Rung 6: KVL — passive lines under Kirchhoff's voltage law¶
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 23962.0 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;transmission_expansion_cost_limit2 vs 1+1 — one block per sense — ==, <=, >= — where PyPSA writes one row per labelled constraint whatever its sense;transmission_volume_expansion_limit2 vs 1+1 — one block per sense — ==, <=, >= — where PyPSA writes one row per labelled constraint whatever its sense; size ✔ 123 rows · ≠ 42 vs 45 columns · ✔ 204 nonzeros; duals ✔ 123 rows; model for model: 19 blocks equal, 2 documented splits, 4 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
20 | 20 |
Generator-fix-p-lower |
16 | 16 |
Generator-fix-p-upper |
16 | 16 |
Kirchhoff-Voltage-Law |
12 | 12 |
Line-ext-s-lower |
8 | 8 |
Line-ext-s-upper |
8 | 8 |
Line-ext-s_nom-lower |
2 | 2 |
Line-ext-s_nom-upper |
2 | 2 |
Line-fix-s-lower |
12 | 12 |
Line-fix-s-upper |
12 | 12 |
Line-s_nom_set |
1 | 1 |
Line-s_set |
1 | 1 |
Link-fix-p-lower |
4 | 4 |
Link-fix-p-upper |
4 | 4 |
tech_capacity_expansion_limit |
1 | 1 |
transmission_expansion_cost_limit |
2 | ≠ 1+1 |
transmission_volume_expansion_limit |
2 | ≠ 1+1 |
| column | PyPSA | specsolve |
|---|---|---|
CVaR |
0 | ≠ 1 |
CVaR-a |
0 | ≠ 1 |
CVaR-theta |
0 | ≠ 1 |
Generator-p |
16 | 16 |
Line-s |
20 | 20 |
Line-s_nom |
2 | 2 |
Link-p |
4 | 4 |
The model¶
The same model, as math
A plain n.optimize(), and its multi-period and stochastic classes, in one file. Every second-stage quantity spans a scenario (a future dispatch is chosen in) and every asset stands in the investment periods its build year and lifetime span. A parameter spans scenario exactly when PyPSA reads it per scenario. Capacity is chosen once, before the future is known, and paid once per active period at its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights, with a share priced at the tail through the CVaR rows, which stand only where 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 \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N},\ \mathrm{Line\_bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{Line\_bus1}: \mathcal{K} \to \mathcal{N}\) — 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 |
| \(\mathcal{K}\) | index \(k\) — line with \(\mathrm{Line\_bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{Line\_bus1}: \mathcal{K} \to \mathcal{N}\) — passive branches, each between two buses, their flow set by impedance |
| \(\mathcal{C}\) | index \(c\) — cycle — independent cycles of the passive network graph — the cycle basis, data prep |
| \(\mathcal{B}\) | 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 |
| \(\mathrm{f}^{\mathrm{nom}}\) | 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 |
| \(\mathrm{load}\) | 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 |
| \(\mathrm{on}^{s}\) | Line_active over \(\mathcal{T} \times \mathcal{K}\) — whether a line stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{W}^{s}\) | Line_capital_weight over \(\mathcal{K}\) — the sum of period weights a line stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{s}^{\mathrm{nom}}\) | Line_s_nom over \(\Xi \times \mathcal{K}\) — nominal apparent power |
| \(\mathrm{ext}^{s}\) | Line_s_nom_extendable over \(\mathcal{K}\) — whether the nominal apparent power is a decision |
| \(\overline{\mathrm{s}}\) | Line_s_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — most flow either way, per unit of nominal apparent power |
| \(\underline{\mathrm{s}}^{\mathrm{nom}}\) | Line_s_nom_min over \(\Xi \times \mathcal{K}\) — least nominal apparent power an extendable line may be built at |
| \(\overline{\mathrm{s}}^{\mathrm{nom}}\) | Line_s_nom_max over \(\Xi \times \mathcal{K}\) — most nominal apparent power an extendable line may be built at |
