ooo!L1098(95)00184-0 Auromorico. Vol. 32, No. 4, pp. 519-532, 19%

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Optimal Strategies for the Control of a Train*

PHIL HOWLETTt

A Lagrangian analysis is used to find fuel-eficient driving strategies for a long-haul freight train with only discrete levels of control. Key equations are used to define strategies of optimal type and optimal switching points.

Key Words-Train control; optimal control; discrete control; optimal switching; numerical algorithms.

Abstract-This paper describes a method for the calculation of optimal control strategies in an important engineering application. A train travels from one station to the next along a track with non-constant gradient. The journey must be completed within a given time, and it is desirable to minimise the fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. This assumption is based on the control mechanism for a typical diesel-electric locomo- tive. For each fixed finite sequence of control settings, we show that fuel consumption is minimised only if the settings are changed when certain key equations are satisfied. The strategy determined by these equations is called a strategy of optimal type. Several realistic examples are given with the results of associated numerical calculations. The examples demonstrate the profound effect of even a small gradient on a strategy of optimal type. We show that a strategy of optimal type with alternate phases of coust and maximum power can be used to approximate the idealised minimum cost strategy.

1. INTRODUCTION

1.1. An outline of the problem A train travels from one station to the next along a track with non-constant gradient. The journey must be completed within a given time, and it is desirable to minimise the fuel consumption.

We assume that only certain discrete control settings are possible. This conforms to the most common situation in railway locomotives. In particular, the GM diesel-electric locomotives (see the GM Locomotive service manual for the GM JT26C-2S.S) and many other long-haul locomotives have eight traction control settings. The JT26C-2SS and other similar models are currently used by Australian National to haul freight trains some 4000 km across the Austra- lian continent between Sydney and Perth. Each traction control setting determines a constant

* Received 14 July 1994; revised 16 March 1995; received in final form 25 August 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor John N. Tsitsiklis under the direction of Editor Tamer Basar. Corresponding author Dr Phil Howlett. Tel. +61 8 302 3804; Fax +61 8 302 3381; E-mail p.howlett@unisa.edu.au.

t Scheduling and Control Group, Centre for Industrial and Applied Mathematics, University of South Australia, The Levels, 5095, Australia.

519

rate of fuel supply. The power developed by the locomotive is directly proportional to the rate of fuel supply, and the corresponding acceleration is non-negative. We shall assume that a single brake control gives constant negative accelera- tion. In current operations dynamic braking is used only to control the speed of the train. None of the recovered energy is used to drive the train.

In our model the train will be controlled using a finite sequence of traction phases and a final brake phase. During each phase, the control setting is constant. A traction phase will be classified as a power phase or, if the fuel supply rate is zero, as a coast phase. For each control strategy there is a uniquely defined speed profile determined by the equations of motion. We shall say that a strategy is feasible if the distance and time constraints are satisfied.

We wish to find a feasible strategy that minimises fuel consumption.

1.2. Applications of the work The cost of fuel is the most significant

recurrent expenditure in many large rail organisations. It is estimated that the cost for purchase and lifetime maintenance of a locomo- tive is approximately equal to the total cost of fuel consumed by the locomotive. Reduction of fuel consumption using energy-efficient driving strategies is therefore a primary concern, and consequently the development of a suitable driver advice system would be most beneficial.

The work in this paper is part of a long-term project carried out by the Scheduling and Control Group at the University of South Australia. The aim of the project is to produce an on-board advice unit that can be used by train drivers to ensure energy-efficient driving strategies.

The first such device is now commercially available under the name Metromiser. Met- romiser is an on-board computer with an

P. Howlett

associated driver advice display unit designed for use on metropolitan trains. Extensive in-service trials with this unit (Benjamin et al., 1987) have shown improved time-keeping and a reduction in fuel consumption of between ten and twenty percent. It has been clearly demonstrated that the timing of control decisions is significantly improved by Metromiser. This comment applies particularly to the timing of the crucial switch into the semi-final coast phase. Even for relatively simple journeys it is not possible for a driver to perform the necessary calculations or to intuitively drive with near optimal control.

A more comprehensive device designed for use on long-haul trains is currently being developed. The algorithms used in the new device will be based on the work described in this paper. In developing the new device, the Group is collaborating closely with railway operators and system suppliers.

1.3. Previous studies The train control problem has been studied by

a number of authors. There are two widely used models: a model with continuous control (Milroy, 1980; Kraft and Schneider, 1981; Howlett, 1984, 1988a, b, 1990; Asnis et al., 1985; Benjamin et al., 1987) in which the control variable is the applied acceleration and the cost of the strategy is the mechanical energy required to drive the train; and a model with discrete control (Benjamin et al., 1989; Cheng and Howlett, 1990a, b, 1992, 1993; Howlett et al., 1992, 1994a, b) in which the control variable is a throttle setting and the cost of the strategy is the total fuel consumption.

The current paper is one of a sequence of papers that have investigated the practical implementation of energy-efficient driving strat- egies on metropolitan trains and on long-haul freight trains. Because the implementation of these strategies on real trains is a fundamental concern, it has been necessary to construct a model that describes the real control mechanism. This is the so-called fuel consumption model. The model has been studied systematically on level track, where a complete solution has been obtained. No previous solution has been obtained for non-level track. For application of this work to real freight trains, it is essential that the effects of track gradient are properly understood.

1.4. A driver perspective on the problem To understand the essence of the problem, it is

helpful to consider a simplified model. Suppose the locomotive has only three

discrete control settings: power, coast and brake. Consider the possible control options for the driver. The strategy must begin with a power phase and follow with alternate phases of coast and power. The strategy must end with a semi-final coast phase and a final brake phase, although it is possible that one or other of these latter two phases could become degenerate. We assume that braking will be used only to stop the train.

