Rung 25: committable links — a link held on by the up time it brought in, one held off by its down time, one kept on by its own up time, and committable builds that are extendable, modular or both¶
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 13982.0 on both sides; structure ✔ 34 constraints · 7 variables, name for name; size ✔ 227 rows · ✔ 95 columns · ✔ 491 nonzeros; duals — integer model, no duals; model for model: 41 blocks equal, 0 documented splits, 1 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
12 | 12 |
Generator-fix-p-lower |
8 | 8 |
Generator-fix-p-upper |
8 | 8 |
Link-com-down-time |
6 | 6 |
Link-com-ext-p-lower |
4 | 4 |
Link-com-ext-p-lower-nonneg |
4 | 4 |
Link-com-ext-p-upper-bigM |
4 | 4 |
Link-com-ext-p-upper-cap |
4 | 4 |
Link-com-mod-p-lower |
8 | 8 |
Link-com-mod-p-upper |
8 | 8 |
Link-com-p-lower |
8 | 8 |
Link-com-p-upper |
8 | 8 |
Link-com-status-min_down_time_must_stay_up |
2 | 2 |
Link-com-status-min_up_time_must_stay_up |
2 | 2 |
Link-com-transition-shut-down |
20 | 20 |
Link-com-transition-start-up |
20 | 20 |
Link-com-up-time |
6 | 6 |
Link-ext-p_nom-lower |
2 | 2 |
Link-ext-p_nom-upper |
2 | 2 |
Link-fix-p-lower |
4 | 4 |
Link-fix-p-upper |
4 | 4 |
Link-p-ramp_limit_down |
3 | 3 |
Link-p-ramp_limit_down-run-bigM |
4 | 4 |
Link-p-ramp_limit_down-shut-bigM |
4 | 4 |
Link-p-ramp_limit_up |
3 | 3 |
Link-p-ramp_limit_up-run-bigM |
4 | 4 |
Link-p-ramp_limit_up-start-bigM |
4 | 4 |
Link-p_nom_modularity |
1 | 1 |
Link-shut_down-p-fixed-upper |
16 | 16 |
Link-shut_down-p_nom-variable-upper |
4 | 4 |
Link-start_up-p-fixed-upper |
16 | 16 |
Link-start_up-p_nom-variable-upper |
4 | 4 |
Link-status-p-fixed-upper |
16 | 16 |
Link-status-p_nom-variable-upper |
4 | 4 |
| column | PyPSA | specsolve |
|---|---|---|
Generator-p |
8 | 8 |
Link-n_mod |
1 | 1 |
Link-p |
24 | 24 |
Link-p_nom |
2 | 2 |
Link-shut_down |
20 | 20 |
Link-start_up |
20 | 20 |
Link-status |
20 | 20 |
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 \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{O}\) | index \(o\) — link_output with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep |
| \(\mathcal{J}\) | index \(j\) — process with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J}\) — 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 \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N}\) — 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 \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters¶
| Symbol | Meaning |
|---|---|
| \(\mathrm{w}\) | snapshot_weightings_objective over \(\mathcal{T}\) — PyPSA's snapshot_weightings.objective — hours a snapshot stands for in the cost |
| \(\mathrm{p}^{\mathrm{nom}}\) | Generator_p_nom over \(\Xi \times \mathcal{G}\) — nominal power |
| \(\mathrm{ext}\) | Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{p}}\) | Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power |
| \(\overline{\mathrm{p}}\) | Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile |
| \(\mathrm{c}\) | Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output |
| \(\mathrm{c}^{(2)}\) | Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output |
| \(\mathrm{sgn}\) | Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (constants.py:43) |
| \(\mathrm{com}\) | Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision |
| \(\mathrm{ru}^{f}\) | Link_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\), neutral where the data has no row — most a link may raise its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{rd}^{f}\) | Link_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\), neutral where the data has no row — most a link may lower its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{f}^{\mathrm{nom}}\) | Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power |
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{f}}\) | Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways |
| \(\overline{\mathrm{f}}\) | Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power |
| \(\eta\) | Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py: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 |
| \(\mathrm{ru}^{f,\mathrm{up}}\) | Link_ramp_limit_start_up over \(\Xi \times \mathcal{L}\), neutral where the data has no row — most flow in the snapshot a link starts, per unit of nominal power; no value means no limit |
| \(\mathrm{rd}^{f,\mathrm{dn}}\) | Link_ramp_limit_shut_down over \(\Xi \times \mathcal{L}\), neutral where the data has no row — most flow in the snapshot before a link stops, per unit of nominal power; no value means no limit |
| \(\mathrm{UT}^{f}\) | Link_min_up_time over \(\Xi \times \mathcal{L}\) — least snapshots a link stays on once started |
| \(\mathrm{DT}^{f}\) | Link_min_down_time over \(\Xi \times \mathcal{L}\) — least snapshots a link stays off once stopped |
| \(\mathrm{u}^{f,0}\) | Link_status_initial over \(\Xi \times \mathcal{L}\) — one where the link was on before the first snapshot, zero where off — PyPSA's up_time_before > 0, data prep |
| \(\mathrm{f}^{0}\) | Link_p_init over \(\Xi \times \mathcal{L}\), neutral where the data has no row — the flow a link brought into the horizon — PyPSA's p_init, read only where the link came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot |
| \(\mathrm{hold}^{f}\) | Link_must_stay_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — true while the up time a link brought into the horizon still binds — data prep, since position() compares against a literal rather than a parameter |
| \(\mathrm{rest}^{f}\) | Link_must_stay_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — true while the down time a link brought into the horizon still binds — PyPSA's min_down_time - down_time_before snapshots, where down_time_before > 0, data prep for the same reason |
| \(\mathrm{c}^{f,\mathrm{up}}\) | Link_start_up_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one start in this snapshot |
| \(\mathrm{c}^{f,\mathrm{dn}}\) | Link_shut_down_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one stop in this snapshot |
| \(\mathrm{c}^{f,\mathrm{on}}\) | Link_stand_by_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one snapshot spent on |
| \(\mathrm{f}^{\mathrm{mod}}\) | Link_p_nom_mod over \(\mathcal{L}\), neutral where the data has no row — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{f,\mathrm{fix}}\) | Link_modules_installed over \(\Xi \times \mathcal{L}\) — how many whole modules a committable build has in place: Link_p_nom / Link_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 |
| \(\mathrm{M}^{f}\) | Link_big_m over \(\Xi \times \mathcal{L}\), neutral where the data has no row — the bound a committed extendable link's big-M rows release it by — the build cap p_nom_max times the highest p_max_pu, where the cap is finite and positive. Elsewhere it is committable_big_m times the highest p_max_pu, and where that keyword is not given, ten times the largest of the peak total load and the component's largest finite p_nom and p_nom_max, or 1e6 where there is none of them (components.py:1050-1121). Below the flow a solve wants, it caps that flow; data prep |
| \(\mathrm{nonneg}^{f}\) | Link_p_min_pu_nonneg over \(\mathcal{L}\) — true where none of the link's own minimums-per-unit is negative — PyPSA's per-unit (p_min_pu >= 0).all() over every snapshot and scenario, data prep |
| \(\alpha\) | 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 |
| \(\mathrm{act}^{f}\) | Link_active_asset over \(\mathcal{L}\) — whether a link 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) |
| \(\mathrm{W}^{f}\) | Link_capital_weight over \(\mathcal{L}\) — the sum of period weights a link stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\underline{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_min over \(\Xi \times \mathcal{L}\) — least nominal power an extendable link may be built at |
| \(\overline{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_max over \(\Xi \times \mathcal{L}\) — most nominal power an extendable link may be built at |
| \(\mathrm{c}^{\mathrm{cap},f}\) | Link_capital_cost over \(\Xi \times \mathcal{L}\) — cost of one unit of nominal 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 |
Conditions¶
| Symbol | Meaning |
|---|---|
