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Rung 32: storage that stands in one period only — two built in the later period open at its first snapshot, and a cyclic one that retires closes on its own last snapshot

One rung of the PyPSA corpus: the file pypsa.yaml projected onto what this network builds, attached to that network, and held to what PyPSA solves it to.

✔ Verified against pypsa 1.3.0 — objective 7230.486698 on both sides; structure ≠ CVaR 0 vs 1 — the file declares the tail's average on every run; PyPSA adds it only under a risk preference, and without one the objective prices it at zero and no row reads it; CVaR-a 0 vs 1 — the file declares each scenario's excess on every run; PyPSA adds it only under a risk preference, and without one no row reads it; CVaR-theta 0 vs 1 — the file declares the tail's start on every run; PyPSA adds it only under a risk preference, and without one no row reads it; size ✔ 92 rows · ≠ 44 vs 47 columns · ✔ 143 nonzeros; duals ✔ 92 rows, 2 negated; model for model: 19 blocks equal, 0 documented splits, 4 recorded deviations.

Rows and columns, PyPSA against specsolve, name for name
row PyPSA specsolve
Bus-nodal_balance 8 8
Generator-fix-p-lower 16 16
Generator-fix-p-upper 16 16
StorageUnit-energy_balance 4 4
StorageUnit-fix-p_dispatch-lower 4 4
StorageUnit-fix-p_dispatch-upper 4 4
StorageUnit-fix-p_store-lower 4 4
StorageUnit-fix-p_store-upper 4 4
StorageUnit-fix-state_of_charge-lower 4 4
StorageUnit-fix-state_of_charge-upper 4 4
Store-energy_balance 8 8
Store-fix-e-lower 8 8
Store-fix-e-upper 8 8
column PyPSA specsolve
CVaR 0 ≠ 1
CVaR-a 0 ≠ 1
CVaR-theta 0 ≠ 1
Generator-p 16 16
StorageUnit-p_dispatch 4 4
StorageUnit-p_store 4 4
StorageUnit-state_of_charge 4 4
Store-e 8 8
Store-p 8 8

The model

The same model, as math

A plain n.optimize(), and its multi-period and stochastic classes, in one file. Every second-stage quantity spans a scenario (a future dispatch is chosen in) and every asset stands in the investment periods its build year and lifetime span. A parameter spans scenario exactly when PyPSA reads it per scenario. Capacity is chosen once, before the future is known, and paid once per active period at its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights, with a share priced at the tail through the CVaR rows, which stand only where that share is positive. A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses to the standard one. A security-constrained run copies each branch flow limit once per outage in an outage set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight, and the outage factors are data prep.

Sets

Symbol Meaning
\(\Xi\) index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight
\(\mathcal{T}\) index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods
\(\mathcal{N}\) index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N},\ \mathrm{StorageUnit\_bus}: \mathcal{S} \to \mathcal{N},\ \mathrm{Store\_bus}: \mathcal{V} \to \mathcal{N}\) — network nodes
\(\mathcal{G}\) index \(g\) — generator with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus
\(\mathcal{D}\) index \(d\) — load with \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus
\(\mathcal{S}\) index \(s\) — storage_unit with \(\mathrm{StorageUnit\_bus}: \mathcal{S} \to \mathcal{N}\) — storage units, dispatch and store behind one bus connection
\(\mathcal{V}\) index \(v\) — store with \(\mathrm{Store\_bus}: \mathcal{V} \to \mathcal{N}\) — pure energy stores, each on one bus
\(\mathcal{Y}\) index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods

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 \(\Xi \times \mathcal{G}\) — nominal power
\(\mathrm{ext}\) Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision
\(\underline{\mathrm{p}}\) Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power
\(\overline{\mathrm{p}}\) Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile
\(\mathrm{c}\) Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output
\(\mathrm{c}^{(2)}\) Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output
\(\mathrm{sgn}\) Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (consistency.py:1187)
\(\mathrm{com}\) Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision
\(\mathrm{load}\) Load_p_set over \(\Xi \times \mathcal{T} \times \mathcal{D}\) — demand
\(\mathrm{sgn}^{\mathrm{load}}\) Load_sign over \(\mathcal{D}\) — the sign a load's demand enters its bus's balance with — PyPSA's sign, -1 unless given, 1 for a load that feeds its bus. PyPSA refuses one that differs by scenario (consistency.py:1187)
\(\mathrm{on}^{\mathrm{load}}\) Load_active over \(\mathcal{D}\) — whether a load stands in the model — PyPSA's active. A load has no build year and no lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (consistency.py:1195)
\(\pi\) scenario_weight over \(\Xi\) — PyPSA's scenario_weightings.weight — the probability of a future
\(\omega\) CVaR_omega (scalar) — PyPSA's risk_preference['omega'] — the share of operating cost priced at the tail rather than in expectation; zero recovers the risk-neutral model
\(\mathrm{w}^{y}\) period_weight_objective over \(\mathcal{Y}\) — PyPSA's investment_period_weightings.objective — what a period's cost weighs
\(\mathrm{on}\) Generator_active over \(\mathcal{T} \times \mathcal{G}\) — whether a generator stands in a snapshot's period — PyPSA's active, from build year and lifetime, data prep
\(\mathrm{on}^{h}\) StorageUnit_active over \(\mathcal{T} \times \mathcal{S}\) — whether a storage unit stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{on}^{e}\) Store_active over \(\mathcal{T} \times \mathcal{V}\) — whether a store stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{w}^{\mathrm{sto}}\) snapshot_weightings_stores over \(\mathcal{T}\) — PyPSA's snapshot_weightings.stores — hours a snapshot stands for in a storage balance
\(\mathrm{h}^{\mathrm{nom}}\) StorageUnit_p_nom over \(\Xi \times \mathcal{S}\) — nominal power
\(\mathrm{ext}^{h}\) StorageUnit_p_nom_extendable over \(\mathcal{S}\) — whether the nominal power is a decision
\(\underline{\mathrm{h}}\) StorageUnit_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — most storing, per unit of nominal power and negated
\(\overline{\mathrm{h}}\) StorageUnit_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — most dispatch, per unit of nominal power
\(\mathrm{T}^{h}\) StorageUnit_max_hours over \(\Xi \times \mathcal{S}\) — energy capacity, as hours of dispatch at nominal power
\(\eta^{-}\) StorageUnit_efficiency_store over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — share of the power drawn from the bus that becomes charge
\(\eta^{+}\) StorageUnit_efficiency_dispatch over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — share of the charge drawn down that reaches the bus
\(\mathrm{sgn}^{h}\) StorageUnit_sign over \(\mathcal{S}\) — the sign net dispatch enters its bus's balance with — PyPSA's sign, 1 unless given. PyPSA refuses one that differs by scenario (consistency.py:1187)
\(\rho\) StorageUnit_retention over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — share of charge kept over a snapshot — PyPSA's (1 - standing_loss) ** elapsed hours, data prep
\(\mathrm{inflow}\) StorageUnit_inflow over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — energy arriving per hour, a river into a reservoir
\(\mathrm{soc}^{0}\) StorageUnit_state_of_charge_initial over \(\Xi \times \mathcal{S}\) — charge held before the first snapshot
\(\mathrm{cyc}\) StorageUnit_cyclic_state_of_charge over \(\Xi \times \mathcal{S}\) — whether the horizon closes on itself instead of opening on the initial charge
\(\mathrm{cyc}^{y}\) StorageUnit_cyclic_state_of_charge_per_period over \(\Xi \times \mathcal{S}\) — whether each investment period closes on itself instead of carrying its charge on to the next; it overrides cyclic_state_of_charge and state_of_charge_initial_per_period. PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise
\(\mathrm{reset}\) StorageUnit_state_of_charge_initial_per_period over \(\Xi \times \mathcal{S}\) — whether each investment period opens on the initial charge instead of carrying the previous period's; PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise
\(\mathrm{open}\) StorageUnit_opens_late over \(\mathcal{T} \times \mathcal{S}\) — whether a snapshot is the first a storage unit stands in, where that is not the first of the horizon — PyPSA's active.cumsum() == 1 over the snapshots it stands in, past the first snapshot, data prep; false in a run where every unit stands throughout
\(\mathrm{idle}\) StorageUnit_inactive_snapshots over \(\mathcal{S}\) — how many snapshots a storage unit does not stand in — PyPSA's (~active).sum(), data prep. A cyclic unit reaches back this many snapshots further, so it closes on the last snapshot it stands in
\(\mathrm{c}^{h}\) StorageUnit_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — cost of one unit of dispatch
\(\mathrm{c}^{h,(2)}\) StorageUnit_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — cost of the square of one unit of dispatch; storing is not charged
\(\mathrm{c}^{\mathrm{soc}}\) StorageUnit_marginal_cost_storage over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — cost of one unit of charge held over one snapshot
\(\mathrm{e}^{\mathrm{nom}}\) Store_e_nom over \(\Xi \times \mathcal{V}\) — nominal energy capacity
\(\mathrm{ext}^{e}\) Store_e_nom_extendable over \(\mathcal{V}\) — whether the nominal energy capacity is a decision
\(\underline{\mathrm{e}}\) Store_e_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — least energy held, per unit of nominal capacity — negative for a store that may go short
\(\overline{\mathrm{e}}\) Store_e_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — most energy held, per unit of nominal capacity
\(\mathrm{sgn}^{q}\) Store_sign over \(\mathcal{V}\) — the sign the power a store delivers enters its bus's balance with — PyPSA's sign, 1 unless given. PyPSA refuses one that differs by scenario (consistency.py:1187)
\(\rho^{e}\) Store_retention over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — share of energy kept over a snapshot — PyPSA's (1 - standing_loss) ** elapsed hours, data prep
\(\mathrm{e}^{0}\) Store_e_initial over \(\Xi \times \mathcal{V}\) — energy held before the first snapshot
\(\mathrm{cyc}^{e}\) Store_e_cyclic over \(\Xi \times \mathcal{V}\) — whether the horizon closes on itself instead of opening on the initial energy
\(\mathrm{cyc}^{e,y}\) Store_e_cyclic_per_period over \(\Xi \times \mathcal{V}\) — whether each investment period closes on itself instead of carrying its energy on to the next; it overrides e_cyclic and e_initial_per_period. PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise
\(\mathrm{reset}^{e}\) Store_e_initial_per_period over \(\Xi \times \mathcal{V}\) — whether each investment period opens on the initial energy instead of carrying the previous period's; PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise
\(\mathrm{open}^{e}\) Store_opens_late over \(\mathcal{T} \times \mathcal{V}\) — whether a snapshot is the first a store stands in, where that is not the first of the horizon — PyPSA's active.cumsum() == 1 over the snapshots it stands in, past the first snapshot, data prep; false in a run where every store stands throughout
\(\mathrm{idle}^{e}\) Store_inactive_snapshots over \(\mathcal{V}\) — how many snapshots a store does not stand in — PyPSA's (~active).sum(), data prep. A cyclic store reaches back this many snapshots further, so it closes on the last snapshot it stands in
\(\mathrm{c}^{q}\) Store_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — cost of one unit of power delivered
\(\mathrm{c}^{q,(2)}\) Store_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — cost of the square of the net power delivered, so charging costs as much as delivering
\(\mathrm{c}^{e}\) Store_marginal_cost_storage over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — cost of one unit of energy held over one snapshot

Variables

Symbol Meaning
\(p\) Generator_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-p — output of a generator in a snapshot
\(h^{+}\) StorageUnit_p_dispatch over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — StorageUnit-p_dispatch — power delivered to the bus
\(h^{-}\) StorageUnit_p_store over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — StorageUnit-p_store — power drawn from the bus into charge
\(\mathit{soc}\) StorageUnit_state_of_charge over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — StorageUnit-state_of_charge — energy held at the end of a snapshot
\(e\) Store_e over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — Store-e — energy held at the end of a snapshot
\(q\) Store_p over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — Store-p — power delivered to the bus; charging is negative
\(a\) CVaR_a over \(\Xi\) — 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

