195-200 Algorithm for Optimizing Train Operation Modes Based on Timetable Parameters


195			Journal	of	Engineering,	Mechanics	and	Architecture															 	 	 					www.	grnjournal.us		
 

 

AMERICAN	Journal	of	Engineering,		
Mechanics	and	Architecture 

Volume	3,	Issue	5,	2025				ISSN	(E):	2993-2637	
 

  
Algorithm for Optimizing Train Operation Modes Based on 

Timetable Parameters 
 

Lesov A., Adilov N., Miryakubov A., Usmonov K. 

Tashkent State Transport University (Tashkent, Uzbekistan) 

 
Abstract: The article focuses on developing an algorithm to optimize the switching points of 
train movement modes, taking into account scheduled time, movement resistance, gradients, and 
rolling stock characteristics. The primary emphasis is on modeling the movement process, 
calculating speed and time increments, and considering the impact of gradients on train 
movement. The developed mathematical model and algorithm aim to minimize deviations from 
the planned arrival time and optimize train movement modes. The methodology described in the 
article can be applied to enhance railway transportation efficiency, improve travel time 
calculation accuracy, and reduce energy consumption during train operation.  
Keywords: rolling stock, movement optimization, scheduled time, traction modes, switching 
algorithm, energy efficiency, adaptive algorithm. 
 
 
Introduction. Optimizing train movement is a key aspect of increasing railway transport 
efficiency, especially in conditions of tight schedules and the need for precise adherence to 
arrival times. One of the crucial stages in this process is accurately identifying the points for 
changing movement modes such as traction, coasting, and braking, as these directly impact 
travel time and energy efficiency [6,9,10]. 
The aim of this scientific work is to develop an algorithm that enables optimization of the change 
points based on the graphical time of train modes, taking into account possible track parameters 
and deviations. The study examines methods for calculating speed, time, and distance increases 
at each stage, as well as the operational characteristics of the regenerative braking and coasting 
systems. 
The article is based on theoretical methods of modeling train movement, considering the actual 
track parameters and acting forces. This allows for proposing a more precise and effective 
approach to calculating traffic schedules and minimizing deviations from planned arrival times. 
The applied approach helps to increase the accuracy of train movement time prediction and 
reduce schedule deviations. 
Model description. The mathematical model of train movement is based on a system of equations 
describing the dynamics of electric rolling stock. The fundamental equation of dynamics 
expresses the change in velocity over time by dividing the difference between the tractive force 
and the resistance to motion by the mass of the train composition. This equation takes into 
account the interaction of various factors, such as resistance to motion and traction 
characteristics, which allows for a more accurate modeling of the movement process [1,2,8]. 

The fundamental equation of dynamics is written as follows: 



196			Journal	of	Engineering,	Mechanics	and	Architecture															 	 	 					www.	grnjournal.us		
 

 

𝑑𝑉
𝑑𝑡 =

𝐹!	(𝑉) −𝑊(𝑉, 𝑆)
(𝑃 + 𝑄) , 

here, 𝐹!	(𝑉)	 is the traction characteristic depending on speed, and W(V,S) is the total resistance, 
which includes all types of resistances dependent on road conditions and external factors, such as 
air resistance, rolling resistance, and other elements. The mass of the train (P+Q) is taken into 
account to ensure the accuracy of calculations, as it directly affects the acceleration and braking 
characteristics. 
To account for the specific characteristics of electric rolling stock, the concept of specific 
accelerating force is introduced [8]. It is calculated using the following formula, which takes into 
account the maximum traction force, adhesion coefficients, and gravitational factors: 

𝑓#	$%& =
1000 ∙ 𝐹!	$%&
(𝑃 + 𝑄) ∙ 𝑔 − 𝑤$ −𝑤% ,	 

here, 𝑤$ represents the main resistance dependent on speed and other factors, while 𝑤% 
represents the additional resistance that accounts for the road's specific characteristics such as 
slopes and unevenness. 
According to this model, velocity changes are calculated for each step of train movement, 
depending on the current and set speeds. These changes are influenced by both the current and 
set speeds. In this process, the following conditions are taken into account: 

𝛥𝑉 = 7

𝑖𝑓	𝑉' < 𝑉((					Þ					𝛥𝑉 > 0
𝑖𝑓	𝑉' > 𝑉((					Þ				𝛥𝑉 < 0

𝑖𝑓	𝑉' ≤ 𝑉) 		Þ		𝛥𝑉 = 10	𝑘𝑚/𝑠
𝑖𝑓	𝑉' > 𝑉) 			Þ		𝛥𝑉 = 5	𝑘𝑚/𝑠

 

where 𝑉' is the current train speed, 𝑉** is the steady-state speed, 𝑉) 	is the limit value for changing 
the speed by 5 km/h, and 𝛥𝑉 is the speed increment for the current step. 
It is noteworthy that for more accurate calculations at closer values, the speed change increment 
can be reduced to 1 km/h [4]. 
Slope and its impact on train movement 
To calculate more accurately the time required to traverse the sloped sections of the track, a 
correction coefficient 𝑘'+, is introduced. Slope affects resistance to movement and traction, 
therefore it is necessary to take this into account when planning the movement regime. The train 
expends more effort to overcome resistance at points of ascent, while at points of descent, 
conversely, the resistance decreases, but problems with speed control may arise. 
The formula for calculating the time required to traverse the sloped section of the track is as 
follows: 

𝑡- = 𝑘'+, ∙
𝑆.
𝑆 ∙ 𝑡/+,$ , 

where 𝑡- is the boundary time on the sloping section of the road, 𝑘'+, is the slope correction 
coefficient, 𝑆. 	is the length of the road section with a slope, S is the total length of the route, and 
𝑡/+,$	is the designated time to cover a section of the road without accounting for the slope. 
This approach helps to calculate more accurately the travel time on inclined sections of the road, 
as well as to adjust the points of mode changes (traction, idling, braking) based on the variations 
in resistance to movement on such sections. 
Basic steps of the algorithm 

The algorithm begins by calculating the maximum specific accelerating force 𝑓!	$%&, taking into 
account the forces of resistance to motion: 



197			Journal	of	Engineering,	Mechanics	and	Architecture															 	 	 					www.	grnjournal.us		
 

 

𝑓#	$%& = 𝑓!	$%& −𝑤$ −𝑤% . 
After that, the increase in time and distance is calculated. The time for each step is determined 
using the following formula: 

∆𝑡. = C
∆𝑉.

