Every number on this page comes from one of these models. Each card gives the equation, the values used and the reason for the choice, with a link to the rule or to the code.
1Minimum power during a §14a reduction
Pmin,14a = max(0.4·ΣPHP, 0.4·ΣPAC) + (n − 1)·GZF(n)·4.2 kW
depot: n = 6 → 0.4·14 + 5·0.6·4.2 = 18.2 kW
n is the number of controllable devices behind the energy manager (4 chargers, the battery and the 14 kW heat pump). GZF is the simultaneity factor of the ruling: 0.8 for 2 devices, falling to 0.45 from 9.
Why. The ruling fixes this floor. The gateway may never grant less, so it is computed from the rule itself and tested against the ruling's own examples.
BK6-22-300, Anlage 1, 4.5.2 · rules.rs
2Power budget while reduced
B = Pmin,14a + SPV − 0.3 kW
controllable devices: Σ PEV + PHP + Pbat,charge ≤ B
SPV is the smallest PV surplus measured over the last 30 s, recomputed every second. The 0.3 kW margin absorbs meter noise and the delay between a setpoint and the device reaching it.
Why. The ruling lets a site use its own PV on top of the floor. Taking the smallest recent value means a passing cloud cannot push the grid draw over the limit, and without the margin measurement noise alone would.
controller.rs
3Gradual release after a reduction
B(t) = B0 + (Prated − B0)·(t − t0) / Tramp, Tramp = 300 s
t0 = tend + U·D, U ~ uniform(0, 1)
B0 is the budget when the reduction ends, Prated the sum of the devices' rated power, D the longest random wait (0, 10 or 30 min in the calculator).
Why. A step back to full power makes every site on a feeder jump at the same second. The ramp is the German rule; the random wait is the idea of the British smart-charging regulation (SI 2021/1467), tested here to see whether it helps.
BK6-22-300, Anlage 1, 4.3 · controller.rs
4Charging current and power
PEV = 3 · 230 V · I
I ∈ {0} ∪ [6 A, 32 A] → P ∈ {0} ∪ [4.14 kW, 22.1 kW]
Three phases at 230 V. The car stops charging below 6 A, so the power is either zero or at least 4.14 kW.
Why. Chargers are controlled by current, as in IEC 61851. The 6 A step is why a floor of 4.2 kW per device makes sense and why the planner leaves minimum powers to the real-time layer.
rules.rs
5Battery
Ek+1 = Ek + η·Pc·Δt − Pd·Δt / η
η = 0.95, E ∈ [0, 100] kWh, |P| ≤ 50 kW
E is the stored energy, Pc and Pd charge and discharge power, η the one-way efficiency.
Why. The simplest model that loses energy in both directions. That is what makes charging and discharging at once never pay, so the planner needs no binary variables.
sim.rs
6Building and heat pump
C·dT/dt = COP(Tout)·PHP + Qgains − UA·(T − Tout)
COP = clamp(3 + 0.08·Tout, 1.8, 5), UA = 1.2 kW/K, C = 18 kWh/K
One thermal mass (first-order RC model). Time constant C/UA = 15 h. Internal gains 3 kW in working hours, 1 kW otherwise. Comfort floor 20 °C from 06:00 to 22:00, 17 °C at night, ceiling 23 °C.
Why. A 15-hour time constant lets the building store heat for a few hours. That is the flexibility the planner uses to heat before a price peak or a reduction.
sim.rs
7Sun and PV output
real day: PPV = 120 kWp · min(1, 0.85·GHI / 1000 W/m²)
study days: PPV = 120 kWp · pclear·sin(π·x)1.4·c, x = (h − hrise)/(hset − hrise)
GHI is the measured global horizontal irradiance of the hour; 0.85 is the performance ratio of a flat rooftop system. On the two study days, c is a random cloudiness per day and the sun's path is a clear-sky curve.
