852
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005
A New Technique for Unbalance Current and Voltage
Estimation With Neural Networks
F. Javier Alcántara and Patricio Salmerón
Abstract—In this paper, a new measurement procedure based
on neural networks for the estimation of harmonic powers and
current/voltage-symmetrical components is presented. The theory
foundation is the Park vectors representation of a three-phase
voltage/current. The measurement system scheme is built with
three neural network blocks. The first block is a feedforward
neural network that computes the Park vectors and the zero-phase
sequence components. The second block is an adaptive linear
neuron (ADALINE) that estimates the harmonic complex coefficients of the current/voltage Park vectors. A third block is another
feedforward neural network that obtains symmetrical components of current/voltage harmonics and harmonic active/reactive
powers. Finally, to check the measurement method performance,
the digital simulation results of a practical case are presented.
Index Terms—Adaptive estimation techniques, artificial neural
networks (ANNs), harmonics, measurements, symmetrical components.
I. INTRODUCTION
T
HE large-scale presence of three-phase and single-phase
nonlinear loads in power systems makes necessary
a system monitoring of harmonic powers and unbalanced
current/voltage. The identification of harmonics produced by
the nonlinear loads and the determination of the degree of
voltage and current unbalance takes part in the evaluation
requirements of electric waveform quality [1], [2]. Thus, the
importance of the electric power quality (PQ) makes having
new methodologies for the analysis and measurement of the
basic electric magnitudes in these conditions necessary. This
task is still not considered well developed. Methods with short
computation time for real-time calculation must be employed.
There are different procedures for the determination of the
harmonic content of an electric signal. Those procedures, which
are derived from the discrete Fourier transform (DFT) as well
as its fast execution algorithm, are usual. If certain undesirable
conditions are given, this method can produce aliasing, leakage,
and picket-fence phenomena. The Kalman filter would constitute another method. This is a minimum square estimator that is
applied to dynamic systems. The main inconvenience is that it
needs the exact definition of the states matrix. Finally, the use
of neural networks appears as an alternative method [3]–[5].
Manuscript received November 12, 2003; revised October 26, 2004. This
work is part of the projects “Study of Electrical Waveform Quality: Their
Measurement and Control” (Grant TIC97-1221-C02-01) and “Electric Power
Quality Control Based in Neural Networks” (Grant DPI2000-1213) supported
by the Ministerio de Ciencia y Tecnología (CICYT), Spain.
The authors are with the Department of Electrical Engineering, Huelva
University, 21819 Palos de la Frontera, Spain (e-mail: benju@uhu.es;
patricio@uhu.es).
Digital Object Identifier 10.1109/TPWRS.2005.846051
The artificial neural networks (ANNs) have been systematically applied to electrical engineering [3]–[7]. Nowadays,
this technique is considered a new tool for digital signal processing. The ANNs present two main characteristics. First, it
is not necessary to set specific input–output relationships, but
they are formulated through a learning process or through an
adaptive algorithm. Second, the parallel computing architecture
increases the system speed and reliability.
In this paper, adaptive and feedforward neural networks techniques have been systematically explored for symmetrical components estimation. Thus, a original method for the measurement of three-phase current/voltage symmetrical components
and harmonic powers in unbalanced systems is presented. The
proposed technique is based on an estimation of the instantaneous time phasors (Park vectors) using the ANN with a reduced computational cost. The performance of the measurement
system increases the speed and reliability of the new parallel
computing architecture. Finally, digital simulation results of a
practical case are presented.
II. THREE-PHASE CIRCUITS: SYMMETRICAL COMPONENTS
AND POWER TERMS
The Park vectors are used to carry out a simplified analysis
of unbalanced three-phase power systems. The current/voltage
Park vectors allow the description of three-phase three-wire systems by means of the use of two complex quantities [8]. In this
section, an analytic procedure to obtain the current/voltage harmonics symmetrical components in a three-phase system from
Park vectors is presented. In the following section, this procedure will be implemented with ANNs.
The Park voltage vector is defined as
(1)
where
and , ,, are the phase voltage.
