EP1719687A2 - Precise determination of travel time of rail vehicles - Google Patents

Abstract

Method involves solving of acceleration phase by differential equation v'(t) = av2>(t)+bv(t)+c, with speed v(t) and parameters a,b,c depending on the time t, by formulae. These formulae are to be used with constant speed or constant braking deceleration in order to determine travel time for train travel. During positive acceleration v'(t) greater than 0, it examines, whether the sufficient distance is there to attain maximum permissible speed. During negative acceleration v'(t) less than 0, it is examined, that whether the train eventually comes to a standstill and the point of time of the standstill and thereby mileage distance is determined. During the steady travel v'(t)= 0, a break-down speed of vo is maintained. Basically a specific limitation is allowed for a exit speed by a timely application of a braking device.

Description

The invention relates to a method for the exact determination of travel times of rail vehicles.

The determination of travel time is the basis of the timetable of rail transport. In this case, an accuracy is required in which a manual calculation because of the required work is practically no longer possible. Therefore, a number of graphic methods were developed with which a driving line could be constructed and subsequently graphically integrated for driving time determination. To support the graphical driving time determination various devices were used, so-called graphic integrators. The best known is the so-called Conzen-Ott device, a mechanical analogue computer that was used until the advent of EDP.

Due to increasing traffic volume and to the optimization of procedures, travel time calculations are only carried out today with the help of EDP procedures.

The results of the travel time calculations are compiled in timetables. In this case, the start-up and brake additional times are shown separately for each possible traffic stop. These surcharge times are the differences from the travel time of a train passing through and the travel time of a stopping train (without holding time). As a result, the timetable processor can use this information to compile a route with any desired stopping sequence.

• Eberhard Jentsch: "Fahrzeitermittlung mit neuen Elementen der Zugfahrtsimulation", ZEVrail 127 (2003) Nr. 2 (Februar), Seite 66-71 ,
• Günter Habich, Friedrich Eickmann: "DYNAMIS - ein Simulationsmodell zur Bearbeitung fahrdynamischer Fragestellungen", Eisenbahntechnische Rundschau 1990, Heft 1/2, Seite 83-87 .

Prior art methods for calculating travel times of rail vehicles by means of EDP methods are so-called Δ-t, Δ-s or Δ-v methods, which are described in the following articles:

• Eberhard Jentsch: "Driving time determination with new elements of the train simulation", ZEVrail 127 (2003) No. 2 (February), page 66-71 .
• Günter Habich, Friedrich Eickmann: "DYNAMIS - a simulation model for handling driving dynamics issues", Eisenbahntechnische Rundschau 1990, Issue 1/2, page 83-87 ,

• Die sich ergebenden Differentialgleichungen werden in Differenzengleichungen umgewandelt.
• Innerhalb eines Δ-t-Intervalls wird der Verlauf der Fahrt durch eine Näherungsfunktion bestimmt, die sich aus der konstanten Beschleunigung ergibt.

In a Δ-t method, the time axis is divided into intervals of the fixed length Δt. Within such an interval, a constant acceleration is assumed. The following two variants are distinguished:

• The resulting differential equations are converted into difference equations.
• Within a Δ-t interval, the course of the journey is determined by an approximation function, which results from the constant acceleration.

A variant of the first method is the Runge-Kutta method, which additionally considers the mid-point of each interval and averages various suggestions for continuing the function.

Δ-s or Δ-v methods are defined accordingly, i. the path or velocity axis are divided into intervals of constant length.

Out Dietrich Wende: "Driving Dynamics of Rail Traffic", Teubner-Verlag, page 51-53 , Further, as the prior art is known that the rolling process can be carried out analytically, but only the above stepping methods are treated. The actual algorithmic consequence of considerably reducing the computation time by analytical procedure has not been drawn.

Another prior art is in the article of Rüdiger Franke, Peter Terwiesch, Markus Meyer: "An algorithm for the optimal control of the driving of trains", Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, December 2000 specified. However, the tensile force in the solution of the differential equation is assumed to be constant. Furthermore, linear terms are not provided in the resistance function. Both represent a simplification and not inconsiderable distortion of the actual circumstances in reality.

It is therefore an object of the invention to provide a method, with the travel times of trains exactly, i. without simplifications of parameters and procedures, and faster, i. without much time.

This object is achieved in connection with the preamble of claim 1 according to the invention by the features specified in claim 1.

Claims 2 to 6 include advantageous embodiments of the inventive solution of claim 1.

The particular advantage of the new driving time calculation is that it is no longer necessary to subdivide the time, path or speed axis into intervals of fixed length and to spend a calculation step for each interval due to the analytical evaluation of the differential equations. Where it is not possible to analytically determine a parameter, the Newton method is used. The latter always converges in a few steps, independent of the axis length. Both circumstances make it possible to significantly shorten the running time of the travel time calculation, whereby the computing speed is increased approximately by a factor of 70. Furthermore, the rounding errors occurring in the iterative methods are eliminated.

• Die Bremsphase eines Zuges wird maßgeblich von der verwendeten Leit- und Sicherungstechnik (Linienzugbeeinflussung (LZB) oder konventionelle Fahrweise) bestimmt und hängt zusätzlich von der zulässigen Höchstgeschwindigkeit, der Zugart und weiteren Parametern ab.
• Für die Höchstgeschwindigkeit eines Zuges gelten unterschiedliche Grenzen bei den Loks, dem Wagenmaterial sowie der Strecke, wobei die maximal zulässige Streckengeschwindigkeit sich ständig (z.B. bei Tunneln) ändern kann und deshalb abschnittsweise festgelegt wird. Ferner muss ein Zug bei einer nachfolgenden niedrigeren Höchstgeschwindigkeit noch rechtzeitig bremsen können. Eine Geschwindigkeitserhöhung wird dagegen im Allgemeinen erst dann wirksam, wenn der Zug vollständig den vorherigen Bereich verlassen hat. Weiter können z.B. auch die sogenannten maßgebenden Neigungen Einfluss auf die Höchstgeschwindigkeit haben.
• Die Formeln für die Beschleunigung eines Zuges benötigen genaue Angaben über die Masse des Zuges, die Beschleunigungskräfte der Loks, Trägheitswiderstände, usw. Für diverse Parameter wird dabei detailliert zwischen Güter- und Personenzügen, zwischen Triebwagen und lokbespannten Zügen usw. unterschieden. Ferner sind Angaben über die aktuellen Neigung (also Steigung oder Gefälle) der Strecke erforderlich (auch diese ändern sich ständig).
• Zum Abbauen von Verspätungen werden auf die eigentlichen Fahrzeiten Zuschläge erhoben. Auch für diese gelten je nach Zugart, Streckenhöchstgeschwindigkeit usw. unterschiedliche Werte.

