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Rung 40: a global constraint takes its own constant and sense in each scenario

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 master at 51986084 — objective 15106.666667 on both sides; structure ≠ primary_energy 2 vs 1+1 — one block per sense — ==, <=, >= — where PyPSA writes one row per labelled constraint whatever its sense; size ✔ 86 rows · ✔ 33 columns · ✔ 140 nonzeros; duals ✔ 86 rows; model for model: 13 blocks equal, 1 documented splits, 1 recorded deviations.

Rows and columns, PyPSA against specsolve, name for name
row PyPSA specsolve
Bus-nodal_balance 16 16
Generator-fix-p-lower 16 16
Generator-fix-p-upper 16 16
Line-ext-s-lower 8 8
Line-ext-s-upper 8 8
Line-ext-s_nom-lower 2 2
Link-fix-p-lower 8 8
Link-fix-p-upper 8 8
primary_energy 2 ≠ 1+1
transmission_volume_expansion_limit 2 2
column PyPSA specsolve
Generator-p 16 16
Line-s 8 8
Line-s_nom 1 1
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 a risk preference is set. 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 \(\mathrm{period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods
\(\mathcal{N}\) index \(n\) — bus with \(\mathrm{bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{bus}: \mathcal{D} \to \mathcal{N},\ \mathrm{bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{bus1}: \mathcal{K} \to \mathcal{N}\) — network nodes
\(\mathcal{G}\) index \(g\) — generator with \(\mathrm{bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus
\(\mathcal{L}\) index \(l\) — link with \(\mathrm{bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to
\(\mathcal{O}\) index \(o\) — link_output with \(\mathrm{link}: \mathcal{O} \to \mathcal{L},\ \mathrm{bus}: \mathcal{O} \to \mathcal{N}\) — 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
\(\mathcal{J}\) index \(j\) — process with \(\mathrm{process}: \mathcal{R} \to \mathcal{J}\) — generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate
\(\mathcal{R}\) index \(r\) — process_output with \(\mathrm{process}: \mathcal{R} \to \mathcal{J},\ \mathrm{bus}: \mathcal{R} \to \mathcal{N}\) — a process's ports, one label per port a process declares — PyPSA's bus0, bus1, … each carry a signed rate, so a process of any number of ports is one term in the balance, data prep
\(\mathcal{D}\) index \(d\) — load with \(\mathrm{bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus
\(\mathcal{K}\) index \(k\) — line with \(\mathrm{bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{bus1}: \mathcal{K} \to \mathcal{N}\) — passive branches, each between two buses, their flow set by impedance
\(\mathcal{B}\) index \(b\) — global_constraint — PyPSA's GlobalConstraint rows, one label per declared limit
\(\mathcal{Y}\) index \(y\) — period with \(\mathrm{period}: \mathcal{T} \to \mathcal{Y}\) — 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 (constants.py:43)
\(\mathrm{com}\) Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision
\(\mathrm{f}^{\mathrm{nom}}\) 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:1498)
\(\mathrm{d}^{f}\) Link_output_delay over \(\mathcal{O}\) — snapshots a port's delivery lags its link's flow — PyPSA's delay, delay2, … read long, divided by the snapshot_weightings.generators value and rounded up in data prep; zero for a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses a delay that differs by scenario (constants.py:52)
\(\mathrm{cyc}^{f}\) Link_output_cyclic_delay over \(\mathcal{O}\) — 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. The flag is the same in every scenario, as the delay is
\(\mathrm{c}^{f}\) 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
\(\alpha\) Process_rate over \(\Xi \times \mathcal{T} \times \mathcal{R}\) — the energy a port draws or delivers per unit of internal power, PyPSA's rate0, rate1, … read long — negative where the port withdraws, positive where it injects; a link is a process whose bus0 rate is minus one and whose output rates are its efficiencies. Read at the snapshot the transfer arrives, so a delayed port transfers at its arrival snapshot's rate (constraints.py:1498)
\(\mathrm{d}^{z}\) Process_output_delay over \(\mathcal{R}\) — snapshots a port's transfer lags its process's internal power — PyPSA's delay0, delay1, … read long, divided by the snapshot_weightings.generators value and rounded up in data prep; zero for a port that transfers at once. A port has the same delay in every scenario, as a link's port has
\(\mathrm{cyc}^{z}\) Process_output_cyclic_delay over \(\mathcal{R}\) — whether a delayed port's transfer wraps from the end of its investment period — PyPSA's cyclic_delay0, cyclic_delay1, …; where it does not, the energy still in transit at each period's first snapshots is lost. The flag is the same in every scenario, as the delay is
