14th PSCC, Sevilla, 24-28 June 2002
Session 24, Paper 4, Page 1
STOCHASTIC PREDICTION OF VOLTAGE DIPS USING AN
ELECTROMAGNETIC TRANSIENTS PROGRAM
Juan A. Martinez-Velasco
Jacinto Martin-Arnedo
Universitat Politècnica de Catalunya
Barcelona, Spain
martinez@ee.upc.es
Abstract –This paper presents a summary of the work
performed by the authors on voltage dip analysis using a
time-domain tool, the ATP version of the EMTP. The
document describes how to adapt this program to this type
of studies and how to perform Monte Carlo-based stochastic predictions of voltage dips. A medium size distribution
network is used to analyze the advantages of this approach, the convergence of the Monte Carlo method, the
influence of the protection system and the importance of
voltage dip indices. The current limitations of this tool are
also discussed.
Keywords: Power Quality, Voltage Dips, Monte
Carlo Analysis, Digital Simulation, Transient Analysis, Stochastic Prediction, ATP/EMTP
1 INTRODUCTION
A voltage dip is a sudden short duration drop of the
rms voltage, followed by a recovery within 1 minute. A
dip is usually characterized by the remaining (retained)
voltage, its duration and the phase jump, that is the
difference between the voltage phase before and after
the dip [1]. The most severe voltage dips are caused by
short-circuits, generally associated to bad weather conditions (i.e. lightning strokes). They can also be caused
by transformer energizing, motor starting and sudden
load changes. Although voltage dips are less severe than
interruptions, they are more frequent; in addition, their
consequences for sensitive equipment, such computers,
adjustable speed drives or control equipment [2], can be
as important as those by an interruption. Given the
diversity of their causes and the difficulty of preventing
all these causes, voltage dips are presently the most
important power quality disturbances. These reasons
have increased the interest on equipment aimed at preventing or reducing their effects [3], [4].
The voltage dip performance of a power network can
be predicted by monitoring the network; however, the
monitoring period that is needed to achieve a reasonable
accuracy is too long [1]. Digital simulation can be then
an alternative to predict this performance. In addition,
other benefits can be derived from simulation, for instance the effect that some mitigation techniques can
have on the network performance. Several methods
have been proposed to predict the number of voltage
dips originated in power networks. The most widely
used are the “Method of Fault Positions” and the
“Method of Critical Distances” [1], [5], [6]. The approach presented in this paper is based on the applica-
tion of the Monte Carlo method and assumes that voltage dips are only caused by faults.
A voltage dip is originated as a consequence of a
transient in an electrical network. Although several type
of tools have been used to simulate voltage dips, only
those based in a time-domain solution can obtain with
high accuracy the main characteristics of voltage dips
and reproduce their effects, include the dynamic behaviour of power components and analyze the performance
of mitigation techniques. This paper presents a new
procedure for the stochastic prediction of voltage dips
in power networks using the ATP (Alternative Transients Program) version of the EMTP (ElectroMagnetic
Transients Program) [7]. The present work has been
based on the development and implementation of new
ATP capabilities created for this specific application. A
stochastic prediction of voltage dips based on a Monte
Carlo method requires several capabilities : a multiple
run option to simulate the test system as many times as
needed, the generation of random numbers that will be
used to obtain the fault characteristics (location, resistance, initial time and duration, type of fault), modules
for monitoring dips and calculating power quality indices [8].
After a short introduction to the Monte Carlo
method, the document presents a summary of the main
solution methods and capabilities of the ATP. All test
studies presented in the paper are based on a distribution network of medium size (27 nodes, 26 lines). They
show the results to be expected from ATP simulations,
how the convergence of the Monte Carlo method is
achieved, the influence of protective devices on the dip
characteristics, and the calculation of voltage dip indices that can be useful to qualify the performance of a
power network. A discussion about the limitations of
the current ATP version for this type of studies is also
included.
