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SENSITIVITY ANALYSIS

An Optimization Model to Choose Bus Fleet with Consideration of Climate Change Impact

1 Upper Great Plains Transportation Institute, North Dakota State University

2 Transportation, Logistics and Finance Department, North Dakota State University

Along with the developments in fuel propulsion and battery technologies

over the last decades, there are much more choices for public transit operators

to choose buses for their fleets. However, there are always tradeoffs between

the costs, capacity, level of service (LOS), and environmental impact. To help

public transit operators to deal with these tradeoffs, our study proposed an

optimization model on minimizing cost with the consideration of environmental

impact under different scenarios. A case study is presented in this study,

followed by sensitivity analysis to identify the impacts of headway and carbon

emissions on internal costs and social cost. Our result shows that the

proposed model does significantly reduce the annualized total cost and carbon

emissions. However, the social cost of carbon emissions does not have much

influence on the choice of bus fleet from an economic standpoint.

ABSTRACT INTRODUCTION

S

Fangzheng Yuan¹, Yuan Xu¹, & Joseph Szmerekovsky²

Fuel Type Description

Gas Gasoline

Diesel Conventional Diesel

Hybrid Hybrid-electric diesel bus

CNG Compressed natural gas

LNG Liquefied natural gas

R-BEB Rapid-charge battery electric bus

S-BEB Slow-charge battery electric bus

B20 A blend of 20% biodiesel and 80% petroleum diesel

B100 Biodiesel (pure)

Bus Length Maximum Capacity

26 feet 22

40 feet 53

60 feet 83

MODEL DEVELOPMENT

Min Total Cost Z = Cc + Co+ CF+ CL+ Cs （1）

s.t.

σ𝑖=1𝑖 σ𝑗=1

𝑗𝑤𝑖𝑗 ∗ 𝑛𝑖𝑗 ≥ 𝑑o ∗ 𝑇c ∀𝑖, 𝑗

σ𝑖=1𝑖 σ𝑗=1

𝑗𝑤𝑖𝑗 ∗ 𝑛𝑖𝑗 ′ ≥𝑑p * 𝑇c ∀𝑖, 𝑗

σ𝑖=1𝑖 σ𝑗=1

𝑗𝑛𝑖𝑗 ≥ 𝑇𝑐/𝑣o ∀𝑖, 𝑗

σ𝑖=1𝑖 σ𝑗=1

𝑗𝑛𝑖𝑗 ′ ≥ 𝑐/𝑣p ∀𝑖, 𝑗

𝑛𝑖𝑗, 𝑛𝑖𝑗 ′∈𝑍

𝑛𝑖𝑗, 𝑛𝑖𝑗 ′ >=0

Where,

Annualized Capital Cost

Cc = σ𝑖=1𝑖 σ𝑗=1

𝑗(𝑛𝑖𝑗 + 𝑛𝑖𝑗′) ∗ (CPij + Ciij ) *

𝛾

1−(1+𝛾)−𝑡(2)

s. t.

If 𝑛𝑖𝑗 >= 𝑛𝑖𝑗′, 𝑛𝑖𝑗′ = 0, else 𝑛𝑖𝑗 = 0 ∀𝑖, 𝑗

Annualized Operation & Maintenance Cost

Co = σ𝑖=1𝑖 σ

𝑗=1𝑗

(𝑛𝑖𝑗∗(𝑛𝑤∗𝑇𝑜+𝑛𝑤′ ∗𝑇𝑜′)+𝑛𝑖𝑗′ ∗ (𝑛𝑤∗𝑇𝑝+𝑛𝑤′∗𝑇𝑝′))

𝑇𝑐* Coij*L (3)

Annual Fuel Cost

CF = σ𝑖=1𝑖 σ

𝑗=1𝑗

(𝑛𝑖𝑗∗(𝑛𝑤∗𝑇𝑜+𝑛𝑤′ ∗𝑇𝑜′)+𝑛𝑖𝑗′ ∗ (𝑛𝑤∗𝑇𝑝+𝑛𝑤′∗𝑇𝑝′))

𝑇𝑐∗CFij *L (4)

Annual Labor Cost

CL = σ𝑖=1𝑖 σ𝑗=1

𝑗(𝑛𝑖𝑗 ∗ (𝑛𝑤 ∗ 𝑇𝑜 + 𝑛𝑤′ ∗ 𝑇𝑜′) + 𝑛𝑖𝑗′ ∗ (𝑛𝑤 ∗ 𝑇𝑝 + 𝑛𝑤′ ∗ 𝑇𝑝′)) ∗CHS (5)

