Stochastic Model to Forecast the Voltage Sags in Real Power Systems

Abstract

The forecast of the occurrence of voltage sags at
the sites of a system is nowadays feasible thanks to the
availability of huge quantity of recorded data. To forecast future
performance from the statistical analysis of recorded sags, the
stochastic modelling of the voltage sags is required since the
events are not statistically time independent. The presence of
groups of sags, named clusters, brings the phenomenon far from
the conditions of Poisson model. This paper proposes the
Gamma distribution to model the sags, which also include the
clusters. Different techniques for assessing the parameters of the
Gamma distribution are presented and applied to forecast the
number of sags expected at selected sites in the year 2018, i.e.,
the year successive to those when the sags were measured. The
outcomes of the forecast are compared with the sags effectively
occurred in those sites in the year 2018, using different criteria
for evaluating the forecast error. The results showed the
viability of the approach and encourage further studies to
improve the accuracy and extend the forecast to entire systems.

Full text

978-88-87237-50-4 ©2021 AEIT
Stochastic Model to Forecast the Voltage Sags
in Real Power Systems
Leonardo Di Stasio, Paola Verde,
Pietro Varilone
Department of Ingegneria elettrica e
dell’informazione “Maurizio Scarano”
Università degli studi di Cassino e del
Lazio Meridionale
Cassino
leonardo.distasio@unicas.it,
verde@unicas.it, varilone@unicas.it
Michele De Santis
Department of Ingegneria
Università Niccolò Cusano di Roma
Roma
michele.desantis@unicusano.it
Christian Noce
Enel X SrL
Roma
christian.noce@enel.com
Abstract— The forecast of the occurrence of voltage sags at
the sites of a system is nowadays feasible thanks to the
availability of huge quantity of recorded data. To forecast future
performance from the statistical analysis of recorded sags, the
stochastic modelling of the voltage sags is required since the
events are not statistically time independent. The presence of
groups of sags, named clusters, brings the phenomenon far from
the conditions of Poisson model. This paper proposes the
Gamma distribution to model the sags, which also include the
clusters. Different techniques for assessing the parameters of the
Gamma distribution are presented and applied to forecast the
number of sags expected at selected sites in the year 2018, i.e.,
the year successive to those when the sags were measured. The
outcomes of the forecast are compared with the sags effectively
occurred in those sites in the year 2018, using different criteria
for evaluating the forecast error. The results showed the
viability of the approach and encourage further studies to
improve the accuracy and extend the forecast to entire systems.
Keywords— Voltage Sag, Voltage sag monitoring, Poisson
process, Gamma distribution
I. INTRODUCTION
he voltage sags are among the most critical power1
quality (PQ) disturbances affecting the voltage [1].
The most frequent causes of a voltage sag in the
networks are the faults. The short circuit current flowing in
the system until the action of the protection system produces
the rapid decrease of the voltage, which can propagate in
large sections of the system.
The relevant literature recognized from several years the
severity of the phenomenon, proposing methods and tools
mainly aimed to estimate the average performance expected
at the system sites in terms of sag frequency, usually
computed on yearly basis. The largest part of the papers of
the literature proposed model-based simulations of the
systems in short circuit conditions, due to both symmetrical
and unsymmetrical faults. The voltages during the faults
obtained by the simulations are then correlated with the
average values of the fault rates of the busses or of the lines
to obtain the expected frequencies of the sags at the sites of
the system [2-12].
The authors L.Di Stasio, P.Varilone, P.Verde acknowledge the financial
support by MIUR through the Special Grant “Dipartimenti di eccellenza”
Lex. N. 232/31.12.2016. The paper was also developed inside the grant
Recently, some papers have approached the problem of
assessing the future occurrence of the sags at network sites
from a completely different point of view [13-15]. The main
novelty was to pass from the estimation of the average
performance of a system obtained from simulations to the
forecast of the number of sags at the sites of a system obtained
from the statistics of recorded sags. Statistical approaches
used the observed history to describe and forecast the future
performance. The statistical models take into account the
effect of external random events on the variable of interest.
Considering the voltage sags as a random phenomenon
which, depends on multiple conditions, also on time, the
statistical approaches are the best way for forecasting voltage
sags as they are built based on previous experiences. This
new approach was made possible, above all, by the ever-
increasing availability of data measured from the field. All
around the world, several campaigns of sag measurement are
active [16-17]. They are aimed to regulate the PQ levels of
the system, driven by rules and resolutions of the national
energy Authorities, and/or to assess the real conditions of
own networks, driven by internal targets of DSOs
(Distribution System Operators) and TSOs (Transmission
System Operators). In Italy, for example, the resolutions [18-
19] required all the DSOs to install at all the MV (Medium
Voltage) busbars of all the HV/MV (High Voltage/Medium
Voltage) stations measurement systems to measure and
record the occurred voltage sags. This produced large data
sets of measured sags in Italy from 2014, which allowed
developing methods to ascertain the origin of voltage sags
[20] also in the presence of GD [21].
The analysis of the sags recorded at the sites of a real
system evidenced that the sags are composed not only of rare
events but also of groups of events close each other in the
time.
Concerning the rare events, they are sags with a frequency
in each site in the range from some sags at year to some sags
at month. All the relevant literature was aimed to estimate the
average performance of a systems, focusing the attention on
rare sags [7-15].
Concerning the groups of sags, we can distinguish them
in two main categories as function of the time between every
“Attività di supporto scientifico sulle tematiche di Power Quality nelle reti
di distribuzione” with E-Distribuzione.
T