Definitions

Symbol Meaning
\(\mathit{StorageUnit\_charge\_carried\_in}\) StorageUnit_charge_carried_in over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — the charge a unit opens a snapshot with — at the first snapshot it stands in, its last such snapshot's less standing loss where it is cyclic and the given initial charge, which no standing loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A unit built in a later period opens in that period, and a cyclic one that retires closes on its own last snapshot. Per period, the same holds with each investment period as the horizon
\(\mathit{Store\_energy\_carried\_in}\) Store_energy_carried_in over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — the energy a store opens a snapshot with — at the first snapshot it stands in, its last such snapshot's less standing loss where it is cyclic and the given initial energy, which no standing loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A store built in a later period opens in that period, and a cyclic one that retires closes on its own last snapshot. Per period, the same holds with each investment period as the horizon
\(\mathit{total\_cost}\) total_cost (scalar) — what the system costs — capacity once per active period at its expected cost over the scenarios, operation in expectation over the scenarios, and a share of it at the tail
\(\mathit{Bus\_injection}\) Bus_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) — what every component puts into a bus, less what it takes out of it; PyPSA writes each term into the balance, and a load on its right-hand side
\(\mathit{risk\_weighted\_opex}\) risk_weighted_opex (scalar)
\(\mathit{Generator\_injection}\) Generator_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathrm{Load\_injection}\) Load_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathit{StorageUnit\_injection}\) StorageUnit_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathit{Store\_injection}\) Store_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathit{scenario\_opex}\) scenario_opex over \(\Xi\) — what a future costs to run — every operating term, weighted by the snapshot's hours and its period, before the scenario's own weight; a start and a stop cost what they cost, unweighted, as PyPSA adds them (optimize.py:414-429)
\(\mathrm{Load\_demand}\) Load_demand over \(\Xi \times \mathcal{T} \times \mathcal{D}\) — what a load draws from its bus's balance — its demand times its sign where it is active, nothing where it is not, since PyPSA drops an inactive load from the balance (constraints.py:1537-1538)
\(\mathit{Generator\_opex}\) Generator_opex over \(\Xi\)
\(\mathit{StorageUnit\_opex}\) StorageUnit_opex over \(\Xi\)
\(\mathit{Store\_opex}\) Store_opex over \(\Xi\)

\(t \ominus k\) denotes cyclic translation: index \(t-k\) taken modulo the size of the dimension (roll). Plain \(t-k\) (shift) has no wraparound — terms translated past the edge are simply absent.

\(t \ominus^{\mathrm{relation}(t)} k\) denotes a translation counted inside the group a relation puts \(t\) in (shift(by=relation)), so a term never crosses out of its own group.

\(\mathrm{pos}(t)\) denotes where index \(t\) sits along its dimension's own order — the order shift steps along, not the order labels sort in — counted from \(0\). The index itself stays the coordinate, so \(t\) compares against labels and \(\mathrm{pos}(t)\) against positions.

\(\mathrm{pos}_{\mathrm{relation}(t)}(t)\) counts within the group a relation puts \(t\) in: the subscript names the map, \(\mathcal{T}_{\mathrm{relation}(t)}\) is the group it lands in, and that group has a first position of its own.

Objective

\[ \min \mathit{total\_cost} \]

Subject to

Generator_fix_p_lower

\[ p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \]

Generator_fix_p_upper

\[ p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \]

StorageUnit_fix_p_dispatch_lower

\[ h^{+}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s} \]

StorageUnit_fix_p_dispatch_upper

\[ h^{+}_{\xi,t,s} \le \overline{\mathrm{h}}_{\xi,t,s} \cdot \mathrm{h}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s} \]

StorageUnit_fix_p_store_lower

\[ h^{-}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s} \]

StorageUnit_fix_p_store_upper

\[ h^{-}_{\xi,t,s} \le -\underline{\mathrm{h}}_{\xi,t,s} \cdot \mathrm{h}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s} \]

StorageUnit_fix_state_of_charge_lower

\[ \mathit{soc}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s} \]

StorageUnit_fix_state_of_charge_upper

\[ \mathit{soc}_{\xi,t,s} \le \mathrm{T}^{h}_{\xi,s} \cdot \mathrm{h}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s} \]

StorageUnit_energy_balance

\[ \mathit{soc}_{\xi,t,s} = \mathit{StorageUnit\_charge\_carried\_in}_{\xi,t,s} + \eta^{-}_{\xi,t,s} \cdot h^{-}_{\xi,t,s} \cdot \mathrm{w}^{\mathrm{sto}}_{t} - \frac{h^{+}_{\xi,t,s} \cdot \mathrm{w}^{\mathrm{sto}}_{t}}{\eta^{+}_{\xi,t,s}} + \mathrm{inflow}_{\xi,t,s} \cdot \mathrm{w}^{\mathrm{sto}}_{t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s} \]

Store_fix_e_lower

\[ e_{\xi,t,v} \ge \underline{\mathrm{e}}_{\xi,t,v} \cdot \mathrm{e}^{\mathrm{nom}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \neg \mathrm{ext}^{e}_{v} \wedge \mathrm{on}^{e}_{t,v} \]

Store_fix_e_upper

\[ e_{\xi,t,v} \le \overline{\mathrm{e}}_{\xi,t,v} \cdot \mathrm{e}^{\mathrm{nom}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \neg \mathrm{ext}^{e}_{v} \wedge \mathrm{on}^{e}_{t,v} \]

Store_energy_balance

\[ e_{\xi,t,v} = \mathit{Store\_energy\_carried\_in}_{\xi,t,v} - q_{\xi,t,v} \cdot \mathrm{w}^{\mathrm{sto}}_{t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{on}^{e}_{t,v} \]

