A mesoscopic approach to model path choice in emergency condition Massimo Di Gangi Dipartimento di Ingegneria Università degli Studi di Messina mdigangi@unime.it

Introduction Evacuation users move away from the evacuation zone users choice the perceived best path users may change their choice according to changes in network conditions

Introduction Presentation features some features of a dynamic traffic assignment model with mesoscopic approach are highlighted; path choice capabilities are considered: en- route (re-routing), implicit; cost functions can take into account of a risk factor; an experimentation on a trial network is presented.

Introduction References Di Gangi M., Velonà P. (2009). Multimodal Mesoscopic Approach in Modeling Pedestrian Evacuation. TRANSPORTATION RESEARCH RECORD, vol. 2090/2009, p. 51-58, ISSN: 0361-1981 doi: 10.3141/2090-06 Di Gangi M. (2011). Modeling evacuation of a transport system: application of a multi-modal mesoscopic dynamic traffic assignment model. IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, vol. 12, p. 1157-1166, ISSN: 1524-9050 doi: 10.1109/TITS.2011.2143408

II. EXTENSION TO EVACUATION TEST APPLICATION Structure I. MESOSCOPIC MODEL II. EXTENSION TO EVACUATION TEST APPLICATION

I. MESOSCOPIC MODEL Mesoscopic model: general demand supply loading model

Mesoscopic model: demand Modal facilities (m) classes (u) parameters ζu ξu εu φu γu car fast 1.0 0.2 1 2 5 slow 0.8 bus small 0.75 0.12 18 large 0.08 3.5 45

Mesoscopic model: demand Modal facilities (m) classes (u) parameters ζu ξu εu φu γu car fast 1.0 0.2 1 2 5 slow 0.8 bus small 0.75 0.12 18 large 0.08 3.5 45

Mesoscopic model: demand Modal facilities (m) classes (u) parameters ζu ξu εu φu γu car fast 1.0 0.2 1 2 5 slow 0.8 bus small 0.75 0.12 18 large 0.08 3.5 45 speed occupancy equivalence filling grouping

Mesoscopic model: demand Modal facilities (m) classes (u) parameters ζu ξu εu φu γu car fast 1.0 0.2 1 2 5 slow 0.8 bus small 0.75 0.12 18 large 0.08 3.5 45

r s Mesoscopic model: demand time Origin Packet [point] P  {h, rs, u} Modal facilities (m) classes (u) parameters ζu ξu εu φu γu h time Origin Packet [point] r P P  {h, rs, u} s Destination

G (N, A) Mesoscopic model: supply A N Nodes Arcs {1, 2, … i, … n} i, j i, j  N

r G (N, A) s3 s1 s4 s2 s5 Mesoscopic model: supply A N Nodes Arcs {1, 2, … i, … n} G (N, A) Arcs i, j i, j  N s3 r s1 s4 s2 s5

r G (N, A) s3 s1 s4 s2 s5 Mesoscopic model: supply A N Nodes Arcs {1, 2, … i, … n} G (N, A) Arcs i, j i, j  N s3 r s1 s4 s2 s5

r G (N, A) rsh ( Nrs N , Ars A) s1 s4 s2 s3 s5 Mesoscopic model: supply G (N, A) r s1 s2 s3 s4 s5 rsh ( Nrs N , Ars A)

Mesoscopic model: supply Arc representation La running S queuing xsa La - xsa

Mesoscopic model: loading model Packets movements Simulation of spillback Respect of capacity constraint Simulation of overtaking opportunities

r s5 Mesoscopic model: loading model Packets movements Arcs Nodes i, j i, j  N Nodes {1, 2, … i, … n} G (N, A) Arc weight wa r s5

Mesoscopic model: loading model Packets movements For each O/D pair rs and departure time h a DAG (Directed Acyclic Graph) sub-graph rsh ( Nrs N , Ars A) of the network is associated to the packet P º {h, rs, u}. Such a sub-graph consists of the set of arcs belonging to the feasible paths connecting the O/D pair rs computed at time h. A choice weight wa’ is associated to each arc a' rsh. r s5 Sub-graph rsh can be generated either by computing explicitly a set of paths (i.e. using a KSP algorithm) or using an implicit algorithm (i.e. Dial's STOCH). In the former case, it is necessary to compute path choice probabilities and wa’ is given by the sum of the probabilities of paths using arc a’ ; in the latter wa’ is directly obtained.