The driver can decide the number of phases and the points at which the control settings will be changed. These points are called the switching points.

The nature of the problem is not changed by allowing a greater range of discrete control settings. The driver must first decide the precise sequence of control settings and then determine the optimal position of the switching points.

The driver must make these decisions in such a way that fuel consumption is minimised.

1.5. Formulation of a well-posed problem Because the control strategy is restricted to a

finite sequence of discrete settings, the problem described above is not well posed. In general, there will be no admissible strategy for which the minimum fuel consumption is achieved, and hence no definitive conditions for optimality.

We overcome this difficulty in the following way. Consider the set of all possible control strategies. We divide the set into disjoint subsets, and seek the best strategy from each subset. Each subset is defined by a fixed finite sequence of control settings. There are many different strategies in each subset; all strategies use the same sequence of control settings, but each one is determined by different switching points.

To find a feasible strategy that minimises fuel consumption within the given control subset, we need to find the optimal locations for the switching points subject to the distance and time constraints. A strategy satisfying these require- ments will be called a strategy of optimal type. This problem is a standard finite-dimensional constrained optimisation problem. Because we are seeking the best location for a finite number of points within a closed and bounded interval the existence of such a strategy is guaranteed.

The original problem can now be regarded as a problem in comparison of the various strategies of optimal type.

1.6. A preview of the new results We consider the train control problem on a

track with non-constant gradient. There has been no practical solution to this problem. In this paper

Train control

we find key equations that determine neces- sary conditions for the strategies of optimal type;

we show that the key equations can be used to describe the nature of the strategies of optimal type;

each setting except at very low speeds. A typical graph of observed acceleration against speed for each traction control setting is shown in Fig.. 1. Figure 2 shows a graph of power against rate of fuel supply.

we show that the minimum fuel consumption can be almost achieved using .a strategy of optimal type and

we illustrate our results with several realistic examples.

On level track it has been shown (Cheng and Howlett, 1992) that the strategies of optimal type are determined by key equations that provide necessary and sufficient conditions. In this paper the key equations of Cheng and Howlett are generalised to give necessary conditions for a strategy of optimal type on a track with non-constant gradient. Although the existence of a strategy of optimal type is clear, the existence and uniqueness of solutions to the key equations is no longer guaranteed. However, we do know that the equations can often be solved and that the solutions could be calculated in real time by a small on-board computer. We believe that these solutions can be used to provide practical advice to drivers about energy-efficient driving strategies.

Dynamic braking is an important aspect of control in a typical diesel-electric locomotive. A minimal rate of fuel supply is normally required to control the effective braking effort. Energy regained by the locomotive from the dynamic braking process is not stored for subsequent use, and is normally lost as heat energy. The dynamic braking performance of a typical locomotive is shown on the graph in Fig. 3. The dynamic braking control mechanism is designed so that the negative acceleration is approximately constant over the normal speed range. The dynamic brakes are supplemented by air brakes, which are used predominantly at low speed to increase the braking capability to an acceptable level. Because energy is lost during braking, it is intuitively obvious that an optimal strategy will use a relatively small braking time and that braking will take place at relatively low speeds. On the basis of the above argument we believe that the precise form of the braking effort will have little impact on our conclusions. Therefore in our model we assume that the negative acceleration during braking is constant.

We shall use the examples to emphasise that there is no significant disadvantage in having only a limited number of control settings. The driver simply alternates between a level of control above the desired value and a level below the desired value.

The examples will also be used to show that an inappropriate choice of switching points can disrupt schedules and increase fuel consumption. The examples will demonstrate that even very small changes in gradient can cause a dramatic change in the position of the optimal switching points. We shall also show that our methods can be used to calculate energy-efficient driving strategies on steep track.

2. FORMULATION OF THE PROBLEM

2.1. Locomotive performance characteristics It is useful to begin by considering the traction

and braking characteristics of the GM JT26C- 2% diesel-electric locomotive. The locomotive has eight traction control settings. Each setting determines a constant rate of fuel supply to the diesel motors, and the observed acceleration is inversely proportional to the speed, provided that the speed exceeds a known small value. This is consistent with a constant power output in

521

2.2. The control strategies To describe the control mechanism we

introduce a control variable j. Each non-negative value of the control variable determines a traction control, and the single negative value determines a braking control. Let

Ce = (-1, 0, 1,2,. . . , m} (1)

denote the set of all possible values for j. Let 6 be the fuel supply rate corresponding to the control setting j, and assume that

O=f-,=fo

522 P. Howlett

Fig. 2. Power against fuel flow rate for the GM JT26C-2SS locomotive.

brake phase with j(n + 1) = -1. This sequence defines a subset

y({j(k + l)]k=o.,.. .J (3)

of control strategies, each with II + 1 distinct control phases. Let us define

o=x~x,~x~I...~x,+,, (4)

where x0 is the starting point, {x~}~=,,~,...,~ are the switching points, x, + , is the stopping point and j(k + 1) is the control setting on the interval

( xk, xk+,). We use &+, to denote the length of the interval (xk, xk+,). The control strategy from the given subset and with the above switching points is denoted by

N[j(k + 1); (xk, ~~+,)l}k=~.,....,n). (5)

2.3. The equations of motion When j z-0, the power developed by the

locomotive is directly proportional to the rate of fuel supply. When j ~0, there is a constant negative acceleration applied to the train. If we write Kj for the braking acceleration then Kj = 0 for j 2 0, and K-, = -K is a negative constant. The equations of motion for a point mass train are

dx -$= n, (6)

du HJ -ET+ Kj- P(V) i-g(X), dt

(7)

where x is the distance along the track, u is the speed of the train, p(v) is the resistive acceleration due to friction, g(x) is the gravitational acceleration due to the track gradient and H is a constant. We assume that p(O) > 0, that p(u) is strictly increasing and that the graph y = VP(U) is strictly convex. Although

Fig. 3. Dynamic braking curves for the GM JT26C-2% locomotive.