| \(\mathrm{c}^{\mathrm{cap},s}\) | Line_capital_cost over \(\Xi \times \mathcal{K}\) — cost of one unit of nominal apparent power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\mathrm{s}^{\mathrm{nom,set}}\) | Line_s_nom_set over \(\Xi \times \mathcal{K}\) — a given nominal apparent power for an extendable line; one without a value has no row here |
| \(\mathrm{s}^{\mathrm{set}}\) | Line_s_set over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — a given flow schedule; a line without one has no row here |
| \(\mathrm{x}\) | Line_cycle_weight over \(\mathcal{K} \times \mathcal{C}\) — the line's series impedance, signed by its orientation in the cycle — the cycle basis, data prep; a line in no cycle has no row. PyPSA builds the cycle basis from the first scenario only (networks.py:1354-1361) |
| \(\mathrm{type}\) | 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) |
| \(\mathrm{len}\) | Line_volume_weight over \(\Xi \times \mathcal{B} \times \mathcal{K}\) — the line'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 line outside it, or one that does not stand in the row's investment_period, has no row |
| \(\mathrm{cc}\) | Line_expansion_cost_weight over \(\Xi \times \mathcal{B} \times \mathcal{K}\) — the line'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 line outside the set, or one that does not stand in the row's period, has no row |
| \(\mathrm{m}^{l}\) | Line_tech_capacity_weight over \(\mathcal{B} \times \mathcal{K}\) — one where the line 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 |
| \(s\) | Line_s over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — Line-s — PyPSA's p0, the flow measured at the Line_bus0 end: a positive value withdraws there and injects at Line_bus1, lossless |
| \(S\) | Line_s_nom_ext over \(\mathcal{K}\) — Line-s_nom — nominal apparent power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
| \(a\) | CVaR_a over \(\Xi\) — CVaR-a — how far a scenario's operating cost exceeds the tail's start; nothing where it does not |
| \(\theta\) | CVaR_theta (scalar) — CVaR-theta — where the tail starts, the value at risk |
| \(CVaR\) | CVaR (scalar) — CVaR — the tail's average cost, what the objective prices at omega |
Definitions¶
| Symbol | Meaning |
|---|---|
| \(\mathit{transmission\_volume\_expansion}\) | 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 |
| \(\mathit{Cycle\_angle\_sum}\) | Cycle_angle_sum over \(\Xi \times \mathcal{T} \times \mathcal{C}\) — the voltage angle differences around a cycle: every branch flow times its cycle weight, and every transformer phase shift |
| \(\mathit{Line\_transmission\_volume\_expansion}\) | Line_transmission_volume_expansion over \(\Xi \times \mathcal{B}\) |
| \(\mathit{Line\_transmission\_expansion\_cost}\) | Line_transmission_expansion_cost over \(\Xi \times \mathcal{B}\) |
| \(\mathit{Line\_tech\_capacity\_expansion}\) | Line_tech_capacity_expansion over \(\mathcal{B}\) |
| \(\mathit{Line\_capex}\) | Line_capex (scalar) |
| \(\mathit{risk\_weighted\_opex}\) | risk_weighted_opex (scalar) |
| \(\mathit{Generator\_injection}\) | Generator_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Line\_injection}\) | Line_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 \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Line\_angle\_sum}\) | Line_angle_sum over \(\Xi \times \mathcal{T} \times \mathcal{C}\) |
| \(\mathit{Link\_output\_arrival}\) | 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 \(\Xi\) |
\(t \ominus k\) denotes cyclic translation: index \(t-k\) taken modulo the size of the dimension (roll). Plain \(t-k\) (shift) has no wraparound — terms translated past the edge are simply absent.
\(t \boxminus_{v} k\) denotes translation with \(v\) standing where index \(t-k\) leaves the dimension (shift(edge=v)), so the row at that boundary is built and carries \(v\) rather than being dropped.
