linopf.py

# -*- coding: utf-8 -*-

## Copyright 2015-2021 PyPSA Developers

## You can find the list of PyPSA Developers at
## https://pypsa.readthedocs.io/en/latest/developers.html

## PyPSA is released under the open source MIT License, see
## https://github.com/PyPSA/PyPSA/blob/master/LICENSE.txt

"""
Build optimisation problems from PyPSA networks without Pyomo.

Originally retrieved from nomopyomo ( -> 'no more Pyomo').
"""

__author__ = (
    "PyPSA Developers, see https://pypsa.readthedocs.io/en/latest/developers.html"
)
__copyright__ = (
    "Copyright 2015-2021 PyPSA Developers, see https://pypsa.readthedocs.io/en/latest/developers.html, "
    "MIT License"
)

from distutils.version import LooseVersion

import numpy as np
import pandas as pd
from numpy import inf

from pypsa.descriptors import (
    Dict,
    additional_linkports,
    expand_series,
    get_active_assets,
    get_activity_mask,
    get_bounds_pu,
    get_extendable_i,
    get_non_extendable_i,
    nominal_attrs,
)
from pypsa.linopt import (
    align_with_static_component,
    define_binaries,
    define_constraints,
    define_variables,
    get_con,
    get_var,
    join_exprs,
    linexpr,
    run_and_read_cbc,
    run_and_read_cplex,
    run_and_read_glpk,
    run_and_read_gurobi,
    run_and_read_highs,
    run_and_read_xpress,
    set_conref,
    set_varref,
    write_bound,
    write_constraint,
    write_objective,
)
from pypsa.pf import _as_snapshots
from pypsa.pf import get_switchable_as_dense as get_as_dense

pd_version = LooseVersion(pd.__version__)
agg_group_kwargs = dict(numeric_only=False) if pd_version >= "1.3" else {}

import gc
import logging
import os
import re
import shutil
import time
from tempfile import mkstemp

logger = logging.getLogger(__name__)


lookup = pd.read_csv(
    os.path.join(os.path.dirname(__file__), "variables.csv"),
    index_col=["component", "variable"],
)


def define_nominal_for_extendable_variables(n, c, attr):
    """
    Initializes variables for nominal capacities for a given component and a
    given attribute.

    Parameters
    ----------
    n : pypsa.Network
    c : str
        network component of which the nominal capacity should be defined
    attr : str
        name of the variable, e.g. 'p_nom'
    """
    ext_i = get_extendable_i(n, c)
    if ext_i.empty:
        return
    lower = n.df(c)[attr + "_min"][ext_i]
    upper = n.df(c)[attr + "_max"][ext_i]
    define_variables(n, lower, upper, c, attr)


def define_dispatch_for_extendable_and_committable_variables(n, sns, c, attr):
    """
    Initializes variables for power dispatch for a given component and a given
    attribute.

    Parameters
    ----------
    n : pypsa.Network
    c : str
        name of the network component
    attr : str
        name of the attribute, e.g. 'p'
    """
    ext_i = get_extendable_i(n, c)
    if c == "Generator":
        ext_i = ext_i.union(n.generators.query("committable").index)
    if ext_i.empty:
        return
    active = get_activity_mask(n, c, sns)[ext_i] if n._multi_invest else None
    define_variables(n, -inf, inf, c, attr, axes=[sns, ext_i], spec="ext", mask=active)


def define_dispatch_for_non_extendable_variables(n, sns, c, attr):
    """
    Initializes variables for power dispatch for a given component and a given
    attribute.

    Parameters
    ----------
    n : pypsa.Network
    c : str
        name of the network component
    attr : str
        name of the attribute, e.g. 'p'
    """
    fix_i = get_non_extendable_i(n, c)
    if c == "Generator":
        fix_i = fix_i.difference(n.generators.query("committable").index)
    if fix_i.empty:
        return
    nominal_fix = n.df(c)[nominal_attrs[c]][fix_i]
    min_pu, max_pu = get_bounds_pu(n, c, sns, fix_i, attr)
    lower = min_pu.mul(nominal_fix)
    upper = max_pu.mul(nominal_fix)
    axes = [sns, fix_i]

    active = get_activity_mask(n, c, sns)[fix_i] if n._multi_invest else None
    kwargs = dict(spec="non_ext", mask=active)

    dispatch = define_variables(n, -inf, inf, c, attr, axes=axes, **kwargs)
    dispatch = linexpr((1, dispatch))
    define_constraints(n, dispatch, ">=", lower, c, "mu_lower", **kwargs)
    define_constraints(n, dispatch, "<=", upper, c, "mu_upper", **kwargs)


def define_dispatch_for_extendable_constraints(n, sns, c, attr):
    """
    Sets power dispatch constraints for extendable devices for a given
    component and a given attribute.

    Parameters
    ----------
    n : pypsa.Network
    c : str
        name of the network component
    attr : str
        name of the attribute, e.g. 'p'
    """
    ext_i = get_extendable_i(n, c)
    if ext_i.empty:
        return
    min_pu, max_pu = get_bounds_pu(n, c, sns, ext_i, attr)
    operational_ext_v = get_var(n, c, attr)[ext_i]
    nominal_v = get_var(n, c, nominal_attrs[c])[ext_i]
    rhs = 0

    active = get_activity_mask(n, c, sns)[ext_i] if n._multi_invest else None
    kwargs = dict(spec=attr, mask=active)

    lhs, *axes = linexpr((max_pu, nominal_v), (-1, operational_ext_v), return_axes=True)
    define_constraints(n, lhs, ">=", rhs, c, "mu_upper", axes=axes, **kwargs)

    lhs, *axes = linexpr((min_pu, nominal_v), (-1, operational_ext_v), return_axes=True)
    define_constraints(n, lhs, "<=", rhs, c, "mu_lower", axes=axes, **kwargs)


def define_fixed_variable_constraints(n, sns, c, attr, pnl=True):
    """
    Sets constraints for fixing variables of a given component and attribute to
    the corresponding values in n.df(c)[attr + '_set'] if pnl is True, or
    n.pnl(c)[attr + '_set']

    Parameters
    ----------
    n : pypsa.Network
    c : str
        name of the network component
    attr : str
        name of the attribute, e.g. 'p'
    pnl : bool, default True
        Whether variable which should be fixed is time-dependent
    """

    if pnl:
        if attr + "_set" not in n.pnl(c):
            return

        fix = n.pnl(c)[attr + "_set"].loc[sns]
        if fix.empty:
            return

        if n._multi_invest:
            active = get_activity_mask(n, c, sns)
            fix = fix.where(active)

        fix = fix.stack()
        lhs = linexpr((1, get_var(n, c, attr).stack()[fix.index]), as_pandas=False)
        constraints = write_constraint(n, lhs, "=", fix).unstack().T
    else:
        if attr + "_set" not in n.df(c):
            return
        fix = n.df(c)[attr + "_set"].dropna()
        if fix.empty:
            return
        lhs = linexpr((1, get_var(n, c, attr)[fix.index]), as_pandas=False)
        constraints = write_constraint(n, lhs, "=", fix)
    set_conref(n, constraints, c, f"mu_{attr}_set")


def define_generator_status_variables(n, sns):
    c = "Generator"
    com_i = n.generators.query("committable").index
    ext_i = get_extendable_i(n, c)
    if not (ext_i.intersection(com_i)).empty:
        logger.warning(
            "The following generators have both investment optimisation"
            f" and unit commitment:\n\n\t{', '.join((ext_i.intersection(com_i)))}\n\nCurrently PyPSA cannot "
            "do both these functions, so PyPSA is choosing investment optimisation "
            "for these generators."
        )
        com_i = com_i.difference(ext_i)
    if com_i.empty:
        return
    active = get_activity_mask(n, c, sns)[com_i] if n._multi_invest else None
    define_binaries(n, (sns, com_i), "Generator", "status", mask=active)


