Computational Methods for Macroscopic Model Validation and Model Predictive Control
Computational Methods for Macroscopic Model Validation and Model Predictive Control Apostolos Kotsialos & Adam Poole School of Engineering and Computing Sciences Durham University Tel.: +44 (0)191 33 42399 E-mail:
[email protected] TRAWS 3 Presentation, IPAM, UCLA School of Engineering and Computing Sciences, Durham University, Durham, DH1 3LE, UK
October 29th, 2015
Computational Methods for Macroscopic Model Validation and Model Predictive Control
Outline 1
Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation Small sites Evolutionary algorithms Gradient based algorithms A larger network
2
Algorithms for Road Network Traffic Model Predictive Control Hierarchical model predictive control Optimisation layer for coordinated ramp metering Rolling horizon
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Macroscopic modelling of road networks Macroscopic description of traffic flow in a network. Density: ρm,i (k ) (veh/km/lane) the number of vehicles in segment i of link m at time k · T divided by the number of lanes in the link λm and by the segment length Lm . Mean speed: vm,i (k ) (km/h) which is the space mean speed of the vehicle flow in segment i at the current time instant k · T . Traffic flow (or volume): qm,i (k ) (veh/h) the number of vehicles leaving the segment i of link m during the interval [k · T , (k + 1) · T ], over T .
Time and space discretization of the conservation and speed equations x(k + 1) = f [x(k ), d(k ); z] , x(0) = x0 x state vector d disturbance vector z vector of model parameters. Our purpose is to find ways of determining an optimal vector of parameters z∗ that minimises the model error. Model error J x(k ), b x(k )
∂ρ(x, t) ∂t ∂v (x, t)
+
∂q(x, t) ∂x
=0
where b x(k ) is a set of measurements at different locations in the network over a specific time period. Typically this is the total square error.
∂v (x, t) + v (x, t) + ∂t ∂x Model spatio-temporal distribution profile of density, flow and speed vs measurements. 1 ∂P(x, t) 1 = {V [ρ(x, t)] − v (x, t)} ρ(x, t) ∂x τ
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Research on this problem Based on the previous work...
Criticism Most of the model validation studies are performed on an ad hoc basis. The sites used are usually unidirectional, i.e. a single motorway stretch. The length of the stretch is relatively small (about 5km to 6km). No general methodology that can be applied to all simulators. Main objective to validate a numerical scheme and support the results with some data. Overparametrization. Requirements Large scale and realistic size sites need to be developed. We need to consider networks rather than just unidirectional road parts. Avoid overparametrization. Generic design, simulator independent if possible. Automating the process.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Validating a motorway network Basic issues... Different models for modelling motorway stretch dynamics (1st, 2nd or higher order). Which one? Flow assignment models at junctions (network topology requirement). How? Real data required. Where, what and how?
Addressing those issues...(continued) Flow assignment model. Introduction of the turning rates at junctions.
Addressing those issues... Data. MIDAS data available from the Highways Agency for selected UK sites. Archived in database managed by Mott-McDonald. Loop detector technology. Traffic counts (flows) and speeds for every lane. Daily minute by minute data. Some uncertainty about the exact location of the loops. Loops not always functioning. Journalistic data missing.
Qn (k ) =
X
qµ,Nµ (k )
∀n
µ∈In
The turning rate βnm (k ) is defined as the percentage of Qn (k ) that leaves through out link m ∈ On during period k . m
qm,0 (k ) = βn (k ) · Qn (k )
∀m ∈ On .
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Validating a motorway network (cont’ed) Addressing those issues... Model used. Second order model METANET. Simulator package providing easy way of set up and configuration. C code available; easy to integrate with developed optimisation algorithms. No change at the core of the simulation program. C code availability allows the use of automatic differentiation (ADOLC package) for gradient based optimisation. Scripts for proper interfacing when the simulator is invoked from the optimisation algorithm. Motorway traffic flow model equations.
Model equations
ρm,i (k + 1)
=
qm,i (k ) vm,i (k + 1)
= =
ρm,i (k ) + L Tλ [qm,i−1 (k ) − qm,i (k )] m m ρm,i (k )vm,i (k )λm vm,i (k ) T {V [ρ
(k )]−v
(k )}
m,i m,i + τ + LT vm,i (k )[vm,i−1 (k ) − vm,i (k )] m
− τνT L
m
ρm,i+1 (k )−ρm,i (k ) ρm,i (k )+κ
−δTqµ (k )vm,1 (k )/(Lm λm (ρm,1 (k ) + κ)) 2 −φT ∆λρm,Nm (k )vm,Nm (k ) /(Lm λm ρcr ,m ) ρm,i (k ) αm V [ρm,i (k )] = vf ,m exp − α1 ρ m
cr ,m
ν (anticipation constant), κ (numerical stability constant), ρcr ,m (critical density), and αm are model parameters to be determined.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Spatial extension of the model parameters Fundamental diagram. Global network-wide parameters for the speed equation.
