20151216 convergence of quasi dynamic assignment models

1Challenge the future
Convergence of (quasi) dynamic
assignment models
Luuk Brederode
Consultant DAT.mobility /
PhD student Delft University

Contents
1. Convergence? What and why?
2. Generic description of equilibria in strategic transport models
3. How do we reach convergence?
-break-
4. Example: convergence of quasi dynamic assignment model STAQ
within BBMA (Brabant Brede Model Aanpak)
5. Example: convergence of mode/destination choice model,
departure time model and STAQ (assignment) within BBMA
6. General conclusions

1. Convergence? What and
why?

Context: strategic transport
models
• Strategic models are used for long term decisions we are
interested in the long term effects of measures.
• Long term effects of measures can be (totally) different than
short term effects! This was already observed in the 1950’s:

Example of ‘induced demand’:
Sydney Harbour Tunnel

• Change in behavior is triggered by change of circumstances (e.g.
travel time on a route decreases; train ticket prices increase,…)
• Each change has its own response time
• One cannot change departure time within the day
• One does not change his mode or destination overnight (mainly due to mode and activity
availability and habit)
• Changing origin mostly involves relocation…
Seconds Minutes Hours Days Weeks Months Years Decades
Response time to changed circumstances
Change of Origin
Change of destination
Change of mode
Change of departure time
Why are long term effects
different? (1/2)
Change of route

• Because these choice types are strongly related
• Because choices of different people are interrelated
• In peoples minds route, departure time, mode and destination are often chosen simultaneously
• Location choice is often related to the accessibility of a location
• Attractiveness of mode, destination, departure time and route alternatives depends on their
usage by other people.
Why are long term effects
different? (2/2)
Change of Origin
Change of mode
Change of route

Choices and interactions in
strategic transport models
• In transport models we assume sequential choices
• Not all transport models have a departure time choice model
• Mode and destination choice models are usually combined
• Most modern transport models have feedback loops
• Note that the assignment model itself contains a loop
between route demand and infrastructural supply
Change of Origin
Change of mode
Change of route
Trip Generation model
Mode choice model
Destination choice model
Departure time choice model
Assignment model

Feedback loops and iterations in
strategic transport models
• Usually models are equilibrated ‘from the inside out’:
Route Choice Model
Route demand
Network loading model
Route travel times
Assignment model
OD travel times
(skim matrix)
Departure Time
Choice Model
OD demand
(OD-matrix)
Mode/destination
Choice Model
To reach route choice equilibrium: x times assignment
To reach departure time choice equilibrium: y times departure time choice +
y*x times assignment
To reach total equilibrium: z times mode/destination choice +
z*y times departure time choice +
z*y*x times assignment

Convergence – what and why?
Conclusions
• Convergence is the extent to which we have reached
equilibrium
• In strategic transport models we want to reach equilibrium
because we are interested in the long term effects of
measures
• Finding the equilibrium is computationally expensive because
we need to take into account:
• Interaction effects between different choices (mode, destination,
departure time and route)
• Interaction effects between people making these different
choices

2. Generic description of
equilibria in strategic
transport models

Demand and supply models
• Demand models determine demand on OD or route level,
given travel times on all ODpairs, routes, modes and
departure times.
• An ‘economist look’ on demand models:
Route Choice modelRoute travel times Route demand
Departure Time
Choice Model
Travel times per
departure time
(skim matrices)
Demand per departure
time (OD matrices)
Mode/destination
Choice Model
Modal travel times
(skim matrices)
Demand per mode
(OD matrices)
D
c Analogy with
distribution functions of
a gravity model ZKM
D=f1(c)

Demand and supply models
• Supply models determine travel times on route or OD level,
given demand on all routes/ODpairs, departure times and
modes.
• An ‘economist look’ on supply models:
Network loading modelRoute demand Route travel time
Assignment model
for departure time t
Demand for
departure time t
(ODmatrix)
Travel times per
departure time (skim
matrices)
Assignment model
for mode m and
departure time t
Demand per mode
m, dep time t and
OD (skim matrices)
Demand per mode
(OD matrices)
D
c
Analogy with BPR
functions in a static
assignment model
c=f2(D)