def define_committable_generator_constraints(n, sns):
    c, attr = "Generator", "status"
    com_i = n.df(c).query("committable and not p_nom_extendable").index
    if com_i.empty:
        return
    nominal = n.df(c)[nominal_attrs[c]][com_i]
    min_pu, max_pu = get_bounds_pu(n, c, sns, com_i, "p")
    lower = min_pu.mul(nominal)
    upper = max_pu.mul(nominal)

    status = get_var(n, c, attr)
    p = get_var(n, c, "p")[com_i]

    lhs = linexpr((lower, status), (-1, p))
    active = get_activity_mask(n, c, sns)[com_i] if n._multi_invest else None
    define_constraints(n, lhs, "<=", 0, "Generators", "committable_lb", mask=active)

    lhs = linexpr((upper, status), (-1, p))
    define_constraints(n, lhs, ">=", 0, "Generators", "committable_ub", mask=active)


def define_ramp_limit_constraints(n, sns, c):
    """
    Defines ramp limits for a given component with valid ramplimit.
    """
    rup_i = n.df(c).query("ramp_limit_up == ramp_limit_up").index
    rdown_i = n.df(c).query("ramp_limit_down == ramp_limit_down").index
    if rup_i.empty & rdown_i.empty:
        return
    fix_i = get_non_extendable_i(n, c)
    ext_i = get_extendable_i(n, c)
    if "committable" in n.df(c):
        com_i = n.df(c).query("committable").index.difference(ext_i)
    else:
        com_i = []

    # Check if ramping is not at start of n.snapshots
    start_i = n.snapshots.get_loc(sns[0]) - 1
    pnl = n.pnl(c)
    # get dispatch for either one or two ports
    attr = ({"p", "p0"} & set(pnl)).pop()
    p_prev_fix = pnl[attr].iloc[start_i]
    is_rolling_horizon = (sns[0] != n.snapshots[0]) and not p_prev_fix.empty

    if is_rolling_horizon:
        active = get_activity_mask(n, c, sns)
        p = get_var(n, c, "p")
        p_prev = get_var(n, c, "p").shift(1, fill_value=-1)
        rhs_prev = pd.DataFrame(0, *p.axes)
        rhs_prev.loc[sns[0]] = p_prev_fix
    else:
        active = get_activity_mask(n, c, sns[1:])
        p = get_var(n, c, "p").loc[sns[1:]]
        p_prev = get_var(n, c, "p").shift(1, fill_value=-1).loc[sns[1:]]
        rhs_prev = pd.DataFrame(0, *p.axes)

    # fix up
    gens_i = rup_i.intersection(fix_i)
    if not gens_i.empty:
        lhs = linexpr((1, p[gens_i]), (-1, p_prev[gens_i]))
        rhs = rhs_prev[gens_i] + n.df(c).loc[gens_i].eval("ramp_limit_up * p_nom")
        kwargs = dict(spec="nonext.", mask=active[gens_i])
        define_constraints(n, lhs, "<=", rhs, c, "mu_ramp_limit_up", **kwargs)

    # ext up
    gens_i = rup_i.intersection(ext_i)
    if not gens_i.empty:
        limit_pu = n.df(c)["ramp_limit_up"][gens_i]
        p_nom = get_var(n, c, "p_nom")[gens_i]
        lhs = linexpr((1, p[gens_i]), (-1, p_prev[gens_i]), (-limit_pu, p_nom))
        rhs = rhs_prev[gens_i]
        kwargs = dict(spec="ext.", mask=active[gens_i])
        define_constraints(n, lhs, "<=", rhs, c, "mu_ramp_limit_up", **kwargs)

    # com up
    gens_i = rup_i.intersection(com_i)
    if not gens_i.empty:
        limit_start = n.df(c).loc[gens_i].eval("ramp_limit_start_up * p_nom")
        limit_up = n.df(c).loc[gens_i].eval("ramp_limit_up * p_nom")
        status = get_var(n, c, "status").loc[p.index, gens_i]
        status_prev = (
            get_var(n, c, "status").shift(1, fill_value=-1).loc[p.index, gens_i]
        )
        lhs = linexpr(
            (1, p[gens_i]),
            (-1, p_prev[gens_i]),
            (limit_start - limit_up, status_prev),
            (-limit_start, status),
        )
        rhs = rhs_prev[gens_i]
        if is_rolling_horizon:
            status_prev_fix = n.pnl(c)["status"][com_i].iloc[start_i]
            rhs.loc[sns[0]] += (limit_up - limit_start) * status_prev_fix
        kwargs = dict(spec="com.", mask=active[gens_i])
        define_constraints(n, lhs, "<=", rhs, c, "mu_ramp_limit_up", **kwargs)

    # fix down
    gens_i = rdown_i.intersection(fix_i)
    if not gens_i.empty:
        lhs = linexpr((1, p[gens_i]), (-1, p_prev[gens_i]))
        rhs = rhs_prev[gens_i] + n.df(c).loc[gens_i].eval(
            "-1 * ramp_limit_down * p_nom"
        )
        kwargs = dict(spec="nonext.", mask=active[gens_i])
        define_constraints(n, lhs, ">=", rhs, c, "mu_ramp_limit_down", **kwargs)

    # ext down
    gens_i = rdown_i.intersection(ext_i)
    if not gens_i.empty:
        limit_pu = n.df(c)["ramp_limit_down"][gens_i]
        p_nom = get_var(n, c, "p_nom")[gens_i]
        lhs = linexpr((1, p[gens_i]), (-1, p_prev[gens_i]), (limit_pu, p_nom))
        rhs = rhs_prev[gens_i]
        kwargs = dict(spec="ext.", mask=active[gens_i])
        define_constraints(n, lhs, ">=", rhs, c, "mu_ramp_limit_down", **kwargs)

    # com down
    gens_i = rdown_i.intersection(com_i)
    if not gens_i.empty:
        limit_shut = n.df(c).loc[gens_i].eval("ramp_limit_shut_down * p_nom")
        limit_down = n.df(c).loc[gens_i].eval("ramp_limit_down * p_nom")
        status = get_var(n, c, "status").loc[p.index, gens_i]
        status_prev = (
            get_var(n, c, "status").shift(1, fill_value=-1).loc[p.index, gens_i]
        )
        lhs = linexpr(
            (1, p[gens_i]),
            (-1, p_prev[gens_i]),
            (limit_down - limit_shut, status),
            (limit_shut, status_prev),
        )
        rhs = rhs_prev[gens_i]
        if is_rolling_horizon:
            status_prev_fix = n.pnl(c)["status"][com_i].iloc[start_i]
            rhs.loc[sns[0]] += -limit_shut * status_prev_fix
        kwargs = dict(spec="com.", mask=active[gens_i])
        define_constraints(n, lhs, ">=", rhs, c, "mu_ramp_limit_down", **kwargs)


def define_nominal_constraints_per_bus_carrier(n, sns):
    for carrier in n.carriers.index:
        for bound, sense in [("max", "<="), ("min", ">=")]:

            col = f"nom_{bound}_{carrier}"
            if col not in n.buses.columns:
                continue
            rhs = n.buses[col].dropna()
            lhs = pd.Series("", rhs.index)

