Rung 26: committable processes — rung 25's committable links restated as processes that draw a quarter more than they deliver¶
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 1.3.0 — objective 15956.125 on both sides; structure ≠
CVaR0 vs 1 — the file declares the tail's average on every run; PyPSA adds it only under a risk preference, and without one the objective prices it at zero and no row reads it;CVaR-a0 vs 1 — the file declares each scenario's excess on every run; PyPSA adds it only under a risk preference, and without one no row reads it;CVaR-theta0 vs 1 — the file declares the tail's start on every run; PyPSA adds it only under a risk preference, and without one no row reads it; size ✔ 235 rows · ≠ 95 vs 98 columns · ✔ 507 nonzeros; duals — integer model, no duals; model for model: 42 blocks equal, 0 documented splits, 4 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
12 | 12 |
Generator-fix-p-lower |
8 | 8 |
Generator-fix-p-upper |
8 | 8 |
Link-fix-p-lower |
4 | 4 |
Link-fix-p-upper |
4 | 4 |
Process-com-down-time |
6 | 6 |
Process-com-ext-p-lower |
4 | 4 |
Process-com-ext-p-lower-nonneg |
4 | 4 |
Process-com-ext-p-upper-bigM |
4 | 4 |
Process-com-ext-p-upper-cap |
4 | 4 |
Process-com-mod-p-lower |
8 | 8 |
Process-com-mod-p-upper |
8 | 8 |
Process-com-p-lower |
12 | 12 |
Process-com-p-upper |
12 | 12 |
Process-com-status-min_down_time_must_stay_up |
2 | 2 |
Process-com-status-min_up_time_must_stay_up |
2 | 2 |
Process-com-transition-shut-down |
20 | 20 |
Process-com-transition-start-up |
20 | 20 |
Process-com-up-time |
6 | 6 |
Process-ext-p_nom-lower |
2 | 2 |
Process-ext-p_nom-upper |
2 | 2 |
Process-p-ramp_limit_down |
3 | 3 |
Process-p-ramp_limit_down-run-bigM |
4 | 4 |
Process-p-ramp_limit_down-shut-bigM |
4 | 4 |
Process-p-ramp_limit_up |
3 | 3 |
Process-p-ramp_limit_up-run-bigM |
4 | 4 |
Process-p-ramp_limit_up-start-bigM |
4 | 4 |
Process-p_nom_modularity |
1 | 1 |
Process-shut_down-p-fixed-upper |
16 | 16 |
Process-shut_down-p_nom-variable-upper |
4 | 4 |
Process-start_up-p-fixed-upper |
16 | 16 |
Process-start_up-p_nom-variable-upper |
4 | 4 |
Process-status-p-fixed-upper |
16 | 16 |
Process-status-p_nom-variable-upper |
4 | 4 |
| column | PyPSA | specsolve |
|---|---|---|
CVaR |
0 | ≠ 1 |
CVaR-a |
0 | ≠ 1 |
CVaR-theta |
0 | ≠ 1 |
Generator-p |
8 | 8 |
Link-p |
4 | 4 |
Process-n_mod |
1 | 1 |
Process-p |
20 | 20 |
Process-p_nom |
2 | 2 |
Process-shut_down |
20 | 20 |
Process-start_up |
20 | 20 |
Process-status |
20 | 20 |
The model¶
The same model, as math
A plain n.optimize(), and its multi-period and stochastic classes, in one file. Every second-stage quantity spans a scenario (a future dispatch is chosen in) and every asset stands in the investment periods its build year and lifetime span. A parameter spans scenario exactly when PyPSA reads it per scenario. Capacity is chosen once, before the future is known, and paid once per active period at its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights, with a share priced at the tail through the CVaR rows, which stand only where that share is positive. A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses to the standard one. A security-constrained run copies each branch flow limit once per outage in an outage set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight, and the outage factors are data prep.
Sets¶
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{O}\) | index \(o\) — link_output with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep |
| \(\mathcal{J}\) | index \(j\) — process with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J}\) — generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate |
| \(\mathcal{R}\) | index \(r\) — process_output with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N}\) — a process's ports, one label per port a process declares — PyPSA's bus0, bus1, … each carry a signed rate, so a process of any number of ports is one term in the balance, data prep |
| \(\mathcal{D}\) | index \(d\) — load with \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters¶
| Symbol | Meaning |
|---|---|
| \(\mathrm{w}\) | snapshot_weightings_objective over \(\mathcal{T}\) — PyPSA's snapshot_weightings.objective — hours a snapshot stands for in the cost |
| \(\mathrm{p}^{\mathrm{nom}}\) | Generator_p_nom over \(\Xi \times \mathcal{G}\) — nominal power |
| \(\mathrm{ext}\) | Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{p}}\) | Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power |
| \(\overline{\mathrm{p}}\) | Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile |
| \(\mathrm{c}\) | Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output |
| \(\mathrm{c}^{(2)}\) | Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output |
| \(\mathrm{sgn}\) | Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (consistency.py:1187) |
| \(\mathrm{com}\) | Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision |
| \(\mathrm{f}^{\mathrm{nom}}\) | Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power |
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{f}}\) | Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways |
| \(\overline{\mathrm{f}}\) | Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power |
| \(\eta\) | Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py:1522) |
| \(\mathrm{d}^{f}\) | Link_output_delay over \(\Xi \times \mathcal{O}\) — snapshots a port's delivery lags its link's flow — PyPSA's delay, delay2, … read long, in snapshot_weightings.generators units, which the file states as whole snapshots; zero for a port that delivers at once. Each scenario takes its own. PyPSA 1.3.0 groups the ports by delay over all scenarios and shifts each group in every one, so a delay that differs by scenario delivers the flow twice (constraints.py:1269-1276, PyPSA/PyPSA#1941) |
| \(\mathrm{cyc}^{f}\) | Link_output_cyclic_delay over \(\Xi \times \mathcal{O}\) — 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. Each scenario takes its own, as the delay |
| \(\mathrm{c}^{f}\) | 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 |
| \(\mathrm{z}^{\mathrm{nom}}\) | 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:1522) |
| \(\mathrm{d}^{z}\) | Process_output_delay over \(\Xi \times \mathcal{R}\) — snapshots a port's transfer lags its process's internal power — PyPSA's delay0, delay1, … read long, in snapshot_weightings.generators units, which the file states as whole snapshots; zero for a port that transfers at once. Each scenario takes its own, as a link's |
| \(\mathrm{cyc}^{z}\) | Process_output_cyclic_delay over \(\Xi \times \mathcal{R}\) — 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. Each scenario takes its own, as the delay |
| \(\mathrm{c}^{z}\) | 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 |
| \(\mathrm{ru}^{z}\) | Process_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — most a process may raise its internal power 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 |
| \(\mathrm{rd}^{z}\) | Process_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — most a process may lower its internal power 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 |
| \(\underline{\mathrm{z}}^{\mathrm{nom}}\) | Process_p_nom_min over \(\Xi \times \mathcal{J}\) — least nominal power an extendable process may be built at |
| \(\overline{\mathrm{z}}^{\mathrm{nom}}\) | Process_p_nom_max over \(\Xi \times \mathcal{J}\) — most nominal power an extendable process may be built at |
