- Email: [email protected]
Improved train simulation with speed control algorithm
Available online at www.sciencedirect.com Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect Transportation Research Procedia 40 (2019) 1563–1570 Transportation Research Procedia 00 (2019) 000–000 Transportation Research Procedia 00 (2019) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
13th 13th International International Scientific Scientific Conference Conference on on Sustainable, Sustainable, Modern Modern and and Safe Safe Transport Transport (TRANSCOM 2019), High Tatras, Novy Smokovec Grand Hotel Bellevue, (TRANSCOM 2019), High Tatras, Novy Smokovec - Grand Hotel Bellevue, Slovak Slovak Republic, Republic, May May 29-31, 29-31, 2019 2019
Improved Improved train train simulation simulation with with speed speed control control algorithm algorithm
a,∗ a a a a Pavel Pavel Sovicka Sovickaa,∗,, Matej Mateja Pacha Pachaa ,, Pavol Pavol Rafajdus Rafajdusa ,, Patrik Patrik Varecha Varechaa ,, Simon Simon Zossak Zossaka a University University
of Zilina, Univerzitna 1, Zilina 010 26, Slovakia of Zilina, Univerzitna 1, Zilina 010 26, Slovakia
Abstract Abstract This paper describes an advanced train simulation based on a classical mathematical model, which was modified to increase This paper describes an advanced train simulation based on a classical mathematical model, which was modified to increase simulation accuracy, especially for simulating run of long trains. The simulation also includes a train speed controller and a power simulation accuracy, especially for simulating run of long trains. The simulation also includes a train speed controller and a power consumption calculation. The speed controller can also calculate an optimal speed for a desired run time between stops. These consumption calculation. The speed controller can also calculate an optimal speed for a desired run time between stops. These modifications enable accurate train simulation, which can be used during locomotive development or during operational changes modifications enable accurate train simulation, which can be used during locomotive development or during operational changes for given trains on selected routes. To improve usability a simple graphical user interface is used. This interface allows simple for given trains on selected routes. To improve usability a simple graphical user interface is used. This interface allows simple addition of input data, for example new locomotive characteristics or another track. addition of input data, for example new locomotive characteristics or another track. c 2019 2019 The Authors. Authors. Published by by Elsevier B.V. B.V. © c 2019 The The Authors. Published Published by Elsevier Elsevier B.V. Peer-review under responsibility of of thethe scientific committee of the International Scientific Conference on Sustainable, Modern Peer-review under responsibility scientific committee of13th the 13th International Scientific Conference on Sustainable, Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe and Transport (TRANSCOM 2019). Modern Safe Transport (TRANSCOM 2019). and Safe Transport (TRANSCOM 2019). Keywords: train simulation; train length; speed control Keywords: train simulation; train length; speed control
1. Introduction 1. Introduction Accurate train simulations are necessary for development purposes, but also as a tool to improve train schedulAccurate train simulations are necessary for development purposes, but also as a tool to improve train scheduling and optimize fuel consumption. However a train is a very complex mechanical system. Especially if wheel/rail ing and optimize fuel consumption. However a train is a very complex mechanical system. Especially if wheel/rail contact, suspension, mechanical stresses, track layout (curves, slope, tunnels) and other factors would be accurately contact, suspension, mechanical stresses, track layout (curves, slope, tunnels) and other factors would be accurately represented. Furthermore several parameters such as load, weather and wear and tear change over time and are difrepresented. Furthermore several parameters such as load, weather and wear and tear change over time and are difficult to predict. Modeling the entire system would have significant computational requirements and would still be ficult to predict. Modeling the entire system would have significant computational requirements and would still be burdened by errors in input parameters. Results would not achieve the accuracy proportional to complexity of the apburdened by errors in input parameters. Results would not achieve the accuracy proportional to complexity of the approach. Also, the need for train modeling was recognized many decades ago, when powerful digital computers were proach. Also, the need for train modeling was recognized many decades ago, when powerful digital computers were not available Danzer (2008); Drabek (2010). not available Danzer (2008); Drabek (2010). Therefore a simplified approach was developed. This method is based heavily on measurements and empirical Therefore a simplified approach was developed. This method is based heavily on measurements and empirical equations. The entire train mass is concentrated into a center of gravity moving along one axis. Input parameters, such equations. The entire train mass is concentrated into a center of gravity moving along one axis. Input parameters, such as vehicle resistance or track resistance force, are expressed by simple equations with coefficients based on the above as vehicle resistance or track resistance force, are expressed by simple equations with coefficients based on the above mentioned measurements. Traction force is calculated from locomotive parameters. The mentioned complexity and mentioned measurements. Traction force is calculated from locomotive parameters. The mentioned complexity and parameter uncertainty is accounted for in the measured input data. For example there are known adhesion coefficients parameter uncertainty is accounted for in the measured input data. For example there are known adhesion coefficients ∗ ∗
Pavel Sovicka Tel.: +421 41 513 2270 Pavel Sovicka Tel.: +421 41 513 2270 E-mail address: [email protected] E-mail address: [email protected]
c 2019 The Authors. Published by Elsevier B.V. 2352-1465 c 2019 The Authors. Published by Elsevier B.V. 2352-1465 2352-1465 2019responsibility The Authors.ofPublished by Elsevier Peer-reviewunder the scientific committeeB.V. of the 13th International Scientific Conference on Sustainable, Modern and Safe Peer-review under of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Peer-review underresponsibility responsibility Transport (TRANSCOM 2019). of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Transport (TRANSCOM 2019). 2019). Safe Transport (TRANSCOM 10.1016/j.trpro.2019.07.216