(6) and (7) are directly applicable to a point-mass train, it is important to remark that similar equations with a modified average gradient acceleration can be used to describe a train with distributed mass. This issue is discussed in detail elsewhere (Howlett et al., 1994a). It is often convenient to rewrite the equations of motion in the form

dt 1

dx=v (8)

u$=?+K,-p(v)+&). (9)

The solution of these equations is discussed in Section A.1 of the Appendix.

2.4. The cost function For the control strategy S({[j(k + 1);

( xk! Xk+l)l)k=O,L. .n), we use rk+i to denote the time taken to traverse the interval (xk, xk+,). The total fuel consumption is given by

J = 3 fi(k+l)rk+l. k=O

(10)

2.5. A precise statement of the problem The problem can now be stated precisely. Let

X be the length of the journey and let T be the time allowed for the journey. For a fixed control sequence {j(k + l)}k=O,I....,nr we wish to choose the switching points {x~}~=,,~,.,.,, to define a control strategy

S({[j(k + 1); (xk, Xk+I)l}k=O.I....,N)

E W(k + l)}k=O.l....,n)t (11)

with u(xo) = 0, u(x,,+,) = 0, cz=o rk+, = T and c;=O tkt, =X in such a way that J = C:=Of,(k+l)~k+l is minimised. The correspond- ing strategy is called a strategy of optimal type.

3. TRACK GRADIENT ANALYSIS AND TERMINOLOGY

Track gradient data is normally stored in digital form, and hence the true gradient is represented as a piecewise-constant gradient. For both theoretical and computational reasons, it is convenient to use the gradient data in this form. It has been shown (Howlett et al., 1992) that solutions to the equation of motion can be computed to any desired accuracy using a piecewise-constant approximation to the true gradient.

Subdivide the interval [0, X] by setting

O=ho

Train control 523

on each subinterval (h,, h,+,). We consider a fixed control sequence {j(k + l)}k=O,l,...,n and a general strategy

S(C(k + 1); (x/C, &+l)lL=O,l,...,n)

E W(k + l)L=w,...,n). (13)

Equation (9) is now written in the following way. When x E (xk, xk+,) tl (h,, h,+l), we have

$, fG+v ; K, dx u I(k+l)

-P(u) + &

= F[k,r](u), (14)

where g, is the acceleration due to the gradient on the interval (h,, h,+l). We could, for example, assume that g, =g(h,), where g(x) is the acceleration due to the known track gradient. Ultimately we shall need to specify the locations of the variable points {xk}k=l,Z,...,n in relation to the fixed points {hr}r~0,1,2,...,p+l. We use the notation {r(k)}k=O,l,...,n to denote the essentially unknown sequence with r(0) = 0 and r(n + 1) = p and with

(15)

for each k = 1,2, . . . , n.

4. THE MAIN RESULTS

4.1. The key equations To explain the main results, it is necessary to

introduce some more terminology. Use u, = u(h,) to denote the speed at x = h, and vk = u(xk) to denote the speed at x = xk. For each k=1,2,... , n and for r(k) 0. let

qJ) = f+ P(U), (22)

and define a function {8&}&x) in the region

524 P. Howlett

x E [xk, ++,I by the formula {%~}&) = E,[u(x)] for x E [hrtktl), Q+,] and by

r(k+l)-1

+ ,;+, &F&4 - ~,(Us+~)l + EI*(Ur(k+IJ .s

(23)

for x E [h,, h,+l] and r(k) 5 r < r(k + 1). The function {8+},(x) can be regarded as the effective energy density for the control strategy. The key equations can now be rewritten as follows. There are non-negative constants A and p such that x, < X is the solution to

x,_ 1 < x, is the solution to

and in general xk < & +, iS the SOhtiOn t0

@+.)k(X) = &tVk+,) (26)

(24)

(25)

for each k = 1,2, . . . , II - 2. On level track we note that vk(x) = u and {%p}k(X) = E,(u) where u = u(x). Since g,(,, = 0, it can be seen that our key equations generalise the key equations for level track (Cheng and Howlett, 1992).

4.3. The strategies of optimal type It can be shown that the function {%p}k(r) is

well defined and continuous with a continuous derivative at all points in the region x E

[ xk, xk+,]. In particular, the derivative is given by the formula

{up}; =&&y@, (27) F

when x E [h,, h,+,] and where u = u(x). From this formula and the convexity of the graph y = E,(u), it can be seen that the function {CG;r}k(~) has a local turning point only when u = We, where w, is the unique minimum turning point for the graph y = E,(u). Note that the critical speed wfi satisfies the equation

++$L) = p. (28)

We shall show that We defines an approximate holding speed. These observations are crucial to our understanding of the solutions to the key equations. A detailed solution algorithm is reviewed in Section A.3 of the Appendix.

If power is applied to the train on the interval

txk> xk+l) and if we assume that the speed increases throughout the interval then there is precisely one solution xk < xk+* to (26) with u(xk) = vk < wp < I$+, = i+&+l). If the coast control is used and we assume that the speed decreases throughout the interval then once again there is precisely One SdUtiOn xk < xk+, , but in this case u(xk) = uk > w, > v,,, = u(&+,). In each of the above cases the interval (xk, xk+,) is uniquely determined.

Therefore, when the gradient is not too large, we can expect the speed to oscillate about the critical value w, as the control is switched between power and coast. As the number of phases is increased it is intuitively reasonable to suggest that the length of each phase is reduced and hence the critical speed w, can be interpreted as an approximate holding speed.

