Rung 64: commitment edges — a committable generator, link and process built in the later period start up when they open, and a committable unit without a build cap is released by an inferred big M¶
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 9245.0 on both sides; structure ✔ 29 constraints · 13 variables, name for name; size ✔ 205 rows · ✔ 97 columns · ✔ 361 nonzeros; duals — integer model, no duals; model for model: 33 blocks equal, 0 documented splits, 10 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
16 | 16 |
Generator-com-ext-p-lower |
8 | 8 |
Generator-com-ext-p-lower-nonneg |
8 | 8 |
Generator-com-ext-p-upper-bigM |
8 | 8 |
Generator-com-ext-p-upper-cap |
8 | 8 |
Generator-com-p-lower |
4 | 4 |
Generator-com-p-upper |
4 | 4 |
Generator-com-transition-shut-down |
12 | 12 |
Generator-com-transition-start-up |
12 | 12 |
Generator-ext-p_nom-lower |
1 | 1 |
Generator-fix-p-lower |
16 | 16 |
Generator-fix-p-upper |
16 | 16 |
Generator-shut_down-p-fixed-upper |
12 | 12 |
Generator-start_up-p-fixed-upper |
12 | 12 |
Generator-status-p-fixed-upper |
12 | 12 |
Link-com-p-lower |
4 | 4 |
Link-com-p-upper |
4 | 4 |
Link-com-transition-shut-down |
4 | 4 |
Link-com-transition-start-up |
4 | 4 |
Link-shut_down-p-fixed-upper |
4 | 4 |
Link-start_up-p-fixed-upper |
4 | 4 |
Link-status-p-fixed-upper |
4 | 4 |
Process-com-p-lower |
4 | 4 |
Process-com-p-upper |
4 | 4 |
Process-com-transition-shut-down |
4 | 4 |
Process-com-transition-start-up |
4 | 4 |
Process-shut_down-p-fixed-upper |
4 | 4 |
Process-start_up-p-fixed-upper |
4 | 4 |
Process-status-p-fixed-upper |
4 | 4 |
| column | PyPSA | specsolve |
|---|---|---|
Generator-p |
28 | 28 |
Generator-p_nom |
1 | 1 |
Generator-shut_down |
12 | 12 |
Generator-start_up |
12 | 12 |
Generator-status |
12 | 12 |
Link-p |
4 | 4 |
Link-shut_down |
4 | 4 |
Link-start_up |
4 | 4 |
Link-status |
4 | 4 |
Process-p |
4 | 4 |
Process-shut_down |
4 | 4 |
Process-start_up |
4 | 4 |
Process-status |
4 | 4 |
The model¶
The same model, as math
A plain n.optimize(), and its multi-period and stochastic classes, in one file. Every second-stage quantity spans a scenario (a future dispatch is chosen in) and every asset stands in the investment periods its build year and lifetime span. A parameter spans scenario exactly when PyPSA reads it per scenario. Capacity is chosen once, before the future is known, and paid once per active period at its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights, with a share priced at the tail through the CVaR rows, which stand only where a risk preference is set. A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses to the standard one. A security-constrained run copies each branch flow limit once per outage in an outage set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight, and the outage factors are data prep.
Sets¶
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{bus}: \mathcal{D} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{O}\) | index \(o\) — link_output with \(\mathrm{link}: \mathcal{O} \to \mathcal{L},\ \mathrm{bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep |
| \(\mathcal{J}\) | index \(j\) — process with \(\mathrm{process}: \mathcal{R} \to \mathcal{J}\) — generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate |
| \(\mathcal{R}\) | index \(r\) — process_output with \(\mathrm{process}: \mathcal{R} \to \mathcal{J},\ \mathrm{bus}: \mathcal{R} \to \mathcal{N}\) — a process's ports, one label per port a process declares — PyPSA's bus0, bus1, … each carry a signed rate, so a process of any number of ports is one term in the balance, data prep |
| \(\mathcal{D}\) | index \(d\) — load with \(\mathrm{bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters¶
| Symbol | Meaning |
|---|---|
| \(\mathrm{w}\) | snapshot_weightings_objective over \(\mathcal{T}\) — PyPSA's snapshot_weightings.objective — hours a snapshot stands for in the cost |
| \(\mathrm{p}^{\mathrm{nom}}\) | Generator_p_nom over \(\Xi \times \mathcal{G}\) — nominal power |
| \(\mathrm{ext}\) | Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{p}}\) | Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power |
| \(\overline{\mathrm{p}}\) | Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile |
| \(\mathrm{c}\) | Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output |
| \(\mathrm{c}^{(2)}\) | Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output |
| \(\mathrm{sgn}\) | Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (constants.py:43) |
| \(\mathrm{com}\) | Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision |
| \(\mathrm{u}^{0}\) | 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 |
| \(\mathrm{c}^{\mathrm{up}}\) | Generator_start_up_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one start in this snapshot |
| \(\mathrm{c}^{\mathrm{dn}}\) | Generator_shut_down_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one stop in this snapshot |
| \(\mathrm{c}^{\mathrm{on}}\) | Generator_stand_by_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one snapshot spent on |
| \(\mathrm{p}^{\mathrm{mod}}\) | Generator_p_nom_mod over \(\mathcal{G}\), absent where the data has no row — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{\mathrm{fix}}\) | Generator_modules_installed over \(\Xi \times \mathcal{G}\) — how many whole modules a committable build has in place: Generator_p_nom / Generator_p_nom_mod where a fixed build is modular, one where it is not, data prep. PyPSA refuses a fixed modular build whose nominal power is not a whole number of modules |
| \(\mathrm{M}\) | Generator_big_m over \(\Xi \times \mathcal{G}\), \(0\) where the data has no row — the bound a committed extendable generator'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 output a solve wants, it caps that output; data prep |
| \(\mathrm{nonneg}\) | Generator_p_min_pu_nonneg over \(\mathcal{G}\) — true where none of the generator's own minimums-per-unit is negative — PyPSA's per-unit (p_min_pu >= 0).all() over every snapshot and scenario, data prep |
| \(\mathrm{f}^{\mathrm{nom}}\) | 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 |
| \(\mathrm{u}^{f,0}\) | 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 |
| \(\mathrm{c}^{f,\mathrm{up}}\) | 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 \(\mathcal{L}\), absent where the data has no row — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{f,\mathrm{fix}}\) | 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 |
