Optimal dynamic route guidance: A model predictive approach using the macroscopic fundamental diagram

Macroscopic Modeling of Urban Networks Multi-region MFD-based Model Dynamic Route Guidance Case Study Concluding Remarks and Future Research

Optimal Dynamic Route Guidance: A Model Predictive Approach

Using Macroscopic Fundamental Diagram

Mohammad Hajiahmadi

_{, Victor L. Knoop}

_{, Bart De Schutter}

_{, Hans Hellendoorn}

1_{Delft Center for Systems and Control, Delft University of Technology, The Netherlands} 2_{Transport and Planning Department, Delft University of Technology, The Netherlands}

2

Multi-region MFD-based Model

3

Dynamic Route Guidance

High-level Optimal Routing Scheme

Objective Function

Model Predictive Control Framework

4

Case Study

Set-up

Results

Macroscopic Modeling of Urban Networks

Multi-region MFD-based Model Dynamic Route Guidance Case Study Concluding Remarks and Future Research

Modeling large-scale urban networks would be a complex task if

one wants to study and model dynamics of every single element

Control using such detailed modeling approach would be a

tedious task

Urban network partitioned into multiple regions, each represented by

an MFD

Macroscopic Modeling of Urban Networks

Multi-region MFD-based Model

Dynamic Route Guidance Case Study Concluding Remarks and Future Research

J

: set of neighboring regions of region i

Flow from region i to region j

_{∈ J}

is min. of 3 elements:

1

_{Demand from region i to region j , D}

2

_{Supply in region j , S}

Accumulation (veh/km/lane)

Production (veh/h/lane)

Macroscopic Fundamental Diagram

0 50 100 150 0 500 1000 1500 2000 2500 (a) Accumulation (veh/km/lane) Demand (veh/h/lane) Demand 0 50 100 150 0 500 1000 1500 2000 2500 (b) Accumulation (veh/km/lane) Supply (veh/h/lane) Supply 0 50 100 150 0 500 1000 1500 2000 2500 (c)

Fig. 2. The factors determining the flow, (a) Macroscopic Fundamental Diagram, (b) demand, and (c) supply.

all the demands from regioni to every neighbouring region j, the same fraction (8) will be applied. Hence, the outflow from regioni to region j∈ Jiis formulated as:

qi,j(k) = φi(k)· ˜Di,j(k) (9)

The flow can be separated per destination. So, similar to reducing the overall flow (9), we can modulate the flow per destination (5) as:

qi,j,d(k) = φi(k)· ˜Di,j,d(k) (10)

Therefore, the accumulation in any regioni towards destina-tiond can now be updated as follows:

ni,d(k+1) = ni,d(k)+ Ts P λ∈Λi κλLλ X j∈Ji qj,i,d(k)− X j∈Ji qi,j,d(k)
, (11)

withTsthe sample time. Hence the total accumulation in

regioni will be: ni(k + 1) =

X

d∈D

ni,d(k + 1) (12)

In the next section, we use the presented model for predic-tion of accumulapredic-tions in the network in order to determine optimal routes.

III. HIGH-LEVEL OPTIMAL ROUTE GUIDANCE

In this section, we develop a route guidance scheme based on the high-level MFD-based model derived in the previous section. In the proposed framework, we solve the dynamic routing problem on a macroscopic level. This means that instead of taking into account individual roads and intersections, we deal with regional destinations and the way that traffic flow should be splitted towards the neighboring regions in order to avoid congestion in the intermediate regions, to decrease the overall travel time and consequently, to improve the arrival rates at the destinations. We assume a two-level structure as depicted in Fig. 3. At the top level, the optimal route guidance problem is solved based on the aggregate model presented in the previous section. At the lower level, the optimal variables (the splitting rates) that are obtained from the high-level optimization problem are taken as references, i.e. local controllers in the lower level aim at realizing the optimal splitting rates for (destination

dependent) flows of vehicles that want to travel across the regions. In the following, we elaborate on the type of optimization problem that has to be solved in the highest level in order to achieve the aforementioned goals.

