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Revert "Introduce variable for storage losses"

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/src/oemof/solph/components/_generic_storage.py

# -*- coding: utf-8 -

"""
GenericStorage and associated individual constraints (blocks) and groupings.

SPDX-FileCopyrightText: Uwe Krien <krien@uni-bremen.de>
SPDX-FileCopyrightText: Simon Hilpert
SPDX-FileCopyrightText: Cord Kaldemeyer
SPDX-FileCopyrightText: Patrik Schönfeldt
SPDX-FileCopyrightText: FranziPl
SPDX-FileCopyrightText: jnnr
SPDX-FileCopyrightText: Stephan Günther
SPDX-FileCopyrightText: FabianTU
SPDX-FileCopyrightText: Johannes Röder
SPDX-FileCopyrightText: Ekaterina Zolotarevskaia
SPDX-FileCopyrightText: Johannes Kochems
SPDX-FileCopyrightText: Johannes Giehl
SPDX-FileCopyrightText: Raul Ciria Aylagas

SPDX-License-Identifier: MIT

"""
import numbers
from warnings import warn

import numpy as np
from oemof.network import Node
from oemof.tools import debugging
from oemof.tools import economics
from pyomo.core.base.block import ScalarBlock
from pyomo.environ import Binary
from pyomo.environ import BuildAction
from pyomo.environ import Constraint
from pyomo.environ import Expression
from pyomo.environ import NonNegativeReals
from pyomo.environ import Set
from pyomo.environ import Var

from oemof.solph._helpers import check_node_object_for_missing_attribute
from oemof.solph._options import Investment
from oemof.solph._plumbing import sequence as solph_sequence


class GenericStorage(Node):
    r"""
    Component `GenericStorage` to model with basic characteristics of storages.

    The GenericStorage is designed for one input and one output.

    Parameters
    ----------
    nominal_storage_capacity : numeric, :math:`E_{nom}` or
            :class:`oemof.solph.options.Investment` object
        Absolute nominal capacity of the storage, fixed value or
        object describing parameter of investment optimisations.
    invest_relation_input_capacity : numeric or None, :math:`r_{cap,in}`
        Ratio between the investment variable of the input Flow and the
        investment variable of the storage:
        :math:`\dot{E}_{in,invest} = E_{invest} \cdot r_{cap,in}`
    invest_relation_output_capacity : numeric or None, :math:`r_{cap,out}`
        Ratio between the investment variable of the output Flow and the
        investment variable of the storage:
        :math:`\dot{E}_{out,invest} = E_{invest} \cdot r_{cap,out}`
    invest_relation_input_output : numeric or None, :math:`r_{in,out}`
        Ratio between the investment variable of the output Flow and the
        investment variable of the input flow. This ratio used to fix the
        flow investments to each other.
        Values < 1 set the input flow lower than the output and > 1 will
        set the input flow higher than the output flow. If None no relation
        will be set:
        :math:`\dot{E}_{in,invest} = \dot{E}_{out,invest} \cdot r_{in,out}`
    initial_storage_level : numeric, :math:`c(-1)`
        The relative storage content in the timestep before the first
        time step of optimization (between 0 and 1).

        Note: When investment mode is used in a multi-period model,
        `initial_storage_level` is not supported.
        Storage output is forced to zero until the storage unit is invested in.
    balanced : boolean
        Couple storage level of first and last time step.
        (Total inflow and total outflow are balanced.)
    loss_rate : numeric (iterable or scalar)
        The relative loss of the storage content per hour.
    fixed_losses_relative : numeric (iterable or scalar), :math:`\gamma(t)`
        Losses per hour that are independent of the storage content but
        proportional to nominal storage capacity.

        Note: Fixed losses are not supported in investment mode.
    fixed_losses_absolute : numeric (iterable or scalar), :math:`\delta(t)`
        Losses per hour that are independent of storage content and independent
        of nominal storage capacity.

        Note: Fixed losses are not supported in investment mode.
    inflow_conversion_factor : numeric (iterable or scalar), :math:`\eta_i(t)`
        The relative conversion factor, i.e. efficiency associated with the
        inflow of the storage.
    outflow_conversion_factor : numeric (iterable or scalar), :math:`\eta_o(t)`
        see: inflow_conversion_factor
    min_storage_level : numeric (iterable or scalar), :math:`c_{min}(t)`
        The normed minimum storage content as fraction of the
        nominal storage capacity or the capacity that has been invested into
        (between 0 and 1).
        To set different values in every time step use a sequence.
    max_storage_level : numeric (iterable or scalar), :math:`c_{max}(t)`
        see: min_storage_level
    investment : :class:`oemof.solph.options.Investment` object
        Object indicating if a nominal_value of the flow is determined by
        the optimization problem. Note: This will refer all attributes to an
        investment variable instead of to the nominal_storage_capacity. The
        nominal_storage_capacity should not be set (or set to None) if an
        investment object is used.
    storage_costs : numeric (iterable or scalar), :math:`c_{storage}(t)`
        Cost (per energy) for having energy in the storage.
    lifetime_inflow : int, :math:`n_{in}`
        Determine the lifetime of an inflow; only applicable for multi-period
        models which can invest in storage capacity and have an
        invest_relation_input_capacity defined
    lifetime_outflow : int, :math:`n_{in}`
        Determine the lifetime of an outflow; only applicable for multi-period
        models which can invest in storage capacity and have an
        invest_relation_output_capacity defined