| \(\mathrm{c}^{\mathrm{cap},z}\) | Process_capital_cost over \(\Xi \times \mathcal{J}\) — cost of one unit of nominal power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\mathrm{com}^{z}\) | Process_committable over \(\mathcal{J}\) — whether internal power is gated by an on/off status decision |
| \(\mathrm{ru}^{z,\mathrm{up}}\) | Process_ramp_limit_start_up over \(\Xi \times \mathcal{J}\) — most internal power in the snapshot a process starts, per unit of nominal power |
| \(\mathrm{rd}^{z,\mathrm{dn}}\) | Process_ramp_limit_shut_down over \(\Xi \times \mathcal{J}\) — most internal power in the snapshot before a process stops, per unit of nominal power |
| \(\mathrm{UT}^{z}\) | Process_min_up_time over \(\Xi \times \mathcal{J}\) — least snapshots a process stays on once started |
| \(\mathrm{DT}^{z}\) | Process_min_down_time over \(\Xi \times \mathcal{J}\) — least snapshots a process stays off once stopped |
| \(\mathrm{u}^{z,0}\) | 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 |
| \(\mathrm{z}^{0}\) | Process_p_init over \(\Xi \times \mathcal{J}\) — the internal power a process brought into the horizon — PyPSA's p_init, read only where the process came in running; no value means it is unknown, so the process carries no ramp row at the first snapshot |
| \(\mathrm{hold}^{z}\) | Process_must_stay_up over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — true while the up time a process brought into the horizon still binds — data prep, since position() compares against a literal rather than a parameter |
| \(\mathrm{rest}^{z}\) | Process_must_stay_down over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — true while the down time a process 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 |
| \(\mathrm{c}^{z,\mathrm{up}}\) | Process_start_up_cost over \(\Xi \times \mathcal{J}\) — cost of one start |
| \(\mathrm{c}^{z,\mathrm{dn}}\) | Process_shut_down_cost over \(\Xi \times \mathcal{J}\) — cost of one stop |
| \(\mathrm{c}^{z,\mathrm{on}}\) | 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 \(\mathcal{J}\) — 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 |
| \(\mathrm{M}^{z}\) | Process_big_m over \(\Xi \times \mathcal{J}\) — a bound safely above any feasible internal power — the build cap at full availability, data prep |
| \(\mathrm{nonneg}^{z}\) | Process_p_min_pu_nonneg over \(\mathcal{J}\) — true where none of the process's own minimums-per-unit is negative — PyPSA's per-unit (p_min_pu >= 0).all() over every snapshot and scenario, data prep |
| \(\mathrm{load}\) | 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 (consistency.py:1187) |
| \(\mathrm{on}^{\mathrm{load}}\) | 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 (consistency.py:1195) |
| \(\pi\) | scenario_weight over \(\Xi\) — PyPSA's scenario_weightings.weight — the probability of a future |
| \(\omega\) | CVaR_omega (scalar) — PyPSA's risk_preference['omega'] — the share of operating cost priced at the tail rather than in expectation; zero recovers the risk-neutral model |
| \(\mathrm{w}^{y}\) | period_weight_objective over \(\mathcal{Y}\) — PyPSA's investment_period_weightings.objective — what a period's cost weighs |
| \(\mathrm{on}\) | 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 |
| \(\mathrm{W}^{z}\) | Process_capital_weight over \(\mathcal{J}\) — the sum of period weights a process stands in — PyPSA's active * period_weighting, summed, data prep |
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 |
| \(N^{z}\) | Process_n_mod over \(\mathcal{J}\) — Process-n_mod — how many modules of an extendable modular build |
| \(u^{z}\) | 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 |
| \(Z\) | Process_p_nom_ext over \(\mathcal{J}\) — Process-p_nom — nominal internal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
| \(a\) | CVaR_a over \(\Xi\) — CVaR-a — how far a scenario's operating cost exceeds the tail's start; nothing where it does not |
| \(\theta\) | CVaR_theta (scalar) — CVaR-theta — where the tail starts, the value at risk |
| \(CVaR\) | CVaR (scalar) — CVaR — the tail's average cost, what the objective prices at omega |
Definitions¶
| Symbol | Meaning |
|---|---|
| \(\mathit{Process\_previous\_status}\) | Process_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the commitment state a process carries into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that |
| \(\mathit{Process\_previous\_p}\) | Process_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the internal power a process carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that |
| \(\mathrm{Process\_ramp\_up\_rate}\) | Process_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the ramp limit a process'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 |
| \(\mathrm{Process\_ramp\_down\_rate}\) | Process_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the ramp limit a process'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 |
| \(\mathrm{Process\_start\_up\_rate}\) | Process_start_up_rate over \(\Xi \times \mathcal{J}\) — the start-up ramp a process's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\mathrm{Process\_shut\_down\_rate}\) | Process_shut_down_rate over \(\Xi \times \mathcal{J}\) — the shut-down ramp a process's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\mathit{Process\_ramp\_up\_allowance}\) | Process_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — how far a process may raise internal power between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\mathit{Process\_ramp\_down\_allowance}\) | Process_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — how far a process may lower internal power between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off |
| \(\mathit{total\_cost}\) | 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 |
| \(\mathit{Bus\_injection}\) | 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 |
| \(\mathit{Process\_p\_nom\_effective}\) | Process_p_nom_effective over \(\Xi \times \mathcal{J}\) — the build a process's limits are taken against — the chosen one where it is extendable, the given one otherwise |
| \(\mathrm{Process\_p\_nom\_committed}\) | Process_p_nom_committed over \(\Xi \times \mathcal{J}\) — the build a committed process's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise |
| \(\mathit{Process\_capex}\) | Process_capex (scalar) |
| \(\mathit{risk\_weighted\_opex}\) | risk_weighted_opex (scalar) |
| \(\mathit{Generator\_injection}\) | Generator_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Link\_injection}\) | Link_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathrm{Load\_injection}\) | Load_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Process\_injection}\) | Process_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Link\_output\_arrival}\) | 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 |
| \(\mathit{scenario\_opex}\) | 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:414-429) |
| \(\mathrm{Load\_demand}\) | 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:1537-1538) |
| \(\mathit{Generator\_opex}\) | Generator_opex over \(\Xi\) |
| \(\mathit{Link\_opex}\) | Link_opex over \(\Xi\) |
| \(\mathit{Process\_opex}\) | Process_opex over \(\Xi\) |
| \(\mathit{Process\_commitment\_opex}\) | Process_commitment_opex over \(\Xi\) |
\(t \ominus k\) denotes cyclic translation: index \(t-k\) taken modulo the size of the dimension (roll). Plain \(t-k\) (shift) has no wraparound — terms translated past the edge are simply absent.
