AI example · E-mobility

Park or charge?

Is it worth leaving your EV at an 11 kW AC charger for, say, 2 hours instead of buying a parking ticket? The calculator compares the parking fee with the charging cost, including the idle fee after 3 hours.

This example uses real rates from Bottrop, Germany: prices in euros, charging on the European three-phase grid (230/400 V).

Calculator

Parking ticket or charger?

Calculation: locally in your browser

Parking duration 2:00 h

15 min2 h4 h6 h8 h

Parking fee, regular space

€/h

Charging current per phase 16 A

6 A8101316 A

Usable battery capacity

kWh

Charging price

39 ct^*per kWh · ELE charger 59 ct^*per kWh · third-party charger

Idle fee of 10 ct/min from the 181st minute of connection time.

^* These kWh prices apply only with the “ELE Autostrom mobil” plan for ELE customers. The plan costs €3.99 per month; this amount is not included in the calculation.

Parked between 8 p.m. and 8 a.m. (no idle fee at ELE chargers)

More assumptions

Home electricity price (ct/kWh) Parking fee at the charger (€/h) Start SoC (%) Target SoC (%) Charging losses (%) Phases used cos φ Consumption (kWh/100 km)

The calculator credits the charged energy at your home electricity price, since you would otherwise have to charge it at home. With 0 ct it compares the full cost at the charger with the parking ticket.

Result

–

Parking ticket – –

Charger, minus home electricity – –

Cost at the charger

Electricity –

Idle fee –

Parking fee at the charger–

Total at the charger –

Home electricity credit –

Effective cost of parking at the charger–

^* Applies only with the “ELE Autostrom mobil” plan for ELE customers (€3.99/month, not included).

Charging power––

Charged––

Range––

Battery full after––

Trend

Cost over parking duration

–

Parking ticket Charger, minus home electricity Charger total (dashed)

Diagram 1

Phasor diagram: how 230 V becomes 400 V

The three phase-to-neutral voltages are each 230 V long and shifted by 120°. The line-to-line voltage U₁₂ = U₁ − U₂ connects the tips of two phasors.

ωt Auxiliary vector −U₂ all line-to-line voltages

Diagram 2

Instantaneous power with a balanced load

Each phase pulsates at twice the grid frequency. The sum of the three phases stays constant. The diagram takes current and cos φ from the calculator.

p₁(t) L1 p₂(t) L2 p₃(t) L3 p₁+p₂+p₃ = 3 · U · I · cos φ

Math

Step-by-step derivation

From complex phasors to the formula P = √3 · U · I · cos φ.

Phase voltages as phasors

U₁ = U·e^(j0°) · U₂ = U·e^(−j120°) · U₃ = U·e^(−j240°)

All three phasors have the same length (U = 230 V) and are 120° apart.

Difference of two phases

U₁₂ = U₁ − U₂ = U·[1 − (−½ − j·√3/2)] = U·(3/2 + j·√3/2)

The line-to-line voltage is the difference of two phase-to-neutral voltages.

Magnitude and angle

|U₁₂| = U·√(9/4 + 3/4) = U·√3

arg U₁₂ = arctan((√3/2)/(3/2)) = 30°

Geometrically: the triangle has a 120° angle at the apex, and the base angles are 30° each. It follows that U_L = 2·U·cos 30° = √3·U.

U_L = 1.732 · 230 V ≈ 398 V ≈ 400 V

Instantaneous power per phase

p₁(t) = u·i = U_Ph·I·[cos φ − cos(2ωt − φ)]

The power of each phase oscillates at 100 Hz around the mean value U_Ph·I·cos φ.

Sum of the three phases

Σ cos(2ωt − φ − k·240°) = 0 (k = 0, 1, 2)

The alternating components form a balanced three-phase system and cancel each other out.

P = p₁ + p₂ + p₃ = 3·U_Ph·I·cos φ

Substituting U_Ph = U_L/√3

P = 3·(U_L/√3)·I·cos φ = (3/√3)·U_L·I·cos φ

P = √3 · U_L · I · cos φ

At 16 A: √3 · 400 V · 16 A · 1 ≈ 11.1 kW. At 32 A: ≈ 22.2 kW.

In practice at the charger

φ is the angle between phase voltage and phase current, not between U_L and I. Modern on-board chargers reach cos φ ≈ 0.99.
3 · 230 V and √3 · 400 V give slightly different values because 400 V / √3 = 230.9 V. Both nominal values are rounded.
The √3 formula only applies to a balanced load. Many vehicles charge on one or two phases. Then P = n · U_Ph · I · cos φ with n phases in use.
Billing is based on the energy measured at the charger’s meter, so you also pay for the charging losses in the vehicle.