PyPSA, the quadratic class#

Rung 10 of PyPSA in one file: PyPSA's marginal_cost_quadratic, stated on rung 1's transport surface in a file of its own. Its network is the shared spine, shown once on the rung ladder's page.

Rung 10 — quadratic costs#

PyPSA	status	note
`marginal_cost_quadratic`	done	degree 2 in the objective; Generator and Link here — PyPSA also carries it on storage units and stores, one more term each of the same shape

✔ pypsa 1.3.0 solves this rung's network at objective 12587.437500000098, 60 rows.

The network, as PyPSA code

rung_10_quadratic_costs.py

"""Rung 10: quadratic costs — a marginal cost quadratic in output, stated by `pypsa_quadratic.yaml`."""

from __future__ import annotations

import spine

#: This rung binds a file of its own.
MODEL = 'pypsa_quadratic.yaml'


def build():
    """The spine plus this rung's additions, as a ``pypsa.Network``."""
    n = spine.build()
    n.add('Bus', 'village')
    n.add('Generator', 'steam', bus='north', p_nom=80, marginal_cost=5, marginal_cost_quadratic=0.08)
    n.add('Generator', 'engine', bus='north', p_nom=80, marginal_cost=20, marginal_cost_quadratic=0.01)
    n.add(
        'Link',
        'wire2',
        bus0='north',
        bus1='village',
        p_nom=40,
        p_min_pu=-1,
        efficiency=0.9,
        marginal_cost=1,
        marginal_cost_quadratic=0.02,
    )
    n.add('Load', 'village_load', bus='village', p_set=15)
    n.add('Load', 'extra10', bus='north', p_set=[30, 50, 40, 60])
    return n

The file#

The quadratic class of a plain n.optimize(): PyPSA's marginal_cost_quadratic, stated on rung 1's transport surface in a file of its own. One model cannot carry a quadratic objective beside commitment's integer variables and still solve on HiGHS, because degree is the model's property and not the data's. So the class a free solver takes as a QP lives here, and examples/pypsa.yaml stays the mixed-integer one. PyPSA also carries the attribute on storage units and stores; each is one more term of the same shape.

Sets#

Symbol	Meaning
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods
\(\mathcal{N}\)	index \(n\) — `bus` with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — network nodes
\(\mathcal{G}\)	index \(g\) — `generator` with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus
\(\mathcal{L}\)	index \(l\) — `link` with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to
\(\mathcal{O}\)	index \(o\) — `link_output` with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's `bus1`, `bus2`, … columns read long, so a link of any number of output ports is one term in the balance, data prep
\(\mathcal{D}\)	index \(d\) — `load` with \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus

Parameters#

Symbol	Meaning
\(\mathrm{w}\)	`snapshot_weightings_objective` over \(\mathcal{T}\) — PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
\(\mathrm{p}^{\mathrm{nom}}\)	`Generator_p_nom` over \(\mathcal{G}\) — nominal power
\(\underline{\mathrm{p}}\)	`Generator_p_min_pu` over \(\mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power
\(\overline{\mathrm{p}}\)	`Generator_p_max_pu` over \(\mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile
\(\mathrm{c}\)	`Generator_marginal_cost` over \(\mathcal{T} \times \mathcal{G}\) — cost of one unit of output
\(\mathrm{c}^{(2)}\)	`Generator_marginal_cost_quadratic` over \(\mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output
\(\mathrm{f}^{\mathrm{nom}}\)	`Link_p_nom` over \(\mathcal{L}\) — nominal power
\(\underline{\mathrm{f}}\)	`Link_p_min_pu` over \(\mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways
\(\overline{\mathrm{f}}\)	`Link_p_max_pu` over \(\mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power
\(\eta\)	`Link_efficiency` over \(\mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's `efficiency`, `efficiency2`, … read long — negative where that port consumes rather than delivers
\(\mathrm{c}^{f}\)	`Link_marginal_cost` over \(\mathcal{T} \times \mathcal{L}\) — cost of one unit of flow
\(\mathrm{c}^{f,(2)}\)	`Link_marginal_cost_quadratic` over \(\mathcal{T} \times \mathcal{L}\) — cost of the square of one unit of flow
\(\mathrm{load}\)	`Load_p_set` over \(\mathcal{T} \times \mathcal{D}\) — demand

