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Rung 7: commitment — committable units with up and down times and ramp limits at the transitions

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 7775.0 on both sides; structure ✔ 17 constraints · 5 variables, name for name; size ✔ 116 rows · ✔ 44 columns · ✔ 237 nonzeros; duals — integer model, no duals; model for model: 19 blocks equal, 0 documented splits, 4 recorded deviations.

Rows and columns, PyPSA against specsolve, name for name
row PyPSA specsolve
Bus-nodal_balance 8 8
Generator-com-down-time 6 6
Generator-com-p-lower 8 8
Generator-com-p-upper 8 8
Generator-com-status-min_up_time_must_stay_up 2 2
Generator-com-transition-shut-down 8 8
Generator-com-transition-start-up 8 8
Generator-com-up-time 6 6
Generator-fix-p-lower 8 8
Generator-fix-p-upper 8 8
Generator-p-ramp_limit_down 7 7
Generator-p-ramp_limit_up 7 7
Generator-shut_down-p-fixed-upper 8 8
Generator-start_up-p-fixed-upper 8 8
Generator-status-p-fixed-upper 8 8
Link-fix-p-lower 4 4
Link-fix-p-upper 4 4
column PyPSA specsolve
Generator-p 16 16
Generator-shut_down 8 8
Generator-start_up 8 8
Generator-status 8 8
Link-p 4 4

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 \(\mathrm{bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{bus}: \mathcal{D} \to \mathcal{N}\) — 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
\(\mathcal{Y}\) 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
\(\mathrm{ru}\) Generator_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\), \(0\) where the data has no row — 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 \(\Xi \times \mathcal{T} \times \mathcal{G}\), \(0\) where the data has no row — 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 \(\Xi \times \mathcal{G}\), \(0\) where the data has no row — most output in the snapshot a unit starts, per unit of nominal power; no value means no limit
\(\mathrm{rd}^{\mathrm{dn}}\) Generator_ramp_limit_shut_down over \(\Xi \times \mathcal{G}\), \(0\) where the data has no row — most output in the snapshot before a unit stops, per unit of nominal power; no value means no limit
\(\mathrm{UT}\) Generator_min_up_time over \(\Xi \times \mathcal{G}\) — least snapshots a unit stays on once started
\(\mathrm{DT}\) Generator_min_down_time over \(\Xi \times \mathcal{G}\) — least snapshots a unit stays off once stopped
\(\mathrm{u}^{0}\) 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 \(\Xi \times \mathcal{G}\), \(0\) where the data has no row — 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
\(\mathrm{hold}\) Generator_must_stay_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — true while the up time a unit brought into the horizon still binds — data prep, since position() compares against a literal rather than a parameter
\(\mathrm{c}^{\mathrm{up}}\) Generator_start_up_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one start in this snapshot
\(\mathrm{c}^{\mathrm{dn}}\) Generator_shut_down_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one stop in this snapshot
\(\mathrm{c}^{\mathrm{on}}\) Generator_stand_by_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one snapshot spent on
\(\mathrm{p}^{\mathrm{mod}}\) Generator_p_nom_mod over \(\mathcal{G}\), absent where the data has no row — the module size a build comes in whole numbers of; no value means the build is continuous
\(\mathrm{N}^{\mathrm{fix}}\) Generator_modules_installed over \(\Xi \times \mathcal{G}\) — how many whole modules a committable build has in place: Generator_p_nom / Generator_p_nom_mod where a fixed build is modular, one where it is not, data prep. PyPSA refuses a fixed modular build whose nominal power is not a whole number of modules
\(\mathrm{f}^{\mathrm{nom}}\) 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)
\(\mathrm{on}\) 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
\(\mathrm{on}^{z}\) Process_active over \(\mathcal{T} \times \mathcal{J}\) — whether a process stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{act}\) Generator_active_asset over \(\mathcal{G}\) — whether a generator 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)

Conditions

Symbol Meaning
\(\mathrm{Generator\_committed}\) Generator_committed over \(\mathcal{G} \times \mathcal{T}\) — a committable generator that stands in the snapshot's period — every unit-commitment row's set
\(\mathrm{Generator\_ramps\_from\_previous}\) Generator_ramps_from_previous over \(\mathcal{G} \times \Xi \times \mathcal{T}\) — a snapshot whose ramp reads a previous output — any snapshot but its period's first, and the horizon's first where the generator comes in off or carries an initial output

