Rung 3: expansion — extendable capacity, energy-sum bounds, fixed and set nominal capacities¶
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 7633.908502024291 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;tech_capacity_expansion_limit5 vs 1+2+2 — one block per sense — ==, <=, >= — where PyPSA writes one row per labelled constraint whatever its sense;transmission_expansion_cost_limit2 vs 1+1 — one block per sense — ==, <=, >= — where PyPSA writes one row per labelled constraint whatever its sense; size ✔ 184 rows · ≠ 73 vs 76 columns · ✔ 328 nonzeros; duals ✔ 184 rows, 4 negated; model for model: 57 blocks equal, 2 documented splits, 4 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
12 | 12 |
Generator-e_sum_max |
1 | 1 |
Generator-e_sum_min |
1 | 1 |
Generator-ext-p-lower |
8 | 8 |
Generator-ext-p-upper |
8 | 8 |
Generator-ext-p_nom-lower |
2 | 2 |
Generator-ext-p_nom-upper |
2 | 2 |
Generator-fix-p-lower |
12 | 12 |
Generator-fix-p-upper |
12 | 12 |
Generator-p-ramp_limit_down |
3 | 3 |
Generator-p-ramp_limit_up |
3 | 3 |
Generator-p_nom_set |
1 | 1 |
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 |
4 | 4 |
Link-fix-p-upper |
4 | 4 |
Link-p-ramp_limit_down |
3 | 3 |
Link-p-ramp_limit_up |
3 | 3 |
Link-p_nom_set |
1 | 1 |
StorageUnit-energy_balance |
8 | 8 |
StorageUnit-ext-p_dispatch-lower |
4 | 4 |
StorageUnit-ext-p_dispatch-upper |
4 | 4 |
StorageUnit-ext-p_nom-lower |
1 | 1 |
StorageUnit-ext-p_nom-upper |
1 | 1 |
StorageUnit-ext-p_store-lower |
4 | 4 |
StorageUnit-ext-p_store-upper |
4 | 4 |
StorageUnit-ext-state_of_charge-lower |
4 | 4 |
StorageUnit-ext-state_of_charge-upper |
4 | 4 |
StorageUnit-fix-p_dispatch-lower |
4 | 4 |
StorageUnit-fix-p_dispatch-upper |
4 | 4 |
StorageUnit-fix-p_store-lower |
4 | 4 |
StorageUnit-fix-p_store-upper |
4 | 4 |
StorageUnit-fix-state_of_charge-lower |
4 | 4 |
StorageUnit-fix-state_of_charge-upper |
4 | 4 |
StorageUnit-p_nom_set |
1 | 1 |
Store-e_nom_set |
1 | 1 |
Store-energy_balance |
8 | 8 |
Store-ext-e-lower |
4 | 4 |
Store-ext-e-upper |
4 | 4 |
Store-ext-e_nom-lower |
1 | 1 |
Store-ext-e_nom-upper |
1 | 1 |
Store-fix-e-lower |
4 | 4 |
Store-fix-e-upper |
4 | 4 |
tech_capacity_expansion_limit |
5 | ≠ 1+2+2 |
transmission_expansion_cost_limit |
2 | ≠ 1+1 |
transmission_volume_expansion_limit |
1 | 1 |
| column | PyPSA | specsolve |
|---|---|---|
CVaR |
0 | ≠ 1 |
CVaR-a |
0 | ≠ 1 |
CVaR-theta |
0 | ≠ 1 |
Generator-p |
20 | 20 |
Generator-p_nom |
2 | 2 |
Link-p |
8 | 8 |
Link-p_nom |
1 | 1 |
StorageUnit-p_dispatch |
8 | 8 |
StorageUnit-p_nom |
1 | 1 |
StorageUnit-p_store |
8 | 8 |
StorageUnit-state_of_charge |
8 | 8 |
Store-e |
8 | 8 |
Store-e_nom |
1 | 1 |
Store-p |
8 | 8 |
The model¶
The same model, as math
A plain n.optimize(), and its multi-period and stochastic classes, in one file. Every second-stage quantity spans a scenario (a future dispatch is chosen in) and every asset stands in the investment periods its build year and lifetime span. A parameter spans scenario exactly when PyPSA reads it per scenario. Capacity is chosen once, before the future is known, and paid once per active period at its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights, with a share priced at the tail through the CVaR rows, which stand only where that share is positive. A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses to the standard one. A security-constrained run copies each branch flow limit once per outage in an outage set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight, and the outage factors are data prep.
Sets¶
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N},\ \mathrm{StorageUnit\_bus}: \mathcal{S} \to \mathcal{N},\ \mathrm{Store\_bus}: \mathcal{V} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{O}\) | index \(o\) — link_output with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep |
| \(\mathcal{D}\) | index \(d\) — load with \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus |
| \(\mathcal{S}\) | index \(s\) — storage_unit with \(\mathrm{StorageUnit\_bus}: \mathcal{S} \to \mathcal{N}\) — storage units, dispatch and store behind one bus connection |
| \(\mathcal{V}\) | index \(v\) — store with \(\mathrm{Store\_bus}: \mathcal{V} \to \mathcal{N}\) — pure energy stores, each on one bus |
| \(\mathcal{B}\) | index \(b\) — global_constraint — PyPSA's GlobalConstraint rows, one label per declared limit |
| \(\mathcal{Y}\) | 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 |
| \(\mathrm{ru}\) | Generator_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most a generator may raise its output between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{rd}\) | Generator_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most a generator may lower its output between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{ru}^{\mathrm{up}}\) | Generator_ramp_limit_start_up over \(\Xi \times \mathcal{G}\) — most output in the snapshot a unit starts, per unit of nominal power |
| \(\mathrm{rd}^{\mathrm{dn}}\) | Generator_ramp_limit_shut_down over \(\Xi \times \mathcal{G}\) — most output in the snapshot before a unit stops, per unit of nominal power |
| \(\mathrm{u}^{0}\) | Generator_status_initial over \(\Xi \times \mathcal{G}\) — one where the unit was on before the first snapshot, zero where off — PyPSA's up_time_before > 0, data prep |
| \(\mathrm{p}^{0}\) | Generator_p_init over \(\Xi \times \mathcal{G}\) — the output a unit brought into the horizon — PyPSA's p_init, read only where the unit came in running; no value means it is unknown, so the unit carries no ramp row at the first snapshot |
| \(\mathrm{p}^{\mathrm{mod}}\) | Generator_p_nom_mod over \(\mathcal{G}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{ru}^{f}\) | Link_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most a link may raise its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{rd}^{f}\) | Link_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most a link may lower its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{f}^{\mathrm{nom}}\) | Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power |
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{f}}\) | Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways |
| \(\overline{\mathrm{f}}\) | Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power |
| \(\eta\) | Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py: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 |
| \(\mathrm{ru}^{f,\mathrm{up}}\) | Link_ramp_limit_start_up over \(\Xi \times \mathcal{L}\) — most flow in the snapshot a link starts, per unit of nominal power |
| \(\mathrm{rd}^{f,\mathrm{dn}}\) | Link_ramp_limit_shut_down over \(\Xi \times \mathcal{L}\) — most flow in the snapshot before a link stops, per unit of nominal power |
| \(\mathrm{u}^{f,0}\) | Link_status_initial over \(\Xi \times \mathcal{L}\) — one where the link was on before the first snapshot, zero where off — PyPSA's up_time_before > 0, data prep |
| \(\mathrm{f}^{0}\) | Link_p_init over \(\Xi \times \mathcal{L}\) — the flow a link brought into the horizon — PyPSA's p_init, read only where the link came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot |
| \(\mathrm{f}^{\mathrm{mod}}\) | Link_p_nom_mod over \(\mathcal{L}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{load}\) | 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 |
| \(\mathrm{on}^{h}\) | StorageUnit_active over \(\mathcal{T} \times \mathcal{S}\) — whether a storage unit stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{on}^{e}\) | Store_active over \(\mathcal{T} \times \mathcal{V}\) — whether a store 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 |
| \(\mathrm{W}^{h}\) | StorageUnit_capital_weight over \(\mathcal{S}\) — the sum of period weights a storage unit stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{W}^{e}\) | Store_capital_weight over \(\mathcal{V}\) — the sum of period weights a store stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{w}^{\mathrm{sto}}\) | snapshot_weightings_stores over \(\mathcal{T}\) — PyPSA's snapshot_weightings.stores — hours a snapshot stands for in a storage balance |
| \(\mathrm{w}^{\mathrm{gen}}\) | snapshot_weightings_generators over \(\mathcal{T}\) — PyPSA's snapshot_weightings.generators — hours a snapshot stands for in an energy total |
| \(\underline{\mathrm{p}}^{\mathrm{nom}}\) | 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 |
| \(\mathrm{p}^{\mathrm{nom,set}}\) | Generator_p_nom_set over \(\Xi \times \mathcal{G}\) — a given nominal power for an extendable generator; one without a value has no row here |
| \(\underline{\mathrm{E}}\) | Generator_e_sum_min over \(\Xi \times \mathcal{G}\) — least energy over the horizon; minus infinity where no floor is meant |
| \(\overline{\mathrm{E}}\) | Generator_e_sum_max over \(\Xi \times \mathcal{G}\) — most energy over the horizon — a fuel or emission budget in energy terms; infinity where no cap is meant |
| \(\underline{\mathrm{f}}^{\mathrm{nom}}\) | 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 |
| \(\mathrm{f}^{\mathrm{nom,set}}\) | Link_p_nom_set over \(\Xi \times \mathcal{L}\) — a given nominal power for an extendable link; one without a value has no row here |
| \(\underline{\mathrm{h}}^{\mathrm{nom}}\) | StorageUnit_p_nom_min over \(\Xi \times \mathcal{S}\) — least nominal power an extendable storage unit may be built at |
| \(\overline{\mathrm{h}}^{\mathrm{nom}}\) | StorageUnit_p_nom_max over \(\Xi \times \mathcal{S}\) — most nominal power an extendable storage unit may be built at |
| \(\mathrm{c}^{\mathrm{cap},h}\) | StorageUnit_capital_cost over \(\Xi \times \mathcal{S}\) — cost of one unit of nominal power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\mathrm{h}^{\mathrm{nom,set}}\) | StorageUnit_p_nom_set over \(\Xi \times \mathcal{S}\) — a given nominal power for an extendable storage unit; one without a value has no row here |
| \(\underline{\mathrm{e}}^{\mathrm{nom}}\) | Store_e_nom_min over \(\Xi \times \mathcal{V}\) — least nominal capacity an extendable store may be built at |
| \(\overline{\mathrm{e}}^{\mathrm{nom}}\) | Store_e_nom_max over \(\Xi \times \mathcal{V}\) — most nominal capacity an extendable store may be built at |
| \(\mathrm{c}^{\mathrm{cap},e}\) | Store_capital_cost over \(\Xi \times \mathcal{V}\) — cost of one unit of nominal capacity — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\mathrm{e}^{\mathrm{nom,set}}\) | Store_e_nom_set over \(\Xi \times \mathcal{V}\) — a given nominal capacity for an extendable store; one without a value has no row here |
| \(\mathrm{h}^{\mathrm{nom}}\) | StorageUnit_p_nom over \(\Xi \times \mathcal{S}\) — nominal power |
| \(\mathrm{ext}^{h}\) | StorageUnit_p_nom_extendable over \(\mathcal{S}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{h}}\) | StorageUnit_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — most storing, per unit of nominal power and negated |
| \(\overline{\mathrm{h}}\) | StorageUnit_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — most dispatch, per unit of nominal power |
| \(\mathrm{T}^{h}\) | StorageUnit_max_hours over \(\Xi \times \mathcal{S}\) — energy capacity, as hours of dispatch at nominal power |
| \(\eta^{-}\) | StorageUnit_efficiency_store over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — share of the power drawn from the bus that becomes charge |
| \(\eta^{+}\) | StorageUnit_efficiency_dispatch over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — share of the charge drawn down that reaches the bus |
| \(\mathrm{sgn}^{h}\) | StorageUnit_sign over \(\mathcal{S}\) — the sign net dispatch enters its bus's balance with — PyPSA's sign, 1 unless given. PyPSA refuses one that differs by scenario (consistency.py:1187) |
| \(\rho\) | StorageUnit_retention over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — share of charge kept over a snapshot — PyPSA's (1 - standing_loss) ** elapsed hours, data prep |
| \(\mathrm{inflow}\) | StorageUnit_inflow over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — energy arriving per hour, a river into a reservoir |
| \(\mathrm{soc}^{0}\) | StorageUnit_state_of_charge_initial over \(\Xi \times \mathcal{S}\) — charge held before the first snapshot |
| \(\mathrm{cyc}\) | StorageUnit_cyclic_state_of_charge over \(\Xi \times \mathcal{S}\) — whether the horizon closes on itself instead of opening on the initial charge |
| \(\mathrm{cyc}^{y}\) | StorageUnit_cyclic_state_of_charge_per_period over \(\Xi \times \mathcal{S}\) — whether each investment period closes on itself instead of carrying its charge on to the next; it overrides cyclic_state_of_charge and state_of_charge_initial_per_period. PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise |
| \(\mathrm{reset}\) | StorageUnit_state_of_charge_initial_per_period over \(\Xi \times \mathcal{S}\) — whether each investment period opens on the initial charge instead of carrying the previous period's; PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise |
| \(\mathrm{open}\) | StorageUnit_opens_late over \(\mathcal{T} \times \mathcal{S}\) — whether a snapshot is the first a storage unit stands in, where that is not the first of the horizon — PyPSA's active.cumsum() == 1 over the snapshots it stands in, past the first snapshot, data prep; false in a run where every unit stands throughout |
| \(\mathrm{idle}\) | StorageUnit_inactive_snapshots over \(\mathcal{S}\) — how many snapshots a storage unit does not stand in — PyPSA's (~active).sum(), data prep. A cyclic unit reaches back this many snapshots further, so it closes on the last snapshot it stands in |
| \(\mathrm{c}^{h}\) | StorageUnit_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — cost of one unit of dispatch |
| \(\mathrm{c}^{h,(2)}\) | StorageUnit_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — cost of the square of one unit of dispatch; storing is not charged |
| \(\mathrm{c}^{\mathrm{soc}}\) | StorageUnit_marginal_cost_storage over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — cost of one unit of charge held over one snapshot |
| \(\mathrm{e}^{\mathrm{nom}}\) | Store_e_nom over \(\Xi \times \mathcal{V}\) — nominal energy capacity |
| \(\mathrm{ext}^{e}\) | Store_e_nom_extendable over \(\mathcal{V}\) — whether the nominal energy capacity is a decision |
