WO2012119197A1

WO2012119197A1 - Improving timekeeping and energy efficiency for trains

Info

Publication number: WO2012119197A1
Application number: PCT/AU2012/000232
Authority: WO
Inventors: Peter John Pudney; Philip George Howlett; Amie Renee ALBRECHT
Original assignee: Ausrail Technologies Pty Limited
Priority date: 2011-03-08
Filing date: 2012-03-08
Publication date: 2012-09-13

Abstract

This invention relates to a method and system for the operation of trains on a rail network. The invention provides a method and system which monitors the progress of a train on a network, calculates efficient control profiles for the train, and displays driving advice to the train crew. The system calculates and provides driving advice that assists to keep the train on time and reduce the energy used by the train by: (i) Monitoring the progress of a journey to determine the current location and speed of the train; (ii) Estimating some parameters of a train performance model; (iii) Calculating or selecting an energy-efficient driving strategy that will get the train to the next key location as close as possible to the desired time; and (iv) Generating and providing driving advice for the driver.

Description

IMPROVING TIMEKEEPING AND ENERGY EFFICIENCY FOR TRAINS FIELD OF THE INVENTION

[0001 ] This invention relates to a method and system for the operation of trains on a rail network.

BACKGROUND OF THE INVENTION

[0002] Any discussion of the prior art throughout the specification should in no way be considered as an admission that such prior art is widely known or forms part of common general knowledge in the field.

[0003] The energy costs of operating trains on a rail network are significant. By driving efficiently, these costs can be significantly reduced.

[0004] Fuel or energy savings of 10% or greater can be achieved by providing train drivers with information on energy-efficient driving strategies.

[0005] The present applicant has developed a method and system employing optimal control theory to calculate a control strategy which minimizes the mechanical work done by the traction system to move a train from its current location and speed to the next scheduled stop. This method and system is disclosed in International Patent Application PCT/AU03/00604, the contents of which are incorporated herein by way of cross-reference.

[0006] The method and system disclosed in PCT/AU03/00604 provides an in-cab system that displays driving advice to help train drivers improve timekeeping and reduce energy consumption. It uses optimal control theory to calculate a control strategy which minimizes the mechanical work done by the traction system to move a train from its current location and speed to the next scheduled stop. The calculation of the optimal control strategy takes into account the required arrival time at the stop as well as train performance parameters and track gradients, curves and speed limits. [0007] However, the method does not calculate optimal driving strategies for situations where the train must pass an intermediate timing location, without stopping, at a specified time.

[0008] It is an object of the present invention to provide a method for operating trains on a rail network which overcomes or ameliorates at least one of the

disadvantages of the prior art, or at least provides a useful alternative.

SUMMARY OF THE INVENTION

[0009] According to a first aspect of the present invention, there is provided a method of monitoring the progress of a train on a rail network and providing driving advice in real time to an operator of said train, said method comprising:

(i) estimating or determining parameters of said train;

(ii) determining, by an optimal control algorithm employing an adjoint variable, an optimal journey profile for a journey from said train's current location to one or more target locations that results in said train arriving at said target locations as close as possible to the desired times and with minimum energy usage; said optimal journey profile including a speed profile for the train, sequence of discrete control modes for said train, and associated switching points between said control modes; said optimal journey profile being determined by solving a system of differential equations for said speed profile of the train and for the value of said adjoint variable, said control modes being determined from the value of said adjoint variable, such that said sequence of control modes is determined as said speed profile is calculated;

(iii) monitoring the current state of said train as it progresses to said target locations; and

(iv) generating said driving advice for the train operator by comparing the current state of the train to a corresponding state on said optimal journey profile and displaying said advice for the train operator that will keep the train close to said optimal journey profile.

[0010] According to a second aspect of the present invention there is provided a method of monitoring the progress of a train on a rail network and providing information on the progress of the train in real time to an operator of said train, said method comprising:

(i) estimating or determining parameters of said train;

(iv) generating said information for the train operator by comparing the current state of the train to a corresponding state on said optimal journey profile and displaying said information for the train operator to assist in keeping the train close to said optimal journey profile.

[001 1 ] According to a third aspect of the present invention, there is provided a method of controlling the progress of a train on a rail network, said method

comprising:

(i) estimating or determining parameters of said train;

(ii) determining, by an optimal control algorithm employing an adjoint variable, an optimal journey profile for a journey from said train's current location to one or more target locations that results in said train arriving at said target locations as close as possible to the desired times and with minimum energy usage; said optimal journey profile including a speed profile for the train, sequence of discrete control modes for said train, and associated switching points between said control modes; said optimal journey profile being determined by solving a system of differential equations for said speed profile of the train and for the value of said adjoint variable, said control modes being determined from the value of said adjoint - ⁱn variable, such that said sequence of control modes is determined as said speed profile is calculated;

(iv) comparing the current state of the train to a corresponding state on said optimal journey profile and then controlling said train to keep the train close to said optimal journey profile.

[0012] Preferably, said adjoint variable evolves according to a differential equation along with the position and speed of the train.

[0013] Preferably, the value of the adjoint variable is calculated directly from the speed of the train.

[0014] Preferably, a numerical method is used to solve a system of differential equations for said speed profile of the train and for the value of said adjoint variable.

[0015] Preferably, steps (i) to (iv) are performed as required so that said driving advice automatically adjusts to compensate for any operational disturbances encountered by the train.