\(t \boxminus_{v} k\) denotes translation with \(v\) standing where index \(t-k\) leaves the dimension (shift(edge=v)), so the row at that boundary is built and carries \(v\) rather than being dropped.
\(t \ominus^{\mathrm{relation}(t)} k\) denotes a translation counted inside the group a relation puts \(t\) in (shift(by=relation)), so a term never crosses out of its own group. The two modifiers take different slots — the group above, the fill below — so \(t \boxminus_{v}^{\mathrm{relation}(t)} k\) is both at once.
\(\mathrm{pos}(t)\) denotes where index \(t\) sits along its dimension's own order — the order shift steps along, not the order labels sort in — counted from \(0\). The index itself stays the coordinate, so \(t\) compares against labels and \(\mathrm{pos}(t)\) against positions.
\(\mathrm{pos}_{\mathrm{relation}(t)}(t)\) counts within the group a relation puts \(t\) in: the subscript names the map, \(\mathcal{T}_{\mathrm{relation}(t)}\) is the group it lands in, and that group has a first position of its own.
Objective¶
\[
\min \mathit{total\_cost}
\]
Subject to¶
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Link_fix_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_fix_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Process_ext_p_nom_lower
\[
Z_{j} \ge \underline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j}
\]
Process_ext_p_nom_upper
\[
Z_{j} \le \overline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j} \wedge \overline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \text{ is defined}
\]
Process_com_p_lower
\[
z_{\xi,t,j} \ge \underline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{nom}}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \neg \mathrm{ext}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_p_upper
\[
z_{\xi,t,j} \le \overline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{nom}}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \neg \mathrm{ext}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_transition_start_up
\[
\mathit{up}^{z}_{\xi,t,j} \ge u^{z}_{\xi,t,j} - \mathit{Process\_previous\_status}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_transition_shut_down
\[
\mathit{dn}^{z}_{\xi,t,j} \ge \mathit{Process\_previous\_status}_{\xi,t,j} - u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_up_time
\[
\sum_{t' \in \mathcal{T} \,:\, 0 \le t - t' < \mathrm{UT}^{z}} \mathit{up}^{z}_{\xi,t',j} \le u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{UT}^{z}_{\xi,j} > 0 \wedge \mathrm{pos}(t) > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_down_time
\[
\sum_{t' \in \mathcal{T} \,:\, 0 \le t - t' < \mathrm{DT}^{z}} \mathit{dn}^{z}_{\xi,t',j} \le 1 - u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{DT}^{z}_{\xi,j} > 0 \wedge \mathrm{pos}(t) > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_status_must_stay_up
\[
u^{z}_{\xi,t,j} = 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{hold}^{z}_{\xi,t,j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_status_must_stay_down
\[
u^{z}_{\xi,t,j} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{rest}^{z}_{\xi,t,j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_up_run_big_m
\[
z_{\xi,t,j} - \mathit{Process\_previous\_p}_{\xi,t,j} \le \mathrm{Process\_ramp\_up\_rate}_{\xi,t,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot \mathit{Process\_previous\_status}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_up_start_big_m
\[
z_{\xi,t,j} - \mathit{Process\_previous\_p}_{\xi,t,j} \le \mathrm{Process\_start\_up\_rate}_{\xi,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot \mathit{up}^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_down_run_big_m
\[
\mathit{Process\_previous\_p}_{\xi,t,j} - z_{\xi,t,j} \le \mathrm{Process\_ramp\_down\_rate}_{\xi,t,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_down_shut_big_m
\[
\mathit{Process\_previous\_p}_{\xi,t,j} - z_{\xi,t,j} \le \mathrm{Process\_shut\_down\_rate}_{\xi,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot \mathit{dn}^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_nom_modularity
\[
Z_{j} = \mathrm{z}^{\mathrm{mod}}_{j} \cdot N^{z}_{j} \qquad \forall\, j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0
\]
Process_com_ext_p_upper_cap
\[
z_{\xi,t,j} \le \overline{\mathrm{z}}_{\xi,t,j} \cdot Z_{j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_ext_p_upper_big_m
\[
z_{\xi,t,j} \le \mathrm{M}^{z}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_ext_p_lower
\[
z_{\xi,t,j} \ge \underline{\mathrm{z}}_{\xi,t,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} \cdot u^{z}_{\xi,t,j} - \mathrm{M}^{z}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_ext_p_lower_nonneg
\[
z_{\xi,t,j} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \mathrm{nonneg}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_mod_p_lower
\[
z_{\xi,t,j} \ge \underline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{mod}}_{j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_com_mod_p_upper
\[
z_{\xi,t,j} \le \overline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{mod}}_{j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_status_p_fixed_upper
\[
u^{z}_{\xi,t,j} \le \mathrm{N}^{z,\mathrm{fix}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \neg \left( \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_start_up_p_fixed_upper
\[
\mathit{up}^{z}_{\xi,t,j} \le \mathrm{N}^{z,\mathrm{fix}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \neg \left( \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_shut_down_p_fixed_upper
\[
\mathit{dn}^{z}_{\xi,t,j} \le \mathrm{N}^{z,\mathrm{fix}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \neg \left( \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_status_p_nom_variable_upper
\[
u^{z}_{\xi,t,j} \le N^{z}_{j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_start_up_p_nom_variable_upper
\[
\mathit{up}^{z}_{\xi,t,j} \le N^{z}_{j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_shut_down_p_nom_variable_upper
\[
\mathit{dn}^{z}_{\xi,t,j} \le N^{z}_{j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_up
\[
z_{\xi,t,j} - \mathit{Process\_previous\_p}_{\xi,t,j} \le \mathit{Process\_ramp\_up\_allowance}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_down
\[
\mathit{Process\_previous\_p}_{\xi,t,j} - z_{\xi,t,j} \le \mathit{Process\_ramp\_down\_allowance}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \left( \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Bus_nodal_balance
\[
\mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Definitions¶