| \(\mathrm{Link\_com\_ext}\) | Link_com_ext over \(\mathcal{L} \times \mathcal{T}\) — a committable link with an extendable, non-modular build that stands in the snapshot's period — PyPSA's com-ext rows, whose status is relaxed against the chosen build |
| \(\mathrm{Link\_committed}\) | Link_committed over \(\mathcal{L} \times \mathcal{T}\) — a committable link that stands in the snapshot's period — every unit-commitment row's set |
| \(\mathrm{Link\_ramps\_from\_previous}\) | Link_ramps_from_previous over \(\mathcal{L} \times \Xi \times \mathcal{T}\) — a snapshot whose ramp reads a previous output — any snapshot but its period's first, and the horizon's first where the link 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 |
| \(N^{f}\) | Link_n_mod over \(\mathcal{L}\) — Link-n_mod — how many modules of an extendable modular build; none for a unit that is not active |
| \(u^{f}\) | Link_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-status — how much of a committable link is on: an integer the rows below cap at one, or at the module count where the build is modular |
| \(\mathit{up}^{f}\) | Link_start_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-start_up — how much of a committable link turns on this snapshot, capped as the status is |
| \(\mathit{dn}^{f}\) | Link_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-shut_down — how much of a committable link turns off this snapshot, capped as the status is |
| \(F\) | Link_p_nom_ext over \(\mathcal{L}\) — Link-p_nom — nominal power where it is a decision, for a link 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 |
|---|---|
| \(\mathit{Link\_p\_nom\_effective}\) | Link_p_nom_effective over \(\Xi \times \mathcal{L}\) — the build a link's limits are taken against — the chosen one where it is extendable, the given one otherwise |
| \(\mathit{Link\_previous\_status}\) | Link_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the commitment state a link carries into a snapshot — 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 |
| \(\mathit{Link\_status\_carried\_over}\) | Link_status_carried_over over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the state a link carries over into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that |
| \(\mathit{Link\_previous\_p}\) | Link_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the flow a link carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that |
| \(\mathrm{Link\_ramp\_up\_rate}\) | Link_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's up row reads — PyPSA's ramp_limit_up, or the full build where it has none, since a start-up ramp alone builds the row |
| \(\mathrm{Link\_ramp\_down\_rate}\) | Link_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's down row reads — PyPSA's ramp_limit_down, or the full build where it has none, since a shut-down ramp alone builds the row |
| \(\mathrm{Link\_start\_up\_rate}\) | Link_start_up_rate over \(\Xi \times \mathcal{L}\) — the start-up ramp a link's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\mathrm{Link\_shut\_down\_rate}\) | Link_shut_down_rate over \(\Xi \times \mathcal{L}\) — the shut-down ramp a link's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\mathrm{Link\_p\_nom\_committed}\) | Link_p_nom_committed over \(\Xi \times \mathcal{L}\) — the build a committed link's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise |
| \(\mathit{Link\_ramp\_up\_allowance}\) | Link_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may raise flow between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\mathit{Link\_ramp\_down\_allowance}\) | Link_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may lower flow between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off |
| \(\mathit{Link\_output\_arrival}\) | 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 \(\Xi\) |
| \(\mathit{Link\_opex}\) | Link_opex over \(\Xi\) |
| \(\mathit{Link\_commitment\_opex}\) | Link_commitment_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) |
| \(\mathit{Link\_capex}\) | Link_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 \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Link\_injection}\) | 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 |
| \(t \ominus^{\mathrm{relation}(t)} k\) | within=relation[...] — \(t-k\) inside the group of \(t\) |
| \(\mathrm{pos}(t)\) | position() — the position of \(t\), counted from \(0\); \(t\) is the label |
| \(\mathrm{pos}_{\mathrm{relation}(t)}(t)\) | position(within=relation[...]) — the position of \(t\) inside its group |
Conditions¶
Link_com_ext
\[
\mathrm{Link\_com\_ext}_{l,t} \iff \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \mathrm{on}^{f}_{t,l} \qquad \forall\, l \in \mathcal{L},\ t \in \mathcal{T}
\]
Link_committed
\[
\mathrm{Link\_committed}_{l,t} \iff \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l} \qquad \forall\, l \in \mathcal{L},\ t \in \mathcal{T}
\]
Link_ramps_from_previous
\[
\mathrm{Link\_ramps\_from\_previous}_{l,\xi,t} \iff \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \qquad \forall\, l \in \mathcal{L},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
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
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
Link_n_mod
\[
N^{f}_{l} \ge 0, N^{f}_{l} \in \mathbb{Z} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \wedge \mathrm{act}^{f}_{l}
\]
Link_status
\[
u^{f}_{\xi,t,l} \ge 0, u^{f}_{\xi,t,l} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t}
\]
Link_start_up
\[
\mathit{up}^{f}_{\xi,t,l} \ge 0, \mathit{up}^{f}_{\xi,t,l} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t}
\]
Link_shut_down
\[
\mathit{dn}^{f}_{\xi,t,l} \ge 0, \mathit{dn}^{f}_{\xi,t,l} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t}
\]
Link_p_nom_ext
\[
F_{l} \in \mathbb{R} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{act}^{f}_{l}
\]
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_p_nom_effective
\[
\mathit{Link\_p\_nom\_effective}_{\xi,l} = \begin{cases} F_{l} & \text{if } \mathrm{ext}^{f}_{l} \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_previous_status
\[
\mathit{Link\_previous\_status}_{\xi,t,l} = \begin{cases} 0 & \text{if } \mathrm{pos}(t) > 0 \wedge \neg \mathrm{on}^{f}_{t - 1,l} \\ \mathit{Link\_status\_carried\_over}_{\xi,t,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_status_carried_over
\[
\mathit{Link\_status\_carried\_over}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ u^{f}_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_previous_p
\[
\mathit{Link\_previous\_p}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} \cdot \mathrm{f}^{0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ f_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_up_rate
\[
\mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} = \begin{cases} \mathrm{ru}^{f}_{\xi,t,l} & \text{if } \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_down_rate
\[
\mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} = \begin{cases} \mathrm{rd}^{f}_{\xi,t,l} & \text{if } \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_start_up_rate
\[
\mathrm{Link\_start\_up\_rate}_{\xi,l} = \begin{cases} \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} & \text{if } \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_shut_down_rate
\[
\mathrm{Link\_shut\_down\_rate}_{\xi,l} = \begin{cases} \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} & \text{if } \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_p_nom_committed
\[
\mathrm{Link\_p\_nom\_committed}_{\xi,l} = \begin{cases} \mathrm{f}^{\mathrm{mod}}_{l} & \text{if } \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_ramp_up_allowance
\[
\mathit{Link\_ramp\_up\_allowance}_{\xi,t,l} = \begin{cases} \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \mathit{Link\_previous\_status}_{\xi,t,l} + \mathrm{Link\_start\_up\_rate}_{\xi,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \left( u^{f}_{\xi,t,l} - \mathit{Link\_previous\_status}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot \mathit{Link\_p\_nom\_effective}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_down_allowance
\[
\mathit{Link\_ramp\_down\_allowance}_{\xi,t,l} = \begin{cases} \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot u^{f}_{\xi,t,l} + \mathrm{Link\_shut\_down\_rate}_{\xi,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \left( \mathit{Link\_previous\_status}_{\xi,t,l} - u^{f}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot \mathit{Link\_p\_nom\_effective}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_output_arrival
