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Rung 15: two investment periods — build years, lifetimes, period weights and a carrier's growth limit

One rung of the PyPSA corpus: the file pypsa.yaml projected onto what this network builds, attached to that network, and held to what PyPSA solves it to.

✔ Verified against pypsa master at 51986084 — objective 12747.191096 on both sides; structure ✔ 10 constraints · 3 variables, name for name; size ✔ 80 rows · ✔ 31 columns · ✔ 117 nonzeros; duals ✔ 80 rows; model for model: 13 blocks equal, 0 documented splits, 1 recorded deviations.

Rows and columns, PyPSA against specsolve, name for name
row PyPSA specsolve
Bus-nodal_balance 16 16
Carrier-growth_limit 2 2
Generator-ext-p-lower 16 16
Generator-ext-p-upper 16 16
Generator-ext-p_nom-lower 3 3
Generator-ext-p_nom-upper 3 3
Generator-fix-p-lower 4 4
Generator-fix-p-upper 4 4
Link-fix-p-lower 8 8
Link-fix-p-upper 8 8
column PyPSA specsolve
Generator-p 20 20
Generator-p_nom 3 3
Link-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 a risk preference is set. A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses to the standard one. A security-constrained run copies each branch flow limit once per outage in an outage set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight, and the outage factors are data prep.

Sets

Symbol Meaning
\(\Xi\) index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight
\(\mathcal{T}\) index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods
\(\mathcal{N}\) index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — network nodes
\(\mathcal{G}\) index \(g\) — generator with \(\mathrm{Generator\_carrier}: \mathcal{G} \to \mathcal{I},\ \mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus
\(\mathcal{L}\) index \(l\) — link with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to
\(\mathcal{O}\) index \(o\) — link_output with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep
\(\mathcal{J}\) index \(j\) — process with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J}\) — generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate
\(\mathcal{R}\) index \(r\) — process_output with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N}\) — a process's ports, one label per port a process declares — PyPSA's bus0, bus1, … each carry a signed rate, so a process of any number of ports is one term in the balance, data prep
\(\mathcal{D}\) index \(d\) — load with \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus
\(\mathcal{Y}\) index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods
\(\mathcal{I}\) index \(i\) — carrier with \(\mathrm{Generator\_carrier}: \mathcal{G} \to \mathcal{I}\) — energy carriers, what a growth limit is set per