Thus, for non-steep track, the strategy of optimal type will contain an initial power phase, followed by an approximate speedhold phase near the critical speed We, a semi-final coast phase and a final brake phase.

In practice, the track gradient may be sufficiently large to disrupt this general pattern. Thus we may have situations where maximum power is applied to the train but the speed decreases. On the other hand, there may be situations where the coast control is used and the speed increases. When these situations occur, we shall say that the track is steep. Note that this definition is relative to the actual speed of the train, and whether or not a track is classified as steep will depend on the desired holding speed. In such cases it is possible that there will be more than one solution to the key equations, and consequently it is possible that the interval

( xk, xk+ I) Will not be UniqUdy determined. For this reason, we reiterate that the key equations give necessary conditions for a strategy of optimal type but that these conditions may not be sufficient. In practice, it is often relatively easy to decide which solutions are superfluous.

Nevertheless, on steep track the strategy of optimal type will have the same form as on non-steep track, except that the approximate speedhold phase may be disrupted by phases of maximum power in the vicinity of steep inclines and by phases of coast in the vicinity of steep declines. These disruptions can be interpreted as necessary adjustments to keep the speed as near as possible to the desired holding speed. The theoretical difficulties associated with steep track are not discussed in this paper.

We shall illustrate our work with several realistic examples that give some indication of the general nature of a strategy of optimal type.

In Section A.5 of the Appendix it is shown

Train control 525

that any segment of non-negative measurable control can be approximated by a sequence of coast-power pairs. Thus we can construct a strategy of optimal type with approximately minimum fuel consumption.

5. EXAMPLES FOR NON-STEEP TRACK

The following examples are based on data obtained from train models used by the Scheduling and Control Group at the University of South Australia. Length is measured in metres and time in seconds. We consider a journey with X = 18 000 and T = 1500. We assume that

p(u) = a + bu + cu*, (29)

where a=1.5~10-*, b=3X10P5 and C=~X lo-. We take H = 1.5 and K = 1, and assume only two allowable rates of fuel supply with f0 = 0 and fi = 1. For each x E [0, X], we shall take

g(x) = ax(X - 2x)(X - x), (30)

where (Y is a constant. Outside the interval [0, X] we assume that g(x) = 0. In all of the examples the value of (Y is so small that the gradient would not be readily apparent to the naked eye. In fact, the height of the track is determined by the formula

h(x)=-gJ 5(X - 25)(X - 5) d5 . 0

= -&*(x-x) (31)

for x E [0, X], with h(x) = 0 otherwise. Hence, if la1 = 2 x 10-14, we have @(x)1 ~6.7 for all x. Thus in a total journey of 18 km there is a rise or fall of only 6.7 m. The strategies of optimal type in these examples all have the basic power, approximate speedhold, coast, brake form, with the approximate speedhold segment constructed using alternate phases of coast and power. Nevertheless, we shall show that even these very small gradients have a dramatic effect on the position of the switching points, and in particular that the extent of the crucial semi-final coast phase is drastically changed.

There are some important general observa- tions that should be made at this stage. There is only a small decrease in the fuel consumption when a strategy of optimal type using only one coast-power pair is replaced by a strategy of optimal type using nine coast-power pairs. Thus a seemingly rudimentary approximation to the idealised minimum cost strategy can be very energy-efficient. Hence, in practice, we do not need a large number of coast-power control pairs to construct an approximate speedhold

strategy. Indeed, the examples confirm that in practical terms coast-poiver control is almost as close as we please to continuous control.

In our examples we have used very simple gradient profiles. There are two critical factors that determine an energy-efficient strategy. Firstly, it is necessary to keep the train speed close to the selected holding speed during the approximate speedhold phase, and secondly it is important to choose the correct switching point to begin the semi-final coust phase. We have found that manual selection of the switching points is difficult because of extreme sensitivity to small changes in gradient. In our simple examples selection of the correct switching point to begin the final phase is extremely important. On metropolitan railcars with apparently flat track and relatively small distances between stations we have found that even the most experienced drivers could not choose this point effectively. This was demonstrated in our Metromiser trials, where audited fuel savings of fourteen percent and improved timekeeping were achieved when Metromiser was used to select the point where the semi-final coust phase begins.

5.1. Strategies of optimal type on level track We begin by considering a level track. Thus

we let (Y = 0. We consider a strategy with one coast-power pair and a strategy with nine coast-power pairs. It is significant for practical purposes that fuel consumption is reduced only marginally by the more elaborate strategy.

Example 1. (One coast-power pair on level truck.) The detailed results are shown in Table 1. We set A = 2.05969 X lop2 and p = 5.51350 X 10w2. The critical speed is given by wfl = 15.832 and the fuel consumption by J = 202.15.

Example 2. (Nine coast-power pairs on level truck.) The detailed results are shown in Table 2. We set h = 2.06798 X 10e2 and p = 5.85768 X lo-*. The critical speed is given by w, = 16.170 and the fuel consumption by J = 201.89. The speed profile is sketched in Fig. 4.

Table 1. Strategy of optimal type with one COW-power pair on level track

k j(k + 1) tk xk vk

0 1 0.000 0.000 0.000 1 0 134.060 16%.581 18.588 2 1 440.430 6 583.762 13.374 3 0 508.525 7 684.230 18.588 4 -1 1497.363 17 996.470 2.677 5 1500.000 18000.000 0.000

526 P. Howlett

Table 2. Strategy of optimal type with nine coasf-power pairs on level track

k i(k + 1) fh J-k Vh

i8 19 20 21

1 0.000 0 104.513 1 152.528 0 163.348 I 211.364

i 623.214 0 634.034

-1 1 497.209 15oo.ooo

0.000 0.000 1 176.226 16.583 1 952.765 15.764 2 127.808 16.583 2 904.347 15.764

9 585.425 15.764 9 740.468 16.583

17 996.048 2.833 18000.000 0.00

5.2. Level track strategies applied to non-level track with small gradients

We consider essentially the same problem on non-level track, but with very small gradients. The gradients are so small that they would not be apparent to the naked eye. We shall show what happens if the driver attempts to use a strategy designed to be optimal on a level track. There are three alternative ways in which the level track strategy could be implemented. The most natural way is to select the same switching points, but it is also possible to select the same switching times or the same switching speeds. In all cases the strategies can be shown to be incorrect.