| \(\underline{\mathrm{e}}\) | Store_e_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — least energy held, per unit of nominal capacity — negative for a store that may go short |
| \(\overline{\mathrm{e}}\) | Store_e_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — most energy held, per unit of nominal capacity |
| \(\mathrm{sgn}^{q}\) | Store_sign over \(\mathcal{V}\) — the sign the power a store delivers enters its bus's balance with — PyPSA's sign, 1 unless given. PyPSA refuses one that differs by scenario (consistency.py:1187) |
| \(\rho^{e}\) | Store_retention over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — share of energy kept over a snapshot — PyPSA's (1 - standing_loss) ** elapsed hours, data prep |
| \(\mathrm{e}^{0}\) | Store_e_initial over \(\Xi \times \mathcal{V}\) — energy held before the first snapshot |
| \(\mathrm{cyc}^{e}\) | Store_e_cyclic over \(\Xi \times \mathcal{V}\) — whether the horizon closes on itself instead of opening on the initial energy |
| \(\mathrm{cyc}^{e,y}\) | Store_e_cyclic_per_period over \(\Xi \times \mathcal{V}\) — whether each investment period closes on itself instead of carrying its energy on to the next; it overrides e_cyclic and e_initial_per_period. PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise |
| \(\mathrm{reset}^{e}\) | Store_e_initial_per_period over \(\Xi \times \mathcal{V}\) — whether each investment period opens on the initial energy instead of carrying the previous period's; PyPSA reads it only under multi_investment_periods, so data prep feeds false otherwise |
| \(\mathrm{open}^{e}\) | Store_opens_late over \(\mathcal{T} \times \mathcal{V}\) — whether a snapshot is the first a store stands in, where that is not the first of the horizon — PyPSA's active.cumsum() == 1 over the snapshots it stands in, past the first snapshot, data prep; false in a run where every store stands throughout |
| \(\mathrm{idle}^{e}\) | Store_inactive_snapshots over \(\mathcal{V}\) — how many snapshots a store does not stand in — PyPSA's (~active).sum(), data prep. A cyclic store reaches back this many snapshots further, so it closes on the last snapshot it stands in |
| \(\mathrm{c}^{q}\) | Store_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — cost of one unit of power delivered |
| \(\mathrm{c}^{q,(2)}\) | Store_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — cost of the square of the net power delivered, so charging costs as much as delivering |
| \(\mathrm{c}^{e}\) | Store_marginal_cost_storage over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — cost of one unit of energy held over one snapshot |
| \(\mathrm{type}\) | GlobalConstraint_type over \(\mathcal{B}\) — which formula the row takes — primary_energy, operational_limit, transmission_volume_expansion_limit, transmission_expansion_cost_limit or tech_capacity_expansion_limit |
| \(\mathrm{sense}\) | GlobalConstraint_sense over \(\Xi \times \mathcal{B}\) — which way the row binds in each scenario — <=, >= or ==; PyPSA reads a row's sense per scenario (global_constraints.py:556, :748, :860) |
| \(\mathrm{K}\) | GlobalConstraint_constant over \(\Xi \times \mathcal{B}\) — the constant the total is held against; what a variable cannot carry — an initial charge, times its period's years for each counted period where the storage reopens per period, or a non-extendable build — is folded in here by data prep. PyPSA reads it per scenario (global_constraints.py:557, :749, :861) |
| \(\mathrm{len}^{f}\) | Link_volume_weight over \(\Xi \times \mathcal{B} \times \mathcal{L}\) — the link's length where its carrier is in the row's set, the first scenario's length as PyPSA reads it (global_constraints.py:835-836) — data prep; a link outside it, or one that does not stand in the row's investment_period, has no row |
| \(\mathrm{cc}^{f}\) | Link_expansion_cost_weight over \(\Xi \times \mathcal{B} \times \mathcal{L}\) — the link's capital cost where its carrier is in the row's set, times the objective weights of the periods it stands in where the row names no investment_period under multi_investment_periods — data prep; a link outside the set, or one that does not stand in the row's period, has no row |
| \(\mathrm{m}\) | Generator_tech_capacity_weight over \(\mathcal{B} \times \mathcal{G}\) — one where the generator is in the row's carrier-and-bus set — data prep; one outside it, or one that does not stand in the row's investment_period, has no row |
| \(\mathrm{m}^{f}\) | Link_tech_capacity_weight over \(\mathcal{B} \times \mathcal{L}\) — one where the link is in the row's carrier-and-bus set — data prep; one outside it, or one that does not stand in the row's investment_period, has no row |
| \(\mathrm{m}^{h}\) | StorageUnit_tech_capacity_weight over \(\mathcal{B} \times \mathcal{S}\) — one where the storage unit is in the row's carrier-and-bus set — data prep; one outside it, or one that does not stand in the row's investment_period, has no row |
| \(\mathrm{m}^{e}\) | Store_tech_capacity_weight over \(\mathcal{B} \times \mathcal{V}\) — one where the store is in the row's carrier-and-bus set — data prep; one outside it, or one that does not stand in the row's investment_period, has no row |
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 |
| \(h^{+}\) | StorageUnit_p_dispatch over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — StorageUnit-p_dispatch — power delivered to the bus |
| \(h^{-}\) | StorageUnit_p_store over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — StorageUnit-p_store — power drawn from the bus into charge |
| \(\mathit{soc}\) | StorageUnit_state_of_charge over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — StorageUnit-state_of_charge — energy held at the end of a snapshot |
| \(e\) | Store_e over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — Store-e — energy held at the end of a snapshot |
| \(q\) | Store_p over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — Store-p — power delivered to the bus; charging is negative |
| \(P\) | 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 |
| \(H\) | StorageUnit_p_nom_ext over \(\mathcal{S}\) — StorageUnit-p_nom — nominal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
| \(E\) | Store_e_nom_ext over \(\mathcal{V}\) — Store-e_nom — nominal capacity 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 |
| \(u\) | Generator_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-status — how much of a committable unit is on: an integer the rows below cap at one, or at the module count where the build is modular |
| \(u^{f}\) | Link_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-status — how much of a committable link is on: an integer the rows below cap at one, or at the module count where the build is modular |
Definitions¶
| Symbol | Meaning |
|---|---|
| \(\mathit{Generator\_previous\_p}\) | Generator_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the output a generator carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that |
| \(\mathit{Generator\_ramp\_up\_allowance}\) | Generator_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — how far a generator may raise output between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\mathit{Generator\_ramp\_down\_allowance}\) | Generator_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — how far a generator may lower output between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off |
| \(\mathit{Link\_previous\_p}\) | Link_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the flow a link carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that |
| \(\mathit{Link\_ramp\_up\_allowance}\) | Link_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may raise flow between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\mathit{Link\_ramp\_down\_allowance}\) | Link_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may lower flow between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off |
| \(\mathit{StorageUnit\_charge\_carried\_in}\) | StorageUnit_charge_carried_in over \(\Xi \times \mathcal{T} \times \mathcal{S}\) — the charge a unit opens a snapshot with — at the first snapshot it stands in, its last such snapshot's less standing loss where it is cyclic and the given initial charge, which no standing loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A unit built in a later period opens in that period, and a cyclic one that retires closes on its own last snapshot. Per period, the same holds with each investment period as the horizon |
| \(\mathit{Store\_energy\_carried\_in}\) | Store_energy_carried_in over \(\Xi \times \mathcal{T} \times \mathcal{V}\) — the energy a store opens a snapshot with — at the first snapshot it stands in, its last such snapshot's less standing loss where it is cyclic and the given initial energy, which no standing loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A store built in a later period opens in that period, and a cyclic one that retires closes on its own last snapshot. Per period, the same holds with each investment period as the horizon |
| \(\mathit{transmission\_volume\_expansion}\) | transmission_volume_expansion over \(\Xi \times \mathcal{B}\) — what a transmission_volume_expansion_limit row totals — length times the chosen build of the row's branches |
| \(\mathit{transmission\_expansion\_cost}\) | transmission_expansion_cost over \(\Xi \times \mathcal{B}\) — what a transmission_expansion_cost_limit row totals — capital cost times the chosen build of the row's branches |
| \(\mathit{tech\_capacity\_expansion}\) | tech_capacity_expansion over \(\mathcal{B}\) — what a tech_capacity_expansion_limit row totals — the chosen build of the row's carrier-and-bus set |
| \(\mathit{total\_cost}\) | 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 |
| \(\mathit{Generator\_previous\_status}\) | Generator_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the commitment state a generator carries into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that |
| \(\mathit{Generator\_p\_nom\_effective}\) | Generator_p_nom_effective over \(\Xi \times \mathcal{G}\) — the build a generator's limits are taken against — the chosen one where it is extendable, the given one otherwise |
| \(\mathrm{Generator\_ramp\_up\_rate}\) | Generator_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the ramp limit a unit's up row reads — PyPSA's ramp_limit_up, or the full build where it has none, since a start-up ramp alone builds the row |
| \(\mathrm{Generator\_ramp\_down\_rate}\) | Generator_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the ramp limit a unit's down row reads — PyPSA's ramp_limit_down, or the full build where it has none, since a shut-down ramp alone builds the row |
| \(\mathrm{Generator\_start\_up\_rate}\) | Generator_start_up_rate over \(\Xi \times \mathcal{G}\) — the start-up ramp a unit's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\mathrm{Generator\_shut\_down\_rate}\) | Generator_shut_down_rate over \(\Xi \times \mathcal{G}\) — the shut-down ramp a unit's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\mathrm{Generator\_p\_nom\_committed}\) | Generator_p_nom_committed over \(\Xi \times \mathcal{G}\) — the build a committed unit's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise |
| \(\mathit{Link\_p\_nom\_effective}\) | Link_p_nom_effective over \(\Xi \times \mathcal{L}\) — the build a link's limits are taken against — the chosen one where it is extendable, the given one otherwise |
| \(\mathit{Link\_previous\_status}\) | Link_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the commitment state a link carries into a snapshot — the state it brought into the horizon at the first, the previous snapshot's after that |
| \(\mathrm{Link\_ramp\_up\_rate}\) | Link_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's up row reads — PyPSA's ramp_limit_up, or the full build where it has none, since a start-up ramp alone builds the row |
| \(\mathrm{Link\_ramp\_down\_rate}\) | Link_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's down row reads — PyPSA's ramp_limit_down, or the full build where it has none, since a shut-down ramp alone builds the row |
| \(\mathrm{Link\_start\_up\_rate}\) | Link_start_up_rate over \(\Xi \times \mathcal{L}\) — the start-up ramp a link's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\mathrm{Link\_shut\_down\_rate}\) | Link_shut_down_rate over \(\Xi \times \mathcal{L}\) — the shut-down ramp a link's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\mathrm{Link\_p\_nom\_committed}\) | Link_p_nom_committed over \(\Xi \times \mathcal{L}\) — the build a committed link's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise |
| \(\mathit{Link\_transmission\_volume\_expansion}\) | Link_transmission_volume_expansion over \(\Xi \times \mathcal{B}\) |
| \(\mathit{Link\_transmission\_expansion\_cost}\) | Link_transmission_expansion_cost over \(\Xi \times \mathcal{B}\) |
| \(\mathit{Generator\_tech\_capacity\_expansion}\) | Generator_tech_capacity_expansion over \(\mathcal{B}\) |
| \(\mathit{Link\_tech\_capacity\_expansion}\) | Link_tech_capacity_expansion over \(\mathcal{B}\) |
| \(\mathit{StorageUnit\_tech\_capacity\_expansion}\) | StorageUnit_tech_capacity_expansion over \(\mathcal{B}\) |
| \(\mathit{Store\_tech\_capacity\_expansion}\) | Store_tech_capacity_expansion over \(\mathcal{B}\) |
| \(\mathit{Generator\_capex}\) | Generator_capex (scalar) |
| \(\mathit{Link\_capex}\) | Link_capex (scalar) |
| \(\mathit{StorageUnit\_capex}\) | StorageUnit_capex (scalar) |
| \(\mathit{Store\_capex}\) | Store_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 \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{StorageUnit\_injection}\) | StorageUnit_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Store\_injection}\) | Store_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 |
| \(\mathit{scenario\_opex}\) | 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 \(\Xi\) |
| \(\mathit{StorageUnit\_opex}\) | StorageUnit_opex over \(\Xi\) |
| \(\mathit{Store\_opex}\) | Store_opex over \(\Xi\) |
\(t \ominus k\) denotes cyclic translation: index \(t-k\) taken modulo the size of the dimension (roll). Plain \(t-k\) (shift) has no wraparound — terms translated past the edge are simply absent.
