PyPSA, the two-stage class#

Rung 14 of PyPSA in one file: n.set_scenarios(...) with n.set_risk_preference(alpha, omega), stated on rungs 1 and 3 in a file of its own — the model's description below says why. Its network is the spine plus the script's own additions.

Rung 14 — two-stage stochastic, with CVaR#

PyPSA	status	note
`Generator-p`, `Link-p`	done	over `scenario`; `Generator-p_nom` is not — chosen once
`Generator-fix-p-`, `-ext-p-`, `Link-fix-p-*`, `Bus-nodal_balance`	done	rungs 1 and 3, over `scenario`
`CVaR-a`, `CVaR-theta`, `CVaR`	done
`CVaR-excess-{s}`	split	PyPSA names a row per scenario; one block over the dimension
`CVaR-def`	done	`1 / (1 - alpha)` is data prep
objective	done	capacity once; operation `(1 - omega)` in expectation, `omega` at the tail

✔ pypsa 1.3.0 solves this rung's network at objective 9267.386666666665, 87 rows.

The network, as PyPSA code

rung_14_stochastic.py

# SPDX-FileCopyrightText: math-spec Contributors
#
# SPDX-License-Identifier: MIT

"""Rung 14: two futures and a risk preference — capacity chosen once, dispatch per scenario, stated by `pypsa_stochastic.yaml`."""

from __future__ import annotations

import spine

MODEL = 'pypsa_stochastic.yaml'


def build():
    """The spine plus an extendable wind unit whose availability and the south's load differ between a calm and a stormy future."""
    n = spine.build()
    n.add('Generator', 'wind14', bus='south', p_nom_extendable=True, p_nom_max=100, marginal_cost=1, capital_cost=20)
    n.add('Load', 'port14', bus='south')
    n.set_scenarios({'calm': 0.6, 'stormy': 0.4})
    n.c.loads.dynamic.p_set[('calm', 'port14')] = [10, 20, 15, 10]
    n.c.loads.dynamic.p_set[('stormy', 'port14')] = [40, 60, 50, 30]
    n.c.generators.dynamic.p_max_pu[('calm', 'wind14')] = [0.9, 0.7, 0.8, 0.6]
    n.c.generators.dynamic.p_max_pu[('stormy', 'wind14')] = [0.3, 0.2, 0.4, 0.1]
    n.set_risk_preference(alpha=0.5, omega=0.3)
    return n

The file#

The two-stage class of a plain n.optimize(): a network with scenarios, stated on rung 1's transport and rung 3's expansion in a file of its own. Everything over a snapshot spans a scenario as well; capacity does not — it is chosen once, before the future is known — and the cost is the expectation over the scenarios' weights. With a risk preference PyPSA adds the CVaR rows: an excess per scenario and the tail's average, blended into the objective. A dimension a run may not have cannot ride on examples/pypsa.yaml, so this class lives here.

Sets#

Symbol	Meaning
\(\mathcal{S}\)	index \(s\) — `scenario` — the futures dispatch is chosen in, each with a weight
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods
\(\mathcal{N}\)	index \(n\) — `bus` — 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 \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N}\) — controllable connections, each from one bus to the buses it delivers to
\(\mathcal{O}\)	index \(o\) — `link_output` with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\enspace \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — 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
\(\pi\)	`scenario_weight` over \(\mathcal{S}\) — PyPSA's `scenario_weightings.weight` — the probability of a future
\(\omega\)	`CVaR_omega` (scalar) — PyPSA's `risk_preference['omega']` — the share of the operating cost priced at the tail rather than in expectation
\(\mathrm{v}\)	`CVaR_inv_tail` (scalar) — PyPSA's `1 / (1 - alpha)` — the tail's own probability, inverted in data prep because a divisor is one factor
\(\mathrm{w}\)	`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
\(\mathrm{ext}\)	`Generator_p_nom_extendable` over \(\mathcal{G}\) — whether the nominal power is a decision
\(\underline{\mathrm{p}}^{\mathrm{nom}}\)	`Generator_p_nom_min` over \(\mathcal{G}\) — least nominal power an extendable generator may be built at
\(\overline{\mathrm{p}}^{\mathrm{nom}}\)	`Generator_p_nom_max` over \(\mathcal{G}\) — most nominal power an extendable generator may be built at
\(\mathrm{c}^{\mathrm{cap}}\)	`Generator_capital_cost` over \(\mathcal{G}\) — cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity in data prep
\(\underline{\mathrm{p}}\)	`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 \(\mathcal{S} \times \mathcal{T} \times \mathcal{G}\) — 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
\(\mathrm{f}^{\mathrm{nom}}\)	`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
\(\mathrm{load}\)	`Load_p_set` over \(\mathcal{S} \times \mathcal{T} \times \mathcal{D}\) — demand