Parameters

Symbol Meaning
\(\mathrm{w}\) snapshot_weightings_objective over \(\mathcal{T}\) — PyPSA's snapshot_weightings.objective — hours a snapshot stands for in the cost
\(\mathrm{p}^{\mathrm{nom}}\) Generator_p_nom over \(\Xi \times \mathcal{G}\) — nominal power
\(\mathrm{ext}\) Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision
\(\underline{\mathrm{p}}\) Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power
\(\overline{\mathrm{p}}\) Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile
\(\mathrm{c}\) Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output
\(\mathrm{c}^{(2)}\) Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output
\(\mathrm{sgn}\) Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (constants.py:43)
\(\mathrm{com}\) Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision
\(\mathrm{f}^{\mathrm{nom}}\) Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power
\(\mathrm{ext}^{f}\) Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision
\(\underline{\mathrm{f}}\) Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways
\(\overline{\mathrm{f}}\) Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power
\(\eta\) Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py:1498)
\(\mathrm{d}^{f}\) Link_output_delay over \(\mathcal{O}\) — snapshots a port's delivery lags its link's flow — PyPSA's delay, delay2, … read long, divided by the snapshot_weightings.generators value and rounded up in data prep; zero for a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses a delay that differs by scenario (constants.py:52)
\(\mathrm{cyc}^{f}\) Link_output_cyclic_delay over \(\mathcal{O}\) — whether a delayed port's flow wraps from the end of its investment period — PyPSA's cyclic_delay, cyclic_delay2, …; where it does not, the flow still in transit at each period's first snapshots is lost. The flag is the same in every scenario, as the delay is
\(\mathrm{c}^{f}\) Link_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one unit of flow
\(\mathrm{c}^{f,(2)}\) Link_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of the square of one unit of flow
\(\mathrm{com}^{f}\) Link_committable over \(\mathcal{L}\) — whether flow is gated by an on/off status decision
\(\alpha\) Process_rate over \(\Xi \times \mathcal{T} \times \mathcal{R}\) — the energy a port draws or delivers per unit of internal power, PyPSA's rate0, rate1, … read long — negative where the port withdraws, positive where it injects; a link is a process whose bus0 rate is minus one and whose output rates are its efficiencies. Read at the snapshot the transfer arrives, so a delayed port transfers at its arrival snapshot's rate (constraints.py:1498)
\(\mathrm{d}^{z}\) Process_output_delay over \(\mathcal{R}\) — snapshots a port's transfer lags its process's internal power — PyPSA's delay0, delay1, … read long, divided by the snapshot_weightings.generators value and rounded up in data prep; zero for a port that transfers at once. A port has the same delay in every scenario, as a link's port has
\(\mathrm{cyc}^{z}\) Process_output_cyclic_delay over \(\mathcal{R}\) — whether a delayed port's transfer wraps from the end of its investment period — PyPSA's cyclic_delay0, cyclic_delay1, …; where it does not, the energy still in transit at each period's first snapshots is lost. The flag is the same in every scenario, as the delay is
\(\mathrm{load}\) Load_p_set over \(\Xi \times \mathcal{T} \times \mathcal{D}\) — demand
\(\mathrm{sgn}^{\mathrm{load}}\) Load_sign over \(\mathcal{D}\) — the sign a load's demand enters its bus's balance with — PyPSA's sign, -1 unless given, 1 for a load that feeds its bus. PyPSA refuses one that differs by scenario (constants.py:43)
\(\mathrm{on}^{\mathrm{load}}\) Load_active over \(\mathcal{D}\) — whether a load stands in the model — PyPSA's active. A load has no build year and no lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (constants.py:51)
\(\pi\) scenario_weight over \(\Xi\) — PyPSA's scenario_weightings.weight — the probability of a future
\(\mathrm{w}^{y}\) period_weight_objective over \(\mathcal{Y}\) — PyPSA's investment_period_weightings.objective — what a period's cost weighs; PyPSA reads it only under multi_investment_periods, so data prep feeds one otherwise, whatever the column holds (optimize.py:205-207, :264-266)
\(\mathrm{on}\) Generator_active over \(\mathcal{T} \times \mathcal{G}\) — whether a generator stands in a snapshot's period — PyPSA's active, from build year and lifetime, data prep
\(\mathrm{on}^{f}\) Link_active over \(\mathcal{T} \times \mathcal{L}\) — whether a link stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{on}^{z}\) Process_active over \(\mathcal{T} \times \mathcal{J}\) — whether a process stands in a snapshot's period — PyPSA's active, data prep
\(\mathrm{act}\) Generator_active_asset over \(\mathcal{G}\) — whether a generator is one of PyPSA's active_assets — its active flag alone, without build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA refuses one that differs by scenario (constants.py:51)
\(\mathrm{W}\) Generator_capital_weight over \(\mathcal{G}\) — the sum of period weights a generator stands in — PyPSA's active * period_weighting, summed, data prep
\(\mathrm{new}\) Generator_first_active over \(\mathcal{Y} \times \mathcal{G}\) — one in the first period a generator stands in, zero elsewhere, data prep. PyPSA takes active & (active.cumsum() == 1) (global_constraints.py:265)
\(\overline{\Delta}\) Carrier_max_growth over \(\mathcal{I}\), neutral where the data has no row — most capacity of a carrier that may be added in a period; no value means no limit. The least over the scenarios, as PyPSA takes it (global_constraints.py:227-231), data prep. PyPSA reads it only under multi_investment_periods (global_constraints.py:220-221), so data prep feeds no value otherwise
\(\mathrm{r}\) Carrier_max_relative_growth over \(\mathcal{I}\) — share of the previous period's additions that may be added on top — the least over the scenarios, as PyPSA takes it, data prep
\(\underline{\mathrm{p}}^{\mathrm{nom}}\) Generator_p_nom_min over \(\Xi \times \mathcal{G}\) — least nominal power an extendable generator may be built at
\(\overline{\mathrm{p}}^{\mathrm{nom}}\) Generator_p_nom_max over \(\Xi \times \mathcal{G}\) — most nominal power an extendable generator may be built at
\(\mathrm{c}^{\mathrm{cap}}\) Generator_capital_cost over \(\Xi \times \mathcal{G}\) — cost of one unit of nominal power for the modelled horizon — PyPSA's periodized_cost: overnight_cost as an annuity over lifetime at discount_rate, times nyears, where it is given, and capital_cost where it is not, plus fom_cost (components.py:1126-1147, costs.py:102-203), data prep

