Rung 19: transmission losses in secant form — the same loss per line, its cuts placed by PyPSA's tolerance loop¶

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'}
  segment: {description: 'the cuts a passive branch''s loss curve is held above — PyPSA''s tangents, as
      many as its `segments` count, or its secants, as many as its tolerance loop places; none in a lossless
      run', dtype: int}
  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_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]
  transmission_losses:
    description: 'whether the network dissipates transmission losses — PyPSA''s `transmission_losses`
      read as a flag; its mode, tangents or secants, only decides how data prep fills the `segment` axis,
      the rows are the same; false with no segments is a lossless run. A security-constrained run over
      a network with passive branches builds no loss: PyPSA does not hand the keyword to `create_model`
      (`abstract.py:437-441`) but to the solver (`:491`), so data prep feeds false there'
    dims: []
    dtype: bool
  Line_loss_max:
    description: the loss at a line's rating — PyPSA's `r_pu_eff * (s_max_pu * s_nom_max)**2`, data prep
    dims: [scenario, snapshot, line]
  Line_loss_slope:
    description: the slope of a cut to the loss curve — a tangent's `2 * r_pu_eff * p_k` at its segment's
      flow, a secant's `r_pu_eff * (p_k + p_k+1)` between consecutive breakpoints, data prep
    dims: [scenario, snapshot, line, segment]
  Line_loss_offset:
    description: where that cut meets the loss axis — a tangent's `loss_k - slope_k * p_k`, a secant's
      `-r_pu_eff * p_k * p_k+1`, negative, data prep
    dims: [scenario, snapshot, line, segment]
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_loss:
    description: '`Line-loss` — what a line dissipates carrying its flow, pushed down by the cost and
      held up by the cuts; absent, and zero in the balance, where the network is lossless'
    dims: [scenario, snapshot, line]
    where: transmission_losses AND Line_active
    absence: zero
    bounds: {lower: 0}
  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_loss >= -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_loss <= 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_loss >= -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_loss <= 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_loss_upper:
    description: '`Line-loss_upper` — a line dissipates at most the loss at its rating'
    dims: [scenario, snapshot, line]
    where: transmission_losses AND Line_active
    expression: Line_loss <= Line_loss_max
  Line_loss_tangents_forward:
    description: '`Line-loss_tangents-{k}-1`, `Line-loss_secants-pos` — the loss sits above every cut
      to its curve for flow one way; PyPSA names one row per tangent `k`, or one row stacked over its
      `secant` axis, and this block states them all over the segment dimension'
    dims: [scenario, snapshot, line, segment]
    where: transmission_losses AND Line_active
    expression: Line_loss + Line_loss_slope * Line_s >= Line_loss_offset
  Line_loss_tangents_reverse:
    description: '`Line-loss_tangents-{k}--1`, `Line-loss_secants-neg` — the same fan mirrored, the loss
      depending on the flow''s magnitude'
    dims: [scenario, snapshot, line, segment]
    where: transmission_losses AND Line_active
    expression: Line_loss - Line_loss_slope * Line_s >= Line_loss_offset
  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
  Bus_nodal_balance:
    description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
      less what the links take away, plus what arrives over them after losses and any delay at every port
      they deliver to, each process port drawing or delivering at its own rate and each passive branch
      carrying its flow, meets the load there, less half of every incident line''s and transformer''s
      loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
      and load term enters with its component''s `sign` (`constraints.py:1428-1429`, `:1538`), and an
      inactive load not at all. A bus nothing is attached to has no row; PyPSA refuses one that carries
      load, and this file does not yet.'
    dims: [scenario, snapshot, bus]
    expression: Bus_injection == 0
expressions:
  total_cost:
    dims: []
    expression: 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_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) - (0.5 * sum(Line_loss, by=Line_bus0, over=line, into=bus)) - (0.5 * sum(Line_loss,
      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}

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_19_losses_secants.yaml', sources) as solution:
    solution.objective  # 11000.926895

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

"""Rung 19: transmission losses in secant form — the same loss per line, its cuts placed by PyPSA's tolerance loop."""

from __future__ import annotations

import spine

OPTIMIZE = {'transmission_losses': {'mode': 'secants', 'atol': 1, 'rtol': 0.1, 'max_segments': 20}}


def build():
    """Rung 13's 110 kV triangle, unchanged, so the two modes differ only in the cuts."""
    n = spine.build()
    n.add('Bus', ['a', 'b', 'c'], v_nom=110)
    n.add('Generator', 'hydro19', bus='a', p_nom=80, marginal_cost=10)
    n.add('Generator', 'diesel19', bus='b', p_nom=80, marginal_cost=50)
    n.add('Line', 'ab19', bus0='a', bus1='b', carrier='AC', x=30, r=6, s_nom=60)
    n.add('Line', 'bc19', bus0='b', bus1='c', carrier='AC', x=60, r=9.7, s_nom=60)
    n.add(
        'Line',
        'ca19',
        bus0='c',
        bus1='a',
        carrier='AC',
        x=45,
        r=6,
        s_nom=40,
        s_nom_extendable=True,
        s_nom_max=90,
        capital_cost=4,
    )
    n.add('Load', 'town19', bus='c', p_set=[35, 55, 15, 45])
    return n

n = build()
n.optimize(solver_name='highs')
n.objective  # 11000.926895