Finally, the case where the random variables
are uniform instead of Gaussian was
investigated. Fig. 10 shows the behavior of the PDF
of w by assuming uniform with ,
and three different values for
. The figure
should be compared with the analogous Fig. 7
where the same standard deviations were
considered, but the underlying random variables
were Gaussian. Notice that, contrary to
Fig. 7, in Fig. 10 even the case
is
deviating from the ideal Rayleigh behavior (solid
line) because of the uniform instead of Gaussian
distribution of .
5 Conclusions
In the paper it was shown that by assuming
Gaussian distribution for the parameters of an
unbalanced three-phase component, the current
injected in the negative-sequence circuit is
approximately distributed as a Rayleigh variable.
The degree of this approximation is strictly related
to the correlation coefficients between the real and
the imaginary parts of the injected current.
Therefore, the evaluation of the correlation
coefficient is crucial in the estimation of the
accuracy of the Rayleigh distribution. Moreover,
approximate expressions for the mean value and the
variance of the injected current have been derived.
The analytical result for the mean value is
satisfactory, whereas the analytical results for the
variance is affected by a larger error since only the
first derivatives are retained in the Taylor
expansion. Using higher order derivatives, however,
would lead to much more complicated analytical
expressions.
Future work will be devoted to extending the
derivations to a more general unbalanced three-
phase component instead of a simple resistive
component. Moreover, different kinds of statistical
distributions of the unbalanced component
parameters will be considered beyond the Gaussian
distribution. Future research will be also conducted
in order to extend the proposed approach to typical
power system loads characterized in terms of active
and reactive power instead of impedance.
Fig. 10: Probability density function of the random
variable w given by (20) for and three
different values of
, assuming uniform
distributions for instead of Gaussian as in
Fig. 7.
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