Rung 44: linearized commitment with the rows the integer file has — a unit that must stay down, ramps read at the full build where a limit is missing, and maintenance¶
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 7400.0 on both sides; structure ✔ 21 constraints · 8 variables, name for name; size ✔ 191 rows · ✔ 100 columns · ✔ 378 nonzeros; duals — integer model, no duals; model for model: 29 blocks equal, 0 documented splits, 1 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
8 | 8 |
Generator-com-down-time |
3 | 3 |
Generator-com-p-lower |
16 | 16 |
Generator-com-p-upper |
16 | 16 |
Generator-com-status-min_down_time_must_stay_up |
2 | 2 |
Generator-com-transition-shut-down |
16 | 16 |
Generator-com-transition-start-up |
16 | 16 |
Generator-fix-p-lower |
12 | 12 |
Generator-fix-p-upper |
12 | 12 |
Generator-maint-event-count |
2 | 2 |
Generator-maint-status-lb |
4 | 4 |
Generator-maint-status-le-maint |
4 | 4 |
Generator-maint-status-le-status |
4 | 4 |
Generator-maint-window |
8 | 8 |
Generator-p-ramp_limit_down |
4 | 4 |
Generator-p-ramp_limit_up |
8 | 8 |
Generator-shut_down-p-fixed-upper |
16 | 16 |
Generator-start_up-p-fixed-upper |
16 | 16 |
Generator-status-p-fixed-upper |
16 | 16 |
Link-fix-p-lower |
4 | 4 |
Link-fix-p-upper |
4 | 4 |
| column | PyPSA | specsolve |
|---|---|---|
Generator-maintenance |
8 | 8 |
Generator-maintenance_start |
8 | 8 |
Generator-maintenance_status |
4 | 4 |
Generator-p |
28 | 28 |
Generator-shut_down |
16 | 16 |
Generator-start_up |
16 | 16 |
Generator-status |
16 | 16 |
Link-p |
4 | 4 |
The model¶
The same model, as math
n.optimize(linearized_unit_commitment=True), as a patch over examples/pypsa.yaml. The status, its starts and its stops of a committable generator, link and process are shares in [0, 1] rather than integers, and four rows PyPSA adds only under the keyword tighten the relaxation of a unit with a fixed build whose start and stop cost the same in every snapshot and every scenario (constraints.py:599-699). A modular committable unit is refused, as PyPSA refuses it. Every other row is the integer run's.
Sets¶
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario with \(\mathrm{Generator\_maintenance\_cover} \subseteq \Xi \times \mathcal{G} \times \mathcal{T} \times \mathcal{T}\), Generator_maintenance_cover is absent where the data has no row — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y},\ \mathrm{Generator\_maintenance\_cover} \subseteq \Xi \times \mathcal{G} \times \mathcal{T} \times \mathcal{T}\), Generator_maintenance_cover is absent where the data has no row — 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 \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Generator\_maintenance\_cover} \subseteq \Xi \times \mathcal{G} \times \mathcal{T} \times \mathcal{T}\), Generator_maintenance_cover is absent where the data has no row — 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 |
| \(\mathrm{ru}\) | Generator_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\), neutral 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 \(\Xi \times \mathcal{T} \times \mathcal{G}\), neutral 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 \(\Xi \times \mathcal{G}\), neutral 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 \(\Xi \times \mathcal{G}\), neutral 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 |
| \(\mathrm{DT}\) | 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 \(\Xi \times \mathcal{G}\), neutral 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 |
| \(\mathrm{rest}\) | Generator_must_stay_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — true while the down time a unit 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 |
| \(\mathrm{c}^{\mathrm{up}}\) | 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 \(\mathcal{G}\), neutral where the data has no row — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{\mathrm{fix}}\) | 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 |
| \(\mathrm{mnt}\) | Generator_maintainable over \(\mathcal{G}\) — whether a generator must be taken off for maintenance within the horizon — in any scenario, as PyPSA takes the union over them (components.py:1016-1019) |
| \(\gamma\) | Generator_maintenance_pu over \(\Xi \times \mathcal{G}\) — the share of the build a maintenance event takes off |
| \(\mathrm{n}^{\mathrm{mnt}}\) | Generator_maintenance_events over \(\Xi \times \mathcal{G}\) — how many maintenance events the horizon holds |
| \(\mathrm{f}^{\mathrm{nom}}\) | 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, a share in [0, 1] rather than an integer: the relaxation the keyword solves (variables.py:86-87) |
| \(\mathit{up}\) | Generator_start_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-start_up — how much of a committable unit turns on this snapshot, a share in [0, 1] (variables.py:137-138) |
| \(\mathit{dn}\) | Generator_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-shut_down — how much of a committable unit turns off this snapshot, a share in [0, 1] (variables.py:188-189) |