Variables#

Symbol	Meaning
\(p\)	`Generator_p` over \(\mathcal{S} \times \mathcal{T} \times \mathcal{G}\) — `Generator-p` — output of a generator in a snapshot
\(f\)	`Link_p` over \(\mathcal{S} \times \mathcal{T} \times \mathcal{L}\) — `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
\(P\)	`Generator_p_nom_ext` over \(\mathcal{G}\) — `Generator-p_nom` — nominal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime
\(a\)	`CVaR_a` over \(\mathcal{S}\) — `CVaR-a` — how far a scenario's operating cost exceeds the tail's start; nothing where it does not
\(\theta\)	`CVaR_theta` (scalar) — `CVaR-theta` — where the tail starts, the value at risk
\(CVaR\)	`CVaR` (scalar) — `CVaR` — the tail's average cost, what the objective prices at `omega`

Objective#

objective:
  sense: minimize
  description: capacity once, operation in expectation, and a share of it at the tail
  expression: >-
    sum(Generator_p_nom_ext * Generator_capital_cost)
    + (1 - CVaR_omega) * sum(scenario_weight * scenario_opex, over=scenario)
    + CVaR_omega * CVaR

\[\min \sum_{g \in \mathcal{G}} P_{g} \cdot \mathrm{c}^{\mathrm{cap}}_{g} + \left( 1 - \omega \right) \cdot \left( \sum_{s \in \mathcal{S}} \pi_{s} \cdot \left( \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{s,t,g} \cdot \mathrm{c}_{t,g} \cdot \mathrm{w}_{t} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{s,t,l} \cdot \mathrm{c}^{f}_{t,l} \cdot \mathrm{w}_{t} \right) \right) + \omega \cdot CVaR\]

`Generator-fix-p-lower`#

Generator_fix_p_lower

Generator_fix_p_lower:
  description: "`Generator-fix-p-lower` — a generator outputs at least its minimum"
  foreach: [scenario, snapshot, generator]
  where: not Generator_p_nom_extendable
  expression: Generator_p >= Generator_p_min_pu * Generator_p_nom

\[p_{s,t,g} \ge \underline{\mathrm{p}}_{t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{g} \qquad \forall\thinspace s \in \mathcal{S},\enspace t \in \mathcal{T},\enspace g \in \mathcal{G} \thinspace:\thinspace \neg \mathrm{ext}_{g}\]

`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"
  foreach: [scenario, snapshot, generator]
  where: not Generator_p_nom_extendable
  expression: Generator_p <= Generator_p_max_pu * Generator_p_nom

\[p_{s,t,g} \le \overline{\mathrm{p}}_{s,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{g} \qquad \forall\thinspace s \in \mathcal{S},\enspace t \in \mathcal{T},\enspace g \in \mathcal{G} \thinspace:\thinspace \neg \mathrm{ext}_{g}\]

`Generator-ext-p-lower`#

Generator_ext_p_lower

Generator_ext_p_lower:
  description: "`Generator-ext-p-lower` — an extendable generator outputs at least its minimum of the chosen build"
  foreach: [scenario, snapshot, generator]
  where: Generator_p_nom_extendable
  expression: Generator_p >= Generator_p_min_pu * Generator_p_nom_ext

\[p_{s,t,g} \ge \underline{\mathrm{p}}_{t,g} \cdot P_{g} \qquad \forall\thinspace s \in \mathcal{S},\enspace t \in \mathcal{T},\enspace g \in \mathcal{G} \thinspace:\thinspace \mathrm{ext}_{g}\]

`Generator-ext-p-upper`#

Generator_ext_p_upper

Generator_ext_p_upper:
  description: "`Generator-ext-p-upper` — an extendable generator outputs at most what is available of the chosen build"
  foreach: [scenario, snapshot, generator]
  where: Generator_p_nom_extendable
  expression: Generator_p <= Generator_p_max_pu * Generator_p_nom_ext

\[p_{s,t,g} \le \overline{\mathrm{p}}_{s,t,g} \cdot P_{g} \qquad \forall\thinspace s \in \mathcal{S},\enspace t \in \mathcal{T},\enspace g \in \mathcal{G} \thinspace:\thinspace \mathrm{ext}_{g}\]