[0016] Preferably, said discrete control modes for said train include drive, hold, coast and brake modes.

[0017] Preferably, said parameters include train mass and mass distribution.

[0018] Preferably, said parameters further include maximum tractive effort and maximum braking effort as functions of speed.

[0019] Preferably, said parameters further include coefficient(s) of rolling resistance.

[0020] Preferably, said driving advice is generated and displayed by a computer located on the train. [0021 ] Preferably, step (iii) involves processing data from a GPS unit and train controls to determine the location and speed of the train.

[0022] Preferably, said optimal journey profile is precomputed.

[0023] Preferably, a plurality optimal journey profiles corresponding to different journey times are calculated and the profile that has an arrival time at the target location closest to the desired arrival time is selected.

[0024] In a particularly preferred embodiment of the invention the optimal journey profile comprises driving in a hold mode (i.e. at constant speed), calculated by the Pontryagin Principle of optimal control, wherever possible and where it is not possible changing as quickly as is safely possible at exactly the right location, calculated by the Pontryagin Principle of optimal control, to drive (i.e. full power), hold, coast or brake modes as necessary.

[0025] Advantageously the present invention provides an optimal driving strategy in the case where a train must pass an intermediate timing location at a specified time. The method of the present invention can also be used on problems with more than one intermediate timing point.

[0026] Unless the context clearly requires otherwise, throughout the description and the claims, the words "comprise", "comprising", and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is to say, in the sense of "including, but not limited to".

BRIEF DECRIPTION OF THE DRAWINGS

[0027] The invention will now be described in further detail, by way of example only, with reference to the accompanying drawings in which:

[0028] Figure 1 shows a block diagram of the system according to a preferred embodiment of the present invention, illustrating the main data flows between various elements of the system;

[0029] Figure 2 illustrates an optimal speed profile for a train over a fictitious section of track; [0030] Figure 3 illustrates an optimal speed profile for a train over another fictitious section of track;

[0031 ] Figure 4 illustrates an optimal journey for a coal train;

[0032] Figure 5 shows the processing of precomputed speed profiles;

[0033] Figure 6 illustrates a preferred embodiment of the system display which provides the train operator with driving advice in real time.

[0034] Figure 7 illustrates speed graphs for two optimal journeys;

[0035] Figure 8 illustrates speed graphs l/, and l/₂; and

[0036] Figure 9 illustrates journey energy as a function of hold speed change location.

DETAILED DESCRIPTION

[0037] The present invention, in one preferred form, provides a fully automatic system that monitors the progress of a train on a long-haul network, calculates efficient control profiles for the train, and displays driving advice to the train crew. In a further preferred embodiment the system works in conjunction with a dynamic rescheduling tool that coordinates interactions between various trains operating on the network.

[0038] The system assists the crew of a long-haul train by calculating and providing driving advice that assists to keep the train on time and reduce the energy used by the train. The system performs four main tasks:

(i) state estimation: monitors the progress of a journey to determine the current location and speed of the train;

(ii) train parameter estimation: estimates some parameters of a train

performance model;

(iii) journey optimisation: calculates or selects an energy-efficient driving strategy that will get the train to the next key location as close as possible to the desired time; and

(iv) advice generation: generates and provides driving advice for the driver. [0039] These tasks are performed continually so that the driving advice automatically adjusts to compensate for any operational disturbances encountered by the train.

[0040] The system includes:

• data communications between on-board units and a central control

system;

• automatic estimation of train performance parameters;

• automatic re-optimisation of optimal journey profiles;

• interaction with a manual or automatic train rescheduling system;

• ergonomic driver interfaces.

[0041 ] Each of these four aspects of the methodology and system will now be discussed in further detail:

State estimation

[0042] The station estimation task processes observations from a GPS unit and the train controls to determine the location and speed of the train and the current control setting.

[0043] Location is the position of the train on a given route, and is used to look up track gradient, curvature and speed limits. The state estimation task uses absolute and relative position data to determine the location of the train.

[0044] Control setting is required for train parameter estimation, and for estimating the energy use of the train if direct measurement of energy use is not available.

Train parameter estimation

[0045] The train parameter estimation task estimates parameters of a train performance model from the sequence of observed journey states.

[0046] The train model used by the in-cab system has the following train parameters:

• train mass and mass distribution;

• maximum tractive effort and maximum braking effort as functions of

speed; and coefficients of rolling resistance.

[0047] Any of these parameters that are not known with sufficient accuracy before the journey commences must be estimated during the journey. The unknown parameters can be estimated using a Kalman filter.

[0048] If mass is to be estimated, the mass distribution is assumed to be uniform. If tractive effort is to be estimated it is assumed to take the form

where P is the maximum power of the train and v₀ is the speed below which maximum tractive effort is assumed to be constant.

[0049] In the simplest implementation, all train model parameters are known in advance and parameter estimation is not required.

Journey Optimisation

[0050] The optimal journey profile between a given journey state and a target journey state is found by solving a set of differential equations for the motion of the train and an additional differential equation that determines the optimal control. The optimal journey profile specifies the time, speed and control at each location of the track between the current train location and the next target.

[0051 ] Journey profiles can be precomputed or else calculated during the journey. If precomputed, several different journeys corresponding to different journey times are used on the train and the journey optimisation task then simply selects the precomputed profile that has the arrival time at the target closest to the desired arrival time.