ν – anticipation constant τ – relaxation constant ρmax – maximum density vmin – minimum speed δ – on-ramp term φ – lane drop term κ – numerical stability Fundamental diagram parameters for a motorway link m (uniform geometry). Qe [ρm,i (k )] = ρm,i (k ) αm vf ,m ρm,i (k ) exp − α1 ρ m
cr ,m
free speed vf ,m exponent αm critical density ρcr ,m (the most important one) Overparametrization: try to use as few as possible FDs. How to decide the number, limits and spatial extension of the fundamental diagrams used?
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Fundamental diagram decisions and validation process... Two-phase approach in previous work for the large scale motorway network of Amsterdam Phase 1: quantitative validation. Split the network into representative motorway stretches and perform model calibration for each one. Outcome: estimate the optimal parameter set for each part of the network.
Phase 2: qualitative validation. Use optimal parameter sets to similar motorways and manual tuning of turning rates. In this work: only quantitative validation based on data; reduced need for qualitative validation.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
Available MIDAS data Locations of loop detectors. Entry and exit points. They provide the boundary conditions (demand, speed and density). At bifurcations. Turning rate trajectories. Along the motorway. Comparison points; data are used for calculating the error.
Data available from loop detectors. Used data: traffic counts per minute and time mean speeds. yj,q (k ) the flow measurement at location j at model time step k . Pλm yj,q (k ) = y (k ) `=1 j,`,q
N (k ) `=1 j,` Pλm Nj,` (k )
0
20
40
110 100 90 80 70 60 50 40 30 20 60 80 100 120 140 160 180 Time (mins)
Flow
Turning Rate
`=1 vj,` (k )
where Nj,` is the number of vehicle counts at lane ` of location j.
6500 6000 5500 5000 4500 4000 3500 3000 2500
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0
20
40
Velocity (km/h)
Pλm
yj,v (k ) =
Flow (veh/h)
Lane time mean speeds are measured vj,` (k ). The time mean speeds are used to estimate the space mean speeds. Assumption: there are homogenous traffic conditions along the lane length. Hence, the lane’s space mean speed is equal to the lane’s time mean speed. The cross-lane segment’s space mean speed is estimated as
Measurements are organised in vector b x.
Velocity
60
80 100 120 Time (mins)
140
160
180
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
The model calibration optimisation problem For a motorway network with: M1 links M2 entry points M3 exit points M4 junctions requiring turning rates M5 measurement locations used for the error calculation The optimisation problem at hand is
The vector of parameters z is a more complicated story... If we impose a single fundamental diagram on the whole network z = [τ κ ν ρmax vmin δ φ vf α ρcr ]T . b different fundamental diagrams to be If we allow up to N used b N b N b T 1 1 1 N z = τ κ ν ρmax vmin δ φ vf α ρcr . . . vf α ρcr .
min J x(k ), b x(k ) z
subject to
This needs to be extended by assigning to each fundamental diagram a starting link l. Hence the parameter’s vector becomes
x(k + 1) = f [x(k ), d(k ); z] , x(0) = x0 zmin ≤ z ≤ zmax x=
T
ρ1,1 v1,1 . . . ρ1,N v1,N 1
1
. . . ρ1,M v1,1 . . . ρ1,N 1
M1
v1,N
M1
b N b N b N b T 1 1 1 1 N z = τ κ ν ρmax vmin δ φ vf α ρcr l . . . vf α ρcr l
T µM µ d = q1 v1 . . . qM vM ρ1 . . . ρM β1 1 βM 4
b where l ι ∈ [1, M1 + 1], ι = 1, . . . , N.
h iT b x = y1,q y1,v . . . yM ,q yM ,v 5 5
Algorithm for splitting the whole network into large sections.
2
2
3
4
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Problem formulation
The problem’s objective function Flow square error Jj,q (x, b x) from sensor j is given by K h i2 X Jj,q (x, b x) = yj,q (k ) − qm ,i (k ) j j k =1
Automatic assignment of FDs; penalty terms Jp (z) included Jp (z) =
PN−1 PN b b ι=1
r =ι+1
Speed square error
Jj,v (x, b x) =
K h i2 X yj,v (k ) − vm ,i (k ) . j j
k =1
h wv vfι − vfr 2 ι +wρ ρcr − ρrcr 2i ι r 2 +wα α − α
where wv , wρ and wα are weights penalising FD parameters’ variance. The objective function becomes
Weighted total error Je is J x, b x, z = Je (x, b x) + wp Jp (z) M4
Je (x, b x) =
i Xh Aq Jj,q (x, b x) + Av Jj,v (x, b x) j=1
where Aq and Av are scaling factors.
where wp a total penalty weight. But we are also using Aq = 0 and Av = 1 and different weights for gradient based algorithm.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Small sites
Two simple sites
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
System structure
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
Particle Swarm Optimisation – basic structure ξ
ξ
ξ
At iteration ξ particle ι is at position zξ ι = [zι,1 , zι,2 , . . . , zι,Γ ] within C and has directional velocity θιξ
=
ξ ξ ξ [θι,1 , θι,2 , . . . , θι,Γ ].