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
\(P\) Generator_p_nom_ext over \(\mathcal{G}\) — Generator-p_nom — nominal power where it is a decision, for a generator that is active; the parameter of the same PyPSA name carries the fixed regime
\(z\) Process_p over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-p — PyPSA's internal power p: a positive value drives every port at its own rate, withdrawing where the rate is negative and injecting where it is positive

Definitions

Symbol Meaning
\(\mathit{Link\_output\_arrival}\) Link_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — what a link delivers to an output port at a snapshot — its flow delayed by the port's delay within its investment period, times the port's efficiency at the snapshot the flow arrives; where the port is cyclic_delay the delayed flow wraps from the period's end, and where it is not the flow still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) delivers its flow unshifted, cyclic or not
\(\mathit{Process\_output\_arrival}\) Process_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{R}\) — what a process transfers at a port at a snapshot — its internal power delayed by the port's delay within its investment period, times the port's rate at the snapshot the transfer arrives; where the port is cyclic_delay the delayed transfer wraps from the period's end, and where it is not the energy still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) transfers at once, cyclic or not
\(\mathit{Generator\_opex}\) Generator_opex over \(\Xi\)
\(\mathit{Link\_opex}\) Link_opex over \(\Xi\)
\(\mathit{scenario\_opex}\) scenario_opex over \(\Xi\) — what a future costs to run — every operating term, weighted by the snapshot's hours and its period, before the scenario's own weight; a start and a stop cost what they cost, unweighted, as PyPSA adds them (optimize.py:415-432)
\(\mathit{Generator\_capex}\) Generator_capex (scalar)
\(\mathit{risk\_weighted\_opex}\) risk_weighted_opex (scalar)
\(\mathit{total\_cost}\) total_cost (scalar) — what the system costs — capacity once per active period at its expected cost over the scenarios, operation in expectation over the scenarios, and a share of it at the tail
\(\mathit{Generator\_additions}\) Generator_additions over \(\mathcal{I} \times \mathcal{Y}\)
\(\mathit{Carrier\_additions}\) Carrier_additions over \(\mathcal{Y} \times \mathcal{I}\) — what a carrier adds in a period — every extendable component of that carrier, counting each build in the first period it stands in. Like PyPSA, it sums only the components that carry a carrier attribute, so a transformer, which has none, counts in no carrier
\(\mathrm{Carrier\_relative\_growth}\) Carrier_relative_growth over \(\mathcal{I}\) — the share of the previous period's additions a carrier's growth limit reads — PyPSA's max_relative_growth clipped at zero, so a negative share adds nothing and never tightens the limit
\(\mathrm{Load\_demand}\) Load_demand over \(\Xi \times \mathcal{T} \times \mathcal{D}\) — what a load draws from its bus's balance — its demand times its sign where it is active, nothing where it is not, since PyPSA drops an inactive load from the balance (constraints.py:1513-1514)
\(\mathit{Generator\_injection}\) Generator_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Link\_injection}\) Link_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathrm{Load\_injection}\) Load_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Process\_injection}\) Process_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\)
\(\mathit{Bus\_injection}\) Bus_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) — what every component puts into a bus, less what it takes out of it; PyPSA writes each term into the balance, and a load on its right-hand side

Notation

Symbol Meaning
\(t \ominus k\) edge='wrap' — \(t-k\), cyclic
\(t \boxminus_{v} k\) edge=v — \(t-k\), and \(v\) past the edge
\(t \ominus^{\mathrm{relation}(t)} k\) within=relation[...] — \(t-k\) inside the group of \(t\)

Variable domains

Generator_p

\[ p_{\xi,t,g} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{on}_{t,g} \]

Link_p

\[ f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l} \]

Generator_p_nom_ext

\[ P_{g} \in \mathbb{R} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{act}_{g} \]

Process_p

\[ z_{\xi,t,j} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{on}^{z}_{t,j} \]

Definitions

Link_output_arrival

\[ \mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O} \]

Process_output_arrival

\[ \mathit{Process\_output\_arrival}_{\xi,t,r} = \begin{cases} z_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{if } \mathrm{cyc}^{z}_{r} \\ z_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ r \in \mathcal{R} \]

