A GEAR-BASED VEHICLE EMISSION MODEL FOR CO2, CO AND NOX ESTIMATION
Hajar Hajmohammadi
Centre for Transport Studies, UCL
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
19
Thank You !