2023 · Problem B — Charging Ahead with E-buses

NPV / finance Optimization Scenario planning

The problem in one paragraph

Help a city convert its bus fleet to all-electric. (1) Build an ecological-impact model. (2) Build a financial model that factors in up to 50% external funding. (3) Build a 10-year roadmap to full electrification by 2033 for one chosen metropolitan area (population ≥ 500K). Apply to two additional cities. (4) Write a one-page recommendation letter.

Read the official PDF →

Solution outline

Part 1: Ecological model

Per-bus emissions delta:

$\Delta\text{CO}_2 = M_{\text{annual}} \cdot \left( e_{\text{diesel}} - e_{\text{grid}}(t) \cdot \eta_{\text{kWh/mi}} \right)$

$M$ is annual miles per bus, $e_{\text{diesel}}$ is kgCO₂/mi for diesel (~2.7), $e_{\text{grid}}(t)$ is grid intensity (kgCO₂/kWh, time-varying as renewables grow), $\eta$ is kWh per mile (~2.0 for modern e-buses). Sum across the fleet.

Part 2: Financial model

Compute Net Present Value over a 15-year horizon:

$\text{NPV} = -C_0 (1 - s) + \sum_{t=1}^{T} \dfrac{B_t - O_t}{(1+r)^t}$

Where $C_0$ is upfront cost per bus + depot/charger, $s$ is subsidy share (≤ 0.5), $B_t$ is annual fuel savings + maintenance savings + carbon-credit revenue, $O_t$ is annual operating cost delta, $r$ is discount rate (3–5%). Apply this across the full fleet replacement schedule.

Part 3: 10-year roadmap (optimization)

Decision: how many buses to electrify per year $x_t$, subject to:

$\sum_t x_t = N_{\text{total}}$ (full conversion by 2033)
$x_t \le c_{\text{depot}}(t)$ (charging capacity ramp-up constraint)
$x_t \ge 0$ integer

Objective: maximize NPV minus penalty for delayed CO₂ reduction. Solve as MIP. Output is a year-by-year procurement plan.

Three cities to compare

City	Why it's interesting
Bogotá, Colombia	Already piloting e-buses; high pollution baseline; tropical climate
Berlin, Germany	Mature grid, high renewables share, expensive labor
Houston, USA	Sprawling network, fossil-heavy grid, low subsidies historically

The model should produce different optimal roadmaps for these three — that's the value.

Pitfalls

Treating grid emissions as constant — they should decline over the 10-year horizon.
Forgetting depot/charger capex (it's often as large as bus purchase cost).
One-size-fits-all roadmap across all three cities.
Letter to transportation officials that doesn't quantify the recommendation.

Next: 2022 Problem A → All past problems