Bus_nodal_balance

\[ \mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

Definitions

StorageUnit_charge_carried_in

\[ \mathit{StorageUnit\_charge\_carried\_in}_{\xi,t,s} = \begin{cases} \rho_{\xi,t,s} \cdot \mathit{soc}_{\xi,\left( t \ominus \mathrm{idle} \right) \ominus 1,s} & \text{if } \mathrm{cyc}_{\xi,s} \wedge \neg \mathrm{cyc}^{y}_{\xi,s} \wedge \neg \mathrm{reset}_{\xi,s} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}_{t,s} \right) \\ \mathrm{soc}^{0}_{\xi,s} & \text{if } \neg \mathrm{cyc}_{\xi,s} \wedge \neg \mathrm{cyc}^{y}_{\xi,s} \wedge \neg \mathrm{reset}_{\xi,s} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}_{t,s} \right) \\ \rho_{\xi,t,s} \cdot \mathit{soc}_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} 1,s} & \text{if } \mathrm{cyc}^{y}_{\xi,s} \\ \mathrm{soc}^{0}_{\xi,s} & \text{if } \mathrm{reset}_{\xi,s} \wedge \neg \mathrm{cyc}^{y}_{\xi,s} \wedge \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) = 0 \\ \rho_{\xi,t,s} \cdot \mathit{soc}_{\xi,t - 1,s} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \]

Store_energy_carried_in

\[ \mathit{Store\_energy\_carried\_in}_{\xi,t,v} = \begin{cases} \rho^{e}_{\xi,t,v} \cdot e_{\xi,\left( t \ominus \mathrm{idle}^{e} \right) \ominus 1,v} & \text{if } \mathrm{cyc}^{e}_{\xi,v} \wedge \neg \mathrm{cyc}^{e,y}_{\xi,v} \wedge \neg \mathrm{reset}^{e}_{\xi,v} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}^{e}_{t,v} \right) \\ \mathrm{e}^{0}_{\xi,v} & \text{if } \neg \mathrm{cyc}^{e}_{\xi,v} \wedge \neg \mathrm{cyc}^{e,y}_{\xi,v} \wedge \neg \mathrm{reset}^{e}_{\xi,v} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}^{e}_{t,v} \right) \\ \rho^{e}_{\xi,t,v} \cdot e_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} 1,v} & \text{if } \mathrm{cyc}^{e,y}_{\xi,v} \\ \mathrm{e}^{0}_{\xi,v} & \text{if } \mathrm{reset}^{e}_{\xi,v} \wedge \neg \mathrm{cyc}^{e,y}_{\xi,v} \wedge \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) = 0 \\ \rho^{e}_{\xi,t,v} \cdot e_{\xi,t - 1,v} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \]

total_cost

\[ \mathit{total\_cost} = \mathit{risk\_weighted\_opex} \]

Bus_injection

\[ \mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{\xi,t,n} + \mathrm{Load\_injection}_{\xi,t,n} + \mathit{StorageUnit\_injection}_{\xi,t,n} + \mathit{Store\_injection}_{\xi,t,n} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

risk_weighted_opex

\[ \mathit{risk\_weighted\_opex} = \left( 1 - \omega \right) \cdot \left( \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi} \right) + \omega \cdot CVaR \]

Generator_injection

\[ \mathit{Generator\_injection}_{\xi,t,n} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

Load_injection

\[ \mathrm{Load\_injection}_{\xi,t,n} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

StorageUnit_injection

\[ \mathit{StorageUnit\_injection}_{\xi,t,n} = \sum_{s \in \mathcal{S} \,:\, \mathrm{StorageUnit\_bus}(s) = n} \mathrm{sgn}^{h}_{s} \cdot \left( h^{+}_{\xi,t,s} - h^{-}_{\xi,t,s} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

Store_injection

\[ \mathit{Store\_injection}_{\xi,t,n} = \sum_{v \in \mathcal{V} \,:\, \mathrm{Store\_bus}(v) = n} \mathrm{sgn}^{q}_{v} \cdot q_{\xi,t,v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

scenario_opex

\[ \mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{StorageUnit\_opex}_{\xi} + \mathit{Store\_opex}_{\xi} \qquad \forall\, \xi \in \Xi \]

Load_demand

\[ \mathrm{Load\_demand}_{\xi,t,d} = \begin{cases} \mathrm{sgn}^{\mathrm{load}}_{d} \cdot \mathrm{load}_{\xi,t,d} & \text{if } \mathrm{on}^{\mathrm{load}}_{d} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ d \in \mathcal{D} \]

Generator_opex

\[ \mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi \]

StorageUnit_opex

\[ \mathit{StorageUnit\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{s \in \mathcal{S}} h^{+}_{\xi,t,s} \cdot \mathrm{c}^{h}_{\xi,t,s} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{s \in \mathcal{S}} h^{+}_{\xi,t,s} \cdot h^{+}_{\xi,t,s} \cdot \mathrm{c}^{h,(2)}_{\xi,t,s} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{s \in \mathcal{S}} \mathit{soc}_{\xi,t,s} \cdot \mathrm{c}^{\mathrm{soc}}_{\xi,t,s} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi \]

Store_opex

\[ \mathit{Store\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{v \in \mathcal{V}} q_{\xi,t,v} \cdot \mathrm{c}^{q}_{\xi,t,v} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{v \in \mathcal{V}} q_{\xi,t,v} \cdot q_{\xi,t,v} \cdot \mathrm{c}^{q,(2)}_{\xi,t,v} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{v \in \mathcal{V}} e_{\xi,t,v} \cdot \mathrm{c}^{e}_{\xi,t,v} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi \]

Variable domains

Generator_p

\[ p_{\xi,t,g} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{on}_{t,g} \]

StorageUnit_p_dispatch

\[ h^{+}_{\xi,t,s} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s} \]

StorageUnit_p_store

\[ h^{-}_{\xi,t,s} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s} \]

StorageUnit_state_of_charge

\[ \mathit{soc}_{\xi,t,s} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s} \]

Store_e

\[ e_{\xi,t,v} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{on}^{e}_{t,v} \]

Store_p

\[ q_{\xi,t,v} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{on}^{e}_{t,v} \]

CVaR_a

\[ a_{\xi} \ge 0 \qquad \forall\, \xi \in \Xi \]

