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Rung 4: ramps — ramp limits on fixed and extendable generators and links

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 8785.0 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 ✔ 64 rows · ≠ 20 vs 23 columns · ✔ 92 nonzeros; duals ✔ 64 rows, 2 negated; model for model: 11 blocks equal, 0 documented splits, 4 recorded deviations.

Rows and columns, PyPSA against specsolve, name for name
row PyPSA specsolve
Bus-nodal_balance 12 12
Generator-fix-p-lower 12 12
Generator-fix-p-upper 12 12
Generator-p-ramp_limit_down 3 3
Generator-p-ramp_limit_up 3 3
Link-fix-p-lower 8 8
Link-fix-p-upper 8 8
Link-p-ramp_limit_down 3 3
Link-p-ramp_limit_up 3 3
column PyPSA specsolve
CVaR 0 ≠ 1
CVaR-a 0 ≠ 1
CVaR-theta 0 ≠ 1
Generator-p 12 12
Link-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 \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — 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
\(\mathcal{Y}\) 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
\(\mathrm{ru}\) Generator_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most a generator may raise its output between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time
\(\mathrm{rd}\) Generator_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most a generator may lower its output between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time
\(\mathrm{ru}^{\mathrm{up}}\) Generator_ramp_limit_start_up over \(\Xi \times \mathcal{G}\) — most output in the snapshot a unit starts, per unit of nominal power
\(\mathrm{rd}^{\mathrm{dn}}\) Generator_ramp_limit_shut_down over \(\Xi \times \mathcal{G}\) — most output in the snapshot before a unit stops, per unit of nominal power
\(\mathrm{u}^{0}\) Generator_status_initial over \(\Xi \times \mathcal{G}\) — one where the unit was on before the first snapshot, zero where off — PyPSA's up_time_before > 0, data prep
\(\mathrm{p}^{0}\) Generator_p_init over \(\Xi \times \mathcal{G}\) — the output a unit brought into the horizon — PyPSA's p_init, read only where the unit came in running; no value means it is unknown, so the unit carries no ramp row at the first snapshot
\(\mathrm{p}^{\mathrm{mod}}\) Generator_p_nom_mod over \(\mathcal{G}\) — the module size a build comes in whole numbers of; no value means the build is continuous
\(\mathrm{ru}^{f}\) Link_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most a link may raise its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time
\(\mathrm{rd}^{f}\) Link_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most a link may lower its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time
\(\mathrm{f}^{\mathrm{nom}}\) Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power
\(\mathrm{ext}^{f}\) Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision
\(\underline{\mathrm{f}}\) Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways
\(\overline{\mathrm{f}}\) Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power
\(\eta\) Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py:1522)
\(\mathrm{d}^{f}\) Link_output_delay over \(\Xi \times \mathcal{O}\) — snapshots a port's delivery lags its link's flow — PyPSA's delay, delay2, … read long, in snapshot_weightings.generators units, which the file states as whole snapshots; zero for a port that delivers at once. Each scenario takes its own. PyPSA 1.3.0 groups the ports by delay over all scenarios and shifts each group in every one, so a delay that differs by scenario delivers the flow twice (constraints.py:1269-1276, PyPSA/PyPSA#1941)
\(\mathrm{cyc}^{f}\) Link_output_cyclic_delay over \(\Xi \times \mathcal{O}\) — whether a delayed port's flow wraps from the end of its investment period — PyPSA's cyclic_delay, cyclic_delay2, …; where it does not, the flow still in transit at each period's first snapshots is lost. Each scenario takes its own, as the delay
\(\mathrm{c}^{f}\) Link_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one unit of flow
\(\mathrm{c}^{f,(2)}\) Link_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of the square of one unit of flow
\(\mathrm{com}^{f}\) Link_committable over \(\mathcal{L}\) — whether flow is gated by an on/off status decision
\(\mathrm{ru}^{f,\mathrm{up}}\) Link_ramp_limit_start_up over \(\Xi \times \mathcal{L}\) — most flow in the snapshot a link starts, per unit of nominal power
\(\mathrm{rd}^{f,\mathrm{dn}}\) Link_ramp_limit_shut_down over \(\Xi \times \mathcal{L}\) — most flow in the snapshot before a link stops, per unit of nominal power
\(\mathrm{u}^{f,0}\) Link_status_initial over \(\Xi \times \mathcal{L}\) — one where the link was on before the first snapshot, zero where off — PyPSA's up_time_before > 0, data prep
\(\mathrm{f}^{0}\) Link_p_init over \(\Xi \times \mathcal{L}\) — the flow a link brought into the horizon — PyPSA's p_init, read only where the link came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot
\(\mathrm{f}^{\mathrm{mod}}\) Link_p_nom_mod over \(\mathcal{L}\) — the module size a build comes in whole numbers of; no value means the build is continuous
\(\mathrm{load}\) 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
\(\mathrm{on}^{f}\) Link_active over \(\mathcal{T} \times \mathcal{L}\) — whether a link stands in a snapshot's period — PyPSA's active, data prep

