A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
A New Technique for Unbalance Current and Voltage Measurement with
Neural Networks
F. Javier Alcántara, Patricio Salmerón, Jaime Prieto
Electrical Engineering Department, Escuela Politécnica Superior, Huelva University
Ctra. Palos de la Frontera s/n
E-21819 Palos de la Frontera, Huelva, Spain
Phone: 00.34.959.017576, Fax: 00.34.959.017304
E-mail: FcoJavier.Alcantara@dfaie.uhu.es; Patricio.Salmeron@dfaie.uhu.es;
Jaime.Prieto@dfaie.uhu.es
Acknowledgments
This work is part of the projects, “Study of Electrical Waveform Quality: Their measurement and
Control” TIC97-1221-C02-01 and “Electric Power Quality Control based in Neural Networks”
DPI2000-1213 financed by the CICYT( Ministerio de Ciencia y Tecnología, Spain)
Keywords
Harmonics, Measurements, Neural Networks, Three-phase systems, Estimation techniques
Abstract
In this paper a new measurement procedure based in neural networks for the estimation of the current
and voltage symmetrical components is presented. The theory foundations are the Park Vectors
representation for a three-phase voltage/current. The measurement system scheme is built with three
neural network blocks. The first block is a feddforward neural network that computes the Park vectors
and the zero phase sequence components. The second block is an adaptative linear neuron
(ADALINE) that estimates the harmonic complex coeficients of the current/voltage Park vectors. A
third block is another feedforward neural network that obtains the symmetrical components of each
individual voltage and current harmonic. Finally, the estimation procedure of the symmetrical
components of a three-phase, unbalanced, nonlinear load current was applied to a practical case.
1. Introduction
Recently, the importance of the electric power quality (PQ) makes necessary to have new
methodologies for the analysis and measurement of the basic electric magnitudes in unbalance and
distorted situations. The identification of harmonics produced by the nonlinear loads and the
determination of the degree of voltage and current unbalance takes part of the requirements for the
evaluation of the electric waveform quality [1,2]. Thus, it is necessary a power system monitoring to
measure harmonic and unbalance current/voltage. This task is still considered not well developed.
Methods with short time of computation for real-time calculation must be employed.
There are diverse procedures for the determination of the harmonic content of an electric signal. They
are usual those derived from the Discrete Fourier Transform (DFT) and their fast execution algorithm
Fast Fourier Transform (FFT). If certain conditions are not desired, this method can produce aliasing,
leakage and picket-fence phenomenons. The Kalman filter would constitute another method. It is an
minimum square estimator that is applied to dynamic systems. It has the inconvenience that needs an
exact definition of the states matrix. Lastly, the use of the neural networks appears as an alternative
method [3-5].
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
In this paper, feedforward neural networks techniques have been systematically explored for the
symmetrical components estimation by the authors. Thus, a new method is presented for the
measurement of three-phase current and voltage symmetrical components in unbalanced sytems. The
proposed technique is based in an estimation of the instantaneous time-phasors (Park vectors) using
Artificial Neural Networks (ANN) with a reduced computacional cost. The performance of the
measurement system based in the neural principle was found to be excellent.
2. Three-Phase Circuits Analysis with Park Vectors
The Park Vectors are used for the analysis of three-phase power systems in unbalanced and distorted
conditions. The current/voltage Park Vectors allows the analysis of three-wire systems by means of
the use of some complex quantities [6,7]. The Park voltage vector is defined as:
2
( va + γ vb + γ 2 vc )
3
vp =
(1)
where γ=exp(j 2π/3) and va, vb,, vc are the phase voltage. Identical relationship is possible for the
three-phase line currents. The Park vectors can be related with the phasors of the symmetrical
components. In fact, if a sinusoidal voltage with pulsation ω1 is supossed, follows:
v p = V + e jω1t + V −∗e − jω1t
(2)
where V+ is the positive sequence phasor and V- is the negative sequence phasor. This relationship can
be extended to non sinusoidal situations by expressing the Park vector in terms of their Fourier series
expansion.
vp =
k =∞
∑Y e
jkω1t
k
(3)
k = −∞
where the coefficients Yk are the complex amplitudes. Thus, this Fourier series expansion provides
two phasors at the pulsation hω1 (h=|k|). These phasors are different magnitudes and rotates with the
same pulsation but in the opposite direction.