events. The first category encloses groups of sags close each
other in the time with a distance less than few seconds. They
are due to automatic closure maneuvers of the protection
systems and/or to incorrect recordings. This typology of
groups of sags are well known and are manipulated by the
time aggregation. All the sags, grouped within a given time
interval, are considered as a single event, which is the first of
the group [22-23]. The second category of groups of sags,
called sag clusters includes groups of sags close each other in
the time with a distance larger than few seconds up to some
hours. They can be due to exogenic factors, such as rain,
lightning strikes, wind, and other adverse weather conditions.
The sag clusters are not affected by the time aggregation and,
consequently, they are present in all the recorded sags even
after the time aggregation was carried out.
The presence of the sag clusters [13-16] makes the sag
frequency dependent on the time. The sags, which contain
both rare sags and sag clusters, present the features of a
stochastic process. They do not have the characteristics of
memoryless events and, consequently, cannot be modeled as
a Poisson process. To make successful forecast, filtering the
sag clusters out from all the measured sags is crucial [14-15].
The use of adequate intermittent indices to detect and filter
out the sag clusters [15] leaves only rare voltage sags which
can be used to forecast the future occurrence at the nodes of
a real system with an accuracy up to 10%. This important
result was obtained also thanks to the choice of the time to
next event, namely ttne, as the statistical variable to describe
the phenomenon.
In this paper, we present the first approach aimed to
predict all the sags, which contain both rare sags and sag
clusters. The statistical model of such time-dependent events
must respect the stochastic nature of the phenomenon. It is
possible by means of the Gamma distribution of ttne. This
model was first proposed in [16] to describe the recorded
data. In this paper, we extend this model to the statistical
analysis of the sags recorded from 1 January 2015 to 31
December 2017 with the specific target to forecast the
expected number of voltage sags in 2018. The sags were
recorded at a regional electric system, whose main
characteristics are reported in [15]. The availability of the
sags really occurred in the year 2018 allowed us to evaluate
the errors of the forecast.
Modelling a process by the Gamma distribution requires
to select two parameters, that is the shape parameter α, and
the scale parameter β. In this paper, we present the application
of both analytical and graphical methods to establish the
values of
α
and
β
, which statistically describe the random
variable ttne with minimum error.
The paper is organized as follows: Section II presents the
approach that was used to describe the stochastic model of all
the measured sags. Section III illustrates the methods adopted
for establishing the parameters of the Gamma distribution.
Section IV describes the methods used to assess theoretical
hypothesis. Section V shows the results of the statistical
validation and of the forecasting activity.
II. STOCHASTIC MODELING THE SAGS BY THE GAMMA
DISTRIBUTION
Most of the papers dealing with the estimation of average
performance of a system in terms of voltage sags proposed
methods and tools based on the simulation of the system in
short circuit conditions [2-7]. The basic assumption is that the
sags are due to faults in the nodes and/or along the lines. The
expected annual frequency of the sags is successively derived
correlating the frequency of the causes, the faults, with the
frequency of the effects, the sags. The subjects of all these
methods are the rare voltage sags, i.e. sudden under-voltages
lower that the 90% threshold that occur with a temporal
distance ranging from a few hours to a few months. The rare
voltage sags can be statistically modelled as a Poisson process,
a series of discrete memory-less and time independent events.
For this type of sags, the distribution of the random variable
ttne, which measured the waiting time until the occurrence of
the next event, follows the exponential density function, that
is:
 (1)
where λ is the sag rate constant with the time.
The measurements of the sags in real systems reveal that
the sags are not only rare, but there are also clusters of sags,
i.e. groups of sags close each other in time with a distance
between them from some seconds to some hours, as said in the
Introduction. As an example, Fig.1 shows the scatter plot of
the voltage sags measured on 17 January 2017 from 00:00 am
to 24:00 pm at the busbar #38 of the regional system described
in [15]. Every blue dot represents the time of occurrence of the
measured voltage sags. The rare sags are highlighted with
black circles and the sag clusters with red circles.
The sags belonging to clusters do not have the statistical
characteristics of a Poisson process since they are events
related to each other, often they have a common origin, for
example an adverse atmospheric phenomenon or fires, and for
these reasons, their time independence cannot be assumed. If
the statistical analysis is conducted on the measured sags, the
only way to apply the distribution given by (1) is to filter and
remove the clusters of sags, leaving only the rare sags in the
data to be analysed, as done in [13-15]. Without applying
these filters, the voltage sags present the characteristics of a
stochastic process, that is a collection of indexed random
variables, ttne in this case, defined on a common probability
sample space which depends on the time. A stochastic model
of voltage sag occurrences is mandatory to describe their
random appearance in the time, with large periods of absences,
when only rare sags occur, and others when sags appear as
clusters. To model all the recorded sags, the assumption of
constant sag rate is not valid, and the distribution of the
random variable ttne is given by [15-16]:


 (2)
where λ(t) is the sag rate variable with the time.
Fig.2 shows the probability density function (pdf), of ttne of
the voltage sags recorded at the site of example #38. It is
Fig. 1. Scatter plot of the voltage sags measured on 17/01/2017 from
00:00 am to 24 pm at the site #38.

Fig. 2. Histogram of probability density function of the random variable
ttne measured in the site #38.
evident a thickening of the ttne values in the first class. The
width of the first class is equal to 2.25 × 105s (62.5 hours), so
the probability that the value of the random variable ttne is
between 3s2 and 2.25 × 105s is equal to 81%. The first class of
the histogram in Fig. 2 is linked to the presence of clusters of
sags [15]. For successive classes, there is a decay of the pdf
values. Fig.3 shows the empirical cumulative density function
(ecdf) corresponding to the pdf of Fig.2; the plot of Fig.3 (b)
is zoomed in the range of low ttne.
From Fig. 3 (a), it is evident that the values of ttne range
from 3 s to 2.3 × 106 s (639 hours), but Fig. 3 (b) revealed that
the greatest probability, up to 60%, is already reached by ttne
less than 5 × 103s (1.4 hours), leaving the remaining 40% to
the ttne up to 2.3 × 106 s. This feature of the recorded data,
evident also for other sites beside site #38, proved that the sag
rate is not constant.
As proposed in [16], the random variable ttne of all the
sags, both rare and clusters, can be described by the Gamma
distribution as:
 !""#$!%&$%'(()*
+, (3)
where α and β are the shape and the scale parameters,
respectively.
The corresponding cdf is given by:

-./0%&
1023456
17

(4)

831
23
where γ (
α
,
β,
ttne) is the incomplete Gamma function and
Γ(
α
) is the Gamma function. The mean value of the Gamma
distribution is equal to:
9
:
;
<


