Acta Polytechnica
doi:10.14311/AP.2018.58.0370
Acta Polytechnica 58(6):370–377, 2018 © Czech Technical University in Prague, 2018
available online at http://ojs.cvut.cz/ojs/index.php/ap
A METHOD OF COMPLEX CALCULATION OF RATIONAL
STRUCTURAL PARAMETERS OF RAILWAY HUMPS
Sergii Panchenkoa, Oleksandr Oharb, Maksym Kutsenkoc,
Julia Smachiloc, ∗
a Rector, Ukrainian State University of Railway Transport, Feyerbach square 7, Kharkiv, Ukraine
b Head of Department of Railway Station and Junctions, Ukrainian State University of Railway Transport,
Feyerbach square 7, Kharkiv, Ukraine
c Department of Railway Station and Junctions, Ukrainian State University of Railway Transport, Feyerbach
square 7, Kharkiv, Ukraine
∗ corresponding author: smachilo.julia@gmail.com
Abstract. The article deals with a method of complex calculation of rational structural parameters
of humps at classification yards. Unlike existing methods, this method allows an implementation of
the technology of guided gravity regulation of the cut speed by applying a special layout and profile
arrangement. The authors believe that it will decrease the maintenance costs to refund damaged car
and cargo, costs on electricity needed for the cut speed regulation and some extra charges due to
demurrages caused by waiting for breaking-up at arrival yards.
Keywords: railway transport, railway humps, optimization.
1. Introduction
Due to certain trends in the world energy resources market and severe competition in transportation, the
research of optimization of rail transportation charges is urgent. Besides, the problem is also of importance for
sorting operations at railway stations, as their parameters are greatly conditioned by structural parameters of
the humps. In article [1], the authors pay an attention to the fact that one of the components considerably
influencing the total costs of transportation are the car processing costs on humps. Cars can be processed
several times on their way from departure stations to destination stations. According to [1], a lot of factors
influence the processing costs, among which are the costs of depreciation, spare parts and maintenance of cut
speed regulators (the maintenance costs are proportional to the costs of the regulators).
Over a long period of time a lot of scientists have paid their attention to an improved layout and profile
structure of a hump, cut rolling speed regulators, systems of automated hump technological processes and cut
braking modes to increase the breaking-up efficiency [2–7]. Thus, in research [2, 3], the aim is achieved by the
optimized structures of hump necks, in [4] — by defining the optimal parameters of the longitudinal hump
profile, in [5] — by developing new and improved existing structures for cut rolling speed regulators, in [6] —
by forming approaches to the automated braking regulation, and in [7] — by optimizing cut braking modes.
The analysis of the above-mentioned sources testifies that many scientists theoretically prove the possibility to
increase the breaking-up efficiency with the methods proposed by the authors for humps.
Research [8] implies that there are some factors of the sorting process substantially influencing the breaking-up
efficiency; thought they are rather difficult to consider, forecast or formalize. The operational condition of
cut rolling speed regulators and automation devices, the degree of consideration and presentation method of
a random nature, the state of wheelsets and other factors are among them. The research concluded that the
reduction of influence of a human factor on the breaking-up efficiency is still a problem. Thus, consideration,
forecasting, or formalization of the above-mentioned factors is currently a very difficult challenge, and the
solution is not found yet.
Also, in research [8], the authors substantiate the importance to implement the technology of guided gravity
braking for cuts. The special layout and profile arrangement for classification yards proposed by the authors
(Figure 1) can be used to implement the given technology.
A special feature of such an arrangement is the location of the switching area (SA) (either in part or in
whole) at the origin of sorting tracks up to the yard retarder position (YRP) on the ascent. The other elements
on the section from the hump crest (HC) to the design point (DP) were located on the descent. The height and
longitudinal profile of the arrangement provide, firstly, the rolling of slow light car (SL) in unfavourable winter
conditions from HC to DP, the most difficult track in terms of resistance, and, secondly, sufficient intervals at
dividing switching points in the SL-FH link (where FH- fast heavy car).