FROM G (N, A) TO rsh ( Nrs N , Ars A) Mesoscopic model: loading model Packets movements FROM G (N, A) TO rsh ( Nrs N , Ars A) r La 2 10 3 12 19 62 54 63 s5

Mesoscopic model: loading model Packets movements d t P  {h, rs, u} time Xsa -x (δ - t)∙vat x space x < XSa

Mesoscopic model: loading model Packets movements d t P  {h, rs, u} time Max density Xsa -x (La − XSa)· k 𝑎𝑚𝑎𝑥 Qa (δ - t)∙vat Capacity x space x < XSa

Mesoscopic model: loading model Packets movements d t P  {h, rs, u} time Max density (La − XSa)· k 𝑎𝑚𝑎𝑥 Qa Capacity x space x  XSa

Mesoscopic model: loading model Simulation of spillback P  {h, rs, u} running

Mesoscopic model: loading model Simulation of spillback P  {h, rs, u} queuing

Mesoscopic model: loading model Respect of capacity constraint Exiting users Ua (t)− ne (P) t < 𝑄 𝑎 If: P  {h, rs, u} Capacity

Mesoscopic model: loading model Respect of capacity constraint Exiting users Ua (t)− ne (P) t ≥ 𝑄 𝑎 If: P  {h, rs, u} Capacity

Mesoscopic model: loading model Simulation of overtaking opportunities Faster class vehicle can overtake slower one P  {h, rs, u} R  {h, rs, u1} running S queuing xsa La - xsa

Mesoscopic model: loading model Simulation of overtaking opportunities Faster class vehicle remains behind slower one P  {h, rs, u} R  {h, rs, u1} running S queuing xsa La - xsa

II. EXTENSION TO EVACUATION TEST APPLICATION Structure I. MESOSCOPIC MODEL II. EXTENSION TO EVACUATION TEST APPLICATION

II. EXTENSION TO EVACUATION Mesoscopic model: loading model II. EXTENSION TO EVACUATION Arc risk function Choice of the next arc Re-routing

Extension to evacuation Arc risk function P  {h, rs, u} running S queuing xsa La - xsa risk level ra()  [0,1] Depending on the dangerous event safety probability sa() = 1 - ra()

Extension to evacuation Arc risk function P  {h, rs, u} running S queuing xsa La - xsa TT a (t)= XSa (t) v a (t) + (La − XSa∙(t))∙ k amax Qa(t) Travel time TW a (t)= TT a (t)∙ 1+α∙ 𝑙𝑛 1 sa(t) 𝛽 Weighted travel time

Extension to evacuation Arc risk function TW a (t)= TT a (t)∙ 1+α∙ 𝑙𝑛 1 sa(t) 𝛽 250 200 150 100 50 sa() = 1 - ra()

Extension to evacuation Choice of the next arc r a s5 Partial path

FROM G (N, A) TO rsh ( Nrs N , Ars A) Mesoscopic model: supply Choice of the next arc FROM G (N, A) TO rsh ( Nrs N , Ars A) a3 a2 a1 w p + = 31 wa1 a1 wa3 a3 wa2 22 a2 40 32

r s5 Extension to evacuation a1 a a+2 a2 a+3 a3 Partial path Choice of the next arc r   a1 a a+2 a2 a+3 a3 s5 Partial path

FROM G (N, A) TO rsh ( Nrs N , Ars A) Mesoscopic model: supply Choice of the next arc FROM G (N, A) TO rsh ( Nrs N , Ars A) 31 wa1 a1 wa3 a3 * wa2 22 a2 40 32