| \(\mathrm{z}^{\mathrm{nom}}\) | Process_p_nom over \(\Xi \times \mathcal{J}\) — nominal internal power |
| \(\mathrm{ext}^{z}\) | Process_p_nom_extendable over \(\mathcal{J}\) — whether the nominal internal power is a decision |
| \(\underline{\mathrm{z}}\) | Process_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — least internal power, per unit of nominal power — negative for a process that runs both ways |
| \(\overline{\mathrm{z}}\) | Process_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — most internal power, per unit of nominal power |
| \(\alpha\) | 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 |
| \(\mathrm{c}^{z}\) | Process_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — cost of one unit of internal power |
| \(\mathrm{c}^{z,(2)}\) | Process_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — cost of the square of one unit of internal power |
| \(\mathrm{com}^{z}\) | Process_committable over \(\mathcal{J}\) — whether internal power is gated by an on/off status decision |
| \(\mathrm{u}^{z,0}\) | Process_status_initial over \(\Xi \times \mathcal{J}\) — one where the process was on before the first snapshot, zero where off — PyPSA's up_time_before > 0, data prep |
| \(\mathrm{c}^{z,\mathrm{up}}\) | Process_start_up_cost over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — cost of one start in this snapshot |
| \(\mathrm{c}^{z,\mathrm{dn}}\) | Process_shut_down_cost over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — cost of one stop in this snapshot |
| \(\mathrm{c}^{z,\mathrm{on}}\) | Process_stand_by_cost over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — cost of one snapshot spent on |
| \(\mathrm{z}^{\mathrm{mod}}\) | Process_p_nom_mod over \(\mathcal{J}\), absent where the data has no row — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{z,\mathrm{fix}}\) | Process_modules_installed over \(\Xi \times \mathcal{J}\) — how many whole modules a committable build has in place: Process_p_nom / Process_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 |
| \(\mathrm{load}\) | 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) |
| \(\mathrm{W}\) | Generator_capital_weight over \(\mathcal{G}\) — the sum of period weights a generator stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\underline{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_min over \(\Xi \times \mathcal{G}\) — least nominal power an extendable generator may be built at |
| \(\mathrm{c}^{\mathrm{cap}}\) | Generator_capital_cost over \(\Xi \times \mathcal{G}\) — 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 |
|---|---|
| \(\mathrm{Generator\_com\_ext}\) | Generator_com_ext over \(\mathcal{G} \times \mathcal{T}\) — a committable generator 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 |
| \(\mathrm{Generator\_committed}\) | Generator_committed over \(\mathcal{G} \times \mathcal{T}\) — a committable generator that stands in the snapshot's period — every unit-commitment row's set |
| \(\mathrm{Link\_committed}\) | Link_committed over \(\mathcal{L} \times \mathcal{T}\) — a committable link that stands in the snapshot's period — every unit-commitment row's set |
| \(\mathrm{Process\_committed}\) | Process_committed over \(\mathcal{J} \times \mathcal{T}\) — a committable process that stands in the snapshot's period — every unit-commitment row's set |
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 |
| \(z\) | 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 |
| \(u\) | Generator_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-status — how much of a committable unit is on: an integer the rows below cap at one, or at the module count where the build is modular |
| \(\mathit{up}\) | Generator_start_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-start_up — how much of a committable unit turns on this snapshot, capped as the status is |
| \(\mathit{dn}\) | Generator_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-shut_down — how much of a committable unit turns off this snapshot, capped as the status is |
| \(u^{f}\) | 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 |
| \(u^{z}\) | Process_status over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-status — how much of a committable process is on: an integer the rows below cap at one, or at the module count where the build is modular |
| \(\mathit{up}^{z}\) | Process_start_up over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-start_up — how much of a committable process turns on this snapshot, capped as the status is |
| \(\mathit{dn}^{z}\) | Process_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-shut_down — how much of a committable process turns off this snapshot, capped as the status is |
| \(P\) | Generator_p_nom_ext over \(\mathcal{G}\) — Generator-p_nom — nominal power where it is a decision, for a generator that is active; the parameter of the same PyPSA name carries the fixed regime |
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 |
| \(\mathit{Link\_previous\_status}\) | 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 |
| \(\mathit{Process\_previous\_status}\) | Process_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the commitment state a process 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 |
| \(\mathit{Process\_status\_carried\_over}\) | Process_status_carried_over over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the state a process carries over into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that |
| \(\mathit{Link\_output\_arrival}\) | 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 \(\Xi\) |
| \(\mathit{Link\_commitment\_opex}\) | Link_commitment_opex over \(\Xi\) |
| \(\mathit{Process\_opex}\) | Process_opex over \(\Xi\) |
| \(\mathit{Process\_commitment\_opex}\) | Process_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) |
| \(\mathit{Generator\_capex}\) | Generator_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 |
| \(t \ominus^{\mathrm{property}(t)} k\) | within=subject[...] — \(t-k\) inside the group of \(t\) |
| \(\mathrm{pos}(t)\) | position() — the position of \(t\), counted from \(0\); \(t\) is the label |
Conditions¶
Generator_com_ext
\[
\mathrm{Generator\_com\_ext}_{g,t} \iff \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \mathrm{on}_{t,g} \qquad \forall\, g \in \mathcal{G},\ t \in \mathcal{T}
\]
Generator_committed