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for different weather (sunny, rain, snow, etc.) and therefore boundary conditions of operation are known. Jansa (1980); Pilmann (2000); Danzer (2008). Several computing programs were developed at the predecessor of University of Zilina since 1960s. These were implemented on MEDA and Amsler analog computers. Later they were replaced by digital computers running ”Metro” software. Czech Railway Infrastructure Administration (SZDC) is using its own information software called ”SENA”, which was originally developed for calculating train timetables. However the option to calculate energy consumption based on track profile and train characteristics was later added to enable prediction of electricity or fuel consumption Drabek (2010); Siman (2006); Hlava and Hruby (2011). This paper proposes improvement in track resistance force calculation. First, the mass point is situated in front of the train, not in the center of mass. Secondly, the algorithm calculates specific track resistance incrementally as the train moves through different sections of the track profile based on the proportion of train already present in this new section. Uniform distribution of mass through entire length of the train is assumed. This approach allows easy modification of existing models. By considering train length, the simulation accuracy is improved for long trains or when short significant track sections, such as small curves, are present. A speed control algorithm, which can drive the train correctly at maximum speed, is also presented. The most important part of this algorithm is calculation of necessary braking distance based on jerk and acceleration limits. This enables the train to correctly respect speed profile of selected track. The entire simulation is implemented in Matlab and Simulink enviroments with a Graphical User Interface (GUI) created in the GUIDE development enviroment. 2. Mathematical model of train and track parameters Most of characteristics and equations used in mathematical modeling of trains do not use basic units for the respective variables. Instead units such as km/h, kN and tonnes are used. This is due to practical reasons, as these units are much more common on railways. For consistency with the theory and with cited sources, this work will use the same units. Therefore great care must be taken to correctly input values into the presented mathematical equations according to units shown in the respective description. The basic motion equation of train movement is (1). Traveled distance and speed can be calculated by modifying this equation to (2) to gain train acceleration and by using integration according to (3) and (4). Fa = a ·
n i=1
(mi · ξi ) = Ft − Fv − Fe − Fc − Fb
(1)
where Fa [N] is acceleration force, a [m/s] is acceleration, n is number of vehicles, mi kg is mass of i-th vehicle, ξi [−] is i-th coefficient of rotational parts, Ft [N] is traction force, Fv [N] is vehicle resistance, Fe [N] is track elevation resistance, Fc [N] is track curve resistance and Fb [N] is an artificial resistance created by braking. Ft − Fv − Fe − Fc − Fb n i=1 (mi · ξi ) t a (t) dt v=
a=
0
where v [m/s] is train speed and t [s] is time since the beginning of the train run. t l= v (t) dt 0
(2) (3)
(4)
where l [m] is traveled distance. Maximum traction force is limited by adhesion. A formula known as ”Curtius-Kniffler” is used in this work (5), although a different one can also be used, e.g. adhesion limit defined by TSI EC (2014). Actual available traction force for the given speed is dependent on the characteristics of the studied locomotive. This adhesion coefficient is then compensated according to (6) to represent adhesion changes in the demanded weather. µa =
7500 + 161 V + 44
(5)
Pavel Sovicka et al. / Transportation Research Procedia 40 (2019) 1563–1570 Sovicka et al. / Transportation Research Procedia 00 (2019) 000–000
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where µa [N/kN] is the adhesion coefficient and V [km/h] is train speed. µcond (6) µacomp = µa · 330 where µacomp [N/kN] is the adhesion coefficient compensated for chosen conditions and µcond [N/kN] is the adhesion coefficient selected to represent demanded weather conditions. For normal (average) weather conditions, µcond = 330◦/◦◦ , for poor conditions, µcond is lower (e.g. 150 ◦/◦◦ ). This value is related to the zero-speed adhesion conditions. This adhesion coefficients is then used to calculate the maximum traction force limit according to (7). Ftlim = µacomp · G = µacomp · M · g
(7)
where F tlim [N] is a maximum traction force limit, G [kN] is the gravitational force, M [t] is the train mass the and g m/s2 is a gravitational constant. As was shown in (1), resistance force acting on the train can be divided into three categories: • Vehicle resistance Fv - Resistance force created by bearing friction, wheel rolling resistance and aerodynamic resistance. • Track resistances Fe and Fc - Additional resistances created by track elements, mostly slopes and curves. This resistance can sometimes be combined into ”reduced slope”, which is a sum of all specific resistance components represented as a slope resistance to ease further calculations by providing only one resistance term in the equations. • Braking Fb can be represented as negative traction force, but also separately as an artificially created resistance to lower train speed. Fv = A + B · V + C · V 2
(8)
where A [N] is bearing resistance coefficient, B [Ns/m] is wheel rolling resistance coefficient and C Ns2 /m2 is aerodynamic resistance coefficient. While absolute values of resistances are necessary for final computation, specific resistances allow dynamic implementation for different train consists. The ratio between specific resistance and absolute resistance is determined by the gravitation force as shown in (9). F (9) p= G where p [N/kN] is the specific resistance. Therefore specific vehicle resistance can be calculated by combining (8) and (10). Usually an opposite calculation is done as the specific resistance for different train types has been measured and is known. This approach allows to accurately calculate resistance force acting on the train using known specific resistance characteristics and the train mass. pv = a + b · V + c · V 2
(10)
where a, b and c are specific vehicle resistance coefficients corresponding to (8). By traveling on a sloped track, part of the gravitational force is acting in the axis of movement. Adhesion railway permits only small slopes (up to several units %). Therefore a simplification shown in (11) can be used. The resulting specific slope resistance is equal to the slope value in represented in per-mille. g · m · sinα ht Fe = ≈ tgα = =s (11) pe = g·m g·m lt where pe [N/kN] is the specific elevation resistance, Fe [N] is elevation resistance, ht [m] is track height difference on a given length lt [m] and s [◦/◦◦ ] is the track slope. Specific curve resistance can also be calculated by several different empiric equations. Equation (12) applies for 1435 mm gauge main lines. 650 (12) pc = r − 55 where pc [N/kN] is the specific curve resistance and r [m] is the curve radius.