[0052] If we use distance travelled, x, as the independent variable then the journey trajectory is described by the state equations dt

1/ v (1)

dx

dv u - R(v) + G(x)

(2)

dx mv

dJ

U₊ + 77_RU (3)

dx where t is elapsed time, v is the speed of the train, J is energy use, u is the controlled driving or braking force, R(v) is the resistive force on the train at speed i and G(x) is force on the train due to track gradient and curvature at location x, and m is the mass of the train. We assume that R and the derivative R' are both increasing functions.

[0053] This model is based on simple physics. It does not model the complexities of traction motors, braking systems, in-train forces or wheel-rail interations. Nor does it need to; in practice, the driving advice derived from this simple model is both realistic and effective.

[0054] The state equations describe the motion of a point mass. In practice the length of a long-haul train can be significant. However, a long train can be treated as a point mass by transforming the track force function. Suppose the train has length L and that the density of the train at distance /from the front of the train is p(l). If we define

G(x) = f (l)G(x- l)dl

1=0 where G is the real track force then the motion of a point mass train on a track with track force G is equivalent to the motion of the long train on the real track.

[0055] The force u is controlled by the driver, and satisfies the constraints F_B(v)≤ u ≤ F_D(v) where F_D(v) > 0 is the maximum drive force that can be achieved at speed i and F_B(v) > 0 is the maximum braking force that can be achieved at speed v.

[0056] For most train journeys the speed of the train is constrained by speed limits that depend on location, and so the optimal journey must satisfy the constraint v≤ V_L(x).

[0057] The optimal control is founded by forming the Hamiltonian function ₁ u - R(v) + G(x)

1 , v: + 7Γ , — — + π₃ [ ₊ + 77_Ru

mv

«B [^PB (v) - u] - a_D [u - F_D (v)] - a_v [v where π_ί are multipliers associated with the state equations and a_i are Lagrange multipliers associated with the control and speed constraints. The complementary slackness conditions are oc [F ( ) -u] = a_D[u - F_D(v)] = a_v[v-V_L(x)] = 0

[0058] There are three adjoint equations. The first and third adjoint equations are άπ_ι άπ₃

— = 0 and — - dx dx

If we let π₃ = -1 and

μ

mv then the second adjoint equation can be written as

1 ΐ ^πι

mv + μΚ (v) + «_v + (1 - μ)¥ '_D (v)] u = F_D (v)

F_B (v) < u < F_D (v) (4)

^[^ ₊ R'(v) + «_v + (_¾ - )F' u = F_B (v)

[0059] This equation is found by substituting each of the three control conditions into the Hamiltonian and then differentiating. The Lagrange multiplier a_v is zero when the train is travelling at a speed less than the speed limit.

[0060] The optimal control maximises the Hamiltonian, and so the optimal control depends on the value of the adjoint variable μ. An optimal strategy has five possible control modes: drive 1 < μ maximum drive force u = F_D(v) hold μ = 1 speed hold with 0 < u < F_D(v) coast η_Η < μ < 1 coast with u = 0 regen speed hold with F_B(v) < u < 0 brake brake with u = F_B(v)

[0061 ] The hold mode is singular. For this driving mode to be maintained on a non- trivial interval requires άμ/άχ = 0. If we are not constrained by a speed limit then we have v²R'(v) = -^-₁

[0062] But 7ti is a constant and the graph y =

is strictly increasing, so there is a unique hold speed V satisfying this equation.

[0063] Maintaining a speed limit also requires μ = 1 . When a speed limit is encountered the adjoint variable // jumps to μ = 1 and at the same time the Lagrange multiplier a_v jumps from zero to a positive value.

[0064] On a track with sufficiently small gradients and no speed limits the optimal trajectory is mainly speed holding at speed V. On most tracks, however, the track gradients disrupt this simple strategy. Track intervals can be divided into four speed- dependent classes:

(i) steep incline: if the maximum drive force is not sufficient to maintain the desired speed;

(ii) not steep: if the desired speed can be maintained using a non -negative drive force;

(iii) steep decline: if braking is required to maintain the desired speed; and

(iv) nasty decline: if even maximum brake force is insufficient to maintain the

desired speed. [0065] The optimal strategy anticipates steep gradients by speeding up before a steep incline and slowing down before a steep decline.

[0066] An optimal trajectory with a given hold speed l/can be found by setting m = VR'(V) and then solving the differential equations (1 ) and (2) while using (4) and the optimal control modes to determine the control. These differential equations are solved using a numerical method such as a Runge-Kutta method. In practice, however, the adjoint equation is unstable. To overcome this difficulty we instead search for a pair of adjacent adjoint trajectories that are lower and upper bounds for the true adjoint trajectory. The lower and upper bounds start close together, but the adjoint values eventually diverge. This does not matter while they are both indicating the same control mode, but as soon as one of the bounds indicates a control change we re-search at that location to find new adjacent bounds that extend the journey.

[0067] The optimal journey trajectory can be constructed in this way as a sequence of trajectory segments between speed-holding phases, where speed holding can occur at the hold speed l/or at a speed limit.

[0068] There are two ways a non-holding optimal trajectory segment can start:

1 . Drive or coast with (x₀, v₀) known and μ₀ unknown. This occurs at the beginning of the journey or at the end of a low speed limit. Calculating an initial upper bound for μ is not usually possible, so instead we search for the location of the next control change.