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vol. 58 no. 6/2018 Example of an Article with xa Long Title
Figure 1. The arrangement of the guided gravity regulation of cut braking.
Therefore, intervals between cuts, sufficient for throwing over points from one position to another, are only
provided due to a special structure of the descent part profile, and the location of certain profile elements on the
ascent allows slowing the cars. In other words, such an arrangement generates the gravity braking effect. When
applying this sorting arrangement, YRP does not change its functionality.
In [8], the authors put forward the hypothesis that under automated car processing, the accrued economic
benefit over the calculation operational period for the arrangement will exceed the accrued economic benefit
over the similar operational period for a conventional automated sorting hump. Even if the costs of cut rolling
speed regulators can be twice as higher when applying the technology of guided gravity regulation of cut
braking (additional investment into automated devices for conventional humps can compensate the difference of
investment into car retarders).
2. A method of complex calculation of rational structural
parameters of railway humps
Therefore, the most efficient structural variant for a hump design with the guided gravity regulation of cut
braking is the variant of the least needed capacity of the YRP (HYRP) in terms of meeting requirements for
safety and fail-safe sorting operations. Thus, the needed capacity of the YRP is the criterion for rationalized
structural parameters of a hump with the guided gravity braking technology for cuts. As far as for a certain
hump, if the position of its crest and profile element length are constants, then:
HYRP = f(I1,I2, . . . ,In), (1)
where I1,I2, . . . ,In — the slopes of section elements from the hump crest to the origin of the switching area.
Let us develop the objective function to define the rational values of structural parameters of a hump.
Under favourable rolling conditions, the needed capacity of the yard retarder position, according to [1], is
defined as follows:
HYRP = ken
(( n∑
r=1
LrIr + Lswitchiswitch + 4Lstist
)
· 10−3 +
V 20
2 ·g′
)
=
(
w0LHC-SA + V 2mid(HC-SA) ·
(
0.56 ·nHC-SA + 0.23 ·
∑
αHC-SA
))
· 10−3 − (Lswitchiswitch + 4Lstist) · 10−3),
(2)
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S. Panchenko, O. Ohar, M. Kutsenko, J. Smachilo Acta Polytechnica
where ken – the enlargement factor of the minimal design capacity of retarder positions on the hump descent; n
– the number of profile elements of the descend (from the hump crest to the origin of the switching area); Lr,Ir –
the length, m and the slope, %0 of the profile elements of a hump from the crest to the origin of the switching
area), respectively; Lswitch,Iswitch – the length, m and the slope, ‰ of the switching area, respectively; 4Lst, ist
– the length, m and the slope, ‰ of the sorting track section from the switching area to the origin of YRP; V0 –
the initial speed of the consist shunting on the hump crest, m/s; g′ – the acceleration of the gravity force of a
heavy car with consideration of the rotating wheelset masses, m/s2; w0 – the basic specific resistance to a heavy
car, N/kN; LHC-SA – the section length from the hump crest to the origin of the switching area, m; V 2mid(HC-SA)
– the average speed of a heavy car on the section from the hump crest to the origin of the switching area, m/s;
nHC-SA – the number of switching points on the section from the hump crest to the origin to the switching area;∑
αHC-SA – the sum of the rotation angles on the section from the hump crest to the origin of the switching
area.
Since each profile element consists of technological elements:
n∑
r=1
LrIr =
m∑
i=1
Liii,
V 2mid(HC-SA)
(
0.56 ·nHC-SA + 0.23 ·
∑
αHC-SA
)
=
m∑
i=1
(
0.56 ·ni + 0.23 ·
∑
ai
)
·V 2mid(i), (3)
where m – the number of technological elements.
The values V0,g′,w0,Li,ni,
∑
αi for i = 1, . . . ,m are constants.