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(a)
(b)
Fig. 1: (a) Gradual slope change algorithm flowchart. Condition ”1” is ”Has the train reached a new elevation section?”; (b) Difference between classical theory and proposed incremental change algorithm shown on a 264 m long freight train with a total mass of 1508 t.
2.1. Incremental specific resistance calculation algorithm This algorithm implements train length into the simulation to improve accuracy without significant increase in computational requirements. The algorithm is used both for slope and elevation profiles. Fig. 1a shows a flowchart of this algorithm. First the current position of train front is loaded. This position is then compared to the track slope profile. If a new slope section has been reached, parameters scurr and s prev are updated to new values. A slope increment is calculated according to (13) and added to the last slope value (14). Finally the limit of the value is checked so that the new value does not exceed the slope limits. ds =
scurr − s prev · lcurr − l prev ltrain
(13)
where ds [N/kN] is the slope increment, scurr [◦/◦◦ ] is current slope loaded from the track profile, s prev [◦/◦◦ ] is previous slope loaded from the track profile, ltrain[m] is length of the train, lcurr [m] is current position of train front and l prev [m] is a position from last calculation step. sk = sk−1 + ds
(14)
where sk [◦/◦◦ ] is the slope value used for resistance force calculation and sk−1 [◦/◦◦ ] is the value from previous calculation step. Difference between classical ”step” and the proposed incremental approach for ”smooth” transition between track section is shown in Fig. 1b. Here a part of a simulation run of a 264 m long train with a mass of 1508 t from Pardubice to Zdarec is shown. 3. Control algorithm Acceleration to a demanded speed is simple as the acceleration must start after the train consist is entirely within a higher speed limit. However braking must start in advance so that the train is moving the correct speed once a new speed limit is reached. The implemented control algorithm calculates the distance required to decrease actual train speed to the new limit. The new speed is then demanded in advance when the train reaches this distance to the new limit. To simplify the analysis the speed profile during braking was divided into 3 parts according to Fig. 2:
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Fig. 2: An example of train deceleration from a given speed to a new demanded speed with constant jerk. Note that for better clarity values are not in correct respective scale.
• During interval (1) the train begins to increase its negative acceleration to a limit value with a constant jerk. Time required to reach this acceleration limit can be calculated by (15). Speed and covered distance from start of braking to end of this first interval is shown in (16) and (17). t1 =
alim blim
(15)
where t1 [s] is time required to reach acceleration limit, alim m/s2 is the acceleration limit and blim m/s3 is a maximum allowed jerk. v1 = v0 −
1 · blim · t1 2 2
(16)
where v1 [m/s] is speed at the end of this interval and v0 [m/s] is an initial speed. l1 = v0 · t1 −
1 · blim · t1 3 6
(17)
where l1 [m] is distance covered during first interval. • Time interval (2) is the main braking interval, where the train has constant negative acceleration. Time length of this interval is calculated by (18). To find the necessary speed difference for this calculation, speed differences from intervals 1 and 3 must be known. Speed and covered distance from the end of first interval to end of this interval is shown in (19) and (20). t2 =
∆v2 ∆v − ∆v1 − ∆v3 vcurrent − vnext − = = alim alim
1 2
· blim · t1 2 − alim
1 2
· blim · t3 2
(18)
where t2 [s] is time necessary to sufficiently lower train speed, ∆v2 [m/s] is a speed difference during this interval , ∆v [m/s] is a total speed between current speed and next track speed limit, ∆v1 [m/s] is a speed difference during first interval, ∆v3 [m/s] is a speed difference during third interval, vcurrent [m/s] is the current speed of the train at beginning of braking, vnext [m/s] is the next track speed limit and t3 [s] is time required to reach zero acceleration during the third interval. v2 = v1 − alim · t2
(19)
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where v2 [m/s] is speed at the end of this interval. l2 = v1 · t2 −
1 · alim · t2 2 2
(20)
where l2 [m] is distance covered during second interval. • During time interval (3) a constant positive jerk si applied to reach zero acceleration and new steady state speed, which is equal to the new speed limit. Third time interval (21) is equal to the first time interval if the absolute value of jerk is also equal. t3 =
alim = t1 blim
(21) 1 · blim · t3 2 2
(22)
1 1 · alim · t3 2 + · blim · t3 3 2 6
(23)
v3 = v2 − alim · t3 + where v3 [m/s] is speed. l3 = v2 · t3 −
where l3 [m] is distance covered during third interval. The total braking distance is a sum of distance covered during these three intervals (24). Braking point can than be found by subtracting this distance and the already covered distance from the point, where the new speed limit applies. lbrake = l1 + l2 + l3
(24)
where lbrake is total distance necessary to decrease speed from initial value to the next limit. lvdem change = lnextlimit − lcovered − lbrake
(25)
where lvdem change [m] is distance at which the braking must start to safely decrease train speed, lnextlimit [m] is distance where the next speed limit applies and lcovered [m] is distance already covered by the train. Note that this speed control approach is very basic. Therefore, while sufficient for this simulation, it is not suitable for implementation in actual control algorithms. 4. Implementation Mathematical model implemented was discussed in chapter 2 and is controlled according to algorithm described in chapter 3. Information about vehicle resistances can be found in Jansa (1980); Drabek (2010). Additional vehicle information, especially traction characteristic is provided from the respective manufacturer documentation. Finally track tables created by the respective railway operator were used for track profiles. Most of the mentioned algorithms were implemented in Matlab, but are called through a block diagram implemented in Simulink. This approach improves clarity and enables easy modifications to the simulation. For ease of use, the entire control is done through a Graphical User Interface (GUI), which also displays results of the simulation. As can be seen in Fig. 3 it is be divided into three parts: • Parameter input is situated in the top left corner. Here the user can select locomotive and train type, track layout and other parameter. These values are loaded from a database provided with the simulation. The ”Load values” button loads all data from the database files into base workspace for simulation while the ”Run simulation” activates a Simulink model.