CVaR_theta

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

CVaR

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

The spec, differential/pypsa/rungs/rung_32_storage_later_period.yaml — the file projected onto what this rung builds:

description: A plain `n.optimize()`, and its multi-period and stochastic classes, in one file. Every second-stage
  quantity spans a `scenario` (a future dispatch is chosen in) and every asset stands in the investment
  `period`s its build year and lifetime span. A parameter spans `scenario` exactly when PyPSA reads it
  per scenario. Capacity is chosen once, before the future is known, and paid once per active period at
  its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights,
  with a share priced at the tail through the CVaR rows, which stand only where that share is positive.
  A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses
  to the standard one. A security-constrained run copies each branch flow limit once per outage in an
  `outage` set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight,
  and the outage factors are data prep.
dimensions:
  scenario: {description: 'the futures dispatch is chosen in, each with a weight'}
  snapshot: {description: dispatch periods, dtype: datetime}
  bus: {description: network nodes}
  generator: {description: 'generating units, each on one bus'}
  load: {description: 'demands, each on one bus'}
  storage_unit: {description: 'storage units, dispatch and store behind one bus connection'}
  store: {description: 'pure energy stores, each on one bus'}
  period: {description: investment periods — PyPSA's `investment_periods`, dtype: int}
relations:
  snapshot_period: {description: the investment period a snapshot falls in, key: snapshot, values: period}
  Generator_bus: {description: the bus a generator sits on, key: generator, values: bus}
  Load_bus: {description: the bus a load sits on, key: load, values: bus}
  StorageUnit_bus: {description: the bus a storage unit sits on, key: storage_unit, values: bus}
  Store_bus: {description: the bus a store sits on, key: store, values: bus}
parameters:
  snapshot_weightings_objective:
    description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
    dims: [snapshot]
  Generator_p_nom:
    description: nominal power
    dims: [scenario, generator]
  Generator_p_nom_extendable:
    description: whether the nominal power is a decision
    dims: [generator]
    dtype: bool
  Generator_p_min_pu:
    description: least output, per unit of nominal power
    dims: [scenario, snapshot, generator]
  Generator_p_max_pu:
    description: most output, per unit of nominal power — an availability profile
    dims: [scenario, snapshot, generator]
  Generator_marginal_cost:
    description: cost of one unit of output
    dims: [scenario, snapshot, generator]
  Generator_marginal_cost_quadratic:
    description: cost of the square of one unit of output
    dims: [scenario, snapshot, generator]
  Generator_sign:
    description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
      for a unit that draws power. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [generator]
  Generator_committable:
    description: whether output is gated by an on/off status decision
    dims: [generator]
    dtype: bool
  Load_p_set:
    description: demand
    dims: [scenario, snapshot, load]
  Load_sign:
    description: the sign a load's demand enters its bus's balance with — PyPSA's `sign`, `-1` unless
      given, `1` for a load that feeds its bus. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [load]
  Load_active:
    description: whether a load stands in the model — PyPSA's `active`. A load has no build year and no
      lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (`consistency.py:1195`)
    dims: [load]
    dtype: bool
  scenario_weight:
    description: PyPSA's `scenario_weightings.weight` — the probability of a future
    dims: [scenario]
  CVaR_omega:
    description: PyPSA's `risk_preference['omega']` — the share of operating cost priced at the tail rather
      than in expectation; zero recovers the risk-neutral model
    dims: []
  period_weight_objective:
    description: PyPSA's `investment_period_weightings.objective` — what a period's cost weighs
    dims: [period]
  Generator_active:
    description: whether a generator stands in a snapshot's period — PyPSA's `active`, from build year
      and lifetime, data prep
    dims: [snapshot, generator]
    dtype: bool
  StorageUnit_active:
    description: whether a storage unit stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, storage_unit]
    dtype: bool
  Store_active:
    description: whether a store stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, store]
    dtype: bool
  snapshot_weightings_stores:
    description: PyPSA's `snapshot_weightings.stores` — hours a snapshot stands for in a storage balance
    dims: [snapshot]
  StorageUnit_p_nom:
    description: nominal power
    dims: [scenario, storage_unit]
  StorageUnit_p_nom_extendable:
    description: whether the nominal power is a decision
    dims: [storage_unit]
    dtype: bool
  StorageUnit_p_min_pu:
    description: most storing, per unit of nominal power and negated
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_p_max_pu:
    description: most dispatch, per unit of nominal power
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_max_hours:
    description: energy capacity, as hours of dispatch at nominal power
    dims: [scenario, storage_unit]
  StorageUnit_efficiency_store:
    description: share of the power drawn from the bus that becomes charge
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_efficiency_dispatch:
    description: share of the charge drawn down that reaches the bus
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_sign:
    description: the sign net dispatch enters its bus's balance with — PyPSA's `sign`, `1` unless given.
      PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [storage_unit]
  StorageUnit_retention:
    description: share of charge kept over a snapshot — PyPSA's `(1 - standing_loss) ** elapsed hours`,
      data prep
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_inflow:
    description: energy arriving per hour, a river into a reservoir
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_state_of_charge_initial:
    description: charge held before the first snapshot
    dims: [scenario, storage_unit]
  StorageUnit_cyclic_state_of_charge:
    description: whether the horizon closes on itself instead of opening on the initial charge
    dims: [scenario, storage_unit]
    dtype: bool
  StorageUnit_cyclic_state_of_charge_per_period:
    description: whether each investment period closes on itself instead of carrying its charge on to
      the next; it overrides `cyclic_state_of_charge` and `state_of_charge_initial_per_period`. PyPSA
      reads it only under `multi_investment_periods`, so data prep feeds false otherwise
    dims: [scenario, storage_unit]
    dtype: bool
  StorageUnit_state_of_charge_initial_per_period:
    description: whether each investment period opens on the initial charge instead of carrying the previous
      period's; PyPSA reads it only under `multi_investment_periods`, so data prep feeds false otherwise
    dims: [scenario, storage_unit]
    dtype: bool
  StorageUnit_opens_late:
    description: whether a snapshot is the first a storage unit stands in, where that is not the first
      of the horizon — PyPSA's `active.cumsum() == 1` over the snapshots it stands in, past the first
      snapshot, data prep; false in a run where every unit stands throughout
    dims: [snapshot, storage_unit]
    dtype: bool
  StorageUnit_inactive_snapshots:
    description: how many snapshots a storage unit does not stand in — PyPSA's `(~active).sum()`, data
      prep. A cyclic unit reaches back this many snapshots further, so it closes on the last snapshot
      it stands in
    dims: [storage_unit]
    dtype: int
  StorageUnit_marginal_cost:
    description: cost of one unit of dispatch
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_marginal_cost_quadratic:
    description: cost of the square of one unit of dispatch; storing is not charged
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_marginal_cost_storage:
    description: cost of one unit of charge held over one snapshot
    dims: [scenario, snapshot, storage_unit]
  Store_e_nom:
    description: nominal energy capacity
    dims: [scenario, store]
  Store_e_nom_extendable:
    description: whether the nominal energy capacity is a decision
    dims: [store]
    dtype: bool
  Store_e_min_pu:
    description: least energy held, per unit of nominal capacity — negative for a store that may go short
    dims: [scenario, snapshot, store]
  Store_e_max_pu:
    description: most energy held, per unit of nominal capacity
    dims: [scenario, snapshot, store]
  Store_sign:
    description: the sign the power a store delivers enters its bus's balance with — PyPSA's `sign`, `1`
      unless given. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [store]
  Store_retention:
    description: share of energy kept over a snapshot — PyPSA's `(1 - standing_loss) ** elapsed hours`,
      data prep
    dims: [scenario, snapshot, store]
  Store_e_initial:
    description: energy held before the first snapshot
    dims: [scenario, store]
  Store_e_cyclic:
    description: whether the horizon closes on itself instead of opening on the initial energy
    dims: [scenario, store]
    dtype: bool
  Store_e_cyclic_per_period:
    description: whether each investment period closes on itself instead of carrying its energy on to
      the next; it overrides `e_cyclic` and `e_initial_per_period`. PyPSA reads it only under `multi_investment_periods`,
      so data prep feeds false otherwise
    dims: [scenario, store]
    dtype: bool
  Store_e_initial_per_period:
    description: whether each investment period opens on the initial energy instead of carrying the previous
      period's; PyPSA reads it only under `multi_investment_periods`, so data prep feeds false otherwise
    dims: [scenario, store]
    dtype: bool
  Store_opens_late:
    description: whether a snapshot is the first a store stands in, where that is not the first of the
      horizon — PyPSA's `active.cumsum() == 1` over the snapshots it stands in, past the first snapshot,
      data prep; false in a run where every store stands throughout
    dims: [snapshot, store]
    dtype: bool
  Store_inactive_snapshots:
    description: how many snapshots a store does not stand in — PyPSA's `(~active).sum()`, data prep.
      A cyclic store reaches back this many snapshots further, so it closes on the last snapshot it stands
      in
    dims: [store]
    dtype: int
  Store_marginal_cost:
    description: cost of one unit of power delivered
    dims: [scenario, snapshot, store]
  Store_marginal_cost_quadratic:
    description: cost of the square of the net power delivered, so charging costs as much as delivering
    dims: [scenario, snapshot, store]
  Store_marginal_cost_storage:
    description: cost of one unit of energy held over one snapshot
    dims: [scenario, snapshot, store]
variables:
  Generator_p:
    description: '`Generator-p` — output of a generator in a snapshot'
    dims: [scenario, snapshot, generator]
    where: Generator_active
  StorageUnit_p_dispatch:
    description: '`StorageUnit-p_dispatch` — power delivered to the bus'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
  StorageUnit_p_store:
    description: '`StorageUnit-p_store` — power drawn from the bus into charge'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
  StorageUnit_state_of_charge:
    description: '`StorageUnit-state_of_charge` — energy held at the end of a snapshot'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
  Store_e:
    description: '`Store-e` — energy held at the end of a snapshot'
    dims: [scenario, snapshot, store]
    where: Store_active
  Store_p:
    description: '`Store-p` — power delivered to the bus; charging is negative'
    dims: [scenario, snapshot, store]
    where: Store_active
  CVaR_a:
    description: '`CVaR-a` — how far a scenario''s operating cost exceeds the tail''s start; nothing where
      it does not'
    dims: [scenario]
    bounds: {lower: 0}
  CVaR_theta:
    description: '`CVaR-theta` — where the tail starts, the value at risk'
    dims: []
  CVaR:
    description: '`CVaR` — the tail''s average cost, what the objective prices at `omega`'
    dims: []
constraints:
  Generator_fix_p_lower:
    description: '`Generator-fix-p-lower` — a fixed generator outputs at least its minimum'
    dims: [scenario, snapshot, generator]
    where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
    expression: Generator_p >= Generator_p_min_pu * Generator_p_nom
  Generator_fix_p_upper:
    description: '`Generator-fix-p-upper` — a fixed generator outputs at most what is available'
    dims: [scenario, snapshot, generator]
    where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
    expression: Generator_p <= Generator_p_max_pu * Generator_p_nom
  StorageUnit_fix_p_dispatch_lower:
    description: '`StorageUnit-fix-p_dispatch-lower` — dispatch is non-negative'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_dispatch >= 0
  StorageUnit_fix_p_dispatch_upper:
    description: '`StorageUnit-fix-p_dispatch-upper` — a fixed unit dispatches at most its nominal power'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_dispatch <= StorageUnit_p_max_pu * StorageUnit_p_nom
  StorageUnit_fix_p_store_lower:
    description: '`StorageUnit-fix-p_store-lower` — storing is non-negative'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_store >= 0
  StorageUnit_fix_p_store_upper:
    description: '`StorageUnit-fix-p_store-upper` — a fixed unit stores at most its nominal power, the
      minimum-per-unit column carrying that cap negated'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_store <= -StorageUnit_p_min_pu * StorageUnit_p_nom
  StorageUnit_fix_state_of_charge_lower:
    description: '`StorageUnit-fix-state_of_charge-lower` — charge is non-negative'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_state_of_charge >= 0
  StorageUnit_fix_state_of_charge_upper:
    description: '`StorageUnit-fix-state_of_charge-upper` — a fixed unit holds at most its hours at nominal
      power'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_state_of_charge <= StorageUnit_max_hours * StorageUnit_p_nom
  StorageUnit_energy_balance:
    description: '`StorageUnit-energy_balance` — the charge carried in, plus what is stored after its
      efficiency, less what dispatch draws down before its own, plus inflow not spilled'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
    expression: StorageUnit_state_of_charge == ((StorageUnit_charge_carried_in + ((StorageUnit_efficiency_store
      * StorageUnit_p_store) * snapshot_weightings_stores)) - ((StorageUnit_p_dispatch * snapshot_weightings_stores)
      / StorageUnit_efficiency_dispatch)) + (StorageUnit_inflow * snapshot_weightings_stores)
  Store_fix_e_lower:
    description: '`Store-fix-e-lower` — a fixed store holds at least its floor'
    dims: [scenario, snapshot, store]
    where: not Store_e_nom_extendable AND Store_active
    expression: Store_e >= Store_e_min_pu * Store_e_nom
  Store_fix_e_upper:
    description: '`Store-fix-e-upper` — a fixed store holds at most its nominal capacity'
    dims: [scenario, snapshot, store]