The invention describes a fundamentally new way of determining and calculating the travel times for trains. Part of this calculation - namely the acceleration phase - is based on the solution of a differential equation, which is numerically approximated in all previous methods. According to the invention, it is shown that this differential equation can also be represented exactly, ie expressed by means of a closed formula. This increases the accuracy of the determined travel times; above all, however, the calculation can be done much faster, because instead of iterative approximations (Runge-Kutta, etc.) only relatively simple formulas have to be evaluated. For a complete travel time calculation, however, the processing of purely physical equations of motion is not sufficient. Rather, numerous regulations and framework conditions must be observed. Extracts may be mentioned at this point:

• The braking phase of a train is largely determined by the control and safety technology used (train control (LZB) or conventional driving style) and also depends on the maximum speed, the type of train and other parameters.
• For the maximum speed of a train different limits apply to the locomotives, the car material and the track, the maximum permissible line speed can change constantly (eg in tunnels) and is therefore set in sections. Furthermore, a train must be able to brake in time at a subsequent lower maximum speed. In contrast, a speed increase generally only takes effect when the train has completely left the previous area. Furthermore, for example, the so-called decisive inclinations can influence the maximum speed.
• The formulas for the acceleration of a train require precise information about the mass of the train, the acceleration forces of the locomotive, inertia, etc. For various parameters, a distinction is made in detail between freight and passenger trains, between railcar and locomotive-drawn trains etc. In addition, information about the current slope (ie slope or gradient) of the route is required (these also change constantly).
• To reduce delays surcharges will be charged on the actual travel times. Also for these, different values apply depending on the type of train, maximum route speed, etc.

1 Physical basics of a train journey

First, the equations of motion that play an essential role in the movement of a train are briefly explained. Some influences such as bow resistances are not considered here; Also, only general values are assumed for certain characteristics such as the braking deceleration. However, all the parameters and formulas required for the travel time calculation for the different trains will be specified later.

• beschleunigt so früh wie möglich,
• fährt so lange es geht mit der maximal zulässigen Höchstgeschwindigkeit,
• und bremst so spät wie möglich.

For the driving time calculation, it is first simplified assuming that the train consists only of a "mass point", which combines the entire train mass in itself; the train has therefore the length zero. Basically, he is always in one of three phases during a trip: either he accelerates, maintains his current speed (steady drive), or he brakes. It is assumed that a tight driving style, ie the train

• accelerate as early as possible,
• drives as long as possible with the maximum permissible speed,
• and brake as late as possible.

As a result, the highest possible average speed is achieved overall. It should be noted, however, that the train e.g. with too high gradients despite maximum acceleration can also slow down. However, this is still called the acceleration phase, i. it is determined:

Acceleration phase: = The train travels using its maximum traction.

The individual phases will now be examined more closely. The inertia ride is characterized in that the tractive forces just exactly compensate for the frictional and path resistances, i. the train runs at a constant speed and then covers a distance of s = v · t meters in t seconds at a speed of v meters per second. If two quantities are given, the third can be calculated from this.

Even when braking from a speed v to a target speed w (w ≦ v), the relationships are simple. It is assumed here that the driver does not take advantage of the maximum braking capacity of the train, but instead seeks for comfort reasons a constant braking deceleration vzg, which is estimated for example in local trains with the value vzg = 0.7 m / s ² . Thus, t = (v-w) / vzg seconds are required for the braking operation, and the distance s traveled is s $${\displaystyle {\int }_0^t\left(v-vzG\cdot r\right)dr}={vr-\frac{1}{2}vzG\cdot r^2|}_0^t=\frac{v^2-vw}{vzG}-\frac{{\left(v-w\right)}^2}{2vzG}=\frac{2v^2-2vw-v^2+2vw-w^2}{2vzG}=\frac{v^2-w^2}{2vzG}$$

festgelegt werden, dass s Meter vor dem Erreichen z.B. einer Langsamfahrstelle, wo nur w m/s erlaubt sind, höchstens √(2s - vzg + w²) m/s gefahren werden dürfen, damit der Zug noch rechtzeitig bremsen kann. Weiter oben wurde dieser Aspekt bereits angerissen.Meter. Thus, the brake application point can be easily determined, and vice versa, because of $$s=\frac{v^2-w^2}{2vzG}\iff v=\sqrt{2s\cdot vzG+w^2}$$

It is stipulated that, for example, s meters may be driven at most √ (2s - vzg + w ² ) m / s before reaching eg a slow driving location where only wm / s are allowed, so that the train can brake in time. Earlier, this aspect has already been touched on.

The acceleration of a turn is more complicated and depends on its current speed. First of all, the tractive force of the locomotive or the railcar (measured in kilonewtons or in short kN) is described in pieces by train parabolas of the form F = k ₀ + k ₁ .V + k ₂ .V ² , where V indicates the speed in km / h and the coefficients k ₀ , k ₁ and k _{2 are taken} from a table that depends on the type of loco and the current speed range.

beschrieben (das Ergebnis ist in der Einheit Newton), wobei g = 9,81 m/s² die Erdbeschleunigung und m die Masse des Zuges in Tonnen bezeichnen. Die Konstanten f₁, f₂ sowie f₃ sind ebenfalls wieder vom konkreten Zug abhängig. Wichtig ist auch der Streckenwiderstand, der sich hemmend (Steigung) bzw. begünstigend (Gefälle) auf die Zugkraft auswirkt.The tensile force counteracts running resistance of the train (eg due to friction losses, air resistance, etc.). These become quite similar to the tensile forces of an equation of form $${\mathrm{f}}_1\mathrm{m}\mathrm{G}+{\mathrm{f}}_2\mathrm{m}\mathrm{G}\mathrm{V}+{\mathrm{f}}_3\mathrm{G}{\mathrm{V}}^2$$

described (the result is in the unit Newton), where g = 9.81 m / s ² denote the acceleration of gravity and m the mass of the train in tons. The constants f ₁ , f ₂ and f ₃ are also again dependent on the actual train. Another important factor is the track resistance, which has an inhibiting (gradient) or favoring (downward gradient) effect on the traction.

die Beschleunigung acc berechnet werden (wegen der Angabe der Masse in Tonnen ist der Faktor 1000 notwendig). Die Zugkraft F wird allerdings nicht nur für die Beschleunigung der Zugmasse, sondern auch für die Rotation der bewegten inneren Bauteile (Getriebe usw.) benötigt. Dieser Verlust wird durch einen sogenannten Massezuschlag berücksichtigt, der pauschal als prozentualer Wert p für jeden Zugtyp vorgegeben ist.If F denotes the remaining tractive force after deduction of the running and distance resistances, then now in principle by means of the basic formula $$F=1000\cdot m\cdot acc\iff acc=\frac{F}{1000\cdot m}$$

the acceleration acc is calculated (because of the mass in tons the factor 1000 is necessary). However, the tensile force F is needed not only for the acceleration of the train mass but also for the rotation of the moving internal components (gears, etc.). This loss is taken into account by a so-called mass penalty, which is given as a percentage as a percentage value p for each type of train.

