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 line reaches speeds of 270 mph.

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

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

The Italian ETR 200 in 1939

It achieved the world mean speed record in 1939, reaching 127 mph near The Express, currently the only high-speed rail line in the U.S., with a top speed of 150 mph 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 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, Traffic OD Demand interview samples Estimation

Traffic OD Demand Matrix

Infrastructure Service Network Resources Design Yard and Terminal (yards and terminals) Blocking Management Line Plan Plan

Route and Frequency Settings

Train Scheduling

TrainTrain Dispatching Dispatching, empty Train Timetables Emptycar Cardistribution Distribution Resources and Locomotive, Car and Crew Policies 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 OD Demand -> Routes-> Blocks-> Trains destination origin b c d a 100 100 500 b 150 200 c 50

a b c d b a d Candidate blocks

c

b

a d Blocking Plan 1 Train schedule Da c + Da d Dad+Dbd Time Dab+Dac+Dad d + Db c + Db d + Dc d c

b c

Blocking Plan 2 a d Da d

Dab+Dac Da c + Db c + Db d Db d + Dc d Terminals c b Station Block 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  Real-time Applications – Satisfy passenger and Adjust the daily and hourly train freight traffic demand operation schedules – Minimize the overall – Improve on-time performance operational costs and reliability Demand Rolling stock Crew Line Planning Timetabling estimation scheduling 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 delay for interline travel 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 p d q(i), k-1 q(i), k-1 τ q(i),k −1 section k

station k section k-1 a τ q(i),k pq(i), k p station k-1 q(i), k bypass station k stop at station k 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

pu,k : pure running time for train type u at section k without acceleration and deceleration times a τ u,k : acceleration time loss at the upstream station of section k with respect to train type u d τ u,k : deceleration time loss at the downstream station of section k with respect to train type u e l h u,v,k h u,v,k : minimum headway between train types u and v entering/leaving section k s i,k : scheduled minimum stop time for train i at station k ~ d : preferred departure time for train i at its origin, i.e. the preferred i release time for job i. Decision Variables

di : departure time for train i at its origin

yi : interdeparture time between train i and train i+1 e : entering time for train i to section k xi,k l : leaving time for train i from section k xi,k t a : actual acceleration time for train i at the upstream station of i,k section k d ti,k : actual deceleration time for train i at the downstream station of section k : total travel time for train i Ci

Bi, j,k : 0 or 1, indicating if train i enters section k earlier or later than train j, respectively a Bi,k : 0 or 1, indicating if train i bypasses/stops at the upstream station of section k, respectively d Bi,k : 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 are entering times and leaving times for each section  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)

l l pt(q(j),m,k) h q(j),q(i), k h q(i),q(j), k Time axis

station k+1  pu,k if m = 00 xl l i,k x j,k  d section k p +τ u,k if m = 01 e e  u,k x i,k x = j,k pt(u,m,k)  a station k pu,k +τ u,k if m = 10 l  x j,k-1  a d section k-1  pu,k +τ u,k +τ u,k if m = 11 he e q(j),q(i), k h q(i),q(j), k  Apply the algorithm proposed by Patterson et al. (1989) for solving station k-1 multi-mode resource constrained project schedulingj problemi Integer Programming Formulation

 Allowable adjustment for departure time: (2N constraints) ~ ~ di − gi ≤ di ≤ di + gi ∀i ∈ I

 Interdeparture time: (Nh -1 constraints for high-speed trains)

yii= d+1 − d i ∀∈ iI h\{ N h }  Departure time: (N constraints) e xi,1 = di ∀i ∈ I  Total travel time: (N constraints) l e Ci = xi,K − xi,1 ∀i ∈ I  Dwell time: (N*(K-1) constraints) e l xi,k − xi,k −1 ≥ si,k ∀i ∈ I,∀k ∈V \{k = 1}  Travel time on sections: (N×K constraints) l e a d xi,k − xi,k = pq(i),k + ti,k + ti,k ∀i ∈ I,∀k ∈V  Acceleration time: (N×K constraints) a e l a a a a Bi,k × M ≥ xi,k − xi,k −1 ∀i ∈ I,∀k ∈V \{k = 1}, Bi,1 = 1 ti,k = Bi,k ×τ q(i),k ∀i ∈ I,∀k ∈V  Deceleration time: (N×K constraints) d e l d Bi,k * M ≥ xi,k +1 − xi,k ∀i ∈ I,∀k ∈V \{k = K}, Bi,K = 1 d d d ti,k = Bi,k ×τ q(i),k ∀i ∈ I,∀k ∈V Integer Programming Formulation (Cont’)