5.2.1. A small valley. In the first case we consider a small valley. To define a track with an initial downhill section and a final uphill section we let (Y = 2 X 10-14. The switching points will be the same points used in Example 2.

Example 3. (Level track strategy over a small valley). The strategy is not feasible and the final phase is degenerate. The train stops at time t = 1363.733 and at distance x = 16 988.205, and the cost of the journey is J = 19058. To make this strategy feasible, it would be necessary to extend the time spent under power. The detailed results are given in Table 3.

5.2.2. A small hill. We now consider a small hill. To define a track with an initial uphill section and a final downhill section, we let

16.58 15.76

0 x 1

Fig. 4. Stylised speed profile of optimal type on level track.

Table 3. Level track strategy over a small valley

k

0 l 2 3 4

lb 19 20 21

i(k + 1)

1 0 I 0 1

i 0

-1

th xh Vk

0.000 0.000 o.ooo 104.085 1 176.226 16.778 151.024 1 952.765 16.331 161.468 2 127.808 17.181 207.116 2 904.347 16.853

567.857 9 585.425 18.477 577.188 9 740.468 19.038

1 363.733 16 988.205 0.000 1 363.733 16 988.205 0.000

ff = (-2) x lo-14. The switching points will be the same points used in Example 2.

Example 4. (Level track strategy over a small hill.) The strategy is not feasible and is not energy-efficient. The train stops at time t = 1461.125 and at distance x = 18 028.070, and the cost of the strategy is J = 218.18. To make this strategy feasible, it would be necessary to decrease the time spent under power. The detailed results are given in Table 4.

5.3. Strategies of optimal type on non-level track with small gradients

Once again, we consider essentially the same problem on non-level track. This time, however, we calculate strategies of optimal type.

5.3.1. A small valley. As before, we let cr = 2 x 10p14. We consider a strategy with nine coast-power pairs. Although we have not presented the detailed calculations, it is sig- nificant for practical purposes that fuel con- sumption is reduced only marginally by using a large number of coast-power pairs.

Example 5. (Strategy of optimal type over a small valley.) The results are shown in Table 5. We set A = 2.25335 X lo-* and p = 4.76702 X lo-. The critical speed is given by We = 15.047 and the fuel consumption by J = 200.18. The speed profile is sketched in Fig. 5.

5.3.2. A small hill. As before, we let (Y = (-2) x 1op4. We consider a strategy with nine

Table 4. Level track strategy over a small hill

k i(k + 1) 1, -r!f vh ~~___

~- 0 1 0.000 0.000 0.000 I 0 104.948 1 176.226 16.386 2 1 154.124 I 952.765 15.177 3 0 165.361 2 127.808 15.967 4 1 216.156 2 904.347 14.599

18 i 704.273 9 565.425 12.480 19 0 717.137 9 740.468 14.214 20 -1 1453.182 17 996.048 8.064 21 1461.125 18 028.070 0.000

Train control 527

Table 5. Strategy of optimal type over a small valley Table 6. Strategy of optimal type over a small hill

k j(k + 1) tk x/e vk

0 1 0.000 0.000 0.000 1 0 88.986 931.336 15.533 2 1 185.874 2 389.459 14.647 3 0 194.573 2 520.418 15.454 4 1 328.719 4 539.073 14.666

i8 i ssi.105 12 853.961 14.k 19 0 897.321 13 098.264 15.642 20 -1 1497.917 17 997.797 2.116 21 1500.000 18000.000 0.000

k i(k + 1) tk xk vk

0.000 0.000 1952.211 19.011 2 194.708 18.667 2 318.567 19.012 2 5!4.841 18.666

is 19 20 21

1 0.000 0 148.575 1 161.447 0 168.021 1 180.563

i 313.289 0 320.002

-1 1494.409 1500.000

5 055.356 18.k66 5 181.827 19.013

17 984.142 5.673 18 OOO.ooo 0.000

coast-power pairs. Although we have not presented the detailed calculations, we again note for practical purposes that fuel consumption is reduced only marginally by using a large number of coast-power pairs.

6. AN EXAMPLE WITH LARGE TRACK GRADIENTS

Example 6. (Strategy of optimal type over a small hill). The results are shown in Table 6. We set A = 1.60209 X lo-* and p = 9.08786 X lo-. The critical speed is given by wP = 18.839 and the fuel consumption by J = 209.52. The speed profile is sketched in Fig. 6.

5.3.3. A brief comparison of the strategies of optimal type. The most significant difference in the strategies of optimal type described in the above examples is the length of the semi-final coast phase. In a total journey of 18 km we have a semi-final coast phase of 4.3 km when travelling over the small valley and a semi-final coast phase of 12.7 km when travelling over the small hill. Since the rise and fall of the track in each case is less than 6.7 m, it is clear that this difference could not be determined precisely by the driver alone. Because there is no fuel consumption while coasting we can see that an error in estimating the length of the semi-final coast phase can have a significant effect on the cost.

