High-speed Passenger Trains on Freight Tracks: Modeling Issues on Capacity Analysis, Train Timetabling and Real-Time Dispatching

Dr. Xuesong Zhou

Assistant Professor Department of Civil and Environmental Engineering Univ. of Utah [email protected]

In collaboration with Dr. Muhammad Babar Khan (Pakistan), Dr. Lingyun Meng (China)

Prepared for NEXTRANS Seminar Series, Purdue University on May 11, 2010

Definitions



High-speed passenger rail

–

152 mph or faster for upgraded track

–

183 mph or faster for new track



In China, high-speed conventional rail lines operate at top speeds of 220 mph, and one maglev line reaches speeds of 270 mph.

Reference: http://en.wikipedia.org/wiki/High-speed_rail

High-Speed Trains

E5 Series Shinkan sen in Japan World speed record holding (357mph) TGV German designed third generation ICEon Cologne-Frankfurt high-speed rail line

First High-speed Service Train

The Italian ETR 200 in 1939 It achieved the world mean speed record in 1939, reaching 127 mph near Milan

The Acela Express the only high-speed rail line in the U.S., with a top speed of 150 mph , currently

North American Railroad Network

5 major US railroads after years of consolidations: CSX, UP, CR, NS, BNSF

(Planned) High-Speed Rail System in United States

High-speed railway plans in China

17,000 mile national high-speed rail system will be built in 4 phases, for completion by 2030.

Chicago Hub Network

If implemented, the plans could return Chicago to a status it had in the 1930s and 1940s •France has a population distribution similar to that in the Midwest.

•French experiences with TGV trains and other high-speed systems could conceivably be duplicated in the U.S.

• The total cost was projected at

$68.5

billion in 2009 dollars, • Only 54% was projected to need public financing if a public-private partnership was pursued. •The public funds could be recovered from revenues in about

15 years

.

Reference: http://en.wikipedia.org/wiki/Chicago_Hub_Network http://www.midwesthsr.org/docs/SNCF_Midwest.pdf

Operational High-Speed Lines in Europe

High-Speed Lines in East Asia

Concepts of the two modes Operation Mode I (Dedicated Line)

Operation Mode II (High-speed passenger trains running on freight tracks)

+

What We Need to Do in United States?



1. Building Infrastructure

– Class I Railroad mileage shrank from 210K to 94K, from 1956 to 2007 – Railroad ton-miles tripled from 589 billion to 1.772 trillion (thanks to technological advance) 

2. Building Education Infrastructure for Railroad Transportation Engineering

 Employment dropped from 1 million to 167K 

3. Building New Tracks for Research…

Reference: Barkan, C.P.L. 2008.

Building an Education Infrastructure for Railway Transportation Engineering: Renewed Partnerships on New Tracks , TR News 257: 18-23, Transportation Research Board of the National Academies, Washington, DC.

Railroad Planning and Operations

Socio-economic data, interview samples Infrastructure Resources (yards and terminals) Traffic OD Demand Estimation Traffic OD Demand Matrix Service Network Design Blocking Plan Line Plan Route and Frequency Settings Yard and Terminal Management Resources and Policies Train Scheduling Train Timetables Locomotive, Car and Crew Scheduling

Railroad Network Capacity

 Line capacity – Single or double-track -> meet-pass plans – – – Signal control type -> minimal headways Locomotive power -> speed, acceleration/deceleration time loss Train schedules -> overall throughput  Node capacity (yards, terminals / sidings) – – – Track configuration Locomotive power-> car processing time Yard make-up plans, terminal operating plans -> overall throughput

a

OD Demand -> Routes-> Blocks-> Trains

destination origin

a b c b 100 c 100 150 d 500 200 50

b c d b a d Candidate blocks c b a d Train schedule Blocking Plan 1

Dab+Dac+Dad Dac+Dad +Dbc+Dbd Dad+Dbd +Dcd

b c d c Time Blocking Plan 2

Dab+Dac Dad Dac+Dbc+Dbd

Station

Block Dbd+Dcd

a c d b a

Background on Train Scheduling

Important role in railroad management:  Determine the level-of-service of train timetables  Serve as the basis for locomotive and crew scheduling  Planning Applications – – Satisfy passenger and freight traffic demand Minimize the overall operational costs  Real-time Applications Adjust the daily and hourly train operation schedules – Improve on-time performance and reliability