Process_previous_status
\[
\mathit{Process\_previous\_status}_{\xi,t,j} = \begin{cases} \mathrm{u}^{z,0}_{\xi,j} & \text{if } \mathrm{pos}(t) = 0 \\ u^{z}_{\xi,t - 1,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_previous_p
\[
\mathit{Process\_previous\_p}_{\xi,t,j} = \begin{cases} \mathrm{u}^{z,0}_{\xi,j} \cdot \mathrm{z}^{0}_{\xi,j} & \text{if } \mathrm{pos}(t) = 0 \\ z_{\xi,t - 1,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_ramp_up_rate
\[
\mathrm{Process\_ramp\_up\_rate}_{\xi,t,j} = \begin{cases} \mathrm{ru}^{z}_{\xi,t,j} & \text{if } \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_ramp_down_rate
\[
\mathrm{Process\_ramp\_down\_rate}_{\xi,t,j} = \begin{cases} \mathrm{rd}^{z}_{\xi,t,j} & \text{if } \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_start_up_rate
\[
\mathrm{Process\_start\_up\_rate}_{\xi,j} = \begin{cases} \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} & \text{if } \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J}
\]
Process_shut_down_rate
\[
\mathrm{Process\_shut\_down\_rate}_{\xi,j} = \begin{cases} \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} & \text{if } \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J}
\]
Process_ramp_up_allowance
\[
\mathit{Process\_ramp\_up\_allowance}_{\xi,t,j} = \begin{cases} \mathrm{Process\_ramp\_up\_rate}_{\xi,t,j} \cdot \mathrm{Process\_p\_nom\_committed}_{\xi,j} \cdot \mathit{Process\_previous\_status}_{\xi,t,j} + \mathrm{Process\_start\_up\_rate}_{\xi,j} \cdot \mathrm{Process\_p\_nom\_committed}_{\xi,j} \cdot \left( u^{z}_{\xi,t,j} - \mathit{Process\_previous\_status}_{\xi,t,j} \right) & \text{if } \mathrm{com}^{z}_{j} \\ \mathrm{Process\_ramp\_up\_rate}_{\xi,t,j} \cdot \mathit{Process\_p\_nom\_effective}_{\xi,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_ramp_down_allowance
\[
\mathit{Process\_ramp\_down\_allowance}_{\xi,t,j} = \begin{cases} \mathrm{Process\_ramp\_down\_rate}_{\xi,t,j} \cdot \mathrm{Process\_p\_nom\_committed}_{\xi,j} \cdot u^{z}_{\xi,t,j} + \mathrm{Process\_shut\_down\_rate}_{\xi,j} \cdot \mathrm{Process\_p\_nom\_committed}_{\xi,j} \cdot \left( \mathit{Process\_previous\_status}_{\xi,t,j} - u^{z}_{\xi,t,j} \right) & \text{if } \mathrm{com}^{z}_{j} \\ \mathrm{Process\_ramp\_down\_rate}_{\xi,t,j} \cdot \mathit{Process\_p\_nom\_effective}_{\xi,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
total_cost
\[
\mathit{total\_cost} = \mathit{Process\_capex} + \mathit{risk\_weighted\_opex}
\]
Bus_injection
\[
\mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{\xi,t,n} + \mathit{Link\_injection}_{\xi,t,n} + \mathrm{Load\_injection}_{\xi,t,n} + \mathit{Process\_injection}_{\xi,t,n} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Process_p_nom_effective
\[
\mathit{Process\_p\_nom\_effective}_{\xi,j} = \begin{cases} Z_{j} & \text{if } \mathrm{ext}^{z}_{j} \\ \mathrm{z}^{\mathrm{nom}}_{\xi,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J}
\]
Process_p_nom_committed
\[
\mathrm{Process\_p\_nom\_committed}_{\xi,j} = \begin{cases} \mathrm{z}^{\mathrm{mod}}_{j} & \text{if } \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \\ \mathrm{z}^{\mathrm{nom}}_{\xi,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J}
\]
Process_capex
\[
\mathit{Process\_capex} = \sum_{\xi \in \Xi,\ j \in \mathcal{J}} \pi_{\xi} \cdot Z_{j} \cdot \mathrm{c}^{\mathrm{cap},z}_{\xi,j} \cdot \mathrm{W}^{z}_{j}
\]
risk_weighted_opex
\[
\mathit{risk\_weighted\_opex} = \left( 1 - \omega \right) \cdot \left( \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi} \right) + \omega \cdot CVaR
\]
Generator_injection
\[
\mathit{Generator\_injection}_{\xi,t,n} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Link_injection
\[
\mathit{Link\_injection}_{\xi,t,n} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Load_injection
\[
\mathrm{Load\_injection}_{\xi,t,n} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Process_injection
\[
\mathit{Process\_injection}_{\xi,t,n} = \sum_{r \in \mathcal{R} \,:\, \mathrm{Process\_output\_bus}(r) = n} \mathit{Process\_output\_arrival}_{\xi,t,r} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Link_output_arrival
\[
\mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{\xi,o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
Process_output_arrival
\[
\mathit{Process\_output\_arrival}_{\xi,t,r} = \begin{cases} z_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{if } \mathrm{cyc}^{z}_{\xi,r} \\ z_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ r \in \mathcal{R}
\]
scenario_opex
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} + \mathit{Process\_opex}_{\xi} + \mathit{Process\_commitment\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
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_opex
\[
\mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Link_opex
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Process_opex
\[
\mathit{Process\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} z_{\xi,t,j} \cdot \mathrm{c}^{z}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} z_{\xi,t,j} \cdot z_{\xi,t,j} \cdot \mathrm{c}^{z,(2)}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Process_commitment_opex
\[
\mathit{Process\_commitment\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} u^{z}_{\xi,t,j} \cdot \mathrm{c}^{z,\mathrm{on}}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} \mathit{up}^{z}_{\xi,t,j} \cdot \mathrm{c}^{z,\mathrm{up}}_{\xi,j} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} \mathit{dn}^{z}_{\xi,t,j} \cdot \mathrm{c}^{z,\mathrm{dn}}_{\xi,j} \qquad \forall\, \xi \in \Xi
\]
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
\[
z_{\xi,t,j} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{on}^{z}_{t,j}
\]
Process_n_mod
\[
N^{z}_{j} \ge 0, N^{z}_{j} \in \mathbb{Z} \qquad \forall\, j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0
\]
Process_status
\[
u^{z}_{\xi,t,j} \ge 0, u^{z}_{\xi,t,j} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_start_up
\[
\mathit{up}^{z}_{\xi,t,j} \ge 0, \mathit{up}^{z}_{\xi,t,j} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_shut_down
\[
\mathit{dn}^{z}_{\xi,t,j} \ge 0, \mathit{dn}^{z}_{\xi,t,j} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_nom_ext
\[
Z_{j} \in \mathbb{R} \qquad \forall\, j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j}
\]
CVaR_a
\[
a_{\xi} \ge 0 \qquad \forall\, \xi \in \Xi
\]
CVaR_theta
\[
\theta \in \mathbb{R}
\]
CVaR
\[
CVaR \in \mathbb{R}
\]
The spec, differential/pypsa/rungs/rung_26_committable_process.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 that share is positive.