\[
\mathrm{Generator\_committed}_{g,t} \iff \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \qquad \forall\, g \in \mathcal{G},\ t \in \mathcal{T}
\]
Link_committed
\[
\mathrm{Link\_committed}_{l,t} \iff \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l} \qquad \forall\, l \in \mathcal{L},\ t \in \mathcal{T}
\]
Process_committed
\[
\mathrm{Process\_committed}_{j,t} \iff \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j} \qquad \forall\, j \in \mathcal{J},\ t \in \mathcal{T}
\]
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
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
Process_p
\[
z_{\xi,t,j} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{on}^{z}_{t,j}
\]
Generator_status
\[
u_{\xi,t,g} \ge 0, u_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_start_up
\[
\mathit{up}_{\xi,t,g} \ge 0, \mathit{up}_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_shut_down
\[
\mathit{dn}_{\xi,t,g} \ge 0, \mathit{dn}_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
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
\[
\mathit{dn}^{f}_{\xi,t,l} \ge 0, \mathit{dn}^{f}_{\xi,t,l} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t}
\]
Process_status
\[
u^{z}_{\xi,t,j} \ge 0, u^{z}_{\xi,t,j} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t}
\]
Process_start_up
\[
\mathit{up}^{z}_{\xi,t,j} \ge 0, \mathit{up}^{z}_{\xi,t,j} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t}
\]
Process_shut_down
\[
\mathit{dn}^{z}_{\xi,t,j} \ge 0, \mathit{dn}^{z}_{\xi,t,j} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t}
\]
Generator_p_nom_ext
\[
P_{g} \in \mathbb{R} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{act}_{g}
\]
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
\[
\mathit{Generator\_status\_carried\_over}_{\xi,t,g} = \begin{cases} \mathrm{u}^{0}_{\xi,g} & \text{if } \mathrm{pos}(t) = 0 \\ u_{\xi,t - 1,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
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
\[
\mathit{Link\_status\_carried\_over}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ u^{f}_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Process_previous_status
\[
\mathit{Process\_previous\_status}_{\xi,t,j} = \begin{cases} 0 & \text{if } \mathrm{pos}(t) > 0 \wedge \neg \mathrm{on}^{z}_{t - 1,j} \\ \mathit{Process\_status\_carried\_over}_{\xi,t,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_status_carried_over
\[
\mathit{Process\_status\_carried\_over}_{\xi,t,j} = \begin{cases} \mathrm{u}^{z,0}_{\xi,j} & \text{if } \mathrm{pos}(t) = 0 \\ u^{z}_{\xi,t - 1,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Link_output_arrival
\[
\mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{period}(t)} \mathrm{d}^{f},\mathrm{link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{period}(t)} \mathrm{d}^{f},\mathrm{link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
Process_output_arrival
\[
\mathit{Process\_output\_arrival}_{\xi,t,r} = \begin{cases} z_{\xi,t \ominus^{\mathrm{period}(t)} \mathrm{d}^{z},\mathrm{process}(r)} \cdot \alpha_{\xi,t,r} & \text{if } \mathrm{cyc}^{z}_{r} \\ z_{\xi,t \boxminus_{0}^{\mathrm{period}(t)} \mathrm{d}^{z},\mathrm{process}(r)} \cdot \alpha_{\xi,t,r} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ r \in \mathcal{R}
\]
Generator_opex
\[
\mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} \qquad \forall\, \xi \in \Xi
\]
Generator_commitment_opex
\[
\mathit{Generator\_commitment\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} u_{\xi,t,g} \cdot \mathrm{c}^{\mathrm{on}}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} \mathit{up}_{\xi,t,g} \cdot \mathrm{c}^{\mathrm{up}}_{\xi,t,g} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} \mathit{dn}_{\xi,t,g} \cdot \mathrm{c}^{\mathrm{dn}}_{\xi,t,g} \qquad \forall\, \xi \in \Xi
\]
Link_opex
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} \qquad \forall\, \xi \in \Xi
\]
Link_commitment_opex
\[
\mathit{Link\_commitment\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} u^{f}_{\xi,t,l} \cdot \mathrm{c}^{f,\mathrm{on}}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} \mathit{up}^{f}_{\xi,t,l} \cdot \mathrm{c}^{f,\mathrm{up}}_{\xi,t,l} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} \mathit{dn}^{f}_{\xi,t,l} \cdot \mathrm{c}^{f,\mathrm{dn}}_{\xi,t,l} \qquad \forall\, \xi \in \Xi
\]
Process_opex
\[
\mathit{Process\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} z_{\xi,t,j} \cdot \mathrm{c}^{z}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} z_{\xi,t,j} \cdot z_{\xi,t,j} \cdot \mathrm{c}^{z,(2)}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} \qquad \forall\, \xi \in \Xi
\]
Process_commitment_opex
\[
\mathit{Process\_commitment\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} u^{z}_{\xi,t,j} \cdot \mathrm{c}^{z,\mathrm{on}}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{period}(t)} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} \mathit{up}^{z}_{\xi,t,j} \cdot \mathrm{c}^{z,\mathrm{up}}_{\xi,t,j} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} \mathit{dn}^{z}_{\xi,t,j} \cdot \mathrm{c}^{z,\mathrm{dn}}_{\xi,t,j} \qquad \forall\, \xi \in \Xi
\]
scenario_opex
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Generator\_commitment\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} + \mathit{Link\_commitment\_opex}_{\xi} + \mathit{Process\_opex}_{\xi} + \mathit{Process\_commitment\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
Generator_capex
\[
\mathit{Generator\_capex} = \sum_{\xi \in \Xi,\ g \in \mathcal{G}} \pi_{\xi} \cdot P_{g} \cdot \mathrm{c}^{\mathrm{cap}}_{\xi,g} \cdot \mathrm{W}_{g}
\]
risk_weighted_opex
\[
\mathit{risk\_weighted\_opex} = \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi}
\]
total_cost
\[
\mathit{total\_cost} = \mathit{Generator\_capex} + \mathit{risk\_weighted\_opex}
\]
Load_demand
\[
\mathrm{Load\_demand}_{\xi,t,d} = \begin{cases} \mathrm{sgn}^{\mathrm{load}}_{d} \cdot \mathrm{load}_{\xi,t,d} & \text{if } \mathrm{on}^{\mathrm{load}}_{d} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ d \in \mathcal{D}
\]
Generator_injection
\[
\mathit{Generator\_injection}_{n,\xi,t} = \sum_{g \in \mathcal{G} \,:\, \mathrm{bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Link_injection
\[
\mathit{Link\_injection}_{n,\xi,t} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Load_injection
\[