Demand estimation Line Planning Timetabling Rolling stock scheduling Crew scheduling  Railway Planning Process Sequential scheduling – – Stage 1: Line planning Determine the routes, frequencies, preferred departure times, and stop schedules – Stage 2: Schedule generation  Construct the arrival and departure times for each train at passing stations  Job-shop scheduling formulation and branch-and-bound solution algorithm (Szpigel, 1973) »

Minimize a weighted sum of train delays (Kraft, 1987)

 Multi-criteria scheduling (e.g. Higgins and Kozan, 1998) »

Mainly focus on the supply side, such as fuel costs for locomotives, labor costs for crews

»

Simplify multiple objectives as a weighted linear combination

Train Scheduling on Beijing-Shanghai High-Speed Passenger Railroad in China

    Around 900 miles High-speed trains (200 mile/h) – Provide direct service for inter city travel in this corridor Medium-speed trains (150 mile/h) – Run on both high-speed line and adjacent regular rail lines in order to  Serve the large volume of traffic passing through this corridor Reduce connecting interline travel delay for

Illustration

    From Shanghai to Xuzhou 17 segments, 385 miles Morning period (6:00 am 12:00 am) 24 high-speed trains and 12 medium-speed trains  Preferred departure time interval for high-speed trains is 30 minutes

Part I: Balancing Two Conflicting Objectives

   Two conflicting objectives – (High-speed trains) Expect a “perfect” schedule with high frequency and even departure time intervals – (Medium-speed trains) Reduce total travel time Operational policies – – High-speed trains hold higher priority, i.e. medium-speed trains have to yield to high-speed trains, if possible conflict exists A “perfect” high-speed train timetable might result in extremely long waiting times for medium-speed trains Need for – – Obtain non-dominated solutions for bicriteria scheduling problem Retrieve the trade-offs between two conflicting objectives Reference: Zhou, X. and Zhong, M. (2005) Bicriteria Train Scheduling for High Speed Passenger Railroad Planning Applications. European Journal of Operational Research Vol 167/3 pp.752-771.

Challenge I

Challenge II: Model Acceleration and Deceleration Time Losses

 Acceleration and deceleration time losses  High-speed trains: 3 minutes  Medium-speed trains: 2 minutes

Time axis

station k+1 p q(i), k-1 station k station k-1 p q(i), k bypass station k p q(i), k-1 

d q

(

i

),

k

 1 

a q

(

i

),

k

p q(i), k stop at station k section k section k-1

  

Formulating Train Timetabling and Dispatching Problem

Given – Line track configuration – – Minimum segment and station headways # of trains and their arrival times at origin stations Find – Timetable: Arrival and departure times of each train at each station Objectives – (Planning) Minimize the transit times and overall operational costs, performance and reliability – (Dispatching) Minimize the deviation between actual schedules and planned schedule

Notations

i: subscript of trains j: subscript of sections u: train types , 0: high-speed train, 1: medium-speed train 

p



d u a

,

u u k

,

k

: and deceleration times : pure running time for train type u at section k without acceleration acceleration time loss at the upstream station of section k with ,

k

respect to train type u : deceleration time loss respect to train type u at the downstream station of section k with

h e u

,

v

,

k s i

,

k

~

d i h l u

,

v

,

k

: minimum headway between train types u and v entering/leaving section k : scheduled minimum stop time for train i at station k : preferred departure time release time for job i.

for train i at its origin, i.e. the preferred

Decision Variables

d i y i x i e

,

k x i l

,

k

: : interdeparture time between train i and train i+1 : departure time entering time for train i at its origin for train i to section k : leaving time for train i from section k

a t i

,

k d t i

,

k C i

: actual acceleration time for train i at the upstream station of section k : actual deceleration time for train i at the downstream station of section k : total travel time for train i

B i

,

j

,

k B i B i a d

, ,

k k

: 0 or 1, indicating if train i enters section k earlier or later than train j, respectively : 0 or 1, indicating if train i bypasses/stops at the upstream station of section k, respectively : 0 or 1, indicating if train i bypasses/stops at the downstream station of section k, respectively