Let:
ken = A,
V 20
2 ·g′
−LHC-SAw0 · 10−3 = B, (0.56 ·ni + 0.23 ·
∑
αi) · 10−3 = Ci, (4)
then
HYRP = A
( m∑
i=1
(Liii) · 10−3 + B +
m∑
i=1
Ci ·V 2mid(i)
)
→ HYRP(min). (5)
The average rolling speed of a heavy car on ith technological element is:
Vmid(i) =
V ′i + Vi−1
2
, (6)
where V ′i – the speed of a heavy car at the end of ith element in a first approximation (calculation of V ′i considers
only those specific resistances, which do not depend on the average rolling speed on a technological element:
basic (w0(i)), on snow and frost (wsn(i)) and on braking (wb(i))), m/s:
V ′i =
√
V 2i−1 + 2 ·g′Li(ii −w0(i) −wsn(i) −wb(i)) · 10−3, (7)
where Li, ii – the length and slope of ith technological element, respectively. Since the HYRP is defined under
favourable rolling conditions, wsn(i) and, besides, according to [1] wb(i) = 0.
Thus,
V ′i =
√
V 2i−1 + 2 ·g′Li(ii −w0(i)) · 10−3, (8)
where Vi−1 – the heavy car speed at the end of i− 1 element in a second approximation (with consideration of
wrol(i − 1) and wb(i − 1)).
The heavy car speed at the end of mth element:
Vm =
(
V 20 + 2 ·g
′L1(i1 −w0 −wrol(1) −wb(1)) · 10−3
+ 2 ·g′L2(i2 −w0 −wrol(2) −wb(2)) · 10−3 + · · · + 2 ·g′Lm(im −w0 −wrol(m) −wb(m)) · 10−3
)1/2
=
√
V 20 + 2 ·g′
∑m
s=1 Ls(is −w0 −wrol(s) −wb(s)) · 10
−3. (9)
Therefore, the heavy car speed at the origin of (i− 1)th element is:
Vi−1 =
√
V 20 + 2 ·g′
∑i−1
s=1 Ls(is −w0 −wrol(s) −wb(s)) · 10
−3. (10)
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According to [1], the actual average rolling speed of a heavy car on the technological element is:
V amid =
Vst +
∑Z
i=1
√
V 2st + 2g′Li(ii −w0)/(1000Z)
Z + 1
, (11)
where Z – the number of elemental sections as components of the technological element (for calculation of
V amid , the elemental section length is taken 0.5 m, i.e. L/Z = 0.5). According to [1], the authors propose to
define the average speed on a technological element by the formula:
Vmid(i) = kiV ′m(i), (12)
where V ′m(i) – the heavy car’s speed in a first approximation in the middle of a technological element:
Vm(i) =
√
V 2i−1 + g′Li(ii −wVRG0 ) · 10−3, (13)
and ki – the correction index.