2 THE MONTE CARLO METHOD
The Monte Carlo method is a widely used technique
for analyzing multidimensional complex systems. It can
be used for solving both stochastic and deterministic
problems, although the first type is the most usual one.
This technique is based on a iterative procedure that is
repeated using in every new step a set of values of the
random variables involved in the process, being these
values generated according to the probability density
function associated to each variable.
14th PSCC, Sevilla, 24-28 June 2002
Session 24, Paper 4, Page 2
The following paragraphs are aimed at clarifying this
question, presenting some basics of the method and
introducing some definitions related to its application
[9]. Although all the comments are based on the assumption that only one variable is involved, the conclusions are also applicable to multidimensional systems.
• A random variable, yn, converges in probability to Y,
if for every γ >0, the probability that yn is outside an
γ interval of Y converges to zero when n → ∞.
• The sample average of a random variable converges
in probability to the mean value of the variable.
• A proper sample procedure for a given Probability
Density Function (PDF), f(x), and a Cumulative
Density Function (CDF), F(x), is an algorithm which
generates an outcome x such that for any value x0 the
condition Pr[x ≤ x0]= F(x0) is fulfilled.
• The realization of a random variable generated by a
proper sampling procedure cannot be distinguished
from a random variable generated in a process controlled by the CDF itself.
• The realization of a random variable, x, obtained
from the solution of
x
∫ f ( x')dx = F ( x) = ξ
(1)
−∞
being ξ a random number generated from a uniform
distribution defined in the interval [0,1], is a proper
sampling procedure from the original CDF.
• The statistical error of a Monte Carlo solution converges to zero at a rate ( n ) −1 , and the PDF of the
variable converges to a normal distribution. This
means that the limit distribution of the sample average is independent of the estimator distribution.
• A Monte Carlo solution has a slower convergence
that a numerical solution, i.e. for calculation of integrals, but the convergence is independent of the dimension of the phase space.
The goal of a Monte Carlo solution is to derive the
response/performance of a system as a function of some
stochastic input variables. Sampling is iteratively repeated until convergence is achieved. If the realization
of the random input variables is generated by a proper
sampling procedure, the solution converges as the number of samples n → ∞, being ( n ) −1 the rate of the
statistical error convergence to zero. The convergence is
independent of the phase space dimension.
3
ATP SOLUTION METHODS AND
CAPABILITIES
ATP is a time-domain circuit oriented tool based on
the trapezoidal rule. This rule converts equations of the
network components into algebraic equations involving
voltages, currents and past values. These equations are
assembled using a nodal approach. Although this tool is
mainly intended for transients simulations, it can also be
used in frequency-domain simulations. Steady-state
phasor solutions can be carried out to establish initial
conditions for transient simulations, to analyze harmonic propagation or to obtain a system impedance as a
function of frequency.
The trapezoidal integration rule is very simple and
numerically stable; however, it uses a fixed time-step
size and can originate numerical oscillations. Although
ATP users can incorporate additional damping or snubber circuits to avoid these oscillations, both techniques
do introduce simulation errors.
In addition, the steady-state solution can be applied
only to linear networks at a single frequency. Many
procedures have been proposed up to date to obtain the
steady state of nonlinear power systems, but no solution
method has been yet implemented in the ATP. These
drawbacks and the fact that a very small time step size
is needed when variable topology converters are included in the test system represent serious limitations
for some studies, for instance a stochastic prediction of
voltage dips when custom power equipment is used to
mitigate their effects.
The ATP package is made of at least three types of
tools [10] : a preprocessor (ATPDraw), the main processor (TPBIG), and a postprocessor. In general, the
ATPDraw provides an interactive, mouse-driven,
graphical user interface, with very flexible options for
incorporating custom-made modules and new tools to
the package.
Some new TPBIG capabilities allow users to create
very powerful modules by adding calculations with
module arguments and internal variables. They have
also expanded the applications of this tool, which can
be currently used for performing sensitivity analysis and
any type of statistical studies.