Relationship between demand and
supply
• Thus, we are looking for the intersection of supply and
demand curves:
• This can be formulated as a fixed point problem in which we
look for equilibrium Demand D* and cost c*:
• where: f1: demand model as function of cost
f2: supply model as function of demand
c
D=f1(c) c=f2(D)
D*
c*
* *
1
*
1 2
( )
( ( ))
D f c
f f D



Relationship between demand and
supply
• However, our demand supply functions (f1 and f2):
• Cannot be expressed as analytical function;
• Are multidimensional (each route and OD pair have their own)
• Are non-separable (demand and cost on a route or OD pair can
be (and mostly are) dependent on cost and demand on other
routes and ODpairs
• So….

3. How do we reach
convergence?

Methods to reach convergence
• The most basic method is the method of repeated
approximations (MRA) simply puts demand and supply
models in a loop:
• As a formula (one Odpair/route):
• As a formula (matrixNotation):
• It has been proved that MRA only converges when the
combined supply and demand function is a
contraction map (see next slides)
ciDemandModel Di SupplyModel
 1 2 1 1 1 2 1 1( ( )) ( ( ))i i i i iD f f D D f f D D      
 1 2 1 2( ( )) ( ( ))f f f f   i i-1 i-1 i-1 i-1D D D I D D
1 2( ( ))f f i-1D

Contraction map?? an example
• Assume single dimensional curves:
• This is a contraction map! MRA iterations (green dotted lines)
converges to the equilibrium, because:
D0
c0
D1
c1
D2
c2
D3
c3
D4
c4
c=f2(D)
(supply Model)
D=f1(c)
(Demand Model)
D
2 ( )f D
1( )f c
c
1
1 1
2
( ) 1.1 40 ( ) 1/ 1.1( 40)
( ) 0.75 0
D f c c f D D
c f D D

       
  
1
2 2 1
1
( ) ( ) ( )
( )
0.75 0.909
f D f D f Dc
D f c D D

  
    
   


Non-contraction maps
1
1 1
2
( ) 1.1 40 ( ) 1/ 1.1( 40)
( ) 1.0 0
D f c c f D D
c f D D

       
  
1
1 1
2
( ) 1.0 40 ( ) 1/ 1.0( 40)
( ) 1.0 0
D f c c f D D
c f D D

       
  
1
2 2 1
1
( ) ( ) ( )
( )
1 0.909
f D f D f Dc
D f c D D

  
    
   

1
2 2 1
1
( ) ( ) ( )
( )
1 1
f D f D f Dc
D f c D D

  
    
   
Diverges! Cyclic unstable!

Improving convergence
• Method of successive averages (MSA)
• Averages demand over all previous iterations
• As a formula (one Odpair/route):
• As a formula (matrixNotation):
• Virtually no extra calculation time needed per iteration
ciDemand model Di Supply ModelMSA iD
1iDi+=1
 1 1 2 1 1
1
( ( ))i i i iD D f f D D
i
    
 1 2
1
( ( ))f f
i
  i i-1 i-1 i-1D D I D D

MSA speeds up if contraction
1 0
1
( )
2
D D
0D1D
2 1
1
( )
3
D D
3 2
1
( )
4
D D
4 3
1
( )
5
D D
5 4
1
( )
6
D D

MSA enforces contraction!