\(t \boxminus_{v} k\) denotes translation with \(v\) standing where index \(t-k\) leaves the dimension (shift(edge=v)), so the row at that boundary is built and carries \(v\) rather than being dropped.
\(t \ominus^{\mathrm{relation}(t)} k\) denotes a translation counted inside the group a relation puts \(t\) in (shift(by=relation)), so a term never crosses out of its own group. The two modifiers take different slots — the group above, the fill below — so \(t \boxminus_{v}^{\mathrm{relation}(t)} k\) is both at once.
\(\mathrm{pos}(t)\) denotes where index \(t\) sits along its dimension's own order — the order shift steps along, not the order labels sort in — counted from \(0\). The index itself stays the coordinate, so \(t\) compares against labels and \(\mathrm{pos}(t)\) against positions.
\(\mathrm{pos}_{\mathrm{relation}(t)}(t)\) counts within the group a relation puts \(t\) in: the subscript names the map, \(\mathcal{T}_{\mathrm{relation}(t)}\) is the group it lands in, and that group has a first position of its own.
Objective¶
\[
\min \mathit{total\_cost}
\]
Subject to¶
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Link_fix_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_fix_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Generator_ext_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot P_{g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot P_{g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
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
\[
P_{g} \le \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \text{ is defined}
\]
Generator_p_nom_set
\[
P_{g} = \mathrm{p}^{\mathrm{nom,set}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{nom,set}}_{\xi,g} \text{ is defined}
\]
Generator_e_sum_min
\[
\sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathrm{w}^{\mathrm{gen}}_{t} \ge \underline{\mathrm{E}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \underline{\mathrm{E}}_{\xi,g} \text{ is defined}
\]
Generator_e_sum_max
\[
\sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathrm{w}^{\mathrm{gen}}_{t} \le \overline{\mathrm{E}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \overline{\mathrm{E}}_{\xi,g} \text{ is defined}
\]
Link_ext_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot F_{l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot F_{l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
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
\[
F_{l} \le \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \text{ is defined}
\]
Link_p_nom_set
\[
F_{l} = \mathrm{f}^{\mathrm{nom,set}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{nom,set}}_{\xi,l} \text{ is defined}
\]
StorageUnit_fix_p_dispatch_lower
\[
h^{+}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_fix_p_dispatch_upper
\[
h^{+}_{\xi,t,s} \le \overline{\mathrm{h}}_{\xi,t,s} \cdot \mathrm{h}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_fix_p_store_lower
\[
h^{-}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_fix_p_store_upper
\[
h^{-}_{\xi,t,s} \le -\underline{\mathrm{h}}_{\xi,t,s} \cdot \mathrm{h}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_fix_state_of_charge_lower
\[
\mathit{soc}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_fix_state_of_charge_upper
\[
\mathit{soc}_{\xi,t,s} \le \mathrm{T}^{h}_{\xi,s} \cdot \mathrm{h}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \neg \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
Generator_p_ramp_limit_up
\[
p_{\xi,t,g} - \mathit{Generator\_previous\_p}_{\xi,t,g} \le \mathit{Generator\_ramp\_up\_allowance}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \left( \mathrm{ru}_{\xi,t,g} \text{ is defined} \vee \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Generator_p_ramp_limit_down
\[
\mathit{Generator\_previous\_p}_{\xi,t,g} - p_{\xi,t,g} \le \mathit{Generator\_ramp\_down\_allowance}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \left( \mathrm{rd}_{\xi,t,g} \text{ is defined} \vee \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Link_p_ramp_limit_up
\[
f_{\xi,t,l} - \mathit{Link\_previous\_p}_{\xi,t,l} \le \mathit{Link\_ramp\_up\_allowance}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_down
\[
\mathit{Link\_previous\_p}_{\xi,t,l} - f_{\xi,t,l} \le \mathit{Link\_ramp\_down\_allowance}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
StorageUnit_ext_p_dispatch_lower
\[
h^{+}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_ext_p_dispatch_upper
\[
h^{+}_{\xi,t,s} \le \overline{\mathrm{h}}_{\xi,t,s} \cdot H_{s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_ext_p_store_lower
\[
h^{-}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_ext_p_store_upper
\[
h^{-}_{\xi,t,s} \le -\underline{\mathrm{h}}_{\xi,t,s} \cdot H_{s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_ext_state_of_charge_lower
\[
\mathit{soc}_{\xi,t,s} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_ext_state_of_charge_upper
\[
\mathit{soc}_{\xi,t,s} \le \mathrm{T}^{h}_{\xi,s} \cdot H_{s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{on}^{h}_{t,s}
\]
StorageUnit_ext_p_nom_lower
\[
H_{s} \ge \underline{\mathrm{h}}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s}
\]
StorageUnit_ext_p_nom_upper
\[
H_{s} \le \overline{\mathrm{h}}^{\mathrm{nom}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \overline{\mathrm{h}}^{\mathrm{nom}}_{\xi,s} \text{ is defined}
\]
StorageUnit_p_nom_set
\[
H_{s} = \mathrm{h}^{\mathrm{nom,set}}_{\xi,s} \qquad \forall\, \xi \in \Xi,\ s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s} \wedge \mathrm{h}^{\mathrm{nom,set}}_{\xi,s} \text{ is defined}
\]
StorageUnit_energy_balance
\[
\mathit{soc}_{\xi,t,s} = \mathit{StorageUnit\_charge\_carried\_in}_{\xi,t,s} + \eta^{-}_{\xi,t,s} \cdot h^{-}_{\xi,t,s} \cdot \mathrm{w}^{\mathrm{sto}}_{t} - \frac{h^{+}_{\xi,t,s} \cdot \mathrm{w}^{\mathrm{sto}}_{t}}{\eta^{+}_{\xi,t,s}} + \mathrm{inflow}_{\xi,t,s} \cdot \mathrm{w}^{\mathrm{sto}}_{t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s}
\]
Store_fix_e_lower
\[
e_{\xi,t,v} \ge \underline{\mathrm{e}}_{\xi,t,v} \cdot \mathrm{e}^{\mathrm{nom}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \neg \mathrm{ext}^{e}_{v} \wedge \mathrm{on}^{e}_{t,v}
\]
Store_fix_e_upper
\[
e_{\xi,t,v} \le \overline{\mathrm{e}}_{\xi,t,v} \cdot \mathrm{e}^{\mathrm{nom}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \neg \mathrm{ext}^{e}_{v} \wedge \mathrm{on}^{e}_{t,v}
\]
Store_ext_e_lower
\[
e_{\xi,t,v} \ge \underline{\mathrm{e}}_{\xi,t,v} \cdot E_{v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{ext}^{e}_{v} \wedge \mathrm{on}^{e}_{t,v}
\]
Store_ext_e_upper
\[
e_{\xi,t,v} \le \overline{\mathrm{e}}_{\xi,t,v} \cdot E_{v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{ext}^{e}_{v} \wedge \mathrm{on}^{e}_{t,v}
\]
Store_ext_e_nom_lower
\[
E_{v} \ge \underline{\mathrm{e}}^{\mathrm{nom}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ v \in \mathcal{V} \,:\, \mathrm{ext}^{e}_{v}
\]
Store_ext_e_nom_upper
\[
E_{v} \le \overline{\mathrm{e}}^{\mathrm{nom}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ v \in \mathcal{V} \,:\, \mathrm{ext}^{e}_{v} \wedge \overline{\mathrm{e}}^{\mathrm{nom}}_{\xi,v} \text{ is defined}
\]
Store_e_nom_set
\[
E_{v} = \mathrm{e}^{\mathrm{nom,set}}_{\xi,v} \qquad \forall\, \xi \in \Xi,\ v \in \mathcal{V} \,:\, \mathrm{ext}^{e}_{v} \wedge \mathrm{e}^{\mathrm{nom,set}}_{\xi,v} \text{ is defined}
\]
Store_energy_balance
\[
e_{\xi,t,v} = \mathit{Store\_energy\_carried\_in}_{\xi,t,v} - q_{\xi,t,v} \cdot \mathrm{w}^{\mathrm{sto}}_{t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{on}^{e}_{t,v}
\]
GlobalConstraint_transmission_volume_expansion_limit_ub
\[
\mathit{transmission\_volume\_expansion}_{\xi,b} \le \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_volume\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{<=}\text{'}
\]
GlobalConstraint_transmission_expansion_cost_limit_lb
\[
\mathit{transmission\_expansion\_cost}_{\xi,b} \ge \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_expansion\_cost\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{>=}\text{'}
\]
GlobalConstraint_transmission_expansion_cost_limit_eq
\[
\mathit{transmission\_expansion\_cost}_{\xi,b} = \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_expansion\_cost\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{==}\text{'}
\]
GlobalConstraint_tech_capacity_expansion_limit_ub
\[
\mathit{tech\_capacity\_expansion}_{b} \le \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{tech\_capacity\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{<=}\text{'}
\]
GlobalConstraint_tech_capacity_expansion_limit_lb
\[
\mathit{tech\_capacity\_expansion}_{b} \ge \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{tech\_capacity\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{>=}\text{'}
\]
GlobalConstraint_tech_capacity_expansion_limit_eq
\[
\mathit{tech\_capacity\_expansion}_{b} = \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{tech\_capacity\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{==}\text{'}
\]
Bus_nodal_balance
\[
\mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Definitions¶
Generator_previous_p
\[
\mathit{Generator\_previous\_p}_{\xi,t,g} = \begin{cases} \mathrm{u}^{0}_{\xi,g} \cdot \mathrm{p}^{0}_{\xi,g} & \text{if } \mathrm{pos}(t) = 0 \\ p_{\xi,t - 1,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_ramp_up_allowance
\[
\mathit{Generator\_ramp\_up\_allowance}_{\xi,t,g} = \begin{cases} \mathrm{Generator\_ramp\_up\_rate}_{\xi,t,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot \mathit{Generator\_previous\_status}_{\xi,t,g} + \mathrm{Generator\_start\_up\_rate}_{\xi,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot \left( u_{\xi,t,g} - \mathit{Generator\_previous\_status}_{\xi,t,g} \right) & \text{if } \mathrm{com}_{g} \\ \mathrm{Generator\_ramp\_up\_rate}_{\xi,t,g} \cdot \mathit{Generator\_p\_nom\_effective}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_ramp_down_allowance
\[
\mathit{Generator\_ramp\_down\_allowance}_{\xi,t,g} = \begin{cases} \mathrm{Generator\_ramp\_down\_rate}_{\xi,t,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot u_{\xi,t,g} + \mathrm{Generator\_shut\_down\_rate}_{\xi,g} \cdot \mathrm{Generator\_p\_nom\_committed}_{\xi,g} \cdot \left( \mathit{Generator\_previous\_status}_{\xi,t,g} - u_{\xi,t,g} \right) & \text{if } \mathrm{com}_{g} \\ \mathrm{Generator\_ramp\_down\_rate}_{\xi,t,g} \cdot \mathit{Generator\_p\_nom\_effective}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Link_previous_p
\[
\mathit{Link\_previous\_p}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} \cdot \mathrm{f}^{0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ f_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_up_allowance
\[
\mathit{Link\_ramp\_up\_allowance}_{\xi,t,l} = \begin{cases} \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \mathit{Link\_previous\_status}_{\xi,t,l} + \mathrm{Link\_start\_up\_rate}_{\xi,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \left( u^{f}_{\xi,t,l} - \mathit{Link\_previous\_status}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} \cdot \mathit{Link\_p\_nom\_effective}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_down_allowance
\[