The custom-made modules that have been developed
for voltage dip studies include power components,
which simplify the use of basic models and extend the
capabilities of the package to more complex equipment
models, protective devices, such as reclosers and fuses,
and monitoring devices, to characterize voltage dip and
calculate power quality indices.
ATP capabilities have also been used to develop
those modules needed to perform a stochastic prediction
of voltage dips, using an approach based on the Monte
Carlo method.
4 TEST SYSTEM
The scheme of the test system is shown in Figure 1,
it is a medium size distribution network with two radial
feeders. The lower voltage side of the substation transformer is grounded by means of a zig-zag reactor of 75
Ω per phase.
14th PSCC, Sevilla, 24-28 June 2002
2 km
6
Session 24, Paper 4, Page 3
8
5 km
2
0
FEEDER 2
10 km
13
11
24
22
2 km
5 km
21
5 km
5 km
2 km
10
10 km
15
17
2 km
12
2 km
10 km
10 km
FEEDER 1
16
5 km
5 km
23
27
25
5 km
5 km
26
2 km
1
9 14
5 km
2 km
5 km
4
3
2 km
2 km
5 7
2 km
18 20
19
2 km
2 km
HV equivalent : 110 kV, 1500 MVA, X/R = 10
Substation transformer: 110/11 kV, 20 MVA, 8%, Yd
Distribution transformers : 11/0.4 kV, 1 MVA, 6%, Dy
Lines : Z1/2 = 0.61 + j0.39, Z0 = 0.76 + j1.56 Ω/km
Figure 1: Diagram of the test system.
5
SIMULATION RESULTS
5.1 Introduction
The approach presented in this paper is based on the
random generation of disturbances, and assumes that
dips are due only to faults originated within the distribution network. Since load are represented as constant
impedances and they do not include any dynamic behaviour, voltage dips will be rectangular, and characterized by the remaining voltage, the duration and the
phase angle jump. Figure 2 shows the rms voltages due
to a line-to-ground fault produced at a radial distribution network, similar to that shown in Figure 1. The plot
shows a case with a dip and a swell originated at the
same node.
A
20
C
Voltage (kV)
15
10
5
0
20
40
60
80
100
Time (ms)
120
140
Figure 2: Voltage dip simulation.
160
The developed procedure can be summarized as follows. The test system is simulated as many times as
required to achieve the convergence of the Monte Carlo
method. Every time the system is run, faults characteristics are randomly generated using the following parameters
• Fault location : the fault may occur at any point of
the distribution system, so the location is selected by
generating a uniform random number
• Fault resistance: normal distribution, average value =
10 Ω, standard deviation = 1 Ω
• Initial time of the fault: uniform distribution between
0.04 and 0.06 s
• Duration of the fault: normal distribution, average
value = 0.06 s, standard deviation =0.02 s
• Probability of each type of fault: LG: 80%, 2LG:
17%, 3LG: 0%, LL: 2%, 3L: 1%.
After every run, voltage dip characteristics at the nodes
of concern are recorded. At the end of the simulation, this
information is manipulated to obtain the voltage dip density function.
Note that the number of variables involved in any voltage dip simulation is rather high, and they could be increased if the random nature of the loads was also represented. Although for radial networks like that simulated in
this paper, a systematic approach could be used, in general
a Monte Carlo method will be the best approach to solve
this problem.
Some of the characteristics values used in this study are
not very realistic, and not applicable to most actual distribution networks. The fault durations are too short and no
permanent faults are considered; a uniform distribution of
the fault location could be a very crude approach in many
actual networks; the fault resistance distribution could be
far from the normal distribution assumed in this work.
A permanent fault will obviously force the protection
equipment to act, and will originate an interruption, which
is not a subject of this study. However, if reclosers have
been installed, their operation could produce typical reclosing sequences with two or more reclosing intervals,
which could become two or more voltage dips at some
nodes.