            for c, attr in nominal_attrs.items():
                if c not in n.one_port_components:
                    continue
                attr = nominal_attrs[c]
                if (c, attr) not in n.variables.index:
                    continue
                nominals = get_var(n, c, attr)[n.df(c).carrier == carrier]
                if nominals.empty:
                    continue
                per_bus = (
                    linexpr((1, nominals)).groupby(n.df(c).bus).sum(**agg_group_kwargs)
                )
                lhs += per_bus.reindex(lhs.index, fill_value="")

            if bound == "max":
                lhs = lhs[lhs != ""]
                rhs = rhs.reindex(lhs.index)
            else:
                assert (lhs != "").all(), (
                    f"No extendable components of carrier {carrier} on bus "
                    f'{list(lhs[lhs == ""].index)}'
                )
            define_constraints(n, lhs, sense, rhs, "Bus", "mu_" + col)


def define_nodal_balance_constraints(n, sns):
    """
    Defines nodal balance constraint.
    """

    def bus_injection(c, attr, groupcol="bus", sign=1):
        # additional sign only necessary for branches in reverse direction
        if "sign" in n.df(c):
            sign = sign * n.df(c).sign
        expr = linexpr((sign, get_var(n, c, attr))).rename(columns=n.df(c)[groupcol])
        # drop empty bus2, bus3 if multiline link
        if c == "Link":
            expr.drop(columns="", errors="ignore", inplace=True)
        return expr

    # one might reduce this a bit by using n.branches and lookup
    args = [
        ["Generator", "p"],
        ["Store", "p"],
        ["StorageUnit", "p_dispatch"],
        ["StorageUnit", "p_store", "bus", -1],
        ["Line", "s", "bus0", -1],
        ["Line", "s", "bus1", 1],
        ["Transformer", "s", "bus0", -1],
        ["Transformer", "s", "bus1", 1],
        ["Link", "p", "bus0", -1],
        ["Link", "p", "bus1", get_as_dense(n, "Link", "efficiency", sns)],
    ]
    args = [arg for arg in args if not n.df(arg[0]).empty]

    if not n.links.empty:
        for i in additional_linkports(n):
            eff = get_as_dense(n, "Link", f"efficiency{i}", sns)
            args.append(["Link", "p", f"bus{i}", eff])

    lhs = (
        pd.concat([bus_injection(*arg) for arg in args], axis=1)
        .groupby(axis=1, level=0)
        .sum(**agg_group_kwargs)
        .reindex(columns=n.buses.index, fill_value="")
    )

    if (lhs == "").any().any():
        raise ValueError("Empty LHS in nodal balance constraint.")

    sense = "="
    rhs = (
        (-get_as_dense(n, "Load", "p_set", sns) * n.loads.sign)
        .groupby(n.loads.bus, axis=1)
        .sum()
        .reindex(columns=n.buses.index, fill_value=0)
    )
    define_constraints(n, lhs, sense, rhs, "Bus", "marginal_price")


def define_kirchhoff_constraints(n, sns):
    """
    Defines Kirchhoff voltage constraints.
    """
    comps = n.passive_branch_components & set(n.variables.index.levels[0])
    if len(comps) == 0:
        return
    branch_vars = pd.concat({c: get_var(n, c, "s") for c in comps}, axis=1)

    def cycle_flow(ds, sns):
        if sns is None:
            sns = slice(None)
        ds = ds[lambda ds: ds != 0.0].dropna()
        vals = linexpr((ds, branch_vars.loc[sns, ds.index]), as_pandas=False)
        return vals.sum(1)

    constraints = []
    periods = sns.unique("period") if n._multi_invest else [None]
    for period in periods:
        n.determine_network_topology(investment_period=period)
        subconstraints = []
        for sub in n.sub_networks.obj:
            branches = sub.branches()
            C = pd.DataFrame(sub.C.todense(), index=branches.index)
            if C.empty:
                continue
            carrier = n.sub_networks.carrier[sub.name]
            weightings = branches.x_pu_eff if carrier == "AC" else branches.r_pu_eff
            C_weighted = 1e5 * C.mul(weightings, axis=0)
            cycle_sum = C_weighted.apply(cycle_flow, sns=period)
            snapshots = sns if period == None else sns[sns.get_loc(period)]
            cycle_sum.set_index(snapshots, inplace=True)

            con = write_constraint(n, cycle_sum, "=", 0)
            subconstraints.append(con)
        if len(subconstraints) == 0:
            continue
        constraints.append(pd.concat(subconstraints, axis=1, ignore_index=True))
    if constraints:
        constraints = pd.concat(constraints).rename_axis(columns="cycle")
        set_conref(n, constraints, "SubNetwork", "mu_kirchhoff_voltage_law")


def define_storage_unit_constraints(n, sns):
    """
    Defines state of charge (soc) constraints for storage units. In principal
    the constraints states:

    previous_soc + p_store - p_dispatch + inflow - spill == soc
    """
    sus_i = n.storage_units.index
    if sus_i.empty:
        return
    c = "StorageUnit"
    # spillage
    has_periods = isinstance(sns, pd.MultiIndex)
    active = get_activity_mask(n, c, sns)

    upper = get_as_dense(n, c, "inflow", sns).loc[:, lambda df: df.max() > 0]
    spill = define_variables(
        n, 0, upper, "StorageUnit", "spill", mask=active[upper.columns]
    )

    # elapsed hours
    eh = expand_series(n.snapshot_weightings.stores[sns], sus_i)
    # efficiencies
    eff_stand = expand_series(1 - n.df(c).standing_loss, sns).T.pow(eh)
    eff_dispatch = expand_series(n.df(c).efficiency_dispatch, sns).T
    eff_store = expand_series(n.df(c).efficiency_store, sns).T

    soc = get_var(n, c, "state_of_charge")

    if has_periods:
        cyclic_i = (
            n.df(c)
            .query("cyclic_state_of_charge & " "~cyclic_state_of_charge_per_period")
            .index
        )
        cyclic_pp_i = (
            n.df(c)
            .query("cyclic_state_of_charge & " "cyclic_state_of_charge_per_period")
            .index
        )
        noncyclic_i = (
            n.df(c)
            .query("~cyclic_state_of_charge & " "~state_of_charge_initial_per_period")
            .index
        )
        noncyclic_pp_i = (
            n.df(c)
            .query("~cyclic_state_of_charge & " "state_of_charge_initial_per_period")
            .index
        )
    else:
        cyclic_i = n.df(c).query("cyclic_state_of_charge").index
        noncyclic_i = n.df(c).query("~cyclic_state_of_charge ").index

    # cyclic constraint for whole optimization horizon
    previous_soc_cyclic = (
        soc.where(active).ffill().apply(lambda ds: np.roll(ds, 1)).ffill()
    )

    # non cyclic constraint: determine the first active snapshot
    first_active_snapshot = active.cumsum()[noncyclic_i] == 1

    coeff_var = [
        (-1, soc),
        (-1 / eff_dispatch * eh, get_var(n, c, "p_dispatch")),
        (eff_store * eh, get_var(n, c, "p_store")),
    ]

    lhs, *axes = linexpr(*coeff_var, return_axes=True)

    def masked_term(coeff, var, cols):
        return (
            linexpr((coeff[cols], var[cols]))
            .reindex(index=axes[0], columns=axes[1], fill_value="")
            .values
        )

    if (c, "spill") in n.variables.index:
        lhs += masked_term(-eh, get_var(n, c, "spill"), spill.columns)

    lhs += masked_term(eff_stand, previous_soc_cyclic, cyclic_i)
    lhs += masked_term(
        eff_stand[~first_active_snapshot],
        soc.shift()[~first_active_snapshot],
        noncyclic_i,
    )