2. Drive or coast with x₀ unknown but bounded, v₀ known and μ₀ = 1 . This may occur if we are holding at the hold speed or at a speed limit. The lower bound for x₀ is the start of the hold phase. The upper bound for x₀ depends on whether we are holding at the hold speed l or at a speed limit. If we are holding at the hold speed l/then the upper bound for x₀ is the next location where either the track becomes steep or else the speed limit drops below V. If we are holding at a speed limit V_L then the upper bound for x₀ is the next location where either the track becomes steep uphill or else the speed limit drops. If a steep decline is encountered during a speed limit phase then the brakes must be partially applied to hold the train at the speed limit.

[0069] There are three ways a non-holding optimal trajectory segment can finish:

1 . At the end of the journey, with the correct speed.

2. At the hold speed with v = V, μ = 1 and the gradient not steep. The next

trajectory segment will have start type 1 .

3. At a speed limit with v = V_L. The next trajectory segment will have start type 2 with control coast, or else start type 1 with control drive.

[0070] Using these conditions, it is possible to construct a complete journey profile to the next target. This journey profile will be optimal for the resulting arrival time at the target. If the resulting arrival time is beyond the desired arrival time then another journey profile, with a higher hold speed, is calculated; if the arrival time at the target is prior to the desired arrival time then another journey profile is calculated, this time with a lower hold speed. A numerical technique such as Brent's method can be used to find the hold speed that gives the desired arrival time.

[0071 ] As a result, the optimal journey profile comprises driving in a hold mode (i.e. at constant speed), calculated by the Pontryagin Principle of optimal control, wherever possible and where it is not possible changing as quickly as is safely possible at exactly the right location, again calculated by the Pontryagin Principle of optimal control, to drive (i.e. full power), coast or brake modes as necessary.

Advice Generation

[0072] The advice generation task compares the current state of the train to the corresponding state on the optimal journey profile and then generates and displays advice for the train operator that will keep the train close to the optimal profile.

[0073] Brake advice is given if braking is required to avoid exceeding a speed limit or a speed on the journey profile that has braking as the optimal control. [0074] Coast advice is given if:

• the speed of the train is significantly higher than the speed indicated by the optimal journey profile, or

• the speed of the train is near or above the speed indicated by the optimal journey profile and the optimal control is coast.

[0075] Hold advice is given if the speed of the train is near or above a holding speed indicated by the optimal journey profile. The speed to be held will be either a speed limit or the journey holding speed.

[0076] Power advice is given if none of the other driving modes are appropriate.

[0077] These decisions can be made without considering time because the optimal speed profile is automatically adjusted by the journey optimisation task to keep the train on time.

[0078] For each type of trip, the optimisation software is used to calculate optimal speed profiles for six difference total journey times. Each profile is designed to minimise fuel consumption for the given journey time. As the time allowed for the journey decreases the minimum possible fuel consumption increases.

[0079] During the journey the system uses a GPS unit to determine the position of the train. Given the speed and position of the train and the time remaining until the train is due at the next key location, the system selects the most appropriate of the pre- computed profiles. Advice is generated to keep the train as close as possible to the selected profile. The crew will enter necessary information such as the arrival time at the next key location. The advice given to the driver will be one of:

• Drive: drive using maximum power, subject to safety and train handling constraints;

• Hold: vary the power to hold the indicated speed; or

• Coast: set the power to zero subject to safety and train handling constraints.

• Brake: apply normal service braking.

[0080] In some cases, brake advice is not displayed and the driver is solely responsible for deciding when to brake. [0081 ] The system is able to work with pre-computed profiles because, in practice, if the control is changed too early or too late, switching between the different pre- computed profiles will automatically adjust future control changes to compensate. Alternatively, the calculations are generally fast enough that new profiles can be computed in real time.

[0082] Energy savings can be achieved simply by demonstrating efficient control techniques to the train operator. Effective techniques can either be demonstrated onboard or by using simulations. However, because of the relationship between fuel consumption and journey time some form of on-board advice system is required to achieve the best possible fuel consumption, and is the reason why coasting boards by the side of the track do not work.

[0083] For example, if a train is running slowly and behind schedule because of a head wind, and the driver coasts at the usual location, the train will end up even further behind schedule. Of course, drivers will take train performance into account, but it is difficult for them to keep track of time and predict the effect their control decisions will have on the final arrival time.

[0084] The system of the present invention achieves significant fuel savings without increasing running times because the system is an adaptive system based on optimal control theory.

[0085] The system can adjust the driving strategy using the actual observed train performance. All systems that rely on pre-computed profiles must take into account the current state of the train with regard to location, time and speed. Any system of non- adaptive control will give unreliable advice when the train is not in the right place at the right time doing the right speed. Whilst non-adaptive systems could possibly be used on Metropolitan railways with fixed timetables and identical trains or on tightly controlled networks with unit trains carrying consistent loads using dedicated track, they are not suitable on networks where the trains are subjected to unpredictable delays.

EXAMPLE

[0086] In the following discussion of an example of the invention, the following notation is used: Train

m train mass (kg)

F_D(v) maximum drive force at speed ^v (N)

F_B(v) minimum brake force at speed ^v (N)

R(v) resistance force at speed ^v (N)

regenerative brake efficiency

Route

The length and mass distribution of a train can be used with a simple averaging procedure to transform the track gradients and speed limits so that the motion of a point mass train on the transformed track corresponds to the motion of the real train on the real track.