At the pulsation hω1, it follows
Yk = Vh
for k = h
Yk = V-∗h for k = − h
(4)
where Vh and V-h* are the phasors ofpositive and negative sequence symmetrical components of the
voltage at the harmonic index k. This property of the Park vector can be exploited to obtain the
positive and negative sequence symmetrical components of the voltage and current throught their
complex DFT. Besides, the equation (3) can be expressed in a trigonometrical form by expanding the
complex exponential in the real and imaginary parts. That is
∞
v p = A0 + ∑ An cosnω1t + Bn sen nω1t
(5)
n =1
Specifically, for the fundamental harmonic is obtained
v1p = A1 cosω1t + B1 senω1t
(6)
with
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
A1 = A1r + jA1i
(7)
B1 = B1r + jB1i
The expression (6) relates with the positive and negative sequence symmetrical components. In fact,
the equation (2) can be expressed as
v1p = ( V1+ + V1− ) cosω1t + j ( V1+ − V1− )sen ω1t
(8)
where
V1+ = V1+r + jV1+i
(9)
V1− = V1−r + jV1−i
Equations (6) and (8) must be equal. This identification permits to compute the phasors V+ and V- by
means of the Fourier series expansion of the Park vector (A1 and B1).
A1r + B1i
2
A − B1i
V1−r = 1r
2
A1i − B1r
2
A − B1r
V1−i = − 1i
2
V1+r =
V1+i =
(10)
The generalization of equations (6)-(10) can be made for any harmonic. For the current Park vector the
same expressions can be obtained.
A aditional calculus is necessary in three-phase systems where there are zero sequence symmetrical
components. In fact, the analisys of the zero sequence component is realized by means of the real
sequence
1
vo (t) =
3
( v a + vb + v c )
(11)
This expresion relates with the zero sequence component phasor of the voltage. It follows
∞
v 0 (t ) = 2 ∑ Re (Vn e jnω1t )
0
(12)
n =1
From (6), the zero sequence V0 phasor is obtained by means of a mathematical procedure analogous to
that followed for the positive and negative symmetrical components. It starts from the Fourier series
expansion of v0(t),
∞
vo (t ) = A0' + ∑ An' cosnω1t + Bn' sen nω1t
(13)
n =1
The equation (13) is equal to the equation (12) expanded in a trigonometrical form.
∞
vo (t ) = V0 r + ∑Vnr cosnω1t − Vni sen nω1t
0
0
(14)
n =1
Thus, is obtained
An'
V =
2
0
nr
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Bn'
V =−
2
0
ni
(15)
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
The equations (10) and (15) give the symmetrical components Vn+, Vn-, Vn0 by means of the
coeficients An, An’, Bn,, Bn’.
The previous procedure can be likewise applied to the current Park vector to obtain the harmonic
symmetrical components.
3. Neural Network-Based Estimation of Voltage and Current Symmetrical
Components
This work proposes a new aproach based in the application of the neural networks to the estimation of
the magnitude and phase of the symmetrical components of individual harmonics [8]. The figure 1
presents the general scheme of the measurement system.
va
vb
vc
ia
ib
ic
vp
An
v0
A’n
ip
Bn
i0
B’n
Vn
+
VnVn0
In +
In In 0
Neural block 1 Adaline block 2 Neural block 3
Figure 1: Measurement system
In the figure 1 three blocks are distinguished. The block 1 takes charge of obtaining the equations (1)
and (11) (Park vectors and the zero sequence component v0(t)) for the voltage and current. This block
is realized by means of a feedforward neural network with off-line training. The block 2 gives the
complex coeficients of the expansions (5) and (13) (An, An’, Bn, Bn’). The block 2 is built by some
adaptative neurons with on-line training process (ADALINE). Finally, the block 3 permits to obtain
symmetrical components of the three-phase voltage and currents of each individual harmonic,
according to the equations (10) and (15). The detailed analysis of each one of these blocks are
described next one
3.1. Neural Network Principles
The Artificial Neural Networks consits of a large number of hyghly connected elements: the artificial
neurons, an abstraction of biological neurons realized as elements in a computer program. The model
of an artificial neuron by a summerlike configuration is shown in figure 2.
i(1)
Wj
i(2)
Σ(Wj i(j))+b
i(3)
INPUT
DATA
f
.
