Task 2 requires us to construct a model to predict the financial implications of transitioning to e-buses. We model a 10-year horizon to estimate the cost increase needed to acquire and maintain an e-bus fleet compared to the current diesel bus fleet.

In order to make these calculations, we separated the considerations into two categories: Initial and operational prices. Initial prices refer to variables such as the capital cost of one electric bus, or the price of purchasing one electric charger. Operational prices refer to costs such as the annual electricity and diesel for a bus, or the maintenance of the electric chargers.

The variables that will be used in Task 2 are listed in Table 8 .

Table 8

Notations	Descriptions	Notations	Descriptions
	Set of years 2024-2033		Fleet size in year t
	Purchase price of an e-bus		Operation price of an e-bus
	Purchase price of a charger		Purchase price of a diesel bus
	Operation price of a diesel bus		Ridership in year t
	Current amount of buses in NYC		Price of electricity annually needed
	Maintenance price of electric bus		Maintenance price of chargers
	Cost of materials		Cost of labor
	Price of diesel annually needed		Maintenance price of diesel bus
	Maximum ridership of bus in a day		Maximum traffic in a day
	Maximum service years of electric buses		Maximum service years of diesel buses

We construct a Total Cost of Ownership, TCO model to analyze the increased capital needed to maintain and transition to a fully electric bus system. Various factors are considered, including the price of purchasing and maintaining chargers, the slow decrease in the popularity of public transportation, and the prices of electricity and diesel. We calculated the difference between the TCO of a fully electric bus model and the TCO of a fully diesel bus model to analyze this increased capital. This difference can be summarized as

(16)

where represents the difference in TCOs of the electric and diesel bus models, and are the electric bus model’s TCO and the diesel bus model’s TCO in year t, respectively. Furthermore,

(17)

where is the fleet number in year t, is the possible max fleet charging at the same time in year t, is the purchase price of an e-bus, is the operation price of an e-bus, and is the purchase price of the charger. Similarly,

(18)

where is the purchase price of a diesel bus and is the operation price of a diesel bus.

In order to create this model, we analyzed the number of people who used the MTA bus system in years 2012 to 2019. We noticed an overall decrease in the popularity of using buses, which means in the future, it would not be necessary to purchase as many electric buses as the current fleet of buses. We were then able to predict the ridership in Table 9 :

We can express the fleet of buses needed as the amount of people decreases as

(19)

where is the ridership of year t and is the current amount of buses in NYC. As of 2021, there were 5780 buses in the MTA fleet, and due to the aftermath of Covid-19 and lack of public transportation usage we feel it is safe to assume there would be little change in the fleet size as the city finishes its recovery from the pandemic [11] . Applying the above data to this formula provides the predicted bus fleet in Table 10 :

We consider both the operational costs of diesel and e-buses in this model. In order to calculate the operational costs for a year of operation for the e-bus system, we use the following equation:

(20)

where refers to the price of electricity, refers to the price of maintaining an electric buses, and refers to the price of maintaining chargers.

Electricity averages around 0.08 cents per kilowatt, and an electric bus uses approximately 2.8 kilowatts per mile. Therefore, the cost of electricity for one electric bus on average can be represented as 2.8 × 0.08 × 36,500 ≈ 8000 dollars per year [12] .

(21)

where is the cost of materials, and is the cost of labor. The material per charger will cost 11,500 dollars and require 72 hours of labor, which with a fully-loaded electrician salary of 87.85 dollars per hour results that the maintenance per charger is approximately 17,825 dollars. However, each charger can supply electricity to multiple buses around the clock. We assume one charger is needed for every four electric buses. The average cost for charger maintenance is 4456 dollars per bus [12] . Therefore,

(22)

Diesel Buses

The operational costs of a diesel bus per year can be expressed as

(23)

where represents the price of diesel, and represents the price of maintaining a diesel bus.