    where: not Store_e_nom_extendable AND Store_active
    expression: Store_e <= Store_e_max_pu * Store_e_nom
  Store_energy_balance:
    description: '`Store-energy_balance` — the energy carried in, less what is delivered to the bus'
    dims: [scenario, snapshot, store]
    where: Store_active
    expression: Store_e == Store_energy_carried_in - Store_p * snapshot_weightings_stores
  Bus_nodal_balance:
    description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
      less what the links take away, plus what arrives over them after losses and any delay at every port
      they deliver to, each process port drawing or delivering at its own rate and each passive branch
      carrying its flow, meets the load there, less half of every incident line''s and transformer''s
      loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
      and load term enters with its component''s `sign` (`constraints.py:1428-1429`, `:1538`), and an
      inactive load not at all. A bus nothing is attached to has no row; PyPSA refuses one that carries
      load, and this file does not yet.'
    dims: [scenario, snapshot, bus]
    expression: Bus_injection == 0
expressions:
  StorageUnit_charge_carried_in:
    description: the charge a unit opens a snapshot with — at the first snapshot it stands in, its last
      such snapshot's less standing loss where it is cyclic and the given initial charge, which no standing
      loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A unit
      built in a later period opens in that period, and a cyclic one that retires closes on its own last
      snapshot. Per period, the same holds with each investment period as the horizon
    dims: [scenario, snapshot, storage_unit]
    cases:
      cyclic: {when: StorageUnit_cyclic_state_of_charge AND NOT StorageUnit_cyclic_state_of_charge_per_period
          AND NOT StorageUnit_state_of_charge_initial_per_period AND (position(snapshot) == 0 OR StorageUnit_opens_late),
        expression: 'StorageUnit_retention * shift(shift(StorageUnit_state_of_charge, along=snapshot,
          offset=1, edge=''wrap''), along=snapshot, offset=StorageUnit_inactive_snapshots, edge=''wrap'')'}
      opening: {when: NOT StorageUnit_cyclic_state_of_charge AND NOT StorageUnit_cyclic_state_of_charge_per_period
          AND NOT StorageUnit_state_of_charge_initial_per_period AND (position(snapshot) == 0 OR StorageUnit_opens_late),
        expression: StorageUnit_state_of_charge_initial}
      period_cyclic: {when: StorageUnit_cyclic_state_of_charge_per_period, expression: 'StorageUnit_retention
          * shift(StorageUnit_state_of_charge, along=snapshot, offset=1, edge=''wrap'', by=snapshot_period,
          within=period)'}
      period_opening: {when: 'StorageUnit_state_of_charge_initial_per_period AND NOT StorageUnit_cyclic_state_of_charge_per_period
          AND position(snapshot, by=snapshot_period, within=period) == 0', expression: StorageUnit_state_of_charge_initial}
    otherwise: StorageUnit_retention * shift(StorageUnit_state_of_charge, along=snapshot, offset=1)
  Store_energy_carried_in:
    description: the energy a store opens a snapshot with — at the first snapshot it stands in, its last
      such snapshot's less standing loss where it is cyclic and the given initial energy, which no standing
      loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A store
      built in a later period opens in that period, and a cyclic one that retires closes on its own last
      snapshot. Per period, the same holds with each investment period as the horizon
    dims: [scenario, snapshot, store]
    cases:
      cyclic: {when: Store_e_cyclic AND NOT Store_e_cyclic_per_period AND NOT Store_e_initial_per_period
          AND (position(snapshot) == 0 OR Store_opens_late), expression: 'Store_retention * shift(shift(Store_e,
          along=snapshot, offset=1, edge=''wrap''), along=snapshot, offset=Store_inactive_snapshots, edge=''wrap'')'}
      opening: {when: NOT Store_e_cyclic AND NOT Store_e_cyclic_per_period AND NOT Store_e_initial_per_period
          AND (position(snapshot) == 0 OR Store_opens_late), expression: Store_e_initial}
      period_cyclic: {when: Store_e_cyclic_per_period, expression: 'Store_retention * shift(Store_e, along=snapshot,
          offset=1, edge=''wrap'', by=snapshot_period, within=period)'}
      period_opening: {when: 'Store_e_initial_per_period AND NOT Store_e_cyclic_per_period AND position(snapshot,
          by=snapshot_period, within=period) == 0', expression: Store_e_initial}
    otherwise: Store_retention * shift(Store_e, along=snapshot, offset=1)
  total_cost:
    dims: []
    expression: risk_weighted_opex
    description: what the system costs — capacity once per active period at its expected cost over the
      scenarios, operation in expectation over the scenarios, and a share of it at the tail
  Bus_injection:
    dims: [scenario, snapshot, bus]
    expression: ((Generator_injection + Load_injection) + StorageUnit_injection) + Store_injection
    description: what every component puts into a bus, less what it takes out of it; PyPSA writes each
      term into the balance, and a load on its right-hand side
  risk_weighted_opex: {expression: '(1 - CVaR_omega) * sum(scenario_weight * scenario_opex, over=scenario)
      + CVaR_omega * CVaR'}
  Generator_injection: {expression: 'sum(Generator_sign * Generator_p, by=Generator_bus, over=generator,
      into=bus)'}
  Load_injection: {expression: 'sum(Load_demand, by=Load_bus, over=load, into=bus)'}
  StorageUnit_injection: {expression: 'sum(StorageUnit_sign * (StorageUnit_p_dispatch - StorageUnit_p_store),
      by=StorageUnit_bus, over=storage_unit, into=bus)'}
  Store_injection: {expression: 'sum(Store_sign * Store_p, by=Store_bus, over=store, into=bus)'}
  scenario_opex:
    dims: [scenario]
    expression: (Generator_opex + StorageUnit_opex) + Store_opex
    description: what a future costs to run — every operating term, weighted by the snapshot's hours and
      its period, before the scenario's own weight; a start and a stop cost what they cost, unweighted,
      as PyPSA adds them (`optimize.py:414-429`)
  Load_demand:
    description: what a load draws from its bus's balance — its demand times its sign where it is active,
      nothing where it is not, since PyPSA drops an inactive load from the balance (`constraints.py:1537-1538`)
    dims: [scenario, snapshot, load]
    cases:
      active: {when: Load_active, expression: Load_sign * Load_p_set}
    otherwise: 0
  Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=generator),
      over=snapshot) + sum(sum((((Generator_p * Generator_p) * Generator_marginal_cost_quadratic) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=generator),
      over=snapshot)'}
  StorageUnit_opex: {expression: '(sum(sum(((StorageUnit_p_dispatch * StorageUnit_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=storage_unit),
      over=snapshot) + sum(sum((((StorageUnit_p_dispatch * StorageUnit_p_dispatch) * StorageUnit_marginal_cost_quadratic)
      * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
      into=snapshot), over=storage_unit), over=snapshot)) + sum(sum(((StorageUnit_state_of_charge * StorageUnit_marginal_cost_storage)
      * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
      into=snapshot), over=storage_unit), over=snapshot)'}
  Store_opex: {expression: 'sum(sum(((Store_p * Store_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)
      + sum(sum((((Store_p * Store_p) * Store_marginal_cost_quadratic) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)
      + sum(sum(((Store_e * Store_marginal_cost_storage) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)'}
objective: {sense: minimize, expression: total_cost}