\[
\mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
Process_output_arrival
\[
\mathit{Process\_output\_arrival}_{\xi,t,r} = \begin{cases} z_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{if } \mathrm{cyc}^{z}_{r} \\ z_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ r \in \mathcal{R}
\]
Generator_opex
\[
\mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Link_opex
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Link_commitment_opex
\[
\mathit{Link\_commitment\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} u^{f}_{\xi,t,l} \cdot \mathrm{c}^{f,\mathrm{on}}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} \mathit{up}^{f}_{\xi,t,l} \cdot \mathrm{c}^{f,\mathrm{up}}_{\xi,t,l} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} \mathit{dn}^{f}_{\xi,t,l} \cdot \mathrm{c}^{f,\mathrm{dn}}_{\xi,t,l} \qquad \forall\, \xi \in \Xi
\]
scenario_opex
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} + \mathit{Link\_commitment\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
Link_capex
\[
\mathit{Link\_capex} = \sum_{\xi \in \Xi,\ l \in \mathcal{L}} \pi_{\xi} \cdot F_{l} \cdot \mathrm{c}^{\mathrm{cap},f}_{\xi,l} \cdot \mathrm{W}^{f}_{l}
\]
risk_weighted_opex
\[
\mathit{risk\_weighted\_opex} = \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi}
\]
total_cost
\[
\mathit{total\_cost} = \mathit{Link\_capex} + \mathit{risk\_weighted\_opex}
\]
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
\[
\mathit{Generator\_injection}_{n,\xi,t} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Link_injection
\[
\mathit{Link\_injection}_{n,\xi,t} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Load_injection
\[
\mathrm{Load\_injection}_{n,\xi,t} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Process_injection
\[
\mathit{Process\_injection}_{n,\xi,t} = \sum_{r \in \mathcal{R} \,:\, \mathrm{Process\_output\_bus}(r) = n} \mathit{Process\_output\_arrival}_{\xi,t,r} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
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
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_nom_lower
\[
F_{l} \ge \underline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{act}^{f}_{l}
\]
Link_ext_p_nom_upper
\[
F_{l} \le \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{act}^{f}_{l} \wedge \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \text{ is defined}
\]
Link_com_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \neg \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right)
\]
Link_com_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \neg \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right)
\]
Link_com_transition_start_up
\[
\mathit{up}^{f}_{\xi,t,l} \ge u^{f}_{\xi,t,l} - \mathit{Link\_previous\_status}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t}
\]
Link_com_transition_shut_down
\[
\mathit{dn}^{f}_{\xi,t,l} \ge \mathit{Link\_previous\_status}_{\xi,t,l} - u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t}
\]
Link_com_up_time
\[
\sum_{t' \in \mathcal{T} \,:\, 0 \le t - t' < \mathrm{UT}^{f}} \mathit{up}^{f}_{\xi,t',l} \le u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{UT}^{f}_{\xi,l} > 0 \wedge \mathrm{pos}(t) > 0
\]
Link_com_down_time
\[
\sum_{t' \in \mathcal{T} \,:\, 0 \le t - t' < \mathrm{DT}^{f}} \mathit{dn}^{f}_{\xi,t',l} \le 1 - u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{DT}^{f}_{\xi,l} > 0 \wedge \mathrm{pos}(t) > 0
\]
Link_com_status_must_stay_up
\[
u^{f}_{\xi,t,l} = 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{hold}^{f}_{\xi,t,l}
\]
Link_com_status_must_stay_down
\[
u^{f}_{\xi,t,l} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{rest}^{f}_{\xi,t,l}
\]
Link_p_ramp_limit_up_run_big_m
\[
f_{\xi,t,l} - \mathit{Link\_previous\_p}_{\xi,t,l} \le \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot \mathit{Link\_previous\_status}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t} \wedge \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \mathrm{Link\_ramps\_from\_previous}_{l,\xi,t}
\]
Link_p_ramp_limit_up_start_big_m
\[
f_{\xi,t,l} - \mathit{Link\_previous\_p}_{\xi,t,l} \le \mathrm{Link\_start\_up\_rate}_{\xi,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot \mathit{up}^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t} \wedge \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \mathrm{Link\_ramps\_from\_previous}_{l,\xi,t}
\]
Link_p_ramp_limit_down_run_big_m
\[
\mathit{Link\_previous\_p}_{\xi,t,l} - f_{\xi,t,l} \le \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t} \wedge \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \mathrm{Link\_ramps\_from\_previous}_{l,\xi,t}
\]
Link_p_ramp_limit_down_shut_big_m
\[
\mathit{Link\_previous\_p}_{\xi,t,l} - f_{\xi,t,l} \le \mathrm{Link\_shut\_down\_rate}_{\xi,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot \mathit{dn}^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t} \wedge \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \mathrm{Link\_ramps\_from\_previous}_{l,\xi,t}
\]
Link_p_nom_modularity
\[
F_{l} = \mathrm{f}^{\mathrm{mod}}_{l} \cdot N^{f}_{l} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \wedge \mathrm{act}^{f}_{l}
\]
Link_com_ext_p_upper_cap
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot F_{l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t}
\]
Link_com_ext_p_upper_big_m
\[
f_{\xi,t,l} \le \mathrm{M}^{f}_{\xi,l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t}
\]
Link_com_ext_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} \cdot u^{f}_{\xi,t,l} - \mathrm{M}^{f}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t}
\]
Link_com_ext_p_lower_nonneg
\[
f_{\xi,t,l} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_com\_ext}_{l,t} \wedge \mathrm{nonneg}^{f}_{l}
\]
Link_com_mod_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{mod}}_{l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_com_mod_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{mod}}_{l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_status_p_fixed_upper
\[
u^{f}_{\xi,t,l} \le \mathrm{N}^{f,\mathrm{fix}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \neg \left( \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right)
\]
Link_start_up_p_fixed_upper
\[
\mathit{up}^{f}_{\xi,t,l} \le \mathrm{N}^{f,\mathrm{fix}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \neg \left( \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right)
\]
Link_shut_down_p_fixed_upper
\[
\mathit{dn}^{f}_{\xi,t,l} \le \mathrm{N}^{f,\mathrm{fix}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \neg \left( \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right)
\]
Link_status_p_nom_variable_upper
\[
u^{f}_{\xi,t,l} \le N^{f}_{l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_start_up_p_nom_variable_upper
\[
\mathit{up}^{f}_{\xi,t,l} \le N^{f}_{l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_shut_down_p_nom_variable_upper
\[
\mathit{dn}^{f}_{\xi,t,l} \le N^{f}_{l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_p_ramp_limit_up
\[
f_{\xi,t,l} - \mathit{Link\_previous\_p}_{\xi,t,l} \le \mathit{Link\_ramp\_up\_allowance}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \mathrm{Link\_ramps\_from\_previous}_{l,\xi,t} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_down
\[
\mathit{Link\_previous\_p}_{\xi,t,l} - f_{\xi,t,l} \le \mathit{Link\_ramp\_down\_allowance}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \mathrm{Link\_ramps\_from\_previous}_{l,\xi,t} \wedge \mathrm{on}^{f}_{t,l}
\]
Bus_nodal_balance
\[
\mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
The spec, differential/pypsa/rungs/rung_25_committable_link.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}
relations:
snapshot_period: {description: the investment period a snapshot falls in, key: snapshot, values: period}
Generator_bus: {description: the bus a generator sits on, key: generator, values: bus}
Link_bus0: {description: the bus a link leaves, key: link, values: bus}
Link_output_link: {description: the link an output port belongs to, key: link_output, values: link}
Link_output_bus: {description: 'the bus an output port delivers to — PyPSA''s `bus1`, `bus2`, … columns.