\(t \ominus^{\mathrm{relation}(t)} k\) denotes a translation counted inside the group a relation puts \(t\) in (shift(by=relation)), so a term never crosses out of its own group. The two modifiers take different slots — the group above, the fill below — so \(t \boxminus_{v}^{\mathrm{relation}(t)} k\) is both at once.
Objective¶
\[
\min \mathit{total\_cost}
\]
Subject to¶
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
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
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Line_fix_s_lower
\[
s_{\xi,t,k} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot \mathrm{s}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_fix_s_upper
\[
s_{\xi,t,k} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot \mathrm{s}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_ext_s_lower
\[
s_{\xi,t,k} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_ext_s_upper
\[
s_{\xi,t,k} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_ext_s_nom_lower
\[
S_{k} \ge \underline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k}
\]
Line_ext_s_nom_upper
\[
S_{k} \le \overline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \overline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \text{ is defined}
\]
Line_s_nom_set
\[
S_{k} = \mathrm{s}^{\mathrm{nom,set}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{s}^{\mathrm{nom,set}}_{\xi,k} \text{ is defined}
\]
Line_s_set
\[
s_{\xi,t,k} = \mathrm{s}^{\mathrm{set}}_{\xi,t,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{s}^{\mathrm{set}}_{\xi,t,k} \text{ is defined} \wedge \mathrm{on}^{s}_{t,k}
\]
Kirchhoff_Voltage_Law
\[
\mathit{Cycle\_angle\_sum}_{\xi,t,c} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C}
\]
GlobalConstraint_transmission_volume_expansion_limit_lb
\[
\mathit{transmission\_volume\_expansion}_{\xi,b} \ge \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_volume\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{>=}\text{'}
\]
GlobalConstraint_transmission_volume_expansion_limit_eq
\[
\mathit{transmission\_volume\_expansion}_{\xi,b} = \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_volume\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{==}\text{'}
\]
GlobalConstraint_transmission_expansion_cost_limit_ub
\[
\mathit{transmission\_expansion\_cost}_{\xi,b} \le \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{transmission\_expansion\_cost\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{<=}\text{'}
\]
GlobalConstraint_transmission_expansion_cost_limit_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_tech_capacity_expansion_limit_ub
\[
\mathit{tech\_capacity\_expansion}_{b} \le \mathrm{K}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B} \,:\, \mathrm{type}_{b} = \text{'}\mathrm{tech\_capacity\_expansion\_limit}\text{'} \wedge \mathrm{sense}_{\xi,b} = \text{'}\mathrm{<=}\text{'}
\]
Bus_nodal_balance
\[
\mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Definitions¶
transmission_volume_expansion
\[
\mathit{transmission\_volume\_expansion}_{\xi,b} = \mathit{Line\_transmission\_volume\_expansion}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
transmission_expansion_cost
\[
\mathit{transmission\_expansion\_cost}_{\xi,b} = \mathit{Line\_transmission\_expansion\_cost}_{\xi,b} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
tech_capacity_expansion
\[
\mathit{tech\_capacity\_expansion}_{b} = \mathit{Line\_tech\_capacity\_expansion}_{b} \qquad \forall\, b \in \mathcal{B}
\]
total_cost
\[
\mathit{total\_cost} = \mathit{Line\_capex} + \mathit{risk\_weighted\_opex}
\]
Bus_injection
\[
\mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{\xi,t,n} + \mathit{Line\_injection}_{\xi,t,n} + \mathit{Link\_injection}_{\xi,t,n} + \mathrm{Load\_injection}_{\xi,t,n} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Cycle_angle_sum