    # rhs set e at beginning of optimization horizon for noncyclic
    rhs = -get_as_dense(n, c, "inflow", sns).mul(eh).astype(float)

    rhs[noncyclic_i] = rhs[noncyclic_i].where(
        ~first_active_snapshot, rhs - n.df(c).state_of_charge_initial, axis=1
    )

    if has_periods:
        # cyclic constraint for soc per period - cyclic soc within each period
        previous_soc_cyclic_pp = soc.groupby(level=0).transform(
            lambda ds: np.roll(ds, 1)
        )
        lhs += masked_term(eff_stand, previous_soc_cyclic_pp, cyclic_pp_i)

        # set the initial enery at the beginning of each period
        first_active_snapshot_pp = active[noncyclic_pp_i].groupby(level=0).cumsum() == 1

        lhs += masked_term(
            eff_stand[~first_active_snapshot_pp],
            soc.shift()[~first_active_snapshot_pp],
            noncyclic_pp_i,
        )

        rhs[noncyclic_pp_i] = rhs[noncyclic_pp_i].where(
            ~first_active_snapshot_pp, rhs - n.df(c).state_of_charge_initial, axis=1
        )

    define_constraints(n, lhs, "==", rhs, c, "mu_state_of_charge", mask=active)


def define_store_constraints(n, sns):
    """
    Defines energy balance constraints for stores. In principal this states:

    previous_e - p == e
    """
    stores_i = n.stores.index
    if stores_i.empty:
        return
    c = "Store"

    has_periods = isinstance(sns, pd.MultiIndex)
    active = get_activity_mask(n, c, sns)

    define_variables(n, -inf, inf, axes=[sns, stores_i], name=c, attr="p", mask=active)

    # elapsed hours
    eh = expand_series(n.snapshot_weightings.stores[sns], stores_i)  # elapsed hours
    eff_stand = expand_series(1 - n.df(c).standing_loss, sns).T.pow(eh)

    e = get_var(n, c, "e")

    if has_periods:
        cyclic_i = n.df(c).query("e_cyclic & ~e_cyclic_per_period").index
        cyclic_pp_i = n.df(c).query("e_cyclic & e_cyclic_per_period").index
        noncyclic_i = n.df(c).query("~e_cyclic & ~e_initial_per_period").index
        noncyclic_pp_i = n.df(c).query("~e_cyclic & e_initial_per_period").index
    else:
        cyclic_i = n.df(c).query("e_cyclic").index
        noncyclic_i = n.df(c).query("~e_cyclic").index

    # cyclic constraint for whole optimization horizon
    previous_e_cyclic = e.where(active).ffill().apply(lambda ds: np.roll(ds, 1)).ffill()

    # non cyclic constraint: determine the first active snapshot
    first_active_snapshot = active.cumsum()[noncyclic_i] == 1

    coeff_var = [(-eh, get_var(n, c, "p")), (-1, e)]

    lhs, *axes = linexpr(*coeff_var, return_axes=True)

    def masked_term(coeff, var, cols):
        return (
            linexpr((coeff[cols], var[cols]))
            .reindex(index=sns, columns=stores_i, fill_value="")
            .values
        )

    lhs += masked_term(eff_stand, previous_e_cyclic, cyclic_i)
    lhs += masked_term(
        eff_stand[~first_active_snapshot],
        e.shift()[~first_active_snapshot],
        noncyclic_i,
    )

    # rhs set e at beginning of optimization horizon for noncyclic
    rhs = pd.DataFrame(0.0, sns, stores_i)

    rhs[noncyclic_i] = rhs[noncyclic_i].where(
        ~first_active_snapshot, -n.df(c).e_initial, axis=1
    )

    if has_periods:
        # cyclic constraint for soc per period - cyclic soc within each period
        previous_e_cyclic_pp = e.groupby(level=0).transform(lambda ds: np.roll(ds, 1))
        lhs += masked_term(eff_stand, previous_e_cyclic_pp, cyclic_pp_i)

        # set the initial enery at the beginning of each period
        first_active_snapshot_pp = active[noncyclic_pp_i].groupby(level=0).cumsum() == 1

        lhs += masked_term(
            eff_stand[~first_active_snapshot_pp],
            e.shift()[~first_active_snapshot_pp],
            noncyclic_pp_i,
        )

        rhs[noncyclic_pp_i] = rhs[noncyclic_pp_i].where(
            ~first_active_snapshot_pp, -n.df(c).e_initial, axis=1
        )

    define_constraints(n, lhs, "==", rhs, c, "mu_state_of_charge", mask=active)


def define_growth_limit(n, sns, c, attr):
    """
    Constraint new installed capacity per investment period.

    Parameters
    ----------
    n    : pypsa.Network
    c    : str
           network component of which the nominal capacity should be defined
    attr : str
           name of the variable, e.g. 'p_nom'
    """
    if not n._multi_invest:
        return

    ext_i = get_extendable_i(n, c)
    if "carrier" not in n.df(c) or n.df(c).empty:
        return
    with_limit = n.carriers.query("max_growth != inf").index
    limit_i = n.df(c).query("carrier in @with_limit").index.intersection(ext_i)
    if limit_i.empty:
        return

    periods = sns.unique("period")

    v = get_var(n, c, attr)
    carriers = n.df(c).loc[limit_i, "carrier"]
    caps = pd.concat(
        {
            period: linexpr((1, v)).where(n.get_active_assets(c, period), "")
            for period in periods
        },
        axis=1,
    ).T[limit_i]
    lhs = caps.groupby(carriers, axis=1).sum(**agg_group_kwargs)
    rhs = n.carriers.max_growth[with_limit]

    define_constraints(n, lhs, "<=", rhs, "Carrier", "growth_limit_{}".format(c))


def define_global_constraints(n, sns):
    """
    Defines global constraints for the optimization. Possible types are.

    1. primary_energy
        Use this to constraint the byproducts of primary energy sources as
        CO2
    2. transmission_volume_expansion_limit
        Use this to set a limit for line volume expansion. Possible carriers
        are 'AC' and 'DC'
    3. transmission_expansion_cost_limit
        Use this to set a limit for line expansion costs. Possible carriers
        are 'AC' and 'DC'
    4. tech_capacity_expansion_limit
        Use this to se a limit for the summed capacitiy of a carrier (e.g.
        'onwind') for each investment period at choosen nodes. This limit
        could e.g. represent land resource/ building restrictions for a
        technology in a certain region. Currently, only the
        capacities of extendable generators have to be below the set limit.
    """

    if n._multi_invest:
        period_weighting = n.investment_period_weightings["years"]
        weightings = n.snapshot_weightings.mul(period_weighting, level=0, axis=0).loc[
            sns
        ]
    else:
        weightings = n.snapshot_weightings.loc[sns]

    def get_period(n, glc, sns):
        period = slice(None)
        if n._multi_invest and not np.isnan(glc["investment_period"]):
            period = int(glc["investment_period"])
            if period not in sns.unique("period"):
                logger.warning(
                    "Optimized snapshots do not contain the investment "
                    f"period required for global constraint `{glc.name}`."
                )
        return period

    # (1) primary_energy
    glcs = n.global_constraints.query('type == "primary_energy"')
    for name, glc in glcs.iterrows():
        rhs = glc.constant
        lhs = ""
        carattr = glc.carrier_attribute
        emissions = n.carriers.query(f"{carattr} != 0")[carattr]
        period = get_period(n, glc, sns)

        if emissions.empty:
            continue

        # generators
        gens = n.generators.query("carrier in @emissions.index")
        if not gens.empty:
            em_pu = gens.carrier.map(emissions) / gens.efficiency
            em_pu = (
                weightings["generators"].to_frame("weightings")
                @ em_pu.to_frame("weightings").T
            ).loc[period]
            p = get_var(n, "Generator", "p").loc[sns, gens.index].loc[period]

            vals = linexpr((em_pu, p), as_pandas=False)
            lhs += join_exprs(vals)