\mathrm{Load\_injection}_{n,\xi,t} = \sum_{d \in \mathcal{D} \,:\, \mathrm{bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Process_injection
\[
\mathit{Process\_injection}_{n,\xi,t} = \sum_{r \in \mathcal{R} \,:\, \mathrm{bus}(r) = n} \mathit{Process\_output\_arrival}_{\xi,t,r} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Bus_injection
\[
\mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{n,\xi,t} + \mathit{Link\_injection}_{n,\xi,t} + \mathrm{Load\_injection}_{n,\xi,t} + \mathit{Process\_injection}_{n,\xi,t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Objective¶
\[
\min \mathit{total\_cost}
\]
Subject to¶
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_nom_lower
\[
P_{g} \ge \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{act}_{g}
\]
Generator_com_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
Generator_com_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
Generator_com_transition_start_up
\[
\mathit{up}_{\xi,t,g} \ge u_{\xi,t,g} - \mathit{Generator\_previous\_status}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_com_transition_shut_down
\[
\mathit{dn}_{\xi,t,g} \ge \mathit{Generator\_previous\_status}_{\xi,t,g} - u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t}
\]
Generator_com_ext_p_upper_cap
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot P_{g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_com\_ext}_{g,t}
\]
Generator_com_ext_p_upper_big_m
\[
p_{\xi,t,g} \le \mathrm{M}_{\xi,g} \cdot u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_com\_ext}_{g,t}
\]
Generator_com_ext_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot P_{g} + \mathrm{M}_{\xi,g} \cdot u_{\xi,t,g} - \mathrm{M}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_com\_ext}_{g,t}
\]
Generator_com_ext_p_lower_nonneg
\[
p_{\xi,t,g} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_com\_ext}_{g,t} \wedge \mathrm{nonneg}_{g}
\]
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
\[
\mathit{dn}_{\xi,t,g} \le \mathrm{N}^{\mathrm{fix}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{Generator\_committed}_{g,t} \wedge \neg \left( \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right)
\]
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_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
\[
\mathit{dn}^{f}_{\xi,t,l} \le \mathrm{N}^{f,\mathrm{fix}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{Link\_committed}_{l,t} \wedge \neg \left( \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right)
\]
Process_com_p_lower
\[
z_{\xi,t,j} \ge \underline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{nom}}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t} \wedge \neg \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right)
\]
Process_com_p_upper
\[
z_{\xi,t,j} \le \overline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{nom}}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t} \wedge \neg \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right)
\]
Process_com_transition_start_up
\[
\mathit{up}^{z}_{\xi,t,j} \ge u^{z}_{\xi,t,j} - \mathit{Process\_previous\_status}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t}
\]
Process_com_transition_shut_down
\[
\mathit{dn}^{z}_{\xi,t,j} \ge \mathit{Process\_previous\_status}_{\xi,t,j} - u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t}
\]
Process_status_p_fixed_upper
\[
u^{z}_{\xi,t,j} \le \mathrm{N}^{z,\mathrm{fix}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t} \wedge \neg \left( \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right)
\]
Process_start_up_p_fixed_upper
\[
\mathit{up}^{z}_{\xi,t,j} \le \mathrm{N}^{z,\mathrm{fix}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t} \wedge \neg \left( \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right)
\]
Process_shut_down_p_fixed_upper
\[
\mathit{dn}^{z}_{\xi,t,j} \le \mathrm{N}^{z,\mathrm{fix}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{Process\_committed}_{j,t} \wedge \neg \left( \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right)
\]
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_64_commitment_edges.yaml — the file projected onto what this rung builds:
description: A plain `n.optimize()`, and its multi-period and stochastic classes, in one file. Every second-stage
quantity spans a `scenario` (a future dispatch is chosen in) and every asset stands in the investment
`period`s its build year and lifetime span. A parameter spans `scenario` exactly when PyPSA reads it
per scenario. Capacity is chosen once, before the future is known, and paid once per active period at
its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights,
with a share priced at the tail through the CVaR rows, which stand only where a risk preference is set.
A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses
to the standard one. A security-constrained run copies each branch flow limit once per outage in an
`outage` set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight,
and the outage factors are data prep.
dimensions:
scenario: {description: 'the futures dispatch is chosen in, each with a weight'}
snapshot: {description: dispatch periods, dtype: datetime, ordered: true}
bus: {description: network nodes}
generator: {description: 'generating units, each on one bus'}
link: {description: 'controllable connections, each from one bus to the buses it delivers to'}
link_output: {description: 'a link''s output ports, one label per port a link declares — PyPSA''s `bus1`,
`bus2`, … columns read long, so a link of any number of output ports is one term in the balance,
data prep'}
process: {description: 'generalized multi-port converters, each with an internal power that every port
draws or delivers at its own rate'}
process_output: {description: 'a process''s ports, one label per port a process declares — PyPSA''s
`bus0`, `bus1`, … each carry a signed `rate`, so a process of any number of ports is one term in
the balance, data prep'}
load: {description: 'demands, each on one bus'}
period: {description: investment periods — PyPSA's `investment_periods`, dtype: int, ordered: true}
properties:
snapshot:
period:
description: the investment period a snapshot falls in
into: period
generator:
bus:
description: the bus a generator sits on
into: bus
link:
bus0:
description: the bus a link leaves
into: bus
link_output:
link:
description: the link an output port belongs to
into: link
bus:
description: >-
the bus an output port delivers to — PyPSA's `bus1`, `bus2`, … columns.