Pavel Sovicka et al. / Transportation Research Procedia 40 (2019) 1563–1570 Sovicka et al. / Transportation Research Procedia 00 (2019) 000–000
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Fig. 3: Simulation GUI interface after a simulation was completed.
• Graphical output consists from three figures: ”Speed vs. Distance” shows how the train performed on selected route, ”Force vs. Distance” shows traction force and resistance sum and ”Force vs. Speed” shows utilization of the locomotive(s) traction characteristic. • Text output located in the bottom left corner and shows basic information to confirm configuration and also energy values and average speed. Several values are updated every 10 s of simulation time providing basic check of simulation status.
5. Results analysis An example of a freight train run between Pardubice and Zdarec pulled by a pair of class 742 locomotives can be seen in Fig. 3. This track was selected to show impact of the proposed algorithm on simulations of regular trains. Table 1: Total traction effort (A) and average speed (V) comparison of step and incremental resistance algorithms for different freight trains pulled by two class 742 locomotives between Pardubice and Zdarec. Note that weight limit for these locomotives on this track is 1028 t, including the locomotives. Train weight [t]
Train length [m]
A step [kWh]
Aincr [kWh]
∆A [%]
V step [km/h]
Vincr [km/h]
∆V [%]
528 928 1088 1328
94.7 162.2 189.2 229.7
601.56 979.77 1120.5 1319.95
600.77 976.89 1116.78 1312.87
-0.13 -0.29 -0.33 -0.54
52.79 44.45 40.91 36.28
52.81 44.50 40.95 36.38
0.04 0.11 0.12 0.28
As can be seen in Table 1, impact of the proposed method for the given example is small. However it can clearly be seen that for longer trains, the impact increases. This difference will be larger in situations, where the train length is comparable to track features. For example for a train of length 200 m, any curve or slope shorter than 200 m will be calculated differently through the entire track section as the resistance will gradually increase until the train is fully in the section and then gradually decrease. This can clearly be seen in the next example of shunting operation. Fig. 4 shows a speed profile of a shunting train run over a classification yard, where the shunting locomotive must move the freight carriages from the station uphill on a track, then stop and reverse the carriages back over the hill.
1570 8
Pavel Sovicka et al. / Transportation Research Procedia 40 (2019) 1563–1570 Sovicka et al. / Transportation Research Procedia 00 (2019) 000–000
Fig. 4: An example of shunting over with 15 loaded freight cars over a classification yard.
This arrangement can be seen for example at Zilina station. The run consists of a 1 km long section without any slope, then 100m of 60 ◦/◦◦ slope followed by leveled track until the end of the track is reached. It can clearly be seen, that with shorter distance available or higher load, the step change simulation would reach zero speed and stop. This would provide a false result showing the locomotive is not suitable for this operation. Traction effort for the step algorithm was calculated to be 59.81 kWh, while for the incremental algorithm it is only 48.76 kWh. A difference of more than 18 %. Using the proposed algorithm shows the locomotive is capable of carrying out this shunting operation with sufficient safety margins. Also, the lower traction effort means that refueling intervals can be predicted more accurately. 6. Conclusion Mathematical model for simulating train operation was thoroughly explained including a new approach for track resistance calculation. This approach implements train length into the calculations with negligible additional performance requirements. The impact of this change has been analyzed. It was shown, that the highest improvement was during special cases, such as shunting over a hill at a classification yard. Differences during regular track operation were small. However the actual impact is dependent on the given train and track selection. Thanks to low computational requirements, the proposed approach can be implemented regardless if its benefits will have a significant impact on the results. Also a basic speed control algorithm was introduced. This algorithm was used during all simulations and was verified to correctly guide the train so that the track speed limit is not exceeded. Acknowledgements This work is supported by the Slovak Scientific Grant Agency VEGA No. 1/0774/18 and by project ITMS: 26220120046. References Danzer, J., 2008. Elektricka Trakce 2. vydani. Prehled problematiky (etr120), vlastni spotreba a chlazeni (etr600), adheze (etr700), rizeni vozidel (etr900) ed., ZCU FEL. Drabek, J., 2010. Energeticke vlastnosti zeleznicnich dopravnich systemu. Katedra vykonovch elektrotechnickch systemov ZU . EC, 2014. Commission regulation (eu) no 1302/2014 concerning a technical specification for interoperability relating to the rolling stock locomotives and passenger rolling stock subsystem of the rail system in the european union, in: Official Journal of the European Union. Hlava, K., Hruby, J., 2011. Elektricka trakcni energie, in: Vedeckotechnicky sbornik CD, c. 31/2011. Jansa, F., 1980. Dynamika a energetika elektricke trakce. NADAS. Pilmann, L., 2000. Dejiny, soucasnost a budoucnost zeleznicniho vyzkumu, in: Vedeckotechnicky sbornik CD, c. 09/2000. Siman, P., 2006. Moznosti uspory trakcni elektricke energie a motorove nafty zavisle na zeleznicni infrastrukture, in: Vedeckotechnicky sbornik CD, c. 22/2006.
