Rung 2: storage — a cyclic battery, an inflow reservoir with a set state of charge, and a store¶

description: A plain `n.optimize()`, and its multi-period and stochastic classes, in one file. Every second-stage
  quantity spans a `scenario` (a future dispatch is chosen in) and every asset stands in the investment
  `period`s its build year and lifetime span. A parameter spans `scenario` exactly when PyPSA reads it
  per scenario. Capacity is chosen once, before the future is known, and paid once per active period at
  its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights,
  with a share priced at the tail through the CVaR rows, which stand only where that share is positive.
  A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses
  to the standard one. A security-constrained run copies each branch flow limit once per outage in an
  `outage` set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight,
  and the outage factors are data prep.
dimensions:
  scenario: {description: 'the futures dispatch is chosen in, each with a weight'}
  snapshot: {description: dispatch periods, dtype: datetime}
  bus: {description: network nodes}
  generator: {description: 'generating units, each on one bus'}
  link: {description: 'controllable connections, each from one bus to the buses it delivers to'}
  link_output: {description: 'a link''s output ports, one label per port a link declares — PyPSA''s `bus1`,
      `bus2`, … columns read long, so a link of any number of output ports is one term in the balance,
      data prep'}
  load: {description: 'demands, each on one bus'}
  storage_unit: {description: 'storage units, dispatch and store behind one bus connection'}
  store: {description: 'pure energy stores, each on one bus'}
  period: {description: investment periods — PyPSA's `investment_periods`, dtype: int}
relations:
  snapshot_period: {description: the investment period a snapshot falls in, key: snapshot, values: period}
  Generator_bus: {description: the bus a generator sits on, key: generator, values: bus}
  Link_bus0: {description: the bus a link leaves, key: link, values: bus}
  Link_output_link: {description: the link an output port belongs to, key: link_output, values: link}
  Link_output_bus: {description: 'the bus an output port delivers to — PyPSA''s `bus1`, `bus2`, … columns.
      A link of three output ports is three labels here rather than a third relation, so the file states
      any number of them', key: link_output, values: bus}
  Load_bus: {description: the bus a load sits on, key: load, values: bus}
  StorageUnit_bus: {description: the bus a storage unit sits on, key: storage_unit, values: bus}
  Store_bus: {description: the bus a store sits on, key: store, values: bus}
parameters:
  snapshot_weightings_objective:
    description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
    dims: [snapshot]
  Generator_p_nom:
    description: nominal power
    dims: [scenario, generator]
  Generator_p_nom_extendable:
    description: whether the nominal power is a decision
    dims: [generator]
    dtype: bool
  Generator_p_min_pu:
    description: least output, per unit of nominal power
    dims: [scenario, snapshot, generator]
  Generator_p_max_pu:
    description: most output, per unit of nominal power — an availability profile
    dims: [scenario, snapshot, generator]
  Generator_marginal_cost:
    description: cost of one unit of output
    dims: [scenario, snapshot, generator]
  Generator_marginal_cost_quadratic:
    description: cost of the square of one unit of output
    dims: [scenario, snapshot, generator]
  Generator_sign:
    description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
      for a unit that draws power. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [generator]
  Generator_committable:
    description: whether output is gated by an on/off status decision
    dims: [generator]
    dtype: bool
  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
  StorageUnit_active:
    description: whether a storage unit stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, storage_unit]
    dtype: bool
  Store_active:
    description: whether a store stands in a snapshot's period — PyPSA's `active`, data prep
    dims: [snapshot, store]
    dtype: bool
  snapshot_weightings_stores:
    description: PyPSA's `snapshot_weightings.stores` — hours a snapshot stands for in a storage balance
    dims: [snapshot]
  StorageUnit_p_nom:
    description: nominal power
    dims: [scenario, storage_unit]
  StorageUnit_p_nom_extendable:
    description: whether the nominal power is a decision
    dims: [storage_unit]
    dtype: bool
  StorageUnit_p_min_pu:
    description: most storing, per unit of nominal power and negated
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_p_max_pu:
    description: most dispatch, per unit of nominal power
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_max_hours:
    description: energy capacity, as hours of dispatch at nominal power
    dims: [scenario, storage_unit]
  StorageUnit_efficiency_store:
    description: share of the power drawn from the bus that becomes charge
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_efficiency_dispatch:
    description: share of the charge drawn down that reaches the bus
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_sign:
    description: the sign net dispatch enters its bus's balance with — PyPSA's `sign`, `1` unless given.
      PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [storage_unit]
  StorageUnit_retention:
    description: share of charge kept over a snapshot — PyPSA's `(1 - standing_loss) ** elapsed hours`,
      data prep
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_inflow:
    description: energy arriving per hour, a river into a reservoir
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_state_of_charge_initial:
    description: charge held before the first snapshot
    dims: [scenario, storage_unit]
  StorageUnit_cyclic_state_of_charge:
    description: whether the horizon closes on itself instead of opening on the initial charge
    dims: [scenario, storage_unit]
    dtype: bool
  StorageUnit_cyclic_state_of_charge_per_period:
    description: whether each investment period closes on itself instead of carrying its charge on to
      the next; it overrides `cyclic_state_of_charge` and `state_of_charge_initial_per_period`. PyPSA
      reads it only under `multi_investment_periods`, so data prep feeds false otherwise
    dims: [scenario, storage_unit]
    dtype: bool
  StorageUnit_state_of_charge_initial_per_period:
    description: whether each investment period opens on the initial charge instead of carrying the previous
      period's; PyPSA reads it only under `multi_investment_periods`, so data prep feeds false otherwise
    dims: [scenario, storage_unit]
    dtype: bool
  StorageUnit_opens_late:
    description: whether a snapshot is the first a storage unit stands in, where that is not the first
      of the horizon — PyPSA's `active.cumsum() == 1` over the snapshots it stands in, past the first
      snapshot, data prep; false in a run where every unit stands throughout
    dims: [snapshot, storage_unit]
    dtype: bool
  StorageUnit_inactive_snapshots:
    description: how many snapshots a storage unit does not stand in — PyPSA's `(~active).sum()`, data
      prep. A cyclic unit reaches back this many snapshots further, so it closes on the last snapshot
      it stands in
    dims: [storage_unit]
    dtype: int
  StorageUnit_marginal_cost:
    description: cost of one unit of dispatch
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_marginal_cost_quadratic:
    description: cost of the square of one unit of dispatch; storing is not charged
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_marginal_cost_storage:
    description: cost of one unit of charge held over one snapshot
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_spill_cost:
    description: cost of one unit of inflow passed on unused
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_p_set:
    description: a given net dispatch schedule; a unit without one has no row here
    dims: [scenario, snapshot, storage_unit]
  StorageUnit_state_of_charge_set:
    description: a given charge schedule; a unit without one has no row here
    dims: [scenario, snapshot, storage_unit]
  Store_e_nom:
    description: nominal energy capacity
    dims: [scenario, store]
  Store_e_nom_extendable:
    description: whether the nominal energy capacity is a decision
    dims: [store]
    dtype: bool
  Store_e_min_pu:
    description: least energy held, per unit of nominal capacity — negative for a store that may go short
    dims: [scenario, snapshot, store]
  Store_e_max_pu:
    description: most energy held, per unit of nominal capacity
    dims: [scenario, snapshot, store]
  Store_sign:
    description: the sign the power a store delivers enters its bus's balance with — PyPSA's `sign`, `1`
      unless given. PyPSA refuses one that differs by scenario (`consistency.py:1187`)
    dims: [store]
  Store_retention:
    description: share of energy kept over a snapshot — PyPSA's `(1 - standing_loss) ** elapsed hours`,
      data prep
    dims: [scenario, snapshot, store]
  Store_e_initial:
    description: energy held before the first snapshot
    dims: [scenario, store]
  Store_e_cyclic:
    description: whether the horizon closes on itself instead of opening on the initial energy
    dims: [scenario, store]
    dtype: bool
  Store_e_cyclic_per_period:
    description: whether each investment period closes on itself instead of carrying its energy on to
      the next; it overrides `e_cyclic` and `e_initial_per_period`. PyPSA reads it only under `multi_investment_periods`,
      so data prep feeds false otherwise
    dims: [scenario, store]
    dtype: bool
  Store_e_initial_per_period:
    description: whether each investment period opens on the initial energy instead of carrying the previous
      period's; PyPSA reads it only under `multi_investment_periods`, so data prep feeds false otherwise
    dims: [scenario, store]
    dtype: bool
  Store_opens_late:
    description: whether a snapshot is the first a store stands in, where that is not the first of the
      horizon — PyPSA's `active.cumsum() == 1` over the snapshots it stands in, past the first snapshot,
      data prep; false in a run where every store stands throughout
    dims: [snapshot, store]
    dtype: bool
  Store_inactive_snapshots:
    description: how many snapshots a store does not stand in — PyPSA's `(~active).sum()`, data prep.