In practice, there are many sections of a track on a typical long-haul journey where the gradients are classified as steep. Even in an apparently flat continent such as Australia, this is the case. Under these conditions, the pre- dominant speedhold mode is interrupted by segments of coast and power. With complicated track gradient profiles, there may be a number of extended coast or power phases. Selection of appropriate switching points will be important in defining the extent of each of these phases. Inappropriate selection will mean that excessive fuel is consumed in returning the train to the desired holding speed.

The following example shows a strategy of optimal type on a track with an isolated steep incline. We consider a journey with X = 50 000 and T = 3000. As before, we assume that

We have also shown that ad hoc strategies, even those with the correct overall structure, can result in non-feasible journeys or journeys where fuel consumption is increased significantly.

p(v) = a + bv + cv*, (32)

where a=1.5XlO-*, b=3X10P5 and c=6x 10e6. We take H = 1.5 and K = 1, and assume only two allowable rates of fuel supply, with f0 = 0 and f, = 1. For x E [0, X], we take

1

0 if O

528 P. Howlett

g(x) = 0. We consider a strategy with nine coast-power pairs. We observe that the strategy of optimal type contains an approximate speedhold segment that is interrupted by a power phase in the vicinity of the steep incline. At an intuitive level, it seems reasonable to remark that the control segments are chosen so that the average speed over the steep interval is kept as close as possible to the desired holding speed. In the case of a steep incline this is done by increasing the train speed before the climb is commenced. Of course, this may seem intuitively reasonable, but our point once again is that correct selection of the appropriate switching points is very difficult and that small mistakes can result in a dramatic error such as stalling or in the consumption of excess fuel. Example 7. (Chengs climb). The detailed re- sults are shown in Table 7. We set A = 2.3030148 X lo- and p = 10.03085 X 10m2. The critical speed is given by w, = 19.495 and the fuel consumption by J = 924.39. The speed profile is sketched in Fig. 7.

7. CONCLUSIONS AND FUTURE DEVELOPMENTS

For a prescribed sequence of fuel supply rates, it has been shown that a strategy of optimal type depends on two real-number parameters. These determine the switching points and ultimately determine the distance travelled by the train and the time taken for the journey. By adjusting the values of the parameters, it has been possible to construct a feasible strategy in several realistic examples.

We have found a minimum-cost strategy given that a prescribed sequence of fuel supply rates must be used. We have shown that even a small track gradient exerts a dramatic influence on the switching points and on the corresponding switching speeds. The solution given in this

Table 7. Strategy of optimal type for Chenga climb

k ;(k + 1) fh _Lk I/ A

II I 0.000 O.WO 0.000 1 0 16Y.396 2 390.834 20.656 2 I 2Yh.749 4 X75.674 I X.380 3 0 335.367 5 630.6 I3 20.656 4 I 462.72 1 x 115.454 18.380

4 0 x3j.2x.i I5 34Y.95 I 2O.hSh 10 1 960.636 17 834.7) 1 18.380 II 0 1425.214 26 555.060 20.5 15 12 I I S37.356 2X 742.692 IX.51 1

18 i 1975.736 37 247.4 I2 IX.51 1 19 0 2009.721 37.Y61.353 20.5 IS 20 -I 2YYS.709 4Y 990.656 4.356 21 3OOO.000 so 000.000 O.OOO

1 I 23.75

0 x z

Fig. 7. Stylised speed profile of optimal type for Chengs climb.

paper is believed to be the first effective solution of this important practical problem.

Simulated performance of our proposed algorithm shows that the practical impact of our methods will be in the affirmation that, even on undulating track, a speedholding strategy is the most energy-efficient strategy. If the locomotive is unable to maintain a speedholding strategy over steep terrain then correct selection of the regions where power should be applied or where coasting should be used will result in significant fuel savings.

One further point is also worthy of comment. The argument (Cheng and Howlett, 1993) that intermediate throttle settings are less efficient is, in essence, still valid on tracks with non-constant gradient. Although we have assumed a linear relationship between power and fuel supply, we also point out that in practice the maximum fuel supply rate is usually the most power-efficient control setting. Thus intermediate settings should be used only if speed fluctuations are avoided and if the power efficiency of the intermediate setting is comparable with the power efficiency at the maximum setting.

It is important to emphasise that the driving strategies recommended in this paper are not used effectively in practice. Where automatic control systems are used, the strategies are much more concerned with safe driving practices. We believe our strategies can be effectively used in conjunction with such systems to produce significant fuel savings. Where automatic control systems are not used, it is demonstrably wrong to assume that an experienced driver can make effective judgements about the timing of control changes.

Although we have shown that our solution can be efficiently computed, there are several practical problems that remain.

Some of these pose no serious difficulty. For example, in this paper we have nominally considered a point-mass train. By considering a real train as a distributed mass, we can construct a modified gradient proofile that allows us to treat the train as if it were a point mass. This

Train control

matter is discussed in another paper (Howlett et al., 1994a). It is also anticipated that the analysis can be extended to incorporate speed limits. At this stage it is not easy to decide whether the difficulties posed by steep gradients will be a serious compututational problem. It should be pointed out that the difficulties do not relate to isolated steep grades. Difficulties arise when individual steep grades are so close together that it is not easy to decide whether the grades should be treated separately or as a group. The difficulties can usually be resolved by personal intervention, but is not so easy to do this automatically.

The Scheduling and Control Group have developed a prototype computer program that calculates the idealised strategy of optimal type for any given journey. This program has been used with realistic data supplied by Australian National to calculate minimum cost strategies for a typical long-haul journey.

Acknowledgements-I would like to thank my colleague Peter Pudney for his assistance with the numerical calculations required for the examples. I should also like to thank the other members of the Scheduling and Control Group at the University of South Australia.

REFERENCES

Asnis, I. A., A. V. Dmitruk and N. P. Osmolovskii (1985). Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle. USSR Comput. Maths Math. Phys., 25(6), 37-44.

