Local energy minimization in optimal train control

Local energy minimization in optimal train control ? P. G. Howlett a , P. J. Pudney a , Xuan Vu a a

Centre for Industrial and Applied Mathematics, Mawson Lakes Campus, University of South Australia, Mawson Lakes, Australia, 5095

Abstract The calculation of optimal driving strategies for on–board control of freight trains is a challenging task. In this paper we show how to calculate the critical switching points for an optimal strategy on a track with steep gradients using a new local energy minimization principle. The method has been used successfully in Australia to calculate optimal switching points and hence provide in–cab advice to train drivers on long–haul freight trains. Key words: optimal train control; energy minimization; calculus of variations.

1

Introduction

We present a brief summary of the literature on optimal train control which we consider from two distinct viewpoints. The first is the development of the modern theory, mostly in the period from 1985—2004; the second is the development of computational algorithms that can be used on board freight trains to compute optimal speed profiles. We then consider the relationship of these specialist methods to more general techniques. 1.1

Optimal train control

There is a substantial literature on the mathematical basis for optimal train control. The genesis of the currently accepted theory can be found in the papers by Asnis et al [1] and Howlett [5] but these papers use acceleration as the driving control and assume that the acceleration is uniformly bounded. This is not the case in real trains. A systematic study by the Scheduling and Control Group (SCG) at the University of South Australia introduced a more realistic model based on the performance characteristics of a typical diesel–electric locomotive [8] in which the driving control for the locomotive is a throttle setting that determines the rate of fuel supply to the motor. In many cases only a limited number of discrete throttle settings are available and the power developed ? This paper was not presented at any IFAC meeting. Corresponding author P. G. Howlett. Email addresses: [email protected] (P. G. Howlett), [email protected] (P. J. Pudney), [email protected] (Xuan Vu).

Preprint submitted to Automatica

by the locomotive is directly proportional to the rate of fuel supply except at very low speeds. Because u is normally used to denote acceleration and because the driving control in real freight trains is essentially power per unit mass we have used p as the driving control where p = u/v and v is the speed. The SCG study is described in a series of papers, a research monograph and two PhD theses. See [11,18] for a list of references. The study showed [7] that the motion of a train with distributed mass can be modelled as the motion of a point mass train on a track with modified gradient and that any policy of continuous control can be approximated as closely as desired by alternating levels of {0, 1} power control. In practice this allows trains with only discrete control to follow, as closely as desired, the speed profile for an optimal policy with continuous control and indeed to keep the cost of this strategy arbitrarily close to the optimal cost. The theory of discrete control has been extended to continuous control by Khmelnitsky [14] and others [11,12,16] and to control of solar–powered cars [4,10]. In this paper we present new formulae developed by Vu [18] that are used to continually update optimal speed profiles on–board modern long–haul locomotives. The key result is that the switching points in a globally optimal control strategy on steep track can be calculated by a local energy minimization principle. 1.2

Computational methods for optimal control

When the Hamiltonian of the optimal control problem is linear in the control variables and the control variables have simple bounds, it is well known that the optimal

19 June 2009

control is a combination of bang–bang and singular arcs. In such cases the computational problem is reduced to one of finding the optimal switching points efficiently and accurately. More generally it may be possible to approximate the optimal control by a switching sequence using only a limited number of well–defined control regimes.

more accurate and more efficient in practice because we use particular knowledge that is special to the train control problem. The novelty of our method is that we can use a local optimization to find the precise switching points for the globally optimal strategy.