\mathit{Link\_ramp\_down\_allowance}_{\xi,t,l} = \begin{cases} \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot u^{f}_{\xi,t,l} + \mathrm{Link\_shut\_down\_rate}_{\xi,l} \cdot \mathrm{Link\_p\_nom\_committed}_{\xi,l} \cdot \left( \mathit{Link\_previous\_status}_{\xi,t,l} - u^{f}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} \cdot \mathit{Link\_p\_nom\_effective}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
StorageUnit_charge_carried_in
\[
\mathit{StorageUnit\_charge\_carried\_in}_{\xi,t,s} = \begin{cases} \rho_{\xi,t,s} \cdot \mathit{soc}_{\xi,\left( t \ominus \mathrm{idle} \right) \ominus 1,s} & \text{if } \mathrm{cyc}_{\xi,s} \wedge \neg \mathrm{cyc}^{y}_{\xi,s} \wedge \neg \mathrm{reset}_{\xi,s} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}_{t,s} \right) \\ \mathrm{soc}^{0}_{\xi,s} & \text{if } \neg \mathrm{cyc}_{\xi,s} \wedge \neg \mathrm{cyc}^{y}_{\xi,s} \wedge \neg \mathrm{reset}_{\xi,s} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}_{t,s} \right) \\ \rho_{\xi,t,s} \cdot \mathit{soc}_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} 1,s} & \text{if } \mathrm{cyc}^{y}_{\xi,s} \\ \mathrm{soc}^{0}_{\xi,s} & \text{if } \mathrm{reset}_{\xi,s} \wedge \neg \mathrm{cyc}^{y}_{\xi,s} \wedge \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) = 0 \\ \rho_{\xi,t,s} \cdot \mathit{soc}_{\xi,t - 1,s} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S}
\]
Store_energy_carried_in
\[
\mathit{Store\_energy\_carried\_in}_{\xi,t,v} = \begin{cases} \rho^{e}_{\xi,t,v} \cdot e_{\xi,\left( t \ominus \mathrm{idle}^{e} \right) \ominus 1,v} & \text{if } \mathrm{cyc}^{e}_{\xi,v} \wedge \neg \mathrm{cyc}^{e,y}_{\xi,v} \wedge \neg \mathrm{reset}^{e}_{\xi,v} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}^{e}_{t,v} \right) \\ \mathrm{e}^{0}_{\xi,v} & \text{if } \neg \mathrm{cyc}^{e}_{\xi,v} \wedge \neg \mathrm{cyc}^{e,y}_{\xi,v} \wedge \neg \mathrm{reset}^{e}_{\xi,v} \wedge \left( \mathrm{pos}(t) = 0 \vee \mathrm{open}^{e}_{t,v} \right) \\ \rho^{e}_{\xi,t,v} \cdot e_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} 1,v} & \text{if } \mathrm{cyc}^{e,y}_{\xi,v} \\ \mathrm{e}^{0}_{\xi,v} & \text{if } \mathrm{reset}^{e}_{\xi,v} \wedge \neg \mathrm{cyc}^{e,y}_{\xi,v} \wedge \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) = 0 \\ \rho^{e}_{\xi,t,v} \cdot e_{\xi,t - 1,v} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V}
\]
transmission_volume_expansion
\[
\mathit{transmission\_volume\_expansion}_{\xi,b} = \mathit{Link\_transmission\_volume\_expansion}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
transmission_expansion_cost
\[
\mathit{transmission\_expansion\_cost}_{\xi,b} = \mathit{Link\_transmission\_expansion\_cost}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
tech_capacity_expansion
\[
\mathit{tech\_capacity\_expansion}_{b} = \mathit{Generator\_tech\_capacity\_expansion}_{b} + \mathit{Link\_tech\_capacity\_expansion}_{b} + \mathit{StorageUnit\_tech\_capacity\_expansion}_{b} + \mathit{Store\_tech\_capacity\_expansion}_{b} \qquad \forall\, b \in \mathcal{B}
\]
total_cost
\[
\mathit{total\_cost} = \mathit{Generator\_capex} + \mathit{Link\_capex} + \mathit{StorageUnit\_capex} + \mathit{Store\_capex} + \mathit{risk\_weighted\_opex}
\]
Bus_injection
\[
\mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{\xi,t,n} + \mathit{Link\_injection}_{\xi,t,n} + \mathrm{Load\_injection}_{\xi,t,n} + \mathit{StorageUnit\_injection}_{\xi,t,n} + \mathit{Store\_injection}_{\xi,t,n} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Generator_previous_status
\[
\mathit{Generator\_previous\_status}_{\xi,t,g} = \begin{cases} \mathrm{u}^{0}_{\xi,g} & \text{if } \mathrm{pos}(t) = 0 \\ u_{\xi,t - 1,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_p_nom_effective
\[
\mathit{Generator\_p\_nom\_effective}_{\xi,g} = \begin{cases} P_{g} & \text{if } \mathrm{ext}_{g} \\ \mathrm{p}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_ramp_up_rate
\[
\mathrm{Generator\_ramp\_up\_rate}_{\xi,t,g} = \begin{cases} \mathrm{ru}_{\xi,t,g} & \text{if } \mathrm{ru}_{\xi,t,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_ramp_down_rate
\[
\mathrm{Generator\_ramp\_down\_rate}_{\xi,t,g} = \begin{cases} \mathrm{rd}_{\xi,t,g} & \text{if } \mathrm{rd}_{\xi,t,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_start_up_rate
\[
\mathrm{Generator\_start\_up\_rate}_{\xi,g} = \begin{cases} \mathrm{ru}^{\mathrm{up}}_{\xi,g} & \text{if } \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_shut_down_rate
\[
\mathrm{Generator\_shut\_down\_rate}_{\xi,g} = \begin{cases} \mathrm{rd}^{\mathrm{dn}}_{\xi,g} & \text{if } \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_p_nom_committed
\[
\mathrm{Generator\_p\_nom\_committed}_{\xi,g} = \begin{cases} \mathrm{p}^{\mathrm{mod}}_{g} & \text{if } \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \\ \mathrm{p}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Link_p_nom_effective
\[
\mathit{Link\_p\_nom\_effective}_{\xi,l} = \begin{cases} F_{l} & \text{if } \mathrm{ext}^{f}_{l} \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_previous_status
\[
\mathit{Link\_previous\_status}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ u^{f}_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_up_rate
\[
\mathrm{Link\_ramp\_up\_rate}_{\xi,t,l} = \begin{cases} \mathrm{ru}^{f}_{\xi,t,l} & \text{if } \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_down_rate
\[
\mathrm{Link\_ramp\_down\_rate}_{\xi,t,l} = \begin{cases} \mathrm{rd}^{f}_{\xi,t,l} & \text{if } \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_start_up_rate
\[
\mathrm{Link\_start\_up\_rate}_{\xi,l} = \begin{cases} \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} & \text{if } \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_shut_down_rate
\[
\mathrm{Link\_shut\_down\_rate}_{\xi,l} = \begin{cases} \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} & \text{if } \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_p_nom_committed
\[
\mathrm{Link\_p\_nom\_committed}_{\xi,l} = \begin{cases} \mathrm{f}^{\mathrm{mod}}_{l} & \text{if } \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_transmission_volume_expansion
\[
\mathit{Link\_transmission\_volume\_expansion}_{\xi,b} = \sum_{l \in \mathcal{L}} F_{l} \cdot \mathrm{len}^{f}_{\xi,b,l} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
Link_transmission_expansion_cost
\[
\mathit{Link\_transmission\_expansion\_cost}_{\xi,b} = \sum_{l \in \mathcal{L}} F_{l} \cdot \mathrm{cc}^{f}_{\xi,b,l} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
Generator_tech_capacity_expansion
\[
\mathit{Generator\_tech\_capacity\_expansion}_{b} = \sum_{g \in \mathcal{G}} P_{g} \cdot \mathrm{m}_{b,g} \qquad \forall\, b \in \mathcal{B}
\]
Link_tech_capacity_expansion
\[
\mathit{Link\_tech\_capacity\_expansion}_{b} = \sum_{l \in \mathcal{L}} F_{l} \cdot \mathrm{m}^{f}_{b,l} \qquad \forall\, b \in \mathcal{B}
\]
StorageUnit_tech_capacity_expansion
\[
\mathit{StorageUnit\_tech\_capacity\_expansion}_{b} = \sum_{s \in \mathcal{S}} H_{s} \cdot \mathrm{m}^{h}_{b,s} \qquad \forall\, b \in \mathcal{B}
\]
Store_tech_capacity_expansion
\[
\mathit{Store\_tech\_capacity\_expansion}_{b} = \sum_{v \in \mathcal{V}} E_{v} \cdot \mathrm{m}^{e}_{b,v} \qquad \forall\, b \in \mathcal{B}
\]
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
\[
\mathit{Link\_capex} = \sum_{\xi \in \Xi,\ l \in \mathcal{L}} \pi_{\xi} \cdot F_{l} \cdot \mathrm{c}^{\mathrm{cap},f}_{\xi,l} \cdot \mathrm{W}^{f}_{l}
\]
StorageUnit_capex
\[
\mathit{StorageUnit\_capex} = \sum_{\xi \in \Xi,\ s \in \mathcal{S}} \pi_{\xi} \cdot H_{s} \cdot \mathrm{c}^{\mathrm{cap},h}_{\xi,s} \cdot \mathrm{W}^{h}_{s}
\]
Store_capex
\[
\mathit{Store\_capex} = \sum_{\xi \in \Xi,\ v \in \mathcal{V}} \pi_{\xi} \cdot E_{v} \cdot \mathrm{c}^{\mathrm{cap},e}_{\xi,v} \cdot \mathrm{W}^{e}_{v}
\]
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
\[
\mathrm{Load\_injection}_{\xi,t,n} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
StorageUnit_injection
\[
\mathit{StorageUnit\_injection}_{\xi,t,n} = \sum_{s \in \mathcal{S} \,:\, \mathrm{StorageUnit\_bus}(s) = n} \mathrm{sgn}^{h}_{s} \cdot \left( h^{+}_{\xi,t,s} - h^{-}_{\xi,t,s} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Store_injection
\[
\mathit{Store\_injection}_{\xi,t,n} = \sum_{v \in \mathcal{V} \,:\, \mathrm{Store\_bus}(v) = n} \mathrm{sgn}^{q}_{v} \cdot q_{\xi,t,v} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Link_output_arrival
\[
\mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{\xi,o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
scenario_opex
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} + \mathit{StorageUnit\_opex}_{\xi} + \mathit{Store\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
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
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
StorageUnit_opex
\[
\mathit{StorageUnit\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{s \in \mathcal{S}} h^{+}_{\xi,t,s} \cdot \mathrm{c}^{h}_{\xi,t,s} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{s \in \mathcal{S}} h^{+}_{\xi,t,s} \cdot h^{+}_{\xi,t,s} \cdot \mathrm{c}^{h,(2)}_{\xi,t,s} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{s \in \mathcal{S}} \mathit{soc}_{\xi,t,s} \cdot \mathrm{c}^{\mathrm{soc}}_{\xi,t,s} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Store_opex
\[
\mathit{Store\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{v \in \mathcal{V}} q_{\xi,t,v} \cdot \mathrm{c}^{q}_{\xi,t,v} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{v \in \mathcal{V}} q_{\xi,t,v} \cdot q_{\xi,t,v} \cdot \mathrm{c}^{q,(2)}_{\xi,t,v} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{v \in \mathcal{V}} e_{\xi,t,v} \cdot \mathrm{c}^{e}_{\xi,t,v} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
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
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
StorageUnit_p_dispatch
\[
h^{+}_{\xi,t,s} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s}
\]
StorageUnit_p_store
\[
h^{-}_{\xi,t,s} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s}
\]
StorageUnit_state_of_charge
\[
\mathit{soc}_{\xi,t,s} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ s \in \mathcal{S} \,:\, \mathrm{on}^{h}_{t,s}
\]
Store_e
\[
e_{\xi,t,v} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{on}^{e}_{t,v}
\]
Store_p
\[
q_{\xi,t,v} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ v \in \mathcal{V} \,:\, \mathrm{on}^{e}_{t,v}
\]
Generator_p_nom_ext
\[
P_{g} \in \mathbb{R} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g}
\]
Link_p_nom_ext
\[
F_{l} \in \mathbb{R} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l}
\]
StorageUnit_p_nom_ext
\[
H_{s} \in \mathbb{R} \qquad \forall\, s \in \mathcal{S} \,:\, \mathrm{ext}^{h}_{s}
\]
Store_e_nom_ext
\[
E_{v} \in \mathbb{R} \qquad \forall\, v \in \mathcal{V} \,:\, \mathrm{ext}^{e}_{v}
\]
CVaR_a
\[
a_{\xi} \ge 0 \qquad \forall\, \xi \in \Xi
\]
CVaR_theta
\[
\theta \in \mathbb{R}
\]
CVaR
\[
CVaR \in \mathbb{R}
\]
Generator_status
\[
u_{\xi,t,g} \ge 0, u_{\xi,t,g} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Link_status
\[
u^{f}_{\xi,t,l} \ge 0, u^{f}_{\xi,t,l} \in \mathbb{Z} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
The spec, differential/pypsa/rungs/rung_03_expansion.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'}
load: {description: 'demands, each on one bus'}