A. Objective function

In order to formulate the routing problem, an objective needs to be defined. The major aim in an urban network could be maximizing the arrival rate, i.e. the number of vehicles that complete their trips and reach their destinations, or similarly minimizing the total travel delays. Over the (discrete) simulation interval[0,· · · , K − 1], the total delay criterionJTD(veh·s) is formulated as:

JTD= Ts· X i∈R K_X−1 k=0  X λ∈Λi κλLλ
· ni(k)  . (13)

Moreover, one can introduce a penalty term on the differ-ences between average speeds of all regions as follows:

Jv(K− 1) = X i,j∈R ¯ Vi(K− 1) − ¯Vj(K− 1)
2 , (14)

with ¯Vi(K−1) the average speed in region i determined from

the MFD of that region at the end of simulation period (note that one can calculate the differences between speeds of all regions for all time steps, but it may increase the computation time of the corresponding optimization problem. Therefore, we try to normalize the speeds only at the end of the time horizon). Basically, with the values of the accumulations in each region, one can estimate an average speed for that region. Assuming an exponential function for the MFD, the average speed can be determined as follows:

¯ Vi(k) = Vfree,i· exp  −1₂ n_ni(k) crit,i 2 , (15)

withVfree,ithe free-flow speed andncrit,ithe critical

accu-mulation corresponding to the maximum production. Essen-tially when there is no congestion, the average speeds in the regions are high. But in case of congestion in a region, the average speed will decrease and consequently, the travel time for vehicles inside that region will increase. By minimizing (13), the overall travel delay in the network will decrease, but it might be possible that the traffic is not distributed evenly and in some regions the average speed will be high while in others we observe low speeds. The objective function (14) Optimal Dynamic Route Guidance: A Model Predictive Approach Using MFD 6/ 18

Macroscopic Modeling of Urban Networks

Multi-region MFD-based Model

Dynamic Route Guidance Case Study Concluding Remarks and Future Research

Supply function

S

(k) =

(

P

if n

(k)

≤ n

P

(n

(k))

if n

(k)

> n

P

(n

(k)): production determined from MFD

Demand function

D

(k) =

X



α

(k)

·

n

(k)

n

(k)

· P

n

(k)



Update equations

n

(k + 1) = n

(k) +

T

P

κ

L

X

q

(k)

−

X

q

(k)

Total accumulation in region i :

n

(k + 1) =

X

d∈D

n

(k + 1)

Macroscopic Modeling of Urban Networks Multi-region MFD-based Model

Dynamic Route Guidance

Case Study Concluding Remarks and Future Research

High-level Optimal Routing Scheme Objective Function

Model Predictive Control Framework

Regional destinations

Optimal splitting traffic towards neighboring regions

Aims:

avoid congestion in intermediate regions

decrease the overall travel time

Local controller Local controller Local controller LC LC LC LC Local controller Local controller Local controller High−level MPC controller Multi−region MFD−based Model Optimization Objective, constraints ni,d(k) αi,ji ,d(k) 1

Macroscopic Modeling of Urban Networks Multi-region MFD-based Model

Dynamic Route Guidance

Case Study Concluding Remarks and Future Research

High-level Optimal Routing Scheme

Objective Function

Model Predictive Control Framework

Minimizing total travel delay:

J

= T

·

X

_X

 X

κ

L

· n

(k)



Local controllers/

Optimization

Prediction

J

α

(k

)

OD Table

n

(k)

α

(k

)

Urban regions

(multi-region) model

Macroscopic Modeling of Urban Networks Multi-region MFD-based Model

Dynamic Route Guidance

Case Study Concluding Remarks and Future Research

High-level Optimal Routing Scheme Objective Function

Model Predictive Control Framework

J

= T

·

X

_X

 X

κ

L

· n

(k)



– Overall optimization problem:

min

J

subject to:

model equations,

0

≤ α

(k)

≤ 1,

α

(k) =

α

(k

),

if k

∈ {M · k

, . . . , M

· (k

+ 1)

− 1}

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Blue squares: origins

Red circles: destinations

Macroscopic Modeling of Urban Networks Multi-region MFD-based Model Dynamic Route Guidance

Case Study

Concluding Remarks and Future Research

Set-up

Results

For each region, the MFD is approximated by:

P

= n

· V

· exp



−

1

2

 n

n





Table :

Origin-destination demands

(veh/h)