ScienceDirect Transportation Research Procedia 40 (2019) 1563–1570 Transportation Research Procedia 00 (2019) 000–000 Transportation Research Procedia 00 (2019) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
13th 13th International International Scientific Scientific Conference Conference on on Sustainable, Sustainable, Modern Modern and and Safe Safe Transport Transport (TRANSCOM 2019), High Tatras, Novy Smokovec Grand Hotel Bellevue, (TRANSCOM 2019), High Tatras, Novy Smokovec - Grand Hotel Bellevue, Slovak Slovak Republic, Republic, May May 29-31, 29-31, 2019 2019
Improved Improved train train simulation simulation with with speed speed control control algorithm algorithm
a,∗ a a a a Pavel Pavel Sovicka Sovickaa,∗,, Matej Mateja Pacha Pachaa ,, Pavol Pavol Rafajdus Rafajdusa ,, Patrik Patrik Varecha Varechaa ,, Simon Simon Zossak Zossaka a University University
of Zilina, Univerzitna 1, Zilina 010 26, Slovakia of Zilina, Univerzitna 1, Zilina 010 26, Slovakia
Abstract Abstract This paper describes an advanced train simulation based on a classical mathematical model, which was modified to increase This paper describes an advanced train simulation based on a classical mathematical model, which was modified to increase simulation accuracy, especially for simulating run of long trains. The simulation also includes a train speed controller and a power simulation accuracy, especially for simulating run of long trains. The simulation also includes a train speed controller and a power consumption calculation. The speed controller can also calculate an optimal speed for a desired run time between stops. These consumption calculation. The speed controller can also calculate an optimal speed for a desired run time between stops. These modifications enable accurate train simulation, which can be used during locomotive development or during operational changes modifications enable accurate train simulation, which can be used during locomotive development or during operational changes for given trains on selected routes. To improve usability a simple graphical user interface is used. This interface allows simple for given trains on selected routes. To improve usability a simple graphical user interface is used. This interface allows simple addition of input data, for example new locomotive characteristics or another track. addition of input data, for example new locomotive characteristics or another track. c 2019 2019 The Authors. Authors. Published by by Elsevier B.V. B.V. © c 2019 The The Authors. Published Published by Elsevier Elsevier B.V. Peer-review under responsibility of of thethe scientific committee of the International Scientific Conference on Sustainable, Modern Peer-review under responsibility scientific committee of13th the 13th International Scientific Conference on Sustainable, Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe and Transport (TRANSCOM 2019). Modern Safe Transport (TRANSCOM 2019). and Safe Transport (TRANSCOM 2019). Keywords: train simulation; train length; speed control Keywords: train simulation; train length; speed control
1. Introduction 1. Introduction Accurate train simulations are necessary for development purposes, but also as a tool to improve train schedulAccurate train simulations are necessary for development purposes, but also as a tool to improve train scheduling and optimize fuel consumption. However a train is a very complex mechanical system. Especially if wheel/rail ing and optimize fuel consumption. However a train is a very complex mechanical system. Especially if wheel/rail contact, suspension, mechanical stresses, track layout (curves, slope, tunnels) and other factors would be accurately contact, suspension, mechanical stresses, track layout (curves, slope, tunnels) and other factors would be accurately represented. Furthermore several parameters such as load, weather and wear and tear change over time and are difrepresented. Furthermore several parameters such as load, weather and wear and tear change over time and are difficult to predict. Modeling the entire system would have significant computational requirements and would still be ficult to predict. Modeling the entire system would have significant computational requirements and would still be burdened by errors in input parameters. Results would not achieve the accuracy proportional to complexity of the apburdened by errors in input parameters. Results would not achieve the accuracy proportional to complexity of the approach. Also, the need for train modeling was recognized many decades ago, when powerful digital computers were proach. Also, the need for train modeling was recognized many decades ago, when powerful digital computers were not available Danzer (2008); Drabek (2010). not available Danzer (2008); Drabek (2010). Therefore a simplified approach was developed. This method is based heavily on measurements and empirical Therefore a simplified approach was developed. This method is based heavily on measurements and empirical equations. The entire train mass is concentrated into a center of gravity moving along one axis. Input parameters, such equations. The entire train mass is concentrated into a center of gravity moving along one axis. Input parameters, such as vehicle resistance or track resistance force, are expressed by simple equations with coefficients based on the above as vehicle resistance or track resistance force, are expressed by simple equations with coefficients based on the above mentioned measurements. Traction force is calculated from locomotive parameters. The mentioned complexity and mentioned measurements. Traction force is calculated from locomotive parameters. The mentioned complexity and parameter uncertainty is accounted for in the measured input data. For example there are known adhesion coefficients parameter uncertainty is accounted for in the measured input data. For example there are known adhesion coefficients ∗ ∗
Pavel Sovicka Tel.: +421 41 513 2270 Pavel Sovicka Tel.: +421 41 513 2270 E-mail address: [email protected] E-mail address: [email protected]
c 2019 The Authors. Published by Elsevier B.V. 2352-1465 c 2019 The Authors. Published by Elsevier B.V. 2352-1465 2352-1465 2019responsibility The Authors.ofPublished by Elsevier Peer-reviewunder the scientific committeeB.V. of the 13th International Scientific Conference on Sustainable, Modern and Safe Peer-review under of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Peer-review underresponsibility responsibility Transport (TRANSCOM 2019). of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Transport (TRANSCOM 2019). 2019). Safe Transport (TRANSCOM 10.1016/j.trpro.2019.07.216