For instance Lee et al [15] use the Control Parametrization Enhancing Technique (CPET) to show that certain optimal discrete–valued control problems are equivalent to optimal control problems involving a new control function which is piecewise constant with pre–fixed switching points. The transformed problems are essentially combinatorial optimization problems involving optimal parameter selection and can be readily solved by various existing algorithms. Bengea and DeCarlo [2] consider an optimal control problem for a switching system. They make no assumptions about the number of switches nor about the mode sequence. The switching system is embedded into a larger family of systems in such a way that the set of trajectories of the switching system is dense in the set of trajectories for the larger system. They find sufficient and necessary conditions for optimality of the larger system and show, if they exist, that bang-bang–type solutions of the larger problem are also solutions to the original problem. Otherwise suboptimal (approximate) solutions are obtained via a Chattering Lemma. Xu and Antsaklis [19] formulate optimal control problems for both continuous–time and discrete–time switched systems as a two stage optimization problem. They discuss solution algorithms and open problems. Kaya and Noakes [13] propose a time– optimal switching (TOS) algorithm to find the switching control of nonlinear systems with a single control input. A feasible switching control is found by shooting from an initial point to a target point with a given number of switches and then an optimal representative is found by minimizing the sum of the arc times using constrained optimization to preserve initial and final conditions.

2

Model description and review

Howlett and Pudney [7] have shown that the motion of a train with distributed mass can be reduced to the motion of a point mass train. For a point mass train the equation of motion can be written as vv 0 =

p − q − r(v) + g(x) v

where x is the position of the train, v = v(x) is the speed, p = p(x) is the controlled power per unit mass, q = q(x) is the controlled braking force per unit mass, r(v) is the resistance force per unit mass and g(x) is the component of gravitational acceleration due to the track gradient at position x. The elapsed time t = t(x) satisfies the differential equation t0 =

1 v

and the total time taken is ZX t(X) =

1 dx v

0

where v = v(x) is the solution to the equation of motion. It is important to understand that the problem is naturally formulated with x as the independent variable because the track gradient is a function of position.

More recently Maurer et al, [17] consider second order sufficient conditions (SSC) which are particularly suited for numerical verification. They present optimization methods and describe a numerical scheme for finding optimal bang–bang controls and verifying SSC.

2.1

A note about the power control

The equation of motion can be re–written as vv 0 = u(x, v) − q − r(v) + g(x)

In optimal train control Howlett and Leizarowitz [12] show that optimal controls for problems with continuous control can be realized by a chattering {0, 1} control or approximated by a suboptimal control with a finite sequence of alternating {0, 1} controls. More generally it is known [11,14,16] that on track with steep grades the optimal control is a complex sequence that depends on the grades but invokes only four distinct modes—power, hold, coast, brake. Our aim in this paper is to find efficient computational algorithms to determine the precise switching points. There is no doubt that in theory the general methods described above could be used but it is equally true that our proposed method will be quicker,

where the acceleration u = u(x, v) is constrained by the non–uniform bound 0 ≤ u(x, v) ≤ min(A, P/v) where A is the observed maximum acceleration at low speeds and P is the maximum power. Graphs of tractive effort and braking effort against speed for a typical GM locomotive can be found in the paper by Howlett [8]. Our simplified model with u = p/v allows trains to start slightly more quickly but because the models are identical except at very small speeds there is no discernible difference in the results. We use power as the driving control because the train driver controls the level of power.

2

2.2

with a unique point c ∈ (0, X) defined by vP (c) = vQ (c) and controls   [P, 0] for x ∈ (0, c) [p(x), q(x)] =  [0, Q] for x ∈ (c, X).

Frictional resistance

Let r : [0, ∞) 7→ [0, ∞) be the frictional resistance per unit mass. We write ϕ(v) = v · r(v) and ψ(v) = v 2 · r 0 (v). The formula r(v) = a + bv + cv 2 is often used in practice but we assume only that ϕ(v) is non–negative and strictly convex with ϕ(0) = 0 and that ϕ(v)/v → ∞ as v → ∞. We recall an important result [9].

40

35

Lemma 1 Let ϕ : [0, ∞) 7→ [0, ∞) where ϕ is strictly convex with ϕ(0) = 0 and r(v) = ϕ(v)/v → ∞ as v → ∞ and suppose ψ : [0, ∞) 7→ [0, ∞) is defined by the formula ψ(v) = v · ϕ 0 (v) − ϕ(v) = v 2 · r 0 (v). Then ψ(v) is strictly increasing with ψ(0) = 0 and ψ(v) → ∞ as v → ∞.