\(\mathrm{load}\) 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 (constants.py:43)
\(\mathrm{on}^{\mathrm{load}}\) 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 (constants.py:51)
\(\pi\) scenario_weight over \(\Xi\) — PyPSA's scenario_weightings.weight — the probability of a future
\(\mathrm{w}^{y}\) period_weight_objective over \(\mathcal{Y}\) — PyPSA's investment_period_weightings.objective — what a period's cost weighs; PyPSA reads it only under multi_investment_periods, so data prep feeds one otherwise, whatever the column holds (optimize.py:205-207, :264-266)
\(\mathrm{w}^{\mathrm{yr}}\) period_weight_years over \(\mathcal{Y}\) — PyPSA's investment_period_weightings.years — what a period's energy weighs in a primary_energy or operational_limit row; PyPSA reads it only under multi_investment_periods, so data prep feeds one otherwise
\(\mathrm{on}\) 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
\(\mathrm{on}^{s}\) Line_active over \(\mathcal{T} \times \mathcal{K}\) — whether a line stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{on}^{z}\) Process_active over \(\mathcal{T} \times \mathcal{J}\) — whether a process stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{act}^{s}\) Line_active_asset over \(\mathcal{K}\) — whether a line is one of PyPSA's active_assets — its active flag alone, without build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA refuses one that differs by scenario (constants.py:51)
\(\mathrm{W}^{s}\) Line_capital_weight over \(\mathcal{K}\) — the sum of period weights a line stands in — PyPSA's active * period_weighting, summed, data prep
\(\mathrm{w}^{\mathrm{gen}}\) snapshot_weightings_generators over \(\mathcal{T}\) — PyPSA's snapshot_weightings.generators — hours a snapshot stands for in an energy total
\(\mathrm{ext}^{s}\) Line_s_nom_extendable over \(\mathcal{K}\) — whether the nominal apparent power is a decision
\(\overline{\mathrm{s}}\) Line_s_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — most flow either way, per unit of nominal apparent power
\(\underline{\mathrm{s}}^{\mathrm{nom}}\) Line_s_nom_min over \(\Xi \times \mathcal{K}\) — least nominal apparent power an extendable line may be built at
\(\mathrm{c}^{\mathrm{cap},s}\) Line_capital_cost over \(\Xi \times \mathcal{K}\) — cost of one unit of nominal apparent power for the modelled horizon — PyPSA's periodized_cost: overnight_cost as an annuity over lifetime at discount_rate, times nyears, where it is given, and capital_cost where it is not, plus fom_cost (components.py:1126-1147, costs.py:102-203), data prep
\(\mathrm{type}\) GlobalConstraint_type over \(\mathcal{B}\) — which formula the row takes — primary_energy, operational_limit, transmission_volume_expansion_limit, transmission_expansion_cost_limit or tech_capacity_expansion_limit
\(\mathrm{sense}\) GlobalConstraint_sense over \(\Xi \times \mathcal{B}\) — which way the row binds in each scenario — <=, >= or ==; PyPSA reads a row's sense per scenario (global_constraints.py:551, :749, :829)
\(\mathrm{K}\) GlobalConstraint_constant over \(\Xi \times \mathcal{B}\) — the constant the total is held against; what a variable cannot carry — an initial charge, times its period's years for each counted period where the storage reopens per period, or a non-extendable build — is folded in here by data prep. PyPSA reads it per scenario (global_constraints.py:552, :750, :829)
\(\mathrm{in}\) GlobalConstraint_counts_snapshot over \(\Xi \times \mathcal{B} \times \mathcal{T}\), false where the data has no row — whether a row counts a snapshot in a scenario — PyPSA's investment_period: every snapshot where the row names none, and only that period's where it names one, data prep. A row that names a period the run does not model has no label here, as PyPSA skips it (global_constraints.py:372); PyPSA reads the column only under multi_investment_periods, and fails on a row that names a period without it (global_constraints.py:364-369)
\(\mathrm{a}\) Generator_primary_energy_weight over \(\Xi \times \mathcal{B} \times \mathcal{T} \times \mathcal{G}\), \(0\) where the data has no row — the constrained attribute per unit of energy at the bus — the carrier's co2_emissions over the generator's efficiency at the snapshot, data prep; a generator of an unweighted carrier has no row
\(\mathrm{len}\) Line_volume_weight over \(\Xi \times \mathcal{B} \times \mathcal{K}\), \(0\) where the data has no row — the line's length in each scenario where its carrier is in the row's set (global_constraints.py:847-848) — data prep; a line outside it, or one that does not stand in the row's investment_period, has no row