Update of ι’s directional velocities and positions: ξ+1 θι,γ
=
h i h i ξ ξ ξ ξ ξ ωθι,γ + c1 r1 πι,γ − zι,γ + c2 r2 zh ,γ − zι,γ ι
ξ+1 zι,γ
=
ξ ξ+1 zι,γ + θι,γ
ω is the inertia weight, c1 , c2 are acceleration coefficients, r1 , r2 are random numbers in the domain [0, 1], ξ
ξ
ξ
πι ξ = [πι,1 , πι,2 , . . . , πι,Γ ] is the best position previously found by particle ι until iteration ξ and hι is the index of the best particle within the neighbourhood of ι. Many variations exist; we used 7 of those.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
PSO algorithms used for Heathrow and Sheffield
APSO-09 APSO-12 APSO-14 HEPSO CEPSO-12 GPSO LPSO Simple GA (baseline) Heathrow: Monday 8th , 15th and 22nd of February 2010 Sheffield: Monday the 1st , 8th and 15th of June 2009.
Summary of best objective value for each algorithm (H = Heathrow and S = Sheffield) H 8th H 15th H 22nd S 1st S 8th S 15th GA 5384538 5858308 6739297 6595545 7024323 7619637 GPSO 3792653 3142996 2466972 4387156 5644626 3913927 LPSO 3785607 1997553 1893783 3956224 4399481 3902504 APSO-09 5108443 2613920 1882825 4530514 5542426 4414949 APSO-12 6541304 2934669 3478906 4841882 7178995 5257622 APSO-14 4357332 5823782 4222196 4366692 7530088 5295245 HEPSO 2585451 2389230 1928796 3797889 4453520 4061613 CEPSO 3801993 3215278 2092989 4567079 5299468 4881406 CS 5733080 3806055 6468042 6273483 6293645 6649381 MCS 5440922 3988003 6483786 7278220 6840414 7581395
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
7e+06
H8-1 H8-2 H8-3 H15-1 H15-2 H15-3 H22-1 H22-2 H22-3
6e+06 5e+06 4e+06
S1-1 S1-2 S1-3 S8-1 S8-2 S8-3 S15-1 S15-2 S15-3
2e+06 1e+06 0
GPSO
5e+06 4.5e+06 4e+06 3.5e+06 3e+06 2.5e+06 2e+06 1.5e+06 1e+06 500000 0
LPSO
APSO-09
APSO-12 Algorithm
APSO-14
HEPSO
CEPSO
H8 H15 H22 S1 S8 S15
SO GP
O O O 09 12 14 LPS PSO- PSO- PSO- HEPS CEPS A A A Algorithm
Dierence to best found objective value
3e+06
Dierence to best found objective value
Dierence to best found objective value
Algorithm perfromance (difference from the best)
6e+06
S1-1 S1-2 S1-3 S1-4 S1-5 S1-6
5e+06 4e+06 3e+06 2e+06 1e+06 0
SO
GP
O
LPS
O O 09 12 14 SO- PSO- PSO- HEPS CEPS AP A A Algorithm
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
PSO algorithm convergence for Heathrow 1e+07
GA GPSO LPSO APSO-09
Best Objective Value
9e+06
APSO-12 APSO-14 HEPSO CEPSO
CS MCS
8e+06 7e+06 6e+06 5e+06 4e+06
1e+07
3e+06
50000 100000 Function Evaluations
150000
H8th 1e+07
GA GPSO LPSO APSO-09
9e+06
APSO-12 APSO-14 HEPSO CEPSO
CS MCS
Best Objective Value
0
Best Objective Value
GA GPSO LPSO APSO-09
9e+06
2e+06
7e+06 6e+06 5e+06 4e+06 3e+06
7e+06
2e+06 0
50000 100000 Function Evaluations
5e+06
H22nd
4e+06 3e+06 2e+06 0
50000 100000 Function Evaluations
H15
th
150000
CS MCS
8e+06
8e+06
6e+06
APSO-12 APSO-14 HEPSO CEPSO
150000
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
PSO algorithm convergence for Sheffield 9e+06
GA GPSO LPSO APSO-09
Best Objective Value
8e+06
APSO-12 APSO-14 HEPSO CEPSO
CS MCS
7e+06
6e+06
5e+06 9e+06
GA GPSO LPSO APSO-09
4e+06 0
50000 100000 Function Evaluations
150000
S1st 9e+06
GA GPSO LPSO APSO-09
Best Objective Value
8e+06
APSO-12 APSO-14 HEPSO CEPSO
CS MCS
Best Objective Value
8e+06
APSO-12 APSO-14 HEPSO CEPSO
CS MCS
7e+06
6e+06
5e+06
4e+06
7e+06
0 6e+06
50000 100000 Function Evaluations
S15th
5e+06
4e+06 0
50000 100000 Function Evaluations
S8th
150000
150000
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
Solutions for the Sheffield site
2500
φ 0.038 0.360 0.122
2000 Flow (veh/hr/lane)