Variables

Symbol Meaning
\(p\) Generator_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-p — output of a generator in a snapshot
\(f\) Link_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — 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
\(a\) 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
\(u\) Generator_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-status — how much of a committable unit is on: an integer the rows below cap at one, or at the module count where the build is modular
\(u^{f}\) Link_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-status — how much of a committable link is on: an integer the rows below cap at one, or at the module count where the build is modular
\(P\) 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
\(F\) Link_p_nom_ext over \(\mathcal{L}\) — Link-p_nom — nominal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime

Definitions

Symbol Meaning
\(\mathit{Generator\_previous\_p}\) Generator_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the output a generator carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that
\(\mathit{Generator\_ramp\_up\_allowance}\) Generator_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — how far a generator may raise output between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on
\(\mathit{Generator\_ramp\_down\_allowance}\) Generator_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — how far a generator may lower output between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off
\(\mathit{Link\_previous\_p}\) Link_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the flow a link carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that
\(\mathit{Link\_ramp\_up\_allowance}\) Link_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may raise flow between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on
\(\mathit{Link\_ramp\_down\_allowance}\) Link_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may lower flow between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off
\(\mathit{total\_cost}\) 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
\(\mathit{Generator\_previous\_status}\) Generator_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the commitment state a generator carries into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that
\(\mathit{Generator\_p\_nom\_effective}\) Generator_p_nom_effective over \(\Xi \times \mathcal{G}\) — the build a generator's limits are taken against — the chosen one where it is extendable, the given one otherwise
\(\mathrm{Generator\_ramp\_up\_rate}\) Generator_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the ramp limit a unit's up row reads — PyPSA's ramp_limit_up, or the full build where it has none, since a start-up ramp alone builds the row
\(\mathrm{Generator\_ramp\_down\_rate}\) Generator_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the ramp limit a unit's down row reads — PyPSA's ramp_limit_down, or the full build where it has none, since a shut-down ramp alone builds the row
\(\mathrm{Generator\_start\_up\_rate}\) Generator_start_up_rate over \(\Xi \times \mathcal{G}\) — the start-up ramp a unit's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none
\(\mathrm{Generator\_shut\_down\_rate}\) Generator_shut_down_rate over \(\Xi \times \mathcal{G}\) — the shut-down ramp a unit's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none
\(\mathrm{Generator\_p\_nom\_committed}\) Generator_p_nom_committed over \(\Xi \times \mathcal{G}\) — the build a committed unit's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise
\(\mathit{Link\_p\_nom\_effective}\) Link_p_nom_effective over \(\Xi \times \mathcal{L}\) — the build a link's limits are taken against — the chosen one where it is extendable, the given one otherwise
\(\mathit{Link\_previous\_status}\) Link_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the commitment state a link carries into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that
\(\mathrm{Link\_ramp\_up\_rate}\) Link_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's up row reads — PyPSA's ramp_limit_up, or the full build where it has none, since a start-up ramp alone builds the row
\(\mathrm{Link\_ramp\_down\_rate}\) Link_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's down row reads — PyPSA's ramp_limit_down, or the full build where it has none, since a shut-down ramp alone builds the row
\(\mathrm{Link\_start\_up\_rate}\) Link_start_up_rate over \(\Xi \times \mathcal{L}\) — the start-up ramp a link's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none
\(\mathrm{Link\_shut\_down\_rate}\) Link_shut_down_rate over \(\Xi \times \mathcal{L}\) — the shut-down ramp a link's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none
\(\mathrm{Link\_p\_nom\_committed}\) Link_p_nom_committed over \(\Xi \times \mathcal{L}\) — the build a committed link's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise
\(\mathit{risk\_weighted\_opex}\) risk_weighted_opex (scalar)
\(\mathit{Generator\_injection}\) Generator_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathit{Link\_injection}\) Link_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathrm{Load\_injection}\) Load_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\)
\(\mathit{Link\_output\_arrival}\) Link_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — what a link delivers to an output port at a snapshot — its flow delayed by the port's delay within its investment period, times the port's efficiency at the snapshot the flow arrives; where the port is cyclic_delay the delayed flow wraps from the period's end, and where it is not the flow still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) delivers its flow unshifted, cyclic or not
\(\mathit{scenario\_opex}\) 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 \(\Xi\)
\(\mathit{Link\_opex}\) Link_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.

