Linear Conversion¶

Formulation¶

A Converter enforces linear coupling between its input and output flows. Each conversion equation requires:

\[ \sum_{f} a_{f} \cdot P_{f,t} = 0 \quad \forall \, \text{converter}, \; \text{eq\_idx}, \; t \in \mathcal{T} \]

where \(a_f\) is the conversion coefficient for flow \(f\). A converter can have multiple equations (one per row in conversion_factors), allowing multi-output devices like CHP plants.

Parameters¶

Symbol	Description	Reference
\(a_f\)	Conversion coefficient	`Converter.conversion_factors`
\(P_{f,t}\)	Flow rate variable	`flow_rate[flow, time]`

See Notation for the full symbol table.

Examples¶

Boiler¶

A gas boiler with thermal efficiency \(\eta_{\text{th}} = 0.9\):

\[ \eta_{\text{th}} \cdot P_{\text{gas},t} - P_{\text{th},t} = 0 \]

\[ 0.9 \cdot P_{\text{gas},t} = P_{\text{th},t} \]

So 10 MW gas input produces 9 MW thermal output.

Converter.boiler("boiler", thermal_efficiency=0.9, fuel_flow=gas, thermal_flow=th)
# conversion_factors = [{gas.id: 0.9, th.id: -1}]

Power-to-Heat¶

An electric resistance heater with efficiency \(\eta = 0.99\):

\[ \eta \cdot P_{\text{el},t} - P_{\text{th},t} = 0 \]

Converter.power2heat("p2h", efficiency=0.99, electrical_flow=el, thermal_flow=th)
# conversion_factors = [{el.id: 0.99, th.id: -1}]

Heat Pump¶

A heat pump with COP = 3.5 has two conversion equations — COP definition and energy balance:

\[ \text{COP} \cdot P_{\text{el},t} - P_{\text{th},t} = 0 \]

\[ P_{\text{el},t} + P_{\text{src},t} - P_{\text{th},t} = 0 \]

So 1 MW electrical input draws 2.5 MW from the environment and produces 3.5 MW thermal output.

Converter.heat_pump("hp", cop=3.5, electrical_flow=el, source_flow=src, thermal_flow=th)
# conversion_factors = [{el.id: 3.5, th.id: -1}, {el.id: 1, src.id: 1, th.id: -1}]

CHP (Combined Heat and Power)¶

A CHP with \(\eta_{\text{el}} = 0.4\) and \(\eta_{\text{th}} = 0.5\) has two conversion equations:

\[ \eta_{\text{el}} \cdot P_{\text{fuel},t} - P_{\text{el},t} = 0 \]

\[ \eta_{\text{th}} \cdot P_{\text{fuel},t} - P_{\text{th},t} = 0 \]

So 10 MW fuel input produces 4 MW electrical + 5 MW thermal.

Converter.chp("chp", eta_el=0.4, eta_th=0.5,
                     fuel_flow=fuel, electrical_flow=el, thermal_flow=th)
# conversion_factors = [
#     {fuel.id: 0.4, el.id: -1},
#     {fuel.id: 0.5, th.id: -1},
# ]

Time-Varying Coefficients¶

Conversion coefficients can be time-varying (e.g., a heat pump with hourly COP from weather data). Pass a list or array instead of a scalar:

cop_profile = [3.2, 3.5, 3.8, 3.1]  # one value per timestep
Converter.heat_pump("hp", cop=cop_profile, electrical_flow=el, source_flow=src, thermal_flow=th)