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}
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}
relations:
snapshot_period: {description: the investment period a snapshot falls in, key: snapshot, values: period}
Generator_bus: {description: the bus a generator sits on, key: generator, values: bus}
Link_bus0: {description: the bus a link leaves, key: link, values: bus}
Link_output_link: {description: the link an output port belongs to, key: link_output, values: link}
Link_output_bus: {description: 'the bus an output port delivers to — PyPSA''s `bus1`, `bus2`, … columns.
A link of three output ports is three labels here rather than a third relation, so the file states
any number of them', key: link_output, values: bus}
Process_output_process: {description: the process a port belongs to, key: process_output, values: process}
Process_output_bus: {description: 'the bus a port draws from or delivers to — PyPSA''s `bus0`, `bus1`,
… columns. A process of three ports is three labels here rather than a third relation, so the file
states any number of them', key: process_output, values: bus}
Load_bus: {description: the bus a load sits on, key: load, values: bus}
parameters:
snapshot_weightings_objective:
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
dims: [snapshot]
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
for a unit that draws power. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
dims: [generator]
Generator_committable:
description: whether output is gated by an on/off status decision
dims: [generator]
dtype: bool
Link_p_nom:
description: nominal power
dims: [scenario, link]
Link_p_nom_extendable:
description: whether the nominal power is a decision
dims: [link]
dtype: bool
Link_p_min_pu:
description: least flow, per unit of nominal power — negative for a link that carries both ways
dims: [scenario, snapshot, link]
Link_p_max_pu:
description: most flow, per unit of nominal power
dims: [scenario, snapshot, link]
Link_efficiency:
description: share of the flow that arrives at an output port, PyPSA's `efficiency`, `efficiency2`,
… read long — negative where that port consumes rather than delivers. Read at the snapshot the flow
arrives, so a delayed port delivers at its arrival snapshot's efficiency (`constraints.py:1522`)
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, in `snapshot_weightings.generators` units, which the file states as whole snapshots; zero
for a port that delivers at once. Each scenario takes its own. PyPSA `1.3.0` groups the ports by
delay over all scenarios and shifts each group in every one, so a delay that differs by scenario
delivers the flow twice (`constraints.py:1269-1276`, PyPSA/PyPSA#1941)
dims: [scenario, 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. Each scenario takes its own, as the delay
dims: [scenario, link_output]
dtype: bool
Link_marginal_cost:
description: cost of one unit of flow
dims: [scenario, snapshot, link]
Link_marginal_cost_quadratic:
description: cost of the square of one unit of flow
dims: [scenario, snapshot, link]
Link_committable:
description: whether flow is gated by an on/off status decision
dims: [link]
dtype: bool
Process_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:1522`)
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, in `snapshot_weightings.generators` units, which the file states as whole snapshots;
zero for a port that transfers at once. Each scenario takes its own, as a link's
dims: [scenario, 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. Each scenario takes its own, as the delay
dims: [scenario, 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_ramp_limit_up:
description: most a process may raise its internal power between snapshots, per unit of nominal power;
no value means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, process]
Process_ramp_limit_down:
description: most a process may lower its internal power between snapshots, per unit of nominal power;
no value means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, process]
Process_p_nom_min:
description: least nominal power an extendable process may be built at
dims: [scenario, process]
Process_p_nom_max:
description: most nominal power an extendable process may be built at
dims: [scenario, process]
Process_capital_cost:
description: cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity
in data prep
dims: [scenario, process]
Process_committable:
description: whether internal power is gated by an on/off status decision
dims: [process]
dtype: bool
Process_ramp_limit_start_up:
description: most internal power in the snapshot a process starts, per unit of nominal power
dims: [scenario, process]
Process_ramp_limit_shut_down:
description: most internal power in the snapshot before a process stops, per unit of nominal power
dims: [scenario, process]
Process_min_up_time:
description: least snapshots a process stays on once started
dims: [scenario, process]
dtype: int
Process_min_down_time:
description: least snapshots a process stays off once stopped
dims: [scenario, process]
dtype: int
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_p_init:
description: the internal power a process brought into the horizon — PyPSA's `p_init`, read only where
the process came in running; no value means it is unknown, so the process carries no ramp row at
the first snapshot
dims: [scenario, process]
Process_must_stay_up:
description: true while the up time a process brought into the horizon still binds — data prep, since
`position()` compares against a literal rather than a parameter
dims: [scenario, snapshot, process]
dtype: bool
Process_must_stay_down:
description: true while the down time a process brought into the horizon still binds — PyPSA's `min_down_time
- down_time_before` snapshots, where `down_time_before > 0`, data prep for the same reason
dims: [scenario, snapshot, process]
dtype: bool
Process_start_up_cost:
description: cost of one start
dims: [scenario, process]
Process_shut_down_cost:
description: cost of one stop
dims: [scenario, 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]
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]
Process_big_m:
description: a bound safely above any feasible internal power — the build cap at full availability,
data prep
dims: [scenario, process]
Process_p_min_pu_nonneg:
description: true where none of the process's own minimums-per-unit is negative — PyPSA's per-unit
`(p_min_pu >= 0).all()` over every snapshot and scenario, data prep
dims: [process]
dtype: bool
Load_p_set:
description: demand
dims: [scenario, snapshot, load]
Load_sign:
description: the sign a load's demand enters its bus's balance with — PyPSA's `sign`, `-1` unless
given, `1` for a load that feeds its bus. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
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 (`consistency.py:1195`)
dims: [load]
dtype: bool
scenario_weight:
description: PyPSA's `scenario_weightings.weight` — the probability of a future
dims: [scenario]
CVaR_omega:
description: PyPSA's `risk_preference['omega']` — the share of operating cost priced at the tail rather
than in expectation; zero recovers the risk-neutral model
dims: []
period_weight_objective:
description: PyPSA's `investment_period_weightings.objective` — what a period's cost weighs
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
Process_capital_weight:
description: the sum of period weights a process stands in — PyPSA's `active * period_weighting`,
summed, data prep
dims: [process]
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
Process_n_mod:
description: '`Process-n_mod` — how many modules of an extendable modular build'
dims: [process]
where: Process_p_nom_extendable AND Process_p_nom_mod > 0
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_committable AND Process_active
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_committable AND Process_active
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_committable AND Process_active
domain: integer
bounds: {lower: 0}
Process_p_nom_ext:
description: '`Process-p_nom` — nominal internal power where it is a decision; the parameter of the
same PyPSA name carries the fixed regime'
dims: [process]
where: Process_p_nom_extendable
CVaR_a:
description: '`CVaR-a` — how far a scenario''s operating cost exceeds the tail''s start; nothing where
it does not'
dims: [scenario]
bounds: {lower: 0}
CVaR_theta:
description: '`CVaR-theta` — where the tail starts, the value at risk'
dims: []
CVaR:
description: '`CVaR` — the tail''s average cost, what the objective prices at `omega`'
dims: []
constraints:
Generator_fix_p_lower:
description: '`Generator-fix-p-lower` — a fixed generator outputs at least its minimum'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * Generator_p_nom
Generator_fix_p_upper:
description: '`Generator-fix-p-upper` — a fixed generator outputs at most what is available'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * Generator_p_nom
Link_fix_p_lower:
description: '`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other
way'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * Link_p_nom
Link_fix_p_upper:
description: '`Link-fix-p-upper` — a fixed link carries at most its nominal power'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * Link_p_nom
Process_ext_p_nom_lower:
description: '`Process-ext-p_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, process]
where: Process_p_nom_extendable
expression: Process_p_nom_ext >= Process_p_nom_min
Process_ext_p_nom_upper:
description: '`Process-ext-p_nom-upper` — the chosen build is at most its cap in every scenario; a
cap of infinity is no row'
dims: [scenario, process]
where: Process_p_nom_extendable AND Process_p_nom_max
expression: Process_p_nom_ext <= Process_p_nom_max
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_committable AND not Process_p_nom_extendable AND Process_active
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_committable AND not Process_p_nom_extendable AND Process_active
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_committable AND Process_active
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_committable AND Process_active
expression: Process_shut_down >= Process_previous_status - Process_status
Process_com_up_time:
description: '`Process-com-up-time` — a process started within its own minimum up time is still on.