The authors propose to define the correction index on the basis of the conditions of equality of errors in
calculation of the average heavy car speed on technological elements; one of them is of an infinitesimal length,
thus, Vmid on the element can be taken as Vst, the second – as a length of 30 m and located on a slope of 50%0,
thus:
Vst −kV ′st = kV
′
m −V
a
mid. (14)
From the equation obtained:
k =
Vst + V amid
Vst + V ′m
. (15)
The research made by the authors on the basis of calculations of the correction index at various Vst showed
that it can be set as an exponential function:
k = −0.0576191 ·e−0.5710201·Vst + 0.9966873. (16)
Thus,
HYRP = A
( m∑
i=1
(Liii) · 10−3 + B +
m∑
i−1
Cik
2
i V
2
i−1g
′Li(ii −wVGC0 ) · 10
−3
)
→ HYRP(min), (17)
where
V 2i−1 = V
2
0 + 2g
′
VGC
i−1∑
s=1
(
Ls(is −wVGC0 ) · 10
−3
−
(
0.56 ·nswitch(s) + 0.23
∑
as
)
k2s (V
2
s−1 + g
′Lg (ig −wVGC0 ) · 10
−3 − 1000 ·hb(s)
)
. (18)
Let:
Cik
2
i = Di, V
2
i−1 −g
′Liwo · 10−3 = Ei, g′Li · 10−3 = Fi, Li · 10−3 = Gi. (19)
Then:
HYRP = A
( m∑
i−1
Giii + B +
m∑
i=1
Di(Ei + Fiii)
)
→ HYRP(min), (20)
or
HYRP = A
(
B +
(
(G1i1) + D1(E1 + F1i1)
)
+
(
(G2i2) + D2(E2 + F2i2)
)
+ · · · +
(
(Gmim) + Dm(Em + Fmim)
))
→ HYRP(min). (21)
As far as each profile element consists of the several technological elements, let us specify:
I1 = ii, where i = 1, . . . ,X1,
I2 = ii, where i = X + 1, . . . ,X2,
In = ii, where i = Xn−1 + 1, . . . ,Xn. (22)
where n – the number of profile elements of the descent part of a hump, and Xi, i = 1, 2, . . . ,n – the number of
the last technological element of the ith profile element
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And, eventually, the objective function takes the form:
HYRP = A
(
B +
X1∑
i=1
(
GiI1 + Di(Ei + FiI1)
)
+
X2∑
i=X1+1
(
GiI2 + Di(Ei + FiI2)
)
+ · · · +
Xn∑
i=Xn−1+1
(
GiIn + Di(Ei + FiIn)
))
→ HYRP(min). (23)
In order to realize the guided gravity technology for cuts, the objective function (23) should be minimized at
non-linear limitations-equalities:
D1 = fD1 (V0), E1 = fE1 (V0),
D2 = fD2 (V0, i1), E2 = fE2 (V0, i1),
D3 = fD3 (V0, i1, i2), E3 = fE3 (V0, i1, i2),
DXn = fDXn (V0, i1, i2, . . . , iXn−1 ), En = fEXn (V0, i1, i2, . . . , iXn−1 ),
, (24)
linear limitations-inequalities:
0 ≤ I1 ≤ 50,
25 ≤ I2 ≤ 50,
−50 ≤ In ≤ 50,
I1 − I2 ≤ 25,
HYRPb ≤ nrhone,
V YRPen ≤ V YRPen(max),
T0 ≤ T max0 ,
, (25)
and linear limitations-equalities:{
Lunfrun = Lc,
V YHRex = 1.4,
(26)
where I1,I2, . . . ,In – the profile element slopes of a hump, %0; HYRPB – the value of the heavy car braking under
favourable summer conditions at YRP, kJ/kN; nr – the number of retarders installed at YRPs; hone – the
capacity of a retarder installed at YRP, kJ/kN; V YRPen – the entry speed of a heavy car under favourable summer
conditions at YRP, m/s; V YRPen(max) – the maximum admissible entry speed of a heavy car under favourable
summer conditions at YRP, m/s; T0 – the time interval at dividing elements between cars rolling in turn, sec;
T max0 – the maximum accessible time at dividing elements between cars rolling in turn, sec; Lunfrun – the run of
a heavy car under unfavourable winter conditions along a difficult track in terms of resistance, m; Lc – the
calculation length of a difficult track in terms of resistance from the hump crest to the design point, m; V YRPex –
the exit speed of a heavy car from YRP, m/s. According to [8], the task cannot be reduced to an unconditional
extremum task.
Therefore, let us consider other ways to solve the task.
The method of Lagrange multipliers is technically difficult to implement as the number of limitations-equalities
is rather great. However, the method cannot be directly used if limitations are inequalities [8].
Thus, there is a needed to have a method that will allow finding the minimal value of HYRP with the minimal
variant search.
The standard method of Lagrange multipliers supplemented with terms, stem from the duality theory, got its
generalization to the task of non-linear programming of a general kind with limitations of the equality-inequality
type [8].