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
\(s\) Line_s over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — Line-s — PyPSA's p0, the flow measured at the line[bus0] end: a positive value withdraws there and injects at line[bus1], lossless
\(S\) Line_s_nom_ext over \(\mathcal{K}\) — Line-s_nom — nominal apparent power where it is a decision, for a line that is active; the parameter of the same PyPSA name carries the fixed regime
\(z\) Process_p over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-p — PyPSA's internal power p: a positive value drives every port at its own rate, withdrawing where the rate is negative and injecting where it is positive

Definitions

Symbol Meaning
\(\mathit{Link\_output\_arrival}\) 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
\(\mathit{Process\_output\_arrival}\) Process_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{R}\) — what a process transfers at a port at a snapshot — its internal power delayed by the port's delay within its investment period, times the port's rate at the snapshot the transfer arrives; where the port is cyclic_delay the delayed transfer wraps from the period's end, and where it is not the energy still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) transfers at once, cyclic or not
\(\mathrm{GlobalConstraint\_energy\_weight}\) GlobalConstraint_energy_weight over \(\Xi \times \mathcal{B} \times \mathcal{T}\) — what one unit of power at a snapshot counts for in a row — the generator weighting times the years of the snapshot's period, where the row counts the snapshot, and nothing where it does not
\(\mathit{Generator\_primary\_energy}\) Generator_primary_energy over \(\mathcal{B} \times \Xi\)
\(\mathit{primary\_energy}\) primary_energy over \(\Xi \times \mathcal{B}\) — what a primary_energy row totals — weighted generator energy over the snapshots it counts, less the charge left in weighted storage at the close; the initial charge it is compared against is folded into the row's constant
\(\mathit{Line\_transmission\_volume\_expansion}\) Line_transmission_volume_expansion over \(\mathcal{B} \times \Xi\)
\(\mathit{transmission\_volume\_expansion}\) transmission_volume_expansion over \(\Xi \times \mathcal{B}\) — what a transmission_volume_expansion_limit row totals — length times the chosen build of the row's branches
\(\mathit{Generator\_opex}\) Generator_opex over \(\Xi\)
\(\mathit{Link\_opex}\) Link_opex over \(\Xi\)
\(\mathit{scenario\_opex}\) 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:415-432)
\(\mathit{Line\_capex}\) Line_capex (scalar)
\(\mathit{risk\_weighted\_opex}\) risk_weighted_opex (scalar)
\(\mathit{total\_cost}\) 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
\(\mathrm{Load\_demand}\) 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:1513-1514)
\(\mathit{Generator\_injection}\) Generator_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Line\_injection}\) Line_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Link\_injection}\) Link_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathrm{Load\_injection}\) Load_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Process\_injection}\) Process_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Bus\_injection}\) 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

Notation

Symbol Meaning
\(t \ominus k\) edge='wrap' — \(t-k\), cyclic
\(t \boxminus_{v} k\) edge=v — \(t-k\), and \(v\) past the edge
\(t \ominus^{\mathrm{property}(t)} k\) within=subject[...] — \(t-k\) inside the group of \(t\)

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

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

Line_s

\[ s_{\xi,t,k} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{on}^{s}_{t,k} \]