Model Acceleration and Deceleration Time Losses

     Multi-mode resource constrained project scheduling approach Activity (i, k) :the process of train i traveling section k and the project is a sequence of K activities Two sets of renewable resources times for each section are entering times and leaving the minimum headway constraints define the consumption of resources by each activity Processing time of activity (i, k) with train type u=q(i) in mode m (0=no-stop and 1=stop) pt(q(j),m,k) h l q(j),q(i), k h l q(i),q(j), k

Time axis



pt

(

u

,

m

,

k

)     

p u

,

k p p



u u

, ,

k k



p u a

  ,

k u

,  

k d a



u u

, ,

k



k if d u

,

if if k m



if m m

00   01 10

m

 11 station k+1 station k x l j,k-1 h e x l i,k x e i,k h e multi-mode resource constrained project scheduling problem x l j,k x e j,k section k section k-1 Apply the algorithm proposed by Patterson et al. (1989) for solving

Integer Programming Formulation

        Allowable adjustment for departure time: (2N constraints) ~

d i



g i



d i

 ~

d i



g i



i



I

Interdeparture time: (N

h y i



d i

 1 

d i I h

\{

N h

} -1 constraints for high-speed trains) Departure time: (N constraints)

x i e

, 1 

d i



i



I

Total travel time: (N constraints)

C i



x i l

,

K



x i e

, 1 

i



I

Dwell time: (N*(K-1) constraints)

x i e

,

k



x i l

,

k

 1 

s i

,

k



i



I

, 

k



V

\ {

k

 1 } Travel time on sections: (N ×

K constraints) x i l

,

k



x i

,

e k



p q

(

i

),

k



a t i

,

k



d t i

,

k



i



I

, 

k



V

Acceleration time: (N ×

K constraints) B i a

,

k



M



x i e

,

k



x i l

,

k

 1 

i



I

, 

k



V

\ Deceleration time: (N ×

K constraints)

{

k

 1 },

B i

,

d k t d i

,

k

*

M



B i d

,

k



x i e

,

k

 1  

d q

(

i

),

k



x i l

,

k



i



I

, 

k



V



i



I

, 

k



V

\ {

k



K

},

B i B i

,

d K a

, 1   1 1

t i a

,

k



B i a

,

k

 

a q

(

i

),

k



i



I

, 

k



V

Integer Programming Formulation (Cont’)

 Minimum headway: (N ×

(N-1)

×

K

×

4 constraints) either x i e

,

k



x e j

,

k



h e q

(

j

),

q

(

i

),

k or x e j

,

k



x i e

,

k



h e q

(

i

),

q

(

j

),

k



i



j

,

i

,

j



I

,

either x i

,

l k



x l j

,

k



h l q

(

j

),

q

(

i

),

k or x l j

,

k



l x i

,

k



h l q

(

i

),

q

(

j

),

k



i



j

,

i

,

j



I

, To model the above “either-or” type constraints

x i e

,

k



x e j

,

k



h e q

(

j

),

q

(

i

)  ( 1 

B i

,

j

,

k

) 

M



i



j

,

i



I

,

j



I

, 

k



V



k



V



k



V x e j

,

k



x i

,

e k



h e q

(

i

),

q

(

j

) 

B i

,

j

,

k



M x i

,

l k



x l j

,

k



h l q

(

j

),

q

(

i

)  ( 1 

B i

,

j

,

k

) 

M



i



j

,

i



I

,

j



I

, 

k



V



i



j

,

i



I

,

j



I

, 

k



V x l j

,

k



x i l

,

k



h l q

(

i

),

q

(

j

) 

B i

,

j

,

k



M



i



j

,

i



I

,

j



I

, 

k



V

Illustration of a Double-Track Train Schedule

station k+1

Time axis

station k station k-1 station k-2

...

station 1 d

j

x l j,k-1 h l q(j),q(i),k-1 d

i

h e q(i),q(j),k x l j,k x e j,k section k section k-1 section k-2

...

section 1 ~

d i



g i

~

d i d

~

i



g i

Utility Function for High-speed Trains Passengers

– Represent passengers’ preference information as a multi attribute utility function –