The first snapshot''s share of the window is the brought-in up time''s, which the must-stay-up mask
carries'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_min_up_time > 0 AND position(snapshot) > 0 AND Process_active
expression: sum_back(Process_start_up, along=snapshot, window=Process_min_up_time) <= Process_status
Process_com_down_time:
description: '`Process-com-down-time` — a process stopped within its own minimum down time is still
off. The first snapshot''s share of the window is the brought-in down time''s, which the must-stay-down
mask carries'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_min_down_time > 0 AND position(snapshot) > 0 AND Process_active
expression: sum_back(Process_shut_down, along=snapshot, window=Process_min_down_time) <= 1 - Process_status
Process_com_status_must_stay_up:
description: '`Process-com-status-min_up_time_must_stay_up` — a process still serving the up time
it brought in stays on'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_must_stay_up AND Process_active
expression: Process_status == 1
Process_com_status_must_stay_down:
description: '`Process-com-status-min_down_time_must_stay_up` — a process still serving the down time
it brought in stays off; PyPSA names the row `_must_stay_up`'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_must_stay_down AND Process_active
expression: Process_status == 0
Process_p_ramp_limit_up_run_big_m:
description: '`Process-p-ramp_limit_up-run-bigM` — a committed extendable process raises internal
power no faster than its limit of the chosen build; the big M releases the row in the snapshot it
turns on'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND (Process_ramp_limit_up
OR Process_ramp_limit_start_up) AND (position(snapshot, by=snapshot_period, within=period) > 0 OR
(position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init))) AND Process_active
expression: Process_p - Process_previous_p <= Process_ramp_up_rate * Process_p_nom_ext + Process_big_m
- Process_big_m * Process_previous_status
Process_p_ramp_limit_up_start_big_m:
description: '`Process-p-ramp_limit_up-start-bigM` — in the snapshot it turns on, a committed extendable
process ramps no further than its start-up ramp of the chosen build; the big M releases the row
everywhere else'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND (Process_ramp_limit_up
OR Process_ramp_limit_start_up) AND (position(snapshot, by=snapshot_period, within=period) > 0 OR
(position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init))) AND Process_active
expression: Process_p - Process_previous_p <= Process_start_up_rate * Process_p_nom_ext + Process_big_m
- Process_big_m * Process_start_up
Process_p_ramp_limit_down_run_big_m:
description: '`Process-p-ramp_limit_down-run-bigM` — a committed extendable process lowers internal
power no faster than its limit of the chosen build; the big M releases the row in the snapshot it
turns off'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND (Process_ramp_limit_down
OR Process_ramp_limit_shut_down) AND (position(snapshot, by=snapshot_period, within=period) > 0
OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init))) AND Process_active
expression: Process_previous_p - Process_p <= Process_ramp_down_rate * Process_p_nom_ext + Process_big_m
- Process_big_m * Process_status
Process_p_ramp_limit_down_shut_big_m:
description: '`Process-p-ramp_limit_down-shut-bigM` — in the snapshot it turns off, a committed extendable
process ramps no further than its shut-down ramp of the chosen build; the big M releases the row
everywhere else'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND (Process_ramp_limit_down
OR Process_ramp_limit_shut_down) AND (position(snapshot, by=snapshot_period, within=period) > 0
OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init))) AND Process_active
expression: Process_previous_p - Process_p <= Process_shut_down_rate * Process_p_nom_ext + Process_big_m
- Process_big_m * Process_shut_down
Process_p_nom_modularity:
description: '`Process-p_nom_modularity` — the chosen build is a whole number of modules'
dims: [process]
where: Process_p_nom_extendable AND Process_p_nom_mod > 0
expression: Process_p_nom_ext == Process_p_nom_mod * Process_n_mod
Process_com_ext_p_upper_cap:
description: '`Process-com-ext-p-upper-cap` — a committed extendable process runs at most what is
available of the chosen build, whatever its status'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND Process_active
expression: Process_p <= Process_p_max_pu * Process_p_nom_ext
Process_com_ext_p_upper_big_m:
description: '`Process-com-ext-p-upper-bigM` — off, a process does not run; on, the big M is no bound'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND Process_active
expression: Process_p <= Process_big_m * Process_status
Process_com_ext_p_lower:
description: '`Process-com-ext-p-lower` — a committed extendable process runs at least its minimum
of the chosen build; off, the big M releases the row'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0) AND Process_active
expression: Process_p >= ((Process_p_min_pu * Process_p_nom_ext) + (Process_big_m * Process_status))
- Process_big_m
Process_com_ext_p_lower_nonneg:
description: '`Process-com-ext-p-lower-nonneg` — where no minimum-per-unit is negative, internal power
is also plainly non-negative, a row the big-M lower cannot assert while the process is off'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND Process_p_min_pu_nonneg AND NOT (Process_p_nom_mod
> 0) AND Process_active
expression: Process_p >= 0
Process_com_mod_p_lower:
description: '`Process-com-mod-p-lower` — a committed modular process runs at least its minimum of
one module, whether the build is fixed or a decision'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_mod > 0 AND Process_active
expression: Process_p >= (Process_p_min_pu * Process_p_nom_mod) * Process_status
Process_com_mod_p_upper:
description: '`Process-com-mod-p-upper` — a committed modular process runs at most one module''s share,
whether the build is fixed or a decision'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_mod > 0 AND Process_active
expression: Process_p <= (Process_p_max_pu * Process_p_nom_mod) * 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_committable AND NOT (Process_p_nom_extendable AND Process_p_nom_mod > 0) AND Process_active
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_committable AND NOT (Process_p_nom_extendable AND Process_p_nom_mod > 0) AND Process_active
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_committable AND NOT (Process_p_nom_extendable AND Process_p_nom_mod > 0) AND Process_active
expression: Process_shut_down <= Process_modules_installed
Process_status_p_nom_variable_upper:
description: '`Process-status-p_nom-variable-upper` — a modular process is on only where a module
is built'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND Process_p_nom_mod > 0 AND Process_active
expression: Process_status <= Process_n_mod
Process_start_up_p_nom_variable_upper:
description: '`Process-start_up-p_nom-variable-upper` — a modular process starts only where a module
is built'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND Process_p_nom_mod > 0 AND Process_active
expression: Process_start_up <= Process_n_mod
Process_shut_down_p_nom_variable_upper:
description: '`Process-shut_down-p_nom-variable-upper` — a modular process stops only where a module
is built'
dims: [scenario, snapshot, process]
where: Process_committable AND Process_p_nom_extendable AND Process_p_nom_mod > 0 AND Process_active
expression: Process_shut_down <= Process_n_mod
Process_p_ramp_limit_up:
description: '`Process-p-ramp_limit_up` — a process raises internal power 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 process that came into the horizon running carries a row at the first snapshot only where
its `p_init` gives the internal power it brought in, and no process carries one at the start of
a later investment period — nor does any process a big M releases instead'
dims: [scenario, snapshot, process]