A link of three output ports is three labels here rather than a third relation, so the file states
any number of them', key: link_output, values: bus}
Process_output_process: {description: the process a port belongs to, key: process_output, values: process}
Process_output_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 relation, so the file
states any number of them', key: process_output, values: bus}
Load_bus: {description: the bus a load sits on, key: load, values: bus}
parameters:
snapshot_weightings_objective:
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
dims: [snapshot]
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
for a unit that draws power. PyPSA refuses one that differs by scenario (`constants.py:43`)
dims: [generator]
Generator_committable:
description: whether output is gated by an on/off status decision
dims: [generator]
dtype: bool
Link_ramp_limit_up:
description: most a link may raise its flow between snapshots, per unit of nominal power; no value
means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, link]
missing: neutral
Link_ramp_limit_down:
description: most a link may lower its flow between snapshots, per unit of nominal power; no value
means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, link]
missing: neutral
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
Link_ramp_limit_start_up:
description: most flow in the snapshot a link starts, per unit of nominal power; no value means no
limit
dims: [scenario, link]
missing: neutral
Link_ramp_limit_shut_down:
description: most flow in the snapshot before a link stops, per unit of nominal power; no value means
no limit
dims: [scenario, link]
missing: neutral
Link_min_up_time:
description: least snapshots a link stays on once started
dims: [scenario, link]
dtype: int
Link_min_down_time:
description: least snapshots a link stays off once stopped
dims: [scenario, link]
dtype: int
Link_status_initial:
description: one where the link was on before the first snapshot, zero where off — PyPSA's `up_time_before
> 0`, data prep
dims: [scenario, link]
dtype: int
Link_p_init:
description: the flow a link brought into the horizon — PyPSA's `p_init`, read only where the link
came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot
dims: [scenario, link]
missing: neutral
Link_must_stay_up:
description: true while the up time a link brought into the horizon still binds — data prep, since
`position()` compares against a literal rather than a parameter
dims: [scenario, snapshot, link]
dtype: bool
Link_must_stay_down:
description: true while the down time a link brought into the horizon still binds — PyPSA's `min_down_time
- down_time_before` snapshots, where `down_time_before > 0`, data prep for the same reason
dims: [scenario, snapshot, link]
dtype: bool
Link_start_up_cost:
description: cost of one start in this snapshot
dims: [scenario, snapshot, link]
Link_shut_down_cost:
description: cost of one stop in this snapshot
dims: [scenario, snapshot, link]
Link_stand_by_cost:
description: cost of one snapshot spent on
dims: [scenario, snapshot, link]
Link_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [link]
missing: neutral
Link_modules_installed:
description: 'how many whole modules a committable build has in place: `Link_p_nom / Link_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, link]
Link_big_m:
description: the bound a committed extendable link's big-M rows release it by — the build cap `p_nom_max`
times the highest `p_max_pu`, where the cap is finite and positive. Elsewhere it is `committable_big_m`
times the highest `p_max_pu`, and where that keyword is not given, ten times the largest of the
peak total load and the component's largest finite `p_nom` and `p_nom_max`, or 1e6 where there is
none of them (`components.py:1050-1121`). Below the flow a solve wants, it caps that flow; data
prep
dims: [scenario, link]
missing: neutral
Link_p_min_pu_nonneg:
description: true where none of the link's own minimums-per-unit is negative — PyPSA's per-unit `(p_min_pu
>= 0).all()` over every snapshot and scenario, data prep
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
Link_active_asset:
description: whether a link 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: [link]
dtype: bool
Link_capital_weight:
description: the sum of period weights a link stands in — PyPSA's `active * period_weighting`, summed,
data prep
dims: [link]
Link_p_nom_min:
description: least nominal power an extendable link may be built at
dims: [scenario, link]
Link_p_nom_max:
description: most nominal power an extendable link may be built at
dims: [scenario, link]
Link_capital_cost:
description: 'cost of one unit of nominal 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, link]
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
Link_n_mod:
description: '`Link-n_mod` — how many modules of an extendable modular build; none for a unit that
is not active'
dims: [link]
where: Link_p_nom_extendable AND Link_p_nom_mod > 0 AND Link_active_asset
domain: integer
bounds: {lower: 0}
Link_status:
description: '`Link-status` — how much of a committable link is on: an integer the rows below cap
at one, or at the module count where the build is modular'
dims: [scenario, snapshot, link]
where: Link_committed
domain: integer
bounds: {lower: 0}
Link_start_up:
description: '`Link-start_up` — how much of a committable link turns on this snapshot, capped as the
status is'
dims: [scenario, snapshot, link]
where: Link_committed
domain: integer
bounds: {lower: 0}
Link_shut_down:
description: '`Link-shut_down` — how much of a committable link turns off this snapshot, capped as
the status is'
dims: [scenario, snapshot, link]
where: Link_committed
domain: integer
bounds: {lower: 0}
Link_p_nom_ext:
description: '`Link-p_nom` — nominal power where it is a decision, for a link that is active; the
parameter of the same PyPSA name carries the fixed regime'
dims: [link]
where: Link_p_nom_extendable AND Link_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
Link_ext_p_nom_lower:
description: '`Link-ext-p_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, link]
where: Link_p_nom_extendable AND Link_active_asset
expression: Link_p_nom_ext >= Link_p_nom_min
Link_ext_p_nom_upper:
description: '`Link-ext-p_nom-upper` — the chosen build is at most its cap in every scenario; a cap
of infinity is no row'
dims: [scenario, link]
where: Link_p_nom_extendable AND Link_active_asset AND Link_p_nom_max
expression: Link_p_nom_ext <= Link_p_nom_max
Link_com_p_lower:
description: '`Link-com-p-lower` — a committed link flows at least its minimum; off, at least nothing'
dims: [scenario, snapshot, link]
where: Link_committed AND NOT Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0)
expression: Link_p >= (Link_p_min_pu * Link_p_nom) * Link_status
Link_com_p_upper:
description: '`Link-com-p-upper` — a committed link flows at most what is available; off, at most
nothing'
dims: [scenario, snapshot, link]
where: Link_committed AND NOT Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0)
expression: Link_p <= (Link_p_max_pu * Link_p_nom) * Link_status
Link_com_transition_start_up:
description: '`Link-com-transition-start-up` — turning on is a start, counted against the state the
link carried into the snapshot'
dims: [scenario, snapshot, link]
where: Link_committed
expression: Link_start_up >= Link_status - Link_previous_status
Link_com_transition_shut_down:
description: '`Link-com-transition-shut-down` — turning off is a stop, counted against the state the
link carried into the snapshot'
dims: [scenario, snapshot, link]
where: Link_committed
expression: Link_shut_down >= Link_previous_status - Link_status
Link_com_up_time:
description: '`Link-com-up-time` — a link 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, link]
where: Link_committed AND Link_min_up_time > 0 AND position(snapshot) > 0
expression: sum_back(Link_start_up, along=snapshot, window=Link_min_up_time) <= Link_status
Link_com_down_time:
description: '`Link-com-down-time` — a link 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, link]
where: Link_committed AND Link_min_down_time > 0 AND position(snapshot) > 0
expression: sum_back(Link_shut_down, along=snapshot, window=Link_min_down_time) <= 1 - Link_status
Link_com_status_must_stay_up:
description: '`Link-com-status-min_up_time_must_stay_up` — a link still serving the up time it brought
in stays on'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_must_stay_up
expression: Link_status == 1
Link_com_status_must_stay_down:
description: '`Link-com-status-min_down_time_must_stay_up` — a link still serving the down time it
brought in stays off; PyPSA names the row `_must_stay_up`'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_must_stay_down
expression: Link_status == 0
Link_p_ramp_limit_up_run_big_m:
description: '`Link-p-ramp_limit_up-run-bigM` — a committed extendable link raises flow no faster
than its limit of the chosen build; the big M releases the row in the snapshot it turns on'
dims: [scenario, snapshot, link]
where: Link_com_ext AND (Link_ramp_limit_up OR Link_ramp_limit_start_up) AND Link_ramps_from_previous
expression: Link_p - Link_previous_p <= Link_ramp_up_rate * Link_p_nom_ext + Link_big_m - Link_big_m
* Link_previous_status
Link_p_ramp_limit_up_start_big_m:
description: '`Link-p-ramp_limit_up-start-bigM` — in the snapshot it turns on, a committed extendable
link ramps no further than its start-up ramp of the chosen build; the big M releases the row everywhere
else'
dims: [scenario, snapshot, link]
where: Link_com_ext AND (Link_ramp_limit_up OR Link_ramp_limit_start_up) AND Link_ramps_from_previous
expression: Link_p - Link_previous_p <= Link_start_up_rate * Link_p_nom_ext + Link_big_m - Link_big_m
* Link_start_up
Link_p_ramp_limit_down_run_big_m:
description: '`Link-p-ramp_limit_down-run-bigM` — a committed extendable link lowers flow no faster
than its limit of the chosen build; the big M releases the row in the snapshot it turns off'
dims: [scenario, snapshot, link]
where: Link_com_ext AND (Link_ramp_limit_down OR Link_ramp_limit_shut_down) AND Link_ramps_from_previous
expression: Link_previous_p - Link_p <= Link_ramp_down_rate * Link_p_nom_ext + Link_big_m - Link_big_m
* Link_status
Link_p_ramp_limit_down_shut_big_m:
description: '`Link-p-ramp_limit_down-shut-bigM` — in the snapshot it turns off, a committed extendable
link ramps no further than its shut-down ramp of the chosen build; the big M releases the row everywhere
else'
dims: [scenario, snapshot, link]
where: Link_com_ext AND (Link_ramp_limit_down OR Link_ramp_limit_shut_down) AND Link_ramps_from_previous
expression: Link_previous_p - Link_p <= Link_shut_down_rate * Link_p_nom_ext + Link_big_m - Link_big_m
* Link_shut_down
Link_p_nom_modularity:
description: '`Link-p_nom_modularity` — the chosen build is a whole number of modules'
dims: [link]
where: Link_p_nom_extendable AND Link_p_nom_mod > 0 AND Link_active_asset
expression: Link_p_nom_ext == Link_p_nom_mod * Link_n_mod
Link_com_ext_p_upper_cap:
description: '`Link-com-ext-p-upper-cap` — a committed extendable link flows at most what is available
of the chosen build, whatever its status'
dims: [scenario, snapshot, link]
where: Link_com_ext
expression: Link_p <= Link_p_max_pu * Link_p_nom_ext
Link_com_ext_p_upper_big_m:
description: '`Link-com-ext-p-upper-bigM` — off, a link flows nothing; on, the big M is no bound'
dims: [scenario, snapshot, link]
where: Link_com_ext
expression: Link_p <= Link_big_m * Link_status
Link_com_ext_p_lower:
description: '`Link-com-ext-p-lower` — a committed extendable link flows at least its minimum of the
chosen build; off, the big M releases the row'
dims: [scenario, snapshot, link]
where: Link_com_ext
expression: Link_p >= ((Link_p_min_pu * Link_p_nom_ext) + (Link_big_m * Link_status)) - Link_big_m
Link_com_ext_p_lower_nonneg:
description: '`Link-com-ext-p-lower-nonneg` — where no minimum-per-unit is negative, flow is also
plainly non-negative, a row the big-M lower cannot assert while the link is off'
dims: [scenario, snapshot, link]
where: Link_com_ext AND Link_p_min_pu_nonneg
expression: Link_p >= 0
Link_com_mod_p_lower:
description: '`Link-com-mod-p-lower` — a committed modular link flows at least its minimum of one
module, whether the build is fixed or a decision'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_p_nom_mod > 0
expression: Link_p >= (Link_p_min_pu * Link_p_nom_mod) * Link_status
Link_com_mod_p_upper:
description: '`Link-com-mod-p-upper` — a committed modular link flows at most one module''s share,