The main goals of this study are, rather than simulating
actual systems, to check the convergence of the Monte
Carlo approach and test the ATP capabilities developed
for this application. On the other hand, it is important to
emphasize digital simulation advantages. As mentioned in
the Introduction, realistic data can be only obtained by
monitoring; however, a very long period can be needed to
obtain accurate information [1]. Digital simulation can be
useful to determine what is the influence of the most important parameters by performing parametric studies (sensitivity analysis) and find out what system
parts/components do need to be improved.
The following sections are dedicated to analyze the
convergence of the Monte Carlo method; the influence
that the protective devices can have on the voltage dip
density function and the usefulness of voltage dip indices.
14th PSCC, Sevilla, 24-28 June 2002
Session 24, Paper 4, Page 4
• Fault initial time : The time span, 20 ms, was divided
into intervals of 2.5 ms, so each interval represents a
12.5% of the span
After 1000 runs : Standard deviation = 0.9695
Confidence interval = 0.8105
After 5000 runs : Standard deviation = 0.4509
Confidence interval = 0.3770
• Fault type : The distribution of fault types after 1000
and 5000 runs was as shown in the following table
Fault type
L-G
2L-G
3L-G
L-L
3L
1000 runs
80.40%
16.00%
0%
2.00%
1.60%
5000 runs
78.98%
17.52%
0%
2.48%
1.02%
As expected, the accuracy is increasing with the
number of runs, although the percentage of some fault
types is actually closer to the given distribution at 1000
runs.
Figure 4 depicts the cumulative voltage dip density
function which results at phase “A” of Node 4, see
Figure 1, after 1000 and 5000 runs.
Frequency (%)
10
8
6
4
2
0
11.4
22.8
34.2
45.6
57
68.4
79.8
91.2 102.6
114
Distance (km)
a) Fault location
Frequency (%)
25
20
15
10
5
0
6
7
8
9
10
11
12
13
14
Resistance (ohms)
b) Fault resistance
14
Frequency (%)
12
10
8
6
4
2
0
0.04
0.0475
0.055
0.0625
Time (s)
c) Initial fault time
9
8
Frequency (%)
• Fault location : The whole distribution line length was
divided into intervals of 11.4 km, that is each interval is
the 10% of the whole length
After 1000 runs : Standard deviation = 1.0154
Confidence interval = 0.5663
After 5000 runs : Standard deviation = 0.3562
Confidence interval = 0.2229
12
7
6
5
4
3
2
1
0
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Duration (s)
d) Fault duration
90
80
Frequency (%)
5.2 Convergence of the Monte Carlo method
The study was performed using a single input file and
taking advantage of the new ATP multiple run option. The
following information was recorded at every run
• the characteristics of the fault (location; initiation time,
duration, resistance, faulted phases, and fault type)
• the characteristics of the dips (remaining voltage,
duration) at every node and every phase.
Several criteria can be used to decide when the
convergence of the Monte Carlo simulation is achieved.
For instance, by checking the error between a few
points on the cumulative probability curve of every
random variable obtained by simulation, against the
theoretical values, or by checking the confidence level,
see Chapter 2.
The test system was simulated up to 5000 times. If it
is assumed that 12 short-circuits are generated by year
and 100 km of overhead lines, as the network has 114
km, then it equals the performance of the network
during 366 years. Figure 3 shows the probability
distribution of all random variables after the 5000 runs.
The following paragraphs present some of these
results, deduced with a confidence level of 95%.
70
60
50
40
30
20
10
0
L-G
2L-G
3L-G
L-L
3L
e) Fault type
Figure 3: Distribution of random variables.