Line_s_nom_ext

\[ S_{k} \in \mathbb{R} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k} \]

Process_p

\[ z_{\xi,t,j} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{on}^{z}_{t,j} \]

Definitions

Link_output_arrival

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

Process_output_arrival

\[ \mathit{Process\_output\_arrival}_{\xi,t,r} = \begin{cases} z_{\xi,t \ominus^{\mathrm{period}(t)} \mathrm{d}^{z},\mathrm{process}(r)} \cdot \alpha_{\xi,t,r} & \text{if } \mathrm{cyc}^{z}_{r} \\ z_{\xi,t \boxminus_{0}^{\mathrm{period}(t)} \mathrm{d}^{z},\mathrm{process}(r)} \cdot \alpha_{\xi,t,r} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ r \in \mathcal{R} \]

GlobalConstraint_energy_weight

\[ \mathrm{GlobalConstraint\_energy\_weight}_{\xi,b,t} = \begin{cases} \mathrm{w}^{\mathrm{gen}}_{t} \cdot \mathrm{w}^{\mathrm{yr}}_{\mathrm{period}(t)} & \text{if } \mathrm{in}_{\xi,b,t} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B},\ t \in \mathcal{T} \]

Generator_primary_energy

\[ \mathit{Generator\_primary\_energy}_{b,\xi} = \sum_{g \in \mathcal{G}} \sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathrm{GlobalConstraint\_energy\_weight}_{\xi,b,t} \cdot \mathrm{a}_{\xi,b,t,g} \qquad \forall\, b \in \mathcal{B},\ \xi \in \Xi \]

primary_energy

\[ \mathit{primary\_energy}_{\xi,b} = \mathit{Generator\_primary\_energy}_{b,\xi} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \]

Line_transmission_volume_expansion

\[ \mathit{Line\_transmission\_volume\_expansion}_{b,\xi} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{len}_{\xi,b,k} \qquad \forall\, b \in \mathcal{B},\ \xi \in \Xi \]

transmission_volume_expansion

\[ \mathit{transmission\_volume\_expansion}_{\xi,b} = \mathit{Line\_transmission\_volume\_expansion}_{b,\xi} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \]

Generator_opex

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

Link_opex

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

scenario_opex

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

Line_capex

\[ \mathit{Line\_capex} = \sum_{\xi \in \Xi,\ k \in \mathcal{K}} \pi_{\xi} \cdot S_{k} \cdot \mathrm{c}^{\mathrm{cap},s}_{\xi,k} \cdot \mathrm{W}^{s}_{k} \]

risk_weighted_opex

\[ \mathit{risk\_weighted\_opex} = \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi} \]

total_cost

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

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_injection

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

Line_injection

\[ \mathit{Line\_injection}_{n,\xi,t} = -\left( \sum_{k \in \mathcal{K} \,:\, \mathrm{bus0}(k) = n} s_{\xi,t,k} \right) + \sum_{k \in \mathcal{K} \,:\, \mathrm{bus1}(k) = n} s_{\xi,t,k} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Link_injection

\[ \mathit{Link\_injection}_{n,\xi,t} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Load_injection

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

Process_injection

\[ \mathit{Process\_injection}_{n,\xi,t} = \sum_{r \in \mathcal{R} \,:\, \mathrm{bus}(r) = n} \mathit{Process\_output\_arrival}_{\xi,t,r} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Bus_injection

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

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

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

Line_ext_s_lower

\[ s_{\xi,t,k} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k} \]

Line_ext_s_upper

\[ s_{\xi,t,k} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k} \]

Line_ext_s_nom_lower

\[ S_{k} \ge \underline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k} \]

GlobalConstraint_primary_energy_ub

\[ \mathit{primary\_energy}_{\xi,b} \le \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{primary\_energy}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{<=}\text{'} \]

GlobalConstraint_primary_energy_eq

\[ \mathit{primary\_energy}_{\xi,b} = \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{primary\_energy}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{==}\text{'} \]

GlobalConstraint_transmission_volume_expansion_limit_ub

\[ \mathit{transmission\_volume\_expansion}_{\xi,b} \le \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_volume\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{<=}\text{'} \]