\(t \boxminus_{v} k\) denotes translation with \(v\) standing where index \(t-k\) leaves the dimension (shift(edge=v)), so the row at that boundary is built and carries \(v\) rather than being dropped.

\(t \ominus^{\mathrm{relation}(t)} k\) 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. The two modifiers take different slots — the group above, the fill below — so \(t \boxminus_{v}^{\mathrm{relation}(t)} k\) is both at once.

\(\mathrm{pos}(t)\) 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

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

Link_fix_p_lower

\[ f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l} \]

Link_fix_p_upper

\[ f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l} \]

Generator_p_ramp_limit_up

\[ p_{\xi,t,g} - \mathit{Generator\_previous\_p}_{\xi,t,g} \le \mathit{Generator\_ramp\_up\_allowance}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \left( \mathrm{ru}_{\xi,t,g} \text{ is defined} \vee \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g} \]

Generator_p_ramp_limit_down

\[ \mathit{Generator\_previous\_p}_{\xi,t,g} - p_{\xi,t,g} \le \mathit{Generator\_ramp\_down\_allowance}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \left( \mathrm{rd}_{\xi,t,g} \text{ is defined} \vee \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g} \]

Link_p_ramp_limit_up

\[ f_{\xi,t,l} - \mathit{Link\_previous\_p}_{\xi,t,l} \le \mathit{Link\_ramp\_up\_allowance}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l} \]

Link_p_ramp_limit_down

\[ \mathit{Link\_previous\_p}_{\xi,t,l} - f_{\xi,t,l} \le \mathit{Link\_ramp\_down\_allowance}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l} \]

Bus_nodal_balance

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

Definitions

Generator_previous_p

\[ \mathit{Generator\_previous\_p}_{\xi,t,g} = \begin{cases} \mathrm{u}^{0}_{\xi,g} \cdot \mathrm{p}^{0}_{\xi,g} & \text{if } \mathrm{pos}(t) = 0 \\ p_{\xi,t - 1,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_ramp_up_allowance

\[ \mathit{Generator\_ramp\_up\_allowance}_{\xi,t,g} = \begin{cases} \mathrm{Generator\_ramp\_up\_rate}_{\xi,t,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot \mathit{Generator\_previous\_status}_{\xi,t,g} + \mathrm{Generator\_start\_up\_rate}_{\xi,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot \left( u_{\xi,t,g} - \mathit{Generator\_previous\_status}_{\xi,t,g} \right) & \text{if } \mathrm{com}_{g} \\ \mathrm{Generator\_ramp\_up\_rate}_{\xi,t,g} \cdot \mathit{Generator\_p\_nom\_effective}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_ramp_down_allowance

\[ \mathit{Generator\_ramp\_down\_allowance}_{\xi,t,g} = \begin{cases} \mathrm{Generator\_ramp\_down\_rate}_{\xi,t,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot u_{\xi,t,g} + \mathrm{Generator\_shut\_down\_rate}_{\xi,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot \left( \mathit{Generator\_previous\_status}_{\xi,t,g} - u_{\xi,t,g} \right) & \text{if } \mathrm{com}_{g} \\ \mathrm{Generator\_ramp\_down\_rate}_{\xi,t,g} \cdot \mathit{Generator\_p\_nom\_effective}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \]

Link_previous_p

\[ \mathit{Link\_previous\_p}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} \cdot \mathrm{f}^{0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ f_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \]

Link_ramp_up_allowance

\[ \mathit{Link\_ramp\_up\_allowance}_{\xi,t,l} = \begin{cases} \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \mathit{Link\_previous\_status}_{\xi,t,l} + \mathrm{Link\_start\_up\_rate}_{\xi,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \left( u^{f}_{\xi,t,l} - \mathit{Link\_previous\_status}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot \mathit{Link\_p\_nom\_effective}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \]

Link_ramp_down_allowance

\[ \mathit{Link\_ramp\_down\_allowance}_{\xi,t,l} = \begin{cases} \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot u^{f}_{\xi,t,l} + \mathrm{Link\_shut\_down\_rate}_{\xi,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \left( \mathit{Link\_previous\_status}_{\xi,t,l} - u^{f}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot \mathit{Link\_p\_nom\_effective}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \]

total_cost

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

Bus_injection

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

Generator_previous_status

\[ \mathit{Generator\_previous\_status}_{\xi,t,g} = \begin{cases} \mathrm{u}^{0}_{\xi,g} & \text{if } \mathrm{pos}(t) = 0 \\ u_{\xi,t - 1,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_p_nom_effective

\[ \mathit{Generator\_p\_nom\_effective}_{\xi,g} = \begin{cases} P_{g} & \text{if } \mathrm{ext}_{g} \\ \mathrm{p}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \]