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for different weather (sunny, rain, snow, etc.) and therefore boundary conditions of operation are known. Jansa (1980); Pilmann (2000); Danzer (2008). Several computing programs were developed at the predecessor of University of Zilina since 1960s. These were implemented on MEDA and Amsler analog computers. Later they were replaced by digital computers running ”Metro” software. Czech Railway Infrastructure Administration (SZDC) is using its own information software called ”SENA”, which was originally developed for calculating train timetables. However the option to calculate energy consumption based on track profile and train characteristics was later added to enable prediction of electricity or fuel consumption Drabek (2010); Siman (2006); Hlava and Hruby (2011). This paper proposes improvement in track resistance force calculation. First, the mass point is situated in front of the train, not in the center of mass. Secondly, the algorithm calculates specific track resistance incrementally as the train moves through different sections of the track profile based on the proportion of train already present in this new section. Uniform distribution of mass through entire length of the train is assumed. This approach allows easy modification of existing models. By considering train length, the simulation accuracy is improved for long trains or when short significant track sections, such as small curves, are present. A speed control algorithm, which can drive the train correctly at maximum speed, is also presented. The most important part of this algorithm is calculation of necessary braking distance based on jerk and acceleration limits. This enables the train to correctly respect speed profile of selected track. The entire simulation is implemented in Matlab and Simulink enviroments with a Graphical User Interface (GUI) created in the GUIDE development enviroment. 2. Mathematical model of train and track parameters Most of characteristics and equations used in mathematical modeling of trains do not use basic units for the respective variables. Instead units such as km/h, kN and tonnes are used. This is due to practical reasons, as these units are much more common on railways. For consistency with the theory and with cited sources, this work will use the same units. Therefore great care must be taken to correctly input values into the presented mathematical equations according to units shown in the respective description. The basic motion equation of train movement is (1). Traveled distance and speed can be calculated by modifying this equation to (2) to gain train acceleration and by using integration according to (3) and (4). Fa = a ·
n i=1
(mi · ξi ) = Ft − Fv − Fe − Fc − Fb
(1)
where Fa [N] is acceleration force, a [m/s] is acceleration, n is number of vehicles, mi kg is mass of i-th vehicle, ξi [−] is i-th coefficient of rotational parts, Ft [N] is traction force, Fv [N] is vehicle resistance, Fe [N] is track elevation resistance, Fc [N] is track curve resistance and Fb [N] is an artificial resistance created by braking. Ft − Fv − Fe − Fc − Fb n i=1 (mi · ξi ) t a (t) dt v=
a=
0
where v [m/s] is train speed and t [s] is time since the beginning of the train run. t l= v (t) dt 0
(2) (3)
(4)
where l [m] is traveled distance. Maximum traction force is limited by adhesion. A formula known as ”Curtius-Kniffler” is used in this work (5), although a different one can also be used, e.g. adhesion limit defined by TSI EC (2014). Actual available traction force for the given speed is dependent on the characteristics of the studied locomotive. This adhesion coefficient is then compensated according to (6) to represent adhesion changes in the demanded weather. µa =
7500 + 161 V + 44
(5)
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where µa [N/kN] is the adhesion coefficient and V [km/h] is train speed. µcond (6) µacomp = µa · 330 where µacomp [N/kN] is the adhesion coefficient compensated for chosen conditions and µcond [N/kN] is the adhesion coefficient selected to represent demanded weather conditions. For normal (average) weather conditions, µcond = 330◦/◦◦ , for poor conditions, µcond is lower (e.g. 150 ◦/◦◦ ). This value is related to the zero-speed adhesion conditions. This adhesion coefficients is then used to calculate the maximum traction force limit according to (7). Ftlim = µacomp · G = µacomp · M · g
(7)
where F tlim [N] is a maximum traction force limit, G [kN] is the gravitational force, M [t] is the train mass the and g m/s2 is a gravitational constant. As was shown in (1), resistance force acting on the train can be divided into three categories: • Vehicle resistance Fv - Resistance force created by bearing friction, wheel rolling resistance and aerodynamic resistance. • Track resistances Fe and Fc - Additional resistances created by track elements, mostly slopes and curves. This resistance can sometimes be combined into ”reduced slope”, which is a sum of all specific resistance components represented as a slope resistance to ease further calculations by providing only one resistance term in the equations. • Braking Fb can be represented as negative traction force, but also separately as an artificially created resistance to lower train speed. Fv = A + B · V + C · V 2
(8)
where A [N] is bearing resistance coefficient, B [Ns/m] is wheel rolling resistance coefficient and C Ns2 /m2 is aerodynamic resistance coefficient. While absolute values of resistances are necessary for final computation, specific resistances allow dynamic implementation for different train consists. The ratio between specific resistance and absolute resistance is determined by the gravitation force as shown in (9). F (9) p= G where p [N/kN] is the specific resistance. Therefore specific vehicle resistance can be calculated by combining (8) and (10). Usually an opposite calculation is done as the specific resistance for different train types has been measured and is known. This approach allows to accurately calculate resistance force acting on the train using known specific resistance characteristics and the train mass. pv = a + b · V + c · V 2
(10)
where a, b and c are specific vehicle resistance coefficients corresponding to (8). By traveling on a sloped track, part of the gravitational force is acting in the axis of movement. Adhesion railway permits only small slopes (up to several units %). Therefore a simplification shown in (11) can be used. The resulting specific slope resistance is equal to the slope value in represented in per-mille. g · m · sinα ht Fe = ≈ tgα = =s (11) pe = g·m g·m lt where pe [N/kN] is the specific elevation resistance, Fe [N] is elevation resistance, ht [m] is track height difference on a given length lt [m] and s [◦/◦◦ ] is the track slope. Specific curve resistance can also be calculated by several different empiric equations. Equation (12) applies for 1435 mm gauge main lines. 650 (12) pc = r − 55 where pc [N/kN] is the specific curve resistance and r [m] is the curve radius.