A link of three output ports is three labels here rather than a third
property, so the file states any number of them
into: bus
process_output:
process:
description: the process a port belongs to
into: process
bus:
description: >-
the bus a port draws from or delivers to — PyPSA's `bus0`, `bus1`, …
columns. A process of three ports is three labels here rather than a
third property, so the file states any number of them
into: bus
load:
bus:
description: the bus a load sits on
into: bus
parameters:
snapshot_weightings_objective:
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
dims: [snapshot]
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
for a unit that draws power. PyPSA refuses one that differs by scenario (`constants.py:43`)
dims: [generator]
Generator_committable:
description: whether output is gated by an on/off status decision
dims: [generator]
dtype: bool
Generator_status_initial:
description: one where the unit was on before the first snapshot, zero where off — PyPSA's `up_time_before
> 0`, data prep
dims: [scenario, generator]
dtype: int
Generator_start_up_cost:
description: cost of one start in this snapshot
dims: [scenario, snapshot, generator]
Generator_shut_down_cost:
description: cost of one stop in this snapshot
dims: [scenario, snapshot, generator]
Generator_stand_by_cost:
description: cost of one snapshot spent on
dims: [scenario, snapshot, generator]
Generator_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [generator]
missing: absent
Generator_modules_installed:
description: 'how many whole modules a committable build has in place: `Generator_p_nom / Generator_p_nom_mod`
where a fixed build is modular, one where it is not, data prep. PyPSA refuses a fixed modular build
whose nominal power is not a whole number of modules'
dims: [scenario, generator]
Generator_big_m:
description: the bound a committed extendable generator'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 output a solve wants, it
caps that output; data prep
dims: [scenario, generator]
missing: 0
Generator_p_min_pu_nonneg:
description: true where none of the generator's own minimums-per-unit is negative — PyPSA's per-unit
`(p_min_pu >= 0).all()` over every snapshot and scenario, data prep
dims: [generator]
dtype: bool
Link_p_nom:
description: nominal power
dims: [scenario, link]
Link_p_nom_extendable:
description: whether the nominal power is a decision
dims: [link]
dtype: bool
Link_p_min_pu:
description: least flow, per unit of nominal power — negative for a link that carries both ways
dims: [scenario, snapshot, link]
Link_p_max_pu:
description: most flow, per unit of nominal power
dims: [scenario, snapshot, link]
Link_efficiency:
description: share of the flow that arrives at an output port, PyPSA's `efficiency`, `efficiency2`,
… read long — negative where that port consumes rather than delivers. Read at the snapshot the flow
arrives, so a delayed port delivers at its arrival snapshot's efficiency (`constraints.py:1498`)
dims: [scenario, snapshot, link_output]
Link_output_delay:
description: snapshots a port's delivery lags its link's flow — PyPSA's `delay`, `delay2`, … read
long, divided by the `snapshot_weightings.generators` value and rounded up in data prep; zero for
a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses
a delay that differs by scenario (`constants.py:52`)
dims: [link_output]
dtype: int
Link_output_cyclic_delay:
description: whether a delayed port's flow wraps from the end of its investment period — PyPSA's `cyclic_delay`,
`cyclic_delay2`, …; where it does not, the flow still in transit at each period's first snapshots
is lost. The flag is the same in every scenario, as the delay is
dims: [link_output]
dtype: bool
Link_marginal_cost:
description: cost of one unit of flow
dims: [scenario, snapshot, link]
Link_marginal_cost_quadratic:
description: cost of the square of one unit of flow
dims: [scenario, snapshot, link]
Link_committable:
description: whether flow is gated by an on/off status decision
dims: [link]
dtype: bool
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_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: absent
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]
Process_p_nom:
description: nominal internal power
dims: [scenario, process]
Process_p_nom_extendable:
description: whether the nominal internal power is a decision
dims: [process]
dtype: bool
Process_p_min_pu:
description: least internal power, per unit of nominal power — negative for a process that runs both
ways
dims: [scenario, snapshot, process]
Process_p_max_pu:
description: most internal power, per unit of nominal power
dims: [scenario, snapshot, process]
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
Process_marginal_cost:
description: cost of one unit of internal power
dims: [scenario, snapshot, process]
Process_marginal_cost_quadratic:
description: cost of the square of one unit of internal power
dims: [scenario, snapshot, process]
Process_committable:
description: whether internal power is gated by an on/off status decision
dims: [process]
dtype: bool
Process_status_initial:
description: one where the process was on before the first snapshot, zero where off — PyPSA's `up_time_before
> 0`, data prep
dims: [scenario, process]
dtype: int
Process_start_up_cost:
description: cost of one start in this snapshot
dims: [scenario, snapshot, process]
Process_shut_down_cost:
description: cost of one stop in this snapshot
dims: [scenario, snapshot, process]
Process_stand_by_cost:
description: cost of one snapshot spent on
dims: [scenario, snapshot, process]
Process_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [process]
missing: absent
Process_modules_installed:
description: 'how many whole modules a committable build has in place: `Process_p_nom / Process_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, process]
Load_p_set:
description: demand
dims: [scenario, snapshot, load]
Load_sign:
description: the sign a load's demand enters its bus's balance with — PyPSA's `sign`, `-1` unless
given, `1` for a load that feeds its bus. PyPSA refuses one that differs by scenario (`constants.py:43`)
dims: [load]
Load_active:
description: whether a load stands in the model — PyPSA's `active`. A load has no build year and no
lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (`constants.py:51`)
dims: [load]
dtype: bool
scenario_weight:
description: PyPSA's `scenario_weightings.weight` — the probability of a future
dims: [scenario]
period_weight_objective:
description: PyPSA's `investment_period_weightings.objective` — what a period's cost weighs; PyPSA
reads it only under `multi_investment_periods`, so data prep feeds one otherwise, whatever the column
holds (`optimize.py:205-207`, `:264-266`)