      A cyclic store reaches back this many snapshots further, so it closes on the last snapshot it stands
      in
    dims: [store]
    dtype: int
  Store_marginal_cost:
    description: cost of one unit of power delivered
    dims: [scenario, snapshot, store]
  Store_marginal_cost_quadratic:
    description: cost of the square of the net power delivered, so charging costs as much as delivering
    dims: [scenario, snapshot, store]
  Store_marginal_cost_storage:
    description: cost of one unit of energy held over one snapshot
    dims: [scenario, snapshot, store]
  Store_e_set:
    description: a given energy schedule; a store without one has no row here
    dims: [scenario, snapshot, store]
variables:
  Generator_p:
    description: '`Generator-p` — output of a generator in a snapshot'
    dims: [scenario, snapshot, generator]
    where: Generator_active
  Link_p:
    description: '`Link-p` — PyPSA''s `p0`, the flow measured at the `Link_bus0` end: a positive value
      withdraws there and injects at every bus the link''s output ports deliver to'
    dims: [scenario, snapshot, link]
    where: Link_active
  StorageUnit_p_dispatch:
    description: '`StorageUnit-p_dispatch` — power delivered to the bus'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
  StorageUnit_p_store:
    description: '`StorageUnit-p_store` — power drawn from the bus into charge'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
  StorageUnit_state_of_charge:
    description: '`StorageUnit-state_of_charge` — energy held at the end of a snapshot'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
  StorageUnit_spill:
    description: '`StorageUnit-spill` — inflow passed on unused. Zero where there is no inflow, so the
      balance keeps its row there; the bounds are PyPSA''s, on the variable rather than as rows'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_inflow > 0 AND StorageUnit_active
    absence: zero
    bounds: {lower: 0, upper: StorageUnit_inflow}
  Store_e:
    description: '`Store-e` — energy held at the end of a snapshot'
    dims: [scenario, snapshot, store]
    where: Store_active
  Store_p:
    description: '`Store-p` — power delivered to the bus; charging is negative'
    dims: [scenario, snapshot, store]
    where: Store_active
  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
  StorageUnit_fix_p_dispatch_lower:
    description: '`StorageUnit-fix-p_dispatch-lower` — dispatch is non-negative'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_dispatch >= 0
  StorageUnit_fix_p_dispatch_upper:
    description: '`StorageUnit-fix-p_dispatch-upper` — a fixed unit dispatches at most its nominal power'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_dispatch <= StorageUnit_p_max_pu * StorageUnit_p_nom
  StorageUnit_fix_p_store_lower:
    description: '`StorageUnit-fix-p_store-lower` — storing is non-negative'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_store >= 0
  StorageUnit_fix_p_store_upper:
    description: '`StorageUnit-fix-p_store-upper` — a fixed unit stores at most its nominal power, the
      minimum-per-unit column carrying that cap negated'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_p_store <= -StorageUnit_p_min_pu * StorageUnit_p_nom
  StorageUnit_fix_state_of_charge_lower:
    description: '`StorageUnit-fix-state_of_charge-lower` — charge is non-negative'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_state_of_charge >= 0
  StorageUnit_fix_state_of_charge_upper:
    description: '`StorageUnit-fix-state_of_charge-upper` — a fixed unit holds at most its hours at nominal
      power'
    dims: [scenario, snapshot, storage_unit]
    where: not StorageUnit_p_nom_extendable AND StorageUnit_active
    expression: StorageUnit_state_of_charge <= StorageUnit_max_hours * StorageUnit_p_nom
  StorageUnit_energy_balance:
    description: '`StorageUnit-energy_balance` — the charge carried in, plus what is stored after its
      efficiency, less what dispatch draws down before its own, plus inflow not spilled'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_active