The above formulas are now combined, where the factor 1 m / s = 3.6 km / h must take into account what constitutes a conversion factor of 3.6 ² = 12.96 at the square dependent on the speed terms. Furthermore, the acceleration represents precisely the mathematical derivative of the velocity. This yields the following equation of motion for the acceleration of a train traveling at the time t at the speed v (t) m / s: $${\mathrm{v}}^{\text{'}}\left(\mathrm{t}\right)=\mathrm{a}{\mathrm{v}}^2\left(\mathrm{t}\right)+\mathrm{b}\mathrm{v}\left(\mathrm{t}\right)+\mathrm{c},$$

The constants a, b and c result in the following statements as follows: $$a=\frac{12960\cdot K_2-12.96\cdot f_3\cdot G}{m\cdot \left(1000+10\cdot \rho \right)}.b=\frac{3600\cdot K_1-3.6\cdot f_2\cdot m\cdot G}{m\cdot \left(1000+10\cdot \rho \right)}.c=\frac{1000\cdot K_0-f_1\cdot m\cdot G-I\cdot m\cdot G}{m\cdot \left(1000+10\cdot \rho \right)},$$

selbst ändert sich nichts. Wenn es nun gelingt, eine geschlossene Formel für v(t) zu finden, so kann man die Beschleunigung des Zuges stückweise (d.h. für die einzelnen Phasen, bei denen die Werte a, b und c gleich bleiben) direkt ausrechnen. Diese Problematik wird im folgenden behandelt.The parameters a, b and c result from the infrastructure data and the train data. However, only the values of the parameters a, b and c change in the equation of motion $${\mathrm{v}}^{\text{'}}\left(\mathrm{t}\right)=\mathrm{a}{\mathrm{v}}^2\left(\mathrm{t}\right)+\mathrm{b}\mathrm{v}\left(\mathrm{t}\right)+\mathrm{c},$$

nothing changes. If one succeeds in finding a closed formula for v (t), then the acceleration of the train can be calculated piecewise (ie for the individual phases in which the values a, b and c remain the same). This problem will be dealt with below.

2 Solution of the differential equation 2.1 Transfers to the required formulas

zum Zeitpunkt t = 0 eine bestimmte Einbruchsgeschwindigkeit v₀ vor (also v(0) = v₀) so ist die Formel für v(t) eindeutig bestimmt. Im folgenden wird diese Formel hergeleitet, so dass für jeden beliebigen Zeitpunkt die Geschwindigkeit des Zuges und daraus die zugehörige Beschleunigung ausgerechnet werden kann. Die Bewegung des Zuges ist damit korrekt erfasst, bis sich mindestens eine der Konstanten a, b oder c ändert. Dies geschieht in den folgenden Fällen:

• Der Zug erreicht eine bestimmte Geschwindigkeit, bei deren Überschreitung sich die Zugkraftparabeln ändern. Demzufolge wird eine Formel für t(v) gesucht, die bei vorgegebener Geschwindigkeit v angibt, wann diese erreicht wird. Man beachte nochmals, dass der Zug trotz maximaler Zugkraft auch langsamer werden kann (z.B. wenn er sich auf einem zu steilen Streckenabschnitt befindet). Deshalb muss auch überprüft werden, ob der aktuell gültige Geschwindigkeitsbereich nach unten hin durchbrochen wird.
• Die Streckenneigung ändert sich ab einer bestimmten Position. Um diesen Fall bearbeiten zu können, muss bekannt sein, wann der Zug den aktuellen Streckenabschnitt durchfahren hat und in den nächsten mit den veränderten Bedingungen eintritt. Deshalb wird zusätzlich eine Formel t(s) benötigt, die für eine gegebene Streckenlänge s die benötigte Durchfahrtszeit ermittelt. Desweiteren ist auch die Umkehrfunktion s(t) (d.h. welche Streckenlänge legt der Zug in einer bestimmten Zeit zurück) von Interesse.
• Der Zug erreicht die zulässige Höchstgeschwindigkeit vmax. Mit den bisherigen Formeln kann man diesen Fall leicht behandeln. Man bestimmt zunächst mittels t(vmax) den Zeitpunkt, bei dem die Höchstgeschwindigkeit erreicht wird (es kommt dabei nicht darauf an, ob diese durch den Zug oder durch die Strecke bedingt ist), und berechnet anschließend mittels der Formel s(t) die bis dahin zurückgelegte Strecke. Anschließend wird mit konstanter Geschwindigkeit weitergefahren.
• Der Zug muss rechtzeitig bremsen, um eine niedrigere Höchstgeschwindigkeit im herannahenden nächsten Streckenabschnitt einzuhalten. Sollte der Zug in Beharrungsfahrt unterwegs sein, so ist auch dieser Fall kein Problem, weil mittels der oben im Abschnitt "Physikalische Grundlagen einer Zugfahrt" hergeleiteten Formel der benötigte Bremsweg und daraus alle resultierenden Größen berechnet werden können. Schwieriger wird es, wenn der Zug bei laufender Beschleunigung direkt in den Bremsvorgang übergehen muss. Wie später gezeigt wird, erfordert die Ermittlung der Position dieses Übergangs ein Nullstellenverfahren, übrigens ebenso wie die Formel für t(s). Diese Verfahren konvergieren allerdings sehr schnell, insbesondere werden nur zwei bis drei Iterationen benötigt.

Is in addition to the motion condition discussed above $${\mathrm{v}}^{\text{'}}\left(\mathrm{t}\right)=\mathrm{a}{\mathrm{v}}^2\left(\mathrm{t}\right)+\mathrm{b}\mathrm{v}\left(\mathrm{t}\right)+\mathrm{c},$$

at the time t = 0 a certain break-in speed v ₀ before (that is, v (0) = v ₀ ), the formula for v (t) is uniquely determined. In the following this formula is derived, so that for each arbitrary time the speed of the train and from it the appropriate acceleration can be calculated. The movement of the train is thus correctly detected until at least one of the constants a, b or c changes. This happens in the following cases:

• The train reaches a certain speed, beyond which the traction parameters change. As a result, a formula is searched for t (v), which at a given velocity v indicates when it is reached. Note again that the train can also slow down despite maximum traction (eg if it is on a steep section). Therefore, it must also be checked whether the currently valid speed range is broken down.
• The track inclination changes from a certain position. In order to be able to process this case, it must be known when the train has traveled through the current section of the route and enters the next with the changed conditions. Therefore, a formula t (s) is additionally required, which determines the transit time required for a given route length s. Furthermore, the inverse function s (t) (ie, which route length does the train return in a certain time) is also of interest.
• The train reaches the maximum permissible speed vmax. With the previous formulas one can handle this case easily. First, t (vmax) is used to determine the point in time at which the maximum speed is reached (it does not matter whether it is due to the train or the track) and then calculates, by means of the formula s (t) there covered distance. Then it is continued at a constant speed.
• The train must brake in time to maintain a lower maximum speed in the approaching next leg. If the train is in steady drive, then this case is not a problem, because by means of the above in the section "Physical basics of a train journey" derived formula the required braking distance and from this all resulting sizes can be calculated. It becomes more difficult when the train has to go directly into braking during acceleration. As will be shown later, the determination of the position of this transition requires a zeroing procedure, by the way as well as the formula for t (s). However, these methods converge very fast, in particular only two to three iterations are needed.

It now determines which of the cases occurs first, calculates all corresponding quantities by means of the formulas provided below, and then processes the next section until finally the entire desired route has been completed. For each section, only a few formulas need to be evaluated, which makes this method very efficient.

