PyPSA LOPF — rung 1¶
PyPSA linear optimal power flow, first rung: transport model, linear marginal cost, no KVL.
✔ Verified against pypsa 1.2.4 (its own linopy 0.9.0) — objective 22000, matched to
rtol=1e-09.
The model¶
The same model, as math
PyPSA linear optimal power flow, rung 1: a transport model — linear marginal cost, controllable links, no voltage law. Optimum 22000.0, from PyPSA itself.
Sets¶
| Symbol | Meaning |
|---|---|
| \(\mathcal{T}\) | index \(t\) --- snapshot --- dispatch periods |
| \(\mathcal{B}\) | index \(b\) --- bus --- network nodes |
| \(\mathcal{G}\) | index \(g\) --- generator with \(\mathrm{gen\_bus}: \mathcal{G} \to \mathcal{B}\) --- generating units, each sitting on one bus |
| \(\mathcal{L}\) | index \(l\) --- link with \(\mathrm{from}: \mathcal{L} \to \mathcal{B},\enspace \mathrm{to}: \mathcal{L} \to \mathcal{B}\) --- controllable connections, each joining two buses |
Parameters¶
| Symbol | Meaning |
|---|---|
| \(p^{\mathrm{nom}}\) | p_nom over \(\mathcal{G}\) --- installed capacity of a generator |
| \(\mathit{marginal\_cost}\) | marginal_cost over \(\mathcal{G}\) --- cost of one unit of output |
| \(\mathit{rating}\) | rating over \(\mathcal{L}\) --- most a link may carry towards its to bus |
| \(\mathit{neg\_rating}\) | neg_rating over \(\mathcal{L}\) --- most a link may carry the other way, negative by convention |
| \(\mathit{load}\) | load over \(\mathcal{T} \times \mathcal{B}\) --- demand at each bus in each snapshot |
Variables¶
| Symbol | Meaning |
|---|---|
| \(p\) | p over \(\mathcal{T} \times \mathcal{G}\) --- output of a generator in a snapshot |
| \(f\) | f over \(\mathcal{T} \times \mathcal{L}\) --- PyPSA's p0 — flow measured at the link's from end, so a positive value withdraws there and injects at to |
Objective¶
Subject to¶
nodal_balance
Variable domains¶
p
f
The tabs start from the instance’s tables — one frame per parameter.
description: >-
PyPSA linear optimal power flow, rung 1: a transport model — linear marginal
cost, controllable links, no voltage law. Optimum 22000.0, from PyPSA itself.
dimensions:
snapshot:
description: dispatch periods
dtype: int
bus:
description: network nodes
dtype: str
generator:
description: generating units, each sitting on one bus
dtype: str
link:
description: controllable connections, each joining two buses
dtype: str
lookups:
gen_bus:
description: the bus a generator sits on
over: generator
into: bus
from:
description: the bus a link leaves
over: link
into: bus
to:
description: the bus a link arrives at
over: link
into: bus
parameters:
p_nom:
description: installed capacity of a generator
dims: [generator]
marginal_cost:
description: cost of one unit of output
dims: [generator]
rating:
description: most a link may carry towards its `to` bus
dims: [link]
neg_rating:
description: most a link may carry the other way, negative by convention
dims: [link]
load:
description: demand at each bus in each snapshot
dims: [snapshot, bus]
variables:
p:
description: output of a generator in a snapshot
foreach: [snapshot, generator]
bounds:
lower: 0
upper: p_nom
f:
description: >-
PyPSA's p0 — flow measured at the link's `from` end, so a positive value
withdraws there and injects at `to`
foreach: [snapshot, link]
bounds:
lower: neg_rating
upper: rating
constraints:
nodal_balance:
description: what is generated at a bus plus what arrives over the links meets the load there
foreach: [snapshot, bus]
expression: >-
sum(p, by=gen_bus)
+ sum(f, by=to)
- sum(f, by=from)
== load
objective:
sense: minimize
description: total cost of generation; moving power over a link is free here
expression: p * marginal_cost
The model-building half of examples/ports/references/pypsa/pypsa_transport.py:
def build(tables: dict[str, pd.DataFrame]) -> pypsa.Network:
"""The port's tables as a PyPSA network, column for column.
``tables`` is the same mapping the lpspec call binds as ``sources``.
``p_min_pu = -1`` makes a link bidirectional. The port cannot say that in
a bound — bounds take a name or a number, never arithmetic (the declaration rules) — so
it ships ``neg_rating`` as data instead. That is the ledger row.
"""
n = pypsa.Network()
n.set_snapshots(tables['snapshot']['snapshot'])
n.add('Bus', tables['bus']['bus'])
generators: pd.DataFrame = tables['generator'].set_index('generator')
links: pd.DataFrame = tables['link'].set_index('link')
n.add(
'Generator',
generators.index,
bus=generators['gen_bus'],
p_nom=tables['p_nom'].set_index('generator')['value'],
marginal_cost=tables['marginal_cost'].set_index('generator')['value'],
)
n.add(
'Link',
links.index,
bus0=links['from'],
bus1=links['to'],
p_nom=tables['rating'].set_index('link')['value'],
p_min_pu=-1.0,
efficiency=1.0,
)
load: pd.DataFrame = tables['load'].pivot(index='snapshot', columns='bus', values='value')
for bus in tables['bus']['bus']:
n.add('Load', f'load_{bus}', bus=bus, p_set=load[bus])
return n
Read this comparison carefully — it flatters neither side fairly. PyPSA is
a domain package: n.add('Generator', ...) and n.add('Link', ...) carry a
power-systems model inside them, so the reference is short because someone
already wrote the power flow. Against that, the YAML looks more explicit rather
than shorter, and it should — it is stating the constraint PyPSA implies.
The comparison against a general-purpose alternative is on the Dantzig page, where both sides write the maths out.
What it exercises¶
Rung 1 of a ladder. Reproducing a full PyPSA objective means reproducing marginal and capital cost, ramp limits, storage cycling and KVL at once, and a mismatch then implicates five features instead of one. So each feature is switched off in PyPSA and reproduced here separately: 1 transport model (this one) · 2 ramp limits · 3 storage with state of charge · 4 cyclic boundary condition · 5 KVL.
This rung hit the ceiling once, and that is recorded rather than worked
around quietly: PyPSA's p_min_pu = -1 is a bound of -rating, an expression
this language cannot yet put in bounds:. It ships as a neg_rating column
instead, and the gap is issue #31
with the verdict primitive. See
the ledger.