\[
\mathit{Cycle\_angle\_sum}_{\xi,t,c} = \mathit{Line\_angle\_sum}_{\xi,t,c} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C}
\]
Line_transmission_volume_expansion
\[
\mathit{Line\_transmission\_volume\_expansion}_{\xi,b} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{len}_{\xi,b,k} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
Line_transmission_expansion_cost
\[
\mathit{Line\_transmission\_expansion\_cost}_{\xi,b} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{cc}_{\xi,b,k} \qquad \forall\, \xi \in \Xi,\ b \in \mathcal{B}
\]
Line_tech_capacity_expansion
\[
\mathit{Line\_tech\_capacity\_expansion}_{b} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{m}^{l}_{b,k} \qquad \forall\, b \in \mathcal{B}
\]
Line_capex
\[
\mathit{Line\_capex} = \sum_{\xi \in \Xi,\ k \in \mathcal{K}} \pi_{\xi} \cdot S_{k} \cdot \mathrm{c}^{\mathrm{cap},s}_{\xi,k} \cdot \mathrm{W}^{s}_{k}
\]
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
\[
\mathit{Generator\_injection}_{\xi,t,n} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Line_injection
\[
\mathit{Line\_injection}_{\xi,t,n} = -\left( \sum_{k \in \mathcal{K} \,:\, \mathrm{Line\_bus0}(k) = n} s_{\xi,t,k} \right) + \sum_{k \in \mathcal{K} \,:\, \mathrm{Line\_bus1}(k) = n} s_{\xi,t,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
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
\[
\mathrm{Load\_injection}_{\xi,t,n} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Line_angle_sum
\[
\mathit{Line\_angle\_sum}_{\xi,t,c} = \sum_{k \in \mathcal{K}} s_{\xi,t,k} \cdot \mathrm{x}_{k,c} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C}
\]
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
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
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
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
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
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
Line_s
\[
s_{\xi,t,k} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{on}^{s}_{t,k}
\]
Line_s_nom_ext
\[
S_{k} \in \mathbb{R} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k}
\]
CVaR_a
\[
a_{\xi} \ge 0 \qquad \forall\, \xi \in \Xi
\]
CVaR_theta
\[
\theta \in \mathbb{R}
\]
CVaR
\[
CVaR \in \mathbb{R}
\]
The spec, differential/pypsa/rungs/rung_06_kvl.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'}
line: {description: 'passive branches, each between two buses, their flow set by impedance'}
cycle: {description: 'independent cycles of the passive network graph — the cycle basis, data prep'}
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}
Line_bus0: {description: the bus a line's flow is measured at, key: line, values: bus}
Line_bus1: {description: the bus at a line's other end, key: line, values: bus}
parameters:
snapshot_weightings_objective:
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
dims: [snapshot]
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
for a unit that draws power. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
dims: [generator]
Generator_committable:
description: whether output is gated by an on/off status decision
dims: [generator]
dtype: bool
Link_p_nom:
description: nominal power
dims: [scenario, link]
Link_p_nom_extendable:
description: whether the nominal power is a decision
dims: [link]
dtype: bool
Link_p_min_pu:
description: least flow, per unit of nominal power — negative for a link that carries both ways
dims: [scenario, snapshot, link]
Link_p_max_pu:
description: most flow, per unit of nominal power
dims: [scenario, snapshot, link]
Link_efficiency:
description: share of the flow that arrives at an output port, PyPSA's `efficiency`, `efficiency2`,
