Rung 33: maintenance — a fixed and an extendable generator, link and process, each taken off for events that the generator weightings time¶
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 7367.560185 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 ✔ 179 rows · ≠ 103 vs 106 columns · ✔ 398 nonzeros; duals — integer model, no duals; model for model: 55 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-ext-p-lower |
4 | 4 |
Generator-ext-p-upper |
4 | 4 |
Generator-ext-p_nom-lower |
1 | 1 |
Generator-ext-p_nom-upper |
1 | 1 |
Generator-fix-p-lower |
16 | 16 |
Generator-fix-p-upper |
16 | 16 |
Generator-maint-event-count |
2 | 2 |
Generator-maint-start-horizon |
1 | 1 |
Generator-maint-window |
8 | 8 |
Generator-maintcap_lower_nommax |
4 | 4 |
Generator-maintcap_lower_nommin |
4 | 4 |
Generator-maintcap_upper |
4 | 4 |
Generator-maintcap_upper_nommax |
4 | 4 |
Link-ext-p-lower |
4 | 4 |
Link-ext-p-upper |
4 | 4 |
Link-ext-p_nom-lower |
1 | 1 |
Link-ext-p_nom-upper |
1 | 1 |
Link-fix-p-lower |
8 | 8 |
Link-fix-p-upper |
8 | 8 |
Link-maint-event-count |
2 | 2 |
Link-maint-start-horizon |
1 | 1 |
Link-maint-window |
8 | 8 |
Link-maintcap_lower_nommax |
4 | 4 |
Link-maintcap_lower_nommin |
4 | 4 |
Link-maintcap_upper |
4 | 4 |
Link-maintcap_upper_nommax |
4 | 4 |
Process-ext-p-lower |
4 | 4 |
Process-ext-p-upper |
4 | 4 |
Process-ext-p_nom-lower |
1 | 1 |
Process-ext-p_nom-upper |
1 | 1 |
Process-fix-p-lower |
4 | 4 |
Process-fix-p-upper |
4 | 4 |
Process-maint-event-count |
2 | 2 |
Process-maint-start-horizon |
1 | 1 |
Process-maint-window |
8 | 8 |
Process-maintcap_lower_nommax |
4 | 4 |
Process-maintcap_lower_nommin |
4 | 4 |
Process-maintcap_upper |
4 | 4 |
Process-maintcap_upper_nommax |
4 | 4 |
| column | PyPSA | specsolve |
|---|---|---|
CVaR |
0 | ≠ 1 |
CVaR-a |
0 | ≠ 1 |
CVaR-theta |
0 | ≠ 1 |
Generator-maintenance |
8 | 8 |
Generator-maintenance_capacity |
4 | 4 |
Generator-maintenance_start |
8 | 8 |
Generator-p |
20 | 20 |
Generator-p_nom |
1 | 1 |
Link-maintenance |
8 | 8 |
Link-maintenance_capacity |
4 | 4 |
Link-maintenance_start |
8 | 8 |
Link-p |
12 | 12 |
Link-p_nom |
1 | 1 |
Process-maintenance |
8 | 8 |
Process-maintenance_capacity |
4 | 4 |
Process-maintenance_start |
8 | 8 |
Process-p |
8 | 8 |
Process-p_nom |
1 | 1 |
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 with \(\mathrm{Generator\_maintenance\_cover} \subseteq \Xi \times \mathcal{G} \times \mathcal{T} \times \mathcal{T},\ \mathrm{Link\_maintenance\_cover} \subseteq \Xi \times \mathcal{L} \times \mathcal{T} \times \mathcal{T},\ \mathrm{Process\_maintenance\_cover} \subseteq \Xi \times \mathcal{J} \times \mathcal{T} \times \mathcal{T}\) — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y},\ \mathrm{Generator\_maintenance\_cover} \subseteq \Xi \times \mathcal{G} \times \mathcal{T} \times \mathcal{T},\ \mathrm{Link\_maintenance\_cover} \subseteq \Xi \times \mathcal{L} \times \mathcal{T} \times \mathcal{T},\ \mathrm{Process\_maintenance\_cover} \subseteq \Xi \times \mathcal{J} \times \mathcal{T} \times \mathcal{T}\) — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Generator\_maintenance\_cover} \subseteq \Xi \times \mathcal{G} \times \mathcal{T} \times \mathcal{T}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_maintenance\_cover} \subseteq \Xi \times \mathcal{L} \times \mathcal{T} \times \mathcal{T}\) — 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 \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J},\ \mathrm{Process\_maintenance\_cover} \subseteq \Xi \times \mathcal{J} \times \mathcal{T} \times \mathcal{T}\) — 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 |
| \(\mathrm{p}^{\mathrm{mod}}\) | Generator_p_nom_mod over \(\mathcal{G}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{mnt}\) | Generator_maintainable over \(\mathcal{G}\) — whether a generator must be taken off for maintenance within the horizon — in any scenario, as PyPSA takes the union over them (components.py:1016-1019) |
| \(\gamma\) | Generator_maintenance_pu over \(\Xi \times \mathcal{G}\) — the share of the build a maintenance event takes off |
| \(\mathrm{n}^{\mathrm{mnt}}\) | Generator_maintenance_events over \(\Xi \times \mathcal{G}\) — how many maintenance events the horizon holds |
| \(\mathrm{blk}\) | Generator_maintenance_start_blocked over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — true where no maintenance event may start, because the snapshots it would cover run past the end of the horizon or into one the generator does not stand in — PyPSA's active & ~valid, from maintenance_duration and the generator weightings, data prep |
| \(\mathrm{f}^{\mathrm{nom}}\) | Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power |
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{f}}\) | Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways |
| \(\overline{\mathrm{f}}\) | Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power |
| \(\eta\) | Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py: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 |
| \(\mathrm{f}^{\mathrm{mod}}\) | Link_p_nom_mod over \(\mathcal{L}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{mnt}^{f}\) | Link_maintainable over \(\mathcal{L}\) — whether a link must be taken off for maintenance within the horizon — in any scenario, as PyPSA takes the union over them (components.py:1016-1019) |
| \(\gamma^{f}\) | Link_maintenance_pu over \(\Xi \times \mathcal{L}\) — the share of the build a maintenance event takes off |
| \(\mathrm{n}^{f,\mathrm{mnt}}\) | Link_maintenance_events over \(\Xi \times \mathcal{L}\) — how many maintenance events the horizon holds |