Why. Measured irradiance gives real days, real clouds included. The clear-sky model is kept for the forecaster, which, like a real energy manager, does not know the day's weather in advance.
climate.rs
8Prices and tariff
cimport = pDA/1000 + 0.12 €/kWh
cexport = max(0, pDA)/1000
demand charge = 100 €/kW·a · P̂ · h/8760
pDA is the day-ahead price in €/MWh. P̂ is the highest quarter-hour of import in the billing period; the plan carries h/8760 of the yearly charge.
Why. A dynamic tariff indexed to the day-ahead market, as German suppliers must offer. Exported PV earns nothing in negative hours (EEG §51). The demand charge is what makes a smart plan avoid new peaks.
EEG §51 · climate.rs
9Planner (MPC)
min Σk=1..96 [cimport,k·g⁺k − cexport,k·g⁻k + ageing + discomfort + unmet] + demand(P̂) + 10⁻⁴·Σ(ΔPbat)² − V·E96
s.t. g⁺ − g⁻ + PV − ΣEV − HP − c + d = base, battery and building dynamics, g⁺k ≤ P̂
96 steps of 15 minutes, about 1,550 variables with four cars. Ageing 0.03 €/kWh plus a penalty outside 20–80% charge; discomfort 20 €/K·h; energy a car leaves without 1 €/kWh. V values the energy left in the battery at the end.
Why. A convex quadratic program has one optimum, a certificate when there is none, and a predictable solve time of about 16 ms. A mixed-integer model would add exactness only where the real-time layer already handles it.
Clarabel · mpc.md
10Reserve against forecast errors
PVlow = PVexpected − z·σ, z = 1.645 (ε = 5%)
Emin,k = Emin + z·Σj σj (errors summed linearly)
σ is the standard deviation of the PV forecast. The reserve is kept only during and just before an expected reduction, the one time a shortfall cannot be bought from the grid.
Why. Cloudy hours come together, so errors are added linearly. Summing them over the whole day asked for more reserve than the battery holds.
mpc.md
11Feeder and rebound
L(t) = Σi=1..N Li(t)
rebound R = maxt ≥ trel [Lred(t) − Lbase(t)]
pushed energy = Σ max(0, Lred − Lbase)·Δt, minutes over = #{t : L(t) > Ctrafo}
Li is the grid exchange of depot i in one-minute means, Lred the evening with the reduction and Lbase the same evening without it, with the same weather and fleet.
Why. Comparing each evening with itself without a reduction isolates the effect of the reduction. The difference is the rebound, measured, with no need to assume it.
feeder.rs · rebound.rs
12Statistics of the studies
x̄ ± t0.975, n−1·s/√n
paired: dj = xjA − xjB on the same weather and fleet
n runs, sample standard deviation s, Student's t quantile for a 95% interval. Strategies are compared day by day on the same random weather.
Why. Pairing removes the day-to-day variation of the weather from the comparison, so a difference of a few euros is visible with 30 days.
study.rs
13Classifying an evening of 2025
k0: Lbase ≤ C all evening
k2: minutes over after 19:30 with the reduction > without it
k3: still over with the reduction, k1: otherwise
C is the transformer capacity of the feeder. The grid operator is assumed to reduce only on evenings that would otherwise go over it.
Why. It answers the practical question: on how many evenings is a reduction needed, and on how many does it cause a second overload.
year.rs
14Proof of compliance
hk = SHA-256(hk−1 ‖ recordk), h0 = fixed genesis digest
Each reduction event produces a record (start, end, budget, measured draw). Each record's hash includes the previous one.
Why. The ruling asks the operator to be able to prove what happened. A chain of hashes makes any later change to one record visible.
BK6-22-300, Anlage 1, 7.2–7.3 · compliance.rs
15Feed-in limit
Pexport ≤ f·120 kWp
PVmax = Pexport,allowed + Pconsumption − 0.3 kW
f is the limit in force after the country's rules: the grid operator's setpoint (IOA 5002), the Austrian 70% cap, or zero once the Swiss free budget is used up outside an emergency.
Why. Only the export is limited. The depot, the chargers and the battery still use all the PV they can, so only the surplus above the limit is lost. Inverter limits are written without a timeout, so they stay in force if the gateway stops.
controller.rs