=Paper=
{{Paper
|id=Vol-2803/paper12
|storemode=property
|title=Stochastic model of thermal processes in the contact network at
arc discharges occurring at high speeds of movement (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2803/paper12.pdf
|volume=Vol-2803
|authors=Viktoria V. Litvinovа,Vladimir V. Moiseev,Evgeniy V. Runev
}}
==Stochastic model of thermal processes in the contact network at
arc discharges occurring at high speeds of movement (short paper)==
https://ceur-ws.org/Vol-2803/paper12.pdf
Stochastic model of thermal processes in the contact network at
arc discharges occurring at high speeds of movement
Viktoria V. Litvinovаa, Vladimir V. Moiseevb and Evgeniy V. Runevb
a
St. Petersburg Electrotechnical University «LETI», ul. Professora Popova 5, Saint Petersburg, 197376, Russia
b
Emperor Alexander I St. Petersburg State Transport University, 9 Moskovsky pr., Saint Petersburg, 190031, Russia
Abstract
The paper proposes a stochastic model, on the basis of which estimates are given of the
parameters at which extreme situations occur due to the interruption of the electrical
contact between the electro-rolling stock current collector (EPS) and a contact wire for
the wear and tear of the contact network as a result of acts of arcing. The model takes into
account the influence of random factors, which are temporary and sometimes
repetitive. The probabilities of deviating from the coordinates of the breakdowns of the
contact network from the values given in advance as a result of acts of arcs with defined
repeatability periods are obtained.
Keywords
Weak contact, stochastic model, intensity function, repeatability period, probability
asymptotics, characteristic function, multimodal distribution, unimodal distribution
1. Introduction when electric rolling stock moves along high-speed
motorways, there are multiple disconnections of the
A topical problem encountered in the operation of current conductors from the contact wire - a
an electric rolling stock is the reduction of the wear violation of the mechanical and electrical contact,
and tear of the contact network and the extension of which result in high-potential arc
its useful life under the influence of electric arc discharges. Repeated acts of discharges result in
discharges arising from the breakdown of severe wear on the surface of the overhead wire,
mechanical contact1. leading to the breakdown of the electrical contact
In the present operating conditions (soft soils, low and, in the worst case, the breakdown of the contact
air temperature for most of the year, high humidity, network.
icing and insufficient quality of contact suspension It should be noted that the above-mentioned
surface treatment), it is not possible to improve the mechanical and electrical contact defects also lead to
elasticity of the contact suspension. As a result, the deterioration of the traction equipment of the
electric rolling stock [4].
According to the available static data on the
Models and Methods for Researching Information overhaul of the contact suspension at the different
Systems offices of the October Railway, the frequency of
in Transport , December 11–12, 2020, Saint-Petersburg, major repairs for the replacement of the contact
Russia suspension is on average from 1,5 to 3 months
EMAIL: vlitvinova78@gmail.com (V.V. Litvinova); depending on the season of operation and the flow of
moiseev_v_i@list.ru (V.I .Moiseev); jr_2010@mail.ru
trains on the main line.
(E.V. Runev);
ORCID: 0000-0002-0628-0440(V.V. Litvinova); 0000- The paper proposes a stochastic model, which
0003-0558-6242 (V.I. Moiseev);0000-0001-6707- makes it possible to assess the important quantitative
888X(E.V. Runev) characteristics of said extreme situation, which is
©️ 2020 Copyright for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
temporary and sometimes repetitive. These
CEUR Workshop Proceedings (CEUR- characteristics include the intensity function, the
WS.org) repeatability period and the probability of deviation
84
of the disconnect coordinate from the specified value 2.2. Distribution intensity function
[1], [3]. The above characteristics make it possible to
assess the periods of inter-service service service and
properties
the probability of wear on the overhead wires, which
includes breakages from electric arc discharges, to From (2) for the intensity function follows the
p t dF
optimize the time and cost of drip repairs and to expression: t (here p t
adjust the repair plan for both the main lines and the 1 F t dt
sections of the road [9]. probability density function of the distribution ),
2. Problem statement which specifies the following properties:
1. f the current probe position takes a value
The key object in the study of extreme situations greater than a given value s , that is, the
[5],[6],[7] arising from the breakage of the current distribution function less than one:
carrier from the contact wire is the random value
position of the breakpoint on the overhead wire. The P s 0 F s 1 for any end value
model parameter is the pair , T , where –
s , then t dt , would be divergent,
intensity function, аnd T – repeatability period of s
the random distribution . The distances between 1
provided that: t O and
the two disconnections of the current collector and t
the contact wire that exceed this are random. The 1
mean of these distances is the repeatability converging if: t O 1 , 0 ;
period. Another important parameter for the t
2. from equality
distribution of extremes is the intensity function [1].