Benjamin, B. R., A. M. Long, I. P. Milroy, R. L. Payne and P. J. Pudney (1987). Control of railway vehicles for energy conservation and improved timekeeping. In Proc. of Conf on Railway Engineering, Institute of Engineering of Australia, Perth, Western Australia, pp. 41-47.

Benjamin, B. R., I. P. Milroy and P. J. Pudney (1989). Energy-efficient operation of long-haul trains. In Proc. 4th International Heavy Haul Railway Conf, Institute of Engineers of Australia, Brisbane, Queensland, pp. 369-372.

Cheng Jiaxin and P. Howlett (1990a). Critical velocities for the minimisation of fuel consumption in the control of trains. University of South Australia, School of Mathemat- ics, Report 1.

Cheng Jiaxin and P. Howlett (1990b). Optimal strategies for the minimisation of fuel consumption in the control of trains. University of South Australia, School of Mathemat- ics, Report 3.

Cheng Jiaxin and P. Howlett (1992). Application of critical velocities to the minimisation of fuel consumption in the control of trains. Automatica, 28, 165-169.

Cheng Jiaxin and P. Howlett (1993). A note on the calculation of optimal strategies for the minimisation of fuel consumption in the control of trains. IEEE Trans. Autom. Control, AC-38, 1730-1734.

Howlett, P. (1984). The optimal control of a train. University of South Australia, Study Leave Report.

Howlett, P. (1988a). Existence of an optimal strategy for the control of a train. University of South Australia, School of Mathematics, Report 3.

Howlett. P. (1988b). Necessary conditions on an optimal strategy for the control of a train. University of South Australia, School of Mathematics, Report 4.

Howlett, P. (1990). An optimal strategy for the control of a train. J. Aust. Math. Sot., Ser. B, 31,454-471.

Howlett, P. (1993). Determination of optimal strategies for the control of a train on a track with non-constant gradient. University of South Australia, School of Mathematics, Report 7.

Howlett, P., P. Pudney and B. Benjamin (1992). Determina- tion of optimal driving strategies for the control of a train. In B. J. Noye, B. R. Benjamin and L. H. Colgan (Eds), Proc. Computational Techniques and applications, CTAC- 91, pp. 241-248. Computational Mathematics Group, Australia Mathematical Society.

Howlett, P. G., I. P. Milroy and P. J. Pudney (1994a). Energy-efficient train control. Control Engineering Practice, 2, 193-200.

Howlett, P. G., J. Cheng and P. J. Pudney (1994b). Optimal strategies for energy-efficient train control. In Proc. SIAM Symp. on Control Problems in Industry, San Diego, CA, Birkhauser, Boston.

Kraft, K. H. and E. Schnieder (1981). Optimale Trajektorien im spurgebundenen Schnellverkehr (Optimal trajectories for rapid transit systems). Regelungstechnik, 29.

Milroy, I. P. (1980). Aspects of automatic train control. PhD thesis, Loughborough University, UK.

APPENDIX

A.l. The speed profiles In this subsection we define functions that satisfy the

equations of motion. For each j = 0, 1, . , m, let WIj,rl be the unique solution to the equation Hfi - u[ p(u) - gr] = 0. If the fuel supply rate is J and if the acceleration due to the track gradient is g, then W(,.,) is the speed at which the net acceleration is zero. Choose W such that W > WE,,) for all [j, r]. A direct integration of the equation of motion can now be used to define distance as a function of speed.

For k 0, it is easy to show that xIk,,l(v) + m as u+ Wcj(e+t).r). It now follows that if u,< W(++,),,) then v,+, < Wtj(k+lr,rl. Alternatively if u, > WU(k+lj,rl then v,+, > Wl,o+Ij,rl. Whichever of the above definitions is used, we denote the inverse function by uIk,,l.

When k = n, we define

-qn,r,(v) = W(-l)wdw

~ for v E (0, W] %rl(W)

(A.2)

and let ul,,,l denote the inverse function. We assume that K >g, for all r.

For all k, we claim that the actual speed on the interval (xk, xk + i) is given by

u(x) =

-,k.r(k,](x - xk + X[k.r(k)](Vk))

for x E cxk, h,(k)+,)>

yk,r](x - hr + qk.rl(ur))

for X E @,, hr+d f-l bkr Xk+dt (A3)

v[k.,(k+l)]@ - hr(k+,j + X[k.,(k+,)](,(k+l)))

for X E (hr(k+,),Xk+,).

Note that

Xik.,](V) =- F[k.rl(u)'

uik,rl(X) = F,k;;;t$)) (A.4) I

and also that the time taken to traverse the interval

P. Howlett

(A.5)

(A.6)

when k =n. Since VI = V,(.$,, &. , &), it follows that u, = u,(&, &. , &,) for each r with r(k)irsr(k -+ I), and hence r,, , = rk+,(5,, &, , &I.

A.2. Thr construinrs Since the initial and final speeds are zero. it is necessary

that

v, = v,, + / = 0. (A.7)

It is important to realise that the variable t,, i, can be eliminated from the problem using the equation V,, +, = 0. In fact, we deduce that

and hence the distance constraint

To solve the key equations (24)-(26). we begin by noting the useful recursive relationships

can be rewritten in the form

h ,(,, / 1) + q,u(,,, 1 ,,(O) .- xI,,,,(,, , , ,I(ue, + I J = X. (A.101

Since X[,,,,(,, i I ,I(% + I ,I depends only on [ - (5, , !L,. , [,,I, we can define

45) = k,,, +, , + ~~l,,.r,,, + ,,,(O) - .rl,,,r(,, t I,~(+,, + I,). (A.1 1)

in which case the distance constraint becomes

{r,},(h,) =&[JuJ ~ E,(u,+,)l +{%(hri,). (A.211 r

X(5) = x. (A.12)

The time constraint is expressed simply as

&A+& (A.13)

Since r, , , = r, + ,([, , &. , & , , ) and since {,, , , depends only on 5. we can define

t(t)= i r,*,. (A.14) !. -0

The key equations can now be solved as follows. We assume that the parameters A and p are known. To begin the calculation, we consider the final interval [x,,, x,,,,]. We note that x,, +, = X and V,, + , = 0. We then apply the recursive procedure described below.