An identical relationship is possible for the three-phase line
currents
(2)
The Park vectors can be related to the phasors of the
symmetrical components. In fact, if a sinusoidal voltage
with pulsation
is supposed, its Park vector can be expressed
as follows:
(3)
where
is the positive sequence phasor, and
is the negative sequence phasor. This relationship can be extended to non-
0885-8950/$20.00 © 2005 IEEE
ALCÁNTARA AND SALMERÓN: A NEW TECHNIQUE FOR UNBALANCE CURRENT AND VOLTAGE ESTIMATION WITH NEURAL NETWORKS
sinusoidal situations by expressing the Park vector in terms of
its Fourier series expansion
fact, the analysis of the zero-sequence component is obtained
by means of the real sequence
(12)
(4)
where the coefficients
are the complex amplitudes. Thus, this
Fourierseriesexpansionprovidestwophasorsattheharmonicfrequency
(
). These phasors have different magnitudes
and rotate with the same ratio but in the opposite direction.
, these phasors are shown in
At the harmonic frequency
the following equations:
for
for
(5)
where
and
are the phasors of positive and negative sequence symmetrical components at the harmonic voltage index
. This property of the Park vector can be exploited to obtain the
positive and negative sequence symmetrical components of the
voltage and current through their complex DFT. Besides, (4) can
be expressed in a trigonometrical form by expanding the complex exponential in the real and imaginary parts. That is
853
This expression relates with the zero-sequence component
phasor of the voltage as follows:
Re
(13)
The zero-sequence
phasor is obtained by means of a
mathematical procedure analogous to the one followed for the
positive and negative symmetrical components. It comes from
the Fourier series expansion of
(14)
Equation (14) is equal to (13). Expanding (13) in a trigonometrical form
(15)
When (14) and (15) are identified, the following is obtained:
(6)
(16)
In particular, the fundamental harmonic is obtained
(7)
with
(8)
Expression (7) relates to the positive and negative sequence
symmetrical components. By means of the expansion of the two
complex exponentials, (3) can be expressed as
Equations (11) and (16) give the symmetrical components of
a three-phase system
,
,
by means of the coefficients
,
,
,
of the Park vector.
The previous procedure can be likewise applied to the current
Park vector to obtain the harmonic symmetrical components
of the current.
Finally, the active power and reactive power of each harmonic
can be calculated by means of current/voltage-symmetrical
components phasors using the following relationships:
(17)
(9)
where
(18)
,
,
,
,
, and
are the harmonic
where
current/voltage-symmetrical components, respectively.
(10)
Equations (7) and (9) must be equal. This identification makes
it possible to compute the phasors
and
by means of
the Fourier series expansion of the Park vector (
and
).
After the identification is made, the following four equations
are resulted:
(11)
The generalization of (7)–(11) can be made for all harmonic
terms. Besides, the same relationships for the current Park
vector can be obtained.
Additional calculus is necessary in three-phase systems
where there are zero-sequence symmetrical components. In
III. NEURAL MEASUREMENT SYSTEM
This paper proposes a new approach based on the application of neural networks to the estimation of the magnitude and
phase of the voltage/current-symmetrical components of individual harmonics. Fig. 1 presents the general scheme of the
neural measurement system (NMS).
In Fig. 1, three blocks are distinguished. Block 1 takes charge
of obtaining (1) and (11) [Park vectors and the zero-sequence
component
] for the voltage and current. This block is
carried out by means of a feedforward neural network with an
offline training process. Block 2 gives the complex coefficients
of the expansions (5) and (13) ( ,
,
,
). Block 2 is
built by a set of adaptive neurons with an online training process
(ADALINE). Finally, block 3 (a feedforward ANN) permits to
obtain the symmetrical components of the three-phase voltage
and currents of each individual harmonic, according to (10) and
854
Fig. 1.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005
Measurement system based on artificial neural networks.
Fig. 3.
Feedforward neural network architecture.
Fig. 4.
Adaptive linear network structure.
Fig. 2. Artificial neuron model.
(15). In the following subsections, the detailed analysis of each
block is described.
A. Neural Network Principles
The ANNs consist of a large number of strongly connected
elements: the artificial neurons, a biological neuron abstraction
carried out in a computer program. The model of an artificial
neuron in a schematic configuration is shown in Fig. 2.
