Algorithm Overview¶

The Heating Curve Optimizer uses a sophisticated dynamic programming approach to minimize electricity costs while meeting your home's heat demand. This page provides a high-level overview of the optimization algorithm.

The Optimization Problem¶

At its core, we're solving this problem:

Optimization Goal

Minimize: Total electricity cost over the planning horizon

Subject to:

Heat demand must be satisfied (or buffered/deferred within limits)
Supply temperature stays within bounds
Offset changes are limited (±1°C per time step)
Thermal buffer constrained by heat debt limit (configurable)

Key Concepts¶

1. Heat Demand¶

Heat demand varies with weather:

\[ Q_{demand}(t) = Q_{loss}(t) - Q_{solar}(t) \]

Where:

$Q_{loss}(t)$ = Heat loss due to temperature difference
$Q_{solar}(t)$ = Solar gain through windows

Negative Demand

When $Q_{solar} > Q_{loss}$, demand is negative, meaning solar gain exceeds heat loss. This excess heat is stored in the building's thermal mass as a buffer.

2. Heating Curve Offset¶

The heating curve offset adjusts the supply temperature:

graph LR A[Base Heating Curve] -->|+ Offset| B[Adjusted Curve] B --> C[Supply Temperature] D[Outdoor Temperature] --> A style B fill:#ff6b35,stroke:#333,stroke-width:2px

The supply temperature is calculated as:

\[ T_{supply} = T_{base}(T_{outdoor}) + \text{offset} \]

Where $T_{base}$ is your heat pump's base heating curve (typically linear with outdoor temperature).

Offset range: -4°C to +4°C (configurable)

3. Heat Pump COP¶

The Coefficient of Performance (COP) determines how efficiently the heat pump converts electricity to heat:

\[ \text{COP} = \left( \text{COP}_{base} + \alpha \times T_{outdoor} - k \times (T_{supply} - 35) \right) \times f \]

Parameters:

$\text{COP}_{base}$: Base COP at standard conditions
$\alpha = 0.025$: Outdoor temperature coefficient
$k$: K-factor (how COP degrades with supply temp)
$f$: COP compensation factor (system efficiency)

COP Calculation Example

Given:

Base COP = 3.5
Outdoor temp = 5°C
Supply temp = 40°C
K-factor = 0.03
Compensation = 0.9

Calculation: $$ \text{COP} = (3.5 + 0.025 \times 5 - 0.03 \times (40 - 35)) \times 0.9 $$ $$ = (3.5 + 0.125 - 0.15) \times 0.9 = 3.475 \times 0.9 = \mathbf{3.13} $$

4. Electricity Cost¶

The cost to produce heat at time $t$:

\[ \text{Cost}(t) = \frac{Q_{heat}(t)}{\text{COP}(t)} \times P_{electricity}(t) \]

Where:

$Q_{heat}(t)$: Heat delivered (kWh)
$\text{COP}(t)$: Heat pump efficiency
$P_{electricity}(t)$: Electricity price (€/kWh)

Two Levers for Optimization

The optimizer can reduce costs by:

Timing: Heat when electricity is cheap
Efficiency: Use lower supply temps (higher COP) when possible

Algorithm Flow¶

graph TD A[Start] --> B[Fetch Weather Forecast] B --> C[Fetch Price Forecast] C --> D[Calculate Heat Demand Forecast] D --> E[Dynamic Programming Optimization] E --> F[Extract Optimal Offset Sequence] F --> G[Output Current Offset] G --> H{Next Update Cycle} H --> B style E fill:#ff6b35,stroke:#333,stroke-width:3px style G fill:#f7931e,stroke:#333,stroke-width:2px

Optimization Strategy¶

Step 1: Forecast Horizon¶

The algorithm looks ahead for a configurable planning window (default: 6 hours).

gantt title 6-Hour Planning Window dateFormat HH:mm axisFormat %H:%M section Current Now :milestone, m1, 12:00, 0m section Forecast Hour 1 :active, 12:00, 60m Hour 2 :active, 13:00, 60m Hour 3 :active, 14:00, 60m Hour 4 :active, 15:00, 60m Hour 5 :active, 16:00, 60m Hour 6 :active, 17:00, 60m

Step 2: State Space¶

For each time step, the algorithm considers:

Offset: -4°C to +4°C (9 possible values)
Buffer: Cumulative thermal energy stored
Cumulative offset sum: Running total (for smoothness)

This creates a 3-dimensional state space.

