Calculation of optimal driving strategies for freight trains

Transcription

1 Calculation of optimal driving strategies for freight trains P. Howlett, P. Pudney & X. Vu Centre for Industrial and Applied Mathematics (CIAM), University of South Australia. Phil Howlett is Professor of Industrial and Applied Mathematics. Peter Pudney is a Senior Research Fellow. Xuan Vu is a PhD student. 1

2 Abstract The problem of finding a driving strategy that minimises fuel consumption subject to completing the required journey within a given time is now well understood. On level track the solution is a power, speedhold, coast, brake strategy in which the dominant speedhold mode is a singular control. This form of solution is preserved on non-steep track but on steep track it is necessary to switch to power before the train reaches a steep incline and to coast before a steep decline. Elsewhere the speedhold mode is maintained. We will consider the calculation of key switching points and present a new formulation of the problem and a corresponding numerical solution procedure. We hope the new procedure will provide a constructive proof that the optimal strategy is uniquely determined by the holding speed. 2

3 Applications A rather encouraging feature of this work is that Sydney rail consultants, TMG International have used algorithms developed by the Centre for Industrial and Applied Mathematics (CIAM) at UniSA to produce an on-board system, known as FreightMiser, that will provide advice to train drivers about optimal driving strategies. FreightMiser is currently being tested by Pacific National for use on Australian freight trains. In-service trials indicate that fuel savings of more than 10% will be possible. This could amount to many millions of dollars per annum. 3

4 Previous Research A key paper by Asnis et al (1985) found an optimal strategy which minimised mechanical energy subject to uniformly bounded acceleration on a level track. A similar approach was used by Howlett (1984, 1990). A more realistic model which sought to minimise fuel consumption and allowed only discrete throttle settings was formulated by Benjamin et al (1989) and subsequently solved in a sequence of papers by Howlett, Pudney and Cheng. A corresponding formulation and solution with continuous throttle settings was given by Khmelnitsky (1994, 2000). Howlett (2000) gave a systematic review of the topic while Howlett and Leizarowitz (2001) showed that discrete control could mimic the optimal continuous control strategy by chattering. 4

5 Key references # 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 Phil. Howlett, 1984, The optimal control of a train, Study Leave Report, School of Mathematics, University of South Australia. Phil. Howlett, 1990, An optimal strategy for the control of a train, J. Aust. Math. Soc. Ser. B 31, pp Benjamin, B.R., Milroy, I.P. and Pudney, P.J., 1989, Energy efficient operation of long haul trains, Proceedings of the Fourth International Heavy Haul Railway Conference, IE Aust., Brisbane, pp P.G.Howlett, I.P.Milroy and P.J.Pudney, 1994, Energyefficient train control, Control Engineering Practice, 2, No.2, pp Phil. Howlett, 1996, Optimal Strategies for the Control of a Train, Automatica, 32, No. 4, pp

6 Key references # 2 Cheng, J., 1997, Analysis of optimal driving strategies for train control problems, PhD thesis, University of South Australia. Phil. Howlett and Cheng Jiaxing, 1997, Optimal Driving Strategies for a Train on a Track with Continuously Varying Gradient, J. Aust. Math. Soc. Ser. B, 38, pp 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 Journal of Mathematics Applied in Business and Industry, 10, pp Khmelnitsky, E., 1994, On an optimal control problem of train operation, Report for the Faculty of Engineering, Department of Industrial Engineering, Tel-Aviv University. Eugene Khmelnitsky, 2000, On an Optimal Control Problem of Train Operation, IEEE Transactions on Automatic Control, 45, No. 7, pp Phil Howlett, 2000, The optimal control of a train, Annals of Operations Research, 98, pp P. G. Howlett and A. Leizarowitz, Optimal strategies for vehicle control problems with finite control sets, Dynamics of Continuous, Discrete and Impulsive Systems, B: Applications & Algorithms, 8, pp ,