whether the build is fixed or a decision'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_p_nom_mod > 0
expression: Link_p <= (Link_p_max_pu * Link_p_nom_mod) * Link_status
Link_status_p_fixed_upper:
description: '`Link-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, link]
where: Link_committed AND NOT (Link_p_nom_extendable AND Link_p_nom_mod > 0)
expression: Link_status <= Link_modules_installed
Link_start_up_p_fixed_upper:
description: '`Link-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, link]
where: Link_committed AND NOT (Link_p_nom_extendable AND Link_p_nom_mod > 0)
expression: Link_start_up <= Link_modules_installed
Link_shut_down_p_fixed_upper:
description: '`Link-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, link]
where: Link_committed AND NOT (Link_p_nom_extendable AND Link_p_nom_mod > 0)
expression: Link_shut_down <= Link_modules_installed
Link_status_p_nom_variable_upper:
description: '`Link-status-p_nom-variable-upper` — a modular link is on only where a module is built'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_p_nom_extendable AND Link_p_nom_mod > 0
expression: Link_status <= Link_n_mod
Link_start_up_p_nom_variable_upper:
description: '`Link-start_up-p_nom-variable-upper` — a modular link starts only where a module is
built'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_p_nom_extendable AND Link_p_nom_mod > 0
expression: Link_start_up <= Link_n_mod
Link_shut_down_p_nom_variable_upper:
description: '`Link-shut_down-p_nom-variable-upper` — a modular link stops only where a module is
built'
dims: [scenario, snapshot, link]
where: Link_committed AND Link_p_nom_extendable AND Link_p_nom_mod > 0
expression: Link_shut_down <= Link_n_mod
Link_p_ramp_limit_up:
description: '`Link-p-ramp_limit_up` — a link raises flow no faster than its ramp limit of the build,
and a committed one no further than its start-up ramp in the snapshot it turns on. A link that came
into the horizon running carries a row at the first snapshot only where its `p_init` gives the flow
it brought in, and no link carries one at the start of a later investment period — nor does any
link a big M releases instead'
dims: [scenario, snapshot, link]
where: (Link_ramp_limit_up OR Link_ramp_limit_start_up) AND NOT (Link_committable AND Link_p_nom_extendable
AND NOT (Link_p_nom_mod > 0)) AND Link_ramps_from_previous AND Link_active
expression: Link_p - Link_previous_p <= Link_ramp_up_allowance
Link_p_ramp_limit_down:
description: '`Link-p-ramp_limit_down` — a link lowers flow no faster than its ramp limit of the build,
and a committed one no further than its shut-down ramp in the snapshot it turns off. A link that
came into the horizon running carries a row at the first snapshot only where its `p_init` gives
the flow it brought in, and no link carries one at the start of a later investment period — nor
does any link a big M releases instead'
dims: [scenario, snapshot, link]
where: (Link_ramp_limit_down OR Link_ramp_limit_shut_down) AND NOT (Link_committable AND Link_p_nom_extendable
AND NOT (Link_p_nom_mod > 0)) AND Link_ramps_from_previous AND Link_active
expression: Link_previous_p - Link_p <= Link_ramp_down_allowance
Bus_nodal_balance:
description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
less what the links take away, plus what arrives over them after losses and any delay at every port
they deliver to, each process port drawing or delivering at its own rate and each passive branch
carrying its flow, meets the load there, less half of every incident line''s and transformer''s
loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
and load term enters with its component''s `sign` (`constraints.py: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_p_nom_effective:
description: the build a link's limits are taken against — the chosen one where it is extendable,
the given one otherwise
dims: [scenario, link]
cases:
extendable: {when: Link_p_nom_extendable, expression: Link_p_nom_ext}
otherwise: Link_p_nom
Link_previous_status:
description: the commitment state a link carries into a snapshot — 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, link]
cases:
opening_late: {when: 'position(snapshot) > 0 AND NOT shift(Link_active, along=snapshot, offset=1)',
expression: 0}
otherwise: Link_status_carried_over
Link_status_carried_over:
description: the state a link carries over into a snapshot — the state it brought into the horizon
at the first, the previous snapshot's after that
dims: [scenario, snapshot, link]
cases:
opening: {when: position(snapshot) == 0, expression: Link_status_initial}
otherwise: shift(Link_status, along=snapshot, offset=1)
Link_previous_p:
description: the flow a link carries into a snapshot — at the first, the `p_init` it brought in where
it came in running and nothing where it came in off; the previous snapshot's after that
dims: [scenario, snapshot, link]
cases:
opening: {when: position(snapshot) == 0, expression: Link_status_initial * Link_p_init}
otherwise: shift(Link_p, along=snapshot, offset=1)
Link_ramp_up_rate:
description: the ramp limit a link's up row reads — PyPSA's `ramp_limit_up`, or the full build where
it has none, since a start-up ramp alone builds the row
dims: [scenario, snapshot, link]
cases:
given: {when: Link_ramp_limit_up, expression: Link_ramp_limit_up}
otherwise: 1
Link_ramp_down_rate:
description: the ramp limit a link's down row reads — PyPSA's `ramp_limit_down`, or the full build
where it has none, since a shut-down ramp alone builds the row
dims: [scenario, snapshot, link]
cases:
given: {when: Link_ramp_limit_down, expression: Link_ramp_limit_down}
otherwise: 1
Link_start_up_rate:
description: the start-up ramp a link's up row reads — PyPSA's `ramp_limit_start_up`, or the full
build where it has none
dims: [scenario, link]
cases:
given: {when: Link_ramp_limit_start_up, expression: Link_ramp_limit_start_up}
otherwise: 1
Link_shut_down_rate:
description: the shut-down ramp a link's down row reads — PyPSA's `ramp_limit_shut_down`, or the full
build where it has none
dims: [scenario, link]
cases:
given: {when: Link_ramp_limit_shut_down, expression: Link_ramp_limit_shut_down}
otherwise: 1
Link_p_nom_committed:
description: the build a committed link's ramp rows are taken against — one module where the build