U= –0.0099

× ( In-vehicle travel time ) –0.0426

× ( Out-of-vehicle waiting time )

– Calibrated by the study for high-speed rail in the Toronto Montreal corridor (KPMG Peat Marwick, Koppelman, 1990) –

In-vehicle travel time:



Out-of-vehicle waiting time

»

Function of variance of inter-departure times for given # of trains

Objectives

First Objective:

Minimize the variation of inter-departure times for high-speed trains

Min Z

1 

Var

(

Y

) 

N i

 1

h

  1 (

y i



y i

) 2 i.e. Minimize the expected waiting time from a passenger arriving at the terminal to the departure time of the next high-speed train If assuming passengers independently and randomly arrive at the terminal, (Random incidence theorem described by Larson and Odoni, 1981)

Second objective:

Minimize the total travel time for medium-speed trains

Min Z

2 

i



N

 

N h

1

C i

Branch-and-Bound Solution Algorithm

   

Step 1: (

(L).

Initialization )

Create a new node, in which contains the first task of all trains. Set the departure time for this train and insert this node into active node list

Step 2: ( Node selection )

Select an active node from L according to a given node selection rule.

Step 3: ( Stopping criterion )

If all of active nodes in L have been visited, then terminate.

Step 4: ( Conflict set construction )

Update the schedulable set in the selected node .

Insert these tasks and task t(i,j) into the current conflict set.

h Conflict i first i j i j j first h j i Additional delay

Branch-and-Bound Algorithm for Generating Non-dominated Solutions

     Step 1: (Initialization) Create a root node into the active node list. i=0.

Step 2: (Branching) Consider high speed train i = i active node list.

*

+1, branch several nodes, each corresponding to different feasible departure time for train i. Insert new nodes into the Step 3: (Evaluation 1) objective function Z 1 Obtain by calculating variance of departure times for existing high-speed trains.

Step 4: (Evaluation 2) objective function Z 2 Obtain by solving subproblem with the fixed departure times for high-speed trains.

Step 5: (Dominance Rule) Apply proposed dominance rules to compare the current node with the other existing nodes, and prune all dominated nodes. Go back to Step 2.

High-speed train 1 High-speed train 2 High-speed train 3 High-speed train 4 High-speed train 5 Subproblem 1: Determine departure time of high-speed trains Subproblem 2: Schedule all medium speed trains

Non-Dominated Schedules

1st objective Z 2 (b) Z 2 (a) Non- Dominated

a

Schedule

b

Dominated Schedule Z 1 (a) Z 1 (b) 2nd objective

First objective: Expected waiting time for high-speed trains at origin Second objective: Average travel time for medium-speed trains

Construction of Non-Dominated Set

Objective 2 Objective 2 Objective 1 Case 1:The new schedule replaces all the schedules in the set Objective 2 Objective 1 Case 2:The new schedule replaces some of the schedules in the set Objective 2 Objective 1 Case 3:The new schedule is added to the set.

Objective 1 Case 4:The new schedule is out of the set.

Illustration of Dominance Rules

 Main Idea: Cut dominated partial schedule at early as possible station 5

Decision point

station 4  Conditions for node a dominating node b station 3 (1) Same set of finished trains (2) Z (a) < Z (b) for finished trains (3) The starting time for each unfinished activity in node a is no later than the counterpart in node b for each feasible mode station 2 station 1 station 5 station 4 station 3 station 2 station 1 8 8 1 2 9 3 Partial schedule at node b 1 2 9 3

Heuristic Algorithm

 Beam search algorithm uses a certain evaluation rule to select the k-best nodes to be computed at next level High-speed train 1 High-speed train 2 High-speed train 3 High-speed train 4 High-speed train 5 High-speed train 1 High-speed train 2 High-speed train 3 High-speed train 4 High-speed train 5 High-speed train 6 High-speed train 7

Limitation of Branch-and-Bound Algorithm

 Remaining non-dominated nodes in the B&B tree still grows rapidly 130000 120000 110000 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 0 Possible Solutions Non-dominated partial schedules 3 4 5