Generator_ramp_up_rate

\[ \mathrm{Generator\_ramp\_up\_rate}_{\xi,t,g} = \begin{cases} \mathrm{ru}_{\xi,t,g} & \text{if } \mathrm{ru}_{\xi,t,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_ramp_down_rate

\[ \mathrm{Generator\_ramp\_down\_rate}_{\xi,t,g} = \begin{cases} \mathrm{rd}_{\xi,t,g} & \text{if } \mathrm{rd}_{\xi,t,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_start_up_rate

\[ \mathrm{Generator\_start\_up\_rate}_{\xi,g} = \begin{cases} \mathrm{ru}^{\mathrm{up}}_{\xi,g} & \text{if } \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \]

Generator_shut_down_rate

\[ \mathrm{Generator\_shut\_down\_rate}_{\xi,g} = \begin{cases} \mathrm{rd}^{\mathrm{dn}}_{\xi,g} & \text{if } \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \]

Generator_p_nom_committed

\[ \mathrm{Generator\_p\_nom\_committed}_{\xi,g} = \begin{cases} \mathrm{p}^{\mathrm{mod}}_{g} & \text{if } \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \\ \mathrm{p}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \]

Link_p_nom_effective

\[ \mathit{Link\_p\_nom\_effective}_{\xi,l} = \begin{cases} F_{l} & \text{if } \mathrm{ext}^{f}_{l} \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \]

Link_previous_status

\[ \mathit{Link\_previous\_status}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ u^{f}_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \]

Link_ramp_up_rate

\[ \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} = \begin{cases} \mathrm{ru}^{f}_{\xi,t,l} & \text{if } \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \]

Link_ramp_down_rate

\[ \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} = \begin{cases} \mathrm{rd}^{f}_{\xi,t,l} & \text{if } \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \]

Link_start_up_rate

\[ \mathrm{Link\_start\_up\_rate}_{\xi,l} = \begin{cases} \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} & \text{if } \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \]

Link_shut_down_rate

\[ \mathrm{Link\_shut\_down\_rate}_{\xi,l} = \begin{cases} \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} & \text{if } \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \]

Link_p_nom_committed

\[ \mathrm{Link\_p\_nom\_committed}_{\xi,l} = \begin{cases} \mathrm{f}^{\mathrm{mod}}_{l} & \text{if } \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \]

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

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

Link_injection

\[ \mathit{Link\_injection}_{\xi,t,n} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

Load_injection

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

Link_output_arrival

\[ \mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{\xi,o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O} \]

scenario_opex

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

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

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

Link_opex

\[ \mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi \]

Variable domains

Generator_p

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

Link_p

\[ f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l} \]

CVaR_a

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

CVaR_theta

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

CVaR

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

Generator_status

\[ u_{\xi,t,g} \ge 0, u_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \]

Link_status

\[ u^{f}_{\xi,t,l} \ge 0, u^{f}_{\xi,t,l} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l} \]

Generator_p_nom_ext

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

Link_p_nom_ext

\[ F_{l} \in \mathbb{R} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \]