is extendable and modular, the given build otherwise
dims: [scenario, link]
cases:
modular_build: {when: Link_p_nom_extendable AND Link_p_nom_mod > 0, expression: Link_p_nom_mod}
otherwise: Link_p_nom
Link_ramp_up_allowance:
description: how far a link may raise flow between two snapshots — its ramp limit of the build while
it stays on, plus its start-up ramp in the snapshot it turns on
dims: [scenario, snapshot, link]
cases:
committed: {when: Link_committable, expression: Link_ramp_up_rate * Link_p_nom_committed * Link_previous_status
+ Link_start_up_rate * Link_p_nom_committed * (Link_status - Link_previous_status)}
otherwise: Link_ramp_up_rate * Link_p_nom_effective
Link_ramp_down_allowance:
description: how far a link may lower flow between two snapshots — its ramp limit of the build while
it stays on, plus its shut-down ramp in the snapshot it turns off
dims: [scenario, snapshot, link]
cases:
committed: {when: Link_committable, expression: Link_ramp_down_rate * Link_p_nom_committed * Link_status
+ Link_shut_down_rate * Link_p_nom_committed * (Link_previous_status - Link_status)}
otherwise: Link_ramp_down_rate * Link_p_nom_effective
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[link]),
along=snapshot, offset=Link_output_delay, edge=''wrap'', within=snapshot_period[period]) * Link_efficiency'}
otherwise: shift(at(Link_p, by=Link_output_link[link]), along=snapshot, offset=Link_output_delay,
edge=0, within=snapshot_period[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[process]),
along=snapshot, offset=Process_output_delay, edge=''wrap'', within=snapshot_period[period])
* Process_rate'}
otherwise: shift(at(Process_p, by=Process_output_process[process]), along=snapshot, offset=Process_output_delay,
edge=0, within=snapshot_period[period]) * Process_rate
Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period[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[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[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[period]), over=link),
over=snapshot)'}
Link_commitment_opex: {expression: 'sum(sum(((Link_status * Link_stand_by_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period[period]), over=link), over=snapshot) + sum(sum(Link_start_up
* Link_start_up_cost, over=link), over=snapshot) + sum(sum(Link_shut_down * Link_shut_down_cost,
over=link), over=snapshot)'}
scenario_opex:
dims: [scenario]
expression: (Generator_opex + Link_opex) + Link_commitment_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`)
Link_capex: {expression: sum(scenario_weight * Link_p_nom_ext * Link_capital_cost * Link_capital_weight)}
risk_weighted_opex: {expression: 'sum(scenario_weight * scenario_opex, over=scenario)'}
total_cost:
dims: []
expression: Link_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[bus])'}
Link_injection: {expression: '-sum(Link_p, over=link, by=Link_bus0[bus]) + sum(Link_output_arrival,
over=link_output, by=Link_output_bus[bus])'}
Load_injection: {expression: 'sum(Load_demand, over=load, by=Load_bus[bus])'}
Process_injection: {expression: 'sum(Process_output_arrival, over=process_output, by=Process_output_bus[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:
Link_com_ext: {description: 'a committable link with an extendable, non-modular build that stands in
the snapshot''s period — PyPSA''s `com-ext` rows, whose status is relaxed against the chosen build',
condition: Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0) AND Link_active}
Link_committed: {description: a committable link that stands in the snapshot's period — every unit-commitment
row's set, condition: Link_committable AND Link_active}
Link_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 link comes in off or carries an initial
output', condition: 'position(snapshot, within=snapshot_period[period]) > 0 OR (position(snapshot)
== 0 AND (Link_status_initial == 0 OR Link_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 relation, static, varying, weighting
n = build() # the network from the PyPSA tab
sources = {
'snapshot': pl.Series('snapshot', list(timesteps(n)), dtype=pl.Datetime('us')),
'bus': pl.Series('bus', list(names(n.buses.index).astype(str)), dtype=pl.String),
**{
dim: pl.Series(dim, list(names(n.static(component).index).astype(str)), dtype=pl.String)
for component, dim in DIM.items()
},
**scenarios(n),
**periods(n, multi),
**carriers(n, multi),
'Generator_bus': relation(n, 'Generator', 'bus'),
'Link_bus0': relation(n, 'Link', 'bus0'),
'Load_bus': relation(n, 'Load', 'bus'),
'snapshot_weightings_objective': weighting(n, 'objective'),
'Generator_sign': per_component('Generator', first_scenario(n.generators['sign'])),
'Load_p_set': varying(n, 'Load', 'p_set'),
'Load_sign': per_component('Load', first_scenario(loads['sign'])),
'Load_active': per_component('Load', first_scenario(loads['active']), bool),
}
with sps.solve('differential/pypsa/rungs/rung_25_committable_link.yaml', sources) as solution:
solution.objective # 13982.0
The network, rung_25_committable_link.py in the corpus — the spine plus what this rung adds:
# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT
"""Rung 25: committable links — a link held on by the up time it brought in, one held off by its down time, one kept on by its own up time, and committable builds that are extendable, modular or both."""
from __future__ import annotations
import spine
def build():
"""The spine plus an east bus that only committable links serve."""
n = spine.build()
n.add('Bus', 'east')
n.add(
'Link',
'hvdc',
bus0='north',
bus1='east',
committable=True,
p_nom=60,
p_min_pu=0.3,
marginal_cost=8,
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(
'Link',
'cold_tie',
bus0='north',
bus1='east',
committable=True,
p_nom=40,
p_min_pu=0.2,
min_up_time=2,
min_down_time=3,
up_time_before=0,
down_time_before=1,
start_up_cost=20,
)
n.add(
'Link',
'ext_tie',
bus0='north',
bus1='east',
committable=True,
p_nom_extendable=True,
p_nom_max=30,
capital_cost=5,
p_min_pu=0.2,
marginal_cost=2,
up_time_before=0,
ramp_limit_up=0.5,
ramp_limit_down=0.5,
)
n.add(
'Link',
'mod_tie',
bus0='south',
bus1='east',
committable=True,
p_nom_extendable=True,
p_nom_mod=10,
p_nom_max=40,
capital_cost=3,
p_min_pu=0.5,
)
n.add(
'Link',
'mod_fix',