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

\[ \mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi \]

scenario_opex

\[ \mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} \qquad \forall\, \xi \in \Xi \]

Generator_capex

\[ \mathit{Generator\_capex} = \sum_{\xi \in \Xi,\ g \in \mathcal{G}} \pi_{\xi} \cdot P_{g} \cdot \mathrm{c}^{\mathrm{cap}}_{\xi,g} \cdot \mathrm{W}_{g} \]

risk_weighted_opex

\[ \mathit{risk\_weighted\_opex} = \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi} \]

total_cost

\[ \mathit{total\_cost} = \mathit{Generator\_capex} + \mathit{risk\_weighted\_opex} \]

Generator_additions

\[ \mathit{Generator\_additions}_{i,y} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_carrier}(g) = i} P_{g} \cdot \mathrm{new}_{y,g} \qquad \forall\, i \in \mathcal{I},\ y \in \mathcal{Y} \]

Carrier_additions

\[ \mathit{Carrier\_additions}_{y,i} = \mathit{Generator\_additions}_{i,y} \qquad \forall\, y \in \mathcal{Y},\ i \in \mathcal{I} \]

Carrier_relative_growth

\[ \mathrm{Carrier\_relative\_growth}_{i} = \begin{cases} \mathrm{r}_{i} & \text{if } \mathrm{r}_{i} > 0 \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, i \in \mathcal{I} \]

Load_demand

\[ \mathrm{Load\_demand}_{\xi,t,d} = \begin{cases} \mathrm{sgn}^{\mathrm{load}}_{d} \cdot \mathrm{load}_{\xi,t,d} & \text{if } \mathrm{on}^{\mathrm{load}}_{d} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ d \in \mathcal{D} \]

Generator_injection

\[ \mathit{Generator\_injection}_{n,\xi,t} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Link_injection

\[ \mathit{Link\_injection}_{n,\xi,t} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Load_injection

\[ \mathrm{Load\_injection}_{n,\xi,t} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Process_injection

\[ \mathit{Process\_injection}_{n,\xi,t} = \sum_{r \in \mathcal{R} \,:\, \mathrm{Process\_output\_bus}(r) = n} \mathit{Process\_output\_arrival}_{\xi,t,r} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Bus_injection

\[ \mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{n,\xi,t} + \mathit{Link\_injection}_{n,\xi,t} + \mathrm{Load\_injection}_{n,\xi,t} + \mathit{Process\_injection}_{n,\xi,t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

Objective

\[ \min \mathit{total\_cost} \]

Subject to

Generator_fix_p_lower

\[ p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \]

Generator_fix_p_upper

\[ p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g} \]

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

\[ P_{g} \ge \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{act}_{g} \]

Generator_ext_p_nom_upper

\[ P_{g} \le \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{act}_{g} \wedge \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \text{ is defined} \]

Bus_nodal_balance

\[ \mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N} \]

Carrier_growth_limit

\[ \mathit{Carrier\_additions}_{y,i} - \mathit{Carrier\_additions}_{y \boxminus_{0} 1,i} \cdot \mathrm{Carrier\_relative\_growth}_{i} \le \overline{\Delta}_{i} \qquad \forall\, i \in \mathcal{I},\ y \in \mathcal{Y} \,:\, \overline{\Delta}_{i} \text{ is defined} \]

The spec, differential/pypsa/rungs/rung_15_multi_period.yaml — the file projected onto what this rung builds:

description: A plain `n.optimize()`, and its multi-period and stochastic classes, in one file. Every second-stage
  quantity spans a `scenario` (a future dispatch is chosen in) and every asset stands in the investment
  `period`s its build year and lifetime span. A parameter spans `scenario` exactly when PyPSA reads it
  per scenario. Capacity is chosen once, before the future is known, and paid once per active period at
  its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights,
  with a share priced at the tail through the CVaR rows, which stand only where a risk preference is set.
  A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses
  to the standard one. A security-constrained run copies each branch flow limit once per outage in an
  `outage` set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight,
  and the outage factors are data prep.
dimensions:
  scenario: {description: 'the futures dispatch is chosen in, each with a weight'}
  snapshot: {description: dispatch periods, dtype: datetime, ordered: true}
  bus: {description: network nodes}
  generator: {description: 'generating units, each on one bus'}
  link: {description: 'controllable connections, each from one bus to the buses it delivers to'}
  link_output: {description: 'a link''s output ports, one label per port a link declares — PyPSA''s `bus1`,
      `bus2`, … columns read long, so a link of any number of output ports is one term in the balance,
      data prep'}
  process: {description: 'generalized multi-port converters, each with an internal power that every port
      draws or delivers at its own rate'}
  process_output: {description: 'a process''s ports, one label per port a process declares — PyPSA''s
      `bus0`, `bus1`, … each carry a signed `rate`, so a process of any number of ports is one term in
      the balance, data prep'}
  load: {description: 'demands, each on one bus'}
  period: {description: investment periods — PyPSA's `investment_periods`, dtype: int, ordered: true}
  carrier: {description: 'energy carriers, what a growth limit is set per'}
relations:
  snapshot_period: {description: the investment period a snapshot falls in, key: snapshot, values: period}
  Generator_carrier: {description: the carrier a generator converts from, key: generator, values: carrier}
  Generator_bus: {description: the bus a generator sits on, key: generator, values: bus}
  Link_bus0: {description: the bus a link leaves, key: link, values: bus}
  Link_output_link: {description: the link an output port belongs to, key: link_output, values: link}
  Link_output_bus: {description: 'the bus an output port delivers to — PyPSA''s `bus1`, `bus2`, … columns.
      A link of three output ports is three labels here rather than a third relation, so the file states
      any number of them', key: link_output, values: bus}
  Process_output_process: {description: the process a port belongs to, key: process_output, values: process}
  Process_output_bus: {description: 'the bus a port draws from or delivers to — PyPSA''s `bus0`, `bus1`,
      … columns. A process of three ports is three labels here rather than a third relation, so the file
      states any number of them', key: process_output, values: bus}
  Load_bus: {description: the bus a load sits on, key: load, values: bus}
parameters:
  snapshot_weightings_objective:
    description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
    dims: [snapshot]
  Generator_p_nom:
    description: nominal power
    dims: [scenario, generator]
  Generator_p_nom_extendable:
    description: whether the nominal power is a decision
    dims: [generator]
    dtype: bool
  Generator_p_min_pu:
    description: least output, per unit of nominal power
    dims: [scenario, snapshot, generator]
  Generator_p_max_pu:
    description: most output, per unit of nominal power — an availability profile
    dims: [scenario, snapshot, generator]
  Generator_marginal_cost:
    description: cost of one unit of output
    dims: [scenario, snapshot, generator]
  Generator_marginal_cost_quadratic:
    description: cost of the square of one unit of output
    dims: [scenario, snapshot, generator]
  Generator_sign:
    description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
      for a unit that draws power. PyPSA refuses one that differs by scenario (`constants.py:43`)
    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:1498`)
    dims: [scenario, snapshot, link_output]
  Link_output_delay:
    description: snapshots a port's delivery lags its link's flow — PyPSA's `delay`, `delay2`, … read
      long, divided by the `snapshot_weightings.generators` value and rounded up in data prep; zero for
      a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses
      a delay that differs by scenario (`constants.py:52`)
    dims: [link_output]
    dtype: int
  Link_output_cyclic_delay:
    description: whether a delayed port's flow wraps from the end of its investment period — PyPSA's `cyclic_delay`,
      `cyclic_delay2`, …; where it does not, the flow still in transit at each period's first snapshots
      is lost. The flag is the same in every scenario, as the delay is
    dims: [link_output]
    dtype: bool
  Link_marginal_cost:
    description: cost of one unit of flow
    dims: [scenario, snapshot, link]
  Link_marginal_cost_quadratic:
    description: cost of the square of one unit of flow
    dims: [scenario, snapshot, link]
  Link_committable:
    description: whether flow is gated by an on/off status decision
    dims: [link]
    dtype: bool
  Process_rate:
    description: the energy a port draws or delivers per unit of internal power, PyPSA's `rate0`, `rate1`,
      … read long — negative where the port withdraws, positive where it injects; a link is a process