Optimal solutions found for Sheffield site model calibration. τ κ ν vmin ρmax δ 1st 32.34 28.63 55.90 7.48 183.13 0.844 th 8 10.22 20.06 20.00 7.99 184.24 0.001 15th 19.09 5.89 23.39 8.00 173.38 0.001
1500 1000 500
1st
Sheffield optimal solutions for FD parameters. ρcr vf α Start link
0 0
FD 1 FD 2 FD 3
27.23 30.19 26.68
122.36 105.35 109.75
2.6760 2.3494 1.1386
1 8 10
7 9 10
FD 1 FD 2 FD 3
31.53 28.50 38.02
122.22 113.77 104.72
2.5587 1.8865 1.0724
1 4 10
3 9 10
FD 1 FD 2 FD 3
28.43 31.88 35.17
115.96 103.43 104.73
2.1077 2.0904 1.1459
1 8 10
7 9 10
8th
15th
20
40
60
80
100
120
Density (veh/km/lane)
End link
1: 1–7 1: 8–9 1: 10
8: 1–3 8: 4–9 8: 10
15: 1–7 15: 8–9 15: 10
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 0.9km7.1km1.9km0.8km3.4km 2.4km 1.6km 2.3km 0.7km0.8km Nottingham Leeds Leicester J30
J31
M18
J33
J34
Sheffield verification total square error Calibrated 1 Calibrated 8 Calibrated 15
S1st 3782550 14388131 7578581
S8th 9741511 4382314 10574672
S15th 7764030 20064014 3858505
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
Calibration results – Sheffield site
60
2000
40
1000
20
0 06:00
0 07:00
08:00
09:00
140
6000
120
5000
100
4000
80
3000
60
2000
40
1000
20
0 06:00
Velocity (km/h)
Flow (veh/h)
Link 6: downstream of M18 merge 7000
0 07:00
08:00
09:00
Time (mins)
S1st
7000
140
6000
120
5000
100
4000
80
3000
60
2000
40
1000 08:00
S15th
09:00
7000
140
6000
120
5000
100
4000
80
3000
60
2000
40
1000
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 0.9km7.1km1.9km0.8km3.4km 2.4km 1.6km 2.3km 0.7km0.8km Nottingham
Velocity (km/h)
Flow (veh/h)
140 120 100 80 60 40 20 0
0 07:00
Link 6: downstream of M18 merge
Leeds Leicester
20
0 06:00
140 120 100 80 60 40 20 0
20
0 06:00
Velocity (km/h)
Flow (veh/h)
Model flow Meas. flow Model velocity Meas. velocity Link 5: upstream of M18 merge
Link 5: upstream of M18 merge 7000 6000 5000 4000 3000 2000 1000 0 06:00 07:00 08:00 09:00 Link 6: downstream of M18 merge 7000 6000 5000 4000 3000 2000 1000 0 06:00 07:00 08:00 09:00 Time (mins) Model flow Meas. flow Model velocity Meas. velocity
Velocity (km/h)
80
3000
Velocity (km/h)
100
4000
Flow (veh/h)
120
5000
Flow (veh/h)
140
6000
Velocity (km/h)
Flow (veh/h)
Link 5: upstream of M18 merge 7000
J30
0 07:00
08:00
09:00
Time (mins) Model flow Model velocity
Meas. flow Meas. velocity
S8th
J31
M18
J33
J34
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
Verification results – Sheffield
Flow (veh/h)
140 120 100 80 60 40 20 0
Flow (veh/h)
140 120 100 80 60 40 20 0
140 120 100 80 60 40 20 0
Velocity (km/h)
Flow (veh/h)
140 120 100 80 60 40 20 0
Flow (veh/h)
Link 5: upstream of M18 merge 7000 6000 5000 4000 3000 2000 1000 0 06:00 07:00 08:00 09:00 Link 6: downstream of M18 merge 7000 6000 5000 4000 3000 2000 1000 0 06:00 07:00 08:00 09:00 Time (mins) Model flow Meas. flow Model velocity Meas. velocity Link 5: upstream of M18 merge 7000 6000 5000 4000 3000 2000 1000 0 06:00 07:00 08:00 09:00 Link 6: downstream of M18 merge 7000 6000 5000 4000 3000 2000 1000 0 06:00 07:00 08:00 09:00 Time (mins) Model flow Meas. flow Model velocity Meas. velocity
Velocity (km/h)
Verification of the Sheffield model using input data from set S15th and using the optimal parameter set based on (a) the S1st data set and (b) the S8th data set.
Velocity (km/h)
(a) L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 0.9km7.1km1.9km0.8km3.4km 2.4km 1.6km 2.3km 0.7km0.8km Nottingham Leeds Leicester
Velocity (km/h)
J30
(b)
J31
M18
J33
J34
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Evolutionary algorithms
Evolutionary algorithms...