 Minimum headway: (N× (N-1) ×K× 4 constraints)

e e e e e e either xi,k − x j,k ≥ h q( j),q(i),k or x j,k − xi,k ≥ h q(i),q( j),k ∀i ≠ j ,i, j ∈ I, ∀k ∈V

l l l l l l either xi,k − x j,k ≥ h q( j),q(i),k or x j,k − xi,k ≥ h q(i),q( j),k ∀i ≠ j ,i, j ∈ I, ∀k ∈V To model the above “either-or” type constraints

e e e xi,k − x j,k ≥ h q( j),q(i) − (1− Bi, j,k ) × M ∀i ≠ j,i ∈ I, j ∈ I, ∀k ∈V

e e e x j,k − xi,k ≥ h q(i),q( j) − Bi, j,k × M ∀i ≠ j,i ∈ I, j ∈ I, ∀k ∈V

l l l xi,k − x j,k ≥ h q( j),q(i) − (1− Bi, j,k ) × M ∀i ≠ j,i ∈ I, j ∈ I,∀k ∈V

l l l x j,k − xi,k ≥ h q(i),q( j) − Bi, j,k × M ∀i ≠ j,i ∈ I, j ∈ I,∀k ∈V Illustration of a Double-Track Train Schedule

Time axis station k+1 hl l q(j),q(i),k-1 x j,k section k e x j,k station k l x j,k-1 section k-1 e h q(i),q(j),k station k-1 section k-2 station k-2 ...

... dj di section 1 station 1

~ ~ ~ di − gi di di + gi 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

Nh −1 2 Min Z1 = Var(Y ) = ∑(yi − yi ) i=1 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

N Min Z2 = ∑Ci i=Nh +1 Branch-and-Bound Solution Algorithm

 Step 1: (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 (L).  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- High-speed train 1 speed train i = i*+1, branch several nodes, each corresponding to High-speed train 2 different feasible departure time for High-speed train 3 train i. Insert new nodes into the active node list. High-speed train 4  Step 3: (Evaluation 1) Obtain objective function Z1 by calculating High-speed train 5 variance of departure times for existing high-speed trains.  Step 4: (Evaluation 2) Obtain Subproblem 1: Determine departure objective function Z2 by solving time of high-speed trains subproblem with the fixed departure Subproblem 2: Schedule all medium- times for high-speed trains. 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. Non-Dominated Schedules

1st objective bDominated Schedule Z2(b) a Z2(a) Non- Dominated

Schedule 2nd objective Z1(a) Z1(b) 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 Objective 1 Case 1:The new schedule replaces Case 2:The new schedule replaces all the schedules in the set some of the schedules in the set Objective 2 Objective 2

Objective 1 Objective 1 Case 3:The new schedule is Case 4:The new schedule is added to the set. out of the set. Illustration of Dominance Rules Decision  Main Idea: point Cut dominated partial schedule Parti al schedul e at node a at early as possible station 5 station 4  Conditions for node a

dominating node b station 3 (1) Same set of finished station 2 trains station 1 8 1 2 9 3

Parti al schedul e at node b (2) Z (a) < Z (b) for station 5

finished trains station 4 (3) The starting time for each unfinished station 3 activity in node a is station 2 no later than the station 1 8 1 2 9 3 counterpart in node b for each feasible mode 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 1 High-speed train 2 High-speed train 2 High-speed train 3 High-speed train 3

High-speed train 4 High-speed train 4

High-speed train 5 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 Possible Solutions 100000 90000 Non-dominated 80000 partial schedules 70000 60000 50000 40000 # of solutions of # 30000 20000 10000 0 3 4 5 6 # of high-speed trains to be considered 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 (Out-of- vehicle time) – 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 Beam width = 50 360 Exact solutions