.
/
(5)
For β = 1, and for different values of the shape parameter
α, Fig. 4 shows the theoretical plots of the pdf and of the cdf
of the Gamma distribution of a generic random variable t in
seconds. Fig. 4 shows that, for a given value of
β
, the
variation of
α
implies completely different waveshapes of the
Gamma pdf and cdf.
For
α
from 0.1 to 0.5, the pdf in Fig. 4 (a) presents a
thickening of observations close to zero, as derived by the
2 3 s is the minimum value of ttne after the time aggregation [9].
Fig. 3. Empirical cumulative density function of the random variable ttne
for the site #38; (a): complete plot; (b) plot zoomed in the range [0,5000]s.
Fig. 4. Theoretical probability density function (a), and cumulative
function (b) of the Gamma distribution.
empirical distribution of all the measured sags at the site #38
showed in Fig.2. For
α
equal to 1, the Gamma cdf is equal to
the exponential cdf with mean parameter λ equal to the scale
parameter β of the Gamma distribution.
Table I shows different percentiles of the theoretical
Gamma cdf of a random variable t, with β =1 and
α
ranging
from 0.1 to 2. For
α
equal to 0.1, 10% of t is less than 2.6 × 10 -
5, i.e. very close to zero, and 70% of the values are not greater
than 1.7 h. As the shape parameter
α
increases, the 10th
percentile increases too. For
α
equal to 2, 10% of t is less than
about 42 h, and 70% of the values are not greater than 194.4
h.
In conclusion, the use of Gamma distribution requires the
correct estimation of the parameters
α
and
β
, as shown in the
successive section.
III. STATISTICAL ANALYSIS OF DATA TO DETERMINE THE
PARAMETERS OF THE GAMMA DISTRIBUTION
Modeling ttne of all the measured sags by the Gamma
distribution requires to establish the two parameters
α
and
β
.
TABLE I. PERCENTILE VALUES OF THE GAMMA DISTRIBUTION WITH
β =1 AND α RANGING FROM 0.1 TO 2.
Percentile
α
0.1
0.5
1
1.5
2
P10%
[s]
2.6 × 10-5 2.4 × 103 3.1 × 104 8.4 × 104 1.5 × 105
[h]
7.2 × 10-9 0.7 8.6 23.3 41.7
P30%
[s]
4.3 2.8 × 104 1.2 × 105 2.3 × 105 3.4 × 105
[h]
1.2 × 10-3 7.8 33.3 63.9 94.4
P50%
[s]
303.2 7.3 × 104 2.1 × 105 3.5 × 105 5.0 × 105
[h]
8.4 × 10-2 20.3 58.3 97.2 138.9
P70%
[s]
6.0 × 103 1.6 × 105 3.5 × 105 5.3 × 105 7.0 × 105
[h]
1.7 44.4 97.2 147.2 194.4
P90%
[s]
7.3 × 104 3.8 × 105 6.4 × 105 8.7 × 105 1.1 × 106
[h]
20.3
105.6
177.8
241.7
305.6

Different techniques are available for the optimal selection of
statistical parameters of an assumed pdf [24-27]. The
Maximum Likelihood Estimation (MLE) is a very popular
technique used for determining the parameters of a
distribution chosen to describe observed data. The obtained
parameters obtained by MLE can be successively optimized in
function of an analytical criterium or applying a graphical
method, as done in this paper.
In the following, we first show the application of MLE to
the case of Gamma distribution, and after we describe two
methods, one analytical and one graphical, which we used to
optimize the parameters obtained by MLE. These methods are
the Hill Climbing Algorithm (HCA) and the Quantile-Quantile
plot (Q-Q plot), respectively.
- MLE method
The searched values of the parameters
α
and
β
are found
such that they maximise the likelihood that the process
described by the pdf produced the data that were observed.
Let:
- 4=
> = (xi,1, xi,2, …, xi,N) be the vector of the N observations
at the site i of the random variable xi;
- f0 be the pdf which describes the observations and depends
on the parameters included in the vector ?
>;
- Ө be the space of all the possible values of the vector ?
>,
i.e., ?
> ϵ Ө;
the application of MLE implies that f0 belongs to the family of
Gamma distributions obtainable by different values of the
parameters α and β, which are the components of the vector
?
>.
The likelihood function @4=
>A?
> is the product of the
individual pdf of all the N samples of the random variables,
assumed independent, that is:
@

4
=
>
A

?
>


B
C
D


4
=
>
E
F
G
H

?
>
).
(6)
For the case of Gamma distribution, the likelihood
function is equal to:
@


4
=
>
A

.

/




B
1
0

%
I
J
K

L

M
K

L
0
%
&

2

3

E
F
G
H
.
(7)
The Log-likelihood function, N4O
>A?
>, is the natural
logarithm of @4O
>A?
>, i.e.:
N


4
=
>
A

?
>



PQ

@

4
=
>
A

?
>


.
(8)
Putting equation (7) in (8), the Log-likelihood function for
the Gamma distribution with shape parameter α and scale
parameter βN4=
>A./ is defined as:
N


4
=
>
A

.

/



PQ
B
1
0

%
I
J
K

L

M
K

L
0
%
&

2

3

E
F
G
H
.
(9)
Applying the logarithm properties and after some
simplifications, the final expression N4=
>A./ is equal to:
N


4
=
>
A

.

/


R.
PQ
/

-
/
S
4
T

F
E
F
G
H
+(α-
1)×
S
PQ
4
T

F
E
F
G
H
- N
PQ
U

.

.
(10)
Maximizing equation (10) with respect of the parameters
α and β, we can derive the so called MLE estimators of the
parameters. With this aim, equalling to zero the derivative of
N4=
>A./ in (10) with respect to
β
, the maximum likelihood
estimator of the parameter β is:
β =
E

3
S
M
K

L
V
L
W
&
.
(11)
Substituting equation (11) in (10), we obtain
N

.