Thus, the starting point of a mathematical model t 1 F t p t the differential
describing periodic extremes [15], [16] is the pair equation linking the intensity function and the
, T . density function follows:
1 d t 1 dp t
1 .
2.1. Intensity function of the position of t dt
2
p t t dt
the current probe xmo
3. if distribution fashion, then
The intensity function is defined as [1],[3]. d
Considers the probability that the position of the xmo 2 xmo .
current probe on the overhead wire will be equal to dt
or greater than a certain value s : These properties make it easy to construct an
P s 1 F s ,
intensity function for models with modal
(1)
distributions. It should be noted that the function of
where F random value distribution function . the density of the current probe is usually
Enter the conditional probability that the unimodal. This is because the probability of
position of the current collector’s disconnect on the withdrawal at certain points of the wire is lower
contact wire will lie within the interval s, s , because of mechanical tension and the position of the
centre of mass of the portion of the wire considered.
provided that its meaning is greater than or equals s ,
which is expressed by the following formula:
2.3. Period of repeatability of
P s, s s separation positions
P s, s s :
P s
s If you look at a series of observations in which
: t dt. the position of the current probe is greater than or
s equal to , this deviation of the position to the right is
(2). an event of interest to us. Let’s determine its
The function is called the intensity function probability P s 1 F s for p , and the
or fault intensity function. It determines the probability of the opposite event – P s 1 p .
probability of a current probe being removed at a
In the experiment, we’ll look at observations at
point with a coordinate s more to the right s by
regular intervals, and the experiment will stop as
the amount величину .
85
soon as we have an event of interest, namely the the right, with certain repeatability periods. In
deviation to the right of the current probe s . addition, the formulae provide valid parameter
To interpret the results, consider the random estimates by the maximum likelihood method
value X number of tests up to first to right (i.е. [11],[12],[14] Multimodal distributions should be
s ), it accepts values from set 0,1,2,... . X is
chosen as model distributions:
for partitions located on one block, it is
subject to geometric distribution with parameter p sufficient to choose a unimodal distribution
( X ~ G p ). Properties of this distribution are with density:
known: distribution series P X k p1 p ; k
p t f t t0 10,L (8),
where t 0 0, L , L – length of block - area;
characteristic function t
p
; expected
1 e 1 p
it
it is sufficient to use linear combinations of
value EX i 0 p , second starting point
1 p functions of the form (8) for breaks occurring
on an extended section comprising several
EX 2 0
1 p 2 p
allow to find the
blocks.
p2 2.4. Simulation example: case of one
required model parameters [8],[10]. block - fixed length section
Define the repeatability period as a mathematical
expectation T : EX [3]. The separation density function in this case
It follows from the definition that for a given belongs to a two-parameter family h, t 0
model the repeatability period takes the form: distributions and has the form:
T
F s
(3).
p t; h, t0 С t t0 h 10,L , here t0 L 2 , h –
2
1 F s
the variation of the contact wire from the equilibrium
It follows from formula (3) that for the position (can be determined by statistical
repeatability period a bottom-up assessment is
evaluation). Random density normalization constant
fair: T 1.
, specified by the normalization condition:
Standard deviation from repeatability period is
p t dt 1 , p t; h, L C t L 2 h 10,L
2
given by
0, L
F s
expression: : EX 2 E 2 X , or 12
1 F s and takes on the importance С
L L 12 h
2
.
using formula (2) to accept:
The distribution function has the form:
: Т 2 Т (4).
Then the probability of deviating right from the
1
3
F s; h, L С s L
2
3 3
8
L h s ,
set position of the current probe to the observation
with the number k or in the room k : 0 s L.
P X k 1 1 p 1 Fk s (5)
k Repeatability period in this model:
4s L 2 L3 12 sh
3 3
If in the capacity of k choose T , for probability
(5) there is an expression: T s; h, L 2 (9).