.
i(n)
.
.
NEURON
OUTPUT
TRANSFER
FUNCTION
b
Figure 2. Artificial neuron model
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
The input data i(1), i(2), i(3), …, i(n) flow through the synapsis wheights and then accumulate in the
node represented by a circle. The wheights amplifies or attenuates the imputs signals before the
addition. The summed data flows to the output throught a transfer function, f, that can be the threshold
type, the signum type, the linear threshold type or pure linear type or it can be a continous nonlinear
function such as the sigmoid type, inverse tan type, hyperbolic type or gaussian type.
The neurons are interconected forming different layers. The arquitecture most commonly adopted is
the feddforward arquitecture shown in figure 3.
i(1)
Σ
Σ
Σ
Σ
Σ
Σ
Σ
Σ
Σ
i(2)
i(3)
i(n)
INPUTS
HIDDEN LAYERS
OUTPUT LAYER
Figure 3. Feedforward Neural Network arquitecture
The arquitecture in figure 3 consists of three layers: the input layer, the hidden layer and the output
layer. In the figure 3, the circles represent neurons. The output layer have neurons equal to the number
of outputs. The particular network shown in figure 3 have six hidden layers neurons in two hidden
layers and three output layer neurons. Thus, the feedforward arquitecture 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 ramdom weights selected. 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 is realized by 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.
3.2. Adaptative Estimation
The adaptative neural networks can estimate any function f(t) of a dynamic system whenever this can
be expanded by means of a linear combination of time-dependent functions. x(t)=(f1(t),f2(t), ...
,fN(t))T,.on expressed by a matrix form as,
f (t ) = W T • x (t )
(16)
The imput signals is sampled in the instants k∆t, with a fixed interval ∆t. This network receives at time
index k an input signal vector xk(t) and a desired output fk(t) used to effect the learning of the network.
The components of the input vector are wheighted by the coeficients of the linear combination W=(W1,
W2, ... ,WN) that are the weight vector. The sum of the weighted inputs is then computed, producing a
linear output, the inner product (Figure 4)
∧
f k (t) =W (k ) T • xk (t )
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
xk
vector
Wheight
update
algorithm
Figure 4. Adaptative Linear Network estructure
The network is trained on-line in the learning process, where the input patterns and the corresponding
desired responses are presented to the network. An adaptation algorithm automatically adjusts the
wheights so that the output responses to the imput patterns will be as close as possible to their
respective desired responses. The method for adapting the wheights is the simple LMS (Least Mean
Square) algorithm, often called Widrow-Hoff delta rule. This algorithm is governed by the expression
W(k + 1 ) = W ( k ) +
α ek x k ( t )
x kT (t ) x k (t )
(18)
This algorithm minimizes the sum of squares of the linear errors over the training set. The ∧linear error
ek(t) is defined to be the difference between the desired response fk(t) and the linear output f k (t) . From
equation (18) the error signal is necessary for adapting the weights. This algorithm uses an
instantaneous gradient to follow the path of steepest descent and minimizes the mean square error of
the training set. The α parameter is modified as shown in the following equation.
•
α = α 0 + c1e + c2 e
(19)
Thus, α depends of the linear error and linear error derivative and in this way improves the algorithm
convergence.
This metodology was applicated to estimate the input signals of the block 2. In the special case of the
voltage Park vector, is,
v p = W ( k ) T • x k (t )
where
W (k )T = [A0 (k ) A1 (k ) B1 (k ) ... ... An (k ) Bn ( k )]
A1n (k ) = A1nr ( k ) + jA1ni ( k )
B1n (k ) = B1nr (k ) + jB1ni (k )
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(20)
(21)
(22)
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
1


 cos kω1∆t 
 sen kω ∆t 
1
x k (t ) = 

!