2.2 Approach to solving the differential equation

handelt es sich um eine sog. separable gewöhnliche Differentialgleichung. Sie ist zusammen mit der Anfangswertbedingung v(0) = v₀ eindeutig lösbar.At the motion condition discussed in the previous section $${\mathrm{v}}^{\text{'}}\left(\mathrm{t}\right)=\mathrm{a}{\mathrm{v}}^2\left(\mathrm{t}\right)+\mathrm{b}\mathrm{v}\left(\mathrm{t}\right)+\mathrm{c}$$

it is a so-called separable ordinary differential equation. It is uniquely solvable together with the initial value condition v (0) = v ₀ .

bzw. $$\mathrm{d}\mathrm{v}/\left(\mathrm{a}{\mathrm{v}}^2+\mathrm{b}\mathrm{v}+\mathrm{c}\right)=\mathrm{d}\mathrm{t}.$$

The general case will now be solved. First of all, v ₀ (t) is replaced by the notation dv / dt, resulting in: $$\mathrm{d}\mathrm{v}/\mathrm{d}\mathrm{t}=\mathrm{a}{\mathrm{v}}^2+\mathrm{b}\mathrm{v}+\mathrm{c}$$

respectively. $$\mathrm{d}\mathrm{v}/\left(\mathrm{a}{\mathrm{v}}^2+\mathrm{b}\mathrm{v}+\mathrm{c}\right)=\mathrm{d}\mathrm{t},$$

bzw. $${\displaystyle \int \frac{dv}{a{v}^2+bv+c}}=t+C$$

wobei C eine Konstante ist, die von der Einbruchsgeschwindigkeit v₀ := v(0) abhängt und später genauer bestimmt wird.By integrating both sides arises $${\displaystyle \int \frac{dv}{a{v}^2+bv+c}}={\displaystyle \int dt}$$

respectively. $${\displaystyle \int \frac{dv}{a{v}^2+bv+c}}=t+C$$

where C is a constant that depends on the break-in velocity v ₀ : = v (0) and is determined more accurately later.

hängt die weitere Vorgehensweise davon ab, ob die Diskriminate b² - 4ac größer, gleich, oder kleiner als Null ist.Because of $$a{v}^2+bv+c=a\left(v^2+\frac{b}{a}v+\frac{c}{a}\right)=a\left({\left(v+\frac{b}{2a}\right)}^2-\frac{b^2-4ac}{4a^2}\right)$$

The further procedure depends on whether the discriminates b ² - 4ac is greater than, equal to, or less than zero.

2.3 Case 1: b ² > 4ac

In this case, set D: = √ (b ² - 4ac).

If we first assume 2av + b + D <0 or 2av + b - D> 0 then we get: $$t\left(v\right)=\frac{1}{D}\ln \left(\left(1-\frac{2D}{2av+b+D}\right)/K\right).\mathrm{in\; which}K=1-\frac{2D}{2a{v}_0+b+D}$$

Solving t (v) for v gives the sought equation of motion $$v\left(t\right)=\frac{1}{2a}\left(\frac{2D}{1-K{e}^{Dt}}-b-D\right)$$

If the other case 2av + bD <0 <2av + b + D is assumed, the following formulas result with analogous considerations (again with k> 0): $$K=\frac{2D}{2a{v}_0+b+D}-1.t\left(v\right)=\frac{1}{D}\ln \left(\left(\frac{2D}{2av+b+D}-1\right)/K\right).v\left(t\right)=\frac{1}{2a}\left(\frac{2D}{1+K{e}^{Dt}}-b-D\right)$$

Since these formulas for negative k result in the formulas from the first case, one can restrict oneself to the second formula and at the same time remove the restriction for k. Now a formula is developed, which sets the elapsed time t in relation to the meanwhile traveled distance s. The following applies: $$s\left(t\right)=\frac{1}{a}\left(\ln \left|\frac{1+K}{1+K{e}^{Dt}}\right|+\frac{D-b}{2}t\right)$$

2.4 Case 2: b ² = 4ac

ist er aber auch einfach zu lösen und wird der Vollständigkeit halber hier ebenfalls behandelt. Wird k := -C gesetzt ergibt sich durch Auflösen von $$t\left(v\right)=k-\frac{1}{av+b/2}$$

nach v die Formel $$v\left(t\right)=\frac{1}{a}\left(\frac{1}{k-t}-\frac{b}{2}\right)$$

This case is only of a theoretical nature, since b ² ≠ 4ac always applies because of the usually "crooked" values for a, b, and c. Because of $$\frac{1}{a}{\displaystyle \int \frac{dv}{{\left(v+b/2a\right)}^2}}=t+C\iff -\frac{1}{a}\cdot \frac{1}{v+b/2a}=t+C$$

But it is also easy to solve and will be covered here for the sake of completeness. If k: = -C is set by dissolving $$t\left(v\right)=K-\frac{1}{av+b/2}$$

after v the formula $$v\left(t\right)=\frac{1}{a}\left(\frac{1}{K-t}-\frac{b}{2}\right)$$

Asymptotically, the train approaches the speed limit $$-\mathrm{b}/2\mathrm{a}$$

Finally, as above, a formula for s (t) is determined. It follows: $$s\left(t\right):={\displaystyle {\int }_0^{t}v\left(r\right)dr}=\frac{1}{a}{\displaystyle {\int }_0^t\frac{1}{K-r}-\frac{b}{2}dr}=\frac{1}{a}\left(\ln \left|K\right|-\ln \left|K-t\right|-\frac{b}{2}t\right)=\frac{1}{a}\left(\ln \left|\frac{K}{K-t}\right|-\frac{b}{2}t\right),$$

2.5 Case 3: b ² <4ac

In this case D: = √ (4ac - b ² ) / 2 is set and the integral named before the case distinction is resolved as follows: $$\frac{1}{a}{\displaystyle \int \frac{dv}{{\left(v+b/2a\right)}^2+{\left(D/a\right)}^2}}=\frac{a}{D^2}{\displaystyle \int \frac{dv}{{\left(av/D+b/\left(2D\right)\right)}^2+1}}=\frac{1}{D}\arctan \left(\frac{av}{D}+\frac{b}{2D}\right)$$

mit einer noch zu bestimmenden Konstante C, die wieder durch k := -C ersetzt wird. Das Auflösen nach v ergibtSo it turns out $$t\left(v\right)=\frac{1}{D}\arctan \left(\frac{av}{D}+\frac{b}{2D}\right)+C$$

with a constant C to be determined, which is again replaced by k: = -C. The resolution to v yields

Finally, the formula for the distance covered s is given as a function of the time t: $$s\left(t\right)=\frac{1}{a}(\ln \left|\frac{\cos \left(\mathit{dk}\right)}{\cos \left(D\right(t+K\left)\right)}\right|-\frac{b}{2}t)$$

3 Treatment of not directly dissolvable formulas

• Es wird eine Formel t(s) benötigt, die für eine gegebene Streckenlänge s die dafür benötigte Zeit t bei gegebener Anfangsgeschwindigkeit v₀ = v(0) und angenommener laufender Beschleunigung bestimmt.
• Wir müssen evtl. den Zeitpunkt t ermittelt, ab dem der Zug bei vorheriger Beschleunigung anfangen muss zu bremsen, so dass er die vorgeschriebene Höchstgeschwindigkeit auf dem nachfolgenden Streckenabschnitt einhält.

In the equations of the last section, two issues still to be solved were excluded in all three cases:

• A formula t (s) is needed which, for a given distance s, determines the time t required for a given initial velocity v ₀ = v (0) and assumed running acceleration.
• We may need to determine the time t, from which the train must start to brake at previous acceleration, so that it complies with the prescribed maximum speed on the subsequent section.

Both cases can be attributed to the problem of determining a zero for an explicitly described differentiable function. This is possible in sufficiently good approximation with the Newton method.

