# Subway ride

Subway ride | |
---|---|

State dimension: | 1 |

Differential states: | 2 |

Discrete control functions: | 1 |

Interior point equalities: | 4 |

The *subway ride* optimal control problem goes back to work of [Bock1982]**Address: ** *Taiwan***Author: ** *Bock, H.G.; Longman, R.W.***Booktitle: ** *Proceedings of the American Astronomical Society. Symposium on Engineering Science and Mechanics***Title: ** *Computation of optimal controls on disjoint control sets for minimum energy subway operation***Year: ** *1982*

for the city of New York. In an extension, also velocity limits that lead to path-constrained arcs appear.
The aim is to minimize the energy used for a subway ride from one station to another, taking into account boundary conditions and a restriction on the time.

## Contents

## Mathematical formulation

The MIOCP reads as

The terminal time denotes the time of arrival of a subway train in the next station. The differential states and describe position and velocity of the train, respectively. The train can be operated in one of four different modes, series, parallel, coasting, or braking that accelerate or decelerate the train and have different energy consumption.

Acceleration and energy comsumption are velocity-dependent. Hence, we will need switching functions for given velocities .

The Lagrange term reads as

The right hand side function reads as

The braking deceleration can be varied between and a given . It can be shown that only maximal braking can be optimal, hence we fixed to without loss of generality.

Occurring forces are

Details about the derivation of this model and the assumptions made can be found in [Bock1982]**Address: ** *Taiwan***Author: ** *Bock, H.G.; Longman, R.W.***Booktitle: ** *Proceedings of the American Astronomical Society. Symposium on Engineering Science and Mechanics***Title: ** *Computation of optimal controls on disjoint control sets for minimum energy subway operation***Year: ** *1982*

or in [Kraemer-Eis1985]**Address: ** *Bonn***Author: ** *Kr\"amer-Eis, P.***Publisher: ** *Universit\"at Bonn***Series: ** *Bonner Mathematische Schriften***Title: ** *Ein Mehrzielverfahren zur numerischen Berechnung optimaler Feedback--Steuerungen bei beschr\"ankten nichtlinearen Steuerungsproblemen***Volume: ** *166***Year: ** *1985*

.

## Parameters

Symbol | Value | Unit | Symbol | Value | Unit |
---|---|---|---|---|---|

78000 | lbs | 0.979474 | mph | ||

85200 | lbs | 6.73211 | mph | ||

2112 | ft | 14.2658 | mph | ||

700 | ft | 22.0 | mph | ||

1200 | ft | 24.0 | mph | ||

6017.611205 | lbs | ||||

100 | ft | 12348.34865 | lbs | ||

10 | - | 11124.63729 | lbs | ||

0.045 | - | 4.4 | ft / sec | ||

0.367 | - | 106.1951102 | - | ||

32.2 | 180.9758408 | - | |||

1.0 | - | 354.136479 | - |

Parameter | Value | Parameter | Value |
---|---|---|---|

b_0(1) | -0.1983670410E02 | c_0(1) | 0.3629738340E02 |

b_1(1) | 0.1952738055E03 | c_1(1) | -0.2115281047E03 |

b_2(1) | 0.2061789974E04 | c_2(1) | 0.7488955419E03 |

b_3(1) | -0.7684409308E03 | c_3(1) | -0.9511076467E03 |

b_4(1) | 0.2677869201E03 | c_4(1) | 0.5710015123E03 |

b_5(1) | -0.3159629687E02 | c_5(1) | -0.1221306465E03 |

b_0(2) | -0.1577169936E03 | c_0(2) | 0.4120568887E02 |

b_1(2) | 0.3389010339E04 | c_1(2) | 0.3408049202E03 |

b_2(2) | 0.6202054610E04 | c_1(2) | 0.3408049202E03 |

b_3(2) | -0.4608734450E04 | c_3(2) | 0.8108316584E02 |

b_4(2) | 0.2207757061E04 | c_4(2) | -0.5689703073E01 |

b_5(2) | -0.3673344160E03 | c_5(2) | -0.2191905731E01 |

## Reference Solutions

The optimal trajectory for this problem has been calculated by means of an indirect approach in [Bock1982]**Address: ** *Taiwan***Author: ** *Bock, H.G.; Longman, R.W.***Booktitle: ** *Proceedings of the American Astronomical Society. Symposium on Engineering Science and Mechanics***Title: ** *Computation of optimal controls on disjoint control sets for minimum energy subway operation***Year: ** *1982*