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(a)
(b)
Fig. 1: (a) Gradual slope change algorithm flowchart. Condition ”1” is ”Has the train reached a new elevation section?”; (b) Difference between classical theory and proposed incremental change algorithm shown on a 264 m long freight train with a total mass of 1508 t.
2.1. Incremental specific resistance calculation algorithm This algorithm implements train length into the simulation to improve accuracy without significant increase in computational requirements. The algorithm is used both for slope and elevation profiles. Fig. 1a shows a flowchart of this algorithm. First the current position of train front is loaded. This position is then compared to the track slope profile. If a new slope section has been reached, parameters scurr and s prev are updated to new values. A slope increment is calculated according to (13) and added to the last slope value (14). Finally the limit of the value is checked so that the new value does not exceed the slope limits. ds =
scurr − s prev · lcurr − l prev ltrain
(13)
where ds [N/kN] is the slope increment, scurr [◦/◦◦ ] is current slope loaded from the track profile, s prev [◦/◦◦ ] is previous slope loaded from the track profile, ltrain[m] is length of the train, lcurr [m] is current position of train front and l prev [m] is a position from last calculation step. sk = sk−1 + ds
(14)
where sk [◦/◦◦ ] is the slope value used for resistance force calculation and sk−1 [◦/◦◦ ] is the value from previous calculation step. Difference between classical ”step” and the proposed incremental approach for ”smooth” transition between track section is shown in Fig. 1b. Here a part of a simulation run of a 264 m long train with a mass of 1508 t from Pardubice to Zdarec is shown. 3. Control algorithm Acceleration to a demanded speed is simple as the acceleration must start after the train consist is entirely within a higher speed limit. However braking must start in advance so that the train is moving the correct speed once a new speed limit is reached. The implemented control algorithm calculates the distance required to decrease actual train speed to the new limit. The new speed is then demanded in advance when the train reaches this distance to the new limit. To simplify the analysis the speed profile during braking was divided into 3 parts according to Fig. 2:
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Fig. 2: An example of train deceleration from a given speed to a new demanded speed with constant jerk. Note that for better clarity values are not in correct respective scale.
• During interval (1) the train begins to increase its negative acceleration to a limit value with a constant jerk. Time required to reach this acceleration limit can be calculated by (15). Speed and covered distance from start of braking to end of this first interval is shown in (16) and (17). t1 =
alim blim
(15)
where t1 [s] is time required to reach acceleration limit, alim m/s2 is the acceleration limit and blim m/s3 is a maximum allowed jerk. v1 = v0 −
1 · blim · t1 2 2
(16)
where v1 [m/s] is speed at the end of this interval and v0 [m/s] is an initial speed. l1 = v0 · t1 −
1 · blim · t1 3 6
(17)
where l1 [m] is distance covered during first interval. • Time interval (2) is the main braking interval, where the train has constant negative acceleration. Time length of this interval is calculated by (18). To find the necessary speed difference for this calculation, speed differences from intervals 1 and 3 must be known. Speed and covered distance from the end of first interval to end of this interval is shown in (19) and (20). t2 =
∆v2 ∆v − ∆v1 − ∆v3 vcurrent − vnext − = = alim alim
1 2
· blim · t1 2 − alim
1 2
· blim · t3 2
(18)
where t2 [s] is time necessary to sufficiently lower train speed, ∆v2 [m/s] is a speed difference during this interval , ∆v [m/s] is a total speed between current speed and next track speed limit, ∆v1 [m/s] is a speed difference during first interval, ∆v3 [m/s] is a speed difference during third interval, vcurrent [m/s] is the current speed of the train at beginning of braking, vnext [m/s] is the next track speed limit and t3 [s] is time required to reach zero acceleration during the third interval. v2 = v1 − alim · t2
(19)
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where v2 [m/s] is speed at the end of this interval. l2 = v1 · t2 −
1 · alim · t2 2 2
(20)
where l2 [m] is distance covered during second interval. • During time interval (3) a constant positive jerk si applied to reach zero acceleration and new steady state speed, which is equal to the new speed limit. Third time interval (21) is equal to the first time interval if the absolute value of jerk is also equal. t3 =
alim = t1 blim
(21) 1 · blim · t3 2 2
(22)
1 1 · alim · t3 2 + · blim · t3 3 2 6
(23)
v3 = v2 − alim · t3 + where v3 [m/s] is speed. l3 = v2 · t3 −
where l3 [m] is distance covered during third interval. The total braking distance is a sum of distance covered during these three intervals (24). Braking point can than be found by subtracting this distance and the already covered distance from the point, where the new speed limit applies. lbrake = l1 + l2 + l3
(24)
where lbrake is total distance necessary to decrease speed from initial value to the next limit. lvdem change = lnextlimit − lcovered − lbrake
(25)
where lvdem change [m] is distance at which the braking must start to safely decrease train speed, lnextlimit [m] is distance where the next speed limit applies and lcovered [m] is distance already covered by the train. Note that this speed control approach is very basic. Therefore, while sufficient for this simulation, it is not suitable for implementation in actual control algorithms. 4. Implementation Mathematical model implemented was discussed in chapter 2 and is controlled according to algorithm described in chapter 3. Information about vehicle resistances can be found in Jansa (1980); Drabek (2010). Additional vehicle information, especially traction characteristic is provided from the respective manufacturer documentation. Finally track tables created by the respective railway operator were used for track profiles. Most of the mentioned algorithms were implemented in Matlab, but are called through a block diagram implemented in Simulink. This approach improves clarity and enables easy modifications to the simulation. For ease of use, the entire control is done through a Graphical User Interface (GUI), which also displays results of the simulation. As can be seen in Fig. 3 it is be divided into three parts: • Parameter input is situated in the top left corner. Here the user can select locomotive and train type, track layout and other parameter. These values are loaded from a database provided with the simulation. The ”Load values” button loads all data from the database files into base workspace for simulation while the ”Run simulation” activates a Simulink model.