^GM effective force due to gradient at distance ^x (N)

h^ effective elevation of the track at ^x (m)

ν(^χ) effective speed limit at ^x (ms-1 )

State variables

x distance along the route (m)

f( ) time taken to reach distance ^x (s)

ν(^χ) speed at distance ^x (ms-1 )

JW energy cost at distance ^x (J)

Control and adjoint variable

u applied drive force ^{0≤ u≤ Fo(y} or brake force ^F^^{v)≤ u < 0} (N) μ an adjoint variable that determines the optimal control switching points

[0087] Steep gradients and speed limits mean that travelling at a constant speed for the entire journey is usually not possible. To find the optimal control for real journeys we use Pontryagin's principle, a standard technique of optimal control theory. The method is described for trains with discrete control in the book by Howlett and Pudney (1995), and for continuous control by Howlett and Khmelnitsky. [0088] The continuous control model is easier to work with, and the results from the two models are practically identical. The optimal control at any stage of the journey depends on the value of an adjoint variable ^μ , which evolves as the journey progresses. There are five control modes in an optimal journey: drive 1 < => u = F_D{v) hold μ = => 0 < u < F_D(v) coast ^R u = 0 regen = VR => F_B{v) < u < 0 brake M < ??R => u = F_B{v)

[0089] By analysing the equations for ^μ we can show that the control mode with ^{μ =} corresponds to speed holding. We can also show that during any one optimal journey, speed holding must always occur at the same speed, ^v .

[0090] w > V . The holding speed V and the regen speed w are related by the simple formula

?7_RW²fl'( W) = V²Ft'(V) .

[0091 ] If regeneration is perfectly efficient then the regen speed is the same as the hold speed, and the coast mode never occurs. If the train does not have regenerative braking then the regen mode does not occur.

[0092] Using the same type of analysis we can show that the control mode with ^{Μ = Η}Η requires the use of regenerative braking to maintain a constant speed

[0093] For a given hold speed V we can divide the track into four classes: • steep inclines, where maximum drive force is not sufficient to hold speed V;

• not steep, where a proportion of the maximum drive force is sufficient to hold speed V;

• steep declines, where braking is required to hold speed V; and

• nasty declines, where full brakes are not enough to hold speed V.

[0094] The key to handling steep grades is to anticipate the grade. For steep inclines, the speed of the train should be increased before the start of the incline; for steep declines, speed should be reduced before the start of the decline. Figure 2 shows an optimal journey segment on a fictitious section of track. The holding speed is

70km/h. The steep sections are each 1 % grades. The optimal journey has the train coasting 2km before the start of the decline, and driving 500m before the start of the incline. The grey curve shows the adjoint variable used to determine the optimal control; it has been scaled and shifted to make it easier to see. For both the drive and the coast phases the adjoint variable starts and finishes at μ = 1.

[0095] Where steep grades are close together the correct switching sequence and switching points are more difficult to find, but they can be calculated using the adjoint equation. In Figure 3 the steep sections are once again 1 % grades. The control is switched from power to coast as the adjoint variable μ passes through μ = 1 , before the top of the hill.

[0096] The same principle can be used to find an optimal speed profile for more complex journeys. Figure 4 shows an optimal journey for a coal train. The hold speed is 70km/h. The elevation profile has been smoothed to compensate for the length and mass distribution of the train.

[0097] This is a particularly difficult journey; there is only one short period of speed holding, indicated by the dark shading at 220km. The lighter shading indicates periods of coasting. The dark shading at the end of the journey indicates braking.

[0098] On long journeys the adjoint variable can be difficult to calculate. The light curves show lower and upper bounds for the adjoint variable. We have to search for a more accurate value whenever the bounds become too far apart, or whenever one bound indicates a control change but the other does not. [0099] The method used to calculate an optimal journey is easily extended to handle speed limits (Pudney & Howlett, 1994; Howlett & Pudney, 1995; Cheng et al, 1999; Khmelntisky). Whenever the speed profile meets a speed limit there is no choice but to apply partial braking to hold the speed of the train at the speed limit. At the point where the speed limit is encountered the value of the adjoint variable jumps by an amount that can be calculated. The optimal journey can be found as before, using the adjoint variable to determine the control and calculating the adjoint jump each time a speed limit is encountered.

[0100] To find the optimal strategy for a given journey time we need to find the appropriate hold speed. Simply dividing the journey time by the journey distance gives an initial guess. In most cases this guess will be an underestimate of the holding speed required; speed limits, gradients and the initial and final phases of a journey tend to reduce the actual average speed.

[0101 ] The time taken for an optimal journey with hold speed ^ decreases as V increases. We simply use a numerical search technique to find the hold speed that gives the correct journey time. As a by-product we generate a sequence of points ^^7"'^ that describe the energy cost of an optimal journey that takes time ^T . These points describe a cost-time curve that can be used for calculating timetables that take into account energy costs.

[0102] It may appear that the speed-holding strategy for long-haul trains is different to the drive-coast-brake strategy for suburban trains, but this is not so. On suburban journeys, the hold speed required to achieve the timetable on short journey sections is usually greater than the maximum speed that can be achieved before coasting and braking are required. The suburban drive-coast-brake strategy is simply a subset of the speed holding strategy used on longer journeys.

[0103] The invention is designed to work on a train with optimisation working as a background task continually updating the optimal speed prof ile from the current state of the journey to the next target.