Price of diesel: From our calculations that a bus travels an annual 36,500 miles, we can write that. given today’s rates, the total cost of diesel evaluates to about 28,000 dollars per year [12] . Therefore,

(24)

E-bus The MTA lists the price of a standard E-bus as 1,033,000 dollars. Therefore, the price of purchasing the needed E-buses in year t can simply be written as

(25)

Diesel Bus The MTA lists the price of a standard diesel bus as 635,000 dollars. Therefore, the price of purchasing the needed diesel buses in year t can simply be written as

(26)

Clearly, having the chargers necessary to charge every single one of the buses in our fleet would be far too costly and illogical. We use data from the George Washington Bridge to extrapolate the total bus needs of the city. On the George Washington Bridge, the maximum amount of people on the bridge is 18,500 at 5 p.m, as seen in the following Figure 6 :

As the bridge reportedly accounts for 8 percent of the traffic in the city, we can calculate the total traffic in NYC. Furthermore, a reported 12 percent of vehicles in the city are buses, so we can calculate the maximum amount of people in buses in one hour in NYC as

(27)

where represents the maximum traffic recorded and represents the maximum ridership of buses at the same time. From this, we evaluate that a maximum of 345 buses will be needed at one time since a standard NYC bus has a limit of 80 people [12] .

An MTA report stated that it costed them 50 million dollars for their purchase of charging facilities needed for 60 electric buses [13] . Therefore, the cost of purchasing a charger per bus is approximated at 50,000,000/60 ≈ 833,000 dollars. We then conclude that given the current maximum number of buses that need to be charged at once, i.e., 345 from above estimation, purchasing chargers will cost 287,385,000 dollars.

The general formula of polynomial fitting can be represented with this equation:

(28)

where x is the predictor variable, y is the response variable, is the coefficient, and k is the order of the polynomial.

Finding the best parameters of this polynomial function aims to minimize the residual error between the value estimated by the polynomial and the actual data . norm of this residual error is then optimized by n training samples:

(29)

To obtain the minimum of this objective function, the partial derivatives of R² with respect to the coefficients a_i is derived:

(30)

which is expressed in matrix form as:

(31)

This is a Vandermonde matrix and can be rewritten as:

(32)

which is Solving this equation yields

(33)

The regularization term is applied to avoid the over-fitting problem. Here we use the L₂ norm of the coefficient as the penalty:

(34)

where λ is inferred by sensitivity analysis to control the effects of regularization.

In Figure 7 , it is evident that when , decreasing λ further by factors of 10 has very little impact on the resulting regression line. Even when λ is increased by a factor of 10 several times, the mode is still relatively insensitive to this change. We found that λ is obtained to be 0.1.

The solution to the optimization is calculated by

(35)

In our data, we found that the cost of maintaining a diesel bus and an electric bus were extremely similar. Therefore, it will not impact our calculation of the difference in TCOs. By substituting the values we found into the original TCO model, our findings are summarized in Figure 8 .

Because of the decrease in popularity of buses, the difference in cost of ownership will decrease further every year. While buses may be losing popularity in New York City, we firmly believe that the introduction of completely net-zero emission buses will promote and give New Yorkers a reason to ride.

By joining the ecological benefits and economical costs together, our findings support the conclusion that, after the initial investment of charging facility construction is made, switching to electric bus fleet is more beneficial in the long run. For New York City, the post-transition yearly-average net gain per bus is estimated to be

(36)

where is the estimated ecological benefits per e-bus in year 2033 ( Table 6 ).

This estimation is based on the current purchase price of e-buses and charger maintenance costs. In the long term, we expect these costs will drop significantly with the technological advances and economy of scale following the broader adoption of electric buses. Therefore, we expect the positive gains of future e-bus fleet can only be larger.

Assumption 1. The city has an active diesel bus fleet serving the current public transportation needs. The operational capacity of the bus program and its future growth are exogenous to our transition plan. The in-transition fleet and the eventual fully electric one run the same pre-determined bus program in terms of bus number and capacity, except in a more eco-friendly way.

Justification: Our model targets to build a transition roadmap for an established metropolitan area with a population over half million people. Public bus programs are usually determined by people’s transportation needs and city’s budget constraints. This report does not have the scope of completely redesigning the capacity and routes of the metropolitan’s bus program.

Assumption 2. Each bus has a limit of maximum service year, after which it must be retired and does not have further redemption value. The existing bus fleet has distributed ages that warrants a natural replenishment procedure.

Justification: The decommissioning and replenishing is mandatory for safety and liability reasons.

Assumption 3. The transition plan can early-retire some diesel buses before they reach maximum service year.

Justification: Early retirement on selected buses is necessary to achieve the targeted full transition by the end of 10 years. Early-retired diesel buses are replenished with e-buses. Within the existing fleet, the model can early-retire those worst polluting buses.