Step 3: Dynamic Programming¶

The algorithm uses backward induction:

Start from the final time step
For each state, calculate the minimum cost to reach the end
Work backward to the current time
Choose the path with minimum total cost

graph LR subgraph "Time Step 1" A1[Offset: -4] A2[Offset: 0] A3[Offset: +4] end subgraph "Time Step 2" B1[Offset: -4] B2[Offset: 0] B3[Offset: +4] end subgraph "Time Step 3" C1[Offset: -4] C2[Offset: 0] C3[Offset: +4] end A1 --> B1 A1 --> B2 A2 --> B1 A2 --> B2 A2 --> B3 A3 --> B2 A3 --> B3 B1 --> C1 B1 --> C2 B2 --> C1 B2 --> C2 B2 --> C3 B3 --> C2 B3 --> C3 style A2 fill:#4caf50,stroke:#333,stroke-width:2px style B2 fill:#4caf50,stroke:#333,stroke-width:2px style C2 fill:#4caf50,stroke:#333,stroke-width:2px

Step 4: Constraints¶

The algorithm enforces several constraints:

Offset Change Constraint¶

Maximum change per time step: ±1°C

# Only consider offsets within ±1 of current offset
for next_offset in range(max(-4, offset - 1), min(5, offset + 2)):
    ...

Temperature Bounds¶

Supply temperature must stay within configured min/max:

if not (min_supply_temp <= supply_temp <= max_supply_temp):
    continue  # Skip this state

Buffer Constraint¶

Thermal buffer constrained by heat debt limit:

if new_buffer < -max_buffer_debt:
    continue  # Exceeds maximum allowed heat debt

This allows the optimizer to reduce heating during expensive hours (creating "heat debt") and compensate during cheaper hours, enabling cost optimization through temporal load shifting.

Output¶

The optimization produces:

Primary Output¶

Optimal Heating Curve Offset for the current time step

This offset is immediately applied to your heating system.

Secondary Outputs¶

Optimized supply temperature sequence (6-hour forecast)
Buffer evolution (how thermal storage changes)
Expected electricity cost (total for planning window)

Example Scenario¶

Let's see how the algorithm handles a typical day:

Input Conditions¶

Time	Outdoor Temp	Solar Radiation	Electricity Price
06:00	2°C	0 W/m²	€0.15/kWh
09:00	5°C	200 W/m²	€0.30/kWh
12:00	8°C	600 W/m²	€0.35/kWh
15:00	7°C	300 W/m²	€0.25/kWh
18:00	4°C	0 W/m²	€0.40/kWh

Algorithm Decisions¶

gantt title Optimal Heating Strategy dateFormat HH:mm axisFormat %H:%M section Offset High (+3°C) :done, 06:00, 3h Medium (0°C) :active, 09:00, 3h Low (-2°C) :crit, 12:00, 3h High (+2°C) : 15:00, 3h section Price Low : 06:00, 3h High :crit, 09:00, 6h Medium : 15:00, 3h

Rationale:

06:00 - 09:00: Prices low, outdoor temp cold → Pre-heat with high offset (+3°C)
09:00 - 12:00: Prices high, solar gain increasing → Reduce to medium offset (0°C)
12:00 - 15:00: Peak solar gain → Minimize heating with low offset (-2°C), build thermal buffer
15:00 - 18:00: Solar fading, prices moderate → Moderate heating (+2°C)

Why This Works¶

1. Temporal Shifting¶

Heat when electricity is cheap, reduce when expensive

2. Thermal Mass¶

Buildings store heat, allowing strategic heating

3. COP Optimization¶

Lower supply temps = higher efficiency = lower consumption

4. Solar Integration¶

Free heat from sun reduces need for heat pump operation

Limitations¶

Algorithm Assumptions

Perfect forecasts: Assumes weather and price forecasts are accurate
Simplified building model: Uses steady-state heat loss, not dynamic thermal modeling
Fixed COP model: Assumes COP formula is accurate across all conditions
No occupancy: Doesn't account for internal heat gains from people/appliances

Next Steps¶

Dive deeper into specific components:

Dynamic Programming Details - Math and implementation
COP Calculation - Heat pump efficiency modeling
Buffer System - Thermal storage management
Optimization Strategy - Advanced techniques

Explore: Real-World Examples to see the algorithm in action