| \(\mu\) | Generator_maintenance over \(\Xi \times \mathcal{T} \times \mathcal{G}\), neutral where the mask leaves it out — Generator-maintenance — whether a maintainable generator is in maintenance: continuous, and one exactly where an event covers the snapshot |
| \(\mu^{\mathrm{up}}\) | Generator_maintenance_start over \(\Xi \times \mathcal{T} \times \mathcal{G}\), neutral where the mask leaves it out — Generator-maintenance_start — whether a maintenance event starts in this snapshot |
| \(\mu^{u}\) | Generator_maintenance_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\), neutral where the mask leaves it out — Generator-maintenance_status — the status while in maintenance, zero otherwise: the product the maint-status rows linearize, so a unit in maintenance may also be off |
| \(P\) | 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 |
| \(t \ominus^{\mathrm{relation}(t)} k\) | 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¶
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
\[
\mathrm{Generator\_ramps\_from\_previous}_{g,\xi,t} \iff \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \qquad \forall\, g \in \mathcal{G},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
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
\[
0 \le u_{\xi,t,g} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_start_up
\[
0 \le \mathit{up}_{\xi,t,g} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_shut_down
\[
0 \le \mathit{dn}_{\xi,t,g} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_maintenance
\[
0 \le \mu_{\xi,t,g} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_maintenance_start
\[
\mu^{\mathrm{up}}_{\xi,t,g} \in \{0, 1\} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_maintenance_status
\[
\mu^{u}_{\xi,t,g} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{Generator\_committed}_{g,t} \wedge \neg \left( \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right)
\]
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
\[
\mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
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
\[
\mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Generator_commitment_opex
\[
\mathit{Generator\_commitment\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} u_{\xi,t,g} \cdot \mathrm{c}^{\mathrm{on}}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} \mathit{up}_{\xi,t,g} \cdot \mathrm{c}^{\mathrm{up}}_{\xi,t,g} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} \mathit{dn}_{\xi,t,g} \cdot \mathrm{c}^{\mathrm{dn}}_{\xi,t,g} \qquad \forall\, \xi \in \Xi
\]
Link_opex
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
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
\[
\mathit{Generator\_injection}_{n,\xi,t} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
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
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \gamma_{\xi,g} \cdot \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \gamma_{\xi,g} \cdot \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
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
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( u_{\xi,t,g} - \gamma_{\xi,g} \cdot \mu^{u}_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
Generator_com_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( u_{\xi,t,g} - \gamma_{\xi,g} \cdot \mu^{u}_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
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_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_down
\[
u_{\xi,t,g} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \mathrm{rest}_{\xi,t,g}
\]
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_maint_event_count
\[
\sum_{t \in \mathcal{T}} \mu^{\mathrm{up}}_{\xi,t,g} = \mathrm{n}^{\mathrm{mnt}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g}
\]
Generator_maint_window
\[
\mu_{\xi,t,g} = \sum_{t' \in \mathcal{T} \,:\, \left( \xi,\ g,\ t',\ t \right) \in \mathrm{Generator\_maintenance\_cover}} \mu^{\mathrm{up}}_{\xi,t',g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_maint_status_le_status
\[
\mu^{u}_{\xi,t,g} \le u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
Generator_maint_status_le_maint
\[
\mu^{u}_{\xi,t,g} \le \mu_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
Generator_maint_status_lb
\[
\mu^{u}_{\xi,t,g} \ge u_{\xi,t,g} + \mu_{\xi,t,g} - 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
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_44_linearized_commitment.yaml — the file projected onto what this rung builds:
version: 0
description: '`n.optimize(linearized_unit_commitment=True)`, as a patch over `examples/pypsa.yaml`. The
status, its starts and its stops of a committable generator, link and process are shares in [0, 1] rather
than integers, and four rows PyPSA adds only under the keyword tighten the relaxation of a unit with
a fixed build whose start and stop cost the same in every snapshot and every scenario (`constraints.py:599-699`).