The spec, differential/pypsa/rungs/rung_04_ramps.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'}
  link: {description: 'controllable connections, each from one bus to the buses it delivers to'}
  link_output: {description: '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'}
  load: {description: 'demands, 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}
  Link_bus0: {description: the bus a link leaves, key: link, values: bus}
  Link_output_link: {description: the link an output port belongs to, key: link_output, values: link}
  Link_output_bus: {description: 'the bus an output port delivers to — PyPSA''s `bus1`, `bus2`, … columns.
      A link of three output ports is three labels here rather than a third relation, so the file states
      any number of them', key: link_output, values: bus}
  Load_bus: {description: the bus a load sits on, key: load, 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
  Generator_ramp_limit_up:
    description: most a generator may raise its output between snapshots, per unit of nominal power; no
      value means no limit — read at the later of the two snapshots, so the limit may change over time
    dims: [scenario, snapshot, generator]
  Generator_ramp_limit_down:
    description: most a generator may lower its output between snapshots, per unit of nominal power; no
      value means no limit — read at the later of the two snapshots, so the limit may change over time
    dims: [scenario, snapshot, generator]
  Generator_ramp_limit_start_up:
    description: most output in the snapshot a unit starts, per unit of nominal power
    dims: [scenario, generator]
  Generator_ramp_limit_shut_down:
    description: most output in the snapshot before a unit stops, per unit of nominal power
    dims: [scenario, generator]
  Generator_status_initial:
    description: one where the unit was on before the first snapshot, zero where off — PyPSA's `up_time_before
      > 0`, data prep
    dims: [scenario, generator]
    dtype: int
  Generator_p_init:
    description: the output a unit brought into the horizon — PyPSA's `p_init`, read only where the unit
      came in running; no value means it is unknown, so the unit carries no ramp row at the first snapshot
    dims: [scenario, generator]
  Generator_p_nom_mod:
    description: the module size a build comes in whole numbers of; no value means the build is continuous
    dims: [generator]
  Link_ramp_limit_up:
    description: most a link may raise its flow between snapshots, per unit of nominal power; no value
      means no limit — read at the later of the two snapshots, so the limit may change over time
    dims: [scenario, snapshot, link]
  Link_ramp_limit_down:
    description: most a link may lower its flow between snapshots, per unit of nominal power; no value
      means no limit — read at the later of the two snapshots, so the limit may change over time
    dims: [scenario, snapshot, link]
  Link_p_nom:
    description: nominal power
    dims: [scenario, link]
  Link_p_nom_extendable:
    description: whether the nominal power is a decision
    dims: [link]
    dtype: bool
  Link_p_min_pu:
    description: least flow, per unit of nominal power — negative for a link that carries both ways
    dims: [scenario, snapshot, link]
  Link_p_max_pu:
    description: most flow, per unit of nominal power
    dims: [scenario, snapshot, link]
  Link_efficiency:
    description: share of the flow that arrives at an output port, PyPSA's `efficiency`, `efficiency2`,
      … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow
      arrives, so a delayed port delivers at its arrival snapshot's efficiency (`constraints.py:1522`)
    dims: [scenario, snapshot, link_output]
  Link_output_delay:
    description: snapshots a port's delivery lags its link's flow — PyPSA's `delay`, `delay2`, … read
      long, in `snapshot_weightings.generators` units, which the file states as whole snapshots; zero
      for a port that delivers at once. Each scenario takes its own. PyPSA `1.3.0` groups the ports by
      delay over all scenarios and shifts each group in every one, so a delay that differs by scenario
      delivers the flow twice (`constraints.py:1269-1276`, PyPSA/PyPSA#1941)
    dims: [scenario, link_output]
    dtype: int
  Link_output_cyclic_delay:
    description: whether a delayed port's flow wraps from the end of its investment period — PyPSA's `cyclic_delay`,
      `cyclic_delay2`, …; where it does not, the flow still in transit at each period's first snapshots
      is lost. Each scenario takes its own, as the delay
    dims: [scenario, link_output]
    dtype: bool
  Link_marginal_cost:
    description: cost of one unit of flow
    dims: [scenario, snapshot, link]
  Link_marginal_cost_quadratic:
    description: cost of the square of one unit of flow
    dims: [scenario, snapshot, link]
  Link_committable:
    description: whether flow is gated by an on/off status decision
    dims: [link]
    dtype: bool
  Link_ramp_limit_start_up:
    description: most flow in the snapshot a link starts, per unit of nominal power
    dims: [scenario, link]
  Link_ramp_limit_shut_down:
    description: most flow in the snapshot before a link stops, per unit of nominal power
    dims: [scenario, link]
  Link_status_initial:
    description: one where the link was on before the first snapshot, zero where off — PyPSA's `up_time_before
      > 0`, data prep
    dims: [scenario, link]
    dtype: int
  Link_p_init:
    description: the flow a link brought into the horizon — PyPSA's `p_init`, read only where the link
      came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot
    dims: [scenario, link]
  Link_p_nom_mod:
    description: the module size a build comes in whole numbers of; no value means the build is continuous
    dims: [link]
  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
  Link_active:
    description: whether a link stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, link]
    dtype: bool
variables:
  Generator_p:
    description: '`Generator-p` — output of a generator in a snapshot'
    dims: [scenario, snapshot, generator]
    where: Generator_active
  Link_p:
    description: '`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'
    dims: [scenario, snapshot, link]
    where: Link_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: []
  Generator_status:
    description: '`Generator-status` — how much of a committable unit is on: an integer the rows below
      cap at one, or at the module count where the build is modular'
    dims: [scenario, snapshot, generator]
    where: Generator_committable AND Generator_active
    domain: integer
    bounds: {lower: 0}
  Link_status:
    description: '`Link-status` — how much of a committable link is on: an integer the rows below cap
      at one, or at the module count where the build is modular'
    dims: [scenario, snapshot, link]
    where: Link_committable AND Link_active
    domain: integer
    bounds: {lower: 0}
  Generator_p_nom_ext:
    description: '`Generator-p_nom` — nominal power where it is a decision; the parameter of the same
      PyPSA name carries the fixed regime'
    dims: [generator]
    where: Generator_p_nom_extendable
  Link_p_nom_ext:
    description: '`Link-p_nom` — nominal power where it is a decision; the parameter of the same PyPSA
      name carries the fixed regime'
    dims: [link]
    where: Link_p_nom_extendable
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
  Link_fix_p_lower:
    description: '`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other
      way'
    dims: [scenario, snapshot, link]
    where: not Link_p_nom_extendable AND not Link_committable AND Link_active
    expression: Link_p >= Link_p_min_pu * Link_p_nom
  Link_fix_p_upper:
    description: '`Link-fix-p-upper` — a fixed link carries at most its nominal power'
    dims: [scenario, snapshot, link]
    where: not Link_p_nom_extendable AND not Link_committable AND Link_active
    expression: Link_p <= Link_p_max_pu * Link_p_nom
  Generator_p_ramp_limit_up:
    description: '`Generator-p-ramp_limit_up` — a generator raises output no faster than its ramp limit
      of the build, and a committed one no further than its start-up ramp in the snapshot it turns on.
      A unit that came into the horizon running carries a row at the first snapshot only where its `p_init`
      gives the output it brought in, and no unit carries one at the start of a later investment period
      — nor does any unit a big M releases instead'
    dims: [scenario, snapshot, generator]
    where: (Generator_ramp_limit_up OR Generator_ramp_limit_start_up) AND NOT (Generator_committable AND
      Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period,
      within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
      AND Generator_active
    expression: Generator_p - Generator_previous_p <= Generator_ramp_up_allowance
  Generator_p_ramp_limit_down:
    description: '`Generator-p-ramp_limit_down` — a generator lowers output no faster than its ramp limit
      of the build, and a committed one no further than its shut-down ramp in the snapshot it turns off.
      A unit that came into the horizon running carries a row at the first snapshot only where its `p_init`
      gives the output it brought in, and no unit carries one at the start of a later investment period
      — nor does any unit a big M releases instead'
    dims: [scenario, snapshot, generator]
    where: (Generator_ramp_limit_down OR Generator_ramp_limit_shut_down) AND NOT (Generator_committable
      AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period,
      within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
      AND Generator_active
    expression: Generator_previous_p - Generator_p <= Generator_ramp_down_allowance
  Link_p_ramp_limit_up:
    description: '`Link-p-ramp_limit_up` — a link raises flow no faster than its ramp limit of the build,
      and a committed one no further than its start-up ramp in the snapshot it turns on. A link that came
      into the horizon running carries a row at the first snapshot only where its `p_init` gives the flow
      it brought in, and no link carries one at the start of a later investment period — nor does any
      link a big M releases instead'
    dims: [scenario, snapshot, link]
    where: (Link_ramp_limit_up OR Link_ramp_limit_start_up) AND NOT (Link_committable AND Link_p_nom_extendable
      AND NOT (Link_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period, within=period) > 0 OR
      (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init))) AND Link_active
    expression: Link_p - Link_previous_p <= Link_ramp_up_allowance
  Link_p_ramp_limit_down:
    description: '`Link-p-ramp_limit_down` — a link lowers flow no faster than its ramp limit of the build,
      and a committed one no further than its shut-down ramp in the snapshot it turns off. A link that
      came into the horizon running carries a row at the first snapshot only where its `p_init` gives
      the flow it brought in, and no link carries one at the start of a later investment period — nor
      does any link a big M releases instead'
    dims: [scenario, snapshot, link]