      whose `bus0` rate is minus one and whose output rates are its efficiencies. Read at the snapshot
      the transfer arrives, so a delayed port transfers at its arrival snapshot's rate (`constraints.py:1498`)
    dims: [scenario, snapshot, process_output]
  Process_output_delay:
    description: snapshots a port's transfer lags its process's internal power — PyPSA's `delay0`, `delay1`,
      … read long, divided by the `snapshot_weightings.generators` value and rounded up in data prep;
      zero for a port that transfers at once. A port has the same delay in every scenario, as a link's
      port has
    dims: [process_output]
    dtype: int
  Process_output_cyclic_delay:
    description: whether a delayed port's transfer wraps from the end of its investment period — PyPSA's
      `cyclic_delay0`, `cyclic_delay1`, …; where it does not, the energy still in transit at each period's
      first snapshots is lost. The flag is the same in every scenario, as the delay is
    dims: [process_output]
    dtype: bool
  Load_p_set:
    description: demand
    dims: [scenario, snapshot, load]
  Load_sign:
    description: the sign a load's demand enters its bus's balance with — PyPSA's `sign`, `-1` unless
      given, `1` for a load that feeds its bus. PyPSA refuses one that differs by scenario (`constants.py:43`)
    dims: [load]
  Load_active:
    description: whether a load stands in the model — PyPSA's `active`. A load has no build year and no
      lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (`constants.py:51`)
    dims: [load]
    dtype: bool
  scenario_weight:
    description: PyPSA's `scenario_weightings.weight` — the probability of a future
    dims: [scenario]
  period_weight_objective:
    description: PyPSA's `investment_period_weightings.objective` — what a period's cost weighs; PyPSA
      reads it only under `multi_investment_periods`, so data prep feeds one otherwise, whatever the column
      holds (`optimize.py:205-207`, `:264-266`)
    dims: [period]
  Generator_active:
    description: whether a generator stands in a snapshot's period — PyPSA's `active`, from build year
      and lifetime, data prep
    dims: [snapshot, generator]
    dtype: bool
  Link_active:
    description: whether a link stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, link]
    dtype: bool
  Process_active:
    description: whether a process stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, process]
    dtype: bool
  Generator_active_asset:
    description: whether a generator is one of PyPSA's `active_assets` — its `active` flag alone, without
      build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA
      refuses one that differs by scenario (`constants.py:51`)
    dims: [generator]
    dtype: bool
  Generator_capital_weight:
    description: the sum of period weights a generator stands in — PyPSA's `active * period_weighting`,
      summed, data prep
    dims: [generator]
  Generator_first_active:
    description: one in the first period a generator stands in, zero elsewhere, data prep. PyPSA takes
      `active & (active.cumsum() == 1)` (`global_constraints.py:265`)
    dims: [period, generator]
  Carrier_max_growth:
    description: most capacity of a carrier that may be added in a period; no value means no limit. The
      least over the scenarios, as PyPSA takes it (`global_constraints.py:227-231`), data prep. PyPSA
      reads it only under `multi_investment_periods` (`global_constraints.py:220-221`), so data prep feeds
      no value otherwise
    dims: [carrier]
    missing: neutral
  Carrier_max_relative_growth:
    description: share of the previous period's additions that may be added on top — the least over the
      scenarios, as PyPSA takes it, data prep
    dims: [carrier]
  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 for the modelled horizon — PyPSA''s `periodized_cost`:
      `overnight_cost` as an annuity over `lifetime` at `discount_rate`, times `nyears`, where it is given,
      and `capital_cost` where it is not, plus `fom_cost` (`components.py:1126-1147`, `costs.py:102-203`),
      data prep'
    dims: [scenario, generator]
variables:
  Generator_p:
    description: '`Generator-p` — output of a generator in a snapshot'
    dims: [scenario, snapshot, generator]
    where: Generator_active
  Link_p:
    description: '`Link-p` — PyPSA''s `p0`, the flow measured at the `Link_bus0` end: a positive value
      withdraws there and injects at every bus the link''s output ports deliver to'
    dims: [scenario, snapshot, link]
    where: Link_active
  Generator_p_nom_ext:
    description: '`Generator-p_nom` — nominal power where it is a decision, for a generator that is active;
      the parameter of the same PyPSA name carries the fixed regime'
    dims: [generator]
    where: Generator_p_nom_extendable AND Generator_active_asset
  Process_p:
    description: '`Process-p` — PyPSA''s internal power `p`: a positive value drives every port at its
      own rate, withdrawing where the rate is negative and injecting where it is positive'
    dims: [scenario, snapshot, process]
    where: Process_active
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 AND Generator_active_asset
    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_active_asset AND Generator_p_nom_max
    expression: Generator_p_nom_ext <= Generator_p_nom_max
  Bus_nodal_balance:
    description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
      less what the links take away, plus what arrives over them after losses and any delay at every port
      they deliver to, each process port drawing or delivering at its own rate and each passive branch