… read long — negative where that port consumes rather than delivers. Read at the snapshot the flow
arrives, so a delayed port delivers at its arrival snapshot's efficiency (`constraints.py:1522`)
dims: [scenario, snapshot, link_output]
Link_output_delay:
description: snapshots a port's delivery lags its link's flow — PyPSA's `delay`, `delay2`, … read
long, in `snapshot_weightings.generators` units, which the file states as whole snapshots; zero
for a port that delivers at once. Each scenario takes its own. PyPSA `1.3.0` groups the ports by
delay over all scenarios and shifts each group in every one, so a delay that differs by scenario
delivers the flow twice (`constraints.py:1269-1276`, PyPSA/PyPSA#1941)
dims: [scenario, link_output]
dtype: int
Link_output_cyclic_delay:
description: whether a delayed port's flow wraps from the end of its investment period — PyPSA's `cyclic_delay`,
`cyclic_delay2`, …; where it does not, the flow still in transit at each period's first snapshots
is lost. Each scenario takes its own, as the delay
dims: [scenario, link_output]
dtype: bool
Link_marginal_cost:
description: cost of one unit of flow
dims: [scenario, snapshot, link]
Link_marginal_cost_quadratic:
description: cost of the square of one unit of flow
dims: [scenario, snapshot, link]
Link_committable:
description: whether flow is gated by an on/off status decision
dims: [link]
dtype: bool
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
Line_active:
description: whether a line stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, line]
dtype: bool
Line_capital_weight:
description: the sum of period weights a line stands in — PyPSA's `active * period_weighting`, summed,
data prep
dims: [line]
Line_s_nom:
description: nominal apparent power
dims: [scenario, line]
Line_s_nom_extendable:
description: whether the nominal apparent power is a decision
dims: [line]
dtype: bool
Line_s_max_pu:
description: most flow either way, per unit of nominal apparent power
dims: [scenario, snapshot, line]
Line_s_nom_min:
description: least nominal apparent power an extendable line may be built at
dims: [scenario, line]
Line_s_nom_max:
description: most nominal apparent power an extendable line may be built at
dims: [scenario, line]
Line_capital_cost:
description: cost of one unit of nominal apparent power — PyPSA's `capital_cost`, periodized as an
annuity in data prep
dims: [scenario, line]
Line_s_nom_set:
description: a given nominal apparent power for an extendable line; one without a value has no row
here
dims: [scenario, line]
Line_s_set:
description: a given flow schedule; a line without one has no row here
dims: [scenario, snapshot, line]
Line_cycle_weight:
description: the line's series impedance, signed by its orientation in the cycle — the cycle basis,
data prep; a line in no cycle has no row. PyPSA builds the cycle basis from the first scenario only
(`networks.py:1354-1361`)
dims: [line, cycle]
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]
Line_volume_weight:
description: the line'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 line outside it, or one that
does not stand in the row's `investment_period`, has no row
dims: [scenario, global_constraint, line]
Line_expansion_cost_weight:
description: the line'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 line outside the set, or one that does not stand in the row's period, has no row
dims: [scenario, global_constraint, line]
Line_tech_capacity_weight:
description: one where the line 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, line]
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
Line_s:
description: '`Line-s` — PyPSA''s `p0`, the flow measured at the `Line_bus0` end: a positive value
withdraws there and injects at `Line_bus1`, lossless'
dims: [scenario, snapshot, line]