Variables#

Symbol	Meaning
\(p\)	`Generator_p` over \(\mathcal{T} \times \mathcal{G}\) — `Generator-p` — output of a generator in a snapshot
\(f\)	`Link_p` over \(\mathcal{T} \times \mathcal{L}\) — `Link-p` — PyPSA's `p0`, the flow measured at the `Link_bus0` end: a positive value withdraws there and injects at every bus the link's output ports deliver to

Objective#

objective:
  sense: minimize
  description: operating cost, linear and quadratic, each snapshot weighted by the hours it stands for
  expression: >-
    sum(Generator_p * Generator_marginal_cost * snapshot_weightings_objective)
    + sum(Generator_p * Generator_p * Generator_marginal_cost_quadratic * snapshot_weightings_objective)
    + sum(Link_p * Link_marginal_cost * snapshot_weightings_objective)
    + sum(Link_p * Link_p * Link_marginal_cost_quadratic * snapshot_weightings_objective)

\[ \min \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} p_{t,g} \cdot \mathrm{c}_{t,g} \cdot \mathrm{w}_{t} + \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} p_{t,g} \cdot p_{t,g} \cdot \mathrm{c}^{(2)}_{t,g} \cdot \mathrm{w}_{t} + \sum_{t \in \mathcal{T},\ l \in \mathcal{L}} f_{t,l} \cdot \mathrm{c}^{f}_{t,l} \cdot \mathrm{w}_{t} + \sum_{t \in \mathcal{T},\ l \in \mathcal{L}} f_{t,l} \cdot f_{t,l} \cdot \mathrm{c}^{f,(2)}_{t,l} \cdot \mathrm{w}_{t} \]

`Generator-fix-p-lower`#

Generator_fix_p_lower

Generator_fix_p_lower:
  description: "`Generator-fix-p-lower` — a generator outputs at least its minimum"
  dims: [snapshot, generator]
  expression: Generator_p >= Generator_p_min_pu * Generator_p_nom

\[ p_{t,g} \ge \underline{\mathrm{p}}_{t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

`Generator-fix-p-upper`#

Generator_fix_p_upper

Generator_fix_p_upper:
  description: "`Generator-fix-p-upper` — a generator outputs at most what is available"
  dims: [snapshot, generator]
  expression: Generator_p <= Generator_p_max_pu * Generator_p_nom

\[ p_{t,g} \le \overline{\mathrm{p}}_{t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

`Link-fix-p-lower`#

Link_fix_p_lower

Link_fix_p_lower:
  description: "`Link-fix-p-lower` — a link carries at least its minimum, negative for the other way"
  dims: [snapshot, link]
  expression: Link_p >= Link_p_min_pu * Link_p_nom

\[ f_{t,l} \ge \underline{\mathrm{f}}_{t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{l} \qquad \forall\, t \in \mathcal{T},\ l \in \mathcal{L} \]

`Link-fix-p-upper`#

Link_fix_p_upper

Link_fix_p_upper:
  description: "`Link-fix-p-upper` — a link carries at most its nominal power"
  dims: [snapshot, link]
  expression: Link_p <= Link_p_max_pu * Link_p_nom

\[ f_{t,l} \le \overline{\mathrm{f}}_{t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{l} \qquad \forall\, t \in \mathcal{T},\ l \in \mathcal{L} \]

`Bus-nodal_balance`#

Bus_nodal_balance

Bus_nodal_balance:
  description: >-
    `Bus-nodal_balance` — what is generated at a bus, less what the links
    take away, plus what arrives over them after losses, meets the load
    there
  dims: [snapshot, bus]
  expression: >-
    sum(Generator_p, by=Generator_bus, over=generator, into=bus)
    - sum(Link_p, by=Link_bus0, over=link, into=bus)
    + sum(at(Link_p, by=Link_output_link, over=link, into=link_output) * Link_efficiency, by=Link_output_bus, over=link_output, into=bus)
    == sum(Load_p_set, by=Load_bus, over=load, into=bus)

\[ \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} p_{t,g} - \left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} f_{t,\mathrm{Link\_output\_link}(o)} \cdot \eta_{o} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{load}_{t,d} \qquad \forall\, t \in \mathcal{T},\ n \in \mathcal{N} \]

Variable domains#

Generator_p

\[ p_{t,g} \in \mathbb{R} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Link_p

\[ f_{t,l} \in \mathbb{R} \qquad \forall\, t \in \mathcal{T},\ l \in \mathcal{L} \]

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