Converters

A Converter couples its input and output flows. fluxopt offers two conversion modes: a linear form (coefficients fixed against the operating point — they may still vary in time) and a piecewise-linear form (PiecewiseConversion) for behaviour that depends on the operating point itself — part-load efficiency curves, load-dependent heat-to-power ratios.

Linear Conversion

Formulation

Each conversion equation enforces a linear coupling between flows:

\[ \sum_{f} \mathrm{a}_{f,i} \cdot P_{f,t} = 0 \quad \forall \, \text{converter}, \; i, \; t \in \mathcal{T} \]

where \(\mathrm{a}_{f,i}\) is the conversion coefficient for flow \(f\) in equation \(i\). A converter can have multiple equations (one per row in conversion_factors), allowing multi-output devices like CHP plants. Coefficients may also broadcast over \(t\) in the API — see Indexing Convention.

Parameters

Symbol	Description	Reference
\(\mathrm{a}_{f,i}\)	Conversion coefficient	Converter.conversion_factors
\(i\)	Equation index within a converter	row in 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.

Power-to-Heat

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

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

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.

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.

Piecewise Conversion

When efficiency or coupling depends on the operating point itself — part-load curves, load-dependent heat-to-power ratios — set conversion=PiecewiseConversion(...) on the Converter instead of conversion_factors. (For coefficients that vary with time but not load, use the linear form with time-varying conversion_factors.)

Formulation

A PiecewiseConversion defines breakpoints \(\mathrm{b}_{f,k}\) for each flow \(f\) at \(K\) piece-vertices \(k = 0, \dots, K-1\). At every timestep, a vector of non-negative interpolation weights \(\lambda_{k,t}\) selects the operating point on the curve:

\[ \sum_{k} \lambda_{k,t} = 1, \qquad \lambda_{k,t} \ge 0 \]

with at most two adjacent weights non-zero (SOS2 condition for a contiguous curve, enforced by linopy.add_piecewise_formulation).

For each curve flow \(f\), the rate is the corresponding weighted breakpoint sum:

\[ P_{f,t} \; \diamond_f \; \sum_{k} \lambda_{k,t} \cdot \mathrm{b}_{f,k} \]

where the relation \(\diamond_f \in \{=, \le, \ge\}\) is set per flow via the optional third tuple element. The default is equality (==); at most one flow may carry an inequality sign, and only with exactly two flows.

Methods

linopy auto-dispatches the formulation:

Method	When	Aux variables
lp	Two flows, one bounded, matching-curvature curve	None — pure tangent-line constraints
incremental	Strictly monotonic breakpoints	One binary per piece
sos2	Otherwise	One \(\lambda\) per breakpoint, SOS2

Override with method="sos2" / "incremental" / "lp" if needed.

Status gating

When PiecewiseConversion.status is set, the curve is gated by the converter's on/off binary \(\sigma_{c,t}\) (see Status) passed as active= to the linopy formulation:

• All-equality curves: \(\sigma_{c,t} = 0\) forces every \(\lambda\) to zero, which pins all curve flows to \(\mathrm{b}_{f,0}\) (typically zero).
• Inequality curves: \(\sigma_{c,t} = 0\) drives the bounded side to zero; the output's own lower bound (default \(P_f \ge 0\)) closes the loop for non-negative outputs.

Availability

A separate envelope constraint scales the upper breakpoint by a per-timestep availability \(\alpha_t \in [0, 1]\):

\[ P_{f^{\star},t} \le \alpha_t \cdot \mathrm{b}_{f^{\star},K-1} \cdot \sigma_{c,t} \]

where \(f^{\star}\) is the first flow in the curve.

Parameters

Symbol	Description	Reference
\(\mathrm{b}_{f,k}\)	Breakpoint values per flow	PiecewiseConversion.points
\(\lambda_{k,t}\)	Interpolation weights	linopy auxiliaries
\(\diamond_f\)	Curve relation	tuple bound '==' / '<=' / '>='
\(\sigma_{c,t}\)	On/off binary	PiecewiseConversion.status
\(\alpha_t\)	Availability scaling	PiecewiseConversion.availability