    where: (Link_ramp_limit_down OR Link_ramp_limit_shut_down) AND NOT (Link_committable AND Link_p_nom_extendable
      AND NOT (Link_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period, within=period) > 0 OR
      (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init))) AND Link_active
    expression: Link_previous_p - Link_p <= Link_ramp_down_allowance
  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:
  Generator_previous_p:
    description: the output a generator carries into a snapshot — at the first, the `p_init` it brought
      in where it came in running and nothing where it came in off; the previous snapshot's after that
    dims: [scenario, snapshot, generator]
    cases:
      opening: {when: position(snapshot) == 0, expression: Generator_status_initial * Generator_p_init}
    otherwise: shift(Generator_p, along=snapshot, offset=1)
  Generator_ramp_up_allowance:
    description: how far a generator may raise output between two snapshots — its ramp limit of the build
      while it stays on, plus its start-up ramp in the snapshot it turns on
    dims: [scenario, snapshot, generator]
    cases:
      committed: {when: Generator_committable, expression: Generator_ramp_up_rate * Generator_p_nom_committed
          * Generator_previous_status + Generator_start_up_rate * Generator_p_nom_committed * (Generator_status
          - Generator_previous_status)}
    otherwise: Generator_ramp_up_rate * Generator_p_nom_effective
  Generator_ramp_down_allowance:
    description: how far a generator may lower output between two snapshots — its ramp limit of the build
      while it stays on, plus its shut-down ramp in the snapshot it turns off
    dims: [scenario, snapshot, generator]
    cases:
      committed: {when: Generator_committable, expression: Generator_ramp_down_rate * Generator_p_nom_committed
          * Generator_status + Generator_shut_down_rate * Generator_p_nom_committed * (Generator_previous_status
          - Generator_status)}
    otherwise: Generator_ramp_down_rate * Generator_p_nom_effective
  Link_previous_p:
    description: the flow a link carries into a snapshot — at the first, the `p_init` it brought in where
      it came in running and nothing where it came in off; the previous snapshot's after that
    dims: [scenario, snapshot, link]
    cases:
      opening: {when: position(snapshot) == 0, expression: Link_status_initial * Link_p_init}
    otherwise: shift(Link_p, along=snapshot, offset=1)
  Link_ramp_up_allowance:
    description: how far a link may raise flow between two snapshots — its ramp limit of the build while
      it stays on, plus its start-up ramp in the snapshot it turns on
    dims: [scenario, snapshot, link]
    cases:
      committed: {when: Link_committable, expression: Link_ramp_up_rate * Link_p_nom_committed * Link_previous_status
          + Link_start_up_rate * Link_p_nom_committed * (Link_status - Link_previous_status)}
    otherwise: Link_ramp_up_rate * Link_p_nom_effective
  Link_ramp_down_allowance:
    description: how far a link may lower flow between two snapshots — its ramp limit of the build while
      it stays on, plus its shut-down ramp in the snapshot it turns off
    dims: [scenario, snapshot, link]
    cases:
      committed: {when: Link_committable, expression: Link_ramp_down_rate * Link_p_nom_committed * Link_status
          + Link_shut_down_rate * Link_p_nom_committed * (Link_previous_status - Link_status)}
    otherwise: Link_ramp_down_rate * Link_p_nom_effective
  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 + Link_injection) + Load_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
  Generator_previous_status:
    description: the commitment state a generator carries into a snapshot — the state it brought into
      the horizon at the first, the previous snapshot's after that
    dims: [scenario, snapshot, generator]
    cases:
      opening: {when: position(snapshot) == 0, expression: Generator_status_initial}
    otherwise: shift(Generator_status, along=snapshot, offset=1)
  Generator_p_nom_effective:
    description: the build a generator's limits are taken against — the chosen one where it is extendable,
      the given one otherwise
    dims: [scenario, generator]
    cases:
      extendable: {when: Generator_p_nom_extendable, expression: Generator_p_nom_ext}
    otherwise: Generator_p_nom
  Generator_ramp_up_rate:
    description: the ramp limit a unit's up row reads — PyPSA's `ramp_limit_up`, or the full build where
      it has none, since a start-up ramp alone builds the row
    dims: [scenario, snapshot, generator]
    cases:
      given: {when: Generator_ramp_limit_up, expression: Generator_ramp_limit_up}
    otherwise: 1
  Generator_ramp_down_rate:
    description: the ramp limit a unit's down row reads — PyPSA's `ramp_limit_down`, or the full build
      where it has none, since a shut-down ramp alone builds the row
    dims: [scenario, snapshot, generator]
    cases:
      given: {when: Generator_ramp_limit_down, expression: Generator_ramp_limit_down}
    otherwise: 1
  Generator_start_up_rate:
    description: the start-up ramp a unit's up row reads — PyPSA's `ramp_limit_start_up`, or the full
      build where it has none
    dims: [scenario, generator]
    cases:
      given: {when: Generator_ramp_limit_start_up, expression: Generator_ramp_limit_start_up}
    otherwise: 1
  Generator_shut_down_rate:
    description: the shut-down ramp a unit's down row reads — PyPSA's `ramp_limit_shut_down`, or the full
      build where it has none
    dims: [scenario, generator]
    cases:
      given: {when: Generator_ramp_limit_shut_down, expression: Generator_ramp_limit_shut_down}
    otherwise: 1
  Generator_p_nom_committed:
    description: the build a committed unit's ramp rows are taken against — one module where the build
      is extendable and modular, the given build otherwise
    dims: [scenario, generator]
    cases:
      modular_build: {when: Generator_p_nom_extendable AND Generator_p_nom_mod > 0, expression: Generator_p_nom_mod}
    otherwise: Generator_p_nom
  Link_p_nom_effective:
    description: the build a link's limits are taken against — the chosen one where it is extendable,
      the given one otherwise
    dims: [scenario, link]
    cases:
      extendable: {when: Link_p_nom_extendable, expression: Link_p_nom_ext}
    otherwise: Link_p_nom
  Link_previous_status:
    description: the commitment state a link carries into a snapshot — the state it brought into the horizon
      at the first, the previous snapshot's after that
    dims: [scenario, snapshot, link]
    cases:
      opening: {when: position(snapshot) == 0, expression: Link_status_initial}
    otherwise: shift(Link_status, along=snapshot, offset=1)
  Link_ramp_up_rate:
    description: the ramp limit a link's up row reads — PyPSA's `ramp_limit_up`, or the full build where
      it has none, since a start-up ramp alone builds the row
    dims: [scenario, snapshot, link]
    cases:
      given: {when: Link_ramp_limit_up, expression: Link_ramp_limit_up}
    otherwise: 1
  Link_ramp_down_rate:
    description: the ramp limit a link's down row reads — PyPSA's `ramp_limit_down`, or the full build
      where it has none, since a shut-down ramp alone builds the row
    dims: [scenario, snapshot, link]
    cases:
      given: {when: Link_ramp_limit_down, expression: Link_ramp_limit_down}
    otherwise: 1
  Link_start_up_rate:
    description: the start-up ramp a link's up row reads — PyPSA's `ramp_limit_start_up`, or the full
      build where it has none
    dims: [scenario, link]
    cases:
      given: {when: Link_ramp_limit_start_up, expression: Link_ramp_limit_start_up}
    otherwise: 1
  Link_shut_down_rate:
    description: the shut-down ramp a link's down row reads — PyPSA's `ramp_limit_shut_down`, or the full
      build where it has none
    dims: [scenario, link]
    cases:
      given: {when: Link_ramp_limit_shut_down, expression: Link_ramp_limit_shut_down}
    otherwise: 1
  Link_p_nom_committed:
    description: the build a committed link's ramp rows are taken against — one module where the build
      is extendable and modular, the given build otherwise
    dims: [scenario, link]
    cases:
      modular_build: {when: Link_p_nom_extendable AND Link_p_nom_mod > 0, expression: Link_p_nom_mod}
    otherwise: Link_p_nom
  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)'}
  Link_injection: {expression: '-sum(Link_p, by=Link_bus0, over=link, into=bus) + sum(Link_output_arrival,
      by=Link_output_bus, over=link_output, into=bus)'}
  Load_injection: {expression: 'sum(Load_demand, by=Load_bus, over=load, into=bus)'}
  Link_output_arrival:
    description: what a link delivers to an output port at a snapshot — its flow delayed by the port's
      `delay` within its investment period, times the port's efficiency at the snapshot the flow arrives;
      where the port is `cyclic_delay` the delayed flow wraps from the period's end, and where it is not
      the flow still in transit at the period's first snapshots is lost. A port that does not delay (`delay`
      zero) delivers its flow unshifted, cyclic or not
    dims: [scenario, snapshot, link_output]
    cases:
      wrapping: {when: Link_output_cyclic_delay, expression: 'shift(at(Link_p, by=Link_output_link, over=link,
          into=link_output), along=snapshot, offset=Link_output_delay, edge=''wrap'', by=snapshot_period,
          within=period) * Link_efficiency'}
    otherwise: shift(at(Link_p, by=Link_output_link, over=link, into=link_output), along=snapshot, offset=Link_output_delay,
      edge=0, by=snapshot_period, within=period) * Link_efficiency
  scenario_opex:
    dims: [scenario]
    expression: Generator_opex + Link_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)'}
  Link_opex: {expression: 'sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period, over=period, into=snapshot), over=link), over=snapshot) + sum(sum((((Link_p
      * Link_p) * Link_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period, over=period, into=snapshot), over=link), 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'),
    'Link_bus0': relation(n, 'Link', 'bus0'),
    'Load_bus': relation(n, 'Load', '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),
}

with sps.solve('differential/pypsa/rungs/rung_04_ramps.yaml', sources) as solution:
    solution.objective  # 8785.0

The network, rung_04_ramps.py in the corpus — the spine plus what this rung adds:

"""Rung 4: ramps — ramp limits on fixed and extendable generators and links."""

from __future__ import annotations

import spine


def build():
    """The spine plus this rung's additions, as a ``pypsa.Network``."""
    n = spine.build()
    n.add('Bus', 'east')
    n.add('Generator', 'coal_slow', bus='north', p_nom=80, marginal_cost=8, ramp_limit_up=0.2, ramp_limit_down=0.2)
    n.add('Link', 'tie', bus0='north', bus1='east', p_nom=50, efficiency=1, ramp_limit_up=0.4, ramp_limit_down=0.4)
    n.add('Load', 'east_load', bus='east', p_set=[5, 20, 25, 10])
    n.add('Load', 'swing', bus='north', p_set=[0, 25, 45, 0])
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 8785.0

The data

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