Other averaging schemes based on
the MSA concept
• Apply instead of (when it is called Polyak)
• Reset the iterationnumber at a fixed iteration interval (Rich
and Nielsen (2015) advise resetting every 5 iteraties.
• Reset the iterationnumber using some scheme. Cantarella et
al (2015) advise to start with a reset interval of 2 iterations
and increase this number by 1 after each reset.
• Scaling the weighing factor using a factor <1. Cantarella et
al (2015) advise a scaling factor in the range [0.7,0.8].
• Method of weighed successive averages (MwSA) using
• The self regulating average (SRA): chooses a large or small
step size, based on the level of convergence of the current
iteration
1/i
1/i 2/3 
1..
/d d
i
i i

Weighing factors over iterations
MSA based methods (1/2)
• MRA vs methods compliant with Blum’s theorem

Weighing factors over iterations
MSA based methods (2/2)
• Methods not compliant with Blum’s theorem

Non-MSA based methods
• Newton and Broyden methods work quite well, but require
explicit calculation of Jacobian (matrix of partial derivatives)
and inversion of large matrices
• These methods are therefore not applicable to strategic
transport model systems
• Intersection method does not perform well in
multidimensional situations
• Fixed weighing factors: heuristic that proves to be very
application specific

Comparison of methods on linear
example (contraction map)

Comparison of methods on linear
example (non-contraction map)

Choosing a convergence method /
averaging scheme
• Criteria:
• Level of convergence
• Speed of convergence (e.g. compared to MRA or MSA)
• Usage of heuristic parameters (not preferred)
• Complexity of implementation
• The level and speed of convergence can be determined by:
• The duality gap (preferred, since it is directly derived from the
definition of the user equilibrium)
• For departure time choice and mode destination choice models,
often ‘normal’ distance functions not using the cost are being
used (MSE,RMSE,…,SSIM)

Convergence measure
route vs network loading model
• The duality gap is directly derived from the definition of the
users equilibrium of Wardrop (1952):
The journey times in all routes actually used are equal and less
than those which would be experienced by a single vehicle on any
unused route
Travel times on all used routes between an od pair must be
equal
• The duality gap is the normalized number of vehicle*hours
spent on the network, compared to the fastest route on each
OD pair.
𝑫𝑮 =
𝒐𝒅 𝒓∈𝑹 𝒐𝒅
𝑫 𝒐𝒅 𝒇 𝒓(𝒕 𝒓 − 𝒕 𝒎𝒊𝒏,𝒐𝒅)
𝒐𝒅 𝑫 𝒐𝒅 𝒕 𝒎𝒊𝒏,𝒐𝒅

Convergence measure
route vs network loading model
• The stochastic duality gap is directly derived from the
duality gap and the definition of the stochastic user
equilibrium (Fisk 1980):
The perceived journey times in all routes actually used are equal
and less than those which would be experienced by a single
vehicle on any unused route
Perceived travel times on all used routes between an od pair
must be equal
• Given an Multinomial Logit route choice model, replace 𝒕 𝒓
by 𝒕 𝒓 +
𝟏
𝝁 𝒐𝒅
ln 𝑫 𝒐𝒅 𝒇 𝒓
• Derivation in Bliemer et al (2013)*
*http://atrf.info/papers/2013/2013_bliemer_raadsen_de_romph_smits.pdf

4. Example: convergence of
quasi dynamic assignment
model STAQ

Supply and demand models used
in this example
• We only focus on the assignment model
• We used:
• STAQ as network loading model;
• Multinomial logit as route choice model
• MSA and SRA as averaging schemes
Route Choice Model
Route demand
Route travel times
Assignment model
OD travel times
(skim matrix)
Departure Time
Choice Model
OD demand
(OD-matrix)
Mode/destination
Choice Model

From static (STA) and dynamic (DTA)
assignment models to STAQ
Static models
• Speed-flow curve
• Stationary travel demand
• Single time period
• No hard capacity constraints
due to lack of node model
First order dynamic models
• Fundamental diagram
• Variable travel demand
• Multiple time periods
• Hard capacity constraints
due to explicit node model
STAQ
(Brederode et al. 2010, Bliemer et al. 2012)
• Fundamental diagram
• Stationary travel demand
• Single time period
• Hard capacity constraints
due to explicit node model