    expression: StorageUnit_state_of_charge == StorageUnit_charge_carried_in + StorageUnit_efficiency_store
      * StorageUnit_p_store * snapshot_weightings_stores - StorageUnit_p_dispatch * snapshot_weightings_stores
      / StorageUnit_efficiency_dispatch + (StorageUnit_inflow - StorageUnit_spill) * snapshot_weightings_stores
  Store_fix_e_lower:
    description: '`Store-fix-e-lower` — a fixed store holds at least its floor'
    dims: [scenario, snapshot, store]
    where: not Store_e_nom_extendable AND Store_active
    expression: Store_e >= Store_e_min_pu * Store_e_nom
  Store_fix_e_upper:
    description: '`Store-fix-e-upper` — a fixed store holds at most its nominal capacity'
    dims: [scenario, snapshot, store]
    where: not Store_e_nom_extendable AND Store_active
    expression: Store_e <= Store_e_max_pu * Store_e_nom
  Store_energy_balance:
    description: '`Store-energy_balance` — the energy carried in, less what is delivered to the bus'
    dims: [scenario, snapshot, store]
    where: Store_active
    expression: Store_e == Store_energy_carried_in - Store_p * snapshot_weightings_stores
  StorageUnit_p_set:
    description: '`StorageUnit-p_set` — net dispatch pinned to the given schedule, wherever one is given'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_p_set AND StorageUnit_active
    expression: StorageUnit_p_dispatch - StorageUnit_p_store == StorageUnit_p_set
  StorageUnit_state_of_charge_set:
    description: '`StorageUnit-state_of_charge_set` — charge pinned to the given schedule, wherever one
      is given'
    dims: [scenario, snapshot, storage_unit]
    where: StorageUnit_state_of_charge_set AND StorageUnit_active
    expression: StorageUnit_state_of_charge == StorageUnit_state_of_charge_set
  Store_e_set:
    description: '`Store-e_set` — energy pinned to the given schedule, wherever one is given'
    dims: [scenario, snapshot, store]
    where: Store_e_set AND Store_active
    expression: Store_e == Store_e_set
  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:
  StorageUnit_charge_carried_in:
    description: the charge a unit opens a snapshot with — at the first snapshot it stands in, its last
      such snapshot's less standing loss where it is cyclic and the given initial charge, which no standing
      loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A unit
      built in a later period opens in that period, and a cyclic one that retires closes on its own last
      snapshot. Per period, the same holds with each investment period as the horizon
    dims: [scenario, snapshot, storage_unit]
    cases:
      cyclic: {when: StorageUnit_cyclic_state_of_charge AND NOT StorageUnit_cyclic_state_of_charge_per_period
          AND NOT StorageUnit_state_of_charge_initial_per_period AND (position(snapshot) == 0 OR StorageUnit_opens_late),
        expression: 'StorageUnit_retention * shift(shift(StorageUnit_state_of_charge, along=snapshot,
          offset=1, edge=''wrap''), along=snapshot, offset=StorageUnit_inactive_snapshots, edge=''wrap'')'}
      opening: {when: NOT StorageUnit_cyclic_state_of_charge AND NOT StorageUnit_cyclic_state_of_charge_per_period
          AND NOT StorageUnit_state_of_charge_initial_per_period AND (position(snapshot) == 0 OR StorageUnit_opens_late),
        expression: StorageUnit_state_of_charge_initial}
      period_cyclic: {when: StorageUnit_cyclic_state_of_charge_per_period, expression: 'StorageUnit_retention
          * shift(StorageUnit_state_of_charge, along=snapshot, offset=1, edge=''wrap'', by=snapshot_period,
          within=period)'}
      period_opening: {when: 'StorageUnit_state_of_charge_initial_per_period AND NOT StorageUnit_cyclic_state_of_charge_per_period
          AND position(snapshot, by=snapshot_period, within=period) == 0', expression: StorageUnit_state_of_charge_initial}
    otherwise: StorageUnit_retention * shift(StorageUnit_state_of_charge, along=snapshot, offset=1)
  Store_energy_carried_in:
    description: the energy a store opens a snapshot with — at the first snapshot it stands in, its last