14th PSCC, Sevilla, 24-28 June 2002
Session 24, Paper 4, Page 5
100
90
80
70
60
50
40
30
20
10
t>
0ms
t >2
0ms
t>
40m
s
t >6
0ms
t >8
0ms
0
V
V<
V<
20
10
%
%
<3
0
V<
%
V
V
<9
0%
V < V < 7 < 80
V<
%
0%
60
50
%
40
%
%
a) Dip density function after 1000 runs
100
90
80
70
60
50
40
30
20
10
t>
0ms
t >2
0ms
t>
40m
s
t >6
0ms
t >8
0ms
0
V
V<
<3
V<
V<
0%
20
10
%
%
V<
40
%
V
V < V < 8 < 90
V
%
0%
7
<6
0% 0 %
50
%
b) Dip density function after 5000 runs
detected, and close contacts after a certain reclosing
interval.
Figure 5 shows the dip density function at Node 4
derived with reclosers installed at the head of both
feeders and from different recloser intervals. The charts
are based on two different reclosing intervals : 20 and
60 ms. As expected, one can observe that the reclosing
interval will significantly affect the voltage dip density
on those dip durations shorter than the reclosing
intervals. Since Figure 5 charts show the dip density
function in percentage, it is not easy to deduce how
many voltage dips are produced. Figure 6 compares the
effect of a recloser on the average number of voltage
dips per year, with and without reclosers.
The results presented in Figures 5 and 6 should be
interpreted as follows. A recloser needs some time to
operate after a fault current has been detected, but once
their terminals are open, the dip duration is increased at
nodes located at the feeder where the fault is originated,
and decreased at nodes located at the other feeder. As
for the remaining voltage, it is decreased to zero at
nodes located at the faulted feeder, and recovered to the
pre-fault voltage at the other feeder nodes. This
performance is obvious from Figures 5 and 6. On one
hand, the number of dips with long duration and small
remaining voltage has increased; on the other hand,
there has been an increment in the number of dips with
short duration.
Figure 4: Dip density function at Node 4 – Phase A.
100
90
5.3 Influence of the protective devices
The previous study has been made without including
any feeder protection in the distribution network model.
However, even assuming only short-duration faults, the
protection equipment can significantly modify the
voltage dip density function. A complete model of a
distribution system should include most protection
equipment : breakers, reclosers, and fuses. Even some
protective relays should be included in some cases if an
accurate enough simulation is needed.
The results presented in this document has been
deduced by assuming that only reclosers are installed
and they have only one operation after the contacts
open, so if the fault condition is still present, the
contacts will remain open. The recloser model is not
based on a time-current characteristic, instead the
recloser will operate 10 ms after the fault current is
80
70
60
50
40
30
20
10
t>
0ms
t >2
0ms
t>
40m
s
t >6
0ms
t >8
0ms
0
V<
V<
V<
20
10
%
%
30
V<
%
V<
V
90
V < V < 7 < 80
%
V<
%
0%
60
50
%
40
%
%
a) Reclosing interval of 20 ms
100
90
80
70
60
50
40
30
20
10
0
t>
0ms
t >2
0ms
t>
40m
s
t >6
0ms
t >8
0ms
Both charts give the probability of every
characteristic interval. They show no significant
differences between them. Similar conclusion was
derived for the dip density functions at other phases and
other nodes. So one can conclude that 1000 runs are
enough to obtain accurate enough dip density functions
for the present system. In fact, it is the income
information (fault statistics) used for this study where
error can be more significant.
Therefore, the subsequent studies will be based on
1000 runs, which equal the performance of the network
for 73 years.
V<
V
90
V < V < 7 < 80
%
V<
%
0%
60
V<
50
V<
%
4
%
V<
0%
3
V<
0
20
%
10
%
%
b) Reclosing interval of 60 ms
Figure 5: Dip density function with recloser –Node 4
14th PSCC, Sevilla, 24-28 June 2002
Session 24, Paper 4, Page 6
3
2.5
2
1.5
1
0-2
0ms
20-4
0ms
40-6
0m
s
6080m
s
80100
ms
0.5
0
80
70
%60
%90
%8
%
%
0
70
%
30
-6 0
%%
20
%50
%
10
%40
%
0%
%30
%
2
%
-1 0
0%
%
40
50
a) Without recloser
4
3.5
3
2.5
2
1.5
1
0.5
0-2
0ms
20-4
0ms
40-6
0m
s
6080m
s
80100
ms
0
80
70
%60
%90
%80
%
%70
%
30
%6
%
2
%
0
50
0%
%
1
4
0
%
0
0%
-3 0
%%
20
%
-1 0
%
%
40
50
b) Reclosing interval of 60 ms
Figure 6: Number of dips per year – Node 4.