Bus_nodal_balance

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

The spec, differential/pypsa/rungs/rung_40_scenario_global_constraints.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 a risk preference is set.
  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, ordered: true}
  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'}
  process: {description: 'generalized multi-port converters, each with an internal power that every port
      draws or delivers at its own rate'}
  process_output: {description: 'a process''s ports, one label per port a process declares — PyPSA''s
      `bus0`, `bus1`, … each carry a signed `rate`, so a process of any number of ports is one term in
      the balance, data prep'}
  load: {description: 'demands, each on one bus'}
  line: {description: 'passive branches, each between two buses, their flow set by impedance'}
  global_constraint: {description: 'PyPSA''s `GlobalConstraint` rows, one label per declared limit'}
  period: {description: investment periods — PyPSA's `investment_periods`, dtype: int, ordered: true}
properties:
  snapshot:
    period:
      description: the investment period a snapshot falls in
      into: period
  generator:
    bus:
      description: the bus a generator sits on
      into: bus
  link:
    bus0:
      description: the bus a link leaves
      into: bus
  link_output:
    link:
      description: the link an output port belongs to
      into: link
    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
        property, so the file states any number of them
      into: bus
  process_output:
    process:
      description: the process a port belongs to
      into: process
    bus:
      description: >-
        the bus a port draws from or delivers to — PyPSA's `bus0`, `bus1`, …
        columns. A process of three ports is three labels here rather than a
        third property, so the file states any number of them
      into: bus
  load:
    bus:
      description: the bus a load sits on
      into: bus
  line:
    bus0:
      description: the bus a line's flow is measured at
      into: bus
    bus1:
      description: the bus at a line's other end
      into: 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 (`constants.py:43`)
    dims: [generator]
  Generator_committable:
    description: whether output is gated by an on/off status decision
    dims: [generator]
    dtype: bool
  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:1498`)
    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, divided by the `snapshot_weightings.generators` value and rounded up in data prep; zero for
      a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses
      a delay that differs by scenario (`constants.py:52`)
    dims: [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. The flag is the same in every scenario, as the delay is
    dims: [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
  Process_rate:
    description: the energy a port draws or delivers per unit of internal power, PyPSA's `rate0`, `rate1`,
      … read long — negative where the port withdraws, positive where it injects; a link is a process
      whose `bus0` rate is minus one and whose output rates are its efficiencies. Read at the snapshot
      the transfer arrives, so a delayed port transfers at its arrival snapshot's rate (`constraints.py:1498`)
    dims: [scenario, snapshot, process_output]
  Process_output_delay:
    description: snapshots a port's transfer lags its process's internal power — PyPSA's `delay0`, `delay1`,
      … read long, divided by the `snapshot_weightings.generators` value and rounded up in data prep;
      zero for a port that transfers at once. A port has the same delay in every scenario, as a link's
      port has
    dims: [process_output]
    dtype: int
  Process_output_cyclic_delay:
    description: whether a delayed port's transfer wraps from the end of its investment period — PyPSA's
      `cyclic_delay0`, `cyclic_delay1`, …; where it does not, the energy still in transit at each period's
      first snapshots is lost. The flag is the same in every scenario, as the delay is
    dims: [process_output]
    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 (`constants.py:43`)
    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 (`constants.py:51`)
    dims: [load]
    dtype: bool
  scenario_weight:
    description: PyPSA's `scenario_weightings.weight` — the probability of a future
    dims: [scenario]
  period_weight_objective:
    description: PyPSA's `investment_period_weightings.objective` — what a period's cost weighs; PyPSA
      reads it only under `multi_investment_periods`, so data prep feeds one otherwise, whatever the column
      holds (`optimize.py:205-207`, `:264-266`)
    dims: [period]
  period_weight_years:
    description: PyPSA's `investment_period_weightings.years` — what a period's energy weighs in a `primary_energy`
      or `operational_limit` row; PyPSA reads it only under `multi_investment_periods`, so data prep feeds
      one otherwise
    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
  Line_active:
    description: whether a line stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, line]
    dtype: bool
  Process_active:
    description: whether a process stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, process]
    dtype: bool
  Line_active_asset:
    description: whether a line is one of PyPSA's `active_assets` — its `active` flag alone, without build
      year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA refuses
      one that differs by scenario (`constants.py:51`)
    dims: [line]
    dtype: bool
  Line_capital_weight:
    description: the sum of period weights a line stands in — PyPSA's `active * period_weighting`, summed,
      data prep
    dims: [line]
  snapshot_weightings_generators:
    description: PyPSA's `snapshot_weightings.generators` — hours a snapshot stands for in an energy total
    dims: [snapshot]
  Line_s_nom_extendable:
    description: whether the nominal apparent power is a decision
    dims: [line]
    dtype: bool
  Line_s_max_pu:
    description: most flow either way, per unit of nominal apparent power
    dims: [scenario, snapshot, line]
  Line_s_nom_min:
    description: least nominal apparent power an extendable line may be built at
    dims: [scenario, line]
  Line_capital_cost:
    description: 'cost of one unit of nominal apparent power for the modelled horizon — PyPSA''s `periodized_cost`:
      `overnight_cost` as an annuity over `lifetime` at `discount_rate`, times `nyears`, where it is given,
      and `capital_cost` where it is not, plus `fom_cost` (`components.py:1126-1147`, `costs.py:102-203`),
      data prep'
    dims: [scenario, line]
  GlobalConstraint_type:
    description: which formula the row takes — `primary_energy`, `operational_limit`, `transmission_volume_expansion_limit`,
      `transmission_expansion_cost_limit` or `tech_capacity_expansion_limit`
    dims: [global_constraint]
    dtype: str
  GlobalConstraint_sense:
    description: which way the row binds in each scenario — `<=`, `>=` or `==`; PyPSA reads a row's sense
      per scenario (`global_constraints.py:551`, `:749`, `:829`)
    dims: [scenario, global_constraint]
    dtype: str
  GlobalConstraint_constant:
    description: the constant the total is held against; what a variable cannot carry — an initial charge,
      times its period's years for each counted period where the storage reopens per period, or a non-extendable
      build — is folded in here by data prep. PyPSA reads it per scenario (`global_constraints.py:552`,
      `:750`, `:829`)
    dims: [scenario, global_constraint]
  GlobalConstraint_counts_snapshot:
    description: 'whether a row counts a snapshot in a scenario — PyPSA''s `investment_period`: every
      snapshot where the row names none, and only that period''s where it names one, data prep. A row
      that names a period the run does not model has no label here, as PyPSA skips it (`global_constraints.py:372`);
      PyPSA reads the column only under `multi_investment_periods`, and fails on a row that names a period
      without it (`global_constraints.py:364-369`)'
    dims: [scenario, global_constraint, snapshot]
    dtype: bool
    missing: false
  Generator_primary_energy_weight:
    description: the constrained attribute per unit of energy at the bus — the carrier's `co2_emissions`
      over the generator's efficiency at the snapshot, data prep; a generator of an unweighted carrier
      has no row
    dims: [scenario, global_constraint, snapshot, generator]
    missing: 0
  Line_volume_weight:
    description: the line's length in each scenario where its carrier is in the row's set (`global_constraints.py:847-848`)
      — data prep; a line outside it, or one that does not stand in the row's `investment_period`, has
      no row
    dims: [scenario, global_constraint, line]
    missing: 0
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
  Line_s:
    description: '`Line-s` — PyPSA''s `p0`, the flow measured at the `line[bus0]` end: a positive value
      withdraws there and injects at `line[bus1]`, lossless'
    dims: [scenario, snapshot, line]
    where: Line_active
  Line_s_nom_ext:
    description: '`Line-s_nom` — nominal apparent power where it is a decision, for a line that is active;
      the parameter of the same PyPSA name carries the fixed regime'
    dims: [line]
    where: Line_s_nom_extendable AND Line_active_asset
  Process_p:
    description: '`Process-p` — PyPSA''s internal power `p`: a positive value drives every port at its
      own rate, withdrawing where the rate is negative and injecting where it is positive'
    dims: [scenario, snapshot, process]
    where: Process_active
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
  Line_ext_s_lower:
    description: '`Line-ext-s-lower` — an extendable line carries at least the negative of its rating
      of the chosen build, the loss counted against it'
    dims: [scenario, snapshot, line]
    where: Line_s_nom_extendable AND Line_active
    expression: Line_s >= (-Line_s_max_pu) * Line_s_nom_ext
  Line_ext_s_upper:
    description: '`Line-ext-s-upper` — an extendable line carries at most its rating of the chosen build,
      the loss included'
    dims: [scenario, snapshot, line]
    where: Line_s_nom_extendable AND Line_active
    expression: Line_s <= Line_s_max_pu * Line_s_nom_ext
  Line_ext_s_nom_lower:
    description: '`Line-ext-s_nom-lower` — the chosen build is at least its floor in every scenario'
    dims: [scenario, line]
    where: Line_s_nom_extendable AND Line_active_asset
    expression: Line_s_nom_ext >= Line_s_nom_min
  GlobalConstraint_primary_energy_ub:
    description: '`primary_energy` — its total, at most its constant'