FROM G (N, A) TO rsh ( Nrs N , Ars A) Mesoscopic model: supply Choice of the next arc FROM G (N, A) TO rsh ( Nrs N , Ars A) 31 a1 a3 22 a2 40 32

r s5 Extension to evacuation a a+3 Partial path Choice of the next arc rsh ( Nrs N , Ars A) r a a+3 s5 Partial path

Extension to evacuation Choice of the next arc r a s5 Partial path

r s5 Extension to evacuation Path at interval t Path at interval t1 Re-routing r s5 Path at interval t Path at interval t1

II. EXTENSION TO EVACUATION TEST APPLICATION Structure I. MESOSCOPIC MODEL II. EXTENSION TO EVACUATION TEST APPLICATION

Extension to evacuation Test network

Extension to evacuation Test network Eulerian equation:

A x, y, z Extension to evacuation Atmospheric diffusion time emission wind Puff based solution: space diffusivity

A x, y, z Extension to evacuation S Atmospheric diffusion La Puff based solution:

A x, y, z Extension to evacuation c(t) S Atmospheric diffusion La Puff based solution: c(t)

ò A x, y, z z t d c > = ) ( (c Pr r Extension to evacuation Atmospheric diffusion La A x, y, z S z t d c cr ò ¥ > = ) ( (c Pr r a sa() = 1 - ra()

Extension to evacuation Atmospheric diffusion r s

Extension to evacuation Atmospheric diffusion: re-routing r t0 s

Test application Atmospheric diffusion: re-routing r t1 s

Test application Atmospheric diffusion: re-routing r t2 s

Test application Atmospheric diffusion: re-routing r t3 s

Test application Atmospheric diffusion: re-routing r t4 s

Test application Atmospheric diffusion: re-routing r t5 s

Logit model Test application gk= Sak TWa KSP k = 3 Path cost Atmospheric diffusion: path probability KSP k = 3 Logit model gk= Sak TWa Path cost Link cost

Test application Atmospheric diffusion: demand Simulation has been conducted considering a time interval of 5 minutes (300 s). Demand has been generated for the first 27 intervals Origin Destination No. Trips per interval 1028 2036 2056 2009 2052 2063 2062 1001 2033 2020 1055 2034 2092

Test application A B C D B B C D C C C D Evolution in path choice

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 B, C, D 0.20 A 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 B, C, D 0.20 A 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 B, C, D 0.20 A 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 C, D 0.20 B 0.40 A 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 C, D 0.20 B 0.40 A 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 C, D 0.20 B 0.40 A 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 C, D 0.20 B 0.40 A 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 C, D 0.20 B 0.40 A 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 C, D 0.20 B 0.40 A 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 D 0.20 C 0.40 B 0.60 A 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 D 0.20 C 0.40 B 0.60 A 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 D 0.20 C 0.40 B 0.60 A 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 0.20 D 0.40 C 0.60 B 0.80 A

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 0.20 D 0.40 C 0.60 B 0.80 A

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 0.20 D 0.40 C 0.60 B 0.80 A

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 0.20 D 0.40 C 0.60 B 0.80 A

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 0.20 D 0.40 C 0.60 B 0.80 A

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 0.20 D 0.40 C 0.60 B 0.80 A

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Test application A B C D B B C D C C C D Evolution in path choice Risk probability Sector 0.00 A, B, C, D 0.20 0.40 0.60 0.80

Conclusions and developments A dta with mesoscopic approach to model the path choice was discussed in evacuation conditions considering: implicit path choice, en-route path choice, explicit management of re-routing capabilities, introduction of a risk factor in arc cost function. Mathematical formulation for the model was proposed defining: demand and supply, loading model, arc risk function, re-routing. An application to a test case was proposed.

A mesoscopic approach to model path choice in emergency condition Massimo Di Gangi Dipartimento di Ingegneria Università degli Studi di Messina mdigangi@unime.it Thank you