The simplest variant LPSO performs better... better setting for evaluation may be necessary but not strong evidence against this conclusion. PSO outperforms simple GA (baseline case anyhow) and Cuckoo search. Computation time still an issue, but in principle does not pose a problem since model calibration is “rest and digest” functionality in a TCC’s information ecosystem, so there should be amble time to perform it periodically and for specific days, months, special events etc. A. Poole and A. Kotsialos, “Swarm intelligence algorithms for macroscopic traffic flow model validation with automatic assignment of fundamental diagrams,” Applied Soft Computing, Vol. 38, pp. 134–150, 2016.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Gradient based algorithms
Improve convergence? maybe if we use gradient based optimisation for the calibration problem. We need a way to calculated the gradient ∂J (x,b x,z) ∂z1 ∂J (x,bx,z) b, z = ∂z2 ∇Jz x, x ... ∂J (x,b x,z)
∂zΓ
We can apply then a gradient-based optimisation algorithm. Difficult to obtain analytical expressions. Estimate the derivatives using Automatic Differentiation (AD) b, z software technology to approximate ∇Jz x, x
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Gradient based algorithms
Using AD Taking some liberty with notation we can rewrite the optimisation problem as minz J simulate for k = 0, . . . , K − 1: x(k + 1) = f [x(k ); z] , x(0) = x0 , b x, z subject to zmin ≤ z ≤ zmax Because of the way links are modelled in METANET b b b T b number of links (not of segments). with N= z = τ κ ν ρmax vmin δ φ vf1 α1 ρ1cr . . . vfN αN ρN cr ADOL-C (the AD software) can provide us with the Jacobian matrix ∂x as we perform the simulation ∂z (forward integration), i.e. we get an approximation of ∂x(1) ∂z1 ∂x(2) ∂z1 ∂x = . ∂z . .
∂x(K ) ∂z1
∂x(1) ∂z2
...
∂x(1) ∂zΓ
∂x(2) ∂z2
...
∂x(2) ∂zΓ
. . .
.
∂x(K ) ∂z2
.
. . .
.
...
∂x(K ) ∂zΓ
Using the chain rule ∇Jz = ∂Jv ∂x
and
∂Jp ∂z
∂Jv T ∂x ∂x
∂z
!T +
∂Jp ∂z
=
are calculated analytically (quadratic terms).
The result is a nonlinear simply bounded optimisation problem.
∂x ∂z
T
∂Jv ∂x
+
∂Jp ∂z
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Gradient based algorithms
The RPROP algorithm The ∇Jz is an approximation. We need an algorithm that can accommodate for these errors. RPROP
RPROP algorithm for optimisation A. Kotsialos, “Nonlinear optimisation using directional step lengths based on RPROP,” Optimization Letters, Vol. 8, nr 4, pp. 1401–1415, 2014. A. Kotsialos, “Non-smooth optimization based on resilient backpropagation search for unconstrained and simply bounded problems,” Optimization Methods and Software, Vol. 28, nr 6, pp. 1282–1301, 2013.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation Gradient based algorithms
METANET + ADOL-C + RPROP Parent
Child
Parent
Child
Create SHM x for ∂ ∂z
Create SHM x for x and ∂ ∂z
Create model input files
Create model input files
Create Child
Create Child
Start
Open child read pipe
Open parent write pipe
Read data from pipe
Run model: pipe/SHM
Start
Run model: SHM
1 2 . . . K
Jacobian SHM
Wait for child to complete
k
k
1
1
2 . . . K
2 . . . K Jacobian
∂x ∂z
End
PIPE
SHM
k
SHM
Wait for child to complete
∂x ∂z
End
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
A much larger network at Manchester M66
M62 M61
M60
M62
M602
M60
M56
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
Automatic sectioning of the network M66
M62 M61
M60
M62
M602
M60
M56
Section
Motorways
Origins
Destinations
Links
Length
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
A5103/M56 M61 M60 (+M62,M66) M62/M602 M60 (+M62,M66) M602 A5103/M56 M60 M56 M60/A5103 M61/A580 M61/M60 M60/R’bout M60/M61 A580 M60/M602 M60 M60/A5103 M66/R’bout M62/M60 M66/R’bout M56 M62 M61 M60/R’bout M602/M60 R’bout/M60 M60/M62 R’bout/M60 M62/M60 R’bout/M62 M60/M602 R’bout/M66 M602/M60 R’bout R’bout R’bout R’bout R’bout
5 2 19 1 20 1 4 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 18 1 20 0 5 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
9 4 49 5 51 3 9 1 3 1 3 1 2 1 1 2 1 1 2 2 2 3 3 2 2 2 2 1 2 4 2 1 2 1 8 1 1 1 1