A modular committable unit is refused, as PyPSA refuses it. Every other row is the integer run''s.'
dimensions:
scenario: {dtype: str, description: 'the futures dispatch is chosen in, each with a weight'}
snapshot: {dtype: datetime, ordered: true, description: dispatch periods}
bus: {dtype: str, description: network nodes}
generator: {dtype: str, description: 'generating units, each on one bus'}
link: {dtype: str, description: 'controllable connections, each from one bus to the buses it delivers
to'}
link_output: {dtype: str, 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: {dtype: str, description: 'generalized multi-port converters, each with an internal power that
every port draws or delivers at its own rate'}
process_output: {dtype: str, 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: {dtype: str, description: 'demands, each on one bus'}
period: {dtype: int, ordered: true, description: investment periods — PyPSA's `investment_periods`}
relations:
snapshot_period: {key: snapshot, values: period, description: the investment period a snapshot falls
in}
Generator_bus: {key: generator, values: bus, description: the bus a generator sits on}
Link_bus0: {key: link, values: bus, description: the bus a link leaves}
Link_output_link: {key: link_output, values: link, description: the link an output port belongs to}
Link_output_bus: {key: link_output, values: 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'}
Process_output_process: {key: process_output, values: process, description: the process a port belongs
to}
Process_output_bus: {key: process_output, values: 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'}
Load_bus: {key: load, values: bus, description: the bus a load sits on}
Generator_maintenance_cover:
key:
scenario: scenario
generator: generator
start: snapshot
covered: snapshot
missing: absent
description: the snapshots a maintenance event covers, by the snapshot it starts in — in a scenario,
the start and the snapshots after it, until their generator weightings reach that scenario's `maintenance_duration`,
data prep; no rows for a generator that is not maintainable
parameters:
snapshot_weightings_objective:
dims: [snapshot]
dtype: float
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
Generator_p_nom:
dims: [scenario, generator]
dtype: float
description: nominal power
Generator_p_nom_extendable:
dims: [generator]
dtype: bool
description: whether the nominal power is a decision
Generator_p_min_pu:
dims: [scenario, snapshot, generator]
dtype: float
description: least output, per unit of nominal power
Generator_p_max_pu:
dims: [scenario, snapshot, generator]
dtype: float
description: most output, per unit of nominal power — an availability profile
Generator_marginal_cost:
dims: [scenario, snapshot, generator]
dtype: float
description: cost of one unit of output
Generator_marginal_cost_quadratic:
dims: [scenario, snapshot, generator]
dtype: float
description: cost of the square of one unit of output
Generator_sign:
dims: [generator]
dtype: float
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`)
Generator_committable:
dims: [generator]
dtype: bool
description: whether output is gated by an on/off status decision
Generator_ramp_limit_up:
dims: [scenario, snapshot, generator]
dtype: float
missing: neutral
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
Generator_ramp_limit_down:
dims: [scenario, snapshot, generator]
dtype: float
missing: neutral
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
Generator_ramp_limit_start_up:
dims: [scenario, generator]
dtype: float
missing: neutral
description: most output in the snapshot a unit starts, per unit of nominal power; no value means
no limit
Generator_ramp_limit_shut_down:
dims: [scenario, generator]
dtype: float
missing: neutral
description: most output in the snapshot before a unit stops, per unit of nominal power; no value
means no limit
Generator_min_down_time:
dims: [scenario, generator]
dtype: int
description: least snapshots a unit stays off once stopped
Generator_status_initial:
dims: [scenario, generator]
dtype: int
description: one where the unit was on before the first snapshot, zero where off — PyPSA's `up_time_before
> 0`, data prep
Generator_p_init:
dims: [scenario, generator]
dtype: float
missing: neutral
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
Generator_must_stay_down:
dims: [scenario, snapshot, generator]
dtype: bool
description: true while the down time a unit 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
Generator_start_up_cost:
dims: [scenario, snapshot, generator]
dtype: float
description: cost of one start in this snapshot
Generator_shut_down_cost:
dims: [scenario, snapshot, generator]
dtype: float
description: cost of one stop in this snapshot
Generator_stand_by_cost:
dims: [scenario, snapshot, generator]