4 Calculation of traction energies

However, in addition to the formulas for determining travel times, the traction energies used for the individual journeys are of high interest, both for environmental and monetary reasons. It will now be shown that closed formulas can also be derived for energy consumption.

In principle, the energy used is calculated from the product "force (in Newtons) times distance (in meters)". During a steady drive this force corresponds exactly to the sum of all running resistances of the traction vehicles and the Cart material and the track resistance. This multiplied by the path length gives the desired result.

gelöst werden. Da keine Formel für F(s) zur Verfügung steht, wird deshalb ersatzweise über die Geschwindigkeit v integriert, da zu jeder Streckenlänge s die dann aktuelle Geschwindigkeit v ermittelt werden kann und umgekehrt: $$E={\displaystyle \int F\left(v\right)ds}={\displaystyle \int F\left(v\right)\frac{ds}{dt}\frac{dt}{dv}dv}$$

More complex is to determine the traction energy for an acceleration phase, since in this case the max. Traction is used and this is constantly changing. So it has to be the integral $$e={\displaystyle \int F\left(s\right)ds}$$

be solved. Since no formula is available for F (s), it is therefore integrated as a substitute via the velocity v, since the current velocity v can then be determined for each route length s, and vice versa: $$e={\displaystyle \int F\left(v\right)ds}={\displaystyle \int F\left(v\right)\frac{ds}{dt}\frac{dt}{dv}dv}$$

For the energy consumption between the beginning of the acceleration phase (with break-in velocity v ₀ and entry time t (v ₀ ) = 0) and its end (with exit velocity v and exit time t (v)) one obtains: $$e\left(v\right)-e\left(v_0\right)=\left(\frac{q_2}{2a}\left(v+v_0\right)+\frac{q_{1}a-q_{2}b}{a^2}\right)\left(v-v_0\right)+\frac{A}{2a}\ln \left|\frac{a{v}^2+bv+c}{a{v}_0^2+b{v}_0+c}\right|+\left(B-\frac{Ab}{2a}\right)t\left(v\right)$$

Here are q ₀ , q ₁ q ₂ coefficients of the Zugparparabel, the speed v in kilometers per hour and the tensile force in kilonewtons are given.

5 First summary

• v (t) indicates the achieved speed (in m / s) after t seconds driving time. v (t) can be given by a closed formula, provided that v (t) is within the scope of a given traction parabola.
• t (v) calculates the time the train takes to reach the speed v. t (v) can be specified within a scope of a traction parabola by a closed formula.
• s (t) indicates the distance traveled in meters after t seconds of driving time. s (t) can be given by a closed formula, provided that v (t) is within the scope of a given traction parabola.
• Conversely, t (s) determines the time required to pass through a section of s meters in length. For the determination of t (s) an iterative zeroing method is used.
• Finally, a method is provided which, for a given route length s, determines the point in time t from which braking must be initiated, so that the maximum speed r valid on the next route section is maintained. If the braking process directly follows the acceleration process, an iterative zeroing procedure is used. If the braking operation follows constant speed driving, the time t is given by a closed formula.
• The energy consumption of an acceleration phase can be specified by a closed formula.

6 Calculation of travel times

• Angabe, ob der Zug konventionell oder unter LZB fährt
• Höchstgeschwindigkeiten des Zuges bei konventioneller und LZB-Fahrt
• Höchstgeschwindigkeit, Länge, Bremsweg und maßgebende Neigung der einzelnen Streckenabschnitte
• Höchstgeschwindigkeit jeder beteiligten Lok
• Allgemeine Höchstgeschwindigkeit des Zuges
• Angabe darüber, ob ein Tunnel durchfahren wird
• Bremshundertstel und Bremsstellung (P, R+Mg, R+Mg, R/P, WB oder G) des Zuges
• Passende Bremstafeln, die aus der Bremsstellung, dem Bremsweg, der maßgebenden Neigung und den Bremshundertsteln die zugehörige Höchstgeschwindigkeit ermitteln
• LZB-Bremskurven
• Länge des Zuges

First of all, the sequence of sections to be traveled through is subjected to preprocessing, whereby numerous secondary conditions are taken into account. This applies above all to the validity of the maximum speeds, which are derived from the following parameters:

• Indication of whether the train travels conventionally or under LZB
• Top speeds of the train at conventional and LZB journey
• Top speed, length, braking distance and decisive inclination of the individual sections
• Maximum speed of each participating locomotive
• General maximum speed of the train
• Indication of whether a tunnel is passed through
• Brake hundredth and brake position (P, R + Mg, R + Mg, R / P, WB or G) of the turn
• Matching brake plates, which determine the associated maximum speed from the braking position, the braking distance, the decisive inclination and the braking hundredths
• LZB braking curves
• Length of the train

Furthermore, the influencing factors on the braking process, whereby a distinction must be made between LZB driving and conventional driving, are used to calculate the factors influencing acceleration and the vehicle surcharges before the actual travel time calculation is carried out. In addition, the constant proportions of the parameters a, b and c of the later differential equations (weight of the train, etc.) can already be processed in advance.

Schritt 1:

Initialisierung. Die Fahrzeit setzt sich aus der benötigten Durchfahrtszeit für verschiedene Teilstücke des Streckenabschnitts zusammen, die in t aufsummiert werden. Deshalb wird anfangs t := 0 gesetzt.

Schritt 2:

Berechnung der Parameter a, b und c der Differentialgleichung sowie der aktuellen Beschleunigung acc := av² + bv + c. Falls v genau auf der Grenze zwischen zwei Zugkraftbereichen liegt, werden für beide Bereiche die jeweiligen Parameter sowie die zugehörigen Beschleunigungen berechnet und eine Wahl zwischen beiden Möglichkeiten getroffen. Ist im höheren Geschwindigkeitsbereich die Beschleunigung noch positiv, wird dieser ausgewählt. Ist nur die andere Beschleunigung im unteren Bereich positiv, so wird der Zug nicht langsamer, aber auch nicht schneller, setzt also seine Fahrt mit konstanter Geschwindigkeit fort, was acc := 0 bedeutet. Falls schließlich beide Beschleunigungen negativ sind, so kann der Zug seine aktuelle Geschwindigkeit nicht halten, und es werden die berechneten Werte bzgl. des niedrigeren Geschwindigkeitsbereichs benutzt.

Schritt 3:

Sicherheitsüberprüfungen. Falls der Zug mit der maximal zulässigen Zug- oder Streckengeschwindigkeit unterwegs ist, wird acc = 0 gesetzt. Falls v = 0 und acc# 0 gilt, wird die Berechnung beendet (der Zug steht).

Schritt 4:

Test auf Beharrungsfahrt. Falls acc = 0 gilt, wird t := t + P/v gesetzt, sofern die zulässige Höchstgeschwindigkeit r des nächsten Streckenabschnitts nicht kleiner als v ist. Ansonsten muss der Bremsvorgang berücksichtigt werden, was durch die Zuweisung $$t:=t+\frac{v-r}{acc}+\frac{\ell }{v}-\frac{v^2-r^2}{2v\cdot acc}$$

berücksichtigt wird. Anschließend ist die Berechnung beendet.

Schritt 5:

Fallunterscheidung. Es wird geprüft, ob b² größer, gleich, oder kleiner als 4ac ist und bei den nachfolgenden Schritten die entsprechenden Formeln für v(t), t(v), usw. verwendet.