| \(\mathrm{blk}^{f}\) | Link_maintenance_start_blocked over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — true where no maintenance event may start, because the snapshots it would cover run past the end of the horizon or into one the link does not stand in — PyPSA's active & ~valid, from maintenance_duration and the generator weightings, data prep |
| \(\mathrm{z}^{\mathrm{nom}}\) | 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 |
| \(\underline{\mathrm{z}}^{\mathrm{nom}}\) | 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 |
| \(\mathrm{z}^{\mathrm{mod}}\) | Process_p_nom_mod over \(\mathcal{J}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{mnt}^{z}\) | Process_maintainable over \(\mathcal{J}\) — whether a process must be taken off for maintenance within the horizon — in any scenario, as PyPSA takes the union over them (components.py:1016-1019) |
| \(\gamma^{z}\) | Process_maintenance_pu over \(\Xi \times \mathcal{J}\) — the share of the build a maintenance event takes off |
| \(\mathrm{n}^{z,\mathrm{mnt}}\) | Process_maintenance_events over \(\Xi \times \mathcal{J}\) — how many maintenance events the horizon holds |
| \(\mathrm{blk}^{z}\) | Process_maintenance_start_blocked over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — true where no maintenance event may start, because the snapshots it would cover run past the end of the horizon or into one the process does not stand in — PyPSA's active & ~valid, from maintenance_duration and the generator weightings, 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 |
| \(\mathrm{W}\) | Generator_capital_weight over \(\mathcal{G}\) — the sum of period weights a generator stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{W}^{f}\) | Link_capital_weight over \(\mathcal{L}\) — the sum of period weights a link stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{W}^{z}\) | Process_capital_weight over \(\mathcal{J}\) — the sum of period weights a process stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\underline{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_min over \(\Xi \times \mathcal{G}\) — least nominal power an extendable generator may be built at |
| \(\overline{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_max over \(\Xi \times \mathcal{G}\) — most nominal power an extendable generator may be built at |
| \(\mathrm{c}^{\mathrm{cap}}\) | Generator_capital_cost over \(\Xi \times \mathcal{G}\) — cost of one unit of nominal power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\underline{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_min over \(\Xi \times \mathcal{L}\) — least nominal power an extendable link may be built at |
| \(\overline{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_max over \(\Xi \times \mathcal{L}\) — most nominal power an extendable link may be built at |
| \(\mathrm{c}^{\mathrm{cap},f}\) | Link_capital_cost over \(\Xi \times \mathcal{L}\) — cost of one unit of nominal power — PyPSA's capital_cost, periodized as an annuity in 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 |
| \(\mu\) | Generator_maintenance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-maintenance — whether a maintainable generator is in maintenance: continuous, and one exactly where an event covers the snapshot |
| \(\mu^{\mathrm{up}}\) | Generator_maintenance_start over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-maintenance_start — whether a maintenance event starts in this snapshot |
| \(\mu^{\mathrm{nom}}\) | Generator_maintenance_capacity over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-maintenance_capacity — the chosen build while in maintenance, zero otherwise: the product the maintcap rows linearize |
| \(\mu^{f}\) | Link_maintenance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-maintenance — whether a maintainable link is in maintenance: continuous, and one exactly where an event covers the snapshot |
| \(\mu^{f,\mathrm{up}}\) | Link_maintenance_start over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-maintenance_start — whether a maintenance event starts in this snapshot |
| \(\mu^{f,\mathrm{nom}}\) | Link_maintenance_capacity over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-maintenance_capacity — the chosen build while in maintenance, zero otherwise: the product the maintcap rows linearize |
| \(\mu^{z}\) | Process_maintenance over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-maintenance — whether a maintainable process is in maintenance: continuous, and one exactly where an event covers the snapshot |
| \(\mu^{z,\mathrm{up}}\) | Process_maintenance_start over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-maintenance_start — whether a maintenance event starts in this snapshot |
| \(\mu^{z,\mathrm{nom}}\) | Process_maintenance_capacity over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-maintenance_capacity — the chosen build while in maintenance, zero otherwise: the product the maintcap rows linearize |
| \(P\) | Generator_p_nom_ext over \(\mathcal{G}\) — Generator-p_nom — nominal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
| \(F\) | Link_p_nom_ext over \(\mathcal{L}\) — Link-p_nom — nominal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
| \(Z\) | 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 |
|---|---|
| \(\mathit{total\_cost}\) | 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 |
| \(\mathit{Generator\_capex}\) | Generator_capex (scalar) |
| \(\mathit{Link\_capex}\) | Link_capex (scalar) |
| \(\mathit{Process\_capex}\) | 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 \(\Xi\) |
\(t \ominus k\) 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.