dtype: float
description: cost of one snapshot spent on
Generator_p_nom_mod:
dims: [generator]
dtype: float
missing: neutral
description: the module size a build comes in whole numbers of; no value means the build is continuous
Generator_modules_installed:
dims: [scenario, generator]
dtype: float
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'
Generator_maintainable:
dims: [generator]
dtype: bool
description: whether a generator must be taken off for maintenance within the horizon — in any scenario,
as PyPSA takes the union over them (`components.py:1016-1019`)
Generator_maintenance_pu:
dims: [scenario, generator]
dtype: float
description: the share of the build a maintenance event takes off
Generator_maintenance_events:
dims: [scenario, generator]
dtype: int
description: how many maintenance events the horizon holds
Link_p_nom:
dims: [scenario, link]
dtype: float
description: nominal power
Link_p_nom_extendable:
dims: [link]
dtype: bool
description: whether the nominal power is a decision
Link_p_min_pu:
dims: [scenario, snapshot, link]
dtype: float
description: least flow, per unit of nominal power — negative for a link that carries both ways
Link_p_max_pu:
dims: [scenario, snapshot, link]
dtype: float
description: most flow, per unit of nominal power
Link_efficiency:
dims: [scenario, snapshot, link_output]
dtype: float
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`)
Link_output_delay:
dims: [link_output]
dtype: int
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`)
Link_output_cyclic_delay:
dims: [link_output]
dtype: bool
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
Link_marginal_cost:
dims: [scenario, snapshot, link]
dtype: float
description: cost of one unit of flow
Link_marginal_cost_quadratic:
dims: [scenario, snapshot, link]
dtype: float
description: cost of the square of one unit of flow
Link_committable:
dims: [link]
dtype: bool
description: whether flow is gated by an on/off status decision
Process_rate:
dims: [scenario, snapshot, process_output]
dtype: float
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`)
Process_output_delay:
dims: [process_output]
dtype: int
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
Process_output_cyclic_delay:
dims: [process_output]
dtype: bool
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
Load_p_set:
dims: [scenario, snapshot, load]
dtype: float
description: demand
Load_sign:
dims: [load]
dtype: float
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`)
Load_active:
dims: [load]
dtype: bool
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`)
scenario_weight:
dims: [scenario]
dtype: float
description: PyPSA's `scenario_weightings.weight` — the probability of a future
period_weight_objective:
dims: [period]
dtype: float
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`)
Generator_active:
dims: [snapshot, generator]
dtype: bool
description: whether a generator stands in a snapshot's period — PyPSA's `active`, from build year
and lifetime, data prep
Link_active:
dims: [snapshot, link]
dtype: bool
description: whether a link stands in a snapshot's period — PyPSA's `active`, data prep
Process_active:
dims: [snapshot, process]
dtype: bool
description: whether a process stands in a snapshot's period — PyPSA's `active`, data prep
Generator_active_asset:
dims: [generator]
dtype: bool
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`)
variables:
Generator_p:
dims: [scenario, snapshot, generator]
where: Generator_active
domain: continuous
missing: absent
description: '`Generator-p` — output of a generator in a snapshot'
Link_p:
dims: [scenario, snapshot, link]
where: Link_active
domain: continuous
missing: absent
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'
Generator_status:
dims: [scenario, snapshot, generator]
where: Generator_committed
bounds: {lower: 0.0, upper: 1.0}
domain: continuous
missing: absent
description: '`Generator-status` — how much of a committable unit is on, a share in [0, 1] rather
than an integer: the relaxation the keyword solves (`variables.py:86-87`)'
Generator_start_up:
dims: [scenario, snapshot, generator]
where: Generator_committed
bounds: {lower: 0.0, upper: 1.0}
domain: continuous
missing: absent
description: '`Generator-start_up` — how much of a committable unit turns on this snapshot, a share
in [0, 1] (`variables.py:137-138`)'
Generator_shut_down:
dims: [scenario, snapshot, generator]
where: Generator_committed
bounds: {lower: 0.0, upper: 1.0}
domain: continuous
missing: absent
description: '`Generator-shut_down` — how much of a committable unit turns off this snapshot, a share
in [0, 1] (`variables.py:188-189`)'
Generator_maintenance:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active
bounds: {lower: 0.0, upper: 1.0}
domain: continuous
missing: neutral