L 4s L 23 12hL s
3 3
Т
1
P X Т 1 1
2
(6)
1 Т Probability of right deviation from a specified
value:
Probability asymptotics (6) at large T :
P X Т s; h, L
T has the form: Т
3 3
lim PX T 1 1 0,63212 . (7) L 4s L 2 3
12 hs (10),
e 1 2
T
The formulas (5) and (6) make it possible to 3L L2 4h
estimate the probability of deviation from the
specified position of the current probe detachment to
86
From (11), (12), (13) it follows that when the
4s L 2
3 3
L 12 sh
3
2 bends are small h : h 0 The repeatability period
here T . will accept the following values at the appropriate
L 4s L 2 12hL s
3 3 3
2 points on the overhead wire: T1 h, L 1 2 ,
T2 h, L 1 , T3 h, L 2 .
Formulas (9), (10) express the functional dependence The above dependency graphics are shown on
of the repeatability period and the probability of the figure 1.
right deviation of the current probe from a given Accepted here as L 1200 meters, then L 2 600
position to the observation with the number k or in meters.
the room k from that of the s on a wire, here
0 s L . These functional relationships are
complex and cumbersome for numerical estimates,
which is particularly important for applications, the
type. Given the symmetry in the probabilistic model
described by the unimodal distribution (8), limit
values were found (9), (10).
2.4.1. Limit value of function T s; h, L
when s 0
As a result of the cut-off s 0 repeatability
period for the left end of the block:
T1h, L lim T s; h, L . Figure 1: Repeatability period dependent on
s 0 contact position on overhead wire.
L2
Here T1 h, L
1 Repeatability period limits are tabulated for ease
2 L 12h 2 24h L2
2
(11).
of reference 1.
Table 1 Repeatability period limit values
2.4.2. Limit value of T s; h, L when Limit value Extreme Limit value
sL 2 T s; h, L contact wire when
position h 0
For the middle of the block s L 2 the s 0 T1 0, L 1 2
T1 h, L
1
repeatability period dependent on model parameters 2 24 h L2
will be:
T2 h, L lim T s; h, L 1 (12). T2 h, L 1 sL 2 T2 h, L 1
sL 2
s L0 T3 h, L 2
T3 h, L 2
24h
2.4.3. Limit value of T s; h, L when L2
The asymptotic formulas for the repeatability
s L
period indicate its limitation to a segment s 0; L ,
For the right end of block - section а s L the corresponding to the length of the block - section
( L 1200 meters). This feature of the repeatability
repeatability period of the model will be:
T3 h, L lim T s; h, L
2 L2 12h
period makes it possible to predict the frequency of
major maintenance activities to replace worn-out
s L L2 (13). parts of the network. It should be noted that in the
24h operation of the network, the optimization of the cost
2 2
L of repairs is important, not only at the cost of the
Due to the symmetry of the partitions, the limits work carried out, but also at the cost of the time
of the repeatability period on the right and left ends spent. The latter means that it is more advantageous
are linked by the ratio: T3 h, L T1-1h, L .
to repair several sections in parallel (in one period)
than to replace the overhead wire consecutively (after
some time to return the repair crew to the same
87
section). The above-mentioned mode of repair makes Thus, for this unimodal two-parameter model (8),
it possible to substantially reduce the cost of idling of
probability limit values P X Т j h, L when
electrified rolling stock. Thus, the replacement of the
h 0 is not dependent on parameter L
overhead wire on at least one block - the section
distributions of type (8) and have a uniform form for
leads to the dysfunction of a fairly long stretch of the
the whole family of distributions. Localize the values
of the probability function on a segment 0,42;0,56
railway network, resulting in economic losses for the
enterprises using the company’s services «Russian
Railways» as the main carrier. results in high accuracy forecast of wear periods and
major maintenance of contact suspension.
2.5. Limit values for deviation from a
specified value 2.6. Severance intensity function
Intensity function of the probability of the current
Based on the expression (10) for deviation
probe being removed at a point with a coordinate s
probabilities and repeatability time limits (11), (12),
(13), the probability limits are determined from: to the right s by the amount , introduced in
T h , L paragraph 2.1 for this model is:
P X Т j h, L 1 1
1 j
1 T j h, L s; h, L
C s L 2 h 10, L
2
(12),
In the least case with low bending values h :
1
1 C s L 2 L 8 hs
3
3 3
h 0 refer:
here 10, L – segment indicator function 0, L . Here
PX Т1h, L (3 3) 3 ;
PX Т2 h, L 1 2 ;
12
С
L L2 12 h
– is the standard density constant of
PX Т h, L 5 9 .