cos nkω1∆t 
sen nkω1 ∆t 
(23)
The very interesting outputs of this adaptative neural network are the weight coeficients W(k). When
the network is trained and the error is small enough, this weights agree with the coeficients of the
expansion in the expressions (5) and (13). To makes the algorithm faster a modification is included
[8,9]. This consits in including the vector λk(t) in the weights adaptation equation. Thus
W(k + 1 ) = W(k ) +
α e k λk ( t )
x Tk (t )λk (t )
λk (t ) = 0,5sgn( x k (t )) + 0,5x k (t )
(24)
(25)
In equation (25) the first term that contains sgn[xk(t)] increases the speed of the algorithm because the
elements of λk(t) are equals or major than the elements of xk(t).
3.3. Symmetrical Components Measurement System
This neural network based methodology was applied to estimate the voltage/current symmetrical
components of a three-phase four wire power system. The measurement system shown in figure has
been realized. 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 since the dc component to the 20th harmonic. The first block give, as
output the voltage/current Park vectors as well as the v0(t) and i0(t) sequences. The neural network is
made with a six neurons layer because six output are required. The reason why this arquitechture has
been chosen is the linearity of equations (1) and (10). This election permits to simplify the employed
scheme of the network. Whit the same criterium, a pure linear transfer function is chosen. The input
and output data sets needed to training the network was generated throught a simulation of the power
system in Simulink. To make that it was necessary to aply the equations (1) and (11) with the
Simulink blocksets. The second block of the measurement system is built with a adaptative linear
network. As inputs, it has the voltage/current Park vectors and the v0(t) and i0(t) sequences. At the
output, the Fourier series expansion coeficients of the inputs signals are obtained. This expansion was
truncated in the 20th harmonic. However this don’t lead to any significant precision losses of the
estimations. The inputs of the second block are the adaptative neural network wheigts. When the
wheights are adapted, the linear error is small enough. Then the outputs of the second block
corresponds to the desired coeficients. This coeficients are the inputs of the third block made by a
feedforward neural network. The inputs number is 126, quantity that results when the 21 harmonics
(20 and the dc component) are multiplied by the 6 inputs necessary for each harmonic. The number of
the output is 246 that corresponds to 120 for the symmetrical components of the 20 voltage harmonics,
120 for the current symmetrical components and 6 for the dc components of the phase voltage and line
currents. The neural network is split in 21 independent subnetworks that computes the input data
corresponding to each individual harmonic. This arquitecture makes posible to decrease the
complexity of the network and increase the paralelism and computation speed. This election have a
justification due the equations (10) and (15). In this expression there is no dependence between the
desired results for symmetrical components of each harmonic and the other harmonics. Each
subnetwork can by built with six pure linear neurons due the linearity of the equations (10) and (15).
The power system is simulated with Simulink to obtain the sets of inputs/outputs data essentials for the
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A New Technique for Unbalance Current and Voltage Measurement with Neural Networks
F. Javier Alcántara
off-line training. The training process of the first and third blocks was realized by the use of the
functions incorporated in the Matlab Neural Networks Toolbox.
4. Practical case
As application was considered the simulation of a practical case with a three-phase source of positive
sequence, where the phase 1 is v1=220√2senω1t + 22√2sen2ω1t + 22√2sen5ω1t. A three-phase
nonlinear unbalanced load, with a star four wire configuration is used. The load is a RL branch in
series with two antiparallel SCRs. Since t=0.2 sec. 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
R1 (Ω)
R2 (Ω)
R3 (Ω)
L1 (mH)
L2 (mH)
L3 (mH)
Firing angle β
T<0.2s
100
80
150
2
5
9
45º
T>0.2s
120
100
100
2
5
9
45º
Table 1: Parameters of the load
The simulation of the system was made in Matlab, using the Simulink blocksets. The most relevant
results are presented.