The needed optimality conditions of such tasks are called the Kuhn-Tucker conditions. In order to build the
Kuhn-Tucker tasks, a mathematical model of the non-linear programming task must have a strict limitations-
inequalities recording:
Z = f(X) → min,{
hk(X) = 0,k = 1,s,
gi(X) ≤ 0, i = 1,m.
(27)
The conditions of non-negative variables are included in the task recording as limitations-inequalities:
gi = −xj ≤ 0. (28)
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The Lagrange function of the task is built with s + m of undetermined coefficients:
L(X,V,U) = f(X) +
s∑
k=1
Vkhk(X) +
m∑
i=1
Uigi(X). (29)
The coefficients Vk(k = 1,s), Ui(i = 1,m) are called the Lagrange multipliers. They are unlimited by
sign dual variables corresponding to limitations-equalities (Vk = 0,k = 1,s) and non-negative dual variables
corresponding to limitations-inequalities (Ui ≥ 0, i = 1,m).
The following equation system with the n + s + m unknown variable is called the Kuhn-Tucker task for
minimization:
~5f(X) +
∑s
k=1 Vk
~5hk(X) +
∑m
i=1 Ui
~5gi(X) = 0,
hk(X) = 0,k = 1,s,
gi(X) ≤ 0, i = 1,m,
Uigi(X) = 0, i = 1,m,
Ui ≥ 0, i = 1,m.
(30)
The equations Uigi(X) = 0, i = 1,m are the complementary slackness conditions; they are an analogy of the
second duality theorem of linear programming tasks. If in point X, the limitation gi(X) is inactive (gi(X) > 0)
then Ui = 0, if gi(X) is active (gi(X) = 0), than Ui > 0.
The solution to the Kuhn-Tucker task should start with the analysis of this group of equations searching all
possible combinations of equality to zero Ui or gi(X)in turn. The optimal solution should be sought among points
meeting the Kuhn-Tucker conditions (30). Let us build the Kuhn-Tucker task at linear limitations-equalities for
output task (23): {
Lunfrun −Lc = 0,
V PHRex − 1.4 = 0,
(31)
non-linear limitations-equalities:
D1 −fD1 (V0) = 0,
E1 −fE1 (V0) = 0,
D2 −fD2 (V0,I1) = 0,
E2 −fE2 (V0,I1) = 0,
D3 −fD3 (V0,I1,I2) = 0,
E3 −fE3 (V0,I1,I2) = 0,
...
DZx −fDZx (V0,I1,I2, ...,IZx−1 ) = 0,
EZx −fEZx (V0,I1,I2, ...,IZx−1 ) = 0;
(32)
linear limitations-inequalities:
0 ≤ I1 ≤ 50 →
{
0 − I1 ≤ 0,
I1 − 50 ≤ 0,
25 ≤ I2 ≤ 50 →
{
25 − I2 ≤ 0,
I2 − 50 ≤ 0,
...
−50 ≤ In ≤ 50 →
{
−50 − In ≤ 0,
In − 50 ≤ 0;
(33)
and
I1 − I2 − 25 ≤ 0,
V YRPen −V YRPen(max) ≤ 0,
T0 −T max0 ≤ 0.