    dims: [scenario, global_constraint]
    where: GlobalConstraint_type == 'primary_energy' AND GlobalConstraint_sense == '<='
    expression: primary_energy <= GlobalConstraint_constant
  GlobalConstraint_primary_energy_eq:
    description: '`primary_energy` — its total, at its constant'
    dims: [scenario, global_constraint]
    where: GlobalConstraint_type == 'primary_energy' AND GlobalConstraint_sense == '=='
    expression: primary_energy == GlobalConstraint_constant
  GlobalConstraint_transmission_volume_expansion_limit_ub:
    description: '`transmission_volume_expansion_limit` — its total, at most its constant'
    dims: [scenario, global_constraint]
    where: GlobalConstraint_type == 'transmission_volume_expansion_limit' AND GlobalConstraint_sense ==
      '<='
    expression: transmission_volume_expansion <= GlobalConstraint_constant
  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:1404-1405`, `:1514`), 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:
  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]),
          along=snapshot, offset=Link_output_delay, edge=''wrap'', within=snapshot[period]) * Link_efficiency'}
    otherwise: shift(at(Link_p, by=link_output[link]), along=snapshot, offset=Link_output_delay,
      edge=0, within=snapshot[period]) * Link_efficiency
  Process_output_arrival:
    description: what a process transfers at a port at a snapshot — its internal power delayed by the
      port's `delay` within its investment period, times the port's rate at the snapshot the transfer
      arrives; where the port is `cyclic_delay` the delayed transfer wraps from the period's end, and
      where it is not the energy still in transit at the period's first snapshots is lost. A port that
      does not delay (`delay` zero) transfers at once, cyclic or not
    dims: [scenario, snapshot, process_output]
    cases:
      wrapping: {when: Process_output_cyclic_delay, expression: 'shift(at(Process_p, by=process_output[process]),
          along=snapshot, offset=Process_output_delay, edge=''wrap'', within=snapshot[period])
          * Process_rate'}
    otherwise: shift(at(Process_p, by=process_output[process]), along=snapshot, offset=Process_output_delay,
      edge=0, within=snapshot[period]) * Process_rate
  GlobalConstraint_energy_weight:
    description: what one unit of power at a snapshot counts for in a row — the generator weighting times
      the years of the snapshot's period, where the row counts the snapshot, and nothing where it does
      not
    dims: [scenario, global_constraint, snapshot]
    cases:
      counted: {when: GlobalConstraint_counts_snapshot, expression: 'snapshot_weightings_generators *
          at(period_weight_years, by=snapshot[period])'}
    otherwise: 0
  Generator_primary_energy: {expression: 'sum(sum((Generator_p * GlobalConstraint_energy_weight) * Generator_primary_energy_weight,
      over=snapshot), over=generator)'}
  primary_energy:
    dims: [scenario, global_constraint]
    expression: Generator_primary_energy
    description: what a `primary_energy` row totals — weighted generator energy over the snapshots it
      counts, less the charge left in weighted storage at the close; the initial charge it is compared
      against is folded into the row's constant
  Line_transmission_volume_expansion: {expression: 'sum(Line_s_nom_ext * Line_volume_weight, over=line)'}
  transmission_volume_expansion:
    dims: [scenario, global_constraint]
    expression: Line_transmission_volume_expansion
    description: what a `transmission_volume_expansion_limit` row totals — length times the chosen build
      of the row's branches
  Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot[period]), 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=generator), over=snapshot)'}
  Link_opex: {expression: 'sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot[period]), 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=link),
      over=snapshot)'}
  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:415-432`)
  Line_capex: {expression: sum(scenario_weight * Line_s_nom_ext * Line_capital_cost * Line_capital_weight)}
  risk_weighted_opex: {expression: 'sum(scenario_weight * scenario_opex, over=scenario)'}
  total_cost:
    dims: []
    expression: Line_capex + 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
  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:1513-1514`)
    dims: [scenario, snapshot, load]
    cases:
      active: {when: Load_active, expression: Load_sign * Load_p_set}
    otherwise: 0
  Generator_injection: {expression: 'sum(Generator_sign * Generator_p, over=generator, by=generator[bus])'}
  Line_injection: {expression: '(-sum(Line_s, over=line, by=line[bus0])) + sum(Line_s, over=line,
      by=line[bus1])'}
  Link_injection: {expression: '-sum(Link_p, over=link, by=link[bus0]) + sum(Link_output_arrival,
      over=link_output, by=link_output[bus])'}
  Load_injection: {expression: 'sum(Load_demand, over=load, by=load[bus])'}
  Process_injection: {expression: 'sum(Process_output_arrival, over=process_output, by=process_output[bus])'}
  Bus_injection:
    dims: [scenario, snapshot, bus]
    expression: (((Generator_injection + Line_injection) + Link_injection) + Load_injection) + Process_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
objective: {sense: minimize, expression: total_cost}