        # storage units
        sus = n.storage_units.query(
            "carrier in @emissions.index and " "not cyclic_state_of_charge"
        )
        sus_i = sus.index
        if not sus.empty:
            em_pu = sus.carrier.map(emissions)
            soc = (
                get_var(n, "StorageUnit", "state_of_charge").loc[sns, sus_i].loc[period]
            )
            soc = soc.where(soc != -1).ffill().iloc[-1]
            vals = linexpr((-em_pu, soc), as_pandas=False)
            lhs = lhs + "\n" + join_exprs(vals)
            rhs -= em_pu @ sus.state_of_charge_initial

        # stores
        n.stores["carrier"] = n.stores.bus.map(n.buses.carrier)
        stores = n.stores.query("carrier in @emissions.index and not e_cyclic")
        if not stores.empty:
            em_pu = stores.carrier.map(emissions)
            e = get_var(n, "Store", "e").loc[sns, stores.index].loc[period]
            e = e.where(e != -1).ffill().iloc[-1]
            vals = linexpr((-em_pu, e), as_pandas=False)
            lhs = lhs + "\n" + join_exprs(vals)
            rhs -= stores.carrier.map(emissions) @ stores.e_initial

        define_constraints(
            n,
            lhs,
            glc.sense,
            rhs,
            "GlobalConstraint",
            "mu",
            axes=pd.Index([name]),
            spec=name,
        )

    # (2) transmission_volume_expansion_limit
    glcs = n.global_constraints.query(
        "type == " '"transmission_volume_expansion_limit"'
    )
    substr = lambda s: re.sub(r"[\[\]\(\)]", "", s)
    for name, glc in glcs.iterrows():
        car = [substr(c.strip()) for c in glc.carrier_attribute.split(",")]
        lhs = ""
        period = get_period(n, glc, sns)
        for c, attr in (("Line", "s_nom"), ("Link", "p_nom")):
            if n.df(c).empty:
                continue
            ext_i = n.df(c).query(f"carrier in @car and {attr}_extendable").index
            ext_i = ext_i[get_activity_mask(n, c, sns)[ext_i].loc[period].any()]

            if ext_i.empty:
                continue
            v = linexpr(
                (n.df(c).length[ext_i], get_var(n, c, attr)[ext_i]), as_pandas=False
            )
            lhs += "\n" + join_exprs(v)
        if lhs == "":
            continue
        sense = glc.sense
        rhs = glc.constant
        define_constraints(
            n,
            lhs,
            sense,
            rhs,
            "GlobalConstraint",
            "mu",
            axes=pd.Index([name]),
            spec=name,
        )

    # (3) transmission_expansion_cost_limit
    glcs = n.global_constraints.query("type == " '"transmission_expansion_cost_limit"')
    for name, glc in glcs.iterrows():
        car = [substr(c.strip()) for c in glc.carrier_attribute.split(",")]
        lhs = ""
        period = get_period(n, glc, sns)
        for c, attr in (("Line", "s_nom"), ("Link", "p_nom")):
            ext_i = n.df(c).query(f"carrier in @car and {attr}_extendable").index
            ext_i = ext_i[get_activity_mask(n, c, sns)[ext_i].loc[period].any()]

            if ext_i.empty:
                continue

            v = linexpr(
                (n.df(c).capital_cost[ext_i], get_var(n, c, attr)[ext_i]),
                as_pandas=False,
            )
            lhs += "\n" + join_exprs(v)
        if lhs == "":
            continue
        sense = glc.sense
        rhs = glc.constant
        define_constraints(
            n,
            lhs,
            sense,
            rhs,
            "GlobalConstraint",
            "mu",
            axes=pd.Index([name]),
            spec=name,
        )

    # (4) tech_capacity_expansion_limit
    # TODO: Generalize to carrier capacity expansion limit (i.e. also for stores etc.)
    substr = lambda s: re.sub(r"[\[\]\(\)]", "", s)
    glcs = n.global_constraints.query("type == " '"tech_capacity_expansion_limit"')
    c, attr = "Generator", "p_nom"

    for name, glc in glcs.iterrows():
        period = get_period(n, glc, sns)
        car = glc["carrier_attribute"]
        bus = str(glc.get("bus", ""))  # in pypsa buses are always strings
        ext_i = n.df(c).query("carrier == @car and p_nom_extendable").index
        if bus:
            ext_i = n.df(c).loc[ext_i].query("bus == @bus").index
        ext_i = ext_i[get_activity_mask(n, c, sns)[ext_i].loc[period].any()]

        if ext_i.empty:
            continue

        cap_vars = get_var(n, c, attr)[ext_i]

        lhs = join_exprs(linexpr((1, cap_vars)))
        rhs = glc.constant
        sense = glc.sense

        define_constraints(
            n,
            lhs,
            sense,
            rhs,
            "GlobalConstraint",
            "mu",
            axes=pd.Index([name]),
            spec=name,
        )


def define_objective(n, sns):
    """
    Defines and writes out the objective function.
    """

    if n._multi_invest:
        period_weighting = n.investment_period_weightings.objective[
            sns.unique("period")
        ]
    # constant for already done investment
    nom_attr = nominal_attrs.items()
    constant = 0
    for c, attr in nom_attr:
        ext_i = get_extendable_i(n, c)
        cost = n.df(c)["capital_cost"][ext_i]
        if cost.empty:
            continue

        if n._multi_invest:
            active = pd.concat(
                {
                    period: get_active_assets(n, c, period)[ext_i]
                    for period in sns.unique("period")
                },
                axis=1,
            )
            cost = active @ period_weighting * cost

        constant += cost @ n.df(c)[attr][ext_i]

    object_const = write_bound(n, constant, constant)
    write_objective(n, linexpr((-1, object_const), as_pandas=False)[0])
    n.objective_constant = constant

    # marginal cost
    if n._multi_invest:
        weighting = n.snapshot_weightings.objective.mul(period_weighting, level=0).loc[
            sns
        ]
    else:
        weighting = n.snapshot_weightings.objective.loc[sns]

    for c, attr in lookup.query("marginal_cost").index:
        cost = (
            get_as_dense(n, c, "marginal_cost", sns)
            .loc[:, lambda ds: (ds != 0).all()]
            .mul(weighting, axis=0)
        )
        if cost.empty:
            continue
        terms = linexpr((cost, get_var(n, c, attr).loc[sns, cost.columns]))
        write_objective(n, terms)

    # investment
    for c, attr in nominal_attrs.items():
        ext_i = get_extendable_i(n, c)
        cost = n.df(c)["capital_cost"][ext_i]
        if cost.empty:
            continue

        if n._multi_invest:
            active = pd.concat(
                {
                    period: get_active_assets(n, c, period)[ext_i]
                    for period in sns.unique("period")
                },
                axis=1,
            )
            cost = active @ period_weighting * cost

        caps = get_var(n, c, attr).loc[ext_i]
        terms = linexpr((cost, caps))
        write_objective(n, terms)


def prepare_lopf(
    n,
    snapshots=None,
    keep_files=False,
    skip_objective=False,
    extra_functionality=None,
    solver_dir=None,
):
    """
    Sets up the linear problem and writes it out to a lp file.