where: Line_active
Line_s_nom_ext:
description: '`Line-s_nom` — nominal apparent power where it is a decision; the parameter of the same
PyPSA name carries the fixed regime'
dims: [line]
where: Line_s_nom_extendable
CVaR_a:
description: '`CVaR-a` — how far a scenario''s operating cost exceeds the tail''s start; nothing where
it does not'
dims: [scenario]
bounds: {lower: 0}
CVaR_theta:
description: '`CVaR-theta` — where the tail starts, the value at risk'
dims: []
CVaR:
description: '`CVaR` — the tail''s average cost, what the objective prices at `omega`'
dims: []
constraints:
Generator_fix_p_lower:
description: '`Generator-fix-p-lower` — a fixed generator outputs at least its minimum'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * Generator_p_nom
Generator_fix_p_upper:
description: '`Generator-fix-p-upper` — a fixed generator outputs at most what is available'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * Generator_p_nom
Link_fix_p_lower:
description: '`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other
way'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * Link_p_nom
Link_fix_p_upper:
description: '`Link-fix-p-upper` — a fixed link carries at most its nominal power'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * Link_p_nom
Line_fix_s_lower:
description: '`Line-fix-s-lower` — a fixed line carries at least the negative of its rating, the loss
counted against it'
dims: [scenario, snapshot, line]
where: not Line_s_nom_extendable AND Line_active
expression: Line_s >= (-Line_s_max_pu) * Line_s_nom
Line_fix_s_upper:
description: '`Line-fix-s-upper` — a fixed line carries at most its rating, the loss included'
dims: [scenario, snapshot, line]
where: not Line_s_nom_extendable AND Line_active
expression: Line_s <= Line_s_max_pu * Line_s_nom
Line_ext_s_lower:
description: '`Line-ext-s-lower` — an extendable line carries at least the negative of its rating
of the chosen build, the loss counted against it'
dims: [scenario, snapshot, line]
where: Line_s_nom_extendable AND Line_active
expression: Line_s >= (-Line_s_max_pu) * Line_s_nom_ext
Line_ext_s_upper:
description: '`Line-ext-s-upper` — an extendable line carries at most its rating of the chosen build,
the loss included'
dims: [scenario, snapshot, line]
where: Line_s_nom_extendable AND Line_active
expression: Line_s <= Line_s_max_pu * Line_s_nom_ext
Line_ext_s_nom_lower:
description: '`Line-ext-s_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, line]
where: Line_s_nom_extendable
expression: Line_s_nom_ext >= Line_s_nom_min
Line_ext_s_nom_upper:
description: '`Line-ext-s_nom-upper` — the chosen build is at most its cap in every scenario; a cap
of infinity is no row'
dims: [scenario, line]
where: Line_s_nom_extendable AND Line_s_nom_max
expression: Line_s_nom_ext <= Line_s_nom_max
Line_s_nom_set:
description: '`Line-s_nom_set` — the chosen build pinned, wherever a value is given'
dims: [scenario, line]
where: Line_s_nom_extendable AND Line_s_nom_set
expression: Line_s_nom_ext == Line_s_nom_set
Line_s_set:
description: '`Line-s_set` — flow pinned to the given schedule, wherever one is given'
dims: [scenario, snapshot, line]
where: Line_s_set AND Line_active
expression: Line_s == Line_s_set
Kirchhoff_Voltage_Law:
description: '`Kirchhoff-Voltage-Law` — around every independent cycle the impedance-weighted flows
sum to nothing, which is what makes the linear power flow physical rather than transport. A transformer''s
flow weighs its effective reactance, and its phase shift enters the cycle sum too: a constant where
the shift is fixed, or the shift decision times its cycle weight where the shift is a phase-shifting
transformer''s to choose'
dims: [scenario, snapshot, cycle]
expression: Cycle_angle_sum == 0
GlobalConstraint_transmission_volume_expansion_limit_lb:
description: '`transmission_volume_expansion_limit` — its total, at least its constant'
dims: [scenario, global_constraint]
where: GlobalConstraint_type == 'transmission_volume_expansion_limit' AND GlobalConstraint_sense ==
'>='
expression: transmission_volume_expansion >= GlobalConstraint_constant
GlobalConstraint_transmission_volume_expansion_limit_eq:
description: '`transmission_volume_expansion_limit` — its total, at 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_ub:
description: '`transmission_expansion_cost_limit` — its total, at most 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_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_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
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:
transmission_volume_expansion:
dims: [scenario, global_constraint]
expression: Line_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: Line_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: Line_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: Line_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 + Line_injection) + Link_injection) + Load_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
Cycle_angle_sum:
dims: [scenario, snapshot, cycle]
expression: Line_angle_sum
description: 'the voltage angle differences around a cycle: every branch flow times its cycle weight,
and every transformer phase shift'
Line_transmission_volume_expansion: {expression: 'sum(Line_s_nom_ext * Line_volume_weight, over=line)'}
Line_transmission_expansion_cost: {expression: 'sum(Line_s_nom_ext * Line_expansion_cost_weight, over=line)'}
Line_tech_capacity_expansion: {expression: 'sum(Line_s_nom_ext * Line_tech_capacity_weight, over=line)'}
Line_capex: {expression: sum(scenario_weight * Line_s_nom_ext * Line_capital_cost * Line_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)'}
Line_injection: {expression: '(-sum(Line_s, by=Line_bus0, over=line, into=bus)) + sum(Line_s, by=Line_bus1,
over=line, 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)'}
Line_angle_sum: {expression: 'sum(Line_s * Line_cycle_weight, over=line)'}
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
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)'}
objective: {sense: minimize, expression: total_cost}
The prep — every table the spec declares, from the network — and the solve:
from differential.pypsa.prep import relation, static, varying, weighting
n = build() # the network from the PyPSA tab
sources = {
'snapshot': pl.Series('snapshot', list(timesteps(n)), dtype=pl.Datetime('us')),
'bus': pl.Series('bus', list(names(n.buses.index).astype(str)), dtype=pl.String),
**{
dim: pl.Series(dim, list(names(n.static(component).index).astype(str)), dtype=pl.String)
for component, dim in DIM.items()
},
**scenarios(n),
**periods(n),
**carriers(n, multi),
'Generator_bus': relation(n, 'Generator', 'bus'),
'Link_bus0': relation(n, 'Link', 'bus0'),
'Load_bus': relation(n, 'Load', 'bus'),
'snapshot_weightings_objective': weighting(n, 'objective'),
'Generator_sign': per_component('Generator', first_scenario(n.generators['sign'])),
'Load_p_set': varying(n, 'Load', 'p_set'),
'Load_sign': per_component('Load', first_scenario(loads['sign'])),
'Load_active': per_component('Load', first_scenario(loads['active']), bool),
}
with sps.solve('differential/pypsa/rungs/rung_06_kvl.yaml', sources) as solution:
solution.objective # 23962.0
The network, rung_06_kvl.py in the corpus — the spine plus what this rung adds:
"""Rung 6: KVL — passive lines under Kirchhoff's voltage law."""
from __future__ import annotations
from math import nan
import spine
def build():
"""The spine plus this rung's additions, as a ``pypsa.Network``."""
n = spine.build()
n.add('Bus', 'a')
n.add('Bus', 'b')
n.add('Bus', 'c')