where: (Process_ramp_limit_up OR Process_ramp_limit_start_up) AND NOT (Process_committable AND Process_p_nom_extendable
AND NOT (Process_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period, within=period) > 0
OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init))) AND Process_active
expression: Process_p - Process_previous_p <= Process_ramp_up_allowance
Process_p_ramp_limit_down:
description: '`Process-p-ramp_limit_down` — a process lowers internal power 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 process that came into the horizon running carries a row at the first snapshot only where
its `p_init` gives the internal power it brought in, and no process carries one at the start of
a later investment period — nor does any process a big M releases instead'
dims: [scenario, snapshot, process]
where: (Process_ramp_limit_down OR Process_ramp_limit_shut_down) AND NOT (Process_committable AND
Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period,
within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: Process_previous_p - Process_p <= Process_ramp_down_allowance
Bus_nodal_balance:
description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
less what the links take away, plus what arrives over them after losses and any delay at every port
they deliver to, each process port drawing or delivering at its own rate and each passive branch
carrying its flow, meets the load there, less half of every incident line''s and transformer''s
loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
and load term enters with its component''s `sign` (`constraints.py:1428-1429`, `:1538`), 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:
Process_previous_status:
description: the commitment state a process carries 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)
Process_previous_p:
description: the internal power a process carries into a snapshot — at the first, the `p_init` it
brought in where it came in running and nothing where it came in off; the previous snapshot's after
that
dims: [scenario, snapshot, process]
cases:
opening: {when: position(snapshot) == 0, expression: Process_status_initial * Process_p_init}
otherwise: shift(Process_p, along=snapshot, offset=1)
Process_ramp_up_rate:
description: the ramp limit a process's up row reads — PyPSA's `ramp_limit_up`, or the full build
where it has none, since a start-up ramp alone builds the row
dims: [scenario, snapshot, process]
cases:
given: {when: Process_ramp_limit_up, expression: Process_ramp_limit_up}
otherwise: 1
Process_ramp_down_rate:
description: the ramp limit a process's down row reads — PyPSA's `ramp_limit_down`, or the full build
where it has none, since a shut-down ramp alone builds the row
dims: [scenario, snapshot, process]
cases:
given: {when: Process_ramp_limit_down, expression: Process_ramp_limit_down}
otherwise: 1
Process_start_up_rate:
description: the start-up ramp a process's up row reads — PyPSA's `ramp_limit_start_up`, or the full
build where it has none
dims: [scenario, process]
cases:
given: {when: Process_ramp_limit_start_up, expression: Process_ramp_limit_start_up}
otherwise: 1
Process_shut_down_rate:
description: the shut-down ramp a process's down row reads — PyPSA's `ramp_limit_shut_down`, or the
full build where it has none
dims: [scenario, process]
cases:
given: {when: Process_ramp_limit_shut_down, expression: Process_ramp_limit_shut_down}
otherwise: 1
Process_ramp_up_allowance:
description: how far a process may raise internal power between two snapshots — its ramp limit of
the build while it stays on, plus its start-up ramp in the snapshot it turns on
dims: [scenario, snapshot, process]
cases:
committed: {when: Process_committable, expression: Process_ramp_up_rate * Process_p_nom_committed
* Process_previous_status + Process_start_up_rate * Process_p_nom_committed * (Process_status
- Process_previous_status)}
otherwise: Process_ramp_up_rate * Process_p_nom_effective
Process_ramp_down_allowance:
description: how far a process may lower internal power between two snapshots — its ramp limit of
the build while it stays on, plus its shut-down ramp in the snapshot it turns off
dims: [scenario, snapshot, process]
cases:
committed: {when: Process_committable, expression: Process_ramp_down_rate * Process_p_nom_committed
* Process_status + Process_shut_down_rate * Process_p_nom_committed * (Process_previous_status
- Process_status)}
otherwise: Process_ramp_down_rate * Process_p_nom_effective
total_cost:
dims: []
expression: Process_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
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
Process_p_nom_effective:
description: the build a process's limits are taken against — the chosen one where it is extendable,
the given one otherwise
dims: [scenario, process]
cases:
extendable: {when: Process_p_nom_extendable, expression: Process_p_nom_ext}
otherwise: Process_p_nom
Process_p_nom_committed:
description: the build a committed process's ramp rows are taken against — one module where the build
is extendable and modular, the given build otherwise
dims: [scenario, process]
cases:
modular_build: {when: Process_p_nom_extendable AND Process_p_nom_mod > 0, expression: Process_p_nom_mod}
otherwise: Process_p_nom
Process_capex: {expression: sum(scenario_weight * Process_p_nom_ext * Process_capital_cost * Process_capital_weight)}
risk_weighted_opex: {expression: '(1 - CVaR_omega) * sum(scenario_weight * scenario_opex, over=scenario)
+ CVaR_omega * CVaR'}
Generator_injection: {expression: 'sum(Generator_sign * Generator_p, by=Generator_bus, over=generator,
into=bus)'}
Link_injection: {expression: '-sum(Link_p, by=Link_bus0, over=link, into=bus) + sum(Link_output_arrival,
by=Link_output_bus, over=link_output, into=bus)'}
Load_injection: {expression: 'sum(Load_demand, by=Load_bus, over=load, into=bus)'}
Process_injection: {expression: 'sum(Process_output_arrival, by=Process_output_bus, over=process_output,
into=bus)'}
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, over=link,
into=link_output), along=snapshot, offset=Link_output_delay, edge=''wrap'', by=snapshot_period,
within=period) * Link_efficiency'}
otherwise: shift(at(Link_p, by=Link_output_link, over=link, into=link_output), along=snapshot, offset=Link_output_delay,
edge=0, by=snapshot_period, within=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,
over=process, into=process_output), along=snapshot, offset=Process_output_delay, edge=''wrap'',
by=snapshot_period, within=period) * Process_rate'}
otherwise: shift(at(Process_p, by=Process_output_process, over=process, into=process_output), along=snapshot,
offset=Process_output_delay, edge=0, by=snapshot_period, within=period) * Process_rate
scenario_opex:
dims: [scenario]
expression: ((Generator_opex + Link_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:414-429`)
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:1537-1538`)
dims: [scenario, snapshot, load]
cases:
active: {when: Load_active, expression: Load_sign * Load_p_set}
otherwise: 0
Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), 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=period, into=snapshot), 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=period, into=snapshot), 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=period, into=snapshot), 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=period, into=snapshot), 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=period, into=snapshot), 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=period, into=snapshot), 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)'}
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),
**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_26_committable_process.yaml', sources) as solution:
solution.objective # 15956.125
The network, rung_26_committable_process.py in the corpus — the spine plus what this rung adds:
# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT
"""Rung 26: committable processes — rung 25's committable links restated as processes that draw a quarter more than they deliver."""