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
\(u\) 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
\(\mathit{up}\) Generator_start_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-start_up — how much of a committable unit turns on this snapshot, capped as the status is
\(\mathit{dn}\) Generator_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-shut_down — how much of a committable unit turns off this snapshot, capped as the status is
\(P\) Generator_p_nom_ext over \(\mathcal{G}\) — Generator-p_nom — nominal power where it is a decision, for a generator 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
\(\mathit{Generator\_previous\_status}\) Generator_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the commitment state a generator carries into a snapshot — off at the first snapshot it stands in past the first of the horizon, as PyPSA reads a status it did not build (constraints.py:298), and the state carried over otherwise
\(\mathit{Generator\_status\_carried\_over}\) Generator_status_carried_over over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the state a generator carries over into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that
\(\mathit{Generator\_previous\_p}\) 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
\(\mathit{Generator\_p\_nom\_effective}\) 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
\(\mathit{Generator\_ramp\_up\_allowance}\) 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
\(\mathit{Link\_output\_arrival}\) 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
\(\mathit{Generator\_opex}\) Generator_opex over \(\Xi\)
\(\mathit{Generator\_commitment\_opex}\) Generator_commitment_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)
\(\mathit{risk\_weighted\_opex}\) 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 \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Link\_injection}\) 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 \(t\)
\(\mathrm{pos}(t)\) position() — the position of \(t\), counted from \(0\); \(t\) is the label
\(\mathrm{pos}_{\mathrm{property}(t)}(t)\) position(within=subject[...]) — the position of \(t\) inside its group

Conditions

Generator_committed

\[ \mathrm{Generator\_committed}_{g,t} \iff \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \qquad \forall\, g \in \mathcal{G},\ t \in \mathcal{T} \]

Generator_ramps_from_previous

\[ \mathrm{Generator\_ramps\_from\_previous}_{g,\xi,t} \iff \mathrm{pos}_{\mathrm{period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \qquad \forall\, g \in \mathcal{G},\ \xi \in \Xi,\ t \in \mathcal{T} \]

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

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

Generator_status

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

Generator_start_up

\[ \mathit{up}_{\xi,t,g} \ge 0, \mathit{up}_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \]

Generator_shut_down

\[ \mathit{dn}_{\xi,t,g} \ge 0, \mathit{dn}_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \]

Generator_p_nom_ext

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

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

Generator_previous_status

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

Generator_status_carried_over

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

Generator_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_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

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

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_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

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

Generator_opex

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

Generator_commitment_opex

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

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

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

risk_weighted_opex

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

total_cost

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

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

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

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

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

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

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

Generator_com_p_lower

\[ p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \]

Generator_com_p_upper

\[ p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \]

Generator_com_transition_start_up

\[ \mathit{up}_{\xi,t,g} \ge u_{\xi,t,g} - \mathit{Generator\_previous\_status}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \]

Generator_com_transition_shut_down

\[ \mathit{dn}_{\xi,t,g} \ge \mathit{Generator\_previous\_status}_{\xi,t,g} - u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \]

Generator_com_up_time

\[ \sum_{t' \in \mathcal{T} \,:\, 0 \le t - t' < \mathrm{UT}} \mathit{up}_{\xi,t',g} \le u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \mathrm{UT}_{\xi,g} > 0 \wedge \mathrm{pos}(t) > 0 \]

Generator_com_down_time

\[ \sum_{t' \in \mathcal{T} \,:\, 0 \le t - t' < \mathrm{DT}} \mathit{dn}_{\xi,t',g} \le 1 - u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \mathrm{DT}_{\xi,g} > 0 \wedge \mathrm{pos}(t) > 0 \]

Generator_com_status_must_stay_up

\[ u_{\xi,t,g} = 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \mathrm{hold}_{\xi,t,g} \]

Generator_status_p_fixed_upper

\[ u_{\xi,t,g} \le \mathrm{N}^{\mathrm{fix}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \left( \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \]

Generator_start_up_p_fixed_upper

\[ \mathit{up}_{\xi,t,g} \le \mathrm{N}^{\mathrm{fix}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \left( \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \]

Generator_shut_down_p_fixed_upper

\[ \mathit{dn}_{\xi,t,g} \le \mathrm{N}^{\mathrm{fix}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \left( \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \]

Generator_p_ramp_limit_up

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

Generator_p_ramp_limit_down

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

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_07_commitment.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'}
  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