Region 2

Region 8

Region 9

Region 14

Region 1

1000

1800

1750

3000

Region 4

1900

1400

1000

1400

Region 11

1700

1200

1300

1300

Region 16

2000

1000

1000

1800

Determining splitting rates

Static shortest-path (in time), Floyd-Warshall algorithm

based on average speed of regions

Shortest-path algorithm, updated every 60 seconds

Dynamic, MPC algorithm using multi-region MFD model

16 15 14 16 13 15 12 14 13 12 11 11 10 9 7 8 10 9 8 6 7 5 5 4 6 3 4 2 3 2 1 1 200 s 16 15 14 16 13 15 12 14 13 12 11 11 10 9 9 8 8 10 7 6 7 5 4 6 5 3 4 2 3 1 2 1 200 s 16 15 14 16 13 13 15 12 14 12 11 11 10 9 7 8 8 10 9 6 7 5 4 6 5 3 4 2 3 2 1 1 200 s 16 15 14 16 13 13 15 12 14 12 11 11 10 9 7 8 8 10 9 6 7 5 5 4 4 6 3 2 3 2 1 1 600 s 16 15 14 16 13 15 12 14 13 12 11 11 10 9 7 8 10 9 8 6 7 5 4 4 6 5 3 2 3 1 2 1 600 s 16 15 14 16 13 15 12 14 13 11 12 11 10 9 8 10 9 7 8 6 7 5 4 6 5 3 4 2 2 3 1 1 600s 16 16 15 15 14 13 12 14 13 7 9 8 8 12 11 11 10 10 9 6 7 5 4 6 5 3 4 2 3 2 1 1 1000 s 16 16 15 15 14 13 12 14 13 12 11 11 10 9 7 8 10 9 8 6 7 5 4 6 5 3 4 2 3 1 2 1 1000 s 16 15 15 14 16 13 12 14 13 11 12 10 11 9 7 8 10 9 8 6 7 5 4 6 5 3 4 2 3 1 2 1 1000 s 16 15 14 14 16 13 15 12 13 9 8 12 11 7 11 10 10 9 8 6 7 5 4 6 5 3 4 2 2 3 1 1 1400 s 16 16 15 12 15 14 13 13 14 12 11 11 10 9 8 8 10 9 7 6 7 5 4 4 6 5 3 2 3 1 1 2 1400 s 16 15 14 14 16 13 15 12 13 12 11 11 10 9 7 8 8 10 9 6 7 5 4 6 5 3 4 2 3 1 2 1 1400 s 16 15 15 14 16 13 14 12 13 9 8 12 11 11 10 10 9 8 7 7 6 5 4 6 5 3 4 2 3 2 1 1 1800 s 16 15 15 12 16 14 13 14 13 12 11 7 9 9 8 8 11 10 10 6 7 5 4 6 5 3 4 2 3 1 2 1 1800 s 16 15 14 16 13 15 12 12 14 13 11 11 10 9 8 10 9 7 8 6 7 5 4 6 5 3 4 2 3 1 1 2 1800 s 16 16 15 14 13 15 14 12 13 8 8 9 12 11 11 10 10 9 7 7 6 5 4 6 5 3 4 2 3 2 1 1 2400 s 16 16 15 12 15 14 13 14 13 12 11 8 9 11 10 10 9 8 7 6 7 5 5 4 6 3 4 2 3 1 2 1 2400 s 16 15 14 16 13 15 12 14 13 11 12 11 10 9 7 8 8 10 9 6 7 5 4 4 6 5 3 2 3 1 2 1 2400 s 16 16 15 14 13 15 14 12 13 8 8 9 12 11 11 10 10 9 7 3 7 6 5 5 4 6 4 2 3 2 1 1 2800 s (a) 16 15 12 1516 14 13 14 13 12 11 8 9 11 10 10 9 8 7 6 7 5 4 6 5 3 4 2 3 1 1 2 2800 s (b) 16 16 15 15 14 13 12 14 13 11 12 11 10 9 7 8 8 10 9 6 7 5 5 4 4 6 3 2 3 1 2 1 2800 s (c) Fig. 5. Results for4× 4 network, (a) Uncontrolled (fixed routes), (b) Shortest-path algorithm, (c) Optimal dynamic routing using MPC [13] B. N. Janson, “Dynamic traffic assignment for urban road networks,”

Transportation Research Part B, vol. 25B, no. 2/3, pp. 143–161, 1991.

[17] J. Maciejowski, Predictive Control with Constraints.Harlow, Eng-land: Prentice Hall, 2002.

Results for 4x4 network: (a) Uncontrolled (fixed routes), (b) Shortest-path algorithm, (c) Optimal dynamic routing using MPC

using MFD-based multi-region model

Optimal splitting rates towards neighboring regions

Avoiding detailed modeling and hence decreasing

computational complexity of route guidance

Lower level control should be properly designed &

connected to the high-level scheme

Multi-level scheme needs to be validated using real

networks’ layouts and empirical data