3 a random distribution , specified by the
These limits for probabilities indicate that the normalization condition: p t dt 1 ,
probability of a deviation increases with the 0, L
coordinate of the detachment.
On figure chart of deviation probability from
p t; h, L C t L 2 h 10,L .
2
coordinates s . Following, on figure 3 is the graph of the
intensity function for the following values of the
distributions: h 0,05 meters, L 1200 meters.
Figure 3: Intensity function graph
Figure 2: Probability of deviation chart along the
overhead wire.
This relationship reflects the fact that at the end
of repeatability, as is the case for most contact
networks, the probability of deviation varies on a
subset of the segment 0,1 , but it does not accept
Analyzing the intensity function and its graph,
you can see that it has at least in the middle of the
values close to 0 or 1. Probability as a function s – segment 0, L , which is fully consistent with the
slowly changing in segment 0, L , i.е. by the length type of probability distribution and the presence of
of the wiring function. At the point L 2 probability the latter mode also at a point t 0 L 2 .
function is bent, which means increasing the rate of The expression for the intensity function (12) has
growth of the function when approaching the right a rather cumbersome and difficult form for analysis.
end.
88
Give an asymptotic intensity function in the vicinity conditional distribution P s, s s
of a fashion point t 0 L 2 : probabilities of lead.
t; h, L
24h Note also that in this model the intensity function
L L 12h
2
1
has no property: t O , t . This is due
576 h 2 t
t L 2
L2 L2 12h 2
(1
to the fact that it is set in a non-trivial way on a
compact segment 0, L , outside which it lasts 0: the
3
24 1 1728 h t L 22 breakdowns of the current collector occur in a fixed-
2
L L 12h L L 12h
3 2 3
length block, so the asymptotics at the large
coordinates of the separation points are not
o t L 2
2
meaningful.
3).
The main part of the formula (13) asymptotics
should be designated ~ : 2.7. Conclusion
~ t ; h, L
24 h
L L 12 h
2
The task of estimating the probability
characteristics of an extreme situation occurring
576 h 2
t L 2
during the operation of the contact network as a
(14).
L2 L2 12 h
2 result of the detachments of the current collector
from the contact wire during the movements of high-
3 speed electric rolling stock was defined and solved,
24 1 1728 h t L 22
L L 12 h
2 3 2
L L 12 h
3
as a consequence of electric arc discharges between
the overhead wire and the current collector.
Graph of the main part of the asymptotics of the Precise formulas and their asymptotic
intensity function in the vicinity of the point t 0 L 2 expressions have been found in important extreme
in the figure below 4. cases for the probabilities of deviations from a given
value, repeatability periods and intensity function for
Figure 4: Graph of the main part of the a special type of unimodal distributions, in
asymptotics of the intensity function in the accordance with the load distribution over the length
vicinity of the point t 0 L 2 of the wire. This makes it possible to assess the most
important characteristics and to predict the
occurrence of said extreme situation - breakage of
the wire as a result of heavy heating with an electric
arc. It is equally important to draw up an optimal
plan of work for periodic major maintenance, thus
minimizing the cost.
Note that this type of distribution of possible
straps from the overhead wire, although
approximate, is for evenly stretched contact
suspensions without severe altitude variations and
the absence of soft soils and underground floating
lakes, It describes the processes of cutting off the
current probes with sufficient precision.
The work, according to the idea of the authors,
~ t shows that the
Dependency graph analysis has a natural continuation, where the type of
distribution will be specified according to the
main part of the intensity function in the geometrical characteristics of the contact wire, such
neighborhood of the mode of the distribution has a as the coordinates of the attachment points, the
quadratic view. Minimum value ~ t is at a point
amount of bending, the curvature, the natural profile.
t 0 L 2 . This is fully consistent with the fact that In addition, it is intended to assess the parameters of
the working distributions on the basis of statistical
the intensity function acts as the density of the
data from different offices and company roads
«Russian Railways» [17].
89
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