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

from differential.pypsa.prep import prop, 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, multi),
        **carriers(n, multi),
    'generator.bus': prop(n, 'Generator', 'bus'),
    'link.bus0': prop(n, 'Link', 'bus0'),
    'load.bus': prop(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),
    'snapshot_weightings_generators': weighting(n, 'generators'),
}

with sps.solve('differential/pypsa/rungs/rung_40_scenario_global_constraints.yaml', sources) as solution:
    solution.objective  # 15106.666667

The network, rung_40_scenario_global_constraints.py in the corpus — the spine plus what this rung adds:

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

"""Rung 40: a global constraint takes its own constant and sense in each scenario."""

from __future__ import annotations

import spine

#: each scenario's own constant and sense, per row
PER_SCENARIO = {
    ('calm', 'volume40'): {'constant': 60},
    ('stormy', 'volume40'): {'constant': 20},
    ('calm', 'co2_40'): {'constant': 250, 'sense': '<='},
    ('stormy', 'co2_40'): {'constant': 200, 'sense': '=='},
}


def build():
    """The spine over two futures, with an extendable line under a volume limit and a CO2 row that differ by scenario."""
    n = spine.build()
    n.add('Carrier', 'AC')
    n.add('Carrier', 'coalc', co2_emissions=0.5)
    n.c.generators.static.loc['coal', 'carrier'] = 'coalc'
    n.add(
        'Line',
        'tie40',
        bus0='north',
        bus1='south',
        x=0.1,
        carrier='AC',
        length=2,
        s_nom_extendable=True,
        capital_cost=1,
    )
    n.add('Load', 'port40', bus='south', p_set=30)
    n.add(
        'GlobalConstraint',
        'volume40',
        type='transmission_volume_expansion_limit',
        carrier_attribute='AC',
        sense='<=',
        constant=60,
    )
    n.add(
        'GlobalConstraint', 'co2_40', type='primary_energy', carrier_attribute='co2_emissions', sense='<=', constant=250
    )
    n.set_scenarios({'calm': 0.6, 'stormy': 0.4})
    for row, values in PER_SCENARIO.items():
        for column, value in values.items():
            n.c.global_constraints.static.loc[row, column] = value
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 15106.666667

The data

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