In general, we consider an interval [xk, I~+,], where x/, , , and V, , , are known from the previous stage but where xk and V, are unknown. We use the recursive calculation (54) to find r = r(k) such that

(r;,},(h,)>E,(V,.,)>(~~},(h,+,). (A.22)

If we assume in the first instance that j(k + 1) = m then u,

Train control 531

two intervals [x,, x,+,1 and [xn_, , x,,], where the form of the key equations is atypical.

This calculation determines a strategy of optimal type. It is necessary to adjust the parameters A and p to obtain a feasible strategy. It is anticipated that these adjustment procedures can be adapted from previous work (Cheng and Howlett, 1992). In general, we note that the critical holding speed w& can be estimated approximately from the average speed for the journey. In fact, w& will be greater than the average speed. Thus we could use an initial estimate for p given by

(A.25)

On level track the value of A must be greater than E,(w,), and so we can begin with

A,,, = EJ+S,(W*eS,) + 6. (A.26)

The Scheduling and Control Group has developed a prototype computer program that is designed to perform the above calculations on board a long-haul freight train in real time.

A.4. The properties of { %P}k(~) In this section we simply emphasise that the function

{9}&) is continuous and has a continuous derivative in the region x E [xk, Q+,]. When x E (h,, h,+,), it is obvious that these properties are true and that

(A.27)

where u = u(x). By using this formula and considering the appropriate left- and right-hand limits, it is easy to show that {S&,h(x) and {5$,};(x) are continuous at x = h,.

A.5 Approximate coast-power strategies It has been shown elsewhere (Howlett et al., 19946) that

any time interval [u,, ub] of non-negative measurable control f(t) can be approximated as accurately as we please by a sequence of coast-power controls. It is convenient to review the argument. In essence, we replace the actual speed profile on each small time subinterval [uk, u*+J by the speed profile resulting from a power-coast-power strategy, where the control changes are timed so that the distance travelled in the given interval is the same as the distance travelled with the original control. We need to show that the fuel consumption for the alternative strategy is essentially the same as the fuel consumption for the original strategy.

On each time interval [uk, uk+,] we have

(A.28)

AJO., = (qk - u,J + (uk+] - rk), (A.29)

where [ukr ~~1, [qk, rk] and [rkr uk+i] are the time subintervals for the new power-coast-power controls. By integrating the appropriate equations of motion, we obtain equivalent energy balance equations. If we write

G(x)=-lg&=-Igudt (A.30)

then for the original strategy we have

I

k+I [:u; + G(x~)]~::+ = [Hf -u,&)] dt, (A.31)

t

and for the alternative strategy we have

]:u& * G(x,,)ll::+l = H](qk - UL)

I

u/l+, + (ak+l - r.dl - ~O.d~O.d df. (A.32)

Uk

By comparing the two equations, it follows that

I u*+,

](9k - u,4) + &+I - rk)l - f dt u*

[%.iP(%,i) - UjP(Uf)I dt, (A.33)

and, by choosing a sufficiently fine subdivision, we can ensure that uO., is as close as we please to u, with

(4k - uk) + &+I - rk) - (A.34)

It follows that

and hence the fuel consumption for the alternative strategy can be made arbitrarily close to the fuel consumption for the original strategy.

A.6. Necessary conditions for a strategy of optimal type We define a Lagrangian function of the form

S5 A, IL) = HJ(5) + A]X -r(5)] + ME) - T], (A.36)

where J(t) is the cost of the journey, x(E) is the distance travelled, t(t) is the time taken and A and p are Lagrange multipliers. To find the minimum cost strategy for the given sequence of throttle settings, we apply the Kuhn-Tucker conditions

$=O for all k k

and the complementary slackness conditions

A[X -x(t)] = 0, p[t([) - T] = 0. (A.38)

If we weaken the equality constraints to read x(t) 2 X and t(c)5 T then we can also guarantee that A and p are non-negative. It is intuitively obvious that the weakened problem has the same solution. We shall assume that 5 lies in an open set such that V,(t) # WI,~k+,~,r~k~l and such that u,(.$) # W,,Ck+,j,rl for each r with r(k) + 1 r r 5 (k + 1) and each k = 1,2, . , n.

In order to apply the Kuhn-Tucker equations, it is necessary to calculate a number of partial derivatives. Because the relevant formulas are rather complicated, it has been convenient to introduce a number of gradient-weighted averages that enable us to write these expressions in a reasonably coherent way. Direct application of the Kuhn-Tucker conditions gives a set of equations that can be simplified by an essentially inductive argument. An outline of the argument is given below. Details of the derivation are quite tedious, and are contained in an expanded version of this paper, which is available as a University of South Australia Research Report (Howlett, 1993).

For convenience, we shall use the notation

The condition

atO z-

(A.39)

(A4

can be simplified to give

(-l)A+p+=O. (A.41) n

532 P. Howlett

When the condition

is combined with (A.41) we can deduce that

In similar fashion. we can combine the condition

with (A.41) and (A.43) to deduce that

If we assume that

(A.42)

(A.43)

(1 - Qd%[ ( p +Hf;v+,d(~-+-j

+&r(h+l)lWh+l) = I H[f;(h+t~ -hd

v, (A 46)

for each h with k < h