0.4
4
I3(A)
I1+ 3
(A)
0.3
0.2
0.1
2
1
I30
(A)
1
-
I1 0.8
(A) 0.6
0.4
0.8
0.2
I5(A)
1.5
0
I1
(A)
1.4
1.2
1
0.8
0.6
0.4
0.2
0.6
0.4
0.2
1
0.5
I7+
(A)
0.05 0.1 0.15 0.2 0.25 t (s)
Figure 5. Symmetrical components magnitudes
of the fundamental harmonic
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.05 0.1 0.15 0.2 0.25 t (s)
Figure 6. Symmetrical components magnitudes
of the most relevant harmonics
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F. Javier Alcántara
In figure 5 the magnitudes and phases of the three sequences for the fundamental harmonic of the
current are presented. Is clear in the figure 5 that the estimation of the positive sequence is obtained in
one period. The negative and zero sequence components are obtained in two periods and a half.
In figure 6 the magnitudes and phases of the negative and zero components for the third harmonic of
the current are shown. Also is presented the negative component phasor for the fifth harmonic of the
current and the positive component phasor for the seventh harmonic of the current The necessary time
for those estimations is between one period (negative component of the fifth harmonic and positive
component of the seventh harmonic) and two periods and a half (negative component of the third
harmonic)
5. Conclusions
A new measurement procedure of the symmetrical components in electric power systems is presented.
This method is based in 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 current/voltage symmetrical components in a three-phase four wire unbalanced and
distorted system. The followed procedure consists in the application of the Park vector theory to the
unbalanced three-phase systems analysis. The measurement system 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 the espectral content of the Park vectors. Finally, the third
block computes the symmetrical components of each individual harmonic starting from the complex
amplitudes obtained in the second block. The algorithm was applicated to a nonlinear unbalanced
three-phase system as a practical case.
The obtained results allow to conclude that the adopted focus is viable for the determination of the
symmetrical components of the three-phase waveforms acording to an acceptable commitment
between accuracy and speed. The developed methodology can be used as a basis of a most general
procedure to measure the power terms in unbalanced and distorted conditions.
References
[1] J. Arrillaga, N. R. Watson, S. Chen. Power System Quality Assesment Wiley, 2000.
[2] IEEE Working Group on Nonsinusiodal Situations: Effects on Metter Performance and Definitions on
Power, Practical Definitions for Power in Systems with Nonsinusiodal Waveforms and Unbalanced Loads:
a Discussion, IEEE Trans on Power Delivery, Vol 11, no 1,January 1996
[3] P. K. Dash, S. K. Panda, Baburam Mishra, D. P. Swain. Fast Estimation of Voltage and Current Phasors in
Powers Networks Using an Adaptative Neural Network, IEEE Trans on Power Systems, vol 12, no 4,
november1997
[4] P. K. Dash, S. K. Panda, A. C. Liew, B. Mishra, R. K. Jena. A New Approach to Monitoring Electric Power
Quality, Electric Power System Research, 46, 1998
[5] L. L. Lai, C. T. Tse, W. L. Chan, A.T.P. So. Real-Time frequency and harmonic evaluation using
artifificial neural networks IEEE Trans. On Power Delivery, Vol 14. No 1, January 1999.
[6] F. J. Alcántara, P. Salmerón, S. P. Litrán, J. R. Vázquez. Análisis y Medida de las Componentes de
Corriente y de Potencia en un Circuito Trifásico Desequilibrado y no Senoidal, 6as Jornadas LusoEspañolas de Ingeniería Eléctrica, Volumen 3, Lisboa, July 1999
[7] L. Cristaldi, A. Ferrero. Mathematical Foundations in the Instantaneous Power Concepts: An Algebraic
Approach, ETEP, vol 6, no 5, September/October 1996
[8] F. Javier. Alcántara, Patricio. Salmerón, Jesús Rodríguez. Aplicación de las redes neuronales a la
estimación de las componentes simétricas en sistemas de potencia, SAAEI 2000, Actas pp 659-662,
Terrassa, Spain, September 2000
[9] F. R. Vázquez, P. R. Salmerón,. Three-Phase Active Power Filter Control using Neural Networks, IEEE
Melecon 2000, Proc. Vol III, pp 924-927, Chyprus, May 2000
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