      carrying its flow, meets the load there, less half of every incident line''s and transformer''s
      loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
      and load term enters with its component''s `sign` (`constraints.py:1404-1405`, `:1514`), and an
      inactive load not at all. A bus nothing is attached to has no row; PyPSA refuses one that carries
      load, and this file does not yet.'
    dims: [scenario, snapshot, bus]
    expression: Bus_injection == 0
  Carrier_growth_limit:
    description: '`Carrier-growth_limit` — what a carrier adds across its extendable components in a period,
      counting each build in the first period it stands in, is at most its allowance plus a share of what
      it added the period before; the first period has no predecessor, so `edge=0` leaves it the bare
      allowance'
    dims: [carrier, period]
    where: Carrier_max_growth
    expression: Carrier_additions - shift(Carrier_additions, along=period, offset=1, edge=0) * Carrier_relative_growth
      <= Carrier_max_growth
expressions:
  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[link]),
          along=snapshot, offset=Link_output_delay, edge=''wrap'', within=snapshot_period[period]) * Link_efficiency'}
    otherwise: shift(at(Link_p, by=Link_output_link[link]), along=snapshot, offset=Link_output_delay,
      edge=0, within=snapshot_period[period]) * Link_efficiency
  Process_output_arrival:
    description: what a process transfers at a port at a snapshot — its internal power delayed by the
      port's `delay` within its investment period, times the port's rate at the snapshot the transfer
      arrives; where the port is `cyclic_delay` the delayed transfer wraps from the period's end, and
      where it is not the energy still in transit at the period's first snapshots is lost. A port that
      does not delay (`delay` zero) transfers at once, cyclic or not
    dims: [scenario, snapshot, process_output]
    cases:
      wrapping: {when: Process_output_cyclic_delay, expression: 'shift(at(Process_p, by=Process_output_process[process]),
          along=snapshot, offset=Process_output_delay, edge=''wrap'', within=snapshot_period[period])
          * Process_rate'}
    otherwise: shift(at(Process_p, by=Process_output_process[process]), along=snapshot, offset=Process_output_delay,
      edge=0, within=snapshot_period[period]) * Process_rate
  Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period[period]), over=generator), over=snapshot) + sum(sum((((Generator_p
      * Generator_p) * Generator_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period[period]), over=generator), over=snapshot)'}
  Link_opex: {expression: 'sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period[period]), over=link), over=snapshot) + sum(sum((((Link_p * Link_p) * Link_marginal_cost_quadratic)
      * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period[period]), over=link),
      over=snapshot)'}
  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:415-432`)
  Generator_capex: {expression: sum(scenario_weight * Generator_p_nom_ext * Generator_capital_cost * Generator_capital_weight)}
  risk_weighted_opex: {expression: 'sum(scenario_weight * scenario_opex, over=scenario)'}
  total_cost:
    dims: []
    expression: Generator_capex + risk_weighted_opex
    description: what the system costs — capacity once per active period at its expected cost over the
      scenarios, operation in expectation over the scenarios, and a share of it at the tail
  Generator_additions: {expression: 'sum(Generator_p_nom_ext * Generator_first_active, over=generator,
      by=Generator_carrier[carrier])'}
  Carrier_additions:
    dims: [period, carrier]
    expression: Generator_additions
    description: what a carrier adds in a period — every extendable component of that carrier, counting
      each build in the first period it stands in. Like PyPSA, it sums only the components that carry
      a carrier attribute, so a transformer, which has none, counts in no carrier
  Carrier_relative_growth:
    description: the share of the previous period's additions a carrier's growth limit reads — PyPSA's
      `max_relative_growth` clipped at zero, so a negative share adds nothing and never tightens the limit
    dims: [carrier]
    cases:
      positive: {when: Carrier_max_relative_growth > 0, expression: Carrier_max_relative_growth}
    otherwise: 0
  Load_demand:
    description: what a load draws from its bus's balance — its demand times its sign where it is active,
      nothing where it is not, since PyPSA drops an inactive load from the balance (`constraints.py:1513-1514`)
    dims: [scenario, snapshot, load]
    cases:
      active: {when: Load_active, expression: Load_sign * Load_p_set}
    otherwise: 0
  Generator_injection: {expression: 'sum(Generator_sign * Generator_p, over=generator, by=Generator_bus[bus])'}
  Link_injection: {expression: '-sum(Link_p, over=link, by=Link_bus0[bus]) + sum(Link_output_arrival,
      over=link_output, by=Link_output_bus[bus])'}
  Load_injection: {expression: 'sum(Load_demand, over=load, by=Load_bus[bus])'}
  Process_injection: {expression: 'sum(Process_output_arrival, over=process_output, by=Process_output_bus[bus])'}
  Bus_injection:
    dims: [scenario, snapshot, bus]
    expression: ((Generator_injection + Link_injection) + Load_injection) + Process_injection
    description: what every component puts into a bus, less what it takes out of it; PyPSA writes each
      term into the balance, and a load on its right-hand side
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, multi),
        **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_15_multi_period.yaml', sources) as solution:
    solution.objective  # 12747.191096