    Notes
    -----
    The following sets, variables, constraints and objective parts are created
     * :py:class:`~oemof.solph.components._generic_storage.GenericStorageBlock`
       (if no Investment object present)
     * :py:class:`~oemof.solph.components._generic_storage.GenericInvestmentStorageBlock`
       (if Investment object present)

    Examples
    --------
    Basic usage examples of the GenericStorage with a random selection of
    attributes. See the Flow class for all Flow attributes.

    >>> from oemof import solph

    >>> my_bus = solph.buses.Bus('my_bus')

    >>> my_storage = solph.components.GenericStorage(
    ...     label='storage',
    ...     nominal_storage_capacity=1000,
    ...     inputs={my_bus: solph.flows.Flow(nominal_value=200, variable_costs=10)},
    ...     outputs={my_bus: solph.flows.Flow(nominal_value=200)},
    ...     loss_rate=0.01,
    ...     initial_storage_level=0,
    ...     max_storage_level = 0.9,
    ...     inflow_conversion_factor=0.9,
    ...     outflow_conversion_factor=0.93)

    >>> my_investment_storage = solph.components.GenericStorage(
    ...     label='storage',
    ...     nominal_storage_capacity=solph.Investment(ep_costs=50),
    ...     inputs={my_bus: solph.flows.Flow()},
    ...     outputs={my_bus: solph.flows.Flow()},
    ...     loss_rate=0.02,
    ...     initial_storage_level=None,
    ...     invest_relation_input_capacity=1/6,
    ...     invest_relation_output_capacity=1/6,
    ...     inflow_conversion_factor=1,
    ...     outflow_conversion_factor=0.8)
    """  # noqa: E501

    def __init__(
        self,
        label=None,
        inputs=None,
        outputs=None,
        nominal_storage_capacity=None,
        initial_storage_level=None,
        investment=None,
        invest_relation_input_output=None,
        invest_relation_input_capacity=None,
        invest_relation_output_capacity=None,
        min_storage_level=0,
        max_storage_level=1,
        balanced=True,
        loss_rate=0,
        fixed_losses_relative=0,
        fixed_losses_absolute=0,
        inflow_conversion_factor=1,
        outflow_conversion_factor=1,
        fixed_costs=0,
        storage_costs=None,
        lifetime_inflow=None,
        lifetime_outflow=None,
        custom_attributes=None,
    ):
        if inputs is None:
            inputs = {}
        if outputs is None:
            outputs = {}
        if custom_attributes is None:
            custom_attributes = {}
        super().__init__(
            label,
            inputs=inputs,
            outputs=outputs,
            custom_properties=custom_attributes,
        )
        # --- BEGIN: The following code can be removed for versions >= v0.6 ---
        if investment is not None:
            msg = (
                "For backward compatibility,"
                " the option investment overwrites the option"
                + " nominal_storage_capacity."
                + " Both options cannot be set at the same time."
            )
            if nominal_storage_capacity is not None:
                raise AttributeError(msg)
            else:
                warn(msg, FutureWarning)
            nominal_storage_capacity = investment
        # --- END ---

        self.nominal_storage_capacity = None
        self.investment = None
        self._invest_group = False
        if isinstance(nominal_storage_capacity, numbers.Real):
            self.nominal_storage_capacity = nominal_storage_capacity
        elif isinstance(nominal_storage_capacity, Investment):
            self.investment = nominal_storage_capacity
            self._invest_group = True

        self.initial_storage_level = initial_storage_level
        self.balanced = balanced
        self.loss_rate = solph_sequence(loss_rate)
        self.fixed_losses_relative = solph_sequence(fixed_losses_relative)
        self.fixed_losses_absolute = solph_sequence(fixed_losses_absolute)
        self.inflow_conversion_factor = solph_sequence(
            inflow_conversion_factor
        )
        self.outflow_conversion_factor = solph_sequence(
            outflow_conversion_factor
        )
        self.max_storage_level = solph_sequence(max_storage_level)
        self.min_storage_level = solph_sequence(min_storage_level)
        self.fixed_costs = solph_sequence(fixed_costs)
        self.storage_costs = solph_sequence(storage_costs)
        self.invest_relation_input_output = invest_relation_input_output
        self.invest_relation_input_capacity = invest_relation_input_capacity
        self.invest_relation_output_capacity = invest_relation_output_capacity
        self.lifetime_inflow = lifetime_inflow
        self.lifetime_outflow = lifetime_outflow