storage_unit: {description: 'storage units, dispatch and store behind one bus connection'}
store: {description: 'pure energy stores, each on one bus'}
global_constraint: {description: 'PyPSA''s `GlobalConstraint` rows, one label per declared limit'}
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}
Load_bus: {description: the bus a load sits on, key: load, values: bus}
StorageUnit_bus: {description: the bus a storage unit sits on, key: storage_unit, values: bus}
Store_bus: {description: the bus a store sits on, key: store, values: bus}
parameters:
snapshot_weightings_objective:
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
dims: [snapshot]
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
for a unit that draws power. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
dims: [generator]
Generator_committable:
description: whether output is gated by an on/off status decision
dims: [generator]
dtype: bool
Generator_ramp_limit_up:
description: most a generator may raise its output between snapshots, per unit of nominal power; no
value means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, generator]
Generator_ramp_limit_down:
description: most a generator may lower its output between snapshots, per unit of nominal power; no
value means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, generator]
Generator_ramp_limit_start_up:
description: most output in the snapshot a unit starts, per unit of nominal power
dims: [scenario, generator]
Generator_ramp_limit_shut_down:
description: most output in the snapshot before a unit stops, per unit of nominal power
dims: [scenario, generator]
Generator_status_initial:
description: one where the unit was on before the first snapshot, zero where off — PyPSA's `up_time_before
> 0`, data prep
dims: [scenario, generator]
dtype: int
Generator_p_init:
description: the output a unit brought into the horizon — PyPSA's `p_init`, read only where the unit
came in running; no value means it is unknown, so the unit carries no ramp row at the first snapshot
dims: [scenario, generator]
Generator_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [generator]
Link_ramp_limit_up:
description: most a link may raise its flow between snapshots, per unit of nominal power; no value
means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, link]
Link_ramp_limit_down:
description: most a link may lower its flow between snapshots, per unit of nominal power; no value
means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, link]
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_ramp_limit_start_up:
description: most flow in the snapshot a link starts, per unit of nominal power
dims: [scenario, link]
Link_ramp_limit_shut_down:
description: most flow in the snapshot before a link stops, per unit of nominal power
dims: [scenario, link]
Link_status_initial:
description: one where the link was on before the first snapshot, zero where off — PyPSA's `up_time_before
> 0`, data prep
dims: [scenario, link]
dtype: int
Link_p_init:
description: the flow a link brought into the horizon — PyPSA's `p_init`, read only where the link
came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot
dims: [scenario, link]
Link_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [link]
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
StorageUnit_active:
description: whether a storage unit stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, storage_unit]
dtype: bool
Store_active:
description: whether a store stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, store]
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]
StorageUnit_capital_weight:
description: the sum of period weights a storage unit stands in — PyPSA's `active * period_weighting`,
summed, data prep
dims: [storage_unit]
Store_capital_weight:
description: the sum of period weights a store stands in — PyPSA's `active * period_weighting`, summed,
data prep
dims: [store]
snapshot_weightings_stores:
description: PyPSA's `snapshot_weightings.stores` — hours a snapshot stands for in a storage balance
dims: [snapshot]
snapshot_weightings_generators:
description: PyPSA's `snapshot_weightings.generators` — hours a snapshot stands for in an energy total
dims: [snapshot]
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]
Generator_p_nom_set:
description: a given nominal power for an extendable generator; one without a value has no row here
dims: [scenario, generator]
Generator_e_sum_min:
description: least energy over the horizon; minus infinity where no floor is meant
dims: [scenario, generator]
Generator_e_sum_max:
description: most energy over the horizon — a fuel or emission budget in energy terms; infinity where
no cap is meant
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]
Link_p_nom_set:
description: a given nominal power for an extendable link; one without a value has no row here
dims: [scenario, link]
StorageUnit_p_nom_min:
description: least nominal power an extendable storage unit may be built at
dims: [scenario, storage_unit]
StorageUnit_p_nom_max:
description: most nominal power an extendable storage unit may be built at
dims: [scenario, storage_unit]
StorageUnit_capital_cost:
description: cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity
in data prep
dims: [scenario, storage_unit]
StorageUnit_p_nom_set:
description: a given nominal power for an extendable storage unit; one without a value has no row
here
dims: [scenario, storage_unit]
Store_e_nom_min:
description: least nominal capacity an extendable store may be built at
dims: [scenario, store]
Store_e_nom_max:
description: most nominal capacity an extendable store may be built at
dims: [scenario, store]
Store_capital_cost:
description: cost of one unit of nominal capacity — PyPSA's `capital_cost`, periodized as an annuity
in data prep
dims: [scenario, store]
Store_e_nom_set:
description: a given nominal capacity for an extendable store; one without a value has no row here
dims: [scenario, store]
StorageUnit_p_nom:
description: nominal power
dims: [scenario, storage_unit]
StorageUnit_p_nom_extendable:
description: whether the nominal power is a decision
dims: [storage_unit]
dtype: bool
StorageUnit_p_min_pu:
description: most storing, per unit of nominal power and negated
dims: [scenario, snapshot, storage_unit]
StorageUnit_p_max_pu:
description: most dispatch, per unit of nominal power
dims: [scenario, snapshot, storage_unit]
StorageUnit_max_hours:
description: energy capacity, as hours of dispatch at nominal power
dims: [scenario, storage_unit]
StorageUnit_efficiency_store:
description: share of the power drawn from the bus that becomes charge
dims: [scenario, snapshot, storage_unit]
StorageUnit_efficiency_dispatch:
description: share of the charge drawn down that reaches the bus
dims: [scenario, snapshot, storage_unit]
StorageUnit_sign:
description: the sign net dispatch enters its bus's balance with — PyPSA's `sign`, `1` unless given.
PyPSA refuses one that differs by scenario (`consistency.py:1187`)
dims: [storage_unit]
StorageUnit_retention:
description: share of charge kept over a snapshot — PyPSA's `(1 - standing_loss) ** elapsed hours`,
data prep
dims: [scenario, snapshot, storage_unit]
StorageUnit_inflow:
description: energy arriving per hour, a river into a reservoir
dims: [scenario, snapshot, storage_unit]
StorageUnit_state_of_charge_initial:
description: charge held before the first snapshot
dims: [scenario, storage_unit]
StorageUnit_cyclic_state_of_charge:
description: whether the horizon closes on itself instead of opening on the initial charge
dims: [scenario, storage_unit]
dtype: bool
StorageUnit_cyclic_state_of_charge_per_period:
description: whether each investment period closes on itself instead of carrying its charge on to
the next; it overrides `cyclic_state_of_charge` and `state_of_charge_initial_per_period`. PyPSA
reads it only under `multi_investment_periods`, so data prep feeds false otherwise
dims: [scenario, storage_unit]
dtype: bool
StorageUnit_state_of_charge_initial_per_period:
description: whether each investment period opens on the initial charge instead of carrying the previous
period's; PyPSA reads it only under `multi_investment_periods`, so data prep feeds false otherwise
dims: [scenario, storage_unit]
dtype: bool
StorageUnit_opens_late:
description: whether a snapshot is the first a storage unit stands in, where that is not the first
of the horizon — PyPSA's `active.cumsum() == 1` over the snapshots it stands in, past the first
snapshot, data prep; false in a run where every unit stands throughout
dims: [snapshot, storage_unit]
dtype: bool
StorageUnit_inactive_snapshots:
description: how many snapshots a storage unit does not stand in — PyPSA's `(~active).sum()`, data
prep. A cyclic unit reaches back this many snapshots further, so it closes on the last snapshot
it stands in
dims: [storage_unit]
dtype: int
StorageUnit_marginal_cost:
description: cost of one unit of dispatch
dims: [scenario, snapshot, storage_unit]
StorageUnit_marginal_cost_quadratic:
description: cost of the square of one unit of dispatch; storing is not charged
dims: [scenario, snapshot, storage_unit]
StorageUnit_marginal_cost_storage:
description: cost of one unit of charge held over one snapshot
dims: [scenario, snapshot, storage_unit]
Store_e_nom:
description: nominal energy capacity
dims: [scenario, store]
Store_e_nom_extendable:
description: whether the nominal energy capacity is a decision
dims: [store]
dtype: bool
Store_e_min_pu:
description: least energy held, per unit of nominal capacity — negative for a store that may go short
dims: [scenario, snapshot, store]
Store_e_max_pu:
description: most energy held, per unit of nominal capacity
dims: [scenario, snapshot, store]
Store_sign:
description: the sign the power a store delivers enters its bus's balance with — PyPSA's `sign`, `1`
unless given. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
dims: [store]
Store_retention:
description: share of energy kept over a snapshot — PyPSA's `(1 - standing_loss) ** elapsed hours`,
data prep
dims: [scenario, snapshot, store]
Store_e_initial:
description: energy held before the first snapshot
dims: [scenario, store]
Store_e_cyclic:
description: whether the horizon closes on itself instead of opening on the initial energy
dims: [scenario, store]
dtype: bool
Store_e_cyclic_per_period:
description: whether each investment period closes on itself instead of carrying its energy on to
the next; it overrides `e_cyclic` and `e_initial_per_period`. PyPSA reads it only under `multi_investment_periods`,
so data prep feeds false otherwise
dims: [scenario, store]
dtype: bool
Store_e_initial_per_period:
description: whether each investment period opens on the initial energy instead of carrying the previous
period's; PyPSA reads it only under `multi_investment_periods`, so data prep feeds false otherwise
dims: [scenario, store]
dtype: bool
Store_opens_late:
description: whether a snapshot is the first a store stands in, where that is not the first of the
horizon — PyPSA's `active.cumsum() == 1` over the snapshots it stands in, past the first snapshot,
data prep; false in a run where every store stands throughout
dims: [snapshot, store]
dtype: bool
Store_inactive_snapshots:
description: how many snapshots a store does not stand in — PyPSA's `(~active).sum()`, data prep.