dims: [period]
Generator_active:
description: whether a generator stands in a snapshot's period — PyPSA's `active`, from build year
and lifetime, data prep
dims: [snapshot, generator]
dtype: bool
Link_active:
description: whether a link stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, link]
dtype: bool
Process_active:
description: whether a process stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, process]
dtype: bool
Generator_active_asset:
description: whether a generator is one of PyPSA's `active_assets` — its `active` flag alone, without
build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA
refuses one that differs by scenario (`constants.py:51`)
dims: [generator]
dtype: bool
Generator_capital_weight:
description: the sum of period weights a generator stands in — PyPSA's `active * period_weighting`,
summed, data prep
dims: [generator]
Generator_p_nom_min:
description: least nominal power an extendable generator may be built at
dims: [scenario, generator]
Generator_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, generator]
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
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
Generator_status:
description: '`Generator-status` — how much of a committable unit is on: an integer the rows below
cap at one, or at the module count where the build is modular'
dims: [scenario, snapshot, generator]
where: Generator_committed
domain: integer
bounds: {lower: 0}
Generator_start_up:
description: '`Generator-start_up` — how much of a committable unit turns on this snapshot, capped
as the status is'
dims: [scenario, snapshot, generator]
where: Generator_committed
domain: integer
bounds: {lower: 0}
Generator_shut_down:
description: '`Generator-shut_down` — how much of a committable unit turns off this snapshot, capped
as the status is'
dims: [scenario, snapshot, generator]
where: Generator_committed
domain: integer
bounds: {lower: 0}
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}
Process_status:
description: '`Process-status` — how much of a committable process is on: an integer the rows below
cap at one, or at the module count where the build is modular'
dims: [scenario, snapshot, process]
where: Process_committed
domain: integer
bounds: {lower: 0}
Process_start_up:
description: '`Process-start_up` — how much of a committable process turns on this snapshot, capped
as the status is'
dims: [scenario, snapshot, process]
where: Process_committed
domain: integer
bounds: {lower: 0}
Process_shut_down:
description: '`Process-shut_down` — how much of a committable process turns off this snapshot, capped
as the status is'
dims: [scenario, snapshot, process]
where: Process_committed
domain: integer
bounds: {lower: 0}
Generator_p_nom_ext:
description: '`Generator-p_nom` — nominal power where it is a decision, for a generator that is active;
the parameter of the same PyPSA name carries the fixed regime'
dims: [generator]
where: Generator_p_nom_extendable AND Generator_active_asset
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
Generator_ext_p_nom_lower:
description: '`Generator-ext-p_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, generator]
where: Generator_p_nom_extendable AND Generator_active_asset
expression: Generator_p_nom_ext >= Generator_p_nom_min
Generator_com_p_lower:
description: '`Generator-com-p-lower` — a committed unit outputs at least its minimum; off, at least
nothing'
dims: [scenario, snapshot, generator]
where: Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
expression: Generator_p >= (Generator_p_min_pu * Generator_p_nom) * Generator_status
Generator_com_p_upper:
description: '`Generator-com-p-upper` — a committed unit outputs at most what is available; off, at
most nothing'
dims: [scenario, snapshot, generator]
where: Generator_committed AND NOT Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
expression: Generator_p <= (Generator_p_max_pu * Generator_p_nom) * Generator_status
Generator_com_transition_start_up:
description: '`Generator-com-transition-start-up` — turning on is a start, counted against the state
the unit carried into the snapshot'
dims: [scenario, snapshot, generator]
where: Generator_committed
expression: Generator_start_up >= Generator_status - Generator_previous_status
Generator_com_transition_shut_down:
description: '`Generator-com-transition-shut-down` — turning off is a stop, counted against the state
the unit carried into the snapshot'
dims: [scenario, snapshot, generator]
where: Generator_committed
expression: Generator_shut_down >= Generator_previous_status - Generator_status
Generator_com_ext_p_upper_cap:
description: '`Generator-com-ext-p-upper-cap` — a committed extendable unit outputs at most what is
available of the chosen build, whatever its status'
dims: [scenario, snapshot, generator]
where: Generator_com_ext
expression: Generator_p <= Generator_p_max_pu * Generator_p_nom_ext
Generator_com_ext_p_upper_big_m:
description: '`Generator-com-ext-p-upper-bigM` — off, a unit outputs nothing; on, the big M is no
bound'
dims: [scenario, snapshot, generator]
where: Generator_com_ext
expression: Generator_p <= Generator_big_m * Generator_status
Generator_com_ext_p_lower:
description: '`Generator-com-ext-p-lower` — a committed extendable unit outputs at least its minimum
of the chosen build; off, the big M releases the row'
dims: [scenario, snapshot, generator]
where: Generator_com_ext
expression: Generator_p >= ((Generator_p_min_pu * Generator_p_nom_ext) + (Generator_big_m * Generator_status))
- Generator_big_m
Generator_com_ext_p_lower_nonneg:
description: '`Generator-com-ext-p-lower-nonneg` — where no minimum-per-unit is negative, output is
also plainly non-negative, a row the big-M lower cannot assert while the unit is off'
dims: [scenario, snapshot, generator]
where: Generator_com_ext AND Generator_p_min_pu_nonneg
expression: Generator_p >= 0
Generator_status_p_fixed_upper:
description: '`Generator-status-p-fixed-upper` — a status is at most the modules in place, an explicit
row as PyPSA writes it: one where the build is not modular, and the fixed build''s whole count of
modules where it is'
dims: [scenario, snapshot, generator]
where: Generator_committed AND NOT (Generator_p_nom_extendable AND Generator_p_nom_mod > 0)
expression: Generator_status <= Generator_modules_installed
Generator_start_up_p_fixed_upper:
description: '`Generator-start_up-p-fixed-upper` — a start is at most the modules in place, an explicit
row as PyPSA writes it: one where the build is not modular, and the fixed build''s whole count of
modules where it is'
dims: [scenario, snapshot, generator]
where: Generator_committed AND NOT (Generator_p_nom_extendable AND Generator_p_nom_mod > 0)