      such snapshot's less standing loss where it is cyclic and the given initial energy, which no standing
      loss has touched yet, where it is not; the previous snapshot's less standing loss otherwise. A store
      built in a later period opens in that period, and a cyclic one that retires closes on its own last
      snapshot. Per period, the same holds with each investment period as the horizon
    dims: [scenario, snapshot, store]
    cases:
      cyclic: {when: Store_e_cyclic AND NOT Store_e_cyclic_per_period AND NOT Store_e_initial_per_period
          AND (position(snapshot) == 0 OR Store_opens_late), expression: 'Store_retention * shift(shift(Store_e,
          along=snapshot, offset=1, edge=''wrap''), along=snapshot, offset=Store_inactive_snapshots, edge=''wrap'')'}
      opening: {when: NOT Store_e_cyclic AND NOT Store_e_cyclic_per_period AND NOT Store_e_initial_per_period
          AND (position(snapshot) == 0 OR Store_opens_late), expression: Store_e_initial}
      period_cyclic: {when: Store_e_cyclic_per_period, expression: 'Store_retention * shift(Store_e, along=snapshot,
          offset=1, edge=''wrap'', by=snapshot_period, within=period)'}
      period_opening: {when: 'Store_e_initial_per_period AND NOT Store_e_cyclic_per_period AND position(snapshot,
          by=snapshot_period, within=period) == 0', expression: Store_e_initial}
    otherwise: Store_retention * shift(Store_e, along=snapshot, offset=1)
  total_cost:
    dims: []
    expression: risk_weighted_opex
    description: what the system costs — capacity once per active period at its expected cost over the
      scenarios, operation in expectation over the scenarios, and a share of it at the tail
  Bus_injection:
    dims: [scenario, snapshot, bus]
    expression: (((Generator_injection + Link_injection) + Load_injection) + StorageUnit_injection) +
      Store_injection
    description: what every component puts into a bus, less what it takes out of it; PyPSA writes each
      term into the balance, and a load on its right-hand side
  risk_weighted_opex: {expression: '(1 - CVaR_omega) * sum(scenario_weight * scenario_opex, over=scenario)
      + CVaR_omega * CVaR'}
  Generator_injection: {expression: 'sum(Generator_sign * Generator_p, by=Generator_bus, over=generator,
      into=bus)'}
  Link_injection: {expression: '-sum(Link_p, by=Link_bus0, over=link, into=bus) + sum(Link_output_arrival,
      by=Link_output_bus, over=link_output, into=bus)'}
  Load_injection: {expression: 'sum(Load_demand, by=Load_bus, over=load, into=bus)'}
  StorageUnit_injection: {expression: 'sum(StorageUnit_sign * (StorageUnit_p_dispatch - StorageUnit_p_store),
      by=StorageUnit_bus, over=storage_unit, into=bus)'}
  Store_injection: {expression: 'sum(Store_sign * Store_p, by=Store_bus, over=store, into=bus)'}
  Link_output_arrival:
    description: what a link delivers to an output port at a snapshot — its flow delayed by the port's
      `delay` within its investment period, times the port's efficiency at the snapshot the flow arrives;
      where the port is `cyclic_delay` the delayed flow wraps from the period's end, and where it is not
      the flow still in transit at the period's first snapshots is lost. A port that does not delay (`delay`
      zero) delivers its flow unshifted, cyclic or not
    dims: [scenario, snapshot, link_output]
    cases:
      wrapping: {when: Link_output_cyclic_delay, expression: 'shift(at(Link_p, by=Link_output_link, over=link,
          into=link_output), along=snapshot, offset=Link_output_delay, edge=''wrap'', by=snapshot_period,
          within=period) * Link_efficiency'}
    otherwise: shift(at(Link_p, by=Link_output_link, over=link, into=link_output), along=snapshot, offset=Link_output_delay,
      edge=0, by=snapshot_period, within=period) * Link_efficiency
  scenario_opex:
    dims: [scenario]
    expression: ((Generator_opex + Link_opex) + StorageUnit_opex) + Store_opex
    description: what a future costs to run — every operating term, weighted by the snapshot's hours and
      its period, before the scenario's own weight; a start and a stop cost what they cost, unweighted,