30

25

v 20 15

Later we write LV (v) = ϕ(V ) + ϕ 0 (V )(v − V ) to denote the tangential approximation to ϕ(v) at v = V . The strict convexity means ϕ(v) − LV (v) > 0 when v 6= V .

10

5

0

2.3

0

2000

4000

6000

2.5

and the total cost is

J(X) =

12000

14000

Feasible control strategies

A feasible control strategy is one in which the journey is completed within a prescribed time T . To pose a meaningful optimal control problem we need to know that a feasible strategy exists and hence that the control set is non–empty. A feasible strategy can be constructed whenever T ≥ Tmin . Choose W such that 0 < W < maxx∈[0,c] vP (x) and define a control function

p v

ZX

10000

Fig. 1. The minimum time power–brake journey on flat track and various longer time journeys using power–coast–brake and power–hold–coast–brake

The cost is usually taken to be the total fuel consumed by the train. If we assume that power per unit mass is directly proportional to the rate of fuel supply then the cost J = J(x) satisfies the differential equation J0 =

8000

x

The cost of a control strategy

p dx. v

0

pW (x) = min [ϕ(W ) − W g(ξ), P ] ξ∈(0,X)

The rate of fuel supply is bounded and hence the power per unit mass p = p(x) is bounded with 0 ≤ p ≤ P for some fixed P . We also assume that the braking force per unit mass q = q(x) is bounded with 0 ≤ q ≤ Q for some fixed Q. The cost of braking is generally ignored. 2.4

with a corresponding speed profile vW = vW (x) satisfying the equation of motion with control pW and initial condition vW (0) = W . The speed profile vf = vf (x) defined by vf (x) = min[vP (x), vW (x), vQ (x)],

The minimum time journey

where vP and vQ are the speed profiles described in the previous subsection, corresponds to a feasible control strategy given by   [P, 0] if vf (x) = vP (x)    [p(x), q(x)] = [pW (x), 0] if vf (x) = vW (x)     [0, Q] if vf (x) = vQ (x)

Let v = vP (x) be the unique solution for x ≥ 0 to vv 0 =

P − r(v) + g(x) v

with vP (0) = 0 and let v = vQ (x) be the unique solution for x ≤ X to vv 0 = −Q − r(v) + g(x)

for all x ∈ [0, X]. The time taken for this strategy will be at least Tmin . If the track has no steep uphill sections at speed W then this strategy takes the form of a power– hold–brake strategy with hold speed W . If the track has

with vQ (X) = 0. The minimum time speed profile v = v(x) with v(x) = min[vP (x), vQ (x)] for x ∈ [0, X] and with terminal time T = Tmin is a power–brake strategy

3

steep uphill sections at speed W then the hold phase will be interrupted by one or more power phases. There may also be feasible strategies in the form power–coast–brake and power–hold–coast–brake as shown in Figure 1. 2.6

3

The optimal control problem is formulated as follows. Problem 1 Let T ≥ Tmin . Find controls p = p0 (x) and q = q0 (x) and a corresponding speed profile v = v0 (x) and time function t = t0 (x) satisfying the differential constraints

If possible a hold strategy is best

If a train travels from (x, v) = (a, V ) to (x, v) = (b, V ) at constant speed then it consumes less fuel than any other strategy from (a, V ) to (b, V ) taking the same time. This is shown by integrating the equation of motion to calculate the cost. See Howlett and Pudney [9] for details. 2.7

vv 0 =

Although hold is the most cost–effective form of control it may not be possible when the track becomes steep. The track is steep uphill at speed V on the interval [b, c] if speed V cannot be maintained or increased under full power. Thus ⇔

g(x) <

p − q − r(v) + g v

and

t0 =

1 v

with the additional speed constraints v(0) = v(X) = 0 and time constraints t(0) = 0 and t(X) = T in such a way that the cost ZX p J= dx v

Steep uphill track

P + g(x) − r(V ) < 0 V

Optimal control strategies

0

is minimized. It is possible to incorporate speed limits into the formulation and solution of the problem [3,14]. In essence the optimal strategy is identical to the unlimited strategy except where the bounds are violated in which case the speed must follow the bound.