[0104] Advice is provided from the result of comparing the current state to the optimal journey and generating appropriate control advice. [0105] Figure 5 shows the processing of precomputed speed profiles, and Figure 6 shows a typical advice task provided to a driver.

[0106] Figure 6 shows a preferred embodiment of the driver display 10 for providing real time driving advice to the driver of the train.

[0107] The target location 12 (in this example, "Crystal Brook") is selected by the driver and represents the next destination the train must reach by a certain time. The estimated time of arrival (ETA) 14 is calculated by the system and represents the predicted time the train will reach the target location 12 based on the current location of the train, the distance to the target location, and the selected journey profile (i.e. driving strategy). If the calculated ETA does not satisfy the driver's requirements (i.e. by being too early or too late) the driver can select a "faster" or "slower" journey profile from a series of journey profiles. These profiles may be selected by the driver from a graduated scale 16. In the preferred embodiment depicted the driver has a choice of seven (7) different journey profiles. As may be appreciated, the slower the journey profile the less fuel is used, whilst the faster the journey profile the less fuel-efficient the journey.

[0108] Line 18 on the display illustrates the vertical profile of the section of track on the display, whilst line 20 depicts the track curvature, or horizontal profile, of the section of track on display.

[0109] The line 22 represents the train, with the vertical line 24 denoting the location of the front of the train and vertical line 26 denoting the location of the rear of the train. The train 22 progresses from left to right on the display.

[01 10] Line 28 and associated numbers 30 indicate the speed limits (in km/h) in various zones of the section of track on display. In this example, the speed limit over the first zone is 75km/h, then the speed limit reduces to 50km/h, then increases to 60km/h, then reduces to 50km/h, and finally increases to 55km/h.

[01 1 1 ] The coloured line 32 indicates the recommended driving profile for the train over the various zones of track. The colour of line 32 at any point denotes the control mode the driver is required to use at that point on the track (i.e. brake, coast, or power). In the display depicted, red represents "brake" mode, white represents "coast" mode, and green represents "power" mode. The shade of the colour varies to indicate the degree of braking or power required. The darker the shade of colour, the greater the degree of braking or power required at the particular point on the track. This is particularly useful when the control mode is 'hold' which is, by nature, somewhere between full power and coast modes.

[01 12] Indicator 34 provides a visual indication to the driver as to how the train is progressing against the recommended speed profile. The indicator comprises a pair of spaced apart arrows which move horizontally across the display as the train progresses and vertically to indicate how the train is progressing against the recommended speed profile. Ideally, the pair of arrows will span the recommended speed profile. If the train is travelling too slowly the arrows will fall below the line 32, whilst if the train is travelling too quickly the arrows will lie above the line 32. In the example shown, the arrows lie slightly below the line, indicating that the train is travelling slightly slower then recommended.

[01 13] Advantageously, by using colour to advise the driver of the control mode, and colour shades to advise the driver as to the intensity of power or braking required, it is possible to move beyond a prescriptive and somewhat inefficient "power-hold-coast" form of driving advice and provide more intuitive driving advice to the driver.

[01 14] The method for determining an optimal driving strategy in the case where the train must pass an intermediate timing location at a specified time, without stopping, will now be described.

[01 15] The motion of a train can be described by the differential equation w ^' " = m^ = F(v d - B(v. ) - R(v} + G(x)

ax di where is the position of the train along the track, v = v{x) is the speed of the train, m is the mass of the train, Fis the force on the train due to the traction system, B is the force on the train due to the braking system, R is the sum of the resistance forces acting against the motion of the train and G is the force on the train due to track gradient when the front of the train is at location x. [01 1 6] Distance, rather than time, is used as the independent variable because gradients and speed limits depend on location. Speed is subject to the constraint v(x) ≤W (x) where W(x) is the track speed limit (due to track geometry or condition) at location x. The elapsed time t = t(x) satisfies the differential equation ώ_ 1

ck V where i is the solution to the equation of motion, and so the total time taken to travel from 0 to X is given by

[01 1 7] The cost of the strategy is the mechanical work done by the traction system,

[01 1 8] Traction and braking forces depend on speed ι and on the driver's control u. The control u 6[-1 ,1 ] is such that u > 0 =» f > 0, B = 0

u = 0 =» f = 0, B = 0

ii < 0 = f = 0, B > 0.

[01 1 9] Suppose we wish to drive from location X₀ starting at time t(X₀) = 0 with speed v(Xo) = 0, and finish at location at time t(X) = Twith speed v(X) = 0, with minimum J{X). Pontryagin's Maximum Principle can be used to find necessary conditions for an optimal driving strategy. An optimal strategy has four driving modes:

• power phases with u =1

• hold phases with v = V and u 6(0, 1 )

• coast phases with u = 0

• brake phases with u = -1 . [0120] The sequence of phases and the switching points between phases for an optimal journey are determined from an adjoint variable which in turn depends on the speed and performance parameters of the train, on the gradient and speed limits of the track, and on a constant related to the hold speed V. For any given l/we can calculate a journey of optimal type that satisfies the necessary conditions for an optimal journey, but does not necessarily finish at the desired time T. We can then vary the hold speed V to find the optimal driving strategy with profiles { v₀,t₀,Jo) that satisfies the constraint t₀ (X) = T .