The input data
flow through the
synapsis weights
and then accumulate in the node represented
by a circle. The weights
amplify or attenuate the input signals
before the addition. The summed data flow to the output through a
transfer function , which may be the threshold one, the sign one,
the linear threshold one, or the pure linear one. Otherwise, it may
be a continuous nonlinear function such as the sigmoid one, the
inverse tan one, the hyperbolic one, or the Gaussian one.
The neurons are interconnected, creating different layers. The
architecture most commonly adopted is the feedforward one, as
it is shown in Fig. 3 and [9].
The feedforward architecture consists of three layers: the
input layer, the hidden layer, and the output layer. In Fig. 3, the
circles represent neurons. The output layer has neurons equal
to the number of outputs. The particular network shown in
Fig. 3 has six hidden layers neurons in two hidden layers and
three output layer neurons. Thus, the feedforward architecture
computes the input data in parallel, faster than the sequential
algorithm of the computers. This network can be trained to give
a desired pattern at the output, when the corresponding input set
of data is applied. The training method most commonly used
for the feedforward network is the backpropagation training
algorithm. The network is initially untrained with selected
random weights. The initial output pattern is compared with
the desired output pattern, and the weights are adjusted by the
algorithm until the error becomes small enough. The training
process is carried out by means of a program that uses a large
number of input/target data. The training input/target data can
be derived from a simulation or from experimental results.
B. Adaptive Estimation
The adaptive neural networks can estimate any function
of a dynamic system whenever this can be expanded by means
of a linear combination of time-dependent functions belonging
to an orthonormal basis set. In this paper, the following model
of periodical signals to be estimated is proposed:
(19)
and
are the amplitudes (in general, complex amwhere
plitudes) of cosine and sine components of order- harmonic.
In vectorial notation
(20)
The general scheme of an adaptive neural network can be observed in Fig. 4. The signals are sampled at uniform rate
.
So, time values are discrete
, where
. The dot
product presented in (20) is carried out by an Adaline, where
is the network weights vector. After an initial estimation of
in the case
with random weights, an adaptive algorithm updates the weights, and the estimated signal converges
to the actual one. The final network weights are the expected
cosine and sine harmonic components.
ALCÁNTARA AND SALMERÓN: A NEW TECHNIQUE FOR UNBALANCE CURRENT AND VOLTAGE ESTIMATION WITH NEURAL NETWORKS
855
In a more concrete way, at time
, the network receives
as inputs the vector
and the desired output
(actual
signal). The neurons, taking into account their weights
,
carry out an estimation
To make the algorithm faster, a modification over the original
algorithm is included. This consists of including the vector
in the weight-adaptation equation. Thus
(21)
(28)
is the difference between the actual signal and
The error
its estimation. An algorithm that minimizes the error allows the
. After this
weights to be used in the next iteration
iterative process operates, the estimated signals adapt to the actual signals.
The network is trained on line, where the input signals of
and the corresponding desired output
are presented
to the network. The weight update algorithm automatically adjusts the weights so that the output response will be as close
as possible to the desired output. The method for adapting the
weights is the simple least mean square (LMS) algorithm, often
called the Widrow–Hoff delta rule. This algorithm is governed
by the following expression:
(22)
This algorithm minimizes the sum of squares of the linear errors over the input dimension. The linear error
is defined
as the difference between the desired output
and the linear
output
. This algorithm uses an instantaneous gradient to
follow the path of steepest descent and minimizes the mean
square error (MSE). The parameter is adjusted, as shown in
the following equation, where
is a constant:
(23)
Thus, depends on the linear error and linear error derivative,
and by this way, the algorithm convergence is improved.
This methodology was applied to estimate the output signals
of block 2 of the NMS. In this block, the inputs are the current
and voltage Park vectors and the sequences and . The special case of the voltage Park vector is
(24)
where
(25)
(26)
and
..
.
(27)
in accordance with (6).