R.
N

R.
6

R.

N

S
4
T

F
E
F
G
H
6
R.

+

.

S
N



E
F
G
H
4
T

F

6
S
N

4
T

F


E
F
G
H


6
RN

U

.


.
(12)
Equalling to zero the derivative of l(α) with respect to
α
:
X
Y
X
3


RN

R.
Z
R.

H
3
6
R
N

S
4
T

F
E
F
G
H


6
R
Z
S
N



E
F
G
H
4
T

F

– Nψ(α) = 0
(13)
where ψ (α) 23[
23 is the Digamma function.
After some simplifications on (13), we obtain:
PQ
R.
– ψ(α) =
PQ
S
4
T

F
E
F
G
H
-
S
Y



V
L
W
&
M
K

L

E
\
(14)
Equation (14) gives the optimal value of α in implicit form,
which can be solved by a numerical method.
- Hill Climbing Algorithm (HCA)
HCA is an iterative algorithm that starts with an arbitrary
solution and, in accordance with a function to be maximized
or minimized, attempts to find a better solution by making an
incremental change to the parameters. The optimal value of
the parameters is found by using a test function [27]. In the
case of this study, the arbitrary solution is given by the cdf of
the Gamma distribution with
α
and
β
obtained by MLE, and
the test function is the Chi Squared Test (CS). Let observations
be divided into M bins:
- F is the theoretical cdf;
- Oi,1, Oi,2,…,Oi,N are the N observed values of each
bins,
- the χ2 value is equal to:
]
^
=
S


S
_
K

`
V
K
W
&


S
ab
K

`
V
K
W
&

c
S
ab
K

`
V
K
W
&
d
e
G
H


(15)
where 9fe are the expected values for each bin computed as:
9f
e
=
C

g

h
R
6
C

i

h
R
(16)
where N is the data size, a is the start value of the kth bin and
b is the end value of the kth bin, (b-a) is the length of the kth
bin, F(a) and F(b) are the values of the cdf of the Gamma
distribution computed in a and b, respectively.
The searched optimal values of the parameters
α
and
β
are those corresponding to the minimum value of χ2.
- Quantile-Quantile plots (Q-Q plots)
Q-Q plot is one of the graphical tools used to make
distributional analyses between an empirical distribution
derived from a dataset and a theoretical distribution. It is based
on the plots of the quantiles of an empirical distribution
derived from a dataset against the corresponding quantiles of
the theoretical distribution. The two main analyses allowed by
the Q-Q plots are the distributional comparisons between the
two distributions [14-16], and the estimation of one
distribution parameter of the theoretical distribution. This
second target was used in this study.
The theoretical distribution was the Gamma distribution
with
α
obtained by MLE, and we used the Q-Q plot to estimate
the value of
β
. To this aim, the values of the quantiles of the
data Qd are equal to:
j



k

l
(17)
and the quantiles of the theoretical distribution Qt are equal to:
j







C

H


H
m
(18)
where F-1 is the inverse cdf of the standard Gamma function
with β = 1 and
α
estimated with MLE.
The searched value of the parameter β of the Gamma
distribution is equal to the slope of the regression line:
/

n
m
(19)
Table II shows the results for three sites of example
obtained applying the three different methods above recalled,
that is α and β estimated by MLE, α and β estimated by HCA,
α estimated by MLE and β estimated by Q-Q plot.

TABLE II. PARAMETERS OF THE CDF OF THE GAMMA FUNCTION
ESTIMATED BY MLE, HCA, AND Q-Q PLOT.
Gamma distribution
#
Site
#
o
:
ppqr
<
[s]
s
β
[s-1]
Parameters estimated
by MLE
38 1.44 × 10
5
0.18 1.25 × 10
-
6
39 1.29 × 10
5
0.21 1.63 × 10
-
6
95 2.10 × 10
5
0.20 9.52 × 10
-
7
Parameters estimated
by HCA
38 1.58 × 10
5
0.18 1.14 × 10
-
6
39 2.02 × 10
5
0.23 1.63 × 10
-
6
95 2.32 × 10
5
0.20 8.62 × 10
-
7
Parameters estimated
by Q-Q plot
38 1.52 × 10
5
0.18 1.18 × 10
-
6
39 5.45 × 10
5
0.21 1.45 × 10
-
6
95 2.36 × 10
5
0.20 8.47 × 10
-
7
From Table II, it is evident that the shape parameter
changes from MLE to HCA only for the site #39, instead the
scale parameter appreciably changes for all the sites using the
Q-Q plot. These changes of the parameters provided different
values of the expected value of the Gamma distribution
computed by the relation (5). Fig. 5 presents, for the site #38,
the histogram of the pdf of the measured ttne (blue bars) and
the pdfs with parameters estimated by MLE (red curve), by
HCA (green curve), and by Q-Q plot (blue curve).
From the curves in Fig.5, all the estimation methods
provided very similar plots, analogous results have been
obtained for the other sites of the system (#39 and #95)
considered as example. It is important to evidence that the
resolution of Fig.5 does not allow appreciating the slight
differences among the curves. However, as Table II
evidenced, the curves give different values of
β
, which causes
diverse value of E[ttne] in accordance with the relation (5).
Therefore, it was mandatory to perform a numerical
quantification of the goodness of the assumed fit of the data to
the Gamma distribution. To this aim, we used three statistical
tools: the R2-test, the Root Mean Squared Error (RMSE test),
and Chi-Squared test (χ2 test).
The R2-test is based on the computation of R2 by the
relation:
t
^