Objective¶
\[
\min \mathit{total\_cost}
\]
Subject to¶
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \gamma_{\xi,g} \cdot \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \gamma_{\xi,g} \cdot \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Link_fix_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \cdot \left( 1 - \gamma^{f}_{\xi,l} \cdot \mu^{f}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_fix_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \cdot \left( 1 - \gamma^{f}_{\xi,l} \cdot \mu^{f}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Generator_ext_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \left( P_{g} - \gamma_{\xi,g} \cdot \mu^{\mathrm{nom}}_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \left( P_{g} - \gamma_{\xi,g} \cdot \mu^{\mathrm{nom}}_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_nom_lower
\[
P_{g} \ge \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g}
\]
Generator_ext_p_nom_upper
\[
P_{g} \le \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \text{ is defined}
\]
Link_ext_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \left( F_{l} - \gamma^{f}_{\xi,l} \cdot \mu^{f,\mathrm{nom}}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \left( F_{l} - \gamma^{f}_{\xi,l} \cdot \mu^{f,\mathrm{nom}}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_nom_lower
\[
F_{l} \ge \underline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l}
\]
Link_ext_p_nom_upper
\[
F_{l} \le \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \text{ is defined}
\]
Process_fix_p_lower
\[
z_{\xi,t,j} \ge \underline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{nom}}_{\xi,j} \cdot \left( 1 - \gamma^{z}_{\xi,j} \cdot \mu^{z}_{\xi,t,j} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \neg \mathrm{ext}^{z}_{j} \wedge \neg \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_fix_p_upper
\[
z_{\xi,t,j} \le \overline{\mathrm{z}}_{\xi,t,j} \cdot \mathrm{z}^{\mathrm{nom}}_{\xi,j} \cdot \left( 1 - \gamma^{z}_{\xi,j} \cdot \mu^{z}_{\xi,t,j} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \neg \mathrm{ext}^{z}_{j} \wedge \neg \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_ext_p_lower
\[
z_{\xi,t,j} \ge \underline{\mathrm{z}}_{\xi,t,j} \cdot \left( Z_{j} - \gamma^{z}_{\xi,j} \cdot \mu^{z,\mathrm{nom}}_{\xi,t,j} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j} \wedge \neg \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_ext_p_upper
\[
z_{\xi,t,j} \le \overline{\mathrm{z}}_{\xi,t,j} \cdot \left( Z_{j} - \gamma^{z}_{\xi,j} \cdot \mu^{z,\mathrm{nom}}_{\xi,t,j} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j} \wedge \neg \mathrm{com}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
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
\[
Z_{j} \le \overline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J} \,:\, \mathrm{ext}^{z}_{j} \wedge \overline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \text{ is defined}
\]
Generator_maint_event_count
\[
\sum_{t \in \mathcal{T}} \mu^{\mathrm{up}}_{\xi,t,g} = \mathrm{n}^{\mathrm{mnt}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g}
\]
Generator_maint_window
\[
\mu_{\xi,t,g} = \sum_{t' \in \mathcal{T} \,:\, \left( \xi,\ g,\ t',\ t \right) \in \mathrm{Generator\_maintenance\_cover}} \mu^{\mathrm{up}}_{\xi,t',g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_maint_start_horizon
\[
\mu^{\mathrm{up}}_{\xi,t,g} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g} \wedge \mathrm{blk}_{\xi,t,g}
\]
Generator_maintcap_upper
\[
\mu^{\mathrm{nom}}_{\xi,t,g} \le P_{g} - \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \mathrm{on}_{t,g}
\]
Generator_maintcap_upper_nommax
\[
\mu^{\mathrm{nom}}_{\xi,t,g} \le \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \mu_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \mathrm{on}_{t,g}
\]
Generator_maintcap_lower_nommax
\[
\mu^{\mathrm{nom}}_{\xi,t,g} \ge P_{g} - \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \mathrm{on}_{t,g}
\]
Generator_maintcap_lower_nommin
\[
\mu^{\mathrm{nom}}_{\xi,t,g} \ge \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \mu_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \mathrm{on}_{t,g} \wedge \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} > 0
\]
Link_maint_event_count
\[
\sum_{t \in \mathcal{T}} \mu^{f,\mathrm{up}}_{\xi,t,l} = \mathrm{n}^{f,\mathrm{mnt}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l}
\]
Link_maint_window
\[
\mu^{f}_{\xi,t,l} = \sum_{t' \in \mathcal{T} \,:\, \left( \xi,\ l,\ t',\ t \right) \in \mathrm{Link\_maintenance\_cover}} \mu^{f,\mathrm{up}}_{\xi,t',l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_maint_start_horizon
\[
\mu^{f,\mathrm{up}}_{\xi,t,l} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l} \wedge \mathrm{blk}^{f}_{\xi,t,l}
\]
Link_maintcap_upper
\[
\mu^{f,\mathrm{nom}}_{\xi,t,l} \le F_{l} - \underline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \left( 1 - \mu^{f}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_maintcap_upper_nommax
\[
\mu^{f,\mathrm{nom}}_{\xi,t,l} \le \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \mu^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_maintcap_lower_nommax
\[
\mu^{f,\mathrm{nom}}_{\xi,t,l} \ge F_{l} - \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \left( 1 - \mu^{f}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_maintcap_lower_nommin
\[
\mu^{f,\mathrm{nom}}_{\xi,t,l} \ge \underline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \mu^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \mathrm{on}^{f}_{t,l} \wedge \underline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} > 0
\]
Process_maint_event_count
\[
\sum_{t \in \mathcal{T}} \mu^{z,\mathrm{up}}_{\xi,t,j} = \mathrm{n}^{z,\mathrm{mnt}}_{\xi,j} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j}
\]
Process_maint_window
\[
\mu^{z}_{\xi,t,j} = \sum_{t' \in \mathcal{T} \,:\, \left( \xi,\ j,\ t',\ t \right) \in \mathrm{Process\_maintenance\_cover}} \mu^{z,\mathrm{up}}_{\xi,t',j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_maint_start_horizon
\[
\mu^{z,\mathrm{up}}_{\xi,t,j} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j} \wedge \mathrm{blk}^{z}_{\xi,t,j}
\]
Process_maintcap_upper
\[