      as PyPSA adds them (`optimize.py:414-429`)
  Load_demand:
    description: what a load draws from its bus's balance — its demand times its sign where it is active,
      nothing where it is not, since PyPSA drops an inactive load from the balance (`constraints.py:1537-1538`)
    dims: [scenario, snapshot, load]
    cases:
      active: {when: Load_active, expression: Load_sign * Load_p_set}
    otherwise: 0
  Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=generator),
      over=snapshot) + sum(sum((((Generator_p * Generator_p) * Generator_marginal_cost_quadratic) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=generator),
      over=snapshot)'}
  Link_opex: {expression: 'sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period, over=period, into=snapshot), over=link), over=snapshot) + sum(sum((((Link_p
      * Link_p) * Link_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period, over=period, into=snapshot), over=link), over=snapshot)'}
  StorageUnit_opex: {expression: 'sum(sum(((StorageUnit_p_dispatch * StorageUnit_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=storage_unit),
      over=snapshot) + sum(sum((((StorageUnit_p_dispatch * StorageUnit_p_dispatch) * StorageUnit_marginal_cost_quadratic)
      * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
      into=snapshot), over=storage_unit), over=snapshot) + sum(sum(((StorageUnit_state_of_charge * StorageUnit_marginal_cost_storage)
      * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
      into=snapshot), over=storage_unit), over=snapshot) + sum(sum(((StorageUnit_spill * StorageUnit_spill_cost)
      * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period,
      into=snapshot), over=storage_unit), over=snapshot)'}
  Store_opex: {expression: 'sum(sum(((Store_p * Store_marginal_cost) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)
      + sum(sum((((Store_p * Store_p) * Store_marginal_cost_quadratic) * snapshot_weightings_objective)
      * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)
      + sum(sum(((Store_e * Store_marginal_cost_storage) * snapshot_weightings_objective) * at(period_weight_objective,
      by=snapshot_period, over=period, into=snapshot), over=store), over=snapshot)'}
objective: {sense: minimize, expression: total_cost}

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'),
    'StorageUnit_bus': relation(n, 'StorageUnit', 'bus'),
    'Store_bus': relation(n, 'Store', 'bus'),
    'snapshot_weightings_objective': weighting(n, 'objective'),
    'Generator_sign': per_component('Generator', first_scenario(n.generators['sign'])),
    'Load_p_set': varying(n, 'Load', 'p_set'),
    'Load_sign': per_component('Load', first_scenario(loads['sign'])),
    'Load_active': per_component('Load', first_scenario(loads['active']), bool),
    'snapshot_weightings_stores': weighting(n, 'stores'),
}

with sps.solve('differential/pypsa/rungs/rung_02_storage.yaml', sources) as solution:
    solution.objective  # 4456.659315422355

"""Rung 2: storage — a cyclic battery, an inflow reservoir with a set state of charge, and a store."""

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.generators_t.marginal_cost['gas'] = [15, 15, 60, 60]
    n.add(
        'StorageUnit',
        'battery',
        bus='south',
        p_nom=20,
        max_hours=4,
        efficiency_store=0.95,
        efficiency_dispatch=0.9,
        standing_loss=0.01,
        cyclic_state_of_charge=True,
        marginal_cost=0.5,
        p_set=[0, nan, nan, nan],
    )
    n.add(
        'StorageUnit',
        'reservoir',
        bus='south',
        p_nom=10,
        max_hours=2,
        spill_cost=2,
        state_of_charge_initial=5,
        marginal_cost_storage=0.1,
        inflow=[12, 12, 12, 12],
        state_of_charge_set=[nan, nan, nan, 10],
    )
    n.add(
        'Store',
        'cavern',
        bus='south',
        e_nom=40,
        e_initial=25,
        standing_loss=0.005,
        marginal_cost=0.2,
        e_set=[nan, nan, nan, 20],
    )
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 4456.659315422355