S


C


T
6
C
uv

^
E
T
G
H
S


C


T
6
C
uv

^
E
T
G
H
Z

S


C
T
6
C


T

^
E
T
G
H
(20)
where Fmean = H
E SCT
E
TGH , CTare the values of the cdf of the
Gamma distribution andCT are the values of the ecdf.
The RMSE test provides a term-by-term comparison of the
actual deviation between observed probabilities and
theoretical probabilities; RMSE is computed by the relation:
twx9


y
z
R

{


C
T
E
T
G
H
6
C


T

|
^
(21)
Fig.5
Histogram of the probability density function (blue bars) and
gamma pdfs with parameters estimated by MLE (red curve), by HCA
(green curve), and by Q-Q plot (black curve)– Site #38.
Finally, the χ2 test is based on the computation of χ2 already
described by relation (15). Table III presents the results
obtained applying these three tools on the Gamma distribution
of ttne of the sags measured at sites #38, #39, and #95 having
the parameters estimated by MLE, HCA, and Q-Q plot.
Table III evidenced that the χ2 test for the MLE method has
the highest value for all the three sites considered, instead the
R2 test gives values of errors lowest for the site #38, for any
method, and for the site #39, for the method HCA. RMSE test,
finally, gives the lowest value for the site #95 for any method.
Summarising, the fit of the data to the Gamma distribution
presented an adequate goodness for all the methods used to
estimate the
α
and
β
parameters.
IV. FORECAST OF THE SAGS
The final goal was to forecast the mean number of events
per year at each site k, Nf,k. We obtained this value by
correlating the average sag rate at the site k, 1/E[ttnek], with
the duration of one year, P, by the relation:
R
}

F



~
•

9
:

F
<
(22)
where the average sag rate is expressed in number of sags per
seconds and P is expressed in seconds (31.5 × 106 s).
E[ttnek] in (22) is the expected value of the Gamma
function given by (5). For every method used to estimate the
parameters of the Gamma distribution, i.e. MLE, HCA, and Q-
Q plot, we have a different value of E[ttnek] and, then a
different value of forecasted Nf,k.
Thanks to the availability of the voltage sags really
measured in 2018, we were able to quantify the error of the
forecast in each of the considered sites. Let Nm,k be the number
of voltage sags measured at the site k in the year 2018 and Nf,k
be the number of sags forecasted at the same site k the relation
(22) using the sags measured in the preceding three years
(2015, 2016, 2017). The error of the forecast at each site k, εf,k,
was computed as the percentage value of the difference
between Nf,k and Nm,k, in per unit of Nm,k. For the considered
sites #38, #39, #95, Table IV shows the results of the forecast
and the computed values of the error εf,k for the three presented
methods (MLE, HCA, Q-Q plot).
The forecast error of all the sags comprehensive of the
clusters ranges from 0% to a maximum value lightly greater
than 10%. This performance of the forecast is really close to
the one obtained in [13-15] where only the rare sags were
forecasted. The accuracies of the references in the literature
are those reported in [12] where, with the highest frequency of
sags (1 per day), the error of 2% would require registering sags
for 25 years. The errors in Table IV are of the same order of
magnitude, but they were obtained for only three years.
TABLE III. STATISTICAL ERRORS OF THE CDF OF THE GAMMA
FUNCTION ESTIMATED BY MLE, HCA, AND Q-Q PLOT
Gamma distribution Site #
€
•
test
‚
•
ƒ„…ƒ