\mu^{z,\mathrm{nom}}_{\xi,t,j} \le Z_{j} - \underline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \left( 1 - \mu^{z}_{\xi,t,j} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_maintcap_upper_nommax
\[
\mu^{z,\mathrm{nom}}_{\xi,t,j} \le \overline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \mu^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_maintcap_lower_nommax
\[
\mu^{z,\mathrm{nom}}_{\xi,t,j} \ge Z_{j} - \overline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \left( 1 - \mu^{z}_{\xi,t,j} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_maintcap_lower_nommin
\[
\mu^{z,\mathrm{nom}}_{\xi,t,j} \ge \underline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \mu^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j} \wedge \underline{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} > 0
\]
Bus_nodal_balance
\[
\mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Definitions¶
total_cost
\[
\mathit{total\_cost} = \mathit{Generator\_capex} + \mathit{Link\_capex} + \mathit{Process\_capex} + \mathit{risk\_weighted\_opex}
\]
Bus_injection
\[
\mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{\xi,t,n} + \mathit{Link\_injection}_{\xi,t,n} + \mathrm{Load\_injection}_{\xi,t,n} + \mathit{Process\_injection}_{\xi,t,n} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Generator_capex
\[
\mathit{Generator\_capex} = \sum_{\xi \in \Xi,\ g \in \mathcal{G}} \pi_{\xi} \cdot P_{g} \cdot \mathrm{c}^{\mathrm{cap}}_{\xi,g} \cdot \mathrm{W}_{g}
\]
Link_capex
\[
\mathit{Link\_capex} = \sum_{\xi \in \Xi,\ l \in \mathcal{L}} \pi_{\xi} \cdot F_{l} \cdot \mathrm{c}^{\mathrm{cap},f}_{\xi,l} \cdot \mathrm{W}^{f}_{l}
\]
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
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} + \mathit{Process\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
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
\[
\mathit{Process\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} z_{\xi,t,j} \cdot \mathrm{c}^{z}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{j \in \mathcal{J}} z_{\xi,t,j} \cdot z_{\xi,t,j} \cdot \mathrm{c}^{z,(2)}_{\xi,t,j} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Variable domains¶
Generator_p
\[
p_{\xi,t,g} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{on}_{t,g}
\]
Link_p
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
Process_p
\[
z_{\xi,t,j} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{on}^{z}_{t,j}
\]
Generator_maintenance
\[
0 \le \mu_{\xi,t,g} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_maintenance_start
\[
\mu^{\mathrm{up}}_{\xi,t,g} \in \{0, 1\} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_maintenance_capacity
\[
\mu^{\mathrm{nom}}_{\xi,t,g} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{mnt}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \mathrm{on}_{t,g}
\]
Link_maintenance
\[
0 \le \mu^{f}_{\xi,t,l} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_maintenance_start
\[
\mu^{f,\mathrm{up}}_{\xi,t,l} \in \{0, 1\} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_maintenance_capacity
\[
\mu^{f,\mathrm{nom}}_{\xi,t,l} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{mnt}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Process_maintenance
\[
0 \le \mu^{z}_{\xi,t,j} \le 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_maintenance_start
\[
\mu^{z,\mathrm{up}}_{\xi,t,j} \in \{0, 1\} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{on}^{z}_{t,j}
\]
Process_maintenance_capacity
\[
\mu^{z,\mathrm{nom}}_{\xi,t,j} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{mnt}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Generator_p_nom_ext
\[
P_{g} \in \mathbb{R} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g}
\]
Link_p_nom_ext
\[
F_{l} \in \mathbb{R} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l}
\]
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_33_maintenance.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}
Generator_maintenance_cover:
description: the snapshots a maintenance event covers, by the snapshot it starts in — in a scenario,
the start and the snapshots after it, until their generator weightings reach that scenario's `maintenance_duration`,
data prep; no rows for a generator that is not maintainable
key:
scenario: scenario
generator: generator
start: snapshot
covered: snapshot
Link_maintenance_cover:
description: the snapshots a maintenance event covers, by the snapshot it starts in — in a scenario,
the start and the snapshots after it, until their generator weightings reach that scenario's `maintenance_duration`,
data prep; no rows for a link that is not maintainable
key:
scenario: scenario
link: link
start: snapshot
covered: snapshot
Process_maintenance_cover:
description: the snapshots a maintenance event covers, by the snapshot it starts in — in a scenario,
the start and the snapshots after it, until their generator weightings reach that scenario's `maintenance_duration`,
data prep; no rows for a process that is not maintainable
key:
scenario: scenario
process: process
start: snapshot
covered: snapshot
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
Generator_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [generator]
Generator_maintainable:
description: whether a generator must be taken off for maintenance within the horizon — in any scenario,
as PyPSA takes the union over them (`components.py:1016-1019`)
dims: [generator]
dtype: bool
Generator_maintenance_pu:
description: the share of the build a maintenance event takes off
dims: [scenario, generator]
Generator_maintenance_events:
description: how many maintenance events the horizon holds
dims: [scenario, generator]
dtype: int
Generator_maintenance_start_blocked:
description: true where no maintenance event may start, because the snapshots it would cover run past
the end of the horizon or into one the generator does not stand in — PyPSA's `active & ~valid`,
from `maintenance_duration` and the generator weightings, data prep
dims: [scenario, snapshot, 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
Link_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [link]
Link_maintainable:
description: whether a link must be taken off for maintenance within the horizon — in any scenario,
as PyPSA takes the union over them (`components.py:1016-1019`)
dims: [link]
dtype: bool
Link_maintenance_pu:
description: the share of the build a maintenance event takes off