7 Formulation of the problem We wish to minimise X p J = 0 v dx subject to the equations of motion dt dx = 1 v v dv dx = p q + g(x) r(v). v We define the Hamiltonian H = α p + β [ p q + g(x) r(v) v v v and the adjoint equations dα dx = 0 dβ dx = [2p qv + g(x)v ϕ(v) + ψ(v)] β v 3 + α p v 2 where ϕ(v) = vr(v) and ψ(v) = v 2 r (v). We assume that ϕ(v) is convex and (hence) that ψ(v) is increasing. ] 7

8 The optimal controls To find the optimal strategy we maximise the Hamiltonian subject to the constraints 0 p P, 0 q Q and 0 t T. There are four possible optimal controls. 1. β > v, p = P and q = 0; 2. β = v, p (0, P ) and q = 0; 3. β (0, v), p = 0 and q = 0; and 4. β < 0, p = 0, q = Q. If β = v over a non-zero interval then dβ dx = dv ψ(v) + α = 0. dx Since α is constant we must have v = V. Hence this phase is speedholding with speed V. The other optimal phases are power, coast and brake. 8

9 The optimal strategy On level track the optimal strategy is simply power, hold, coast and brake. This remains true if there are no steep sections. The track is steep uphill at speed V if the speed V cannot be maintained under full power. Thus P ϕ(v ) P + g(x) r(v ) < 0 g(x) <. V V In such cases it is necessary to switch to power before the train reaches the steep section. We want to find the correct switching point. If we define then η = β v 1 dη dx ψ(v) + P ψ(v) ψ(v ) v 3 η = v 3. The equation shows that switches from hold to power and power to hold occur when v = V, η = 0 and dη dx = 0. 9

10 The usual calculation To calculate the switching points for the optimal strategy v 0 (x) we use the conditions on the previous page. If the steep section is the interval [b, c] then we need to find a < b and d > c such that we switch from hold to power at x = a and from power to hold at x = d. We define an integrating factor I 0 (x) = x 0 exp [ ( 1) ψ(v 0) + P v 0 3 where we write v 0 (ξ) = v 0 for convenience. By integrating from x = a to x = d we find the necessary condition [ ] d ψ(v0 ) ψ(v ) I 0 (x)dx = 0 (1) a v 0 3 where this time we have written v 0 (x) = v 0. This condition has been used to calculate the switching points. The usual procedure is to guess the starting point x = a, evaluate the integral and refine the guess. ] dξ 10

11 Infinitessimal variations to the optimal speed If the speed changes from the optimal profile v 0 (x) by an infinitessimal increment δv(x) then by neglecting second order terms it follows from the equation of motion that δv dv [ ] 0 dx + v d(δv) P 0 dx = ( 1) v 2 + r (v 0 ) 0 which is equivalent to [ d(v 0 δv) ψ(v0 ) + P + dx v 3 0 Thus ] (v 0 δv) = 0. (v 0 δv)(x) = V δv(a)i 0 (x) and we can rewrite condition (1) in the alternative form [ d a r (v 0 ) ψ(v ) v 0 2 ] δv dx = 0 (2) where we have written δv(x) = δv for convenience. 11

12 A variational principle for optimal switching For each candidate optimal speed profile v(x) define a special cost functional J 0 (v) = [ d a r(v) + ψ(v ) v ϕ (V ) where a = a(v) and d = d(v) are the switching points. Clearly v(a) = v(d) = V and hence the integrand ψ(v ) + r(v) ϕ (V ) v is zero at the endpoints x = a and x = d. If we consider an infinitessimal variation δv to the optimal profile then it follows that δj 0 = [ d a r (v) ψ(v ) v 2 ] ] dx δv dx (3) Hence condition (2) is a necessary condition for a minimum of J 0. Thus the optimal strategy v = v 0 is the one that minimises J 0. 12