description: '`Generator-maintenance` — whether a maintainable generator is in maintenance: continuous,
and one exactly where an event covers the snapshot'
Generator_maintenance_start:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active
domain: binary
missing: neutral
description: '`Generator-maintenance_start` — whether a maintenance event starts in this snapshot'
Generator_maintenance_status:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_committed AND NOT (Generator_p_nom_extendable AND NOT
(Generator_p_nom_mod > 0))
bounds: {lower: 0.0}
domain: continuous
missing: neutral
description: '`Generator-maintenance_status` — the status while in maintenance, zero otherwise: the
product the `maint-status` rows linearize, so a unit in maintenance may also be off'
Generator_p_nom_ext:
dims: [generator]
where: Generator_p_nom_extendable AND Generator_active_asset
domain: continuous
missing: absent
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'
Process_p:
dims: [scenario, snapshot, process]
where: Process_active
domain: continuous
missing: absent
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'
constraints:
Generator_fix_p_lower:
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 * (1 - Generator_maintenance_pu *
Generator_maintenance)
description: '`Generator-fix-p-lower` — a fixed generator outputs at least its minimum'
Generator_fix_p_upper:
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 * (1 - Generator_maintenance_pu *
Generator_maintenance)
description: '`Generator-fix-p-upper` — a fixed generator outputs at most what is available'
Link_fix_p_lower:
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
description: '`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other
way'
Link_fix_p_upper:
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
description: '`Link-fix-p-upper` — a fixed link carries at most its nominal power'
Generator_com_p_lower:
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_maintenance_pu
* Generator_maintenance_status)
description: '`Generator-com-p-lower` — a committed unit outputs at least its minimum; off, at least
nothing'
Generator_com_p_upper:
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_maintenance_pu
* Generator_maintenance_status)
description: '`Generator-com-p-upper` — a committed unit outputs at most what is available; off, at
most nothing'
Generator_com_transition_start_up:
dims: [scenario, snapshot, generator]
where: Generator_committed
expression: Generator_start_up >= Generator_status - Generator_previous_status
description: '`Generator-com-transition-start-up` — turning on is a start, counted against the state
the unit carried into the snapshot'
Generator_com_transition_shut_down:
dims: [scenario, snapshot, generator]
where: Generator_committed
expression: Generator_shut_down >= Generator_previous_status - Generator_status
description: '`Generator-com-transition-shut-down` — turning off is a stop, counted against the state
the unit carried into the snapshot'
Generator_com_down_time:
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
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'
Generator_com_status_must_stay_down:
dims: [scenario, snapshot, generator]
where: Generator_committed AND Generator_must_stay_down
expression: Generator_status == 0
description: '`Generator-com-status-min_down_time_must_stay_up` — a unit still serving the down time
it brought in stays off; PyPSA names the row `_must_stay_up`'
Generator_status_p_fixed_upper:
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
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'
Generator_start_up_p_fixed_upper:
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
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'
Generator_shut_down_p_fixed_upper:
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
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'
Generator_maint_event_count:
dims: [scenario, generator]
where: Generator_maintainable
expression: sum(Generator_maintenance_start, over=snapshot) == Generator_maintenance_events
description: '`Generator-maint-event-count` — a maintainable generator holds its number of maintenance
events over the horizon'
Generator_maint_window:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active
expression: Generator_maintenance == sum(Generator_maintenance_start, over=start, by=Generator_maintenance_cover[covered])
description: '`Generator-maint-window` — a generator is in maintenance exactly where an event it started
covers the snapshot; two events do not overlap, since the maintenance status is at most one'
Generator_maint_status_le_status:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod
> 0)
expression: Generator_maintenance_status <= Generator_status
description: '`Generator-maint-status-le-status` — the status in maintenance is at most the status'