[Kraemer-Eis1985]**Address: ** *Bonn***Author: ** *Kr\"amer-Eis, P.***Publisher: ** *Universit\"at Bonn***Series: ** *Bonner Mathematische Schriften***Title: ** *Ein Mehrzielverfahren zur numerischen Berechnung optimaler Feedback--Steuerungen bei beschr\"ankten nichtlinearen Steuerungsproblemen***Volume: ** *166***Year: ** *1985*

, and based on the direct multiple shooting method in [Sager2009]**Author: ** *Sager, S.; Reinelt, G.; Bock, H.G.***Journal: ** *Mathematical Programming***Number: ** *1***Pages: ** *109--149***Title: ** *Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2009.pdf***Volume: ** *118***Year: ** *2009*

.

Time | [ft] | [mph] | [ftps] | Energy | ||
---|---|---|---|---|---|---|

1 | 0.0 | 0.0 | 0.0 | 0.0 | ||

1 | 0.453711 | 0.979474 | 1.43656 | 0.0186331 | ||

1 | 10.6776 | 6.73211 | 9.87375 | 0.109518 | ||

2 | 24.4836 | 8.65723 | 12.6973 | 0.147387 | ||

2 | 57.3729 | 14.2658 | 20.9232 | 0.339851 | ||

1 | 277.711 | 25.6452 | 37.6129 | 0.93519 | ||

3 | 1556.5 | 26.8579 | 39.3915 | 1.14569 | ||

3 | 1600 | 26.5306 | 38.9115 | 1.14569 | ||

4 | 1976.78 | 23.5201 | 34.4961 | 1.14569 | ||

- | 2112 | 0.0 | 0.0 | 1.14569 |

## Variants

The given parameters have to be modified to match different parts of the track, subway train types, or amount of passengers. A minimization of travel time might also be considered.

The problem becomes more challenging, when additional point or path constraints are considered.

### Point constraint

We consider the point constraint

for a given distance and velocity . Note that the state is strictly monotonically increasing with time, as for all .

The optimal order of gears for and with the additional interior point constraints (\ref{FASOPOINTCON}) is . The stage lengths between switches are 2.86362, 10.722, 15.3108, 5.81821, 1.18383, 2.72451, 12.917, 5.47402, and 7.98594 with . For different parameters and we obtain the gear choice 1, 2, 1, 3, 2, 1, 3, 4 and stage lengths 2.98084, 6.28428, 11.0714, 4.77575, 6.0483, 18.6081, 6.4893, and 8.74202 with .

### Path constraint

A more practical restriction are path constraints on subsets of the track. We will consider a problem with additional path constraints

The additional path constraint changes the qualitative behavior of the relaxed solution. While all solutions considered this far were bang-bang and the main work consisted in finding the switching points, we now have a path-constrained arc. The optimal solutions for refined grids yield a series of monotonically decreasing objective function values, where the limit is the best value that can be approximated by an integer feasible solution. In our case we obtain 1.33108, 1.31070, 1.31058, 1.31058, ...

The plot shows two possible integer realizations, with a trade-off between energy consumption and number of switches. Note that the solutions approximate the optimal driving behavior (a convex combination of two operation modes) by switching between the two and causing a touching of the velocity constraint from below as many times as we switch.

The differential state *velocity* of a subway train over time. The dotted vertical line indicates the beginning of the path constraint, the horizontal line the maximum velocity. Left: one switch leading to one touch point. Right: optimal solution for three switches. The energy-optimal solution needs to stay as close as possible to the maximum velocity on this time interval to avoid even higher energy-intensive accelerations in the start-up phase to match the terminal time constraint to reach the next station.

## Source Code

Model descriptions are available in

## References

[Bock1982] | Bock, H.G.; Longman, R.W. (1982): Computation of optimal controls on disjoint control sets for minimum energy subway operation. %publisher%, Proceedings of the American Astronomical Society. Symposium on Engineering Science and Mechanics | |

[Kraemer-Eis1985] | Kr\"amer-Eis, P. (1985): Ein Mehrzielverfahren zur numerischen Berechnung optimaler Feedback--Steuerungen bei beschr\"ankten nichtlinearen Steuerungsproblemen. (%edition%). Universit\"at Bonn, Bonn, %pages% | |

[Sager2009] | Sager, S.; Reinelt, G.; Bock, H.G. (2009): Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems. Mathematical Programming, 118, 109--149 |