Annualized Social Cost of Carbon Emissions

Cs = σ𝑖=1𝑖 σ𝑗=1

𝑗(𝑛𝑖𝑗 ∗ (𝑛𝑤 ∗ 𝑇𝑜 + 𝑛𝑤′ ∗ 𝑇𝑜′) + 𝑛𝑖𝑗′ ∗ (𝑛𝑤 ∗ 𝑇𝑝 + 𝑛𝑤′ ∗ 𝑇𝑝′)) *

𝑃𝑖𝑗∗𝐶𝑠𝑐

2205∗𝑀𝑖𝑗*L (6)

CASE STUDY

Route Length (Miles) 11.7

Cycle Time (Minutes) 60

Daily Operation Time (Weekdays/Weekend) 12 hrs/ 8 hrs

Operation Days Per year (Days) 350

Estimated Demands (off-peak hours) 32

Estimated Demands (peak hours) 58

Headway (off-peak hours)(mins) 60

Headway (peak hours)(mins) 30

RESULTS

SENSITIVITY ANALYSIS

Options VehiclesAnnualized Carbon

Emission Cost

Annualized Total

Cost

Original

One 40 feet Diesel bus

(Off-peak hours) $6,190 $252,821

Two 40 feet Diesel bus

(Peak hours)

Optimal

One 40 feet R-BEB bus

(Off-peak hours) $2,718 $208,354

Plus one 26 feet Gas bus

(Peak hours)

Headway Cc Co CF Cs CL Total

60/60 $82,185 $38,610 $9,342 $2,718 $75,500 $208,354

60/45 $82,185 $38,610 $9,342 $2,718 $75,500 $208,354

60/30 $82,185 $38,610 $9,342 $2,718 $75,500 $208,354

60/15 $64,415 $73,593 $38,498 $6,859 $111,740 $295,106

45/45 $21,324 $87,516 $28,672 $5,266 $132,880 $275,658

45/30 $21,324 $87,516 $28,672 $5,266 $132,880 $275,658

45/15 $28,432 $99,450 $32,582 $5,984 $151,000 $317,448

30/30 $21,324 $87,516 $28,672 $5,266 $132,880 $275,658

30/15 $28,432 $99,450 $32,582 $5,984 $151,000 $317,448

CONCLUSION & REFERENCE

In this study, we developed a linear program that seeks to minimize total cost for

adding a new bus route. The main purpose is to identify the number and type of buses

needed for the route, as well as trade-offs involved among the different parameters. A

case study of a route planning in Fargo-Moorhead area is presented, showing that the

linear optimization model reduced the annualized total cost by 17.6% and carbon

emissions by 56.1%. Nevertheless, a lower annualized total cost does not always lead

to a lower carbon emissions due to the low ratio between carbon emission cost and the

total cost. Sensitivity analysis also indicates that the level of service is a much more

important parameter for the annualized total cost rather than the social cost of carbon.

1. Tong, F., Hendrickson, C., Biehler, A., Jaramillo, P., & Seki, S. (2017). Life cycle ownership cost and

environmental externality of alternative fuel options for transit buses. Transportation Research Part

D: Transport and Environment, 57, 287-302. doi:10.1016/j.trd.2017.09.023

2. Alternative Fuels and Advanced Vehicles. (n.d.). Retrieved from https://afdc.energy.gov/fuels

3. FAQ – What is the SCC? (n.d.). Retrieved from https://costofcarbon.org/faq/what-is-the-scc

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4,000

5,000

10 20 30 40 50 60 70 80

Annualiz

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Carb

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Social Cost of Carbon ($/Metric ton CO2)

Carbon Emission Cost Total Cost

Impact of Carbon Social Cost on Annualized Total Cost

Annualized Captial Cost ,

$82,185

Annalized O&M Cost,

$38,610 Annual Fuel Cost, $9,342

Annualized Carbon Emission Cost,

$2,718

Anuual Labor Cost, $75,500

THE SHARE OF OPTIMAL ANNUALIZED TOTAL COST

Annualized Captial Cost ,

$86,183.33

Annalized O&M Cost, $49,725.00

Annual Fuel Cost, $35,221.88

Annualized Carbon Emission Cost ,

$6,190.48

Anuual Labor Cost, $75,500.00

THE SHARE OF ORIGINAL ANNUALIZED TOTAL COST