5.4 Calculation of voltage dip indices
Several voltage dip indices have been proposed up to
date to reflect the behaviour of a system and to assess
the effect of mitigation techniques [8], [11], [12]. They
can characterize either a site or a full system, using
single event or a group of events. The current version of
the module library allows users to obtain site and
system indices.
Some of the indices proposed are based on the
concept of voltage dip energy, which can be calculated
according to [11], [12]
2
W = {1 − V pu } T
only those cases for which the voltage drops below 90%
of the rated voltage.
Voltage dip index values will depend on the fault
statistics and the performance of the protection system.
Based on the same studies described above, several
calculations were made to analyze these influences. To
compare the performance of the system and calculate
the voltage dip index, the same sequence of events
should be used with both configurations. A feature was
implemented in order to record randomly generated
events and use them every time they were needed, as for
this example.
Figure 7 shows the AVSEI values at all load nodes,
see Figure 1, after 500 runs. Initially, the cases were run
without installing any feeder protection; the subsequent
simulations were run with reclosers installed at both
feeders. All simulations were performed without
permanent faults. It is obvious from Figure 7 that the
voltage dip energy is significantly increasing when the
reclosers do act. An explanation for this performance is
shown in Figure 8. This chart compares the voltage dip
energy values that correspond to 10 runs obtained at
Node 4, which is located at Feeder 1. It is easy to
deduce from this chart on which feeder the fault was
originated. For instance, events 15, 16, 17, 21 and 23
were originated at Feeder 1, and the rest were originated
at the other feeder. For those dips originated at Feeder
1, the actuation of the recloser increased the index value
since the dip became a short interruption. For those dips
originated at the other feeder, the index value decreased
when reclosers did act, but the reduction was less
significant.
0.10
0.09
0.08
AVSEI
4
3.5
0.07
0.06
Without Recloser
With Recloser 20 ms
With Recloser 60 ms
0.05
0.04
0.03
0.02
1
2
3
4
5
6
(3)
being V the dip magnitude and T the dip duration. For
three-phase events the dip energy is added for the three
phases.
One of these indices is the so-called Average Voltage
Sag Energy Index (AVSEI)
7
8
9
10
11
12
13
14
Load
Figure 7: Voltage Dip Energy at load nodes.
0.2
Without Recloser
With Recloser 20 ms
With Recloser 60 ms
0.15
0.1
N
AVSEI =
1
⋅ Wk
N k =1
∑
(4)
being Wk the lost energy during the dip event “k”, and
N the number of events. Although this expression does
not differentiate dips from swells, the calculation of this
index will be made taking into account only those
phases in which dips are originated, and considering
0.05
0
15
16
17
18
19
20
21
22
23
24
Case
Figure 8: Voltage Dip Energy at Node 4
25
14th PSCC, Sevilla, 24-28 June 2002
6 CONCLUSIONS
This paper has presented the application of the ATP
package in voltage dip analysis. The capabilities of this
tool have been used for developing new custom-made
modules adapted for this type of studies.
Digital simulation is a very efficient alternative for
predicting the performance of a network and for testing
devices and techniques which could mitigate voltage
dip effects. A tool based on a time-domain has many
advantages, but the limitations of such tools and those
of the procedure developed for this work are still
important. The procedure can be used to analyze the
performance of a system by assuming that dips are
originated only by faults of short duration. Although it
can also be used to compare its performance after
installing some mitigation devices, in reality this second
possibility is not yet realistic if custom power
controllers are used. Since a very small time step size is
needed when variable topology converters are included,
simulating a big network several thousands times would
be extremely long.
