H9MSO: A High-Speed Train as Planned for the HS2 Line from London to Birmingham: Modelling, Simulation and Optimisation Assignment, NCI

The Context

A high-speed train as planned for the HS2 line from London to Birmingham has an acceleration of 0.72m/s2, about the same as a commuter train. While the trains can brake in an emergency at 2.5m/s2, the energy optimal deceleration (using regenerative braking) is 0.36m/s2.

The maximum traveling speed of the train is about 300km/h (83.3m/s). Accelerating from 0 to the maximum speed takes, therefore, 115.7s during which time the train travels about 4,820m. Slow decelerating takes 231.4s and the distance traveled is 9,640m.

A railway line consists of a sequence of signaling blocks. A train is only allowed to enter a block when there is no other train in the block and the entry signal is green. When a train enters a block the entry signal switches to red. The entrance signal switches back to green 5 seconds after the end of the train have left the block.

Depending on the intended maximum traveling speed there is an optimal distance for setting a pre-signal. At a maximum traveling speed of the train of 300km/h (83.3m/s) the slow deceleration takes 231.4s and the train travels 9,640m during this time. In this case, the signal would be 10km before the actual signal at the end of one block and the entry to the next. The length of a block should, therefore, be at least 10km, but to allow a train to achieve and run as long as possible at full speed, the block length should be at least 1.5x this length. The control problem is to keep the trains moving at maximum possible speed. If there is a slight delay in one train it may cause the train in the following block to decelerate, potentially down to a stop, and then reaccelerate. This, in turn, will delay the train thereafter. Just like the “traffic jam” effect you know from the motorway.

A train can run constantly at full speed if there is always at least one free block ahead. With a block length of 15km, that means that the distance between two trains should be about 30km, at full speed a traveling time of about 6min. This would indicate that one could achieve a maximum schedule of 10 trains per hour. The new government under Boris Johnson is pushing the development of HS2. The economic argument is based on a scheduled time of 9 trains per hour.

Part 1: Simulation

Create a simulation for the London-Birmingham section of the high-speed line under the assumption of a number k of signaling blocks. A decision on the number of signaling blocks is required, as the track laying is supposed to start soon. Also, a decision on the number of trains running per hour has to be made.

Depending on the number of signaling blocks simulate the throughput of trains. Take into account that between Birmingham Interchange and Birmingham Curzon street as well as between London Old Oak Common and London Euston there will be a practical speed limit and the traveling times are 5 min between London Euston and London Old Oak Common (one signaling block) and 9 min between Birmingham Interchange and Birmingham Curzon Street station (two signaling blocks). Take into account the variability of achievable speeds due to wind and weather conditions on the stretch between London Old Oak Common and Birmingham Interchange. Take into account the variability in stop times, i.e. minor delays due to passenger movements.

London Euston

5 min

London Old Oak Common

31 min

Birmingham Interchange

9 min

Birmingham Curzon Street

The distance between London Old Oak Common Station and Birmingham Interchange Station is 145km. This could be broken down in up-to 14 blocks. If there are fewer but longer blocks the trains could achieve in theory a higher average speed, however, the throughput in trains per hour is smaller.

Part 2: Optimisation

Optimization For the project you need to create a simulation of the train network which would give for a given train schedule (numbers of trains per hour n ∈ {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20}) and a given break-down of the line in signaling blocks k ∈ {1, …, 15} a distribution of average overall traveling time. The simulation results are fed into an optimization problem to determine (nopt, kopt) to achieve optimal results

Using the simulation described above solve the following optimization problems:

(1) Minimize the overall average traveling time.

The overall traveling time consists of the waiting time for the next train and the actual traveling time until the arrival of the train. The problem can be simplified by assuming a fixed average waiting time (0.5*60/n).

(2) Maximize the throughput of passengers during peak hours.

The trains can be configured to run at a length of 200m (short train, 420 passengers) or 300m (max train, 630 passengers). While longer trains carry potentially more passengers, you need to consider the walking times at the train station. The walking speed of a train passenger-carrying luggage is between 1.0 and 1.2m/s, which impacts on the stop times at the station. Also, more passengers disembarking and embarking may lengthen the stop times.

For a more realistic simulation assume a Poisson-Distribution for passengers arriving at the train station at an average rate of m passengers per hour and how they can be served by n trains per hour. As the trains are on a tight schedule, we cannot assume that everyone gets on board during the short stopping time. Assume that the trains are only filled to 70% of nominal capacity. A backlog of passengers on the departing station will impact on the average traveling time.

Project Report:

Your report should be in the form of a well-formatted Jupyter Notebook. Use section headings as appropriate. References should comprise of a complete list of academic works and/or online materials used in the project and should be in the IEEE citation style. Detailed comments throughout code are essential.