    Returns
    -------
    Tuple (fdp, problem_fn) indicating the file descriptor and the file name of
    the lp file
    """
    n._xCounter, n._cCounter = 1, 1
    n.vars, n.cons = Dict(), Dict()

    cols = ["component", "name", "pnl", "specification"]
    n.variables = pd.DataFrame(columns=cols).set_index(cols[:2])
    n.constraints = pd.DataFrame(columns=cols).set_index(cols[:2])

    snapshots = n.snapshots if snapshots is None else snapshots
    start = time.time()

    tmpkwargs = dict(text=True, dir=solver_dir)
    # mkstemp(suffix, prefix, **tmpkwargs)
    fdo, objective_fn = mkstemp(".txt", "pypsa-objectve-", **tmpkwargs)
    fdc, constraints_fn = mkstemp(".txt", "pypsa-constraints-", **tmpkwargs)
    fdb, bounds_fn = mkstemp(".txt", "pypsa-bounds-", **tmpkwargs)
    fdi, binaries_fn = mkstemp(".txt", "pypsa-binaries-", **tmpkwargs)
    fdp, problem_fn = mkstemp(".lp", "pypsa-problem-", **tmpkwargs)

    n.objective_f = open(objective_fn, mode="w")
    n.constraints_f = open(constraints_fn, mode="w")
    n.bounds_f = open(bounds_fn, mode="w")
    n.binaries_f = open(binaries_fn, mode="w")

    n.objective_f.write("\* LOPF *\n\nmin\nobj:\n")
    n.constraints_f.write("\n\ns.t.\n\n")
    n.bounds_f.write("\nbounds\n")
    n.binaries_f.write("\nbinary\n")

    for c, attr in lookup.query("nominal and not handle_separately").index:
        define_nominal_for_extendable_variables(n, c, attr)
        # define constraint for newly installed capacity per investment period
        define_growth_limit(n, snapshots, c, attr)
        # define_fixed_variable_constraints(n, snapshots, c, attr, pnl=False)
    for c, attr in lookup.query("not nominal and not handle_separately").index:
        define_dispatch_for_non_extendable_variables(n, snapshots, c, attr)
        define_dispatch_for_extendable_and_committable_variables(n, snapshots, c, attr)
        align_with_static_component(n, c, attr)
        define_dispatch_for_extendable_constraints(n, snapshots, c, attr)
        # define_fixed_variable_constraints(n, snapshots, c, attr)
    define_generator_status_variables(n, snapshots)
    define_nominal_constraints_per_bus_carrier(n, snapshots)

    # consider only state_of_charge_set for the moment
    define_fixed_variable_constraints(n, snapshots, "StorageUnit", "state_of_charge")
    define_fixed_variable_constraints(n, snapshots, "Store", "e")

    define_committable_generator_constraints(n, snapshots)
    define_ramp_limit_constraints(n, snapshots, c="Generator")
    define_ramp_limit_constraints(n, snapshots, c="Link")
    define_storage_unit_constraints(n, snapshots)
    define_store_constraints(n, snapshots)
    define_kirchhoff_constraints(n, snapshots)
    define_nodal_balance_constraints(n, snapshots)
    define_global_constraints(n, snapshots)
    if skip_objective:
        logger.info(
            "The argument `skip_objective` is set to True. Expecting a "
            "custom objective to be build via `extra_functionality`."
        )
    else:
        define_objective(n, snapshots)

    if extra_functionality is not None:
        extra_functionality(n, snapshots)

    n.binaries_f.write("end\n")

    # explicit closing with file descriptor is necessary for windows machines
    for f, fd in (
        ("bounds_f", fdb),
        ("constraints_f", fdc),
        ("objective_f", fdo),
        ("binaries_f", fdi),
    ):
        getattr(n, f).close()
        delattr(n, f)
        os.close(fd)

    # concatenate files
    with open(problem_fn, "wb") as wfd:
        for f in [objective_fn, constraints_fn, bounds_fn, binaries_fn]:
            with open(f, "rb") as fd:
                shutil.copyfileobj(fd, wfd)
            if not keep_files:
                os.remove(f)

    logger.info(f"Total preparation time: {round(time.time()-start, 2)}s")
    return fdp, problem_fn


def assign_solution(
    n,
    sns,
    variables_sol,
    constraints_dual,
    keep_references=False,
    keep_shadowprices=None,
):
    """
    Map solution to network components.

    Helper function. Assigns the solution of a succesful optimization to
    the network.
    """

    def set_from_frame(pnl, attr, df):
        if attr not in pnl:  # use this for subnetworks_t
            pnl[attr] = df.reindex(n.snapshots, fill_value=0)
        elif pnl[attr].empty:
            pnl[attr] = df.reindex(n.snapshots, fill_value=0)
        else:
            pnl[attr].loc[sns, df.columns] = df

    pop = not keep_references

    def map_solution(c, attr):
        variables = get_var(n, c, attr, pop=pop)
        predefined = True
        if (c, attr) not in lookup.index:
            predefined = False
            n.sols[c] = n.sols[c] if c in n.sols else Dict(df=pd.DataFrame(), pnl={})
        n.solutions.at[(c, attr), "in_comp"] = predefined
        if isinstance(variables, pd.DataFrame):
            # case that variables are timedependent
            n.solutions.at[(c, attr), "pnl"] = True
            pnl = n.pnl(c) if predefined else n.sols[c].pnl
            variables_sol.loc[-1] = 0
            values = variables.applymap(lambda x: variables_sol.loc[x])
            if c in n.passive_branch_components and attr == "s":
                set_from_frame(pnl, "p0", values)
                set_from_frame(pnl, "p1", -values)
            elif c == "Link" and attr == "p":
                set_from_frame(pnl, "p0", values)
                for i in ["1"] + additional_linkports(n):
                    i_eff = "" if i == "1" else i
                    eff = get_as_dense(n, "Link", f"efficiency{i_eff}", sns)
                    set_from_frame(pnl, f"p{i}", -values * eff)
                    pnl[f"p{i}"].loc[
                        sns, n.links.index[n.links[f"bus{i}"] == ""]
                    ] = n.component_attrs["Link"].loc[f"p{i}", "default"]
            else:
                set_from_frame(pnl, attr, values)
        else:
            # case that variables are static
            n.solutions.at[(c, attr), "pnl"] = False
            sol = variables.map(variables_sol)
            if predefined:
                non_ext = n.df(c)[attr]
                n.df(c)[attr + "_opt"] = sol.reindex(non_ext.index).fillna(non_ext)
            else:
                n.sols[c].df[attr] = sol

    n.sols = Dict()
    n.solutions = pd.DataFrame(index=n.variables.index, columns=["in_comp", "pnl"])
    for c, attr in n.variables.index:
        map_solution(c, attr)

    # if nominal capacity was no variable set optimal value to nominal
    for c, attr in lookup.query("nominal").index.difference(n.variables.index):
        n.df(c)[attr + "_opt"] = n.df(c)[attr]

    # recalculate storageunit net dispatch
    if not n.df("StorageUnit").empty:
        c = "StorageUnit"
        n.pnl(c)["p"] = n.pnl(c)["p_dispatch"] - n.pnl(c)["p_store"]