dims: [scenario, link]
Link_maintenance_events:
description: how many maintenance events the horizon holds
dims: [scenario, link]
dtype: int
Link_maintenance_start_blocked:
description: true where no maintenance event may start, because the snapshots it would cover run past
the end of the horizon or into one the link does not stand in — PyPSA's `active & ~valid`, from
`maintenance_duration` and the generator weightings, data prep
dims: [scenario, snapshot, 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_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_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [process]
Process_maintainable:
description: whether a process must be taken off for maintenance within the horizon — in any scenario,
as PyPSA takes the union over them (`components.py:1016-1019`)
dims: [process]
dtype: bool
Process_maintenance_pu:
description: the share of the build a maintenance event takes off
dims: [scenario, process]
Process_maintenance_events:
description: how many maintenance events the horizon holds
dims: [scenario, process]
dtype: int
Process_maintenance_start_blocked:
description: true where no maintenance event may start, because the snapshots it would cover run past
the end of the horizon or into one the process does not stand in — PyPSA's `active & ~valid`, from
`maintenance_duration` and the generator weightings, data prep
dims: [scenario, snapshot, 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
Generator_capital_weight:
description: the sum of period weights a generator stands in — PyPSA's `active * period_weighting`,
summed, data prep
dims: [generator]
Link_capital_weight:
description: the sum of period weights a link stands in — PyPSA's `active * period_weighting`, summed,
data prep
dims: [link]
Process_capital_weight:
description: the sum of period weights a process stands in — PyPSA's `active * period_weighting`,
summed, data prep
dims: [process]
Generator_p_nom_min:
description: least nominal power an extendable generator may be built at
dims: [scenario, generator]
Generator_p_nom_max:
description: most nominal power an extendable generator may be built at
dims: [scenario, generator]
Generator_capital_cost:
description: cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity
in data prep
dims: [scenario, generator]
Link_p_nom_min:
description: least nominal power an extendable link may be built at
dims: [scenario, link]
Link_p_nom_max:
description: most nominal power an extendable link may be built at
dims: [scenario, link]
Link_capital_cost:
description: cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity
in data prep
dims: [scenario, link]
variables:
Generator_p:
description: '`Generator-p` — output of a generator in a snapshot'
dims: [scenario, snapshot, generator]
where: Generator_active
Link_p:
description: '`Link-p` — PyPSA''s `p0`, the flow measured at the `Link_bus0` end: a positive value
withdraws there and injects at every bus the link''s output ports deliver to'
dims: [scenario, snapshot, link]
where: Link_active
Process_p:
description: '`Process-p` — PyPSA''s internal power `p`: a positive value drives every port at its
own rate, withdrawing where the rate is negative and injecting where it is positive'
dims: [scenario, snapshot, process]
where: Process_active
Generator_maintenance:
description: '`Generator-maintenance` — whether a maintainable generator is in maintenance: continuous,
and one exactly where an event covers the snapshot'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active
absence: zero
bounds: {lower: 0, upper: 1}
Generator_maintenance_start:
description: '`Generator-maintenance_start` — whether a maintenance event starts in this snapshot'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active
absence: zero
domain: binary
Generator_maintenance_capacity:
description: '`Generator-maintenance_capacity` — the chosen build while in maintenance, zero otherwise:
the product the `maintcap` rows linearize'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_p_nom_extendable AND NOT (Generator_committable AND Generator_p_nom_mod
> 0) AND Generator_active
absence: zero
bounds: {lower: 0}
Link_maintenance:
description: '`Link-maintenance` — whether a maintainable link is in maintenance: continuous, and
one exactly where an event covers the snapshot'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_active
absence: zero
bounds: {lower: 0, upper: 1}
Link_maintenance_start:
description: '`Link-maintenance_start` — whether a maintenance event starts in this snapshot'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_active
absence: zero
domain: binary
Link_maintenance_capacity:
description: '`Link-maintenance_capacity` — the chosen build while in maintenance, zero otherwise:
the product the `maintcap` rows linearize'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_p_nom_extendable AND NOT (Link_committable AND Link_p_nom_mod >
0) AND Link_active
absence: zero
bounds: {lower: 0}
Process_maintenance:
description: '`Process-maintenance` — whether a maintainable process is in maintenance: continuous,
and one exactly where an event covers the snapshot'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_active
absence: zero
bounds: {lower: 0, upper: 1}
Process_maintenance_start:
description: '`Process-maintenance_start` — whether a maintenance event starts in this snapshot'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_active
absence: zero
domain: binary
Process_maintenance_capacity:
description: '`Process-maintenance_capacity` — the chosen build while in maintenance, zero otherwise:
the product the `maintcap` rows linearize'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_p_nom_extendable AND NOT (Process_committable AND Process_p_nom_mod
> 0) AND Process_active
absence: zero
bounds: {lower: 0}
Generator_p_nom_ext:
description: '`Generator-p_nom` — nominal power where it is a decision; the parameter of the same
PyPSA name carries the fixed regime'
dims: [generator]
where: Generator_p_nom_extendable
Link_p_nom_ext:
description: '`Link-p_nom` — nominal power where it is a decision; the parameter of the same PyPSA
name carries the fixed regime'
dims: [link]
where: Link_p_nom_extendable
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 * (1 - Generator_maintenance_pu *
Generator_maintenance)