ϕ(V ) − P V

for x ∈ [b, c]. The Hamiltonian is defined by V

  β−p α H= + 2 · p − qv − ϕ(v) + gv v v

V

where the adjoint variables α = α(x) and β = β(x) are solutions to the differential equations α0 =

Fig. 2. Speed decreases on a steep uphill section even if full power is applied

2.8

  β−p α + · 2p − qv − ϕ(v) + ψ(v) + gv v2 v3

β 0 = 0.

Steep downhill track

The second equation shows that β = b for some b ∈ R. The track is steep downhill at speed V on the interval [b, c] if speed V cannot be maintained without braking. Thus g(x) − r(V ) > 0

⇔

3.1

g(x) > r(V )

The Pontryagin principle

We wish to maximize the Hamiltonian subject to the constraints 0 ≤ p ≤ P and 0 ≤ q ≤ Q. Thus we define a Lagrangian function

for x ∈ [b, c].

H = H + ρp + σ(P − p) + τ q + ω(Q − q) V

where ρ, σ, τ and ω are Lagrange multipliers and maximize H pointwise subject to the given constraints. This shows four possible modes of optimal control [11,14,16].

V

(1) (2) (3) (4)

Fig. 3. Speed increases on a steep downhill section even if coasting

4

α > v, p = P and q = 0; α = v, p ∈ (0, P ) and q = 0; α ∈ (0, v), p = 0 and q = 0; and α < 0, p = 0, q = Q.

If we define vmin for x ≥ b by setting vmin (b) = V then a = b and vmin (x) < V for x ∈ (a, d) for some d > c and the integral is negative. If we define vmax for x < c by setting vmax (c) = V then d = c and vmax (x) > V for all x ∈ (a, d) for some a < b and the integral is positive. Thus vmin < v0 < vmax and the optimal speed v0 is found by searching between the two bounds. Because the variation is continuous there must be at least one solution. Note that the graph η = η(x) for x ∈ [a, d] takes a characteristic form for the optimal strategy. Suppose v = v0 (x) is the solution to the equation

If the singular control α = v is maintained over a non trivial interval then α 0 (x) = v 0 (x) and so ψ(v) + b = 0 where b ∈ R is the constant value of β. Since Lemma 1 shows that ψ(v) is strictly increasing it follows that this equation has a unique solution v = V for some V > 0. Hence this is a hold mode with [p, q] = [ϕ(V ) − V g, 0] and constant speed V . The other optimal modes are maximum power with [p, q] = [P, 0], coast with [p, q] = [0, 0] and brake with [p, q] = [0, Q]. 3.2

The optimal strategy on level track

vv 0 =

On level track with g = 0 the optimal strategy is simply power–hold–coast–brake. The speed VB at which braking begins is determined from the hold speed V by the formula VB = ψ(V )/ϕ 0 (V ). If the time is relatively short then the hold phase is omitted. Details can be found in [7,11] and references contained therein.

on [0, X] with v0 (a) = v0 (d) = V and suppose the adjusted adjoint variable satisfies η(a) = η(d) = 0 with η(x) > 0 for x ∈ (a, d). Then η 0 (a) = 0

3.3

η0 −

ψ(v0 ) + P ψ(v0 ) − ψ(V ) ·η = v03 v03



Zx

V

I0 (x) = exp (−1)

3

0

3

v0 (x)