    # duals
    if keep_shadowprices == False:
        keep_shadowprices = []

    sp = n.constraints.index
    if isinstance(keep_shadowprices, list):
        sp = sp[sp.isin(keep_shadowprices, level=0)]

    def map_dual(c, attr):
        # If c is a pypsa component name the dual is stored at n.pnl(c)
        # or n.df(c). For the second case the index of the constraints have to
        # be a subset of n.df(c).index otherwise the dual is stored at
        # n.duals[c].df
        constraints = get_con(n, c, attr, pop=pop)
        is_pnl = isinstance(constraints, pd.DataFrame)
        # TODO: setting the sign is not very clear
        sign = 1 if "upper" in attr or attr == "marginal_price" else -1
        n.dualvalues.at[(c, attr), "pnl"] = is_pnl
        to_component = c in n.all_components
        if is_pnl:
            n.dualvalues.at[(c, attr), "in_comp"] = to_component
            duals = constraints.applymap(
                lambda x: sign * constraints_dual.loc[x]
                if x in constraints_dual.index
                else np.nan
            )
            if c not in n.duals and not to_component:
                n.duals[c] = Dict(df=pd.DataFrame(), pnl={})
            pnl = n.pnl(c) if to_component else n.duals[c].pnl
            set_from_frame(pnl, attr, duals)
        else:
            # here to_component can change
            duals = constraints.map(sign * constraints_dual)
            if to_component:
                to_component = duals.index.isin(n.df(c).index).all()
            n.dualvalues.at[(c, attr), "in_comp"] = to_component
            if c not in n.duals and not to_component:
                n.duals[c] = Dict(df=pd.DataFrame(), pnl={})
            df = n.df(c) if to_component else n.duals[c].df
            df[attr] = duals

    n.duals = Dict()
    n.dualvalues = pd.DataFrame(index=sp, columns=["in_comp", "pnl"])
    # extract shadow prices attached to components
    for c, attr in sp:
        map_dual(c, attr)

    # correct prices with objective weightings
    if n._multi_invest:
        period_weighting = n.investment_period_weightings.objective
        weightings = n.snapshot_weightings.objective.mul(
            period_weighting, level=0, axis=0
        ).loc[sns]
    else:
        weightings = n.snapshot_weightings.objective.loc[sns]

    n.buses_t.marginal_price.loc[sns] = n.buses_t.marginal_price.loc[sns].divide(
        weightings, axis=0
    )

    # discard remaining if wanted
    if not keep_references:
        for c, attr in n.constraints.index.difference(sp):
            get_con(n, c, attr, pop)

    # load
    if len(n.loads):
        set_from_frame(n.pnl("Load"), "p", get_as_dense(n, "Load", "p_set", sns))

    # clean up vars and cons
    for c in list(n.vars):
        if n.vars[c].df.empty and n.vars[c].pnl == {}:
            n.vars.pop(c)
    for c in list(n.cons):
        if n.cons[c].df.empty and n.cons[c].pnl == {}:
            n.cons.pop(c)

    # recalculate injection
    ca = [
        ("Generator", "p", "bus"),
        ("Store", "p", "bus"),
        ("Load", "p", "bus"),
        ("StorageUnit", "p", "bus"),
        ("Link", "p0", "bus0"),
        ("Link", "p1", "bus1"),
    ]
    for i in additional_linkports(n):
        ca.append(("Link", f"p{i}", f"bus{i}"))

    sign = lambda c: n.df(c).sign if "sign" in n.df(c) else -1  # sign for 'Link'
    n.buses_t.p = (
        pd.concat(
            [
                n.pnl(c)[attr].mul(sign(c)).rename(columns=n.df(c)[group])
                for c, attr, group in ca
            ],
            axis=1,
        )
        .groupby(level=0, axis=1)
        .sum()
        .reindex(columns=n.buses.index, fill_value=0)
    )

    def v_ang_for_(sub):
        buses_i = sub.buses_o
        if len(buses_i) == 1:
            return pd.DataFrame(0, index=sns, columns=buses_i)
        sub.calculate_B_H(skip_pre=True)
        Z = pd.DataFrame(np.linalg.pinv((sub.B).todense()), buses_i, buses_i)
        Z -= Z[sub.slack_bus]
        return n.buses_t.p.reindex(columns=buses_i) @ Z

    n.buses_t.v_ang = pd.concat(
        [v_ang_for_(sub) for sub in n.sub_networks.obj], axis=1
    ).reindex(columns=n.buses.index, fill_value=0)


def network_lopf(
    n,
    snapshots=None,
    solver_name="cbc",
    solver_logfile=None,
    extra_functionality=None,
    multi_investment_periods=False,
    skip_objective=False,
    skip_pre=False,
    extra_postprocessing=None,
    formulation="kirchhoff",
    keep_references=False,
    keep_files=False,
    keep_shadowprices=["Bus", "Line", "Transformer", "Link", "GlobalConstraint"],
    solver_options=None,
    warmstart=False,
    store_basis=False,
    solver_dir=None,
):
    """
    Linear optimal power flow for a group of snapshots.

    Parameters
    ----------
    snapshots : list or index slice
        A list of snapshots to optimise, must be a subset of
        network.snapshots, defaults to network.snapshots
    solver_name : string
        Must be a solver name that pyomo recognises and that is
        installed, e.g. "glpk", "gurobi"
    solver_logfile : None|string
        If not None, sets the logfile option of the solver.
    solver_options : dictionary
        A dictionary with additional options that get passed to the solver.
        (e.g. {'threads':2} tells gurobi to use only 2 cpus)
    solver_dir : str, default None
        Path to directory where necessary files are written, default None leads
        to the default temporary directory used by tempfile.mkstemp().
    keep_files : bool, default False
        Keep the files that pyomo constructs from OPF problem
        construction, e.g. .lp file - useful for debugging
    formulation : string
        Formulation of the linear power flow equations to use; must be
        one of ["angles","cycles","kirchhoff","ptdf"]
    extra_functionality : callable function
        This function must take two arguments
        `extra_functionality(network,snapshots)` and is called after
        the model building is complete, but before it is sent to the
        solver. It allows the user to
        add/change constraints and add/change the objective function.
    skip_pre : bool, default False
        Skip the preliminary steps of computing topology.
    skip_objective : bool, default False
        Skip writing the default objective function. If False, a custom
        objective has to be defined via extra_functionality.
    extra_postprocessing : callable function
        This function must take three arguments
        `extra_postprocessing(network,snapshots,duals)` and is called after
        the model has solved and the results are extracted. It allows the user
        to extract further information about the solution, such as additional
        shadow prices.
    warmstart : bool or string, default False
        Use this to warmstart the optimization. Pass a string which gives
        the path to the basis file. If set to True, a path to
        a basis file must be given in network.basis_fn.
    store_basis : bool, default False
        Whether to store the basis of the optimization results. If True,
        the path to the basis file is saved in network.basis_fn. Note that
        a basis can only be stored if simplex, dual-simplex, or barrier
        *with* crossover is used for solving.
    keep_references : bool, default False
        Keep the references of variable and constraint names withing the
        network. These can be looked up in `n.vars` and `n.cons` after solving.
    keep_shadowprices : bool or list of component names
        Keep shadow prices for all constraints, if set to True. If a list
        is passed the shadow prices will only be parsed for those constraint
        names. Defaults to ['Bus', 'Line', 'GlobalConstraint'].
        After solving, the shadow prices can be retrieved using
        :func:`pypsa.linopt.get_dual` with corresponding name
    """
    supported_solvers = ["highs", "cbc", "gurobi", "glpk", "cplex", "xpress"]
    if solver_name not in supported_solvers:
        raise NotImplementedError(
            f"Solver {solver_name} not in " f"supported solvers: {supported_solvers}"
        )

    if formulation != "kirchhoff":
        raise NotImplementedError("Only the kirchhoff formulation is supported")

    if n.generators.committable.any():
        logger.warning(
            "Unit commitment is not yet completely implemented for "
            "optimising without pyomo. Thus minimum up time, minimum down time, "
            "start up costs, shut down costs will be ignored."
        )

    snapshots = _as_snapshots(n, snapshots)

    if multi_investment_periods:
        logger.info("Perform multi-investment optimization.")
        assert not n.investment_periods.empty, "No investment periods defined."
        assert (
            n.snapshots.levels[0].difference(n.investment_periods).empty
        ), "Not all first-level snapshots values in investment periods."
    n._multi_invest = int(multi_investment_periods)

