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Piecewise

Piecewise linearization approximates non-linear relationships using connected linear segments.

Mathematical Formulation

A piecewise linear function with \(n\) segments uses per-segment interpolation:

\[ x = \sum_{i=1}^{n} \left( \lambda_i^0 \cdot x_i^{start} + \lambda_i^1 \cdot x_i^{end} \right) \]

Each segment \(i\) has:

  • \(s_i \in \{0, 1\}\) — binary indicating if segment is active
  • \(\lambda_i^0, \lambda_i^1 \geq 0\) — interpolation weights for segment endpoints

Constraints ensure valid interpolation:

\[ \lambda_i^0 + \lambda_i^1 = s_i \quad \forall i \]
\[ \sum_{i=1}^{n} s_i \leq 1 \]

When segment \(i\) is active (\(s_i = 1\)), the lambdas interpolate between \(x_i^{start}\) and \(x_i^{end}\). When inactive (\(s_i = 0\)), both lambdas are zero.

Implementation Note

This formulation is an explicit binary reformulation of SOS2 (Special Ordered Set Type 2) constraints. It produces identical results but uses more variables. We will migrate to native SOS2 constraints once linopy supports them.

Building Blocks

A linear segment from start to end value:

fx.Piece(start=10, end=50)  # Linear from 10 to 50

Values can be time-varying:

fx.Piece(
    start=np.linspace(5, 6, n_timesteps),
    end=np.linspace(30, 35, n_timesteps)
)

Multiple segments forming a piecewise linear function:

fx.Piecewise([
    fx.Piece(0, 30),   # Segment 1: 0 → 30
    fx.Piece(30, 60),  # Segment 2: 30 → 60
])

Synchronizes multiple flows — all interpolate at the same relative position:

fx.PiecewiseConversion({
    'input_flow':  fx.Piecewise([...]),
    'output_flow': fx.Piecewise([...]),
})

All piecewise functions must have the same number of segments.

Maps a size/capacity variable to effects (costs, emissions):

fx.PiecewiseEffects(
    piecewise_origin=fx.Piecewise([...]),  # Size segments
    piecewise_shares={'costs': fx.Piecewise([...])},  # Effect segments
)

Usage

Converter efficiency that varies with load:

chp = fx.LinearConverter(
    ...,
    piecewise_conversion=fx.PiecewiseConversion({
        'el':   fx.Piecewise([fx.Piece(5, 30), fx.Piece(40, 60)]),
        'heat': fx.Piecewise([fx.Piece(6, 35), fx.Piece(45, 100)]),
        'fuel': fx.Piecewise([fx.Piece(12, 70), fx.Piece(90, 200)]),
    }),
)

Investment cost per unit decreases with size:

fx.InvestParameters(
    piecewise_effects_of_investment=fx.PiecewiseEffects(
        piecewise_origin=fx.Piecewise([
            fx.Piece(0, 100),
            fx.Piece(100, 500),
        ]),
        piecewise_shares={
            'costs': fx.Piecewise([
                fx.Piece(0, 80_000),
                fx.Piece(80_000, 280_000),
            ])
        },
    ),
)

Equipment cannot operate in certain ranges:

fx.PiecewiseConversion({
    'fuel':  fx.Piecewise([fx.Piece(0, 0), fx.Piece(40, 100)]),
    'power': fx.Piecewise([fx.Piece(0, 0), fx.Piece(35, 95)]),
})
# Either off (0,0) or operating above 40%

Reference

Symbol Type Description
\(x\) \(\mathbb{R}\) Interpolated variable value
\(s_i\) \(\{0, 1\}\) Binary: segment \(i\) is active
\(\lambda_i^0\) \([0, 1]\) Interpolation weight for segment start
\(\lambda_i^1\) \([0, 1]\) Interpolation weight for segment end
\(x_i^{start}\) \(\mathbb{R}\) Start value of segment \(i\)
\(x_i^{end}\) \(\mathbb{R}\) End value of segment \(i\)
\(n\) \(\mathbb{Z}_{> 0}\) Number of segments

Classes: Piecewise, Piece, PiecewiseConversion, PiecewiseEffects