4

d

The new results

In this section we show that the necessary conditions defining the optimal switching points for a power phase over a steep uphill section of track or for a coast phase over a steep downhill section of track are also necessary conditions for minimization of local energy usage subject to a weighted time penalty. This minimization has been adopted as a more efficient means to compute optimal switching points for the in–cab driver advice system known as FreightmiserTM . For track with piecewise constant gradient we will also derive an analytic expression for the adjoint variable on each section of constant gradient and show that the local minimization implies continuity of the adjoint variables across discontinuities in the gradient.

dξ 

# I0 (x)dx = 0.

c

A similar analysis can be applied to a coast phase on steep downhill track. In this case the optimal control should switch from hold to coast at some point x = a before the steep section and from coast to hold at some point x = d beyond the steep section.

and integrate the adjusted adjoint equation to find the necessary condition ψ[v0 (x)] − ψ(V )

b

Fig. 4. Optimal speed profile on a steep uphill section.

 v0 (ξ)

V

a

(1)

ψ[v0 (ξ)] + P

ψ 0 (V )v00 (a) >0 V3

and hence the graph η = η(x) has a minimum turning point at x = a. The same argument shows a minimum turning point at x = d too.

with η(x) > 0 for x ∈ (a, d) and η(a) = η(d) = 0. The switches occur when v0 = V and η = 0 which implies η 0 = 0 too. The precise switching points can be found by integrating the equation for the adjusted adjoint variable. Define the integrating factor

a

and η 00 (a) =

The optimal strategy on steep track

On track with steep gradients a power–hold–coast–brake strategy may not be feasible and it will be necessary to interrupt the hold phase with one or more power phases on the steep uphill sections or with one or more coast phases on the steep downhill sections [11,14]. For individual steep uphill sections one might expect that the optimal control should switch from hold to power at some point x = a before the steep section and from power to hold at some point x = d beyond the steep section. This is shown in Figure 4. Consider a power segment with [p, q] = [P, 0] and follow the methodology suggested by Howlett [11] to find the precise switching points. If we define an adjusted adjoint variable η = α/v − 1 and consider an optimal profile v = v0 then

Zd "

P − r(v) + g(x) v

(2)

5

4.1

from which it follows that  Z  ψ(v) + P vv 0 = exp − dx v3

A local optimization to determine the switching points for a steep section

We consider an infinitesimal perturbation δv = δv(x) to the optimal speed profile v0 = v0 (x) during a power phase on a steep uphill section. The perturbation satisfies the differential equation

is an integrating factor for (1). After integration we substitute for vv 0 using the equation of motion.

  d P + ψ(v0 ) v0 δv = (−1) · · v0 δv dx v0 3

Now consider a track with piecewise constant gradient with xr−1 < xr < · · · < xs+1 and g(x) = γj for x ∈ (xj , xj+1 ) and where we suppose there is a steep uphill section (b, c) ⊂ (xr , xs ) with switching points a ∈ (xr−1 , xr ) and d ∈ (xs , xs+1 ). If we write v(xj ) = Vj then J = J(Vr , . . . , Vs ) is given by

and hence we see that v0 δv ∝ I0 . Thus the necessary condition for optimal switching (2) can be rewritten as Zd 

 ψ[v0 ] − ψ(V ) δv dx = 0. v0 2

(3)

ZVr J=

a

V

In this form we can see that the necessary condition for optimal switching is also a necessary condition for a minimum of the functional Zd  J(v) =

=

s−1 X

VZj+1

j=r V j

 ψ(V ) + r(v) − ϕ 0 (V ) dx v

ϕ(v) − LV (v) dx v

ZV

[ϕ(v) − LV (v)] vdv P − ϕ(v) + γs v

Vs

+

[ϕ(v) − LV (v)] vdv . P − ϕ(v) + γj v

We wish to minimize J subject to the distance constraints

a

Zd

[ϕ(v) − LV (v)] vdv + P − ϕ(v) + γr−1 v

VZj+1

(4) xj+1 − xj =

a

v 2 dv P − ϕ(v) + γj v

Vj

where v = v(x) is a solution to the equation of motion in power mode and where a = a(v) < b and d = d(v) > c are chosen so that v(a) = v(d) = V .