358 Utility evaluation rule

Random selection rule 356 (mins) 354

352

350 Average travel time for medium-speed trains 348

346 0 20 40 60 80 100 120 Variance of interdeparture times Trade-Off Curves for Two Conflicting Objectives

Exact solutions for 6 high- 365 speed trains

360 Utility evaluation rule for 24 high-speed trains w ith beam w idth = 50 355 Utility evaluation rule for 24 high-speed trains w ith beam w idth = 100 20 min 350 1 hour optimization horizon (mins)

345

6 hour optimization horizon 340 2 min

Average travel time for medium-speed trains 335 14.5 15 15.5 16 16.5 17 17.5 Expected waiting time for high-speed trains (mins) Part II: Optimizing Slack Time Allocation

 Marketing Slack – concurrent use of critical pointsTime ? (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 Station node Segment arc

J4 5 Delay arc

J3 4 High-speed 3 train

2 J2 Station ( distance ) 1 J1

Time Two-stage Recourse Model

 1st Stage Objective – Minimize total trains’ tripn time Min c( x ) =∑ wi ( e im, − r i ) i=1

High-speed Medium-speed J4 5 train (i) train (j)

e2,4

J3 4 Segment J4

d1,3 3

f3,1 2 Station , ( distance ) J2

r1 b2,1 1 Segment J1 J1

Time Illustration of model variables Two-stage Recourse Model…

 2nd Stage Objective

– Minimize Schedulen deviation =+ −++ −− − gyxweewee(,)()()ωω∑( i im,, im , i im,, ω im , ) i=1 station 4

high-speed train realized schedule segment 4 station 3 e3,3,ω medium-speed train segment 3 d 1,3 b station 2 3,3,ω e3,2 h1,3 segment 2 f s b3,2 f1,1,ω 3,1 3,1 station 1

segment 1

r1 r1,ω r2 r3 station 0 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…

High-speed On timedelay!! J4 5 train ?

J3 4 Segment J4

3 Slack time Medium-speed 2 train J2 Station ( distance ) 1 Segment J1 J1

Time Solution Algorithm…

Many alternative paths J4 5

J3 4 Segment J4

3 Stochastic Time-depende 2 Shortest Path Problem J2 Station ( Distance ) 1 Segment J1 J1

Time Strategies for a Single Train Problem

 Constructing random segment running times – vector with given probability ρ ω fij,,ω

 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

1 1 F '(δ ) 1 F ''(δ ) 3/4 3/4 3/4

1/2 1/2 1/2 F '(δ ) Freq Freq Freq

1/4 1/4 1/4

0 0 0 CDF Delay CDF Delay CDF Delay 1 1 1 0.75×1+0.25×2=1.25 3/4 3/4 3/4 0.5×1+0.5×2=1.5 1/2 1/2 1/2 Freq Freq Freq planning arc 1/4 1/4 1/4 scheduling arc 0 0 0 PDF Delay PDF Delay PDF Delay 0.25 0.5 0.25 0.5 0.5 0.75 0.25 station j eij′, eij′′, +1 segment j station j-1 0.5 0.5 0.5 0.5 +1 segment j-1 station j-2 (a) No slack time (b) Timetable v' (c) Timetable v'' (do-nothing) (slack time on segment j-1) (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 algorithms are critically needed to generate executable, recoverable train timetable with 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 Extracted from The Slotted Railway - slot price system Living With Passenger Trains

 Maximum speed design and Sebastian Schilling Railion Deutschland AG 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.)

Slot price = base price x product factors x special multipliers + special additions x regional factor Scheduling Freight And Passenger Trains Extracted from The Slotted Railway - Basic Living With Passenger Trains Line Capacity Layout

C Sebastian Schilling Railion Deutschland AG B location A* 1. 2. 3. Mixed traffic - Mixed traffic - Network 21 reduced capacity capacity enlargement `Harmonizing` C C C B B B A A A

High-Speed passenger train Additional freight train slots Regional passenger train Connecting passenger services Freight train 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 > 50 > 30 flexible trains*2

 Slot fixed fixed (reserved) on demand

 Days of regular flexible on demand service week before  Ordering service fixed*3 min >24 h date departure

 Price 100% 100% + X 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 passenger rail service in America (April 16, 2009)