# of high-speed trains to be considered

6

Illustration of One Non-Dominated Schedule

Evaluation Rules

  Utility based evaluation rule – Represent passengers’ preference information as a multi attribute utility function – E.g. U= –0.0099

× (In-vehicle time) –0.0426

× vehicle time) (Out-of Calibrated by the study for high-speed rail in the Toronto Montreal corridor (KPMG Peat Marwick, Koppelman, 1990) Random sampling – – Capture the global trade-off information associated with the efficient frontier Randomly sample the nodes in the non-dominated partial solutions at the current level

Exact Algorithm (B&B) vs. Heuristic Algorithm (Beam Search)

362 360 358 356 354 352 350 348 346 0

Beam width = 50

20 40 60

Variance of inte rde parture tim e s

80 Exact solutions Utility evaluation rule Random selection rule 100 120

Trade-Off Curves for Two Conflicting Objectives

365 360 355 350

20 min

345 340 335 14.5

2 min

Exact solutions for 6 high speed trains Utility evaluation rule for 24 high-speed trains w ith beam w idth = 50 Utility evaluation rule for 24 high-speed trains w ith beam w idth = 100 1 hour optimization horizon 15 15.5

16 16.5

17

Expected w aiting tim e for high-speed trains (m ins)

6 hour optimization horizon 17.5

Part II: Optimizing Slack Time Allocation

   Marketing –

Slack Time ?

(e.g. stations, switches and signals) Logistics – Costs, efficient usage of rolling stock and personnel Operating Constraints – passengers ’ travel times, pleasant transfers and waiting times Reference: Muhammad, K, and Zhou, X (2010) Stochastic Optimization Model and Solution Algorithm for Robust Double-Track Train-Timetabling Problem. IEEE Transactions on Intelligent Transportation Systems. Vol. 11. No. 1. pp. 81 – 89

Model Formulation

Space-Time Network Representation J 4 5 Station node Segment arc Delay arc J 3 4 3 J 2 2 1 J 1

High-speed train

Time

Two-stage Recourse Model

 1st Stage Objective – Minimize total trains

’

i

  1

i

J 4 5

High-speed train (i)

e 2,4

Medium-speed train (j)

J 3 4

d 1,3

3

f 3,1

J 2 2 J 1 1

r 1 b 2,1 Illustration of model variables



r

i

)

Segment J 4 Segment J 1 Time

Two-stage Recourse Model

…

 2nd Stage Objective –

g y

 

i

  1 

i

 station 4  

e

high-speed train station 3 station 2 medium-speed train d 1,3

h

1,3 f 1,1,  station 1 b 3,3,  f 3,1 s 3,1 ) + 

i

 e 3,3,  b 3,2 e 3,2  

e

)  realized schedule segment 4 segment 3 segment 2  segment 1 station 0 r 1 r 1,  r 2 r 3 Time

Solution Strategies

   – Sequential Decomposition First plan high-speed trains and then medium-speed trains – Space-time network representation To reformulate the problem as shortest-path problem – – – Stochastic shortest path reformulation a priori stochastic least expected time path problem with the cost function as schedule delay late the recourse decisions taken once random variables are realized

Solution Algorithm

…

J 4 5 J 3 4 J 2 3

Slack time

2

Medium-speed train

J 1 1

High-speed train

Time

?

Segment J 4 Segment J 1

J 4 5 J 3 4 3 J 2 2 J 1 1

Solution Algorithm

… Many alternative paths Segment J 4 Stochastic Time-dependent Shortest Path Problem Segment J 1 Time

Strategies for a Single Train Problem

 – Constructing random segment running times

f

    – Stochastic dominance rules I: Timetable v'' first-order stochastically dominates timetable v', if. the CDF of delay distribution for timetable v'' is above or overlapping with the counterpart in timetable v'.