for each j = r, r + 1, . . . , s − 1. We form a Lagrangian function J = J (Vr , . . . , Vs ) by setting

A similar analysis applies to steep downhill sections with power replaced by coast. 4.2

J =J +

VZj+1

 λj (xj+1 − xj ) −

ηr−1 · vdv +

=

 v 2 dv   P − ϕ(v) + γj v

Vj

ZVr

For a power phase on an interval where g(x) = γ is constant we can integrate both the equation of motion and the adjusted adjoint equations exactly to give

x(W ) − x(V ) =



j=r

Determination of optimal switching points for a track with piecewise constant gradient

ZW

s−1 X

s−1 X

VZj+1

ηj · vdv +

j=r V j

V

+

s−1 X

ZV ηs · vdv Vs

λj (xj+1 − xj )

(5)

j=r

v 2 dv P − ϕ(v) + γv

where we now set λr−1 = λs = 0 and define

V

and

ηj =

ϕ(v) − LV (v) − λj v P − ϕ(v) + γj v

ϕ(v) − LV (v) − λv η= P − ϕ(v) + γv where λ is an arbitrary constant of integration. To solve the adjoint equation we begin by observing that

for each j = r − 1, r, . . . , s. To minimize J subject to the imposed constraints we solve the equations

 0  v 00 v ψ(v) + P = (−1) + v0 v v3

∂J =0 ∂Vj

6

where ∆x = d − a and ∆t = t(d) − t(a). Hence we minimize the difference between the work done by the proposed strategy and an hypothetical hold strategy at speed V subject to a weighted penalty term given by the difference between the time taken for the proposed strategy and the time taken for the hold strategy. This is a natural minimization to expect since the original problem was to find a strategy that minimized energy consumption subject to completion of the journey within a specified time.

to find the necessary conditions ηj−1 (Vj ) − ηj (Vj ) = 0

(6)

for each j = r, r + 1, . . . , s. This means that the adjusted adjoint variable η = η(v) is continuous at the points xj where the gradient changes. We can summarize this result as follows. Theorem 1 Suppose 0 < xr−1 < · · · < xs+1 < X and assume that g(x) = γj for x ∈ (xj , xj+1 ). Let (b, c) ⊂ (xr , xs ) be a steep uphill section of track relative to the desired hold speed V and suppose that v = v(x) is a solution to the equation of motion in the power mode with v(a) = v(d) = V for some points a ∈ (xr−1 , xr ) and d ∈ (xs , xs+1 ). For x ∈ (xj , xj+1 ) and all j = r − 1, r, . . . , s the adjusted adjoint variable η = η(x) is given by the formula η = ηj where ηj =

4.5

We consider a typical train with hold speed V = 20 passing over a steep section of track from x = 5000 to x = 6800. Let x1 = 5000, x2 = 5600, x3 = 6000, x4 = 6500 and x5 = 6800 with constant gradient accelerations on each interval given by γ0 = −0.075 for x < x1 , γ1 = −0.220 for x ∈ (x1 , x2 ), γ2 = −0.270 for x ∈ (x2 , x3 ), γ3 = −0.150 for x ∈ (x3 , x4 ), γ4 = −0.200 for x ∈ (x4 , x5 ) and γ5 = −0.090 for x > x5 . We have used P = 3 and r(v) = r0 + r1 v + r2 v 2 where r0 = 0.00675, r1 = 0 and r2 = 0.00005. These values are typical of the values used in practice. The speed profiles v = v(x) for three possible power phases are shown in Figure 5 with the corresponding profiles for the adjoint variable η = η[v(x)] shown in Figure 6. An optimal phase must start and finish with v = V and η = 0. The first power phase starts too early at (x, v) = (2500, 20) and terminates at (x, v) = (8075, 20) but with η > 0; the second phase starts at (x, v) = (3399, 20) and finishes at (x, v) = (8171, 20) with η = 0 and hence is optimal; and the third phase starts too late at (x, v) = (4400, 20) and terminates at (x, v) = (8293, 20) with η < 0. The local cost function is shown in Figure 7 for various starting points. The corresponding phase plots of η against v, all starting at (v, η) = (20, 0), are shown in Figure 8. We use an adaptive Runge–Kutta scheme to find the speed profile v = v(x) for a selection of possible starting points and the new formula η = η(v) to calculate the modified adjoint variable quickly and accurately at critical points. Our specific procedure finds the optimal strategy more efficiently and accurately than a general scheme.