expression: Generator_start_up <= Generator_modules_installed
Generator_shut_down_p_fixed_upper:
description: '`Generator-shut_down-p-fixed-upper` — a stop is at most the modules in place, an explicit
row as PyPSA writes it: one where the build is not modular, and the fixed build''s whole count of
modules where it is'
dims: [scenario, snapshot, generator]
where: Generator_committed AND NOT (Generator_p_nom_extendable AND Generator_p_nom_mod > 0)
expression: Generator_shut_down <= Generator_modules_installed
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_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
Process_com_p_lower:
description: '`Process-com-p-lower` — a committed process runs at least its minimum; off, at least
nothing'
dims: [scenario, snapshot, process]
where: Process_committed AND NOT Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)
expression: Process_p >= (Process_p_min_pu * Process_p_nom) * Process_status
Process_com_p_upper:
description: '`Process-com-p-upper` — a committed process runs at most what is available; off, at
most nothing'
dims: [scenario, snapshot, process]
where: Process_committed AND NOT Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)
expression: Process_p <= (Process_p_max_pu * Process_p_nom) * Process_status
Process_com_transition_start_up:
description: '`Process-com-transition-start-up` — turning on is a start, counted against the state
the process carried into the snapshot'
dims: [scenario, snapshot, process]
where: Process_committed
expression: Process_start_up >= Process_status - Process_previous_status
Process_com_transition_shut_down:
description: '`Process-com-transition-shut-down` — turning off is a stop, counted against the state
the process carried into the snapshot'
dims: [scenario, snapshot, process]
where: Process_committed
expression: Process_shut_down >= Process_previous_status - Process_status
Process_status_p_fixed_upper:
description: '`Process-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, process]
where: Process_committed AND NOT (Process_p_nom_extendable AND Process_p_nom_mod > 0)
expression: Process_status <= Process_modules_installed
Process_start_up_p_fixed_upper:
description: '`Process-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, process]
where: Process_committed AND NOT (Process_p_nom_extendable AND Process_p_nom_mod > 0)
expression: Process_start_up <= Process_modules_installed
Process_shut_down_p_fixed_upper:
description: '`Process-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, process]
where: Process_committed AND NOT (Process_p_nom_extendable AND Process_p_nom_mod > 0)
expression: Process_shut_down <= Process_modules_installed
Bus_nodal_balance:
description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
less what the links take away, plus what arrives over them after losses and any delay at every port
they deliver to, each process port drawing or delivering at its own rate and each passive branch
carrying its flow, meets the load there, less half of every incident line''s and transformer''s
loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
and load term enters with its component''s `sign` (`constraints.py:1404-1405`, `:1514`), and an
inactive load not at all. A bus nothing is attached to has no row; PyPSA refuses one that carries
load, and this file does not yet.'
dims: [scenario, snapshot, bus]
expression: Bus_injection == 0
expressions:
Generator_previous_status:
description: the commitment state a generator carries into a snapshot — off at the first snapshot
it stands in past the first of the horizon, as PyPSA reads a status it did not build (`constraints.py:298`),
and the state carried over otherwise
dims: [scenario, snapshot, generator]
cases:
opening_late: {when: 'position(snapshot) > 0 AND NOT shift(Generator_active, along=snapshot, offset=1)',
expression: 0}
otherwise: Generator_status_carried_over
Generator_status_carried_over:
description: the state a generator carries over into a snapshot — the state it brought into the horizon
at the first, the previous snapshot's after that
dims: [scenario, snapshot, generator]
cases:
opening: {when: position(snapshot) == 0, expression: Generator_status_initial}
otherwise: shift(Generator_status, along=snapshot, offset=1)
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)
Process_previous_status:
description: the commitment state a process 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, process]
cases:
opening_late: {when: 'position(snapshot) > 0 AND NOT shift(Process_active, along=snapshot, offset=1)',
expression: 0}
otherwise: Process_status_carried_over
Process_status_carried_over:
description: the state a process carries over into a snapshot — the state it brought into the horizon
at the first, the previous snapshot's after that
dims: [scenario, snapshot, process]
cases:
opening: {when: position(snapshot) == 0, expression: Process_status_initial}
otherwise: shift(Process_status, along=snapshot, offset=1)
Link_output_arrival:
description: what a link delivers to an output port at a snapshot — its flow delayed by the port's
`delay` within its investment period, times the port's efficiency at the snapshot the flow arrives;
where the port is `cyclic_delay` the delayed flow wraps from the period's end, and where it is not
the flow still in transit at the period's first snapshots is lost. A port that does not delay (`delay`
zero) delivers its flow unshifted, cyclic or not
dims: [scenario, snapshot, link_output]
cases:
wrapping: {when: Link_output_cyclic_delay, expression: 'shift(at(Link_p, by=link_output[link]),
along=snapshot, offset=Link_output_delay, edge=''wrap'', within=snapshot[period]) * Link_efficiency'}
otherwise: shift(at(Link_p, by=link_output[link]), along=snapshot, offset=Link_output_delay,
edge=0, within=snapshot[period]) * Link_efficiency
Process_output_arrival:
description: what a process transfers at a port at a snapshot — its internal power delayed by the
port's `delay` within its investment period, times the port's rate at the snapshot the transfer
arrives; where the port is `cyclic_delay` the delayed transfer wraps from the period's end, and
where it is not the energy still in transit at the period's first snapshots is lost. A port that
does not delay (`delay` zero) transfers at once, cyclic or not
dims: [scenario, snapshot, process_output]
cases:
wrapping: {when: Process_output_cyclic_delay, expression: 'shift(at(Process_p, by=process_output[process]),
along=snapshot, offset=Process_output_delay, edge=''wrap'', within=snapshot[period])
* Process_rate'}
otherwise: shift(at(Process_p, by=process_output[process]), along=snapshot, offset=Process_output_delay,
edge=0, within=snapshot[period]) * Process_rate
Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot[period]), over=generator), over=snapshot) + sum(sum((((Generator_p
* Generator_p) * Generator_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective,
by=snapshot[period]), over=generator), over=snapshot)'}
Generator_commitment_opex: {expression: 'sum(sum(((Generator_status * Generator_stand_by_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot[period]), over=generator), over=snapshot) + sum(sum(Generator_start_up
* Generator_start_up_cost, over=generator), over=snapshot) + sum(sum(Generator_shut_down * Generator_shut_down_cost,
over=generator), over=snapshot)'}
Link_opex: {expression: 'sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective,
by=snapshot[period]), over=link), over=snapshot) + sum(sum((((Link_p * Link_p) * Link_marginal_cost_quadratic)
* snapshot_weightings_objective) * at(period_weight_objective, by=snapshot[period]), over=link),
over=snapshot)'}
Link_commitment_opex: {expression: 'sum(sum(((Link_status * Link_stand_by_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot[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)'}
Process_opex: {expression: 'sum(sum(((Process_p * Process_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot[period]), over=process), over=snapshot) + sum(sum((((Process_p
* Process_p) * Process_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective,
by=snapshot[period]), over=process), over=snapshot)'}
Process_commitment_opex: {expression: 'sum(sum(((Process_status * Process_stand_by_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot[period]), over=process), over=snapshot) + sum(sum(Process_start_up
* Process_start_up_cost, over=process), over=snapshot) + sum(sum(Process_shut_down * Process_shut_down_cost,
over=process), over=snapshot)'}
scenario_opex:
dims: [scenario]
expression: ((((Generator_opex + Generator_commitment_opex) + Link_opex) + Link_commitment_opex) +
Process_opex) + Process_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`)
Generator_capex: {expression: sum(scenario_weight * Generator_p_nom_ext * Generator_capital_cost * Generator_capital_weight)}
risk_weighted_opex: {expression: 'sum(scenario_weight * scenario_opex, over=scenario)'}
total_cost:
dims: []
expression: Generator_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])'}
Link_injection: {expression: '-sum(Link_p, over=link, by=link[bus0]) + sum(Link_output_arrival,
over=link_output, by=link_output[bus])'}
Load_injection: {expression: 'sum(Load_demand, over=load, by=load[bus])'}
Process_injection: {expression: 'sum(Process_output_arrival, over=process_output, by=process_output[bus])'}
Bus_injection:
dims: [scenario, snapshot, bus]
expression: ((Generator_injection + Link_injection) + Load_injection) + Process_injection
description: what every component puts into a bus, less what it takes out of it; PyPSA writes each
term into the balance, and a load on its right-hand side
conditions:
Generator_com_ext: {description: 'a committable generator 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: Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod
> 0) AND Generator_active}
Generator_committed: {description: a committable generator that stands in the snapshot's period — every
unit-commitment row's set, condition: Generator_committable AND Generator_active}
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}
Process_committed: {description: a committable process that stands in the snapshot's period — every
unit-commitment row's set, condition: Process_committable AND Process_active}
objective: {sense: minimize, expression: total_cost}
The prep — every table the spec declares, from the network — and the solve:
from differential.pypsa.prep import prop, static, varying, weighting
n = build() # the network from the PyPSA tab
sources = {
'snapshot': pl.Series('snapshot', list(timesteps(n)), dtype=pl.Datetime('us')),
'bus': pl.Series('bus', list(names(n.buses.index).astype(str)), dtype=pl.String),
**{
dim: pl.Series(dim, list(names(n.static(component).index).astype(str)), dtype=pl.String)
for component, dim in DIM.items()
},
**scenarios(n),
**periods(n, multi),
**carriers(n, multi),
'generator.bus': prop(n, 'Generator', 'bus'),
'link.bus0': prop(n, 'Link', 'bus0'),
'load.bus': prop(n, 'Load', 'bus'),
'snapshot_weightings_objective': weighting(n, 'objective'),
'Generator_sign': per_component('Generator', first_scenario(n.generators['sign'])),
'Load_p_set': varying(n, 'Load', 'p_set'),
'Load_sign': per_component('Load', first_scenario(loads['sign'])),
'Load_active': per_component('Load', first_scenario(loads['active']), bool),
}
with sps.solve('differential/pypsa/rungs/rung_64_commitment_edges.yaml', sources) as solution:
solution.objective # 9245.0
The network, rung_64_commitment_edges.py in the corpus — the spine plus what this rung adds:
# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT
"""Rung 64: commitment edges — a committable generator, link and process built in the later period start up when they open, and a committable unit without a build cap is released by an inferred big M."""
from __future__ import annotations
from datetime import datetime
import pandas as pd
OPTIMIZE = {'multi_investment_periods': True}
def build():
"""A whole network, not the spine: eight snapshots over two periods, three committable units built in 2030, an uncapped committable build, and a dear peaker."""
import pypsa
n = pypsa.Network()
n.snapshots = pd.MultiIndex.from_tuples(
[(2020, datetime(2020, 1, 1, t)) for t in range(4)] + [(2030, datetime(2030, 1, 1, t)) for t in range(4)]
)
n.investment_periods = [2020, 2030]
n.investment_period_weightings['objective'] = [1.0, 0.5]
n.investment_period_weightings['years'] = [10.0, 10.0]
n.snapshot_weightings['objective'] = [2.0, 1.5, 2.5, 2.0, 2.0, 1.5, 2.5, 2.0]
n.add('Bus', ['hub', 'fuel'])
n.add('Generator', 'well64', bus='fuel', p_nom=200, marginal_cost=1)
n.add('Generator', 'peak64', bus='hub', p_nom=300, marginal_cost=50)
late = {'committable': True, 'p_nom': 30, 'p_min_pu': 0.25, 'build_year': 2030, 'lifetime': 30}
n.add('Generator', 'late_gen64', bus='hub', marginal_cost=2, start_up_cost=100, **late)
n.add('Link', 'late_link64', bus0='fuel', bus1='hub', marginal_cost=1, start_up_cost=70, **late)
n.add('Process', 'late_proc64', bus0='fuel', bus1='hub', rate0=-1.25, start_up_cost=40, **late)
n.add(
'Generator',
'uncapped64',
bus='hub',
committable=True,
p_nom_extendable=True,
capital_cost=20,
marginal_cost=10,
p_min_pu=0.2,
up_time_before=0,
start_up_cost=30,
)
n.add('Load', 'hub_load64', bus='hub', p_set=[40, 60, 70, 50, 120, 140, 150, 130])
return n
The data¶
Every table this spec declares was first declared by a lower rung; its values here are in the prep above.