[0121 ] Figure 7 shows speed graphs for two optimal journeys on the range x

£[120,196]. The shaded area at the bottom of the graph indicates the track altitude. The upper orange curve is the track speed limit. The colours on the two speed profiles indicate control— green is power, grey is coast, and red is brake. The fast journey has V = 250 km/h and a duration of 2960 seconds; the slow journey has V= 97.8 km/h and a duration of 3200 seconds.

[0122] Suppose now that we wish for the train to pass Χ =155 at time 7Ί =1430, but still finish the journey at time T= 3200. The fast journey passes ΛΊ at time 1398; the slow journey passes ΛΊ at time 1465. To meet the new timing constraint, we need to have a higher hold speed on the first part of the journey and a lower hold speed on the second part of the journey.

[0123] The speed at which the train passes the timing location is not constrained. However, some speeds that are achievable at the timing location may be so high that the train is unable to satisfy speed limit constraints on the second part of the journey. To avoid this problem, we first construct a journey (v ,J^) on the entire distance [X₀,X] with a single hold speed l/, such that the journey passes through the timing point ( i , 7i) with speed and is feasible for the remainder of the journey, but does not necessarily finish at the correct time T . We can then construct the second part of the journey (v₂,t₂,J₂) on [Χ^ ,Χ] that starts with v₂(X^) = t₂ = fi( i) and J₂(X ) =

Ji( i), has hold speed V₂ and finishes at (X, ^~f).

[0124] Figure 8 shows these two journey parts. The upper speed profile is V which passes through the desired timing point. The lower speed profile is V₂, which finishes at the correct time. The two hold speeds are l/, =106.1 and V₂ = 93.1 . The composite journey, l/, on [X₀,X ] and V₂ on [Χ^ ,Χ], arrives at both the timing location and at the end of the journey at the correct times, and both parts of the journey are journeys of optimal type.

[0125] This is not the only possible journey that satisfies these conditions. This journey changes hold speed at location X, . In general, we could change hold speed at any location. It is not sensible to change hold speed after the train has passed the timing point; once the train arrives at the timing location the remaining journey has known initial distance, speed and time, and so the optimal strategy has a single hold speed V₂. But changing hold speed before the timing location may be beneficial.

[0126] A procedure for constructing a journey where the hold speed changes at a given location a < ΛΊ is:

• choose a hold speed l/, and construct the corresponding journey (v ,J^) on the interval [X₀,X]

• find the hold speed V₂ for which the journey (v₂,t₂,J₂) on [a,X] with v₂(a) =

1/1 (a), t₂ (a) = fi(a) and J₂(a) = Ji(a) finishes at location with t(X) = T

• if t₂ > 7Ί then increase l/, and try again; if t₂ (ΛΊ) < 7Ί then decrease l i and try again.

[0127] The parameter a can then be varied to find the composite journey with the minimum cost.

[0128] Figure 9 shows how cost J varies with a for our example problem. The data points do not lie exactly on a smooth curve because of inaccuracies in the numerical procedures used to calculate the optimal trajectories. In this case, changing hold speed at the timing location result in energy use that is close to the minimum. In practice, changing hold speed at the timing location is likely to be good enough. Furthermore, the speed profile ν does not need to be calculated all the way to the end of the journey; it only needs to be calculated far enough beyond ΛΊ to ensure that speed limits beyond ΛΊ will not be exceeded. The method can be extended to handle multiple timing points before the next stop.

[0129] The ideal hold speed from any location and speed, taking into account future timing locations where the earliest desired arrival time and the latest desired arrival times are specified, can be found using a numerical search procedure, such as a binary search:

let V_ be the lowest allowable hold speed for the train

let V_H be the highest allowable hold speed for the train

repeat

calculate the speed profile i with hold speed V,

finishing at the first timing location where the train arrives early or late

if i is late then

V_L := V

else

V_H := V

until V_H - V_L is small

[0130] The method of the present invention is typically embodied in software. In one preferred form, the invention provides an automated system that monitors the progress of a train on a long-haul network, calculates efficient control profiles for the train, and displays driving advice to the train crew. In a further preferred embodiment the system works in conjunction with a dynamic rescheduling tool that coordinates interactions between various trains operating on the network.

[0131 ] The invention is designed to work on a train with optimisation working as a background task continually updating the optimal speed profile from the current state of the journey to the next target.

[0132] Advice is provided to the driver from the result of comparing the current state to the optimal journey and generating appropriate control advice.

[0133] Advantageously, the present invention at least in the preferred form provides one or more of the following benefits:

• efficient driving strategies which can reduce energy costs whilst improving time keeping and network performance.

• improved on-time running and shorter waits at crossing loops;

• reduced air braking, lower brake wear, reduced wear on traction motors, extended service life, lower maintenance costs; • improved consistency between drivers;

• accelerated driver training.

[0134] Although the invention has been described with reference to specific examples, it will be appreciated by those skilled in the art that the invention may be embodied in many other forms.

Claims

1 . A method of monitoring the progress of a train on a rail network and providing driving advice in real time to an operator of said train, said method comprising:

(i) estimating or determining parameters of said train;

(v) generating said driving advice for the train operator by comparing the current state of the train to a corresponding state on said optimal journey profile and displaying said advice for the train operator that will keep the train close to said optimal journey profile.

1 . The method as claimed in claim 1 wherein said discrete control modes for said train include drive, hold, coast and brake modes.

3. The method of monitoring the progress of a train on a rail network as claimed in claims 1 or 2 wherein said adjoint variable evolves according to a differential equation along with the position and speed of the train.