ϕ(v) − LV (v) − λj v P − ϕ(v) + γj v

and where v = v(x) and λj ∈ R is a constant. If v = v(x) is an optimal strategy then λr−1 = λs = 0, λj ≥ 0 for all j = r, r + 1, . . . , s − 1, η = η(x) is continuous and non– negative for all x ∈ (a, d) and η has minimum turning points at x = a and x = d with η = 0 in each case. A similar analysis applies to determination of switching points for a coast phase on a steep downhill section. 4.3

The continuity condition

Once again we consider a power phase and impose the requirement that ηj−1 (Vj ) = ηj (Vj ). Thus we have ϕ(Vj ) − LV (Vj ) − λj Vj ϕ(Vj ) − LV (Vj ) − λj−1 Vj = P − ϕ(Vj ) + γj−1 Vj P − ϕ(Vj ) + γj Vj from which we deduce that λj = λj−1 − (γj − γj−1 )ηj (Vj ). Since λr−1 = λs = 0 it follows that s X

(γj−1 − γj )ηj (Vj ) = 0.

Table 1 The optimal speed profile showing critical values. i x γ v η λ

(7)

j=r

4.4

A physical interpretation of the local minimization

The cost function in the local minimization can be written in the form  Zd  ∆x + [r(v) − r(V )] dx J(v) = ψ(V ) ∆t − V

An example

(8)

a

7

N/A

3399

-0.075

20.00

0.000

0.00000

1

5000

-0.220

22.63

0.038

0.00549

2

5600

-0.270

19.60

0.059

0.00843

3

6000

-0.150

16.79

0.060

0.00128

4

6500

-0.200

16.98

0.046

0.00357

5

6800

-0.090

16.26

0.032

0.00000

N/A

8171

-0.090

20.00

0.000

N/A

4.2562

Speed profiles of one optimal and two non−optimal strategies 24

23 4.2562

22

21

4.2562

J

20

v(m/s) 19

4.2562

18 4.2562

17

16

15 2000

4.2562 3392

3000

4000

5000

6000

7000

8000

3394

3396

3398

3400

3402

3404

starting points

9000

x

Fig. 7. The local cost for various starting points.

Fig. 5. The optimal and non–optimal speed profiles.

Phase diagrams of the optimal and two non−optimal strategies 0.4 Adjoint variable profiles of the optimal and two non−optimal strategies

0.3

0.4

0.2

0.3

0.1

0.2

0.1

0

η

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 15

η

16

17

18

19

20

21

22

23

24

v −0.5 2000

3000

4000

5000

6000

7000

8000

9000

Fig. 8. Phase plots for the optimal and non–optimal profiles.

x

References

Fig. 6. The optimal and non optimal adjoint variable profiles.

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[1]

I.A. Asnis, 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, U.S.S.R. Comput.Maths.Math.Phys., 25, No.6, pp. 37–44.

[2]

Sorin C. Bengea and Raymond A. DeCarlo, 2005, Optimal control of switching systems, Automatica, 41, 1, pp. 11–27.

[3]

Cheng, J., Davydova, Y., Howlett, P.G. and Pudney, P.J., 1999, Optimal driving strategies for a train journey with non–zero track gradient and speed limits, IMA J Management Math formerly IMA Journal of Mathematics Applied in Business and Industry, 10, pp. 89–115.