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 * (1 - Generator_maintenance_pu *
Generator_maintenance)
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 * (1 - Link_maintenance_pu * Link_maintenance)
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 * (1 - Link_maintenance_pu * Link_maintenance)
Generator_ext_p_lower:
description: '`Generator-ext-p-lower` — an extendable generator outputs at least its minimum of the
chosen build'
dims: [scenario, snapshot, generator]
where: Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * (Generator_p_nom_ext - Generator_maintenance_pu *
Generator_maintenance_capacity)
Generator_ext_p_upper:
description: '`Generator-ext-p-upper` — an extendable generator outputs at most what is available
of the chosen build'
dims: [scenario, snapshot, generator]
where: Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * (Generator_p_nom_ext - Generator_maintenance_pu *
Generator_maintenance_capacity)
Generator_ext_p_nom_lower:
description: '`Generator-ext-p_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, generator]
where: Generator_p_nom_extendable
expression: Generator_p_nom_ext >= Generator_p_nom_min
Generator_ext_p_nom_upper:
description: '`Generator-ext-p_nom-upper` — the chosen build is at most its cap in every scenario;
a cap of infinity is no row'
dims: [scenario, generator]
where: Generator_p_nom_extendable AND Generator_p_nom_max
expression: Generator_p_nom_ext <= Generator_p_nom_max
Link_ext_p_lower:
description: '`Link-ext-p-lower` — an extendable link carries at least its minimum of the chosen build,
negative for the other way'
dims: [scenario, snapshot, link]
where: Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * (Link_p_nom_ext - Link_maintenance_pu * Link_maintenance_capacity)
Link_ext_p_upper:
description: '`Link-ext-p-upper` — an extendable link carries at most the chosen build'
dims: [scenario, snapshot, link]
where: Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * (Link_p_nom_ext - Link_maintenance_pu * Link_maintenance_capacity)
Link_ext_p_nom_lower:
description: '`Link-ext-p_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, link]
where: Link_p_nom_extendable
expression: Link_p_nom_ext >= Link_p_nom_min
Link_ext_p_nom_upper:
description: '`Link-ext-p_nom-upper` — the chosen build is at most its cap in every scenario; a cap
of infinity is no row'
dims: [scenario, link]
where: Link_p_nom_extendable AND Link_p_nom_max
expression: Link_p_nom_ext <= Link_p_nom_max
Process_fix_p_lower:
description: '`Process-fix-p-lower` — a fixed process runs at least its minimum, negative for the
other way'
dims: [scenario, snapshot, process]
where: not Process_p_nom_extendable AND not Process_committable AND Process_active
expression: Process_p >= Process_p_min_pu * Process_p_nom * (1 - Process_maintenance_pu * Process_maintenance)
Process_fix_p_upper:
description: '`Process-fix-p-upper` — a fixed process runs at most its nominal power'
dims: [scenario, snapshot, process]
where: not Process_p_nom_extendable AND not Process_committable AND Process_active
expression: Process_p <= Process_p_max_pu * Process_p_nom * (1 - Process_maintenance_pu * Process_maintenance)
Process_ext_p_lower:
description: '`Process-ext-p-lower` — an extendable process runs at least its minimum of the chosen
build, negative for the other way'
dims: [scenario, snapshot, process]
where: Process_p_nom_extendable AND not Process_committable AND Process_active
expression: Process_p >= Process_p_min_pu * (Process_p_nom_ext - Process_maintenance_pu * Process_maintenance_capacity)
Process_ext_p_upper:
description: '`Process-ext-p-upper` — an extendable process runs at most the chosen build'
dims: [scenario, snapshot, process]
where: Process_p_nom_extendable AND not Process_committable AND Process_active
expression: Process_p <= Process_p_max_pu * (Process_p_nom_ext - Process_maintenance_pu * Process_maintenance_capacity)
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
Generator_maint_event_count:
description: '`Generator-maint-event-count` — a maintainable generator holds its number of maintenance
events over the horizon'
dims: [scenario, generator]
where: Generator_maintainable
expression: sum(Generator_maintenance_start, over=snapshot) == Generator_maintenance_events
Generator_maint_window:
description: '`Generator-maint-window` — a generator is in maintenance exactly where an event it started
covers the snapshot; two events do not overlap, since the maintenance status is at most one'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active
expression: Generator_maintenance == sum(Generator_maintenance_start, by=Generator_maintenance_cover,
over=start, into=covered)
Generator_maint_start_horizon:
description: '`Generator-maint-start-horizon` — no event starts where it could not run its whole duration'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_active AND Generator_maintenance_start_blocked
expression: Generator_maintenance_start == 0
Generator_maintcap_upper:
description: '`Generator-maintcap_upper` — the build taken off is at most the chosen build in maintenance,
and at most the build less its floor out of it'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_p_nom_extendable AND NOT (Generator_committable AND Generator_p_nom_mod
> 0) AND Generator_active
expression: Generator_maintenance_capacity <= Generator_p_nom_ext - Generator_p_nom_min * (1 - Generator_maintenance)
Generator_maintcap_upper_nommax:
description: '`Generator-maintcap_upper_nommax` — out of maintenance, no build is taken off'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_p_nom_extendable AND NOT (Generator_committable AND Generator_p_nom_mod
> 0) AND Generator_active
expression: Generator_maintenance_capacity <= Generator_p_nom_max * Generator_maintenance
Generator_maintcap_lower_nommax:
description: '`Generator-maintcap_lower_nommax` — in maintenance, the whole chosen build is taken
off'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_p_nom_extendable AND NOT (Generator_committable AND Generator_p_nom_mod
> 0) AND Generator_active
expression: Generator_maintenance_capacity >= Generator_p_nom_ext - Generator_p_nom_max * (1 - Generator_maintenance)
Generator_maintcap_lower_nommin:
description: '`Generator-maintcap_lower_nommin` — in maintenance, at least the floor of the build
is taken off'
dims: [scenario, snapshot, generator]