Both new custom-made modules and ATP
capabilities are still needed to obtain more realistic
results. The aim of a stochastic prediction is not only to
deduce the number of voltage dips but also the number
of trips of sensitive equipment. Therefore, the
representation of equipment sensitivity is also required.
Several approaches can be considered; for instance,
ATP capabilities can be used to include a very detailed
ASD model (rotating machines plus static converters),
when this is the type of affected equipment. However,
for some studies, i.e. voltage dip index calculation, the
model of a big industrial plant can be reduced to a few
linear loads, being represented each of them by a
module that incorporates voltage dip tolerances curves.
An important aspect of any voltage dip analysis is the
influence of protecive equipment. All cases analyzed
with protective devices (reclosers) included in the
system representation were very simple; however, they
have been very useful to prove the influence that these
devices can have on voltage dip characteristics and
indices.
The calculation of voltage dip indices is a rather
recent approach for qualifying the performance of a
power system [8], and no consensus has been reached
about the most appropriate indices. The document has
presented the calculation of only one index, and some
of the influences that protective devices could have on
its values. However, there would be no problem to
improve present custom-made modules that represent
measuring devices in order to calculate some other
voltage dip indices.
Load representation is an important subject in which
capabilities of a tool like ATP have several advantages.
All calculations presented in this work were made
Session 24, Paper 4, Page 7
assuming that loads could be represented as constant
impedances, and peak loads were simultaneous at all
nodes. More realistic representations can be made by
introducing any voltage dependence and assuming
probabilistic models. As mentioned above, voltage
tolerance is another important feature to be considered.
REFERENCES
[1] M.H.J. Bollen, “Understanding Power Quality
Problems. Voltage Sags and Interruptions,” IEEE
Press, 2000, New York.
[2] H. Sarmiento and E. Estrada, “A voltage sag study
in an industry with adjustable speed drives”, IEEE
Industry Applications Magazine, vol. 2, no. 1, pp.
16-19, January/February 1996.
[3] N.G. Hingorani, “Introducing Custom Power”,
IEEE Spectrum, vol. 32, no. 6, pp. 41-48, June
1995.
[4] N. Woodley, L. Morgan and A. Sundaram, “Experience with an inverter based dynamic voltage restorer”, IEEE Trans. on Power Delivery, vol. 14,
no. 3, pp. 1181-1186, July 1999.
[5] M.H.J. Bollen, “Fast assessment methods for voltage sags in distribution systems”, IEEE Trans. on
Industry Applications, vol. 32, no. 6, pp. 14141423, November/December 1996.
[6] M.H. Qader, M.H.J. Bollen and R.N. Allan, “Stochastic prediction of voltage sags in a large transmission system”, IEEE Trans. on Industry Applications, vol. 35, no. 1, pp. 152-162, January/February
1999.
[7] H.W. Dommel, “ElectroMagnetic Transients Program. Reference Manual (EMTP Theory Book)”,
Bonneville Power Administration, Portland, 1986.
[8] IEEE Voltage Quality Working Group, “Recommended practice for the establishment of voltage
sag indices”, IEEE P1564, Draft, March 2001.
[9] A. Dubi, “Monte Carlo Applications in Systems
Engineering”, John Wiley, 1999.
[10] J.A. Martinez, “The ATP package. An environment
for power quality analysis”, 9th International Conference on Harmonics and Quality of Power, October 14-16, 2000, Orlando.
[11] D.L. Brooks, R.C. Dugan, M. Waclawiak and A.
Sundaram, “Indices for assessing utility distribution
system RMS variation performance”, IEEE Trans.
on Power Delivery, vol. 13, no. 1, pp. 254-259,
January 1998.
[12] R.S. Thallam and G.T. Heydt, “Power acceptability
and voltage sag indices in the three phase sense”,
2000 IEEE PES Summer Meeting, July 16-20,
2000, Seattle.