35Challenge the future 35
From static to STAQ (1/2)
Capacity = 4000 veh/h
Demand= 4200 veh/h
A B
What is the queue length and travel time from A to B after one hour?
In traditional static network loading model:
- No physical queue, delay within the bottleneck
- Travel time derived from travel time function:
cap
cap

From static to STAQ (2/2)
Demand= 4200 veh/h
A B
In STAQ: Squeezing…
- Queue length: 1150m
- Travel time: 12 min.
Queuing…
4000
0.95
4200
  
What is the queue length and travel time from A to B after one hour?
State 2 State 1State 3

Options for STAQ
• Different options on how to use STAQ:
• With or without spillback (queuing phase can be disabled)
• Junction modelling:
• None
• Only turndelays (only route choice affected)
• Turncapacities and turndelays (route choice and network loading
affected)
• Averaging scheme
• Method of successive averages (MSA)
• Self Regulating Average (SRA)

Test runs
• How do these options affect convergence?
• What would be a good tolerance for the duality gap value
Run# Title treshold Averaging scheme Spillback JM Iterations
1 MSA no spillback, no JM 1 MSA false False 100
2 MSA no spillback, turndelays 1 MSA false turndelays 100
3 MSA no spillback, JM 1 MSA false true 100
4 MSA spillback, no JM 1 MSA true false 100
5 MSA spillback, turndelays 1 MSA true turndelays 100
6 MSA spillback, JM 1 MSA true true 100
7 SRA no spillback, no JM 1 SRA false false 100
8 SRA no spillback, turndelays 1 SRA false turndelays 100
9 SRA no spillback, JM 1 SRA false true 100
10 SRA spillback, no JM 1 SRA true false 100
11 SRA spillback, turndelays 1 SRA true turndelays 100
12 SRA spillback, JM 1 SRA true true 100

3.321 Centroids
142.336 Links
106.780 Nodes
808.708 Used OD pairs
1.272.330 Routes
Case: BBMA network (Province of
Noord Brabant)

Effect of different STAQ options

Convergence - conclusions
• Runs with spillback do NOT converge:
• interaction effects between users too large, therefore the implicit cost function is no
longer diagonally dominant
• The duality gap never gets below 5.0E-04 and keeps oscillating
• Runs witout spillback do converge sufficiently:
• MSA with full junction modelling: after 80 iterations (2:15 h) duality gap ≈ 1.0E-04
• SRA with full junction modelling: after 32 iterations (0:54 h) duality gap < 1.0E-04
• SRA with full junction modelling: after 40 iterations (1:07 h) duality gap ≈ 5.0E-03
• Run without spillback and junction modelling truly converges:
• SRA after 12 iterations (0:19 h) duality gap < 1.0E-04
• SRA after 19 iterations (0:30 h) duality gap < 1.0E-05
• SRA after 100 iterations (2:37 h) duality gap <5.0E-11 (~1E-14 == machine precision)

Determining tolerance on duality
gap
• According to Boyce, Ralevic-Dekic and Bar-Gera (2004)*, a
duality gap of 1E-04 should be enough.
• Run 9 (SRA, no spillback, JM) proved to be the most realistic,
whilst still converging.
• For this run, we compared the link flows of the following
intermediate iterations with the most converged situation
(iteration 100, DG =3.5E-05)
• We thus used iteration 100 as a benchmark
*David Boyce, Biljana Ralevic-Dekic, and Hillel Bar-Gera - Convergence of Traffic Assignments: How Much Is Enough?
Technical Report Number 155, January 2004 - National Institute of Statistical Sciences

absolute differences of link
flows compared to last iteration
Iteration 2 (DG < 1E-01)

Iteration 10 (DG <5E-04)

Iteration 16 (DG <1E-04)

Determining tolerance on duality
gap - conclusions
• Around Tilburg, a duality gap tolerance of 1E-03 proved to be
low enough
• Network wide, a duality gap tolerance of 5E-04 proved to be
low enough. This is mainly needed in Eindhoven, where there
where some unresolved some issues with network coarsity.
• Similar results on Haaglanden, Amsterdam and Leuven
networks