– II: Timetable v'' second-order stochastically dominates timetable v', if , i.e., the expected delay in timetable v'' is less than its counterpart in timetable v'

Stochastic Dominance Rules

planning arc scheduling arc 1 3/4 1/2 1/4 0 CDF 1 3/4 1/2 1/4 0 PDF 0.25

0.5

0.25

Delay Delay station j station j-1 0.5

0.5

station j-2 (a) No slack time (do-nothing) +1 1 3/4 1/2 1/4 (b) Timetable

v

' (slack time on segment j-1)

F

0 CDF Delay 1 3/4 0.5

 1+0.5

 2=1.5

1/2 1/4

e

 0 PDF 0.5

0.5

Delay 1 3/4 1/2 1/4

F F

0 1 3/4 CDF Delay 0.75

 1+0.25

 2=1.25

1/2 1/4 +1

e

 0 PDF 0.75 0.25

Delay segment j 0.5

0.5

segment j-1 (c) Timetable

v

'' (slack time on segment j)

Other Issues: Estimating Line Capacity

102 train pairs 226 train pairs

Estimating/Simulating Terminal Capacity

Train Routing Problem at Terminals

  Given – Track configuration ( track lengths, switcher engines ) – – Signal configuration Inflow/Outflow (arrival and departure times of trains) Find – Train paths through a terminal – Choke points – System performance of a rail facility under a variety of conditions

Train Routing through Terminals

 Switch Grouping    Train Paths – Train type I: switch groups a, b, d – – Train type II: switch groups c, d, e Train type III: switch groups f, g, h Carey and Lockwood (1995); Carey (1994) – – Mixed integer programming formulation Heuristic solution algorithm Zwaneveld, Kroon, Hoesel (2001); Kroon, Romeijn, Zwaneveld (1997) – Complexity issues – Node packing model

Recommendations

 1. The performance impacts of high-speed passenger trains to freight/ medium-speed trains should be systematically evaluated in all stages of capacity estimation, timetabling and dispatching.

 2. Efficient optimization timetable with algorithms are critically needed to generate executable, recoverable train quality guarantee and balanced performance .

 3. Heuristic algorithms should take into account randomness of train delays , capacity breakdowns to improve the reliability of sub-optimal solutions .

Slot Price System 2001 in Germany

Factors in DB Netz ´s slot price system

 Maximum speed design and capacity of the line  Expected speed of the train  Slot flexibility (special factors for interdependent trains in linked systems)  Gross weight of freight train  Deviation from the standard, (e.g.: dimensional, overweight etc.) Extracted from The Slotted Railway Living With Passenger Trains Sebastian Schilling Railion Deutschland AG

Slot price = base price x product factors x special multipliers + special additions x regional factor

Scheduling Freight And Passenger Trains

Basic Line Capacity Layout

Extracted from The Slotted Railway Living With Passenger Trains Sebastian Schilling Railion Deutschland AG

1.

Mixed traffic reduced capacity 2.

Mixed traffic capacity enlargement 3.

Network 21 `Harmonizing`

High-Speed passenger train Regional passenger train Freight train Additional freight train slots Connecting passenger services

Railion´s Product Design

Extracted from The Slotted Railway Living With Passenger Trains

Products for unit trains (CT & IT *1 )

Sebastian Schilling Railion Deutschland AG

Plantrain Variotrain Flextrain

 Number of trains *2  Slot  Days of service  Ordering date  Price > 50 fixed regular service fixed *3 100% > 30 fixed (reserved) flexible week before departure 100% + X flexible on demand on demand min >24 h 100% + XX *1: CT & IT: conventional & intermodal transport *2: per year *3: regular services; cancellations (< 10% of services) until week before service possible

Research Directions

 Robust schedule design – – Executable vs. recoverable, from planning to real-time decision Improve freight railroad service reliability  Disruption management under real time information – – Service networks (blocking and line planning) Train dispatching – – Rail network and terminal capacity recovery plan Locomotive and crew recovery plan  Integrated pricing and demand management model – Long term and short term pricing schemes and cost structures  Separation of track from traction in Europe – Impact on traffic demand and operating plans (train schedule, fleet sizing and repositioning) – – Shipper logistics modeling Demand estimation and prediction model

New Vision for High-speed and Intercity Passenger Rail Service in America

“Imagine whisking through towns at speeds over 100 miles an hour, walking only a few steps to public transportation, and ending up just blocks from your destination. Imagine what a great project that would be to rebuild America.”

– President Obama announcing a new vision for high-speed and intercity passenger rail service in America (April 16, 2009)