6.90 3.15 62.50 6.55 62.27 1.50 7.00 0.90 3.45 0.70 2.30 1.20 0.90 1.00 0.80 0.90 0.45 1.30 0.90 0.90 0.90 3.55 1.80 1.70 0.90 1.15 0.90 0.80 0.90 2.35 0.90 0.50 0.90 0.50 1.80 0.45 0.45 0.45 0.45
TOTAL
56
55
192
186.92
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
Main sections for Manchester site M66
M66
M62
M62
M61
M61
M60
M62
M602
M60
M60
M62
M602
M60
M56
M56
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
Some of the minor ones M66
M66 S20
S24
S32
M62 S10
S12
M61
S26
M61 S23
S18 S7
M62
S30
M62
S1
S14
S3
M60 M602
S5
S19
M62
S15
M60 M602
S22
M60
S28 S11
S25
M60
S17
S8
S9 S21 S0
S6
M56
M56
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
40 35 30 25 20
Alpha
J18
S24 S30 J17
S22 S19 S23 S1 J14 J15 J16
J7
J6
J9
J8 J9 J10 J11 J12 J13
S9 J5
J2 J3/4
J25 J26 J27 J1
J24
S21
4000 3500 3000 2500 2000 1500 1000 500 0 130 120 110 100 90 80 70 60 50 45 40 35 30 25 20
4
4
3.5
3.5
3
J23
J18 J19 J20 Capacity
Critical Density
(km/h)
Free Speed
(veh/lane)
J18
J19
J20
J21
J22
J23
J25
J24
J27
45
J24 J25 J21 J22
S20 S26
S18 S32 J1
J5
J6
J3/4 J2
J7
J11
J9
J10
J8
J12
J13
J16
J17
J18
S8
3 Alpha
Capacity
S17
(veh/km/lane)
(km/h)
S13 S11 S15 S29
4000 3500 3000 2500 2000 1500 1000 500 0 130 120 110 100 90 80 70 60 50
(veh/km/lane)
Critical Density
Free Speed
(veh/lane)
S12S28
J15 J14
62-J18
LPSO calibration results – sections 2 & 4
2.5 2 1.5
2.5 2 1.5
1
1
0.5
0.5
0
10
20
30
40
50
60
0
10
20
Distance (km) 14
21
28
30
40
Distance (km) 14
21
28
50
60
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
LPSO calibration contours – sections 2 & 4
Data
Model
S32 S18
60
Data
120
100
50
Model
120
60
S30 S24 S1 S23
50
100
S19 S22
20
30
40
20
20
10
S21
J25 J24
S29 S15 S11 S13
10
07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
0
60
40
20
S28 S12 0
80
S9
S26 S20 07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
Velocity (km/h)
60
S17
J9
40 Distance (km)
Distance (km)
S8
30
Velocity (km/h)
80
40
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
40 35 30 25 20
Alpha
J18
S24 S30 J17
S22 S19 S23 S1 J14 J15 J16
J7
J6
J9
J8 J9 J10 J11 J12 J13
S9 J5
J2 J3/4
J25 J26 J27 J1
J24
S21
4000 3500 3000 2500 2000 1500 1000 500 0 130 120 110 100 90 80 70 60 50 45 40 35 30 25 20
4
4
3.5
3.5
3
J23
J18 J19 J20 Capacity
Critical Density
(km/h)
Free Speed
(veh/lane)
J18
J19
J20
J21
J22
J23
J25
J24
J27
45
J24 J25 J21 J22
S20 S26
S18 S32 J1
J5
J6
J3/4 J2
J7
J11
J9
J10
J8
J12
J13
J16
J17
J18
S8
3 Alpha
Capacity
S17
(veh/km/lane)
(km/h)
S13 S11 S15 S29
4000 3500 3000 2500 2000 1500 1000 500 0 130 120 110 100 90 80 70 60 50
(veh/km/lane)
Critical Density
Free Speed
(veh/lane)
S12S28
J15 J14
62-J18
RPROP calibration results - sections 2 & 4
2.5 2 1.5
2.5 2 1.5
1
1
0.5
0.5
0
10
20
30
40
50
60
0
10
20
Distance (km) 14
21
28
30
40
Distance (km) 14
21
28
50
60
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
RPROP calibration contours – sections 2 & 4
Data
Model
S32 S18
60
Data
120
100
50
Model
120
60
S30 S24 S1 S23
50
100
S19 S22
20
30
40
20
20
10
S21
J25 J24
S29 S15 S11 S13
10
07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
0
60
40
20
S28 S12 0
80
S9
S26 S20 07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
Velocity (km/h)
60
S17
J9
40 Distance (km)
Distance (km)
S8
30
Velocity (km/h)
80
40
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
40 35 30 25 20
Alpha
J18
S24 S30 J17
S22 S19 S23 S1 J14 J15 J16
J7
J6
J9
J8 J9 J10 J11 J12 J13
S9 J5
J2 J3/4
J24
J25 J26 J27 J1
S21
4000 3500 3000 2500 2000 1500 1000 500 0 130 120 110 100 90 80 70 60 50 45 40 35 30 25 20
4
4
3.5
3.5
3
J23
J18 J19 J20 Capacity
Critical Density
(km/h)
Free Speed
(veh/lane)
J18
J19
J20
J21
J22
J23
J25
J24
J27
45
J24 J25 J21 J22
S20 S26
S18 S32 J1
J5
J6
J3/4 J2
J7
J11
J9
J10
J8
J12
J13
J16
J17
J18
S8
3 Alpha
Capacity
S17
(veh/km/lane)
(km/h)
S13 S11 S15 S29
4000 3500 3000 2500 2000 1500 1000 500 0 130 120 110 100 90 80 70 60 50
(veh/km/lane)
Critical Density
Free Speed
(veh/lane)
S12S28
J15 J14
62-J18
RPROP, LPSO and HEPSO comparison; data set of the 14th for sections 2 & 4
2.5 2 1.5
2.5 2 1.5
1
1
0.5
0.5
0
10
20
30
40
50
60
0
10
20
Distance (km) RPROP
LSPO
HEPSO
30
40
Distance (km) RPROP
LPSO
HEPSO
50
60
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