Generator_maint_status_le_maint:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod
> 0)
expression: Generator_maintenance_status <= Generator_maintenance
description: '`Generator-maint-status-le-maint` — out of maintenance, the status in maintenance is
zero'
Generator_maint_status_lb:
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod
> 0)
expression: Generator_maintenance_status >= Generator_status + Generator_maintenance - 1
description: '`Generator-maint-status-lb` — on and in maintenance, the status in maintenance is one'
Generator_p_ramp_limit_up:
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
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'
Generator_p_ramp_limit_down:
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
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'
Bus_nodal_balance:
dims: [scenario, snapshot, bus]
expression: Bus_injection == 0
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.'
expressions:
Generator_previous_status:
dims: [scenario, snapshot, generator]
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
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:
dims: [scenario, snapshot, generator]
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
cases:
opening: {when: position(snapshot) == 0, expression: Generator_status_initial}
otherwise: shift(Generator_status, along=snapshot, offset=1)
Generator_previous_p:
dims: [scenario, snapshot, generator]
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
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:
dims: [scenario, generator]
description: the build a generator's limits are taken against — the chosen one where it is extendable,
the given one otherwise
cases:
extendable: {when: Generator_p_nom_extendable, expression: Generator_p_nom_ext}
otherwise: Generator_p_nom
Generator_ramp_up_rate:
dims: [scenario, snapshot, generator]
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
cases:
given: {when: Generator_ramp_limit_up, expression: Generator_ramp_limit_up}
otherwise: '1'
Generator_ramp_down_rate:
dims: [scenario, snapshot, generator]
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
cases:
given: {when: Generator_ramp_limit_down, expression: Generator_ramp_limit_down}
otherwise: '1'
Generator_start_up_rate:
dims: [scenario, generator]
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
cases:
given: {when: Generator_ramp_limit_start_up, expression: Generator_ramp_limit_start_up}
otherwise: '1'
Generator_shut_down_rate:
dims: [scenario, generator]
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
cases:
given: {when: Generator_ramp_limit_shut_down, expression: Generator_ramp_limit_shut_down}
otherwise: '1'
Generator_p_nom_committed:
dims: [scenario, generator]
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
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:
dims: [scenario, snapshot, generator]
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
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:
dims: [scenario, snapshot, generator]
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
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:
dims: [scenario, snapshot, link_output]
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
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:
dims: [scenario, snapshot, process_output]
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
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)'}
Generator_commitment_opex: {expression: 'sum(sum(((Generator_status * Generator_stand_by_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period[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[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)'}
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:
dims: [scenario, snapshot, load]
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`)
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:
Generator_committed: {condition: Generator_committable AND Generator_active, description: a committable
generator that stands in the snapshot's period — every unit-commitment row's set}
Generator_ramps_from_previous: {condition: 'position(snapshot, within=snapshot_period[period]) > 0 OR
(position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init))', 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'}
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_44_linearized_commitment.yaml', sources) as solution:
solution.objective # 7400.0
The network, rung_44_linearized_commitment.py in the corpus — the spine plus what this rung adds:
# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT
"""Rung 44: linearized commitment with the rows the integer file has — a unit that must stay down, ramps read at the full build where a limit is missing, and maintenance."""