2 ∙ 𝑓#	$%&	.
C. 

After calculating the time for each step, the total time ti is determined as follows: 

𝑡. = 𝑡0 + ∆𝑡. , 

where 𝑡0 is the time from the previous step. 

Path gain in current step: 

∆𝑆. = 16,7 ∙ G𝑉' +
∆𝑉.
2 H ∙ ∆𝑡.	, 

𝑆. = 𝑆0 + ∆𝑆. . 

Given the length of the section 𝑆. 	and the travel time 𝑡#, along the section, we obtain the 
following relationship: 

𝑡#,	.(𝑆) = 0,001 ⋅ 𝑆. 	,
 

𝛥𝑡#,	. = 𝑡1,	. − 𝑡-	., 

At this stage, the condition of equality between the travel time 𝑡#, in the section and the time 𝑡. 	at 
the current step is checked. Due to the discrete nature of calculations, it may be impossible to 
achieve exact equality. Therefore, an allowable 𝜖 is applied, and the condition is verified as 
follows: 

𝑡. = 𝑡#, ± 𝜖. 

If the difference between 𝑡. and 𝑡#, is within the \epsilon boundary, it is possible to proceed to 
the next step. However, if the travel time on the section 𝑡#, < 𝑡. 	is smaller than the time at the 
current step by more than 𝜖, it is necessary to adjust parameters such as speed or tractive force to 
eliminate the deviation and bring the calculations to optimal accuracy. 
To correctly account for time and distance, the condition ΔS must remain unchanged. To ensure 
that the distance traveled corresponds to the given speed and time, the new increase in speed, ΔV, 
is recalculated as follows. 

∆𝑉. =
∆𝑆. − 16,7 ∙ 𝑉'	. ∙ ∆𝑡#,	.

8,35 ∙ ∆𝑡#,	.
. 

The specific accelerating traction force 𝑓!	 is recalculated, taking into account all resistances and 
motion characteristics that ensure the necessary dynamics for train movement. 

𝑓!	. =
∆𝑆. − 16,7 ∙ 𝑉'	. ∙ ∆𝑡#,	.

16,7 ∙ ∆𝑡#,	.2 . 

A block diagram is presented in Figure 1 to visualize the computational stages and the algorithm 
for switching modes. This diagram illustrates the main steps, including the calculation of the 
specific accelerating force, as well as the increase in velocity and time. 



198			Journal	of	Engineering,	Mechanics	and	Architecture															 	 	 					www.	grnjournal.us		
 

 

 
Figure 1. Adaptive algorithm for switching train modes based on a time schedule 

The main stages of the flowchart include: 

The initial calculation is to determine the maximum specific accelerating force 𝑓#	$%&	. 
Calculation of velocity and time increases - based on current and steady-state velocity and 
resistance to motion. 
Switching between modes - the algorithm automatically switches traffic modes depending on the 
value of the specific power and speed of the train. 
Limit time - calculation of the limit time and adjustment of the mode switching points, taking 
into account the slopes on the road. 
Train movement modes 

Traction mode 
During the traction mode phase, the train's speed increases, and the time increment is calculated 
taking into account the traction force and resistance to movement. 
Coasting mode 

When the accelerating force becomes zero (𝑓#	 ≤ 0), the train switches to coasting mode. In this 
state, the force acting on the train is defined as the sum of the resistances to motion: 

𝑓#	 = 𝑤$ +𝑤% . 
In this mode, the train moves at a constant speed, and the time at this stage is calculated using the 
following formula: 

∆𝑡 =
∆𝑆

16,7 ∙ 𝑉**
. 



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Recuperative braking mode 

When using recuperative braking, the tractive force 𝑓,	!		is determined as follows: 

𝑓,	!	 = 0,8 ∙ 𝑓!(𝑉), 

where 𝑓!(𝑉) is the traction characteristic of the electric locomotive [7]. In calculations, the 
motion resistances 𝑤$	 and 𝑤%	are also taken into account. 
Mechanical braking 
When the train speed reaches a certain limit, mechanical braking is applied. Calculation of time 
and path increments is carried out in the same way as in previous modes [3,5]. 
Conclusion. In this article, an algorithm for optimizing the switching points of train modes based 
on the time of the graph is proposed. Various factors influencing traffic dynamics, such as road 
slopes and resistance, were taken into account. The described calculation methods allow for 
more accurate modeling of the train movement process and reduce deviations from the 
established arrival time. For accurate adjustment of time indicators in sections where the road 
profile changes, the use of the slope correction coefficient is of great importance. The 
mathematical model, based on the system of equations of motion dynamics, allows for the 
calculation of specific accelerating forces, changes in speed and time. This, in turn, serves to 
increase the accuracy of train traffic management. 
The proposed algorithm can be effectively used to optimize traffic schedules, increase energy 
efficiency, and reduce travel time. As a result, railway transport operates more stably. 
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