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Fig. 3: Simulation GUI interface after a simulation was completed.
• Graphical output consists from three figures: ”Speed vs. Distance” shows how the train performed on selected route, ”Force vs. Distance” shows traction force and resistance sum and ”Force vs. Speed” shows utilization of the locomotive(s) traction characteristic. • Text output located in the bottom left corner and shows basic information to confirm configuration and also energy values and average speed. Several values are updated every 10 s of simulation time providing basic check of simulation status.
5. Results analysis An example of a freight train run between Pardubice and Zdarec pulled by a pair of class 742 locomotives can be seen in Fig. 3. This track was selected to show impact of the proposed algorithm on simulations of regular trains. Table 1: Total traction effort (A) and average speed (V) comparison of step and incremental resistance algorithms for different freight trains pulled by two class 742 locomotives between Pardubice and Zdarec. Note that weight limit for these locomotives on this track is 1028 t, including the locomotives. Train weight [t]
Train length [m]
A step [kWh]
Aincr [kWh]
∆A [%]
V step [km/h]
Vincr [km/h]
∆V [%]
528 928 1088 1328
94.7 162.2 189.2 229.7
601.56 979.77 1120.5 1319.95
600.77 976.89 1116.78 1312.87
-0.13 -0.29 -0.33 -0.54
52.79 44.45 40.91 36.28
52.81 44.50 40.95 36.38
0.04 0.11 0.12 0.28
As can be seen in Table 1, impact of the proposed method for the given example is small. However it can clearly be seen that for longer trains, the impact increases. This difference will be larger in situations, where the train length is comparable to track features. For example for a train of length 200 m, any curve or slope shorter than 200 m will be calculated differently through the entire track section as the resistance will gradually increase until the train is fully in the section and then gradually decrease. This can clearly be seen in the next example of shunting operation. Fig. 4 shows a speed profile of a shunting train run over a classification yard, where the shunting locomotive must move the freight carriages from the station uphill on a track, then stop and reverse the carriages back over the hill.
1570 8
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Fig. 4: An example of shunting over with 15 loaded freight cars over a classification yard.
This arrangement can be seen for example at Zilina station. The run consists of a 1 km long section without any slope, then 100m of 60 ◦/◦◦ slope followed by leveled track until the end of the track is reached. It can clearly be seen, that with shorter distance available or higher load, the step change simulation would reach zero speed and stop. This would provide a false result showing the locomotive is not suitable for this operation. Traction effort for the step algorithm was calculated to be 59.81 kWh, while for the incremental algorithm it is only 48.76 kWh. A difference of more than 18 %. Using the proposed algorithm shows the locomotive is capable of carrying out this shunting operation with sufficient safety margins. Also, the lower traction effort means that refueling intervals can be predicted more accurately. 6. Conclusion Mathematical model for simulating train operation was thoroughly explained including a new approach for track resistance calculation. This approach implements train length into the calculations with negligible additional performance requirements. The impact of this change has been analyzed. It was shown, that the highest improvement was during special cases, such as shunting over a hill at a classification yard. Differences during regular track operation were small. However the actual impact is dependent on the given train and track selection. Thanks to low computational requirements, the proposed approach can be implemented regardless if its benefits will have a significant impact on the results. Also a basic speed control algorithm was introduced. This algorithm was used during all simulations and was verified to correctly guide the train so that the track speed limit is not exceeded. Acknowledgements This work is supported by the Slovak Scientific Grant Agency VEGA No. 1/0774/18 and by project ITMS: 26220120046. References Danzer, J., 2008. Elektricka Trakce 2. vydani. Prehled problematiky (etr120), vlastni spotreba a chlazeni (etr600), adheze (etr700), rizeni vozidel (etr900) ed., ZCU FEL. Drabek, J., 2010. Energeticke vlastnosti zeleznicnich dopravnich systemu. Katedra vykonovch elektrotechnickch systemov ZU . EC, 2014. Commission regulation (eu) no 1302/2014 concerning a technical specification for interoperability relating to the rolling stock locomotives and passenger rolling stock subsystem of the rail system in the european union, in: Official Journal of the European Union. Hlava, K., Hruby, J., 2011. Elektricka trakcni energie, in: Vedeckotechnicky sbornik CD, c. 31/2011. Jansa, F., 1980. Dynamika a energetika elektricke trakce. NADAS. Pilmann, L., 2000. Dejiny, soucasnost a budoucnost zeleznicniho vyzkumu, in: Vedeckotechnicky sbornik CD, c. 09/2000. Siman, P., 2006. Moznosti uspory trakcni elektricke energie a motorove nafty zavisle na zeleznicni infrastrukture, in: Vedeckotechnicky sbornik CD, c. 22/2006.