A cyclic store reaches back this many snapshots further, so it closes on the last snapshot it stands
in
dims: [store]
dtype: int
Store_marginal_cost:
description: cost of one unit of power delivered
dims: [scenario, snapshot, store]
Store_marginal_cost_quadratic:
description: cost of the square of the net power delivered, so charging costs as much as delivering
dims: [scenario, snapshot, store]
Store_marginal_cost_storage:
description: cost of one unit of energy held over one snapshot
dims: [scenario, snapshot, store]
GlobalConstraint_type:
description: which formula the row takes — `primary_energy`, `operational_limit`, `transmission_volume_expansion_limit`,
`transmission_expansion_cost_limit` or `tech_capacity_expansion_limit`
dims: [global_constraint]
dtype: str
GlobalConstraint_sense:
description: which way the row binds in each scenario — `<=`, `>=` or `==`; PyPSA reads a row's sense
per scenario (`global_constraints.py:556`, `:748`, `:860`)
dims: [scenario, global_constraint]
dtype: str
GlobalConstraint_constant:
description: the constant the total is held against; what a variable cannot carry — an initial charge,
times its period's years for each counted period where the storage reopens per period, or a non-extendable
build — is folded in here by data prep. PyPSA reads it per scenario (`global_constraints.py:557`,
`:749`, `:861`)
dims: [scenario, global_constraint]
Link_volume_weight:
description: the link's length where its carrier is in the row's set, the first scenario's length
as PyPSA reads it (`global_constraints.py:835-836`) — data prep; a link outside it, or one that
does not stand in the row's `investment_period`, has no row
dims: [scenario, global_constraint, link]
Link_expansion_cost_weight:
description: the link's capital cost where its carrier is in the row's set, times the objective weights
of the periods it stands in where the row names no `investment_period` under `multi_investment_periods`
— data prep; a link outside the set, or one that does not stand in the row's period, has no row
dims: [scenario, global_constraint, link]
Generator_tech_capacity_weight:
description: one where the generator is in the row's carrier-and-bus set — data prep; one outside
it, or one that does not stand in the row's `investment_period`, has no row
dims: [global_constraint, generator]
Link_tech_capacity_weight:
description: one where the link is in the row's carrier-and-bus set — data prep; one outside it, or
one that does not stand in the row's `investment_period`, has no row
dims: [global_constraint, link]
StorageUnit_tech_capacity_weight:
description: one where the storage unit is in the row's carrier-and-bus set — data prep; one outside
it, or one that does not stand in the row's `investment_period`, has no row
dims: [global_constraint, storage_unit]
Store_tech_capacity_weight:
description: one where the store is in the row's carrier-and-bus set — data prep; one outside it,
or one that does not stand in the row's `investment_period`, has no row
dims: [global_constraint, store]
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
StorageUnit_p_dispatch:
description: '`StorageUnit-p_dispatch` — power delivered to the bus'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_active
StorageUnit_p_store:
description: '`StorageUnit-p_store` — power drawn from the bus into charge'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_active
StorageUnit_state_of_charge:
description: '`StorageUnit-state_of_charge` — energy held at the end of a snapshot'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_active
Store_e:
description: '`Store-e` — energy held at the end of a snapshot'
dims: [scenario, snapshot, store]
where: Store_active
Store_p:
description: '`Store-p` — power delivered to the bus; charging is negative'
dims: [scenario, snapshot, store]
where: Store_active
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
StorageUnit_p_nom_ext:
description: '`StorageUnit-p_nom` — nominal power where it is a decision; the parameter of the same
PyPSA name carries the fixed regime'
dims: [storage_unit]
where: StorageUnit_p_nom_extendable
Store_e_nom_ext:
description: '`Store-e_nom` — nominal capacity where it is a decision; the parameter of the same PyPSA
name carries the fixed regime'
dims: [store]
where: Store_e_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: []
Generator_status:
description: '`Generator-status` — how much of a committable unit is on: an integer the rows below
cap at one, or at the module count where the build is modular'
dims: [scenario, snapshot, generator]
where: Generator_committable AND Generator_active
domain: integer
bounds: {lower: 0}
Link_status:
description: '`Link-status` — how much of a committable link is on: an integer the rows below cap
at one, or at the module count where the build is modular'
dims: [scenario, snapshot, link]
where: Link_committable AND Link_active
domain: integer
bounds: {lower: 0}
constraints:
Generator_fix_p_lower:
description: '`Generator-fix-p-lower` — a fixed generator outputs at least its minimum'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * Generator_p_nom
Generator_fix_p_upper:
description: '`Generator-fix-p-upper` — a fixed generator outputs at most what is available'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * Generator_p_nom
Link_fix_p_lower:
description: '`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other
way'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * Link_p_nom
Link_fix_p_upper:
description: '`Link-fix-p-upper` — a fixed link carries at most its nominal power'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * Link_p_nom
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_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_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
Generator_p_nom_set:
description: '`Generator-p_nom_set` — the chosen build pinned, wherever a value is given'
dims: [scenario, generator]
where: Generator_p_nom_extendable AND Generator_p_nom_set
expression: Generator_p_nom_ext == Generator_p_nom_set
Generator_e_sum_min:
description: '`Generator-e_sum_min` — energy over the horizon is at least its floor; a floor of minus
infinity is no row'
dims: [scenario, generator]
where: Generator_e_sum_min
expression: sum(Generator_p * snapshot_weightings_generators, over=snapshot) >= Generator_e_sum_min
Generator_e_sum_max:
description: '`Generator-e_sum_max` — energy over the horizon is at most its budget; a budget of infinity
is no row'
dims: [scenario, generator]
where: Generator_e_sum_max
expression: sum(Generator_p * snapshot_weightings_generators, over=snapshot) <= Generator_e_sum_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_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_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
Link_p_nom_set:
description: '`Link-p_nom_set` — the chosen build pinned, wherever a value is given'
dims: [scenario, link]
where: Link_p_nom_extendable AND Link_p_nom_set
expression: Link_p_nom_ext == Link_p_nom_set
StorageUnit_fix_p_dispatch_lower:
description: '`StorageUnit-fix-p_dispatch-lower` — dispatch is non-negative'
dims: [scenario, snapshot, storage_unit]
where: not StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_dispatch >= 0
StorageUnit_fix_p_dispatch_upper:
description: '`StorageUnit-fix-p_dispatch-upper` — a fixed unit dispatches at most its nominal power'
dims: [scenario, snapshot, storage_unit]
where: not StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_dispatch <= StorageUnit_p_max_pu * StorageUnit_p_nom
StorageUnit_fix_p_store_lower:
description: '`StorageUnit-fix-p_store-lower` — storing is non-negative'
dims: [scenario, snapshot, storage_unit]
where: not StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_store >= 0
StorageUnit_fix_p_store_upper:
description: '`StorageUnit-fix-p_store-upper` — a fixed unit stores at most its nominal power, the
minimum-per-unit column carrying that cap negated'
dims: [scenario, snapshot, storage_unit]
where: not StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_store <= -StorageUnit_p_min_pu * StorageUnit_p_nom
StorageUnit_fix_state_of_charge_lower:
description: '`StorageUnit-fix-state_of_charge-lower` — charge is non-negative'
dims: [scenario, snapshot, storage_unit]
where: not StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_state_of_charge >= 0
StorageUnit_fix_state_of_charge_upper:
description: '`StorageUnit-fix-state_of_charge-upper` — a fixed unit holds at most its hours at nominal
power'
dims: [scenario, snapshot, storage_unit]
where: not StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_state_of_charge <= StorageUnit_max_hours * StorageUnit_p_nom
Generator_p_ramp_limit_up:
description: '`Generator-p-ramp_limit_up` — a generator raises output no faster than its ramp limit
of the build, and a committed one no further than its start-up ramp in the snapshot it turns on.
A unit that came into the horizon running carries a row at the first snapshot only where its `p_init`
gives the output it brought in, and no unit carries one at the start of a later investment period
— nor does any unit a big M releases instead'
dims: [scenario, snapshot, generator]
where: (Generator_ramp_limit_up OR Generator_ramp_limit_start_up) AND NOT (Generator_committable AND
Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period,
within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: Generator_p - Generator_previous_p <= Generator_ramp_up_allowance
Generator_p_ramp_limit_down:
description: '`Generator-p-ramp_limit_down` — a generator lowers output no faster than its ramp limit
of the build, and a committed one no further than its shut-down ramp in the snapshot it turns off.
A unit that came into the horizon running carries a row at the first snapshot only where its `p_init`
gives the output it brought in, and no unit carries one at the start of a later investment period
— nor does any unit a big M releases instead'
dims: [scenario, snapshot, generator]
where: (Generator_ramp_limit_down OR Generator_ramp_limit_shut_down) AND NOT (Generator_committable
AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period,
within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: Generator_previous_p - Generator_p <= Generator_ramp_down_allowance
Link_p_ramp_limit_up:
description: '`Link-p-ramp_limit_up` — a link raises flow no faster than its ramp limit of the build,
and a committed one no further than its start-up ramp in the snapshot it turns on. A link that came
into the horizon running carries a row at the first snapshot only where its `p_init` gives the flow
it brought in, and no link carries one at the start of a later investment period — nor does any
link a big M releases instead'
dims: [scenario, snapshot, link]
where: (Link_ramp_limit_up OR Link_ramp_limit_start_up) AND NOT (Link_committable AND Link_p_nom_extendable
AND NOT (Link_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period, within=period) > 0 OR
(position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init))) AND Link_active
expression: Link_p - Link_previous_p <= Link_ramp_up_allowance
Link_p_ramp_limit_down:
description: '`Link-p-ramp_limit_down` — a link lowers flow no faster than its ramp limit of the build,
and a committed one no further than its shut-down ramp in the snapshot it turns off. A link that
came into the horizon running carries a row at the first snapshot only where its `p_init` gives
the flow it brought in, and no link carries one at the start of a later investment period — nor
does any link a big M releases instead'
dims: [scenario, snapshot, link]
where: (Link_ramp_limit_down OR Link_ramp_limit_shut_down) AND NOT (Link_committable AND Link_p_nom_extendable
AND NOT (Link_p_nom_mod > 0)) AND (position(snapshot, by=snapshot_period, within=period) > 0 OR
(position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init))) AND Link_active
expression: Link_previous_p - Link_p <= Link_ramp_down_allowance
StorageUnit_ext_p_dispatch_lower:
description: '`StorageUnit-ext-p_dispatch-lower` — dispatch is non-negative'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_dispatch >= 0
StorageUnit_ext_p_dispatch_upper:
description: '`StorageUnit-ext-p_dispatch-upper` — an extendable unit dispatches at most the chosen
build'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_dispatch <= StorageUnit_p_max_pu * StorageUnit_p_nom_ext
StorageUnit_ext_p_store_lower:
description: '`StorageUnit-ext-p_store-lower` — storing is non-negative'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_store >= 0
StorageUnit_ext_p_store_upper:
description: '`StorageUnit-ext-p_store-upper` — an extendable unit stores at most the chosen build,
the minimum-per-unit column carrying that cap negated'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_p_store <= -StorageUnit_p_min_pu * StorageUnit_p_nom_ext
StorageUnit_ext_state_of_charge_lower:
description: '`StorageUnit-ext-state_of_charge-lower` — charge is non-negative'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_state_of_charge >= 0
StorageUnit_ext_state_of_charge_upper:
description: '`StorageUnit-ext-state_of_charge-upper` — an extendable unit holds at most its hours
at the chosen build'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_active
expression: StorageUnit_state_of_charge <= StorageUnit_max_hours * StorageUnit_p_nom_ext
StorageUnit_ext_p_nom_lower:
description: '`StorageUnit-ext-p_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, storage_unit]
where: StorageUnit_p_nom_extendable
expression: StorageUnit_p_nom_ext >= StorageUnit_p_nom_min
StorageUnit_ext_p_nom_upper:
description: '`StorageUnit-ext-p_nom-upper` — the chosen build is at most its cap in every scenario;
a cap of infinity is no row'
dims: [scenario, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_p_nom_max
expression: StorageUnit_p_nom_ext <= StorageUnit_p_nom_max
StorageUnit_p_nom_set:
description: '`StorageUnit-p_nom_set` — the chosen build pinned, wherever a value is given'
dims: [scenario, storage_unit]
where: StorageUnit_p_nom_extendable AND StorageUnit_p_nom_set
expression: StorageUnit_p_nom_ext == StorageUnit_p_nom_set
StorageUnit_energy_balance:
description: '`StorageUnit-energy_balance` — the charge carried in, plus what is stored after its
efficiency, less what dispatch draws down before its own, plus inflow not spilled'
dims: [scenario, snapshot, storage_unit]
where: StorageUnit_active
expression: StorageUnit_state_of_charge == ((StorageUnit_charge_carried_in + ((StorageUnit_efficiency_store
* StorageUnit_p_store) * snapshot_weightings_stores)) - ((StorageUnit_p_dispatch * snapshot_weightings_stores)
/ StorageUnit_efficiency_dispatch)) + (StorageUnit_inflow * snapshot_weightings_stores)
Store_fix_e_lower:
description: '`Store-fix-e-lower` — a fixed store holds at least its floor'
dims: [scenario, snapshot, store]
where: not Store_e_nom_extendable AND Store_active
expression: Store_e >= Store_e_min_pu * Store_e_nom
Store_fix_e_upper:
description: '`Store-fix-e-upper` — a fixed store holds at most its nominal capacity'
dims: [scenario, snapshot, store]
where: not Store_e_nom_extendable AND Store_active
expression: Store_e <= Store_e_max_pu * Store_e_nom
Store_ext_e_lower:
description: '`Store-ext-e-lower` — an extendable store holds at least its floor of the chosen build'
dims: [scenario, snapshot, store]
where: Store_e_nom_extendable AND Store_active
expression: Store_e >= Store_e_min_pu * Store_e_nom_ext
Store_ext_e_upper:
description: '`Store-ext-e-upper` — an extendable store holds at most the chosen build'
dims: [scenario, snapshot, store]
where: Store_e_nom_extendable AND Store_active
expression: Store_e <= Store_e_max_pu * Store_e_nom_ext
Store_ext_e_nom_lower:
description: '`Store-ext-e_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, store]
where: Store_e_nom_extendable
expression: Store_e_nom_ext >= Store_e_nom_min
Store_ext_e_nom_upper:
description: '`Store-ext-e_nom-upper` — the chosen build is at most its cap in every scenario; a cap
of infinity is no row'
dims: [scenario, store]
where: Store_e_nom_extendable AND Store_e_nom_max
expression: Store_e_nom_ext <= Store_e_nom_max
Store_e_nom_set:
description: '`Store-e_nom_set` — the chosen build pinned, wherever a value is given'
dims: [scenario, store]
where: Store_e_nom_extendable AND Store_e_nom_set