The prep — every table the spec declares, from the network — and the solve:

from differential.pypsa.prep import relation, static, varying, weighting


n = build()  # the network from the PyPSA tab

sources = {
    'snapshot': pl.Series('snapshot', list(timesteps(n)), dtype=pl.Datetime('us')),
    'bus': pl.Series('bus', list(names(n.buses.index).astype(str)), dtype=pl.String),
        **{
            dim: pl.Series(dim, list(names(n.static(component).index).astype(str)), dtype=pl.String)
            for component, dim in DIM.items()
        },
        **scenarios(n),
        **periods(n),
        **carriers(n, multi),
    'Generator_bus': relation(n, 'Generator', 'bus'),
    'Load_bus': relation(n, 'Load', 'bus'),
    'StorageUnit_bus': relation(n, 'StorageUnit', 'bus'),
    'Store_bus': relation(n, 'Store', 'bus'),
    'snapshot_weightings_objective': weighting(n, 'objective'),
    'Generator_sign': per_component('Generator', first_scenario(n.generators['sign'])),
    'Load_p_set': varying(n, 'Load', 'p_set'),
    'Load_sign': per_component('Load', first_scenario(loads['sign'])),
    'Load_active': per_component('Load', first_scenario(loads['active']), bool),
    'snapshot_weightings_stores': weighting(n, 'stores'),
}

with sps.solve('differential/pypsa/rungs/rung_32_storage_later_period.yaml', sources) as solution:
    solution.objective  # 7230.486698

The network, rung_32_storage_later_period.py in the corpus — the spine plus what this rung adds:

# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT

"""Rung 32: storage that stands in one period only — two built in the later period open at its first snapshot, and a cyclic one that retires closes on its own last snapshot."""

from __future__ import annotations

from datetime import datetime

import pandas as pd

OPTIMIZE = {'multi_investment_periods': True}


def build():
    """A whole network, not the spine: eight snapshots over two periods, two storages built in 2030, one cyclic store that retires after 2020."""
    import pypsa

    n = pypsa.Network()
    n.snapshots = pd.MultiIndex.from_tuples(
        [(2020, datetime(2020, 1, 1, t)) for t in range(4)] + [(2030, datetime(2030, 1, 1, t)) for t in range(4)]
    )
    n.investment_periods = [2020, 2030]
    n.investment_period_weightings['objective'] = [1.0, 0.5]
    n.investment_period_weightings['years'] = [10.0, 10.0]
    n.snapshot_weightings['objective'] = [2.0, 1.5, 2.5, 2.0, 2.0, 1.5, 2.5, 2.0]
    n.snapshot_weightings['stores'] = [0.5, 2.0, 1.5, 2.5, 0.5, 2.0, 1.5, 2.5]
    n.add('Bus', 'hub')
    n.add('Generator', 'base32', bus='hub', p_nom=100, marginal_cost=10)
    n.add('Generator', 'peak32', bus='hub', p_nom=200, marginal_cost=[80, 20, 90, 30, 80, 20, 90, 30])
    n.add(
        'StorageUnit',
        'su_late',
        bus='hub',
        p_nom=15,
        max_hours=4,
        standing_loss=0.02,
        cyclic_state_of_charge=True,
        build_year=2030,
        lifetime=30,
    )
    n.add('Store', 'e_late', bus='hub', e_nom=30, e_initial=5, build_year=2030, lifetime=30)
    n.add('Store', 'e_retire', bus='hub', e_nom=30, standing_loss=0.01, e_cyclic=True, build_year=2020, lifetime=10)
    n.add('Load', 'hub_load', bus='hub', p_set=[40, 60, 70, 40, 90, 110, 120, 90])
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 7230.486698

The data

Every table this spec declares was first declared by a lower rung; its values here are in the prep above.