where: Generator_maintainable AND Generator_p_nom_extendable AND NOT (Generator_committable AND Generator_p_nom_mod
> 0) AND Generator_active AND Generator_p_nom_min > 0
expression: Generator_maintenance_capacity >= Generator_p_nom_min * Generator_maintenance
Link_maint_event_count:
description: '`Link-maint-event-count` — a maintainable link holds its number of maintenance events
over the horizon'
dims: [scenario, link]
where: Link_maintainable
expression: sum(Link_maintenance_start, over=snapshot) == Link_maintenance_events
Link_maint_window:
description: '`Link-maint-window` — a link is in maintenance exactly where an event it started covers
the snapshot; two events do not overlap, since the maintenance status is at most one'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_active
expression: Link_maintenance == sum(Link_maintenance_start, by=Link_maintenance_cover, over=start,
into=covered)
Link_maint_start_horizon:
description: '`Link-maint-start-horizon` — no event starts where it could not run its whole duration'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_active AND Link_maintenance_start_blocked
expression: Link_maintenance_start == 0
Link_maintcap_upper:
description: '`Link-maintcap_upper` — the build taken off is at most the chosen build in maintenance,
and at most the build less its floor out of it'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_p_nom_extendable AND NOT (Link_committable AND Link_p_nom_mod >
0) AND Link_active
expression: Link_maintenance_capacity <= Link_p_nom_ext - Link_p_nom_min * (1 - Link_maintenance)
Link_maintcap_upper_nommax:
description: '`Link-maintcap_upper_nommax` — out of maintenance, no build is taken off'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_p_nom_extendable AND NOT (Link_committable AND Link_p_nom_mod >
0) AND Link_active
expression: Link_maintenance_capacity <= Link_p_nom_max * Link_maintenance
Link_maintcap_lower_nommax:
description: '`Link-maintcap_lower_nommax` — in maintenance, the whole chosen build is taken off'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_p_nom_extendable AND NOT (Link_committable AND Link_p_nom_mod >
0) AND Link_active
expression: Link_maintenance_capacity >= Link_p_nom_ext - Link_p_nom_max * (1 - Link_maintenance)
Link_maintcap_lower_nommin:
description: '`Link-maintcap_lower_nommin` — in maintenance, at least the floor of the build is taken
off'
dims: [scenario, snapshot, link]
where: Link_maintainable AND Link_p_nom_extendable AND NOT (Link_committable AND Link_p_nom_mod >
0) AND Link_active AND Link_p_nom_min > 0
expression: Link_maintenance_capacity >= Link_p_nom_min * Link_maintenance
Process_maint_event_count:
description: '`Process-maint-event-count` — a maintainable process holds its number of maintenance
events over the horizon'
dims: [scenario, process]
where: Process_maintainable
expression: sum(Process_maintenance_start, over=snapshot) == Process_maintenance_events
Process_maint_window:
description: '`Process-maint-window` — a process is in maintenance exactly where an event it started
covers the snapshot; two events do not overlap, since the maintenance status is at most one'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_active
expression: Process_maintenance == sum(Process_maintenance_start, by=Process_maintenance_cover, over=start,
into=covered)
Process_maint_start_horizon:
description: '`Process-maint-start-horizon` — no event starts where it could not run its whole duration'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_active AND Process_maintenance_start_blocked
expression: Process_maintenance_start == 0
Process_maintcap_upper:
description: '`Process-maintcap_upper` — the build taken off is at most the chosen build in maintenance,
and at most the build less its floor out of it'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_p_nom_extendable AND NOT (Process_committable AND Process_p_nom_mod
> 0) AND Process_active
expression: Process_maintenance_capacity <= Process_p_nom_ext - Process_p_nom_min * (1 - Process_maintenance)
Process_maintcap_upper_nommax:
description: '`Process-maintcap_upper_nommax` — out of maintenance, no build is taken off'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_p_nom_extendable AND NOT (Process_committable AND Process_p_nom_mod
> 0) AND Process_active
expression: Process_maintenance_capacity <= Process_p_nom_max * Process_maintenance
Process_maintcap_lower_nommax:
description: '`Process-maintcap_lower_nommax` — in maintenance, the whole chosen build is taken off'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_p_nom_extendable AND NOT (Process_committable AND Process_p_nom_mod
> 0) AND Process_active
expression: Process_maintenance_capacity >= Process_p_nom_ext - Process_p_nom_max * (1 - Process_maintenance)
Process_maintcap_lower_nommin:
description: '`Process-maintcap_lower_nommin` — in maintenance, at least the floor of the build is
taken off'
dims: [scenario, snapshot, process]
where: Process_maintainable AND Process_p_nom_extendable AND NOT (Process_committable AND Process_p_nom_mod
> 0) AND Process_active AND Process_p_nom_min > 0
expression: Process_maintenance_capacity >= Process_p_nom_min * Process_maintenance
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:
total_cost:
dims: []
expression: ((Generator_capex + Link_capex) + 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
Generator_capex: {expression: sum(scenario_weight * Generator_p_nom_ext * Generator_capital_cost * Generator_capital_weight)}
Link_capex: {expression: sum(scenario_weight * Link_p_nom_ext * Link_capital_cost * Link_capital_weight)}
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
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)'}
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_33_maintenance.yaml', sources) as solution:
solution.objective # 7367.560185
The network, rung_33_maintenance.py in the corpus — the spine plus what this rung adds:
# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT
"""Rung 33: maintenance — a fixed and an extendable generator, link and process, each taken off for events that the generator weightings time."""
from __future__ import annotations
import spine
def build():
"""The spine plus an east bus and six maintainable components, fixed and extendable, with a dear peaker to cover them."""