The mostinteresting outputs of this adaptiveneural network are
theweightcoefficients
.Whenthenetworkistrainedandthe
error is small enough, these weights agree with the coefficients of
the expansion in (6) and (14),
, ,
, and .
where
sgn
In (28), the first term that contains
speed of the algorithm because the elements of
to or greater than the elements of
.
(29)
increases the
are equal
C. NMS
The NMS (see Fig. 1) developed in this paper
was applied to estimate the harmonic power and the
voltage/current-symmetrical components of a three-phase
four-wire power system. The inputs are the three-phase
voltages and the three-line currents corresponding to an
unbalanced and nonsinusoidal three-phase power system.
The outputs are the voltage/current-symmetrical components
phasors from the dc component to the 20th harmonic.
The proposed system NMS has been structured in three
different blocks carried out with the same technology, the
ANNs. The use of oneself technology for the three blocks
decreases the appearance of compatibility problems and simplifies the global design. Blocks 1 and 3 should carry out
simple lineal operations, because of that simple single layer,
neural networks with lineal transfer functions will be used.
This supposes, first, networks of a reduced number of neurons
and, second, networks of very easy previous training. Block 2
constitutes the kernel of the estimation system. For their design,
previous training networks of perceptron multilayer type with
backpropagation training algorithm and adaptive networks
without previous training were attempted. Due to the great
variety of possible electric variables for the estimation, the first
ones showed difficulties in the training process. However, the
second ones, which do not require previous training, were able
to adapt quickly to any change in the system.
The first block gives as output the voltage/current Park
and
zero-phase sequences.
vectors as well as the
The neural network is made of six neurons layer because six
outputs are required. A pure linear function is chosen as a
transfer function (see Fig. 5). The reason that this architecture
has been chosen is the linearity of (1), (2), and (12). This
selection allows the simplification of the employed scheme of
the network. The input data sets needed for training the network were generated through a set of simulations of the power
systems in Matlab/Simulink. In each simulation, the parameters
of the nonlinear load were changed, but it was found that the
neural network resulting from each case was exactly the same.
Because of that, it was decided to end the training process
without simulating more different loads. The explanation of
that fast process is the linearity of (1), (2), and (12). The output
data necessary to give the targets of the training process to the
neural network were calculated by means of (1), (2), and (12),
which were built with the Simulink blocksets.
856
Fig. 5.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005
Artificial neuron model with pure linear transfer function.
The second block of the measurement system is built through
an adaptive linear network. As inputs, this block has the
voltage/current Park vectors and the
and
zero-phase
sequences. At the output, the complex coefficients of the
Fourier series expansion of the inputs signals are obtained.
This expansion was truncated in the 20th harmonic. However,
this does not lead to any significant precision losses of the
estimations. The inputs of this block are the adaptive neural
network weights of the second block.
These coefficients are the inputs of the third block made of a
feedforward neural network. The inputs number is 126, a quantity that results when the 21 harmonics (20 and the dc component) are multiplied by the six inputs necessary for each harmonic. The number of the outputs is 294, which corresponds to
symmetrical components, active and reactive powers of the 20
harmonics, and the dc component.
The last neural network is split in 21 independent subnetworks that compute the input data corresponding to each
individual harmonic. This architecture makes it possible to
decrease the complexity of the network and increase the
parallelism and computation speed. This election has a justification due to (11) and (16), where there is no dependence
between the desired results for symmetrical components of
each harmonic and the other harmonics. Each subnetwork can
be built through six linear neurons due to the linearity of (11)
and (16). Besides, a calculation block computes the active and
reactive powers following (17) and (18). The power system
was simulated with the blocksets of Simulink to obtain the sets
of inputs/outputs data essentials for the offline training. The
training process of this block was realized following an equal
method to that used with the first block, and also, it was as
quick as that.
Fig. 6.
Unbalanced three-phase system used as a comparison example.
Fig. 7.
Symmetrical components of the current.
Fig. 8.
Active power.
IV. PRACTICAL CASE
A simple initial example has been included as a comparison
example constituted by a unbalanced four-wire three-phase
system, as shown in Fig. 6.
The theoretical results of the circuit are given as follows:
(30)
(31)
The most relevant results of the measurements made by the
NMS are presented in Figs. 7 and 8.