Schritt 6:

Überwachung der Geschwindigkeit. Bezeichnet vmax die maximale Geschwindigkeit (eingeschränkt u.a. durch den aktuellen Zugkraftbereich), für die die Konstanten a, b und c noch gültig sind, und gilt acc > 0, wird getestet, ob die Gültigkeit der Formeln durch das Erreichen der maximalen Geschwindigkeit begrenzt wird. Zu diesem Zweck wird geprüft, ob die asymptotische Geschwindigkeit des Zuges vmax übersteigt. (Dies ist im dritten Fall b² < 4ac nicht erforderlich, da der Zug dann immer unendlich schnell wird und folglich jede Geschwindigkeitsgrenze überbietet.) Falls dies zutrifft, so erreicht der Zug die maximale Geschwindigkeit nach t(vmax) Sekunden. Zu diesem Zeitpunkt hat er bereits s := s(t(vmax)) Meter Strecke zurückgelegt. Falls vmax größer als r ist, so muss der Zug später noch bremsen, und deshalb wird s um den Bremsweg (v²max - r²)/(2acc) erhöht. Falls nun s ≤ P gilt, so steht fest, dass der Zug innerhalb des gegenwärtigen Abschnitts auf vmax Meter pro Sekunde beschleunigt. Deshalb wird v := vmax, t := t+t(vmax) sowie P := P-s(t(vmax)) gesetzt und erneut bei Schritt 2 begonnen. Ansonsten wird auf die gleiche Art und Weise für die minimale Geschwindigkeit vmin geprüft, ob einerseits acc < 0 gilt und ferner der Zug von seiner asymptotischen Geschwindigkeit her die Grenze vmin unterbietet (im dritten Fall kann man diesen Test wieder überspringen). Gegebenenfalls werden die obigen Tests mit vmin an Stelle von vmax durchgeführt.

Schritt 7:

Einsatz der Nullstellenverfahren. Wird dieser Schritt erreicht, wird der Zug ohne Wechsel der Zugkraftparabel den Abschnitt durchfahren, wobei am Ende ggfs. noch auf die nächste Höchstgeschwindigkeit r abgebremst werden muss.

In a parameter v, the current speed is always stored and this value is initialized with the predetermined break-in speed v ₀ . Assuming that a link of length P is given and that r denotes the valid maximum speed on the next section, the exit speed (which is stored later in v) and the required travel time t are calculated as follows:

Step 1:

Initialization. The travel time consists of the required transit time for different parts of the route section, which are summed up in t. Therefore, t: = 0 is initially set.

Step 2:

Calculation of the parameters a, b and c of the differential equation and the current acceleration acc: = av ² + bv + c. If v is exactly on the boundary between two tractive force ranges, the respective parameters as well as the associated accelerations are calculated for both ranges and a choice is made between both possibilities. If the acceleration is still positive in the higher speed range, this is selected. If only the other acceleration in the lower range is positive, then the train does not slow down, but not faster, so it continues its motion at a constant speed, which means acc: = 0. Finally, if both accelerations are negative, the train can do its Do not hold the current speed and use the calculated values for the lower speed range.

Step 3:

Security checks. If the train is traveling at the maximum permitted train or line speed, acc = 0 is set. If v = 0 and acc # 0, the calculation ends (the train stops).

Step 4:

Test for steady ride. If acc = 0, then t: = t + P / v is set if the maximum permissible speed r of the next leg is not less than v. Otherwise, the braking process must be taken into account, due to the assignment $$t:=t+\frac{v-r}{acc}+\frac{\ell }{v}-\frac{v^2-r^2}{2v\cdot acc}$$

is taken into account. Then the calculation is finished.

Step 5:

Case distinction. It is checked whether b ^{2 is} greater than, equal to, or smaller than 4ac and uses the corresponding formulas for v (t), t (v), etc. in subsequent steps.

Step 6:

Monitoring the speed. If vmax denotes the maximum speed (limited inter alia by the current traction range) for which the constants a, b and c are still valid, and acc> 0, it is tested whether the validity of the formulas is limited by the maximum speed being reached. For this purpose it is checked whether the asymptotic speed of the train exceeds vmax. (This is not necessary in the third case, b ² <4ac, since the train then always becomes infinitely fast and thus outstrips any speed limit.) If this is the case, the train will reach the maximum speed after t (vmax) seconds. At this time he has already traveled s: = s (t (vmax)) meter distance. If vmax is greater than r, the train must brake later, and therefore s is increased by the braking distance (v ² max - r ² ) / (2acc). If now s ≤ P holds, it is clear that the train within the current section accelerates to vmax meters per second. Therefore, set v: = vmax, t: = t + t (vmax) and P: = Ps (t (vmax)) and start again at step 2. Otherwise it is checked in the same way for the minimum speed vmin whether on the one hand acc <0 and on the other hand the train undercuts the limit vmin from its asymptotic speed (in the third case one can skip this test again). Optionally, the above tests are performed with vmin instead of vmax.

Step 7:

Use of the zeroing method. If this step is reached, the train will pass through the section without changing the traction parabola, whereby at the end it may be necessary to decelerate to the next maximum speed r.

7 advantages of the invention over the prior art

• Von der Genauigkeit her weichen die Fahrzeiten und Zuggeschwindigkeiten, wenn überhaupt, meist nur minimal (etwa um ein Promille) voneinander ab. Die größte beobachtete Zeitabweichung betrug etwa sechs Promille.
• Grund für die Abweichungen sind die Beschleunigungsphasen, bei denen die bisherige Fahrzeitrechnung auf eine Approximation zurückgreift. Die größten Ungenauigkeiten ergeben sich beim Anfahren aus dem Stillstand heraus, hier treten Abweichungen von 20-30 % auf. Diese treten in dieser Größenordnung aber nur während der ersten Sekunden auf (bis die Züge also etwas Fahrt aufgenommen hatten), so dass die resultierenden Abweichungen von z.B. einer halben Sekunde für die Gesamtfahrzeit unwesentlich sind.
• Zumindest theoretisch scheint es aber denkbar, eine Zugfahrt zu konstruieren, bei der sich die Approximationsfehler erheblich auswirken. Man denke z.B. an einen Stop-and-Go-Verkehr, bei dem die Züge häufig wiederanfahren müssen, oder an einen "passend gewählten" Steigungsabschnitt, bei dem es der Zug bei exakter Rechnung gerade noch schafft, über den Berg zu kommen, bei einem eigentlich nur geringem Approximationsfehler aber vorher stehen bleibt.
• Der signifikante Unterschied zwischen beiden Fahrzeitrechnungen ist die Ausführungsgeschwindigkeit. Die geschlossenen Formeln und eine effziente Algorithmik für die Vorverarbeitung ergeben in der Praxis ein um etwa 70-mal schnelleres Verfahren. Dies bedeutet, dass sich alle wenige 100 Meter die Neigung bzw. Höchstgeschwindigkeit ändert oder ein Messpunkt auftaucht. Falls die Abschnitte mit konstanten Bedingungen kürzer werden, schrumpft auch der Geschwindigkeitsvorteil zusammen, da selbst mit den geschlossenen Formeln bei jeder Änderung der Rahmenbedingungen ein weiterer Rechenschritt notwendig wird. Umgekehrt kann man sich auch den Extremfall vorstellen, dass z.B. ein Zug auf einer langen Strecke mit konstanter Steigung beschleunigt, ohne die erlaubte Höchstgeschwindigkeit zu erreichen. In diesem Fall endet die Beschleunigungsphase nicht, und die resultierende Fahrzeit kann mit nur einem Schritt ermittelt werden (unabhängig davon, wie lang die Strecke ist!). Die herkömmliche Approximation kann die Besonderheit dieser Situation dagegen nicht ausnutzen. Der Geschwindigkeitsvorteil kann also theoretisch in beide Richtungen hin variieren.

