A GEAR-BASED VEHICLE EMISSION MODEL FOR CO2 ... faculteit...Transfer Function Model Engine speed Gear =roll speed/ engine speed Gear-memberships 17 Results 0 2000 4000 6000 8000 10000

A GEAR-BASED VEHICLE EMISSION MODEL FOR CO2, CO AND NOX ESTIMATION

Hajar Hajmohammadi

Centre for Transport Studies, UCL

[email protected]

Benjamin Heydecker Professor of Transport studies, UCL

MATTS 2018

Mathematics Applied in Transport and Traffic Systems

17-19 October 2018

Introduction

2

Why we need vehicle emission model?

To evaluate the impacts of traffic policies on the air quality

To develop traffic control policies that can reduce air pollution in critical areas

Influential variables in the vehicle emission modelling:

Vehicle-related parameters: e.g. model, engine size, fuel and catalyst type and technology level

Operational factors: driving cycle

Vehicle type

Measurements

emission: CO2, CO, THC, NOx, PMn

vehicle: Roll speed, Engine speed

3

Dataset: Laboratory vehicle emission tests

Fuel Petrol - Diesel

Transmission Automatic – Manual

Engine Size from 1000 to 3200 cc

4

Real Driving Cycle for London

Traffic Conditions:

Free Flow

AM peak

Inter peak

[SERIES NAME] Suburban [SERIES NAME]

0

20

40

60

80

100

120

0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250

Sp

ee

d (

km

/h)

Time (s)

5

0

0,02

0,04

0,06

0,08

0,1

0,12

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120

NO

x (

mg

/se

c)

CO

2 (

mg

/se

c)

speed (km/h)

CO2 NOx

Statistical modelling

Vehicle B, Diesel, manual, 1400 cc

6

Background

Single model

𝑦 = 𝑐 + 𝛽𝑗𝑥𝑗

𝐽

𝑗=1

+ 𝜀 𝑦= emission

𝑥= Roll Speed, 𝑣

Acceleration, 𝑎 Power demand

𝑉𝑆𝑃 = 𝐴 𝑀 𝑣 + 𝐵𝑀 𝑣2 + 𝐶 𝑀 𝑣

3 + (𝑎 + 𝑔 𝑠𝑖𝑛 𝜃 )𝑣

Classified model: Acceleration, Cruising, Idling, Deceleration (CADI)

𝑦 =

𝑒𝑥𝑝( 𝐿𝑖,𝑗 × 𝑣𝑖 × 𝑎𝑗 ) 𝑓𝑜𝑟 𝑎 ≥ 0

3

𝑗=0

3

𝑖=0

𝑒𝑥𝑝( 𝛾𝑖,𝑗 × 𝑣𝑖 × 𝑎𝑗

3

𝑗=0

3

𝑖=0

𝑓𝑜𝑟 𝑎 < 0

VT-micro emission model

7

0

0,02

0,04

0,06

0,08

0,1

0,12

0

1

2

3

4

5

6

7

8

9

10

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

NO

x (

mg

/se

c)

CO

2 (

mg

/se

c)

gear

CO2 NOx

New latent variable, Gear=roll speed/engine speed

8

Urban Motorway Sub-Urban

0

0,5

1

1,5

2

2,5

3

0 500 1000 1500 2000 2500 3000

g

Time (sec)

New latent variable, Gear=roll speed/engine speed

9

F(𝑔 |π;θ)= 𝜋𝑘 F𝐾𝑘=1 𝑔 θk

𝑤𝑡𝑘 = 𝑓 𝑔𝑡 θk πk

𝑓𝐾𝑗=1 𝑔𝑡 θj πj Bayes’ theorem

𝑙𝑜𝑔 𝑙 π, θ = (𝑙𝑜𝑔 𝜋𝑘𝑓 (𝑔𝑛|θk))𝐾𝑘=1

𝑁𝑛=1 Log-likelihood

Mixed distribution model

Maximization Expectation-Maximization (EM) algorithm

𝒘(𝒕, 𝒌)

𝛑, 𝛍 and 𝛔 for each component 𝑘

Gear-memberships

10

Time (sec)

0 2000 4000 6000 8000 10000 12000

0.0

0.4

0.2

0.8

0.6

1.0 G

ea

r m

em

bers

hip

s

Component 1

Component 2

Component 3

Component 4

Component 5

Component 6

Component 7

Gear memberships

Vehicle B

Diesel – Supermini – Manual

12

𝒚 = 𝐱. β

𝐱: Explanatory variables 𝑣, 𝑎, 𝑎𝑣, 𝑣3

𝒘: Gear membership (latent variable)

𝐱.𝐰: Interaction

𝛆: Residuals

Gear-based emission modelling

𝒚 = 𝐱. β + 𝐰.φ 𝒚 = 𝐱. β + 𝐰.φ + 𝐱.𝐰 . γ + 𝛆

13

Challenge: engine speed as a latent variable

gea𝑟, 𝑔 = 𝑣/𝑒

roll speed, 𝑣

gea𝑟, 𝑔 = 𝑣/𝑒

Latent variable

(estimated) 𝑒𝑛𝑔𝑖𝑛𝑒 𝑠𝑝𝑒𝑒𝑑, 𝑒

𝑒𝑛𝑔𝑖𝑛𝑒 𝑠𝑝𝑒𝑒𝑑, 𝑒 Observation

14

𝑒𝑡 = 𝑡(𝐵)𝑣𝑡 +Nt

𝒆𝒕: engine speed at time 𝑡

𝒗𝒕: roll speed at time 𝑡

𝑵𝒕: noise- can be serially correlated

∅(𝐵)

𝜃 (𝐵) 𝑁𝑡= 𝜀𝑡

𝒕(𝑩): rational polynomial in B: 𝑤(𝐵)𝐵𝑏

𝛿(𝐵)

Systematic part

Stochastic part

Transfer Function Model

15

𝑒𝑡 = c+ 𝑤0+𝑤1𝐵+𝑤2𝐵

2

1−𝛿1𝐵−𝛿2𝐵2 𝑣𝑡 +

1− ∅1𝐵−∅2 𝐵2

1−𝜃1𝐵−𝜃2𝐵2 (1 − 𝐵) 𝜀𝑡

observed

fitte

d

R2 = 0.896

ACF and PACF of 𝜀

Transfer Function Model

16

Driving Cycle

Gear-based emission modelling

𝒚 = 𝐱. β + 𝐰.φ + 𝐱.𝐰 . γ + 𝛆

Mixed distribution model

Explanatory variables 𝑣, 𝑎, 𝑎𝑣, 𝑣3

Transfer Function Model

Engine speed

Gear =roll speed/ engine speed Gear-memberships

17

Results

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

A,Petrol,Manual B, Diesel,Manual C,Diesel,Automatic

D,Diesel,Automatic

E,Petrol,Automatic

BIC

Vehicle type

BIC

Gear-based Single VT-micro

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

A,Petrol,Manual B, Diesel,Manual C,Diesel, Automatic D,Diesel, Automatic E,Petrol,Automatic

Vehicle type

𝑅-squared

𝑅2

18

Conclusion

• Gear for driving mode instead of acceleration

• Introducing gear-membership

• Transfer function model

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Thank You !