expression: Store_e_nom_ext == Store_e_nom_set
Store_energy_balance:
description: '`Store-energy_balance` — the energy carried in, less what is delivered to the bus'
dims: [scenario, snapshot, store]
where: Store_active
expression: Store_e == Store_energy_carried_in - Store_p * snapshot_weightings_stores
GlobalConstraint_transmission_volume_expansion_limit_ub:
description: '`transmission_volume_expansion_limit` — its total, at most its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'transmission_volume_expansion_limit' AND GlobalConstraint_sense ==
'<='
expression: transmission_volume_expansion <= GlobalConstraint_constant
GlobalConstraint_transmission_expansion_cost_limit_lb:
description: '`transmission_expansion_cost_limit` — its total, at least its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'transmission_expansion_cost_limit' AND GlobalConstraint_sense ==
'>='
expression: transmission_expansion_cost >= GlobalConstraint_constant
GlobalConstraint_transmission_expansion_cost_limit_eq:
description: '`transmission_expansion_cost_limit` — its total, at its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'transmission_expansion_cost_limit' AND GlobalConstraint_sense ==
'=='
expression: transmission_expansion_cost == GlobalConstraint_constant
GlobalConstraint_tech_capacity_expansion_limit_ub:
description: '`tech_capacity_expansion_limit` — its total, at most its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'tech_capacity_expansion_limit' AND GlobalConstraint_sense == '<='
expression: tech_capacity_expansion <= GlobalConstraint_constant
GlobalConstraint_tech_capacity_expansion_limit_lb:
description: '`tech_capacity_expansion_limit` — its total, at least its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'tech_capacity_expansion_limit' AND GlobalConstraint_sense == '>='
expression: tech_capacity_expansion >= GlobalConstraint_constant
GlobalConstraint_tech_capacity_expansion_limit_eq:
description: '`tech_capacity_expansion_limit` — its total, at its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'tech_capacity_expansion_limit' AND GlobalConstraint_sense == '=='
expression: tech_capacity_expansion == GlobalConstraint_constant
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:
Generator_previous_p:
description: the output a generator carries into a snapshot — at the first, the `p_init` it brought
in where it came in running and nothing where it came in off; the previous snapshot's after that
dims: [scenario, snapshot, generator]
cases:
opening: {when: position(snapshot) == 0, expression: Generator_status_initial * Generator_p_init}
otherwise: shift(Generator_p, along=snapshot, offset=1)
Generator_ramp_up_allowance:
description: how far a generator may raise output between two snapshots — its ramp limit of the build
while it stays on, plus its start-up ramp in the snapshot it turns on
dims: [scenario, snapshot, generator]
cases:
committed: {when: Generator_committable, expression: Generator_ramp_up_rate * Generator_p_nom_committed
* Generator_previous_status + Generator_start_up_rate * Generator_p_nom_committed * (Generator_status
- Generator_previous_status)}
otherwise: Generator_ramp_up_rate * Generator_p_nom_effective
Generator_ramp_down_allowance:
description: how far a generator may lower output between two snapshots — its ramp limit of the build
while it stays on, plus its shut-down ramp in the snapshot it turns off
dims: [scenario, snapshot, generator]
cases:
committed: {when: Generator_committable, expression: Generator_ramp_down_rate * Generator_p_nom_committed
* Generator_status + Generator_shut_down_rate * Generator_p_nom_committed * (Generator_previous_status
- Generator_status)}
otherwise: Generator_ramp_down_rate * Generator_p_nom_effective
Link_previous_p:
description: the flow a link carries into a snapshot — at the first, the `p_init` it brought in where
it came in running and nothing where it came in off; the previous snapshot's after that
dims: [scenario, snapshot, link]
cases:
opening: {when: position(snapshot) == 0, expression: Link_status_initial * Link_p_init}
otherwise: shift(Link_p, along=snapshot, offset=1)
Link_ramp_up_allowance:
description: how far a link may raise flow between two snapshots — its ramp limit of the build while
it stays on, plus its start-up ramp in the snapshot it turns on
dims: [scenario, snapshot, link]
cases:
committed: {when: Link_committable, expression: Link_ramp_up_rate * Link_p_nom_committed * Link_previous_status
+ Link_start_up_rate * Link_p_nom_committed * (Link_status - Link_previous_status)}
otherwise: Link_ramp_up_rate * Link_p_nom_effective
Link_ramp_down_allowance:
description: how far a link may lower flow between two snapshots — its ramp limit of the build while
it stays on, plus its shut-down ramp in the snapshot it turns off
dims: [scenario, snapshot, link]
cases:
committed: {when: Link_committable, expression: Link_ramp_down_rate * Link_p_nom_committed * Link_status
+ Link_shut_down_rate * Link_p_nom_committed * (Link_previous_status - Link_status)}
otherwise: Link_ramp_down_rate * Link_p_nom_effective
StorageUnit_charge_carried_in:
description: the charge a unit opens a snapshot with — at the first snapshot it stands in, its last
such snapshot's less standing loss where it is cyclic and the given initial charge, which no standing
loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A unit
built in a later period opens in that period, and a cyclic one that retires closes on its own last
snapshot. Per period, the same holds with each investment period as the horizon
dims: [scenario, snapshot, storage_unit]
cases:
cyclic: {when: StorageUnit_cyclic_state_of_charge AND NOT StorageUnit_cyclic_state_of_charge_per_period
AND NOT StorageUnit_state_of_charge_initial_per_period AND (position(snapshot) == 0 OR StorageUnit_opens_late),
expression: 'StorageUnit_retention * shift(shift(StorageUnit_state_of_charge, along=snapshot,
offset=1, edge=''wrap''), along=snapshot, offset=StorageUnit_inactive_snapshots, edge=''wrap'')'}
opening: {when: NOT StorageUnit_cyclic_state_of_charge AND NOT StorageUnit_cyclic_state_of_charge_per_period
AND NOT StorageUnit_state_of_charge_initial_per_period AND (position(snapshot) == 0 OR StorageUnit_opens_late),
expression: StorageUnit_state_of_charge_initial}
period_cyclic: {when: StorageUnit_cyclic_state_of_charge_per_period, expression: 'StorageUnit_retention
* shift(StorageUnit_state_of_charge, along=snapshot, offset=1, edge=''wrap'', by=snapshot_period,
within=period)'}
period_opening: {when: 'StorageUnit_state_of_charge_initial_per_period AND NOT StorageUnit_cyclic_state_of_charge_per_period
AND position(snapshot, by=snapshot_period, within=period) == 0', expression: StorageUnit_state_of_charge_initial}
otherwise: StorageUnit_retention * shift(StorageUnit_state_of_charge, along=snapshot, offset=1)
Store_energy_carried_in:
description: the energy a store opens a snapshot with — at the first snapshot it stands in, its last
such snapshot's less standing loss where it is cyclic and the given initial energy, which no standing
loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A store
built in a later period opens in that period, and a cyclic one that retires closes on its own last
snapshot. Per period, the same holds with each investment period as the horizon
dims: [scenario, snapshot, store]
cases:
cyclic: {when: Store_e_cyclic AND NOT Store_e_cyclic_per_period AND NOT Store_e_initial_per_period
AND (position(snapshot) == 0 OR Store_opens_late), expression: 'Store_retention * shift(shift(Store_e,
along=snapshot, offset=1, edge=''wrap''), along=snapshot, offset=Store_inactive_snapshots, edge=''wrap'')'}
opening: {when: NOT Store_e_cyclic AND NOT Store_e_cyclic_per_period AND NOT Store_e_initial_per_period
AND (position(snapshot) == 0 OR Store_opens_late), expression: Store_e_initial}
period_cyclic: {when: Store_e_cyclic_per_period, expression: 'Store_retention * shift(Store_e, along=snapshot,
offset=1, edge=''wrap'', by=snapshot_period, within=period)'}
period_opening: {when: 'Store_e_initial_per_period AND NOT Store_e_cyclic_per_period AND position(snapshot,
by=snapshot_period, within=period) == 0', expression: Store_e_initial}
otherwise: Store_retention * shift(Store_e, along=snapshot, offset=1)
transmission_volume_expansion:
dims: [scenario, global_constraint]
expression: Link_transmission_volume_expansion
description: what a `transmission_volume_expansion_limit` row totals — length times the chosen build
of the row's branches
transmission_expansion_cost:
dims: [scenario, global_constraint]
expression: Link_transmission_expansion_cost
description: what a `transmission_expansion_cost_limit` row totals — capital cost times the chosen
build of the row's branches
tech_capacity_expansion:
dims: [global_constraint]
expression: ((Generator_tech_capacity_expansion + Link_tech_capacity_expansion) + StorageUnit_tech_capacity_expansion)
+ Store_tech_capacity_expansion
description: what a `tech_capacity_expansion_limit` row totals — the chosen build of the row's carrier-and-bus
set
total_cost:
dims: []
expression: (((Generator_capex + Link_capex) + StorageUnit_capex) + Store_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) + StorageUnit_injection) +
Store_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_previous_status:
description: the commitment state a generator carries into a snapshot — the state it brought into
the horizon at the first, the previous snapshot's after that
dims: [scenario, snapshot, generator]
cases:
opening: {when: position(snapshot) == 0, expression: Generator_status_initial}
otherwise: shift(Generator_status, along=snapshot, offset=1)
Generator_p_nom_effective:
description: the build a generator's limits are taken against — the chosen one where it is extendable,
the given one otherwise
dims: [scenario, generator]
cases:
extendable: {when: Generator_p_nom_extendable, expression: Generator_p_nom_ext}
otherwise: Generator_p_nom
Generator_ramp_up_rate:
description: the ramp limit a unit's up row reads — PyPSA's `ramp_limit_up`, or the full build where
it has none, since a start-up ramp alone builds the row
dims: [scenario, snapshot, generator]
cases:
given: {when: Generator_ramp_limit_up, expression: Generator_ramp_limit_up}
otherwise: 1
Generator_ramp_down_rate:
description: the ramp limit a unit's down row reads — PyPSA's `ramp_limit_down`, or the full build
where it has none, since a shut-down ramp alone builds the row
dims: [scenario, snapshot, generator]
cases:
given: {when: Generator_ramp_limit_down, expression: Generator_ramp_limit_down}
otherwise: 1
Generator_start_up_rate:
description: the start-up ramp a unit's up row reads — PyPSA's `ramp_limit_start_up`, or the full
build where it has none
dims: [scenario, generator]
cases:
given: {when: Generator_ramp_limit_start_up, expression: Generator_ramp_limit_start_up}
otherwise: 1
Generator_shut_down_rate:
description: the shut-down ramp a unit's down row reads — PyPSA's `ramp_limit_shut_down`, or the full
build where it has none
dims: [scenario, generator]
cases:
given: {when: Generator_ramp_limit_shut_down, expression: Generator_ramp_limit_shut_down}
otherwise: 1
Generator_p_nom_committed:
description: the build a committed unit's ramp rows are taken against — one module where the build
is extendable and modular, the given build otherwise
dims: [scenario, generator]
cases:
modular_build: {when: Generator_p_nom_extendable AND Generator_p_nom_mod > 0, expression: Generator_p_nom_mod}
otherwise: Generator_p_nom
Link_p_nom_effective:
description: the build a link's limits are taken against — the chosen one where it is extendable,
the given one otherwise
dims: [scenario, link]
cases:
extendable: {when: Link_p_nom_extendable, expression: Link_p_nom_ext}
otherwise: Link_p_nom
Link_previous_status:
description: the commitment state a link carries into a snapshot — the state it brought into the horizon
at the first, the previous snapshot's after that
dims: [scenario, snapshot, link]
cases:
opening: {when: position(snapshot) == 0, expression: Link_status_initial}
otherwise: shift(Link_status, along=snapshot, offset=1)
Link_ramp_up_rate:
description: the ramp limit a link's up row reads — PyPSA's `ramp_limit_up`, or the full build where
it has none, since a start-up ramp alone builds the row
dims: [scenario, snapshot, link]
cases:
given: {when: Link_ramp_limit_up, expression: Link_ramp_limit_up}
otherwise: 1
Link_ramp_down_rate:
description: the ramp limit a link's down row reads — PyPSA's `ramp_limit_down`, or the full build
where it has none, since a shut-down ramp alone builds the row
dims: [scenario, snapshot, link]
cases:
given: {when: Link_ramp_limit_down, expression: Link_ramp_limit_down}
otherwise: 1
Link_start_up_rate:
description: the start-up ramp a link's up row reads — PyPSA's `ramp_limit_start_up`, or the full
build where it has none
dims: [scenario, link]
cases:
given: {when: Link_ramp_limit_start_up, expression: Link_ramp_limit_start_up}
otherwise: 1
Link_shut_down_rate:
description: the shut-down ramp a link's down row reads — PyPSA's `ramp_limit_shut_down`, or the full
build where it has none
dims: [scenario, link]
cases:
given: {when: Link_ramp_limit_shut_down, expression: Link_ramp_limit_shut_down}
otherwise: 1
Link_p_nom_committed:
description: the build a committed link's ramp rows are taken against — one module where the build
is extendable and modular, the given build otherwise
dims: [scenario, link]
cases:
modular_build: {when: Link_p_nom_extendable AND Link_p_nom_mod > 0, expression: Link_p_nom_mod}
otherwise: Link_p_nom
Link_transmission_volume_expansion: {expression: 'sum(Link_p_nom_ext * Link_volume_weight, over=link)'}
Link_transmission_expansion_cost: {expression: 'sum(Link_p_nom_ext * Link_expansion_cost_weight, over=link)'}
Generator_tech_capacity_expansion: {expression: 'sum(Generator_p_nom_ext * Generator_tech_capacity_weight,
over=generator)'}
Link_tech_capacity_expansion: {expression: 'sum(Link_p_nom_ext * Link_tech_capacity_weight, over=link)'}
StorageUnit_tech_capacity_expansion: {expression: 'sum(StorageUnit_p_nom_ext * StorageUnit_tech_capacity_weight,
over=storage_unit)'}
Store_tech_capacity_expansion: {expression: 'sum(Store_e_nom_ext * Store_tech_capacity_weight, over=store)'}
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)}
StorageUnit_capex: {expression: sum(scenario_weight * StorageUnit_p_nom_ext * StorageUnit_capital_cost
* StorageUnit_capital_weight)}
Store_capex: {expression: sum(scenario_weight * Store_e_nom_ext * Store_capital_cost * Store_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)'}
StorageUnit_injection: {expression: 'sum(StorageUnit_sign * (StorageUnit_p_dispatch - StorageUnit_p_store),
by=StorageUnit_bus, over=storage_unit, into=bus)'}
Store_injection: {expression: 'sum(Store_sign * Store_p, by=Store_bus, over=store, 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
scenario_opex:
dims: [scenario]
expression: ((Generator_opex + Link_opex) + StorageUnit_opex) + Store_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)'}
StorageUnit_opex: {expression: '(sum(sum(((StorageUnit_p_dispatch * StorageUnit_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=storage_unit),
over=snapshot) + sum(sum((((StorageUnit_p_dispatch * StorageUnit_p_dispatch) * StorageUnit_marginal_cost_quadratic)
* snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
into=snapshot), over=storage_unit), over=snapshot)) + sum(sum(((StorageUnit_state_of_charge * StorageUnit_marginal_cost_storage)
* snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
into=snapshot), over=storage_unit), over=snapshot)'}
Store_opex: {expression: 'sum(sum(((Store_p * Store_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)
+ sum(sum((((Store_p * Store_p) * Store_marginal_cost_quadratic) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)
+ sum(sum(((Store_e * Store_marginal_cost_storage) * snapshot_weightings_objective) * at(period_weight_objective,
by=snapshot_period, over=period, into=snapshot), over=store), 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
def _cycle_weights(n: pypsa.Network) -> dict[str, object]:
"""The KVL rows PyPSA itself writes, per branch — ``n.cycle_matrix(apply_weights=True)``, times the 1e5 PyPSA scales every cycle row by for conditioning.