RPROP verification for sections 2 & 4; optimal parameter set of the 14th applied on the 21st data
Data
Model
S32 S18
60
Data
120
100
50
Model
120
60
S30 S24 S1 S23
50
100
S19 S22
20
30
40
20
20
10
S21
J25 J24
S29 S15 S11 S13
10
07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
0
60
40
20
S28 S12 0
80
S9
S26 S20 07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
Velocity (km/h)
60
S17
J9
40 Distance (km)
Distance (km)
S8
30
Velocity (km/h)
80
40
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
LPSO verification for sections 2 & 4; optimal parameter set of the 14th applied on the 21st data
Data
Model
S32 S18
60
Data
120
100
50
Model
120
60
S30 S24 S1 S23
50
100
S19 S22
20
30
40
20
20
10
S21
J25 J24
S29 S15 S11 S13
10
07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
0
60
40
20
S28 S12 0
80
S9
S26 S20 07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
Velocity (km/h)
60
S17
J9
40 Distance (km)
Distance (km)
S8
30
Velocity (km/h)
80
40
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
RPROP and LPSO verification (14th → 21st) RPROP LPSO S32 S18
60
S17
20
80
S8
30
S11 S13
40
20
20
10
40
S29 S15 S11 S13
S28 S12 0
07:00
08:00
09:00
07:00
08:00
Time
Time
Data
Model
0
120
60
S1 S23
07:00
08:00
09:00
07:00
08:00
Time
Time
Data
Model
100
120
S30 S24 S1 S23
50
S19 S22
S21
J25
20
J24
10
0
S26 S20 07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
60
80
S9 30
40
20
20
10
0
J9
40 Distance (km)
Distance (km)
80
S9 30
100
S19 S22
Velocity (km/h)
J9
40
0
09:00
60
S30 S24
50
20
S28 S12 0
09:00
60
S17
S29 S15 10
120
100
40 Distance (km)
30
Velocity (km/h)
S8
S32 S18
50
80
40
Model
60
100
50
Distance (km)
Data
120
Velocity (km/h)
Model
0
S21
J25 J24
60
40
20
S26 S20 07:00
08:00 Time
09:00
07:00
08:00 Time
09:00
0
Velocity (km/h)
Data 60
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Macroscopic Traffic Flow Model Validation A larger network
Conclusions for model calibration algorithms Macroscopic traffic flow model calibration of large scale motorway networks. Optimisation problem formulation very important for obtaining relevant solutions. Solutions generalise well. Automatic assignment of fundamental diagrams over the whole network. Avoid over-parametrization. Gradient based optimisation using RPROP possible thanks to automatic differentiation and RPROP convergence properties. LPSO version of PSO works better than more recent variations of PSO algorithms. Macroscopic simulator can be changed and the methodology still works; preliminary work with the cell transmission model.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Hierarchical model predictive control
Road Network Traffic Model Predictive Control Integrated ramp metering, motorway-to-motorway control, route guidance and variable speed limits control is formulated as an optimal control problem. The discrete time optimal control problem has the form min J = u
K −1 X
ϕ [x(k ), u(k ), d(k )] + ϑ [x(K )]
k =0
subject to x(k + 1) = f [x(k ), u(k ), d(k )] , x(0) = x0 umin ≤ u(k ) ≤ umax where x is the state vector u is the control vector, and d(k ) is the disturbance vector.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Hierarchical model predictive control
...for traffic network models... Second order model ρm,i (k + 1)
=
qm,i (k ) vm,i (k + 1)
= =
ρm,i (k ) + L Tλ [qm,i−1 (k ) − qm,i (k )] m m ρm,i (k )vm,i (k )λm vm,i (k )
T {V [ρm,i (k )]−vm,i (k )} + τ T + L vm,i (k )[vm,i−1 (k ) − vm,i (k )] m ρ m,i+1 (k )−ρm,i (k ) − τνT Lm ρm,i (k )+κ −δTqµ (k )vm,1 (k )/(Lm λm (ρm,1 (k ) + κ)) 2 −φT ∆λρm,Nm (k )vm,Nm (k ) /(Lm λm ρcr ,m ) x= ρm,i (k ) αm V [ρm,i (k )] = vf ,m exp − α1 ρ m cr ,m
Second order model + queue model state h ρ1,1 v1,1 . . . ρ1,N v1,N 1
1
. . . ρM,N
M
vM,N
M
w1 . . . wO
iT
µ Disturbance d = d1 . . . dO βn . . . T Control vector (lets assume all on-ramps are controlled) u = [r1 , . . . rO ]T
Queueing model ˜o (k ) qo (k ) = ro (k )q ˜o (k ) = min q ˜o,1 (k ), q ˜o,2 (k ) q ˜o,1 (k ) = d(k ) + w(k ) q T ρmax −ρµ,1 (k ) ˜o,2 (k ) = Qo min 1, q ρ −ρ max
cr ,µ
Open loop optimal control solution: u∗ (k ) and x∗ (k ), k = 0, . . . , K . control sample time Tc = zc T , zc ∈ N∗ .