n = spine.build()
n.add('Bus', 'east')
n.add('Generator', 'hydro', bus='north', p_nom=50, marginal_cost=2, maintainable=True, maintenance_duration=3)
n.add(
'Generator',
'wind_ext',
bus='south',
p_nom_extendable=True,
p_nom_min=10,
p_nom_max=60,
capital_cost=20,
marginal_cost=1,
p_max_pu=[0.9, 0.6, 0.8, 0.7],
maintainable=True,
maintenance_duration=2,
maintenance_pu=0.5,
)
n.add(
'Link',
'tie_fix',
bus0='north',
bus1='east',
p_nom=30,
efficiency=0.95,
maintainable=True,
maintenance_duration=1,
maintenance_events=2,
)
n.add(
'Link',
'tie_ext',
bus0='south',
bus1='east',
p_nom_extendable=True,
p_nom_min=5,
p_nom_max=40,
capital_cost=4,
efficiency=0.9,
p_min_pu=-1,
maintainable=True,
maintenance_duration=3,
)
n.add(
'Process',
'conv_fix',
bus0='north',
bus1='east',
rate0=-1.25,
p_nom=25,
maintainable=True,
maintenance_duration=3,
maintenance_pu=0.6,
)
n.add(
'Process',
'conv_ext',
bus0='south',
bus1='east',
rate0=-1.25,
p_nom_extendable=True,
p_nom_min=5,
p_nom_max=30,
capital_cost=3,
maintainable=True,
maintenance_duration=1,
)
n.add('Generator', 'east_peak', bus='east', p_nom=100, marginal_cost=60)
n.add('Load', 'east_load', bus='east', p_set=[50, 60, 40, 55])
return n
The data¶
The tables this rung is the first to declare (15), as the prep produced them:
Generator_maintainable.csv
Generator_maintenance_cover.csv
scenario,generator,start,covered
base,hydro,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,hydro,2015-01-01T00:00:00.000000,2015-01-01T01:00:00.000000
base,hydro,2015-01-01T00:00:00.000000,2015-01-01T02:00:00.000000
base,hydro,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,hydro,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,hydro,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,hydro,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
base,wind_ext,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,wind_ext,2015-01-01T00:00:00.000000,2015-01-01T01:00:00.000000
base,wind_ext,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,wind_ext,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,wind_ext,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,wind_ext,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
Generator_maintenance_events.csv
Generator_maintenance_pu.csv
scenario,generator,value
base,coal,1.0
base,east_peak,1.0
base,gas,1.0
base,hydro,1.0
base,wind_ext,0.5
Generator_maintenance_start_blocked.csv
scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,hydro,false
base,2015-01-01T00:00:00.000000,wind_ext,false
base,2015-01-01T01:00:00.000000,hydro,false
base,2015-01-01T01:00:00.000000,wind_ext,false
base,2015-01-01T02:00:00.000000,hydro,false
base,2015-01-01T02:00:00.000000,wind_ext,false
base,2015-01-01T03:00:00.000000,hydro,true
base,2015-01-01T03:00:00.000000,wind_ext,false
Link_maintainable.csv
Link_maintenance_cover.csv
scenario,link,start,covered
base,tie_ext,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,tie_ext,2015-01-01T00:00:00.000000,2015-01-01T01:00:00.000000
base,tie_ext,2015-01-01T00:00:00.000000,2015-01-01T02:00:00.000000
base,tie_ext,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,tie_ext,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,tie_ext,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,tie_ext,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
base,tie_fix,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,tie_fix,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,tie_fix,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,tie_fix,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,tie_fix,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
Link_maintenance_events.csv
Link_maintenance_pu.csv
Link_maintenance_start_blocked.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,tie_ext,false
base,2015-01-01T00:00:00.000000,tie_fix,false
base,2015-01-01T01:00:00.000000,tie_ext,false
base,2015-01-01T01:00:00.000000,tie_fix,false
base,2015-01-01T02:00:00.000000,tie_ext,false
base,2015-01-01T02:00:00.000000,tie_fix,false
base,2015-01-01T03:00:00.000000,tie_ext,true
base,2015-01-01T03:00:00.000000,tie_fix,false
Process_maintainable.csv
Process_maintenance_cover.csv
scenario,process,start,covered
base,conv_ext,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,conv_ext,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,conv_ext,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,conv_ext,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,conv_ext,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
base,conv_fix,2015-01-01T00:00:00.000000,2015-01-01T00:00:00.000000
base,conv_fix,2015-01-01T00:00:00.000000,2015-01-01T01:00:00.000000
base,conv_fix,2015-01-01T00:00:00.000000,2015-01-01T02:00:00.000000
base,conv_fix,2015-01-01T01:00:00.000000,2015-01-01T01:00:00.000000
base,conv_fix,2015-01-01T01:00:00.000000,2015-01-01T02:00:00.000000
base,conv_fix,2015-01-01T02:00:00.000000,2015-01-01T02:00:00.000000
base,conv_fix,2015-01-01T03:00:00.000000,2015-01-01T03:00:00.000000
Process_maintenance_events.csv
Process_maintenance_pu.csv
Process_maintenance_start_blocked.csv
scenario,snapshot,process,value
base,2015-01-01T00:00:00.000000,conv_ext,false
base,2015-01-01T00:00:00.000000,conv_fix,false
base,2015-01-01T01:00:00.000000,conv_ext,false
base,2015-01-01T01:00:00.000000,conv_fix,false
base,2015-01-01T02:00:00.000000,conv_ext,false
base,2015-01-01T02:00:00.000000,conv_fix,false
base,2015-01-01T03:00:00.000000,conv_ext,false
base,2015-01-01T03:00:00.000000,conv_fix,true