A comparison of the driving time calculation for the operating centers, short FZR-BZ, version 1.4, the prior art and the inventive method shows:

• In terms of accuracy, travel times and train speeds differ, if at all, only minimally (by about one thousandth of one) from one another. The largest observed time deviation was about six per thousand.
• Reason for the deviations are the acceleration phases, in which the previous travel time calculation uses an approximation. The greatest inaccuracies arise when starting from a standstill, here occur deviations of 20-30%. However, these occur on this scale only during the first few seconds (until the trains had taken some drive), so that the resulting deviations of eg half a second for the total driving time are insignificant.
• At least theoretically, however, it seems conceivable to construct a train run in which the approximation errors have a significant effect. Consider, for example, a stop-and-go traffic, in which the trains have to restart frequently, or to a "suitably chosen" slope section, in which the train with exact calculation just manages to get over the mountain, at one actually only a small approximation error but stops earlier.
• The significant difference between both travel time calculations is the execution speed. The closed formulas and efficient algorithms for preprocessing result in a process that is about 70 times faster. This means that every few 100 meters the inclination or maximum speed changes or a measuring point appears. If the sections with constant conditions become shorter, the speed advantage also shrinks, since even with the closed formulas, a further computation step becomes necessary whenever the boundary conditions change. Conversely, one can also imagine the extreme case that, for example, a train accelerates on a long distance with a constant gradient without reaching the maximum speed allowed. In this case, the acceleration phase does not end, and the resulting travel time can be determined with just one step (regardless of how long the route is!). By contrast, the conventional approximation can not exploit the peculiarity of this situation. The speed advantage can theoretically vary in both directions.

For example, one millisecond computing time per preprocessing and actual travel time calculation can be estimated on a 500 MHz Intel Pentium 111 computer as a guideline value per 100 km distance.

9 Hyperbolic acceleration

gesucht. Selbst in diesem Fall lassen sich geschlossene Formeln ermitteln.As shown above, the differential equation yields v '(t) = ² av (t) + bv (t) + c from the fact that all forces acting on the train depend quadratically on the instantaneous velocity v. This is particularly true for the tractive force, which is actually hyperbolic, ie proportional to the reciprocal of the speed (apart from low speeds, where the static friction values between wheel and rail limit the maximum traction). Therefore, a hyperbolic component in the tensile force and thus in the differential equation is used, ie it becomes a solution of the equation $${\mathrm{v}}^{\text{'}}\left(\mathrm{t}\right)=\mathrm{a}{\mathrm{v}}^2\left(\mathrm{t}\right)+\mathrm{b}\mathrm{v}\left(\mathrm{t}\right)+\mathrm{c}+\mathrm{d}/\mathrm{v}$$

searched. Even in this case, closed formulas can be determined.

The solution of the differential equation leads analogously to the above-indicated path to the integral $$t\left(v\right)={\displaystyle \int \frac{dv}{a{v}^2+bv+c+d/v}}={\displaystyle \int \frac{vdv}{a{v}^3+b{v}^2+cv+d}}$$

According to the invention, the zeros of the cubic polynomial in the denominator are first determined and, based on these solutions, the resulting parent functions are determined.

Case 1: There exists a triple zero x ₁ . Then: $${\displaystyle \int \frac{vdv}{{\left(v-x_1\right)}^3}}=\frac{x_1-2v}{2{\left(v-x_1\right)}^2}$$

Case 2: There exists a simple zero x ₁ and a double zero x ₂ . Then: $${\displaystyle \int \frac{vdv}{\left(v-x_1\right){\left(v-x_2\right)}^2}}=\frac{x_1}{{\left(x_1-x_2\right)}^2}\ln \left|\frac{v-x_1}{v-x_2}\right|+\frac{x_2}{\left(x_1-x_2\right)\left(v-x_2\right)}$$

Case 3: There are three different zeros x ₁ , x ₂ and x ₃ . Then: $${\displaystyle \int \frac{vdv}{\left(v-x_1\right)\left(v-x_2\right)\left(v-x_3\right)}}=\frac{x_1\ln \left|v-x_1\right|}{\left(x_1-x_2\right)\left(x_1-x_3\right)}+\frac{x_2\ln \left|v-x_2\right|}{\left(x_2-x_1\right)\left(x_2-x_3\right)}+\frac{x_3\ln \left|v-x_3\right|}{\left(x_3-x_1\right)\left(x_3-x_2\right)}$$

Case 4: There is only one simple zero x ₁ ; the remaining polynomial v ² + pv + q can not be factored. Then: $${\displaystyle \int \frac{vdv}{\left(v-x_1\right)\left(v^2+pv+q\right)}}=\frac{1}{x_1^2+p{x}_1+q}\left(x_1\ln \left|\frac{v-x_1}{\sqrt{v^2+pv+q}}\right|+\frac{2q+x_{1}p}{\sqrt{4q-p^2}}\arctan \left(\frac{2v+p}{\sqrt{4q-p^2}}\right)\right)$$

However, these terms can not be reversed (except for the first case), ie in general no closed formula for v (t) can be found and also no formula for s (t) can be formed. However, a closed formula for s (v) can be derived, since: $$s={\displaystyle \int vdt}={\displaystyle \int v\frac{dt}{dv}dv}={\displaystyle \int v\cdot t^{\text{'}}\left(v\right)dv}={\displaystyle \int \frac{v^{2}dv}{a{v}^3+b{v}^2+cv+d}}$$

For the remaining integral, different solutions can be given according to the above explanations depending on the zeros of the denominator polynomial.

8 application examples 8.1 Energy-saving driving style

The calculation of travel times is usually, as mentioned above, based on a strict driving style, i. The trains accelerate as early as possible, drive if possible at top speed and slow down as late as possible. Since the travel times are provided with rule surcharges, you can - if no delays are catch up - use these time buffers for energy-saving driving (ESF), i. E. you let the train roll out in a targeted manner, thus saving the traction energy, and is nevertheless punctually at the finish.

It seems to make sense to roll out, especially if the train later has to reduce the speed anyway or even stop at a signal or train station. In addition to these situations, DB Systems' ESF system for ICE trains also uses downhill sections as coasting sections. The problem with this is, however, that all associated timetable and route data must be recorded before the train ride in the on-board computer. This is partly due to the previously too slow travel time calculation.

With the aid of the much faster travel time calculation according to the invention, it is possible to overcome this problem. In addition, it is not clear from the outset why, for example, it should not be possible to roll out on inclines as long as only the control calculation shows that the momentum will be sufficient. Here we find an even better energy saving, even on non-ICE trains, to be possible. Considering the enormous amounts of energy and costs spent on rail travel each year, there is still considerable potential here.

