Heat Pump Scheduling: Always On, Setback or Cheap-Hours Preheat

The two-node balance is derived here from energy conservation:

Capacitances use kWh/K, conductances kW/K and time h. The room controller and plant update are coupled. For each interval, the unconstrained air target follows the exact first-order response with a 0.5 h response time. The corresponding mass temperature is predicted implicitly, and the heat needed to reach these two end states follows from the discrete energy balances. That heat is clipped at zero, available heat-pump capacity and emitter output evaluated at the planned end temperature. The actual air and mass end states are then solved together. This avoids applying a stale air-to-mass heat-transfer demand over a long interval. When output is limited, the conservative emitter estimate is refined along with the time step.

Emitter output is $Q=Q_r(\Delta T_m/\Delta T_r)^n$, with mean water $T_f-\Delta T_w/2$. Inverting it gives requested flow. The weather curve is linear between the entered flows at +15 °C and −5 °C outdoors, shifted by the scheduled room-target change. Flow is limited by the curve plus recovery boost and the entered maximum, with an assumed 20 °C minimum command. Ideal modulation/throttling may reduce output while keeping that flow command.

Each COP point gives an effective Carnot fraction $\eta_i=COP_i(T_f-T_o)/T_{f,K}$. Separate logarithmic slopes fit flow and outdoor dependence:

The A7/W35 and A7/W55 points determine $a_f$; A−7/W35 determines $a_o$. With two points, $a_o=0$ is an explicit assumption. The fraction is capped at 0.95, and $COP=\max(1,\eta T_{f,K}/\max(T_f-T_o,5\,K))$. These are effective air/water temperatures, not refrigerant saturation temperatures. Input conditions enforce decreasing COP with flow and increasing COP with outdoors over the model's 20–70 °C flow and −15 to +15 °C outdoor range. They apply to entered data, before the optional defrost allowance.

W35 capacity interpolates between entered −5 and +2 °C outputs and is held at endpoints outside that interval. The entered fractional penalty reduces it above W35. Optional extra defrost reduces COP and capacity between −5 and +5 °C; avoid double counting test-data defrost.

Each grid finds its own periodic midnight states by a finite-difference Newton solve. The solver compares 96 with 192 intervals, then refines to 384 or 768 if needed. Acceptance requires changes of at most 0.2% in electricity; the larger of 0.01 currency units or 0.2% of absolute variable charges in cost; 2 minutes in all-day cold, occupied cold and occupied hot duration; and 0.02 K·h in occupied deficit. Both thermal states must repeat within 0.01 K. Failure blocks results. Durations and deficit use linearly interpolated threshold crossings, and schedule/occupancy times use quarter-hour increments.

The final grid supplies all comfort statistics, extrema, limitation hours and energy-balance residuals. Charts store quarter-hour temperature/flow samples and quarter-hour average heat/electrical powers; averaging preserves electricity, delivered heat and cost because tariff changes are hourly. The reported peak is the maximum modelled input over accepted internal intervals, not a compressor-start surge. Grid differences estimate numerical sensitivity, not rigorous error bounds or building-model uncertainty. The recommendation adds the occupied-duration difference as a comfort margin and reports a cost tie when the winning gap is within twice the largest cost difference. Energy-balance residuals are rounded to 10⁻⁹ kWh. Electricity includes the entered extra allowance only while heating. The three scheduling rules are compared; no tariff optimisation is performed.

One well-mixed heated zone; all external heat loss is attached to its air node. The slower mass exchanges heat only with room air. Effective capacitances and coupling are user estimates, not a construction-material catalogue. Internal gains are constant. Every 24-hour weather, tariff and occupancy profile repeats.

The hydronic emitter has no separate water or slab state. Flow-to-return drop and emitter exponent stay constant. Ideal continuous modulation and room feedback are assumed; minimum compressor speed, cycling loss, anti-short-cycle delays, hydraulic buffers and actual manufacturer recovery logic are not modelled. “Always on” describes a constant room target, not a compressor forced to run continuously.

Additional electricity is charged only while useful heat is delivered. Enter only auxiliaries outside the COP test boundary. There is no domestic hot water, immersion/backup heater, cooling, solar PV export, standing charge or demand-charge calculation. At negative electricity prices, a negative variable heating cost is possible. All money is in the currency whose hourly prices you enter.

Quarter-hour schedule and occupancy boundaries coincide with every tested grid. Weather and prices remain hourly steps. Time-step acceptance measures agreement of successive grids; near a comfort or cost decision boundary the recommendation uses the reported grid differences conservatively.

One-zone heating-day screening, with outdoor temperatures from −15 to +15 °C and heat-producing flow commands from 20 to 70 °C. COP is calibrated at W35/W55 and available capacity only at the two entered outdoor points. Beyond those data, the smooth COP surface and endpoint-held capacity are approximations, not a validated manufacturer map. The tool warns when simulated flow exceeds W55. Use real capacity data and check the full operating envelope.

The COP model is checked against a withheld legacy A2/W35 point and a separate VWL 105/6 A7/W45 point. The latter predicts about 3.91 against the published 4.10 within a declared 5% component-check tolerance. These checks do not establish whole-map accuracy or validate daily savings in a real house. Automatic grid refinement tests numerical cost and comfort sensitivity separately from energy conservation and daily repetition; a reproduced stiff-house case is retained as a regression property.

Fraunhofer WPsmart im Bestand and UK Electrification of Heat reports provide field-performance context, not one-day COP targets. Date et al. motivates accounting for thermal mass and recovery; its electric-heater peak reductions are not transferred to heat pumps here.

EN 442 and EN 14511 are named for emitter and test-condition context only; no standards tables or figures are reproduced, and this is not a conformity calculation. Check against the standard in force where applicable. Editable prices are illustrative scenarios dated 2026-10-27, not current supplier quotations or legal entitlements. Installation and electrical work belong to qualified persons.