Assumption 4. The full electric bus fleet has salvage value at the end of 10-year transition period. The salvage value of each bus, after an initial 10% discount, is assumed to be linearly proportional to its remaining service life and purchase cost.

Justification: Even though this planning model focuses on the 10-year transition period, it needs to incorporate the salvage value into consideration as part of the long-term planning. The linear depreciation model is one common method in financial accounting.

Assumption 5. The purchase of new electric buses are spread into years. The purchase in any year cannot be 20% more than the yearly average of the eventual target.

Justification: A practical roadmap needs to smooth the purchase and update into multiple years. The gradual transition helps to mitigate any potential resistance from stakeholders, and gives all teams more time to overcome the learning curves in operation and management.

The model runs constrained optimization with the aid of MATLAB [14] [15] . The concept of constrained optimization is demonstrated in Figure 10 . The variables of the optimization model are summarized in Table 11 . This model optimizes the decision variables , which are the numbers of new e-buses purchased in the coming 10 years.

The optimization objective is to minimize the net present value of the financial and environmental costs in next 10 years, net of the salvage value of the full e-bus fleet at the end of the period.

The optimization is subject to following constraints:

1) Mandatory target of full e-bus fleet at the end of 10 years.

2) Purchase in any year cannot be 20% more than the yearly average of the eventual target.

(37)

(38)

where operation, maintenance and ecological costs of active fleet

(39)

acquisition costs of new buses, including the construction costs of new charging facilities

(40)

and salvage value at the end of year 2033

(41)

With the aid of this optimization model, we craft a roadmap to transition the mostly combustion engine bus fleet to a fully electric one in 10 years.

In this gradual transition plan, diesel buses retire, which are replenished by electric buses over years. The mix of buses in the fleet is illustrated in Figure 11 and Table 12 . The downward trend of total fleet size is extrapolated from the shrinking trend of NYC bus ridership in years 2012 to 2019.

Purchase of new electric buses ( Figure 11 ) started at a relatively lower rate of around 309 vehicles per year in the first four years. They are allocated to partially replenish the retired vehicles as the total fleet size is shrinking.

In these first several years, no buses are retired before their maximum service age. This is due to the expensive purchase price of e-buses available to US cities as well as the high construction cost of charging facilities, which we will analyze in more details in later chapter.

The purchase will accelerate in year 2028 and stabilize at 488 e-buses per year in the last five years. This acceleration is necessary to achieve our target of full transition by year 2033, and it is bounded by our constraint to smooth the purchase budgeting over years.

In this transition period, the metropolitan needs to make additional investment due to the relatively higher prices of e-buses and the extra cost of building charging facilities ( Figure 12 ). This cost increases in year 2028 and 2029 when the purchases accelerate. On the other hand, the lower operation and maintenance cost of e-bus contributes to the downward trend of net cost in the other years when the new purchases are stable.

More importantly, with the expansion of e-buses, the fleet will realize more ecological benefits year over year which is illustrated in the same Figure 12 .

The transition plan takes the working condition of buses and polluting situation of the city into consideration. It manages smart retirement-scheduling among buses of various polluting situations and city boroughs of different polluting levels ( Figure 13 ). Starting from year 2028, it expedites the decommissioning of the heaviest pollution generating group. After its completion in year 2031, the second group is expedited. And then in year 2032, the third group is expedited.

Government subsidies, such as the allocation to e-bus projects from the 2023 US Bipartisan Infrastructure Law will be crucial for the successful transition to an e-bus future. In the 10-year roadmap for New York City, we expected 50% of the transition cost will be shouldered by various government funding. This kind of government support is common among the cities that have been successful in achieving the transition to electric bus fleet, such as Shenzhen, China.

Without the needed government support, the transition cost can become prohibitively high for the transportation enterprise. Figure 14 shows the net cost could be as high as $567M in the peak year of 2029 if there was no extra government support.

Compared with Shenzhen, China, the transition to the e-bus future is financially more challenging for US cities. The main concern will be the significantly higher e-bus purchase price and construction cost of charging facilities.

We encourage transportation officials and private enterprises to be more innovative and competitive in lowering these costs. For instance, cities can import lower-priced e-buses, such as BYD, to reduce the cost and encourage domestic manufacturers to be more competitive. US government can provide more synchronized support to cities and bus manufacturers so that it is easier to achieve economy of scale and lowered costs for all stakeholders.