13 A simple steep incline Suppose that where g(x) = γ 0 for x < b γ 1 for b < x < c γ 2 for c < x γ 1 < ϕ(v ) P V < γ j for j = 0, 2. We wish to find an interval (a, d) with (b, c) (a, d) and a speed profile v 0 (x) defined by a power phase on (a, d) with v 0 (a) = V and v 0 (d) = V that minimises J 0 (v) over all possible choices of (a, d). Note that the speed increases from v a = V to v b on (a, b), decreases from v b to v c on (b, c) and increases from v c to v d = V on (c, d). In this case we will show that the functional J 0 (v) can be expressed in terms of elementary integrals. 13

14 Integrating the equation of motion By integrating the equation of motion directly we have d r(v)dx a = a d P v dx γ 0(b a) γ 1 (c b) γ 2 (d c) where we have written v(x) = v for convenience. On the other hand we can separate the variables to show that d a = c b + vb V v 2 dv P γ 0 v ϕ(v) V v 2 dv + P γ 2 v ϕ(v) v c and hence also deduce that d a dx v = vb V vdv P γ 0 v ϕ(v) vb vdv + v c ϕ(v) + γ 1 v P V vdv + P γ 2 v ϕ(v). v c 14

15 An elementary expression for J 0 The previous equations can be used to write J 0 (v b, v c ) = ( ψ(v ) + P ) [ vb + vb v c vdv ϕ(v) + γ 1 v P + V v c V vdv P γ 0 v ϕ(v) vdv P γ 2 v ϕ(v) γ 0 (b a) γ 1 (c b) γ 2 (d c) ϕ (V ) + [ c b + V v c vb V v 2 dv P γ 2 v ϕ(v) v 2 dv P γ 0 v ϕ(v) where we note that J 0 is now expressed as a function of v b and v c. We wish to minimise J 0 subject to the distance constraint ] ] c b vb v c v 2 dv ϕ(v) + γ 1 v P = 0. 15

16 The Lagrangian We define J 0 = J 0 + λ ( c b vb v c v 2 dv ϕ(v) + γ 1 v P ) and solve the KKT equations J 0 v b = 0 and J 0 v c = 0 subject to the complementary slackness condition ( λ c b vb v c v 2 dv ϕ(v) + γ 1 v P ) = 0. 16

17 Necessary conditions If we make the transformation µ = λ ϕ (V ) γ 1 then the necessary conditions become ϕ(v b ) = L µ,γ0 (v b ) and ϕ(v c ) = L µ,γ2 (v c ) (4) where the linear function L µ,γ (v) has gradient m µ,γ = γ + (ϕ (V ) + γ)(γ 1 γ) µ + (γ 1 γ) passes through the fixed point ( ψ(v ) + P (ψ(v ) + P )γ ϕ, P (V ) + γ ϕ (V ) + γ Since ϕ(v) is convex there are at most two solutions to each equation. When µ = 0 the line y = L 0,γ (v) is the tangent to the curve y = ϕ(v) at the point v = V. When µ > 0 the line y = L µ,γ (v) cuts the curve y = ϕ(v) at two points v 1,γ and v 2,γ with v 1,γ < V < v 2,γ. ). 17

18 The unique solution Since v c < V < v b there is only one possible solution to each equation. Furthermore as µ increases the slopes of the two lines y = L µ,γ0 (v) and y = L µ,γ2 (v) decrease and we can see that the solution v b to the equation ϕ(v b ) = L µ,γ0 (v b ) increases and the solution v c to the equation ϕ(v c ) = L µ,γ2 (v c ) decreases. On the other hand the constraint vb v 2 dv c b v c ϕ(v) + γ 1 v P = 0 means that if we increase v b we must also increase v c. Hence there is precisely one value of µ for which the necessary conditions are satisfied. Thus the solution is unique. Once we know v b and v c we can calculate a = a(v b ) and d = d(v c ) from and a = b d = c + vb V V v c v 2 dv P γ 0 v ϕ(v) v 2 dv P γ 2 v ϕ(v). 18