The network, rung_15_multi_period.py in the corpus — the spine plus what this rung adds:

# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT

"""Rung 15: two investment periods — build years, lifetimes, period weights and a carrier's growth limit."""

from __future__ import annotations

from datetime import datetime

import pandas as pd

OPTIMIZE = {'multi_investment_periods': True}


def build():
    """A whole network, not the spine: eight snapshots over two periods, a unit that retires, two wind builds capped by growth."""
    import pypsa

    n = pypsa.Network()
    n.snapshots = pd.MultiIndex.from_tuples(
        [(2020, datetime(2020, 1, 1, t)) for t in range(4)] + [(2030, datetime(2030, 1, 1, t)) for t in range(4)]
    )
    n.investment_periods = [2020, 2030]
    n.investment_period_weightings['objective'] = [1.0, 0.5]
    n.investment_period_weightings['years'] = [10.0, 10.0]
    n.snapshot_weightings['objective'] = [2.0, 1.5, 2.5, 2.0, 2.0, 1.5, 2.5, 2.0]
    n.add('Bus', ['north', 'south'])
    n.add('Carrier', 'wind', max_growth=50, max_relative_growth=0.5)
    n.add('Carrier', 'gas')
    n.add('Generator', 'old_gas', bus='north', carrier='gas', p_nom=40, marginal_cost=30, build_year=2010, lifetime=15)
    n.add(
        'Generator',
        'wind20',
        bus='north',
        carrier='wind',
        p_nom_extendable=True,
        p_nom_max=200,
        marginal_cost=1,
        capital_cost=100,
        build_year=2020,
        lifetime=30,
        p_max_pu=[0.8, 0.6, 0.7, 0.5, 0.8, 0.6, 0.7, 0.5],
    )
    n.add(
        'Generator',
        'wind30',
        bus='south',
        carrier='wind',
        p_nom_extendable=True,
        p_nom_max=200,
        marginal_cost=1,
        capital_cost=80,
        build_year=2030,
        lifetime=30,
        p_max_pu=[0.9, 0.7, 0.6, 0.8, 0.9, 0.7, 0.6, 0.8],
    )
    n.add(
        'Generator',
        'gas30',
        bus='south',
        carrier='gas',
        p_nom_extendable=True,
        p_nom_max=200,
        marginal_cost=40,
        capital_cost=50,
        build_year=2030,
        lifetime=30,
    )
    n.add('Link', 'wire15', bus0='north', bus1='south', p_nom=60, p_min_pu=-1, efficiency=0.95)
    n.add('Load', 'town15', bus='north', p_set=[20, 30, 25, 20, 35, 45, 40, 30])
    n.add('Load', 'port15', bus='south', p_set=[10, 20, 15, 10, 30, 40, 35, 25])
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 12747.191096

The data

The tables this rung is the first to declare (5), as the prep produced them:

Carrier_max_growth.csv

carrier,value
wind,50.0

Carrier_max_relative_growth.csv

carrier,value
"",0.0
gas,0.0
wind,0.5

Generator_carrier.csv

generator,carrier
gas30,gas
old_gas,gas
wind20,wind
wind30,wind

Generator_first_active.csv

period,generator,value
2020,gas30,0.0
2020,old_gas,1.0
2020,wind20,1.0
2020,wind30,0.0
2030,gas30,1.0
2030,old_gas,0.0
2030,wind20,0.0
2030,wind30,1.0

carrier.csv

carrier
""
gas
wind