A branch whose weight is NaN — an extendable one at zero nominal rating,
whose per-unit reactance is infinite — has no term: linopy drops it.
A transformer's phase shift enters the same rows: a fixed one as a constant
in radians at each snapshot, a varying one as the shift decision times its
cycle sign and π/180. PyPSA builds the basis from the first scenario only.
The constant is written at every snapshot, transformer and cycle, zero
where nothing shifts, because a constant summed over transformers is owed
each one.
"""
n.determine_network_topology()
n.calculate_dependent_values()
weighted = n.cycle_matrix(apply_weights=True) * 1e5
plain = n.cycle_matrix(apply_weights=False)
rows: dict[str, list[dict]] = {'Line': [], 'Transformer': []}
for (kind, name), weights in weighted.iterrows():
rows.setdefault(kind, [])
rows[kind].extend(
{DIM[kind]: str(name), 'cycle': str(cycle), 'value': float(w)}
for cycle, w in weights.items()
if w and not math.isnan(w)
)
transformers = first_scenario(n.static('Transformer'))
varying_ = transformers['phase_shift_min'] < transformers['phase_shift_max']
shift_rows, decision_rows = [], []
signs = plain.loc['Transformer'] if 'Transformer' in plain.index.get_level_values(0) else pd.DataFrame()
dense = get_switchable_as_dense(n, 'Transformer', 'phase_shift').set_axis(timesteps(n), axis=0)
if dense.columns.nlevels > 1:
dense = dense.T.groupby(level='name').first().T
for name in transformers.index:
for cycle in plain.columns:
sign = float(signs.at[name, cycle]) if name in signs.index else 0.0
if varying_.get(name, False):
if sign:
decision_rows.append(
{'transformer': str(name), 'cycle': str(cycle), 'value': sign * math.pi / 180 * 1e5}
)
sign = 0.0
shift_rows.extend(
{
'snapshot': t,
'transformer': str(name),
'cycle': str(cycle),
'value': sign * float(shift) * math.pi / 180 * 1e5,
}
for t, shift in dense[name].items()
)
line = pd.DataFrame(rows['Line'], columns=['line', 'cycle', 'value']).astype({'value': float})
transformer = pd.DataFrame(rows['Transformer'], columns=['transformer', 'cycle', 'value']).astype({'value': float})
cycles = list(pd.unique(pd.concat([line['cycle'], transformer['cycle']])))
return {
'cycle': pl.Series('cycle', cycles, dtype=pl.String),
'Line_cycle_weight': line,
'Transformer_cycle_weight': transformer,
'Transformer_phase_shift_weight': pd.DataFrame(
shift_rows, columns=['snapshot', 'transformer', 'cycle', 'value']
).astype({'value': float}),
'Transformer_phase_shift_cycle_weight': pd.DataFrame(
decision_rows, columns=['transformer', 'cycle', 'value']
).astype({'value': float}),
'Transformer_phase_shift_varying': per_component('Transformer', varying_, bool),
'Transformer_phase_shift_min': per_component('Transformer', transformers['phase_shift_min']),
'Transformer_phase_shift_max': per_component('Transformer', transformers['phase_shift_max']),
}
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'),
'StorageUnit_bus': relation(n, 'StorageUnit', 'bus'),
'Store_bus': relation(n, 'Store', '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),
'snapshot_weightings_stores': weighting(n, 'stores'),
'snapshot_weightings_generators': weighting(n, 'generators'),
'Generator_e_sum_min': static(n, 'Generator', 'e_sum_min'),
'Generator_e_sum_max': static(n, 'Generator', 'e_sum_max'),
**{name: table for component in DISPATCHABLE for name, table in dispatchable(n, component).items()},
**ports(n, 'Link'),
**ports(n, 'Process'),
**storage(n, multi),
**branches(n),
**_cycle_weights(n),
**losses(n, optimize, outages),
**security(n, outages),
**global_constraints(n, multi),
}
with sps.solve('differential/pypsa/rungs/rung_03_expansion.yaml', sources) as solution:
solution.objective # 7633.908502024291
The network, rung_03_expansion.py in the corpus — the spine plus what this rung adds:
"""Rung 3: expansion — extendable capacity, energy-sum bounds, fixed and set nominal capacities."""
from __future__ import annotations
import spine
def build():
"""The spine plus this rung's additions, as a ``pypsa.Network``."""
n = spine.build()
n.add('Bus', 'island')
n.add('Carrier', 'onwind')
n.add('Carrier', 'solarpv')
n.add('Carrier', 'dc')
n.add('Carrier', 'phs')
n.add('Carrier', 'h2')
n.add(
'Generator',
'wind',
bus='north',
carrier='onwind',
p_nom_extendable=True,
capital_cost=50,
p_nom_min=5,
p_nom_max=80,
marginal_cost=0,
e_sum_min=40,
ramp_limit_up=0.4,
ramp_limit_down=0.4,
p_max_pu=[0.3, 0.8, 0.5, 0.9],
)
n.add(
'Generator',
'solar',
bus='north',
carrier='solarpv',
p_nom_extendable=True,
capital_cost=60,
p_nom_max=40,
marginal_cost=0,
p_nom_set=15,
p_max_pu=[0.5, 0.6, 0.4, 0.2],
)
n.add('Generator', 'diesel', bus='island', marginal_cost=40, p_nom=60, e_sum_max=70)
n.add(
'Link',
'cable',
bus0='north',
bus1='island',
carrier='dc',
length=120,
p_nom_extendable=True,
capital_cost=20,
p_nom_max=30,
efficiency=0.95,
p_nom_set=25,
ramp_limit_up=0.3,
ramp_limit_down=0.3,
)
n.add('Load', 'island_load', bus='island', p_set=10)
n.add(
'StorageUnit',
'pump',
bus='north',
carrier='phs',
p_nom_extendable=True,
capital_cost=15,
p_nom_max=30,
max_hours=4,
efficiency_store=0.9,
efficiency_dispatch=0.9,
cyclic_state_of_charge=True,
p_nom_set=20,
)
n.add('StorageUnit', 'ice', bus='island', max_hours=2, p_nom=8, state_of_charge_initial=6)
n.add(
'Store',
'tank',
bus='north',
carrier='h2',
e_nom_extendable=True,
capital_cost=2,
e_nom_max=80,
e_cyclic=True,
e_nom_set=50,
)
n.add('Store', 'keg', bus='island', e_nom=15, e_initial=5)
n.add(
'GlobalConstraint',
'tech_wind',
type='tech_capacity_expansion_limit',
carrier_attribute='onwind',
sense='==',
constant=50,
)
n.add(
'GlobalConstraint',
'tech_solar',
type='tech_capacity_expansion_limit',
carrier_attribute='solarpv',
sense='>=',
constant=10,
)
n.add(
'GlobalConstraint',
'tech_dc',
type='tech_capacity_expansion_limit',
carrier_attribute='dc',
sense='<=',
constant=28,
)
n.add(
'GlobalConstraint',
'tech_phs',
type='tech_capacity_expansion_limit',
carrier_attribute='phs',
sense='<=',
constant=25,
)
n.add(
'GlobalConstraint',
'tech_h2',
type='tech_capacity_expansion_limit',
carrier_attribute='h2',
sense='>=',
constant=30,
)
n.add(
'GlobalConstraint',
'vol_dc',
type='transmission_volume_expansion_limit',
carrier_attribute='dc',
sense='<=',
constant=3500,
)
n.add(
'GlobalConstraint',
'cost_dc',
type='transmission_expansion_cost_limit',
carrier_attribute='dc',
sense='>=',
constant=400,
)
n.add(
'GlobalConstraint',
'cost_dc_exact',
type='transmission_expansion_cost_limit',
carrier_attribute='dc',
sense='==',
constant=500,
)
return n
The data¶
The tables this rung is the first to declare (39), as the prep produced them:
Generator_capital_cost.csv
Generator_capital_weight.csv
Generator_e_sum_max.csv
Generator_e_sum_min.csv
scenario,generator,value
base,coal,-inf
base,diesel,-inf
base,gas,-inf
base,solar,-inf
base,wind,40.0
Generator_p_nom_max.csv
Generator_p_nom_min.csv
Generator_p_nom_set.csv
Generator_ramp_limit_down.csv
scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,wind,0.4
base,2015-01-01T01:00:00.000000,wind,0.4
base,2015-01-01T02:00:00.000000,wind,0.4
base,2015-01-01T03:00:00.000000,wind,0.4
Generator_ramp_limit_up.csv
scenario,snapshot,generator,value
base,2015-01-01T00:00:00.000000,wind,0.4
base,2015-01-01T01:00:00.000000,wind,0.4
base,2015-01-01T02:00:00.000000,wind,0.4
base,2015-01-01T03:00:00.000000,wind,0.4
Generator_status_initial.csv
Generator_tech_capacity_weight.csv
GlobalConstraint_constant.csv
scenario,global_constraint,value
base,cost_dc,400.0
base,cost_dc_exact,500.0
base,tech_dc,28.0
base,tech_h2,30.0
base,tech_phs,25.0
base,tech_solar,10.0
base,tech_wind,50.0
base,vol_dc,3500.0
GlobalConstraint_sense.csv
scenario,global_constraint,value
base,cost_dc,>=
base,cost_dc_exact,==
base,tech_dc,<=
base,tech_h2,>=
base,tech_phs,<=
base,tech_solar,>=
base,tech_wind,==
base,vol_dc,<=
GlobalConstraint_type.csv
global_constraint,value
cost_dc,transmission_expansion_cost_limit
cost_dc_exact,transmission_expansion_cost_limit
tech_dc,tech_capacity_expansion_limit
tech_h2,tech_capacity_expansion_limit
tech_phs,tech_capacity_expansion_limit
tech_solar,tech_capacity_expansion_limit
tech_wind,tech_capacity_expansion_limit
vol_dc,transmission_volume_expansion_limit
Link_capital_cost.csv
Link_capital_weight.csv
Link_expansion_cost_weight.csv
Link_p_nom_max.csv
Link_p_nom_min.csv
Link_p_nom_set.csv
Link_ramp_limit_down.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cable,0.3
base,2015-01-01T01:00:00.000000,cable,0.3
base,2015-01-01T02:00:00.000000,cable,0.3
base,2015-01-01T03:00:00.000000,cable,0.3
Link_ramp_limit_up.csv
scenario,snapshot,link,value
base,2015-01-01T00:00:00.000000,cable,0.3
base,2015-01-01T01:00:00.000000,cable,0.3
base,2015-01-01T02:00:00.000000,cable,0.3
base,2015-01-01T03:00:00.000000,cable,0.3
Link_status_initial.csv
Link_tech_capacity_weight.csv
Link_volume_weight.csv
StorageUnit_capital_cost.csv
StorageUnit_capital_weight.csv
StorageUnit_p_nom_max.csv
StorageUnit_p_nom_min.csv
StorageUnit_p_nom_set.csv
StorageUnit_tech_capacity_weight.csv
Store_capital_cost.csv
Store_capital_weight.csv
Store_e_nom_max.csv
Store_e_nom_min.csv
Store_e_nom_set.csv
Store_tech_capacity_weight.csv
global_constraint.csv
snapshot_weightings_generators.csv