from __future__ import annotations
import spine
PATCH = 'variants/pypsa_linearized_uc.yaml'
OPTIMIZE = {'linearized_unit_commitment': True}
#: a start costs more than a stop, so PyPSA does not tighten these units and the rung isolates the rows it adds
UNTIGHTENED = {'committable': True, 'up_time_before': 0, 'start_up_cost': 1}
def build():
"""The spine plus five cheap units on a north bus with a swinging load: each carries one of the rows under review."""
n = spine.build()
n.add(
'Generator',
'down44',
bus='north',
p_nom=40,
marginal_cost=2,
min_down_time=3,
down_time_before=1,
**UNTIGHTENED,
)
n.add('Generator', 'pulse44', bus='north', p_nom=40, marginal_cost=3, ramp_limit_start_up=0.4, **UNTIGHTENED)
n.add(
'Generator',
'ramp44',
bus='north',
p_nom=40,
marginal_cost=4,
ramp_limit_up=0.5,
ramp_limit_down=0.5,
**UNTIGHTENED,
)
n.add(
'Generator',
'maint44',
bus='north',
p_nom=40,
p_min_pu=0.3,
marginal_cost=5,
maintainable=True,
maintenance_duration=2,
**UNTIGHTENED,
)
n.add('Generator', 'fixed44', bus='north', p_nom=20, marginal_cost=1, maintainable=True, maintenance_duration=1)
n.add('Load', 'swing44', bus='north', p_set=[40, 120, 160, 60])
return n
The data¶
The tables this rung is the first to declare (5), as the prep produced them:
Generator_maintainable.csv
generator,value
coal,false
down44,false
fixed44,true
gas,false
maint44,true
pulse44,false
ramp44,false
Generator_maintenance_cover.csv
scenario,generator,start,covered
base,fixed44,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,fixed44,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,fixed44,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,fixed44,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,fixed44,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
base,maint44,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,maint44,2015-01-01T00:00:00.000000,2015-01-01T01:00:00.000000
base,maint44,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,maint44,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,maint44,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,maint44,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
Generator_maintenance_events.csv
scenario,generator,value
base,coal,1
base,down44,1
base,fixed44,1
base,gas,1
base,maint44,1
base,pulse44,1
base,ramp44,1
Generator_maintenance_pu.csv
scenario,generator,value
base,coal,1.0
base,down44,1.0
base,fixed44,1.0
base,gas,1.0
base,maint44,1.0
base,pulse44,1.0
base,ramp44,1.0
Generator_must_stay_down.csv
scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,coal,false
base,2015-01-01T00:00:00.000000,down44,true
base,2015-01-01T00:00:00.000000,fixed44,false
base,2015-01-01T00:00:00.000000,gas,false
base,2015-01-01T00:00:00.000000,maint44,false
base,2015-01-01T00:00:00.000000,pulse44,false
base,2015-01-01T00:00:00.000000,ramp44,false
base,2015-01-01T01:00:00.000000,coal,false
base,2015-01-01T01:00:00.000000,down44,true
base,2015-01-01T01:00:00.000000,fixed44,false
base,2015-01-01T01:00:00.000000,gas,false
base,2015-01-01T01:00:00.000000,maint44,false
base,2015-01-01T01:00:00.000000,pulse44,false
base,2015-01-01T01:00:00.000000,ramp44,false
base,2015-01-01T02:00:00.000000,coal,false
base,2015-01-01T02:00:00.000000,down44,false
base,2015-01-01T02:00:00.000000,fixed44,false
base,2015-01-01T02:00:00.000000,gas,false
base,2015-01-01T02:00:00.000000,maint44,false
base,2015-01-01T02:00:00.000000,pulse44,false
base,2015-01-01T02:00:00.000000,ramp44,false
base,2015-01-01T03:00:00.000000,coal,false
base,2015-01-01T03:00:00.000000,down44,false
base,2015-01-01T03:00:00.000000,fixed44,false
base,2015-01-01T03:00:00.000000,gas,false
base,2015-01-01T03:00:00.000000,maint44,false
base,2015-01-01T03:00:00.000000,pulse44,false
base,2015-01-01T03:00:00.000000,ramp44,false