As an application, the simulation of a practical case with a
three-phase voltage source of positive sequence, where phase
1 is
, was considered. A three-phase nonlinear
with
unbalanced load with a star four wire configuration is used. The
load is a RL branch in series with two antiparallel SCRs. The
circuit of the practical case is shown in Fig. 9. Since
,
a change in parameters of the load was made to evaluate the
dynamic performance of the measurement system.
All elements values of the power system load are included in
Table I
ALCÁNTARA AND SALMERÓN: A NEW TECHNIQUE FOR UNBALANCE CURRENT AND VOLTAGE ESTIMATION WITH NEURAL NETWORKS
Fig. 9.
857
Circuit of the practical case.
TABLE I
PARAMETERS OF THE LOAD
Fig. 10.
Fig. 11.
Phase angles of the first, second, and fifth current harmonics.
Fig. 12.
Harmonics active power of the first, second, and fifth harmonics.
Amplitudes of the first, second, and fifth current harmonics.
The simulation of the system was made in Matlab, using the
Simulink blocksets. The most relevant results are presented.
In Fig. 10, the three-phase rms magnitudes of the current
positive sequence of first harmonic and the current negative sequences of second and fifth harmonic are presented. Fig. 10
shows that the estimation of those sequences is obtained between two periods and two periods and a half (
).
In Fig. 11, the arguments of the current positive sequence for
the first harmonic and the current negative sequences for the
second and fifth harmonics are shown. The settling time for the
estimations is between two periods (second harmonic) and three
periods (first and fifth harmonics).
Figs. 12 and 13 show the three-phase active and reactive
power of first, second, and fifth harmonics. The settling time of
estimation oscillates between two and three periods. A change
of load is produced in
. In this situation, the initial
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005
The followed procedure consists of the application of the
Park vector theory to the analysis of unbalanced three-phase
systems. The NMS is composed of three blocks. The first block
builds the Park vectors and the zero component of the voltage
and current. The second block obtains the complex amplitudes
of harmonic Park vectors. Finally, the third block computes the
active and reactive powers and the symmetrical components of
each individual harmonic starting from the complex amplitudes
obtained in the second block.
The algorithm was applied to a nonlinear unbalanced threephase system as a practical case. The results allowed to conclude that the adopted approach is viable for the determination
of the harmonic powers and the symmetrical components of the
three-phase waveforms according to an acceptable commitment
between accuracy and speed.
REFERENCES
Fig. 13.
Harmonics reactive power of the first, second, and fifth harmonics.
TABLE II
NUMERICAL RESULTS OF THE MEASUREMENTS
weights make a better weight adaptation time. Table II shows
the numerical results of the measurements in the considered
practical case.
V. CONCLUSION
A novel measurement procedure of symmetrical components
and harmonic powers in electric power systems is presented.
This method is based on the use of artificial neural networks,
which increases the measurement system speed and reliability
by the parallel computing architecture. The measurement
system is designed to estimate the active and reactive
powers and the current/voltage symmetrical components
in a three-phase four-wire unbalanced and distorted system.
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F. Javier Alcántara was born in Bollullos del
Condado (Huelva), Spain, in 1970. He received the
degree in physics from the University of Seville,
Seville, Spain, in 1993. He is working toward the
Ph.D. degree at the Huelva University, Palos de la
Frontera, Spain. His research includes the power
theory and the measurement and quality of the
electrical power using artificial neural networks.
Since 1996, he has been with the Department
of Electrical Engineering of the Huelva University,
where since 2004, he has been a Professor in the
Escuela Politécnica Superior.
Patricio Salmerón was born in Huelva, Spain. He
received the Ph.D. in physics from the University of
Seville, Seville, Spain, in 1993.
From 1983 to 1993, he was with the Department
of Electrical Engineering, University of Seville, and
since 1993, he has been a Professor of Electric Circuits and Power Electronics with the Department of
Electrical Engineering, Huelva University, in the Escuela Politécnica Superior, where at present, he is the
Dean. He has joined various projects in connection
with research in power theory of nonsinusoidal systems and power control in electrical systems. At present, his research includes
electrical power theory, active power filters, and artificial neural networks.