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
  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]
    missing: 0
  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]
    missing: 0
  Generator_ramp_limit_start_up:
    description: most output in the snapshot a unit starts, per unit of nominal power; no value means
      no limit
    dims: [scenario, generator]
    missing: 0
  Generator_ramp_limit_shut_down:
    description: most output in the snapshot before a unit stops, per unit of nominal power; no value
      means no limit
    dims: [scenario, generator]
    missing: 0
  Generator_min_up_time:
    description: least snapshots a unit stays on once started
    dims: [scenario, generator]
    dtype: int
  Generator_min_down_time:
    description: least snapshots a unit stays off once stopped
    dims: [scenario, generator]
    dtype: int
  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]
    missing: 0
  Generator_must_stay_up:
    description: true while the up time a unit brought into the horizon still binds — data prep, since
      `position()` compares against a literal rather than a parameter
    dims: [scenario, snapshot, generator]
    dtype: bool
  Generator_start_up_cost:
    description: cost of one start in this snapshot
    dims: [scenario, snapshot, generator]
  Generator_shut_down_cost:
    description: cost of one stop in this snapshot
    dims: [scenario, snapshot, generator]
  Generator_stand_by_cost:
    description: cost of one snapshot spent on
    dims: [scenario, snapshot, 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]
    missing: absent
  Generator_modules_installed:
    description: 'how many whole modules a committable build has in place: `Generator_p_nom / Generator_p_nom_mod`
      where a fixed build is modular, one where it is not, data prep. PyPSA refuses a fixed modular build
      whose nominal power is not a whole number of modules'
    dims: [scenario, generator]
  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]
  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
  Process_active:
    description: whether a process stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, process]
    dtype: bool
  Generator_active_asset:
    description: whether a generator 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: [generator]
    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
  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_committed
    domain: integer
    bounds: {lower: 0}
  Generator_start_up:
    description: '`Generator-start_up` — how much of a committable unit turns on this snapshot, capped
      as the status is'
    dims: [scenario, snapshot, generator]
    where: Generator_committed
    domain: integer
    bounds: {lower: 0}
  Generator_shut_down:
    description: '`Generator-shut_down` — how much of a committable unit turns off this snapshot, capped
      as the status is'
    dims: [scenario, snapshot, generator]
    where: Generator_committed
    domain: integer
    bounds: {lower: 0}
  Generator_p_nom_ext:
    description: '`Generator-p_nom` — nominal power where it is a decision, for a generator that is active;
      the parameter of the same PyPSA name carries the fixed regime'
    dims: [generator]
    where: Generator_p_nom_extendable AND Generator_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
  Generator_com_p_lower:
    description: '`Generator-com-p-lower` — a committed unit outputs at least its minimum; off, at least
      nothing'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
    expression: Generator_p >= (Generator_p_min_pu * Generator_p_nom) * Generator_status
  Generator_com_p_upper:
    description: '`Generator-com-p-upper` — a committed unit outputs at most what is available; off, at
      most nothing'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
    expression: Generator_p <= (Generator_p_max_pu * Generator_p_nom) * Generator_status
  Generator_com_transition_start_up:
    description: '`Generator-com-transition-start-up` — turning on is a start, counted against the state
      the unit carried into the snapshot'
    dims: [scenario, snapshot, generator]
    where: Generator_committed
    expression: Generator_start_up >= Generator_status - Generator_previous_status
  Generator_com_transition_shut_down:
    description: '`Generator-com-transition-shut-down` — turning off is a stop, counted against the state
      the unit carried into the snapshot'
    dims: [scenario, snapshot, generator]
    where: Generator_committed
    expression: Generator_shut_down >= Generator_previous_status - Generator_status
  Generator_com_up_time:
    description: '`Generator-com-up-time` — a unit started within its own minimum up time is still on.
      The first snapshot''s share of the window is the brought-in up time''s, which the must-stay-up mask
      carries'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND Generator_min_up_time > 0 AND position(snapshot) > 0
    expression: sum_back(Generator_start_up, along=snapshot, window=Generator_min_up_time) <= Generator_status
  Generator_com_down_time:
    description: '`Generator-com-down-time` — a unit stopped within its own minimum down time is still
      off. The first snapshot''s share of the window is the brought-in down time''s, which the must-stay-down
      mask carries'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND Generator_min_down_time > 0 AND position(snapshot) > 0
    expression: sum_back(Generator_shut_down, along=snapshot, window=Generator_min_down_time) <= 1 - Generator_status
  Generator_com_status_must_stay_up:
    description: '`Generator-com-status-min_up_time_must_stay_up` — a unit still serving the up time it
      brought in stays on'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND Generator_must_stay_up
    expression: Generator_status == 1
  Generator_status_p_fixed_upper:
    description: '`Generator-status-p-fixed-upper` — a status is at most the modules in place, an explicit
      row as PyPSA writes it: one where the build is not modular, and the fixed build''s whole count of
      modules where it is'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND NOT (Generator_p_nom_extendable AND Generator_p_nom_mod > 0)
    expression: Generator_status <= Generator_modules_installed
  Generator_start_up_p_fixed_upper:
    description: '`Generator-start_up-p-fixed-upper` — a start is at most the modules in place, an explicit
      row as PyPSA writes it: one where the build is not modular, and the fixed build''s whole count of
      modules where it is'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND NOT (Generator_p_nom_extendable AND Generator_p_nom_mod > 0)
    expression: Generator_start_up <= Generator_modules_installed
  Generator_shut_down_p_fixed_upper:
    description: '`Generator-shut_down-p-fixed-upper` — a stop is at most the modules in place, an explicit
      row as PyPSA writes it: one where the build is not modular, and the fixed build''s whole count of
      modules where it is'
    dims: [scenario, snapshot, generator]
    where: Generator_committed AND NOT (Generator_p_nom_extendable AND Generator_p_nom_mod > 0)
    expression: Generator_shut_down <= Generator_modules_installed
  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 Generator_ramps_from_previous
      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 Generator_ramps_from_previous
      AND Generator_active
    expression: Generator_previous_p - Generator_p <= Generator_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: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:
  Generator_previous_status:
    description: the commitment state a generator carries into a snapshot — off at the first snapshot
      it stands in past the first of the horizon, as PyPSA reads a status it did not build (`constraints.py:298`),
      and the state carried over otherwise
    dims: [scenario, snapshot, generator]
    cases:
      opening_late: {when: 'position(snapshot) > 0 AND NOT shift(Generator_active, along=snapshot, offset=1)',
        expression: 0}
    otherwise: Generator_status_carried_over
  Generator_status_carried_over:
    description: the state a generator carries over 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_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_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
  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_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
  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)'}
  Generator_commitment_opex: {expression: 'sum(sum(((Generator_status * Generator_stand_by_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot[period]), over=generator), over=snapshot) + sum(sum(Generator_start_up
      * Generator_start_up_cost, over=generator), over=snapshot) + sum(sum(Generator_shut_down * Generator_shut_down_cost,
      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 + Generator_commitment_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`)
  risk_weighted_opex: {expression: 'sum(scenario_weight * scenario_opex, over=scenario)'}
  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
  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])'}
  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 + 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
conditions:
  Generator_committed: {description: a committable generator that stands in the snapshot's period — every
      unit-commitment row's set, condition: Generator_committable AND Generator_active}
  Generator_ramps_from_previous: {description: 'a snapshot whose ramp reads a previous output — any snapshot
      but its period''s first, and the horizon''s first where the generator comes in off or carries an
      initial output', condition: 'position(snapshot, within=snapshot[period]) > 0 OR (position(snapshot)
      == 0 AND (Generator_status_initial == 0 OR Generator_p_init))'}
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),
}