8.2 Energy-optimized timetables

With the ideas from the last section, it makes sense to also measure timetables on the energy efficiency of train journeys and to optimize them accordingly. Nowadays the calculations are usually based on a tight driving style. As seen above, however, it makes no sense, e.g. with 160 km / h on a slow driving open. Only a slight loss of time is offset here by a significant energy savings, so that a corresponding timetable also pays off in monetary terms, apart from the lower wear on the brakes, etc. quite apart.

8.3 Calculation of construction surcharges

Construction surcharges are time buffers that are added to the journey time in the event of anticipated obstruction due to construction site work (for example, single track operation) in addition to the standard surcharges. The amount of construction surcharges, however, is based only on the expected loss of time of long-distance trains.

In a first step, the driving time calculation determines the effects of slow driving points etc. on individual trains and thus evaluates a given construction surcharge. The next step is to automatically determine construction surcharges that are dependent on the train and, if necessary, taken into account directly when publishing timetables.

8.4 Real-time Disposition

Rolling trains "skilfully" is not only useful for energy-saving driving styles, but can also shorten the driving time in certain cases. If a train runs in particular on a (still) stop-pointing main signal, it must decelerate to a standstill and then accelerate again from standstill. However, it is faster and also far more energy-efficient to fill in the time until the signal is switched up by energy-saving driving, in order then to pass the signal at a (high) residual speed. The re-acceleration phase is correspondingly shorter.

To calculate the appropriate time from which the roll-out has to take place, data from the control and safety technology are collected, which are essential for the current knowledge of the signal positions. With a corresponding read access, there are corresponding possibilities to optimize the train paths not only before, but even during operation, at the same time to shorten travel times while still saving energy and thus money.

Claims (6)

A method for determining a required travel time of a train when covering a distance with a given slope and maximum permissible speed, characterized in that one of an acceleration phase underlying differential equation v '(t) = av ² (t) + bv (t) + c with one of a Time t dependent speed v (t) and parameters a, b, c is solved by closed formulas and these closed formulas are used in addition to the well-known computational rules for trains with constant speed or constant deceleration to determine the travel time as follows: in the case of positive acceleration v '(t)> 0, it is checked whether there is enough distance available to reach the maximum speed and if there is enough distance, the time and the path required for the acceleration phase are expressed by closed formulas and continue at constant top speed, and otherwise the closed formulas for the time and the way to get to a certain speed will be used to, by a numerical method, in particular the Newton's method, the exit speed and time from the section determine, - If the acceleration is negative v '(t) <0, it is checked whether the train may come to a stop beforehand; if necessary, the time of the standstill and the distance traveled are determined, otherwise the exit speed and the required travel time are determined by the closed formulas . if steady-state travel v '(t) = 0, a break-in velocity v _{0 is} maintained, - In principle, a possibly predetermined limitation of a discharge speeds is taken into account by a timely initiated braking process. A method for determining a required travel time of a train according to claim 1, characterized in that for determining a required acceleration energy of a train an energy consumption of the train underlying product of the applied power of drive units and the path traveled as integral $${\displaystyle \int \mathrm{F}\left(\mathrm{v}\right)\left(\mathrm{d}\mathrm{s}/\mathrm{d}\mathrm{t}\right)\left(\mathrm{d}\mathrm{t}/\mathrm{d}\mathrm{v}\right)\mathrm{d}\mathrm{v}}={\displaystyle \int \left(\mathrm{F}\left(\mathrm{v}\right)\mathrm{v}/\left(\mathrm{a}{\mathrm{v}}^2+\mathrm{b}\mathrm{v}+\mathrm{c}\right)\right)\mathrm{d}\mathrm{v}}={\displaystyle \int \left({\mathrm{q}}_2{\mathrm{v}}^2+{\mathrm{q}}_1\mathrm{v}+{\mathrm{q}}_0\right)\mathrm{v}/\left(\mathrm{a}{\mathrm{v}}^2+\mathrm{b}\mathrm{v}+\mathrm{c}\right)\mathrm{d}\mathrm{v}}=\left({\mathrm{q}}_2\mathrm{v}/2\mathrm{a}+\left({\mathrm{q}}_1\mathrm{a}-{\mathrm{q}}_2\mathrm{b}\right)/{\mathrm{a}}^2\right)\mathrm{v}+\mathrm{A}\ln \left(\mathrm{a}{\mathrm{v}}^2+\mathrm{b}\mathrm{v}+\mathrm{c}\right)+\left(\mathrm{B}-\mathrm{A}\mathrm{b}/2\mathrm{a}\right)\mathrm{t}\left(\mathrm{v}\right)+\mathrm{C}$$

formulated and solved, where a, b and c are the coefficients of the total acceleration, ie the traction acceleration minus drag acceleration, q ₀ , q ₁ and q _{2 are} the coefficients of traction force and A and B are the coefficients of the residual polynomial resulting from the polynomial division for (q ₂ v ² + q ₁ v + q ₀ ) v / (av ² + bv + c), and t (v) is the time to accelerate from zero to v. Method for determining a required travel time of a train according to one or more of claims 1 and 2, characterized in that routes with changing maximum speeds and / or varying inclinations divided into sections with constant maximum speeds and / or constant inclinations in a preprocessing and respective travel times or Traction energies are cumulative to an overall result, ie, the exit speeds, the travel times and the traction energies are calculated for the individual sections successively at predetermined entry speeds and added the travel times and the traction energies and considered railway operating rules and peculiarities. Method for determining a required travel time of a train according to one or more of claims 1 to 3, characterized in that all the tensile and resistive forces acting on the acceleration by preprocessing to a differential equation of the form v '(t) = av ² (t) + bv (t) + c, the tractive force of a traction vehicle is given by a piecewise quadratic polynomial as a function of the velocity v, the total resistance force is given by a quadratic polynomial and the acceleration is divided by the difference between the traction force and the total resistance force Mass of the train is determined and the total resistance is determined from the quadratic velocity dependent air resistance, the rolling friction resistance given by a linear function and the pitch resistance which is constant. Method for determining a required traveling time of a train according to one of Claims 1 to 4, characterized in that a plurality of differential equations for the acceleration phase which are valid depending on the speed range are taken into account by a corresponding catalog of closed formulas, wherein during the determination of the acceleration valid formula set is used: - in the case that b ² > 4ac, D = √ (b ² - 4ac) and k = 1 - 2D / (2av ₀ + b + D), so that t (v) = (1 / D) In | ((1 - 2D / (2av + b + D)) / k) | , v (t) = (2D / (1 + ke ^Dt ) -b -D) / 2a and s (t) = (In | (1 + k) / (1 + ke ^Dt ) | + (Db) t / 2) / a, in the case that b ² <4ac D = √ (4ac - b ² ) is set, so that t (v) = arctan (av / D + b / 2D) + C with a constant, v (t) = (D (t (t + k) - b / 2) / a and s (t) = (In | cos (Dk) / cos (D (t + k) | bt / 2) / a, and v ( t) the time at time t and s (t) is the distance traveled at time t. A method for determining a required travel time of a train according to one of claims 1 to 5, characterized in that the total or partial disregard of the tensile forces of one or more of the trains involved in the train a driving time is based, which is not a technically shortest achievable travel time represents and when multiple traction vehicles are present in a train, it is specified which traction vehicle is active and only the tensile forces of the active traction vehicles are added.