from __future__ import annotations
import spine
def build():
"""The spine plus an east bus that only committable processes serve."""
n = spine.build()
n.add('Bus', 'east')
n.add(
'Process',
'warm_conv',
bus0='north',
bus1='east',
rate0=-1.25,
committable=True,
p_nom=60,
p_min_pu=0.3,
marginal_cost=8,
min_up_time=3,
min_down_time=2,
up_time_before=1,
ramp_limit_up=0.5,
ramp_limit_down=0.5,
ramp_limit_start_up=0.6,
ramp_limit_shut_down=0.6,
start_up_cost=100,
shut_down_cost=50,
stand_by_cost=5,
)
n.add(
'Process',
'cold_conv',
bus0='north',
bus1='east',
rate0=-1.25,
committable=True,
p_nom=40,
p_min_pu=0.2,
min_up_time=2,
min_down_time=3,
up_time_before=0,
down_time_before=1,
start_up_cost=20,
)
n.add(
'Process',
'ext_conv',
bus0='north',
bus1='east',
rate0=-1.25,
committable=True,
p_nom_extendable=True,
p_nom_max=30,
capital_cost=5,
p_min_pu=0.2,
marginal_cost=2,
up_time_before=0,
ramp_limit_up=0.5,
ramp_limit_down=0.5,
)
n.add(
'Process',
'mod_conv',
bus0='south',
bus1='east',
rate0=-1.25,
committable=True,
p_nom_extendable=True,
p_nom_mod=10,
p_nom_max=40,
capital_cost=3,
p_min_pu=0.5,
)
n.add(
'Process',
'mod_fix',
bus0='north',
bus1='east',
rate0=-1.25,
committable=True,
p_nom=20,
p_nom_mod=10,
p_min_pu=0.5,
marginal_cost=1,
)
n.add('Load', 'east_load', bus='east', p_set=[20, 70, 60, 5])
return n
The data¶
The tables this rung is the first to declare (14), as the prep produced them:
Process_big_m.csv
Process_min_down_time.csv
scenario,process,value
base,cold_conv,3
base,ext_conv,0
base,mod_conv,0
base,mod_fix,0
base,warm_conv,2
Process_min_up_time.csv
scenario,process,value
base,cold_conv,2
base,ext_conv,0
base,mod_conv,0
base,mod_fix,0
base,warm_conv,3
Process_modules_installed.csv
scenario,process,value
base,cold_conv,1.0
base,ext_conv,1.0
base,mod_conv,1.0
base,mod_fix,2.0
base,warm_conv,1.0
Process_must_stay_down.csv
scenario,snapshot,process,value
base,2015-01-01T00:00:00.000000,cold_conv,true
base,2015-01-01T01:00:00.000000,cold_conv,true
Process_must_stay_up.csv
scenario,snapshot,process,value
base,2015-01-01T00:00:00.000000,warm_conv,true
base,2015-01-01T01:00:00.000000,warm_conv,true
Process_p_init.csv
Process_p_min_pu_nonneg.csv
Process_p_nom_mod.csv
Process_ramp_limit_shut_down.csv
Process_ramp_limit_start_up.csv
Process_shut_down_cost.csv
scenario,process,value
base,cold_conv,0.0
base,ext_conv,0.0
base,mod_conv,0.0
base,mod_fix,0.0
base,warm_conv,50.0
Process_stand_by_cost.csv
scenario,snapshot,process,value
base,2015-01-01T00:00:00.000000,cold_conv,0.0
base,2015-01-01T00:00:00.000000,ext_conv,0.0
base,2015-01-01T00:00:00.000000,mod_conv,0.0
base,2015-01-01T00:00:00.000000,mod_fix,0.0
base,2015-01-01T00:00:00.000000,warm_conv,5.0
base,2015-01-01T01:00:00.000000,cold_conv,0.0
base,2015-01-01T01:00:00.000000,ext_conv,0.0
base,2015-01-01T01:00:00.000000,mod_conv,0.0
base,2015-01-01T01:00:00.000000,mod_fix,0.0
base,2015-01-01T01:00:00.000000,warm_conv,5.0
base,2015-01-01T02:00:00.000000,cold_conv,0.0
base,2015-01-01T02:00:00.000000,ext_conv,0.0
base,2015-01-01T02:00:00.000000,mod_conv,0.0
base,2015-01-01T02:00:00.000000,mod_fix,0.0
base,2015-01-01T02:00:00.000000,warm_conv,5.0
base,2015-01-01T03:00:00.000000,cold_conv,0.0
base,2015-01-01T03:00:00.000000,ext_conv,0.0
base,2015-01-01T03:00:00.000000,mod_conv,0.0
base,2015-01-01T03:00:00.000000,mod_fix,0.0
base,2015-01-01T03:00:00.000000,warm_conv,5.0
Process_start_up_cost.csv