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Hierarchical model predictive control
Hierarchical control system
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Optimisation layer for coordinated ramp metering
Optimality conditions Let us assume there are p different classes of control measures with sample time Tc,` = z` T , ` = 1, . . . , p. The control time step at model time step k is $ k` =
k
KKT conditions on the Lagrangian for a known u(k ) yield g` (k` ) = Pb ∂ϕ(k )
%
k =a
∂u` (k` )
∂f(k )T
+ ∂u (k ) λ(k + 1) ` `
z` where
h iT u(k ) = u(k1 )T . . . u(kp )T Objective function PK −1 P P ρm,i (k )vm,i (k )λm + wo (k ) n oo Pk =0 2 + o af [ro (k ) − ro (k − 1)] + aw ψ [wo (k )]2
J =
T
where ψ [wo (k )] = max {o, wmax − wo (k )}
a = k` z` b = min {(k` + 1)z` − 1, K − 1} k` = 0, . . . , K` K −1 if K mod z` = 0 jz` k K` = K otherwise z `
and the Lagrange multipliers are obtained by a backwards integration
The problem’s Lagrangian is λ(k ) = KX −1
L = J−
k =0
∂f(k )T ∂x(k )
λ(k + 1) +
λ(k +1) {x(k + 1) − f [x(k ), u(k ), d(k )]} with λ(K ) =
∂ϑ(x(K )) ∂x(K )
∂ϕ(k ) ∂x(k )
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Optimisation layer for coordinated ramp metering
Line search optimisation Step 1 Set iteration index ι = 0. Select initial control trajectory u(0) . Step 2 Perform a forwards integration of x(ι) (k + 1) = f(k ) for known u(ι) to obtain the state x(ι) (k ). Step 3 With known x(ι) u(ι) perform a backwards integration to calculate the Lagrange multipliers λ(ι) (k ). Step 4 With known x(ι) u(ι) and λ(ι) (k ) calculate the gradient g(ι) . Step 5 Solve the line search optimisation subproblem: h i (ι) (ι) α = arg min J sat u + αs where s(ι) is a search direction method calculated from current and previous information collected over the optimisation iterations.
Search direction methods include (Fletcher, Practical Methos of Optimisation) Steepest descent Conjugate gradient: Fletcher-Reeves, Polak-Ribiere. Quasi-Newton methods: BFGS, DFP RPROP Typically, for solving the line search subproblem about 6 to 8 gradient evaluations for the bracketing and sectioning phases. RPROP performs only one gradient evaluation at each iteration. Non-smooth conditions in the model, but continuous.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Optimisation layer for coordinated ramp metering
The A’dam network and search direction methods (4h horizon)
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
The open loop optimal state
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
Rolling into time
30 minutes ahead
60 minutes ahead
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
Impact of prediction and application horizons
30 minutes ahead, application 30 minutes
60 minutes ahead, application 30 minutes
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
Direct control layer
How to do the job? We have u∗ , qo∗ (k ) and x∗ (k ) for the application horizon. Employ local controllers at each on-ramp. Different choices: Realise directly the optimal flows qo∗ (k ). ALINEA with dynamic set points
ρ b(ka ) =
1
(za ka +1)−1 X
za ka
∗
ρµ,1 (k )
k =za ka
and ALINEA applied as ˜o (ka + 1) = min Ro (ka ) + KA ρ q b(ka ) − ρµ,1 (k ) , qo,min Queue control as well.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
Investigations for disturbance prediction errors
scen. nr.
error
1 2 3 4 5
+5% +10% -5 % -10% 0%
AMOC+ALINEA (veh.∗h) 7,841.88 8,486.30 8,443.75 8,722.20 7,971.67
AMOC (veh∗h) 8,834.42 8,903.91 10,956.35 11,098.68 8,244.27
Overestimating demand is good. Similar direction of overestimation for all ramps. What happens if we overestimate the demand? Urban on-ramp storage capacity 100 veh. and motorway 200 veh. METANET has exact turning rates, AMOC their constant average. Horizons: (HP , HA ) = (60, 10) minutes. Implement AMOC (optimal flows) or ALINE+AMOC.
The direct control layer is computationally not intensive (cannot afford it anyhow). Emphasis on heuristic rules for setting up the parameters of the local controllers.
Computational Methods for Macroscopic Model Validation and Model Predictive Control Algorithms for Road Network Traffic Model Predictive Control Rolling horizon
Conclusions for hierarchical control Hierarchical model predictive control requires a very efficient optimisation solver. RPROP is suitable for fast convergence and quality of solutions. Direct control layer improves the solution’s implementation. It is not computationally demanding but needs fast communications and clever design. Overestimation of disturbances is allowable for the optimal control problem, but needs the existence of a suitable direct control layer. Large scale networks like that of Manchester feasible to control.