n.add('Generator', 'hydro', bus='a', p_nom=80, marginal_cost=10)
n.add('Generator', 'diesel6', bus='b', p_nom=80, marginal_cost=50)
n.add('Load', 'town', bus='c', p_set=45)
n.add('Line', 'ab', bus0='a', bus1='b', carrier='AC', length=30, x=0.1, r=0.01, s_nom=60)
n.add('Line', 'bc', bus0='b', bus1='c', carrier='AC', length=40, x=0.2, r=0.01, s_nom=60, s_set=[16, nan, nan, nan])
n.add('Line', 'ca', bus0='c', bus1='a', carrier='AC', length=35, x=0.1, r=0.01, s_nom=60)
n.add(
'Line',
'ca2',
bus0='c',
bus1='a',
carrier='AC',
length=50,
x=0.15,
r=0.01,
s_nom_extendable=True,
capital_cost=10,
s_nom_max=40,
s_nom_set=30,
)
n.add(
'Line',
'ca3',
bus0='c',
bus1='a',
carrier='AC',
length=80,
x=0.12,
r=0.01,
s_nom_extendable=True,
capital_cost=8,
s_nom_max=40,
)
n.add(
'GlobalConstraint',
'vol_ac',
type='transmission_volume_expansion_limit',
carrier_attribute='AC',
sense='==',
constant=2300,
)
n.add(
'GlobalConstraint',
'vol_ac_floor',
type='transmission_volume_expansion_limit',
carrier_attribute='AC',
sense='>=',
constant=1000,
)
n.add(
'GlobalConstraint',
'cost_ac',
type='transmission_expansion_cost_limit',
carrier_attribute='AC',
sense='<=',
constant=500,
)
n.add(
'GlobalConstraint',
'cost_ac_floor',
type='transmission_expansion_cost_limit',
carrier_attribute='AC',
sense='>=',
constant=100,
)
n.add(
'GlobalConstraint',
'tech_ac',
type='tech_capacity_expansion_limit',
carrier_attribute='AC',
sense='<=',
constant=60,
)
return n
The data¶
The tables this rung is the first to declare (18), as the prep produced them:
Line_active.csv
snapshot,line,value
2015-01-01T00:00:00.000000,ab,true
2015-01-01T00:00:00.000000,bc,true
2015-01-01T00:00:00.000000,ca,true
2015-01-01T00:00:00.000000,ca2,true
2015-01-01T00:00:00.000000,ca3,true
2015-01-01T01:00:00.000000,ab,true
2015-01-01T01:00:00.000000,bc,true
2015-01-01T01:00:00.000000,ca,true
2015-01-01T01:00:00.000000,ca2,true
2015-01-01T01:00:00.000000,ca3,true
2015-01-01T02:00:00.000000,ab,true
2015-01-01T02:00:00.000000,bc,true
2015-01-01T02:00:00.000000,ca,true
2015-01-01T02:00:00.000000,ca2,true
2015-01-01T02:00:00.000000,ca3,true
2015-01-01T03:00:00.000000,ab,true
2015-01-01T03:00:00.000000,bc,true
2015-01-01T03:00:00.000000,ca,true
2015-01-01T03:00:00.000000,ca2,true
2015-01-01T03:00:00.000000,ca3,true
Line_bus0.csv
Line_bus1.csv
Line_capital_cost.csv
Line_capital_weight.csv
Line_cycle_weight.csv
line,cycle,value
ab,0,10000.0
bc,0,20000.0
ca,0,10000.0
ca,1,10000.0
ca,2,10000.0
ca2,1,-15000.0
ca3,2,-12000.0
Line_expansion_cost_weight.csv
scenario,global_constraint,line,value
base,cost_ac,ca2,10.0
base,cost_ac,ca3,8.0
base,cost_ac_floor,ca2,10.0
base,cost_ac_floor,ca3,8.0
Line_s_max_pu.csv
scenario,snapshot,line,value
base,2015-01-01T00:00:00.000000,ab,1.0
base,2015-01-01T00:00:00.000000,bc,1.0
base,2015-01-01T00:00:00.000000,ca,1.0
base,2015-01-01T00:00:00.000000,ca2,1.0
base,2015-01-01T00:00:00.000000,ca3,1.0
base,2015-01-01T01:00:00.000000,ab,1.0
base,2015-01-01T01:00:00.000000,bc,1.0
base,2015-01-01T01:00:00.000000,ca,1.0
base,2015-01-01T01:00:00.000000,ca2,1.0
base,2015-01-01T01:00:00.000000,ca3,1.0
base,2015-01-01T02:00:00.000000,ab,1.0
base,2015-01-01T02:00:00.000000,bc,1.0
base,2015-01-01T02:00:00.000000,ca,1.0
base,2015-01-01T02:00:00.000000,ca2,1.0
base,2015-01-01T02:00:00.000000,ca3,1.0
base,2015-01-01T03:00:00.000000,ab,1.0
base,2015-01-01T03:00:00.000000,bc,1.0
base,2015-01-01T03:00:00.000000,ca,1.0
base,2015-01-01T03:00:00.000000,ca2,1.0
base,2015-01-01T03:00:00.000000,ca3,1.0
Line_s_nom.csv
Line_s_nom_extendable.csv
Line_s_nom_max.csv
Line_s_nom_min.csv
Line_s_nom_set.csv
Line_s_set.csv
Line_tech_capacity_weight.csv
Line_volume_weight.csv
scenario,global_constraint,line,value
base,vol_ac,ca2,50.0
base,vol_ac,ca3,80.0
base,vol_ac_floor,ca2,50.0
base,vol_ac_floor,ca3,80.0
cycle.csv
line.csv