(34)
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S. Panchenko, O. Ohar, M. Kutsenko, J. Smachilo Acta Polytechnica
The Lagrange function has the form:
L(I,V,U) = A
(
B +
Z1∑
j=1
(
CjI1 + Dj (Ej + FjI1)
)
+
Z2∑
j=Z1+1
(
CjI2 + Dj (Ej + FjI2)
)
+ · · · +
Zx∑
j=Zx−1+1
(
CjIx + Dj (Ej + FjIx)
))
+ V1(Lunfrun −Lc) + V2(V
YRP
ex − 1.4) + V3(D1 −fD1 (V0))
+V4(E1−fE1 (V0))+V5(D2−fD2 (V0,I1))+V6(E2−fE2 (V0,I1))+V7(D3−fD3 (V0,I1,I2))+V8(E3−fE3 (V0,I1,I2))
+ · · · + V2Zx+2 (DZx −fDZx (V0,I1,I2, . . .IZx−1 )) + V2Zx+3 (EZx −fEZx (V0,I1,I2, . . .IZx−1 ))
+ U1(0 − I1) + U2(I1 − 50) + U3(25 − I2) + U4(I2 − 50) + · · · + U2x−1(−50 − Ix) + U2x(Ix − 50)
+ U2x+1(I1 − I2 − 25) + U2x+2(V YRPen −V
YRP
en(max)) + U2x+3(T0 −T
max
0 ), (35)
and conditions:
(1.)
A
( Z1∑
j=1
(Gj + DjFj )
)
−V6
∂fD2
∂I1
−V7
∂fE2
∂I1
−V8
∂fD3
∂I1
−V9
∂fE3
∂I1
−···−V2Zx+2
∂fDZx
∂I1
−V2Zx+3
∂fEZx
∂I1
−U1 + U2 + U2x+1 = 0,
...
A
( Zx∑
j=Zx−1+1
(Gj + DjFj )
)
−U2x−1 −U2x = 0; (36)
(2.) partial derivatives by dual variables ∂L
∂Vk
and ∂L
∂Ui
:
Lunfrun −Lc = 0,
V YRPex − 1.4 = 0,
D1 −FD1 (V0) = 0,E1 −fE1 (V0) = 0,
D2 −FD2 (V0,I1) = 0,E2 −fD2 (V0,I1) = 0,
D3 −FD3 (V0,I1,I2) = 0,E3 −fE3 (V0,I1,I2) = 0,
...
DZx −fDZx (V0,I1,I2, . . . ,IZx−1 ) = 0,
EZx −fEZx (V0,I1,I2, . . . ,IZx−1 ) = 0,
0 − I1 = 0; I1 − 50 = 0,
25 − I2 = 0,I2 − 50 = 0,
...
−50 − Ix = 0; Ix − 50 = 0; I1 − I2 − 25 = 0,
V YRPen −V
YRP
en(max) = 0; T0 −T
max
0 = 0; (37)
(3.) the complementary slackness condition (2nd duality theorem):
U1(0 − I1) = 0,
U2(I1 − 50) = 0,
U3(25 − I2) = 0,
U4(I2 − 50) = 0,
...
U2x−1(−50 − IX ) − 0 = 0,
U2x(Ix − 50) = 0,
U2x+1(I1 − I2 − 25) = 0,
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U2x+2(V YRPen −V
YRP
en(max)) = 0,
U2x+3(T0 −T max0 ) = 0; (38)
(4.)
Ui ≥ 0. (39)
According to [8], the solution should start with a linear search of all possible combinations of equality to
zero for multipliers of group III. At first, let us suppose that dual variables are equal to zero or not equal to
zero: Ui = 0 (∀i) then one out of Ui = 0 , the others are not equal to zero, etc.
3. Conclusions
The solution to the optimization task with limitations presented in the study will make it possible to minimize
the demand for the capacity of braking facilities and implement the technology of guided gravity regulation
of the cut speed. The authors believe that the implementation of the technology will encourage the decrease
of operational costs for a refund of a car and freight damages (due to better conditions improving the quality
of the cut rolling speed regulation), and for electricity needed for this regulation (a possible decrease of wind
consumption by car retarders). Besides, it will reduce additional charges for demurrages caused by waiting for
breaking-up at arrival yards (due to a possible reduction of hump intervals and lower volume of marshalling
works for pulling cars with a subsequent remedy of the consequences).
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377
Acta Polytechnica 58(6):370–377, 2018
1 Introduction
2 A method of complex calculation of rational structural parameters of railway humps
3 Conclusions
References