5. Example: convergence of
mode/destination choice model,
departure time choice model and
STAQ

Supply and demand models used
in this example
• We only focus on the assignment model
• We used:
• MD-PIT departure time choice model (logit based)
• Doubly constrained multimodal gravity model for
mode/destination choice
Route Choice Model
Route demand
Route travel times
Assignment model
OD travel times
(skim matrix)
Departure Time
Choice Model
OD demand
(OD-matrix)
Mode/destination
Choice Model

Departure Time Choice Model
(TOD)
• Convergence of TOD iterations (using STAQ assignments
converged to 5E-04 )

Departure Time Choice model
• Link flows, speeds and vertical queues over 10 TOD iterations:

Tolerance on the duality gap
(departure time loop)
• After iteration 4, no noticeable changes occur
• Which means –again- a tolerance on the duality gap of 1E-04
• Note: we use an assumption that simplifies the departure
time choice model:
• Travel time in the off peak periods is not affected by the
increase of traffic demand due to departure time shifts
• This means that once demand is in the off peak, it will stay
there over iterations

Mode/Destination choice model
(gravity model)
• Convergentiegrafieken ZKM+TOD

Mode/Destination choice model
(gravity model)

Tolerance on the duality gap
(gravity model loop)
• After iteration 4, no noticeable changes occur (as in TOD)
• Which means –again- a tolerance on the duality gap of 1E-04
• Note:
• Distribution functions where not recalibrated after replacing
static traffic assignment model with STAQ
• As such, the gravity model is too sensitive, and too much
demand was moved to public transport and bike
• Recalibration of the gravity model is likely to lead to a more
subtle equilibrium, and thus more car users, and thus harder to
equilibrate the assignment model (note: duality gaps on sheet 40 won’t
be affected, because these where based on runs without feedback loops to gravity
model and TOD)

Conclusions – assignment model
• Adding spillback to the assignment model causes non-
convergence
• Junction modelling also has a clear impact on convergence of
the assignment model, but acceptable levels of convergence
can still be reached
• The self regulating average improves convergence of
assignment model, making it capable to reach machine
precision

Conclusions – departure time and
mode/destination models
• Departure time choice model converges really fast (3
iterations), mainly due to assumption of fixed travel times in
off peak period
• Mode/destination choice model parameters need to be
recalibrated due to change in assignment method (definition
of travel time has changed!)
• Mode/destination choice loop also converges relatively fast (3
iterations)

6. General conclusions

Conclusions part 1
• In strategic models, (user) equilibrium is an important
concept. Due to interaction effects between choices and
between people making those choices
• Each demand model within strategic transport models
(destination choice, mode choice, departure time choice,
route choice) should be brought in equilibrium with its
respective supply model.
• When assuming that the equilibrium can mathematically be
described as a fixed point problem, we can use MSA based
techniques to enforce and/or speed up convergence towards
it.

Conclusions part 2
• It is worthwhile to spend some time on finding a way to
minimize the number of iterations needed for all equilibria in
the model (z*y*x assignments, z*y TOD’s, z gravity models)
• The duality gap as convergence measure is the most direct
translation of the definition of user equilibrium, and as such
the most suitable indicator
• A duality gap value of 1E-04 seems to be a suited level of
convergence for assignment, TOD and gravity model
• This seems to be a model independent conclusion

Thank you for your attention!
• Looking for a interesting topic for your masters thesis
assignment or internship?
• Equillibria and convergence
• Quasi dynamic assignment models
• Demand matrix estimation using (quasi) dynamic assignment
models
• Network aggregation to improve calculation speed
• Automatic transport model network generation using TomTom
or Here data sources
• Improved bicycle modelling in strategic model systems
• …
• Drop by (room 4.14), or mail me at lbrederode@dat.nl

20151216 convergence of quasi dynamic assignment models

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