`Generator-ext-p_nom-lower`#

Generator_ext_p_nom_lower

Generator_ext_p_nom_lower:
  description: "`Generator-ext-p_nom-lower` — the chosen build is at least its floor"
  foreach: [generator]
  where: Generator_p_nom_extendable
  expression: Generator_p_nom_ext >= Generator_p_nom_min

\[P_{g} \ge \underline{\mathrm{p}}^{\mathrm{nom}}_{g} \qquad \forall\thinspace g \in \mathcal{G} \thinspace:\thinspace \mathrm{ext}_{g}\]

`Generator-ext-p_nom-upper`#

Generator_ext_p_nom_upper

Generator_ext_p_nom_upper:
  description: "`Generator-ext-p_nom-upper` — the chosen build is at most its cap; a cap of infinity is no row"
  foreach: [generator]
  where: Generator_p_nom_extendable AND Generator_p_nom_max
  expression: Generator_p_nom_ext <= Generator_p_nom_max

\[P_{g} \le \overline{\mathrm{p}}^{\mathrm{nom}}_{g} \qquad \forall\thinspace g \in \mathcal{G} \thinspace:\thinspace \mathrm{ext}_{g} \wedge \overline{\mathrm{p}}^{\mathrm{nom}}_{g} \text{ is defined}\]

`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"
  foreach: [scenario, snapshot, link]
  expression: Link_p >= Link_p_min_pu * Link_p_nom

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

`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"
  foreach: [scenario, snapshot, link]
  expression: Link_p <= Link_p_max_pu * Link_p_nom

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

`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
  foreach: [scenario, snapshot, bus]
  expression: >-
    sum(Generator_p, by=Generator_bus)
    - sum(Link_p, by=Link_bus0)
    + sum(at(Link_p, by=Link_output_link) * Link_efficiency, by=Link_output_bus)
    == sum(Load_p_set, by=Load_bus)

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

`CVaR-excess-{s}`#

CVaR_excess

CVaR_excess:
  description: "`CVaR-excess-{s}` — a scenario's operating cost beyond the tail's start is its excess; PyPSA names one row per scenario"
  foreach: [scenario]
  expression: CVaR_a - scenario_opex + CVaR_theta >= 0

\[a_{s} - \left( \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{s,t,g} \cdot \mathrm{c}_{t,g} \cdot \mathrm{w}_{t} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{s,t,l} \cdot \mathrm{c}^{f}_{t,l} \cdot \mathrm{w}_{t} \right) + \theta \ge 0 \qquad \forall\thinspace s \in \mathcal{S}\]

`CVaR-def`#

CVaR_def

CVaR_def:
  description: "`CVaR-def` — the tail's average is at least where it starts plus the expected excess over the tail's probability"
  foreach: []
  expression: CVaR_theta + CVaR_inv_tail * sum(scenario_weight * CVaR_a, over=scenario) <= CVaR

\[\theta + \mathrm{v} \cdot \left( \sum_{s \in \mathcal{S}} \pi_{s} \cdot a_{s} \right) \le CVaR\]

Variable domains#

Generator_p

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

Link_p

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

Generator_p_nom_ext

\[P_{g} \in \mathbb{R} \qquad \forall\thinspace g \in \mathcal{G} \thinspace:\thinspace \mathrm{ext}_{g}\]

CVaR_a

\[a_{s} \ge 0 \qquad \forall\thinspace s \in \mathcal{S}\]

CVaR_theta

\[\theta \in \mathbb{R}\]

CVaR

\[CVaR \in \mathbb{R}\]

Regenerate with pixi run python -m tools.gallery.

PyPSA, the two-stage class#

Rung 14 — two-stage stochastic, with CVaR#

The file#

Sets#

Parameters#

Variables#

Objective#

Generator-fix-p-lower#

Generator-fix-p-upper#

Generator-ext-p-lower#

Generator-ext-p-upper#

Generator-ext-p_nom-lower#

Generator-ext-p_nom-upper#

Link-fix-p-lower#

Link-fix-p-upper#

Bus-nodal_balance#

CVaR-excess-{s}#

CVaR-def#