EV Range Estimation — Tesla Roadster Simulation

This project extends a navigation app and onboard computer for electric vehicles with functionality to estimate driving range along a given route. Range depends strongly on speed, so consumption and travel time vary between routes. We use consumption data for the Tesla Roadster together with GPS-derived speed data, but the methods generalize to other vehicles and routes.

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Part 1a) Energy Consumption Model

Task:
Model energy consumption as a function of speed using the fitted relation

$$ c(v) = a_1 v^{-1} + a_2 + a_3 v + a_4 v^2 $$

with coefficients $a_1=546.8$, $a_2=50.31$, $a_3=0.2584$, $a_4=0.008210$.

Implement consumption(v) that works with scalars and NumPy arrays, and plot $c(v)$ for $v \in [1,200]$ km/h.

How I did it:
Implemented the formula in roadster.py and added a plotting routine in script_part1a.py.

Solution:
Function in roadster.py; plots in script_part1a.py.

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Part 1b) Route Data & Speed Profile

Task:
Load route data (speed_anna.npz, speed_elsa.npz) and plot speed vs. distance. Interpolate a continuous speed profile using velocity(x, route).

How I did it:
Used roadster.load_route to read data and scatter-plot raw samples. Then evaluated velocity densely to plot the interpolated curve.

Solution:
Functions in roadster.py; plots in script_part1b.py.

Note:
velocity uses PCHIP interpolation, which passes through all data points and ensures monotone, smooth behavior between them.

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Part 2a) Travel Time (Trapezoidal Rule)

Task:
Estimate travel time along a route by integrating

$$ T(x) = \int_0^x \frac{1}{v(s)} ds $$

Implement time_to_destination(x, route, n) using the trapezoidal rule.

How I did it:
Coded a manual trapezoidal integration (no NumPy/SciPy integrators) and tested convergence by increasing (n) until results stabilized to the nearest minute.

Solution:
Function in roadster.py.

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Part 2b) Total Energy Consumption (Trapezoidal Rule)

Task:
Estimate total energy usage along a route by integrating

$$ E(x) = \int_0^x c(v(s))ds. $$

Implement total_consumption(x, route, n).

How I did it:
Combined consumption with interpolated speeds from velocity, then applied trapezoidal integration over the route length.

Solution:
Function in roadster.py.

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Part 3a) Distance from Time — Root Finding (Newton’s Method)

Task:

Given $T>0$, find $x\ge 0$ such that

$$ T(x)=T,\qquad T(x)=\int_{0}^{x}\frac{1}{v(s)},ds. $$

Newton’s method: define

$$ F(x)=T(x)-T,\qquad F'(x)=\frac{1}{v(x)}, $$

then iterate

$$ x_{k+1}=x_k-\frac{F(x_k)}{F'(x_k)} =x_k-\big(T(x_k)-T\big)*v(x_k). $$

Implementation notes

• Use time_to_destination(x, route, n) for $T(x)$ and velocity(x, route) for $v(x)$.

• Initial guess:

$$ x_0 = \min\Big(L, L\frac{T}{T(L)}\Big), $$

where $L$ is the route length and $T(L)$ the total travel time.

• Clamp iterates to $[0,L]$ and stop when

$$ |x_{k+1}-x_k| < 10^{-4}\ \text{km}, $$

with a hard cap on iterations.

• Accuracy depends on the trapezoidal resolution $n$ inside time_to_destination and the Newton stopping tolerance.

Solution:
Function distance(T, route) in roadster.py.

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Part 3b) Range from Battery Energy — Newton’s Method

Task. Given battery charge C (Wh), find the range x km along a route.
Total energy up to distance x is:

$$ E(x)=\int_{0}^{x} c\big(v(s)\big)ds \quad [\mathrm{Wh}]. $$

We solve for x such that $E(x)=C$ by Newton’s method. Define

$$ F(x)=E(x)-C, \qquad F'(x)=E'(x)=c\big(v(x)\big), $$

then update

$$ x_{k+1}=x_k-\frac{E(x_k)-C}{c\big(v(x_k)\big)}. $$

Solution:
Function reach(C, route) in roadster.py.

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Part 4a) Time-Dependent Route Simulation

Task:
Simulate travel along a route where speed depends on both position and time of day, based on historical traffic data. The aim is to estimate arrival time given a start time, and generate route files that can be used as input for earlier functions (e.g. reach from Part 3b).

How I did it:
Used the provided function route_nyc(t, x) in route_nyc.py, which returns the expected speed (km/h) at time t [hours of day] and position x [km] along a 60 km route.

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Part 4b) Plotting Route Simulations

Task:
Update the plotting script to show simulated trips along the NYC route with different start times. Each trip is represented by a curve from start time $t_0$ at position $x=0$ to the route end at $x=60$ km.

How I did it:
Modified script_part4b.py to simulate and plot two runs:

• Start at 04:00
• Start at 09:30

The curves illustrate how traffic conditions at different times of day affect travel along the route.

Solution:
Code updates in script_part4b.py.