with sps.solve('differential/pypsa/rungs/rung_07_commitment.yaml', sources) as solution:
    solution.objective  # 7775.0

The network, rung_07_commitment.py in the corpus — the spine plus what this rung adds:

"""Rung 7: commitment — committable units with up and down times and ramp limits at the transitions."""

from __future__ import annotations

import spine


def build():
    """The spine plus this rung's additions, as a ``pypsa.Network``."""
    n = spine.build()
    n.add(
        'Generator',
        'uc',
        bus='north',
        committable=True,
        p_nom=50,
        marginal_cost=5,
        p_min_pu=0.4,
        min_up_time=3,
        min_down_time=2,
        up_time_before=1,
        ramp_limit_up=0.5,
        ramp_limit_down=0.5,
        ramp_limit_start_up=0.6,
        ramp_limit_shut_down=0.6,
        start_up_cost=100,
        shut_down_cost=50,
        stand_by_cost=5,
    )
    n.add(
        'Generator',
        'cold',
        bus='south',
        committable=True,
        p_nom=30,
        marginal_cost=60,
        p_min_pu=0.3,
        min_up_time=2,
        min_down_time=1,
        up_time_before=0,
        ramp_limit_up=0.5,
        ramp_limit_down=0.5,
        start_up_cost=80,
    )
    n.add('Load', 'swing7', bus='north', p_set=[25, 45, 45, 10])
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 7775.0

The data

The tables this rung is the first to declare (10), as the prep produced them:

Generator_min_down_time.csv

scenario,generator,value
base,coal,0
base,cold,1
base,gas,0
base,uc,2

Generator_min_up_time.csv

scenario,generator,value
base,coal,0
base,cold,2
base,gas,0
base,uc,3

Generator_modules_installed.csv

scenario,generator,value
base,coal,1.0
base,cold,1.0
base,gas,1.0
base,uc,1.0

Generator_must_stay_up.csv

scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,coal,false
base,2015-01-01T00:00:00.000000,cold,false
base,2015-01-01T00:00:00.000000,gas,false
base,2015-01-01T00:00:00.000000,uc,true
base,2015-01-01T01:00:00.000000,coal,false
base,2015-01-01T01:00:00.000000,cold,false
base,2015-01-01T01:00:00.000000,gas,false
base,2015-01-01T01:00:00.000000,uc,true
base,2015-01-01T02:00:00.000000,coal,false
base,2015-01-01T02:00:00.000000,cold,false
base,2015-01-01T02:00:00.000000,gas,false
base,2015-01-01T02:00:00.000000,uc,false
base,2015-01-01T03:00:00.000000,coal,false
base,2015-01-01T03:00:00.000000,cold,false
base,2015-01-01T03:00:00.000000,gas,false
base,2015-01-01T03:00:00.000000,uc,false

Generator_p_init.csv

scenario,generator,value
base,cold,0.0

Generator_ramp_limit_shut_down.csv

scenario,generator,value
base,uc,0.6

Generator_ramp_limit_start_up.csv

scenario,generator,value
base,uc,0.6

Generator_shut_down_cost.csv

scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,coal,0.0
base,2015-01-01T00:00:00.000000,cold,0.0
base,2015-01-01T00:00:00.000000,gas,0.0
base,2015-01-01T00:00:00.000000,uc,50.0
base,2015-01-01T01:00:00.000000,coal,0.0
base,2015-01-01T01:00:00.000000,cold,0.0
base,2015-01-01T01:00:00.000000,gas,0.0
base,2015-01-01T01:00:00.000000,uc,50.0
base,2015-01-01T02:00:00.000000,coal,0.0
base,2015-01-01T02:00:00.000000,cold,0.0
base,2015-01-01T02:00:00.000000,gas,0.0
base,2015-01-01T02:00:00.000000,uc,50.0
base,2015-01-01T03:00:00.000000,coal,0.0
base,2015-01-01T03:00:00.000000,cold,0.0
base,2015-01-01T03:00:00.000000,gas,0.0
base,2015-01-01T03:00:00.000000,uc,50.0

Generator_stand_by_cost.csv

scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,coal,0.0
base,2015-01-01T00:00:00.000000,cold,0.0
base,2015-01-01T00:00:00.000000,gas,0.0
base,2015-01-01T00:00:00.000000,uc,5.0
base,2015-01-01T01:00:00.000000,coal,0.0
base,2015-01-01T01:00:00.000000,cold,0.0
base,2015-01-01T01:00:00.000000,gas,0.0
base,2015-01-01T01:00:00.000000,uc,5.0
base,2015-01-01T02:00:00.000000,coal,0.0
base,2015-01-01T02:00:00.000000,cold,0.0
base,2015-01-01T02:00:00.000000,gas,0.0
base,2015-01-01T02:00:00.000000,uc,5.0
base,2015-01-01T03:00:00.000000,coal,0.0
base,2015-01-01T03:00:00.000000,cold,0.0
base,2015-01-01T03:00:00.000000,gas,0.0
base,2015-01-01T03:00:00.000000,uc,5.0

Generator_start_up_cost.csv

scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,coal,0.0
base,2015-01-01T00:00:00.000000,cold,80.0
base,2015-01-01T00:00:00.000000,gas,0.0
base,2015-01-01T00:00:00.000000,uc,100.0
base,2015-01-01T01:00:00.000000,coal,0.0
base,2015-01-01T01:00:00.000000,cold,80.0
base,2015-01-01T01:00:00.000000,gas,0.0
base,2015-01-01T01:00:00.000000,uc,100.0
base,2015-01-01T02:00:00.000000,coal,0.0
base,2015-01-01T02:00:00.000000,cold,80.0
base,2015-01-01T02:00:00.000000,gas,0.0
base,2015-01-01T02:00:00.000000,uc,100.0
base,2015-01-01T03:00:00.000000,coal,0.0
base,2015-01-01T03:00:00.000000,cold,80.0
base,2015-01-01T03:00:00.000000,gas,0.0
base,2015-01-01T03:00:00.000000,uc,100.0