4. The method of monitoring the progress of a train on a rail network as claimed in claims 1 or 2 wherein the value of the adjoint variable is calculated directly from speed of the train.

5. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 4 wherein a numerical method is used to solve said system of differential equations for said speed profile of the train and for the value of said adjoint variable.

6. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 5 wherein steps (i) to (iv) are performed as required so that said driving advice automatically adjusts to compensate for any operational disturbances encountered by the train.

7. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 6 wherein said parameters include train mass and mass distribution.

8. The method of monitoring the progress of a train on a rail network as claimed in claim 7 wherein said parameters further include maximum tractive efforts and maximum braking effort as functions of speed.

9. The method of monitoring the progress of a train on a rail network as claimed in claims 7 and 8 wherein said parameters further include coefficient(s) of rolling resistance.

10. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 9 wherein said driving advice is generated and displayed by a computer located on the train.

1 1 . The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 10 wherein step (iii) involves processing data from a GPS unit and train controls to determine the location and speed of the train.

12. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 1 1 wherein said optimal journey profile specifies the time, speed and control at each location between the current train location and the next target locations on the network.

13. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 12 wherein said optimal journey profile is precomputed.

14. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 1 to 12 wherein a plurality optimal journey profiles corresponding to different journey times are calculated and the profile that has an arrival time at the target locations closest to the desired arrival time is selected.

15. A method of monitoring the progress of a train on a rail network and providing information on the progress of the train in real time to an operator of said train, said method comprising:

(i) estimating or determining parameters of said train;

16. The method as claimed in claim 15 wherein said discrete control modes for said train include drive, hold, coast and brake modes.

17. The method of monitoring the progress of a train on a rail network as claimed in claims 15 or 16 wherein said adjoint variable evolves according to a differential equation along with the position and speed of the train.

18. The method of monitoring the progress of a train on a rail network as claimed in claims 15 or 16 wherein the value of the adjoint variable is calculated directly from speed of the train.

19. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 18 wherein a numerical method is used to solve said system of differential equations for said speed profile of the train and for the value of said adjoint variable.

20. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 19 wherein steps (i) to (iv) are performed as required so that said driving advice automatically adjusts to compensate for any operational disturbances encountered by the train .

21 . The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 20 wherein said parameters include train mass and mass distribution.

22. The method of monitoring the progress of a train on a rail network as claimed in claim 21 wherein said parameters further include maximum tractive efforts and maximum braking effort as functions of speed.

23. The method of monitoring the progress of a train on a rail network as claimed in claims 21 or 22 wherein said parameters further include coefficient(s) of rolling resistance.

24. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 23 wherein said information is generated and displayed by a computer located on the train.

25. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 24 wherein step (iii) involves processing data from a GPS unit and train controls to determine the location and speed of the train.

26. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 25 wherein said optimal journey profile specifies the time, speed and control at each location between the current train location and the next target locations on the network.

27. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 26 wherein said optimal journey profile is precomputed.

28. The method of monitoring the progress of a train on a rail network as claimed in any one of claims 15 to 26 wherein a plurality optimal journey profiles corresponding to different journey times are calculated and the profile that has an arrival time at the target locations closest to the desired arrival time is selected.

29. A method of controlling the progress of a train on a rail network, said method comprising:

(i) estimating or determining parameters of said train;

30. The method as claimed in claim 29 wherein said discrete control modes for said train include drive, hold, coast and brake modes.

31 . The method of controlling the progress of a train on a rail network as claimed in claims 29 or 30 wherein said adjoint variable evolves according to a differential equation along with the position and speed of the train.

32. The method of controlling the progress of a train on a rail network as claimed in claims 29 or 30 wherein the value of the adjoint variable is calculated directly from speed of the train.

33. The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 32 wherein a numerical method is used to solve said system of differential equations for said speed profile of the train and for the value of said adjoint variable.

34. The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 33 wherein steps (i) to (iv) are performed as required so as to automatically adjust to compensate for any operational disturbances encountered by the train.

35. The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 34 wherein said parameters include train mass and mass distribution.

36. The method of controlling the progress of a train on a rail network as claimed in claim 35 wherein said parameters further include maximum tractive efforts and maximum braking effort as functions of speed.

37. The method of controlling the progress of a train on a rail network as claimed in claims 35 or 36 wherein said parameters further include coefficient(s) of rolling resistance.

38. The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 37 wherein step (iii) involves processing data from a GPS unit and train controls to determine the location and speed of the train.

39. The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 38 wherein said optimal journey profile specifies the time, speed and control at each location between the current train location and the next target locations on the network.

40. The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 39 wherein said optimal journey profile is precomputed.

41 . The method of controlling the progress of a train on a rail network as claimed in any one of claims 29 to 39 wherein a plurality optimal journey profiles corresponding to different journey times are calculated and the profile that has an arrival time at the target locations closest to the desired arrival time is selected.

42. A method of monitoring the progress of a train on a rail network and providing driving advice in real time to an operator of said train substantially as herein described with reference to any one of the embodiments of the invention illustrated in the accompanying drawings and/or examples.

43. A method of monitoring the progress of a train on a rail network and providing information on the progress of the train in real time to an operator of said train substantially as herein described with reference to any one of the embodiments of the invention illustrated in the accompanying drawings and/or examples.

44. A method of controlling the progress of a train on a rail network substantially as herein described with reference to any one of the embodiments of the invention illustrated in the accompanying drawings and/or examples.