bus0='north',
bus1='east',
committable=True,
p_nom=20,
p_nom_mod=10,
p_min_pu=0.5,
marginal_cost=1,
)
n.add('Load', 'east_load', bus='east', p_set=[20, 70, 60, 5])
return n
The data¶
The tables this rung is the first to declare (14), as the prep produced them:
Link_big_m.csv
Link_min_down_time.csv
scenario,link,value
base,cold_tie,3
base,ext_tie,0
base,hvdc,2
base,mod_fix,0
base,mod_tie,0
base,wire,0
Link_min_up_time.csv
scenario,link,value
base,cold_tie,2
base,ext_tie,0
base,hvdc,3
base,mod_fix,0
base,mod_tie,0
base,wire,0
Link_modules_installed.csv
scenario,link,value
base,cold_tie,1.0
base,ext_tie,1.0
base,hvdc,1.0
base,mod_fix,2.0
base,mod_tie,1.0
base,wire,1.0
Link_must_stay_down.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cold_tie,true
base,2015-01-01T00:00:00.000000,ext_tie,false
base,2015-01-01T00:00:00.000000,hvdc,false
base,2015-01-01T00:00:00.000000,mod_fix,false
base,2015-01-01T00:00:00.000000,mod_tie,false
base,2015-01-01T00:00:00.000000,wire,false
base,2015-01-01T01:00:00.000000,cold_tie,true
base,2015-01-01T01:00:00.000000,ext_tie,false
base,2015-01-01T01:00:00.000000,hvdc,false
base,2015-01-01T01:00:00.000000,mod_fix,false
base,2015-01-01T01:00:00.000000,mod_tie,false
base,2015-01-01T01:00:00.000000,wire,false
base,2015-01-01T02:00:00.000000,cold_tie,false
base,2015-01-01T02:00:00.000000,ext_tie,false
base,2015-01-01T02:00:00.000000,hvdc,false
base,2015-01-01T02:00:00.000000,mod_fix,false
base,2015-01-01T02:00:00.000000,mod_tie,false
base,2015-01-01T02:00:00.000000,wire,false
base,2015-01-01T03:00:00.000000,cold_tie,false
base,2015-01-01T03:00:00.000000,ext_tie,false
base,2015-01-01T03:00:00.000000,hvdc,false
base,2015-01-01T03:00:00.000000,mod_fix,false
base,2015-01-01T03:00:00.000000,mod_tie,false
base,2015-01-01T03:00:00.000000,wire,false
Link_must_stay_up.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cold_tie,false
base,2015-01-01T00:00:00.000000,ext_tie,false
base,2015-01-01T00:00:00.000000,hvdc,true
base,2015-01-01T00:00:00.000000,mod_fix,false
base,2015-01-01T00:00:00.000000,mod_tie,false
base,2015-01-01T00:00:00.000000,wire,false
base,2015-01-01T01:00:00.000000,cold_tie,false
base,2015-01-01T01:00:00.000000,ext_tie,false
base,2015-01-01T01:00:00.000000,hvdc,true
base,2015-01-01T01:00:00.000000,mod_fix,false
base,2015-01-01T01:00:00.000000,mod_tie,false
base,2015-01-01T01:00:00.000000,wire,false
base,2015-01-01T02:00:00.000000,cold_tie,false
base,2015-01-01T02:00:00.000000,ext_tie,false
base,2015-01-01T02:00:00.000000,hvdc,false
base,2015-01-01T02:00:00.000000,mod_fix,false
base,2015-01-01T02:00:00.000000,mod_tie,false
base,2015-01-01T02:00:00.000000,wire,false
base,2015-01-01T03:00:00.000000,cold_tie,false
base,2015-01-01T03:00:00.000000,ext_tie,false
base,2015-01-01T03:00:00.000000,hvdc,false
base,2015-01-01T03:00:00.000000,mod_fix,false
base,2015-01-01T03:00:00.000000,mod_tie,false
base,2015-01-01T03:00:00.000000,wire,false
Link_p_init.csv
Link_p_min_pu_nonneg.csv
Link_p_nom_mod.csv
Link_ramp_limit_shut_down.csv
Link_ramp_limit_start_up.csv
Link_shut_down_cost.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cold_tie,0.0
base,2015-01-01T00:00:00.000000,ext_tie,0.0
base,2015-01-01T00:00:00.000000,hvdc,50.0
base,2015-01-01T00:00:00.000000,mod_fix,0.0
base,2015-01-01T00:00:00.000000,mod_tie,0.0
base,2015-01-01T00:00:00.000000,wire,0.0
base,2015-01-01T01:00:00.000000,cold_tie,0.0
base,2015-01-01T01:00:00.000000,ext_tie,0.0
base,2015-01-01T01:00:00.000000,hvdc,50.0
base,2015-01-01T01:00:00.000000,mod_fix,0.0
base,2015-01-01T01:00:00.000000,mod_tie,0.0
base,2015-01-01T01:00:00.000000,wire,0.0
base,2015-01-01T02:00:00.000000,cold_tie,0.0
base,2015-01-01T02:00:00.000000,ext_tie,0.0
base,2015-01-01T02:00:00.000000,hvdc,50.0
base,2015-01-01T02:00:00.000000,mod_fix,0.0
base,2015-01-01T02:00:00.000000,mod_tie,0.0
base,2015-01-01T02:00:00.000000,wire,0.0
base,2015-01-01T03:00:00.000000,cold_tie,0.0
base,2015-01-01T03:00:00.000000,ext_tie,0.0
base,2015-01-01T03:00:00.000000,hvdc,50.0
base,2015-01-01T03:00:00.000000,mod_fix,0.0
base,2015-01-01T03:00:00.000000,mod_tie,0.0
base,2015-01-01T03:00:00.000000,wire,0.0
Link_stand_by_cost.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cold_tie,0.0
base,2015-01-01T00:00:00.000000,ext_tie,0.0
base,2015-01-01T00:00:00.000000,hvdc,5.0
base,2015-01-01T00:00:00.000000,mod_fix,0.0
base,2015-01-01T00:00:00.000000,mod_tie,0.0
base,2015-01-01T00:00:00.000000,wire,0.0
base,2015-01-01T01:00:00.000000,cold_tie,0.0
base,2015-01-01T01:00:00.000000,ext_tie,0.0
base,2015-01-01T01:00:00.000000,hvdc,5.0
base,2015-01-01T01:00:00.000000,mod_fix,0.0
base,2015-01-01T01:00:00.000000,mod_tie,0.0
base,2015-01-01T01:00:00.000000,wire,0.0
base,2015-01-01T02:00:00.000000,cold_tie,0.0
base,2015-01-01T02:00:00.000000,ext_tie,0.0
base,2015-01-01T02:00:00.000000,hvdc,5.0
base,2015-01-01T02:00:00.000000,mod_fix,0.0
base,2015-01-01T02:00:00.000000,mod_tie,0.0
base,2015-01-01T02:00:00.000000,wire,0.0
base,2015-01-01T03:00:00.000000,cold_tie,0.0
base,2015-01-01T03:00:00.000000,ext_tie,0.0
base,2015-01-01T03:00:00.000000,hvdc,5.0
base,2015-01-01T03:00:00.000000,mod_fix,0.0
base,2015-01-01T03:00:00.000000,mod_tie,0.0
base,2015-01-01T03:00:00.000000,wire,0.0
Link_start_up_cost.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cold_tie,20.0
base,2015-01-01T00:00:00.000000,ext_tie,0.0
base,2015-01-01T00:00:00.000000,hvdc,100.0
base,2015-01-01T00:00:00.000000,mod_fix,0.0
base,2015-01-01T00:00:00.000000,mod_tie,0.0
base,2015-01-01T00:00:00.000000,wire,0.0
base,2015-01-01T01:00:00.000000,cold_tie,20.0
base,2015-01-01T01:00:00.000000,ext_tie,0.0
base,2015-01-01T01:00:00.000000,hvdc,100.0
base,2015-01-01T01:00:00.000000,mod_fix,0.0
base,2015-01-01T01:00:00.000000,mod_tie,0.0
base,2015-01-01T01:00:00.000000,wire,0.0
base,2015-01-01T02:00:00.000000,cold_tie,20.0
base,2015-01-01T02:00:00.000000,ext_tie,0.0
base,2015-01-01T02:00:00.000000,hvdc,100.0
base,2015-01-01T02:00:00.000000,mod_fix,0.0
base,2015-01-01T02:00:00.000000,mod_tie,0.0
base,2015-01-01T02:00:00.000000,wire,0.0
base,2015-01-01T03:00:00.000000,cold_tie,20.0
base,2015-01-01T03:00:00.000000,ext_tie,0.0
base,2015-01-01T03:00:00.000000,hvdc,100.0
base,2015-01-01T03:00:00.000000,mod_fix,0.0
base,2015-01-01T03:00:00.000000,mod_tie,0.0
base,2015-01-01T03:00:00.000000,wire,0.0