    if not skip_pre:
        n.calculate_dependent_values()
        n.determine_network_topology()

    logger.info("Prepare linear problem")
    fdp, problem_fn = prepare_lopf(
        n, snapshots, keep_files, skip_objective, extra_functionality, solver_dir
    )
    fds, solution_fn = mkstemp(prefix="pypsa-solve", suffix=".sol", dir=solver_dir)

    if warmstart == True:
        warmstart = n.basis_fn
        logger.info("Solve linear problem using warmstart")
    else:
        logger.info(f"Solve linear problem using {solver_name.title()} solver")

    solve = eval(f"run_and_read_{solver_name}")
    res = solve(
        n,
        problem_fn,
        solution_fn,
        solver_logfile,
        solver_options,
        warmstart,
        store_basis,
    )

    status, termination_condition, variables_sol, constraints_dual, obj = res

    if not keep_files:
        os.close(fdp)
        os.remove(problem_fn)
        os.close(fds)
        os.remove(solution_fn)

    if status == "ok" and termination_condition == "optimal":
        logger.info("Optimization successful. Objective value: {:.2e}".format(obj))
    elif status == "warning" and termination_condition == "suboptimal":
        logger.warning(
            "Optimization solution is sub-optimal. "
            "Objective value: {:.2e}".format(obj)
        )
    else:
        logger.warning(
            f"Optimization failed with status {status} and "
            f"termination condition {termination_condition}"
        )
        return status, termination_condition

    n.objective = obj
    assign_solution(
        n,
        snapshots,
        variables_sol,
        constraints_dual,
        keep_references=keep_references,
        keep_shadowprices=keep_shadowprices,
    )
    gc.collect()

    return status, termination_condition


def ilopf(
    n,
    snapshots=None,
    msq_threshold=0.05,
    min_iterations=1,
    max_iterations=100,
    track_iterations=False,
    **kwargs,
):
    """
    Iterative linear optimization updating the line parameters for passive AC
    and DC lines. This is helpful when line expansion is enabled. After each
    sucessful solving, line impedances and line resistance are recalculated
    based on the optimization result. If warmstart is possible, it uses the
    result from the previous iteration to fasten the optimization.

    Parameters
    ----------
    snapshots : list or index slice
        A list of snapshots to optimise, must be a subset of
        network.snapshots, defaults to network.snapshots
    msq_threshold: float, default 0.05
        Maximal mean square difference between optimized line capacity of
        the current and the previous iteration. As soon as this threshold is
        undercut, and the number of iterations is bigger than 'min_iterations'
        the iterative optimization stops
    min_iterations : integer, default 1
        Minimal number of iteration to run regardless whether the msq_threshold
        is already undercut
    max_iterations : integer, default 100
        Maximal number of iterations to run regardless whether msq_threshold
        is already undercut
    track_iterations: bool, default False
        If True, the intermediate branch capacities and values of the
        objective function are recorded for each iteration. The values of
        iteration 0 represent the initial state.
    **kwargs
        Keyword arguments of the lopf function which runs at each iteration
    """

    n.lines["carrier"] = n.lines.bus0.map(n.buses.carrier)
    ext_i = get_extendable_i(n, "Line")
    typed_i = n.lines.query('type != ""').index
    ext_untyped_i = ext_i.difference(typed_i)
    ext_typed_i = ext_i.intersection(typed_i)
    base_s_nom = (
        np.sqrt(3)
        * n.lines["type"].map(n.line_types.i_nom)
        * n.lines.bus0.map(n.buses.v_nom)
    )
    n.lines.loc[ext_typed_i, "num_parallel"] = (n.lines.s_nom / base_s_nom)[ext_typed_i]

    def update_line_params(n, s_nom_prev):
        factor = n.lines.s_nom_opt / s_nom_prev
        for attr, carrier in (("x", "AC"), ("r", "DC")):
            ln_i = n.lines.query("carrier == @carrier").index.intersection(
                ext_untyped_i
            )
            n.lines.loc[ln_i, attr] /= factor[ln_i]
        ln_i = ext_i.intersection(typed_i)
        n.lines.loc[ln_i, "num_parallel"] = (n.lines.s_nom_opt / base_s_nom)[ln_i]

    def msq_diff(n, s_nom_prev):
        lines_err = (
            np.sqrt((s_nom_prev - n.lines.s_nom_opt).pow(2).mean())
            / n.lines["s_nom_opt"].mean()
        )
        logger.info(
            f"Mean square difference after iteration {iteration} is " f"{lines_err}"
        )
        return lines_err

    def save_optimal_capacities(n, iteration, status):
        for c, attr in pd.Series(nominal_attrs)[n.branch_components].items():
            n.df(c)[f"{attr}_opt_{iteration}"] = n.df(c)[f"{attr}_opt"]
        setattr(n, f"status_{iteration}", status)
        setattr(n, f"objective_{iteration}", n.objective)
        n.iteration = iteration
        n.global_constraints = n.global_constraints.rename(
            columns={"mu": f"mu_{iteration}"}
        )

    if track_iterations:
        for c, attr in pd.Series(nominal_attrs)[n.branch_components].items():
            n.df(c)[f"{attr}_opt_0"] = n.df(c)[f"{attr}"]
    iteration = 1
    kwargs["store_basis"] = True
    diff = msq_threshold
    while diff >= msq_threshold or iteration < min_iterations:
        if iteration > max_iterations:
            logger.info(
                f"Iteration {iteration} beyond max_iterations "
                f"{max_iterations}. Stopping ..."
            )
            break

        s_nom_prev = n.lines.s_nom_opt.copy() if iteration else n.lines.s_nom.copy()
        kwargs["warmstart"] = bool(iteration and ("basis_fn" in n.__dir__()))
        status, termination_condition = network_lopf(n, snapshots, **kwargs)
        assert status == "ok", (
            f"Optimization failed with status {status}"
            f"and termination {termination_condition}"
        )
        if track_iterations:
            save_optimal_capacities(n, iteration, status)
        update_line_params(n, s_nom_prev)
        diff = msq_diff(n, s_nom_prev)
        iteration += 1
    logger.info("Running last lopf with fixed branches (HVDC links and HVAC lines)")
    ext_dc_links_b = n.links.p_nom_extendable & (n.links.carrier == "DC")
    s_nom_orig = n.lines.s_nom.copy()
    p_nom_orig = n.links.p_nom.copy()
    n.lines.loc[ext_i, ["s_nom", "s_nom_extendable"]] = (
        n.lines.loc[ext_i, "s_nom_opt"],
        False,
    )
    n.links.loc[ext_dc_links_b, ["p_nom", "p_nom_extendable"]] = (
        n.links.loc[ext_dc_links_b, "p_nom_opt"],
        False,
    )
    kwargs["warmstart"] = False
    network_lopf(n, snapshots, **kwargs)
    n.lines.loc[ext_i, ["s_nom", "s_nom_extendable"]] = s_nom_orig.loc[ext_i], True
    n.links.loc[ext_dc_links_b, ["p_nom", "p_nom_extendable"]] = (
        p_nom_orig.loc[ext_dc_links_b],
        True,
    )
    ## add costs of additional infrastructure to objective value of last iteration
    obj_links = (
        n.links[ext_dc_links_b].eval("capital_cost * (p_nom_opt - p_nom_min)").sum()
    )
    obj_lines = n.lines.eval("capital_cost * (s_nom_opt - s_nom_min)").sum()
    n.objective += obj_links + obj_lines
    n.objective_constant -= obj_links + obj_lines