†‡ˆ‰

ƒ„…ƒ

Parameters estimated
by MLE
38 3.22 0.88 2.19
39 3.59 0.92 1.87
95 3.28 0.93 1.53
Parameters estimated
by HCA
38 2.15 0.87 2.25
39 2.03 0.88 2.30
95 2.32 0.92 1.60
Parameters estimated
by Q-Q plot
38 2.33 0.87 2.29
39 2.68 0.91 1.98
95 2.28 0.92 1.62

TABLE IV. FORECASTED SAGS, MEASURED SAGS, AND FORECAST
ERROR USING THE GAMMA FUNCTION WITH PARAMETERS ESTIMATED BY
MLE, HCA, AND Q-Q PLOT.
Gamma distribution Site #
Š
‹

Œ
Nm,k εf,k

Parameters estimated
by MLE
38 222 211 5.2%
39 248 248 0.0%
95 147 135 8.8%
Parameters estimated
by HCA
38 202 211 4.2%
39 225 248 9.2%
95 134 135 0.7%
Parameters estimated
by Q-Q plot
38 209 211 0.9%
39 220 248 11.3%
95 132 135 2.2%
We cannot ascertain the method, among MLE, HCA and Q-Q
plot, which gives the best performance of forecast in all the
sites. For the site #38, Q-Q plot gives the minimum forecast
error value, instead the minimum forecast error is obtained by
MLE and by HCA for the site #39 and #95, respectively. The
different performances of the methods at the various busses of
example could be linked to a different share of the sags
between rare sags and clusters. This analysis was not
performed in this first stage of the research. However, for the
considered sites of example, the different methods used to
estimate the parameters of the Gamma distribution are
effective to provide the range of the mean number of events
per year with acceptable errors. Further studies are in progress
to extend the forecast to all the regional system in study.
V. CONCLUSIONS
The paper faces the problem of forecasting the voltage sags
at the sites of a real system in terms of number of occurrences
expected in those sites in the successive year. The forecast is
based on the statistics of measured sags in three preceding
years, adopting a data-driven approach rather than model
based one. Since, the sags measured in real systems are
composed not only by rare sags, but also of groups of sags,
the paper proposed the Gamma distribution to properly model
them in view of the forecast. The use of Gamma distribution
requires to estimate two parameters, i.e. the shape parameter
and the scale parameter. The paper shows three alternative
methods to estimate their values, quantifies the estimation
errors and the forecast errors in three sites of example of a
real regional system. For these cases, it is not possible to
establish which method gives the best performance in terms
of forecast. However, even in the worst case, the forecast
presented an error lightly greater than 10%. This value is in
line with the error obtainable when only rare sags were
estimated. It is also considered acceptable, accounting that
only three years of measuring on field were used. Further
studies are ongoing to extend the approach to all the sites of
the regional system, also to verify if a method, among those
proposed in this paper, could be chosen as the one of the best
forecast performances.
REFERENCES
[1] M. H. Bollen, I. Gu, Signal Processing of Power Quality Disturbances.
Hoboken, NJ, USA: Wiley, 2006, pp. 766–771.
[2] G. Carpinelli, P. Caramia, C. Di Perna, P. Varilone, P. Verde,
“Complete matrix formulation of fault-position method for voltage-dip
characterisation,” IET G.T.D. 2007, 1, 56, doi:10.1049/iet-
gtd:20050544.
[3] A. Dos Santos, M.C. De Barros, “Voltage sag prediction for network
planning,” Electr. Power Syst. Res. 2016, 140, 976–983,
doi:10.1016/j.epsr.2016.03.033.
[4] G. Olguin, M. Aedo, M. Arias, A. Ortiz, “A Monte Carlo Simulation
Approach to the Method of Fault Positions for Stochastic Assessment
of Voltage Dips (Sags),” IEEE/PES T & D Conf. & Exp: Asia and
Pacific, Dalian, China, 18 August 2005, pp. 1–6.
[5] P. Caramia, P. Varilone, P. Verde, L. Vitale, “Tools for Assessing the
Robustness of Electrical System against Voltage Dips in terms of