[4]

Gates, D. J., & Westcott, 1996, M. Solar cars and variational problems equivalent to shortest paths, SIAM Journal on Control and Optimization, 34(2), pp. 428–436.

[5]

Phil. Howlett, 1990, An optimal strategy for the control of a train, ANZIAM J. formerly J. Aust. Math. Soc. Ser. B, 31, pp. 454–471.

[6]

P.G. Howlett, I.P. Milroy and P.J. Pudney, 1994, Energy– efficient train control, Control Engineering Practice, 2, No.2, pp. 193–200.

Conclusions and future work

We have extended known work on optimal train control to find a local energy minimization problem and a new formula that can be used to calculate optimal switching points on a track with steep gradients. More details of these methods can be found in the PhD thesis by Xuan Vu [18]. Vu also showed that phase plots of η against v provide a useful graphical tool. Indeed the initial and final conditions (v, η) = (V, 0) mean optimal strategies are described by a sequence of closed curves in the phase plane. It is hoped that the new formula η = η(v) and the phase plane analysis can be used to find a constructive proof that the optimal strategy is unique for each journey time with T ≥ Tmin . Vu has already established uniqueness proofs for certain simple gradient profiles but a general proof of this type is not yet known.

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[7]

Howlett, P.G. and Pudney, P.J., 1995, Energy–Efficient Train Control, Advances in Industrial Control, Springer, London.

[8]

Phil. Howlett, 1996, Optimal Strategies for the Control of a Train, Automatica, 32, No. 4, pp. 519–532.

[9]

Phil. Howlett and Cheng Jiaxing, 1997, Optimal Driving Strategies for a Train on a Track with Continuously Varying Gradient, ANZIAM J. formerly J. Aust. Math. Soc. Ser. B, 38, pp. 388–410.

[10]

P. G. Howlett and P. J. Pudney, 1998, An optimal driving strategy for a solar powered car on an undulating road, Dynamics of Continuous, Discrete and Impulsive Systems, 4, pp. 553–567.

[11]

Phil Howlett, 2000, The optimal control of a train, Annals of Operations Research, 98, pp. 65-87.

[12]

P. G. Howlett and A. Leizarowitz, 2001, Optimal strategies for vehicle control problems with finite control sets, Dynamics of Continuous, Discrete and Impulsive Systems, B: Applications & Algorithms, 8, pp. 41–69.

[13]

Kaya, C.Y. and Noakes J.L., 2003, Computational method for time–optimal switching control, Journal of optimization theory and applications, 117, 1, pp. 69–92.

[14]

Eugene Khmelnitsky, 2000, On an Optimal Control Problem of Train Operation, IEEE Transactions on Automatic Control, 45, No. 7, pp. 1257–1266.

[15]

Lee, H.W.J., Teo, K.L., Rehbock, V., Jennings, L.S., 1999, Control parametrization enhancing technique for optimal discrete–valued control problems, Automatica, 35, pp. 1401–1407.

[16]

R. Liu and I. Golovitcher, 2003, Energy–efficient operation of rail vehicles, Transportation Research Part A: Policy and Practice 37, pp. 917–932.

[17]

H. Maurer, C. B¨ uskens, J–H. R. Kim and C. Y. Kaya, 2005, Optimization methods for the verification of second order sufficient conditions for bang–bang controls, Optimal Control Applications and Methods, 26(3), 129–156, 2005.

[18]

Vu, Xuan, 2006, Analysis of necessary conditions for the optimal control of a train, PhD thesis, University of South Australia (see www.amazon.co.uk/Analysis–necessary– conditions–optimal–control/dp/3639120000 and search.arrow.edu.au/main/results? subject = optimality theory).

[19]

Xuping Xu and Antsaklis, P.J., 2004, Optimal control of switched systems: new results and open problems, IEEE Trans. Automatic Control, 4, pp. 2683–2687.

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