With the optimistic expectation of this success, by assuming the e-bus prices can be lowered by 30% and the charging facility cost can be cut to $150K per new bus, we use the same optimization model to construct an alternative 10-year roadmap, which significantly expedites the transition in the first several years ( Figure 15 ). The purchase of e-bus is not only unequal over years, it is not monotonic. Instead, it started by purchasing 488 new e-buses in the first two years and then slows down to 200+ per year for another three years. Afterwards, it accelerates again to 488 buses per year in the last five years.

The accelerated purchase in the first two years is justified by the fact that the ecological benefits and operation savings can be harvested perennially after the one-time investment is made. As long as the cost is justified, it is beneficial to invest earlier. Meanwhile there is financial incentive to preserve the cash and utilize the existing bus fleet till the end of their service life. These two competing drivers lead to the U-shaped purchasing schedule in this alternative roadmap. The expedited replacement in the early years helps to decommission the worst polluting maker earlier ( Figure 16 ).

Cities have different needs in their bus programs. Each city also has its unique mix of power sources, which contribute to varying ecological benefits from the transition to the e-bus future. Nevertheless, the same analytical methodology can be applied to other metropolitan areas, such as San Diego, CA and Chicago, IL. The roadmap of e-bus transition of each city is driven by its own unique situation of multiple factors ( Figure 17 and Table 13 )

The baseline plan for New York City was built on a decreasing bus ridership trend, which was extrapolated from the surveyed data in years 2012 to 2019. On the other hand, one major benefit of transition to e-bus would be the reduction of air pollution. It is possible that cleaner and more eco-friendly electric buses might attract more bus riders, and reduce the need of private transportation and further reduce greenhouse gas emission and other air pollution. Shenzhen’s data shows, after transition to a full e-bus fleet, the bus ridership increased by 2% [3] .

If the trend of bus ridership deviates from the historical trend, the implementation plan needs to have certain level of flexibility to adapt to the new reality. We analyze the NYC’s transition plan by modelling two different future scenarios of constant ridership and another one with 1% increase per year. The results ( Table 14 ; Figure 18 for constant ridership & Figure 19 for increasing ridership) show that the transition roadmap can be easily adjusted and adapt to potential evolution of bus ridership.

The optimization model assumes one budget smoothing constraint that new purchase in any year cannot be 20% more than the average of the eventual target. This constraint turns out to be binding in the planning. Without this constraint, the model would tend to concentrate more of the purchase to the last year for most US cities. This is mainly driven by the high purchase price of e-bus, the expensive charging-facility construction cost, the preference to utilize existing buses to their maximum service year, as well as the time value of money. On the other hand, we believe this constraint is practically reasonable because significant variations in project budgeting might cause more resistance from stakeholders and even interruption in future execution. We test the sensitivity and robustness of this constraint by varying its tightness to 10% and 30%, which do lead to different roadmaps from the model ( Figure 20 ). For a tightened constraint of 10% on budget variation, the e-bus purchases are spread to more and earlier years. For a relaxed constraint of 30%, the purchase schedule is concentrated more to the later years. These differences are expected and consistent with the intention of the constraint itself.

The model assumes 5% yearly discount rate, which is based on the current interest rate in the US. The mandatory discount rate for public transportation projects may vary according to local laws and drift over time. Our testing of the model with discount rate from 3% to 12% barely yields any difference in the transition roadmap. The tested range covers most reasonable scenarios in foreseeable future.

Our models are built with live logic and data from industry reports and research papers. They are highly relevant to the topic. It helps transportation officials, government planning entities, and general public to gain in-depth understanding of the topic and get guidance on the transition process.

The transition model is highly practical. It optimizes the balance between ecological benefits and net financial costs. The roadmap generally aligns with the natural replenishment process of existing bus fleet. For the purpose of long-term planning, it takes the salvage value into modelling formula. It also mitigates the variation in budgeting over 10 years.

The transition model is robust under various scenarios of analysis and sensitivity tests. This is rooted in the practical approach of the modelling that aligns with the natural replenishment process and constrains on fleet size and budget variation.

We built our models with current data from environmental research and transportation industry reports, and conducted near-term forecasting. On the other hand, we recognize that the design and manufacturing of electric vehicles and charging facilities are at the frontier of technology innovation. The progress in this field can be fast and potentially interrupting. More in-depth and comprehensive investigations into the technological perspective of this subject could further strengthen the report.