Amplitude, Duration and Frequency,” Renew. Energy Power Qual. J.
2014, 1, 177–182, doi:10.24084/repqj12.275.
[6] Y. Lim, G. Strbac, “Analytical approach to probabilistic prediction of
voltage sags on transmission networks,” IEEE Proc. G.T.D. 2002, 149,
7–14, doi:10.1049/ip-gtd:20020008.
[7] E. Espinosa Juarez, A. Hernandez, “An Analytical Approach for
Stochastic Assessment of Balanced and Unbalanced Voltage Sags in
Large Systems,” IEEE Trans.on PD. 2006, 21, 1493–1500,
doi:10.1109/tpwrd.2005.860266.
[8] E. Di Mambro, P. Caramia, P. Varilone and P. Verde, “Voltage sag
estimation of real transmission systems for faults along the lines,” 18th
ICHQP, Ljubljana, 2018, pp. 1-6.
[9] P. Caramia, E. Di Mambro, P. Varilone, P. Verde, “Impact of
Distributed Generation on the Voltage Sag Performance of
Transmission Systems,” Energies. 10 (959), 2017, pp.1-19.
[10] Y. S. Lim and G. Strbac, “Analytical approach to probabilistic
prediction of voltage sags on transmission networks,” IEE Proc. G,T
and D. 149(1), 2002, pp. 7-14.
[11] A. dos Santos and M. T. Correia de Barros, “Predicting Equipment
Outages Due to Voltage Sags,” IEEE Trans. on PD. 31(4), 2016, pp.
1683-1691.
[12] M.R. Qader, M. H. J. Bollen, and R. N. Allan, “Stochastic prediction
of voltage sags in a large transmission system,” IEEE Trans. Ind. Appl.,
vol. 35, no. 1, pp. 152–162, Jan./Feb. 1999.
[13] C. Noce, L. Di Stasio, P. Varilone, P. Verde and M. De Santis, “On the
Forecast of the Voltage Sags: First Stages of Analysis on Real
Systems,” in 55th UPEC, Torino, Italy, 2020, pp. 1-6.
[14] M. De Santis, L. Di Stasio, C. Noce, P. Verde, P. Varilone, “Initial
Results of an Extensive, Long-Term Study of the Forecasting of
Voltage Sags,” Energies, 2021; 14, (5):1264.
https://doi.org/10.3390/en14051264.
[15] M. De Santis, L. Di Stasio, C. Noce, P. Verde and P. Varilone, "Indices
of Intermittence to Improve the Forecasting of the Voltage Sags
Measured in Real Systems," in IEEE Trans on PD, 2021, doi:
10.1109/TPWRD.2021.3082280.
[16] A. d. Santos, T. Rosa and M. T. Correia de Barros, “Stochastic
Characterization of Voltage Sag Occurrence Based on Field Data,”
IEEE Trans on PD. 34 (2), 2019, pp. 496-504.
[17] Council of European Energy Regulators (CEER), 6th CEER
Benchmarking Report on the Quality of Electricity and Gas Supply,
Brussels, August 2016, www.ceer.eu.
[18] ARERA (formerly AEEGSI), Resolution ARG/elt 198/11. Italy, 2011,
www.arera.it/it/docs/11/198-11arg.htm.
[19] ARERA, Resolution 646/2015/R/eel. Italy, 2015.
[20] M. De Santis, C. Noce, P. Varilone, P. Verde, “Analysis of the Origin
of Measured Voltage Sags in InterconnectedNetworks”, Electric Power
Systems Research, vol. 154, 2018, pp. 391–400.
[21] C. Noce, M. De Santis, L. Di Stasio, P. Varilone, P. Verde, “Detecting
the Origin of the Voltage Sags Measured in the Smart Grids”, 7°
ICCEP, Otranto (LE), Italy, 2-4 July 2019.
[22] IEEE Guide for Voltage Sag Indices. In IEEE Std 1564–2014; IEEE:
Piscataway, NJ, USA,2014; pp. 1–59
[23] Joint Working Group Cigre/Cired C4.110. Voltage Dip Immunity of
Equipment and Installations; Report 412; Cigré, Paris, France, 2010.
[24] S. Kadry, Stochastic Methods for Estimation and Problem Solving in
Engineering. IGI Global, March 2018. ISBN: 978-152255046-4; pp.
1-275.
[25] R. J. Rossi, Mathematical Statistics: An Introduction to Likelihood
Based Inference. Wiley, October 2018, ISBN: 978-1-118-77104-4, pp.
1-448.
[26] P. Olofsson, M. Andersson, Probability, Statistics, and Stochastic
Processes. 2nd Edition, John Wiley & Sons, Inc, May 2012, ISBN:
9780470889749, pp. 1-558.
[27] Z. Klaić, D. Šljivac and Z. Baus "Probability Density Functions of
Voltage Sags Measured Indices," Journal of Electrical Engineering,
vol.62, no.6, 2011, pp.335-341.https://doi.org/10.2478/v10187-011-
0053-8