        # Check number of flows.
        self._check_number_of_flows()
        # Check for infeasible parameter combinations
        self._check_infeasible_parameter_combinations()

        if self._invest_group:
            self._check_invest_attributes()

    def _set_flows(self):
        """Define inflow / outflow as investment flows when they are
        coupled with storage capacity via invest relations
        """
        for flow in self.inputs.values():
            if (
                self.invest_relation_input_capacity is not None
                and not isinstance(flow.investment, Investment)
            ):
                flow.investment = Investment(lifetime=self.lifetime_inflow)
        for flow in self.outputs.values():
            if (
                self.invest_relation_output_capacity is not None
                and not isinstance(flow.investment, Investment)
            ):
                flow.investment = Investment(lifetime=self.lifetime_outflow)

    def _check_invest_attributes(self):
        """Raise errors for infeasible investment attribute combinations"""
        if (
            self.invest_relation_input_output is not None
            and self.invest_relation_output_capacity is not None
            and self.invest_relation_input_capacity is not None
        ):
            e2 = (
                "Overdetermined. Three investment object will be coupled"
                "with three constraints. Set one invest relation to 'None'."
            )
            raise AttributeError(e2)
        if (
            self.investment
            and sum(solph_sequence(self.fixed_losses_absolute)) != 0
            and self.investment.existing == 0
            and self.investment.minimum[0] == 0
        ):
            e3 = (
                "With fixed_losses_absolute > 0, either investment.existing "
                "or investment.minimum has to be non-zero."
            )
            raise AttributeError(e3)

        self._set_flows()

    def _check_number_of_flows(self):
        """Ensure that there is only one inflow and outflow to the storage"""
        msg = "Only one {0} flow allowed in the GenericStorage {1}."
        check_node_object_for_missing_attribute(self, "inputs")
        check_node_object_for_missing_attribute(self, "outputs")
        if len(self.inputs) > 1:
            raise AttributeError(msg.format("input", self.label))
        if len(self.outputs) > 1:
            raise AttributeError(msg.format("output", self.label))

    def _check_infeasible_parameter_combinations(self):
        """Check for infeasible parameter combinations and raise error"""
        msg = (
            "initial_storage_level must be greater or equal to "
            "min_storage_level and smaller or equal to "
            "max_storage_level."
        )
        if self.initial_storage_level is not None:
            if (
                self.initial_storage_level < self.min_storage_level[0]
                or self.initial_storage_level > self.max_storage_level[0]
            ):
                raise ValueError(msg)

    def constraint_group(self):
        if self._invest_group is True:
            return GenericInvestmentStorageBlock
        else:
            return GenericStorageBlock


class GenericStorageBlock(ScalarBlock):
    r"""Storage without an :class:`.Investment` object.

    **The following sets are created:** (-> see basic sets at
    :class:`.Model` )

    STORAGES
        A set with all :py:class:`~.GenericStorage` objects, which do not have an
        :attr:`investment` of type :class:`.Investment`.

    STORAGES_BALANCED
        A set of  all :py:class:`~.GenericStorage` objects, with 'balanced' attribute set
        to True.

    STORAGES_WITH_INVEST_FLOW_REL
        A set with all :py:class:`~.GenericStorage` objects with two investment
        flows coupled with the 'invest_relation_input_output' attribute.

    **The following variables are created:**

    storage_content
        Storage content for every storage and timestep. The value for the
        storage content at the beginning is set by the parameter
        `initial_storage_level` or not set if `initial_storage_level` is None.
        The variable of storage s and timestep t can be accessed by:
        `om.GenericStorageBlock.storage_content[s, t]`

    **The following constraints are created:**

    Set storage_content of last time step to one at t=0 if balanced == True
        .. math::
            E(t_{last}) = E(-1)

    Storage balance :attr:`om.Storage.balance[n, t]`
        .. math:: E(t) = &E(t-1) \cdot
            (1 - \beta(t)) ^{\tau(t)/(t_u)} \\
            &- \gamma(t)\cdot E_{nom} \cdot {\tau(t)/(t_u)}\\
            &- \delta(t) \cdot {\tau(t)/(t_u)}\\
            &- \frac{\dot{E}_o(p, t)}{\eta_o(t)} \cdot \tau(t)
            + \dot{E}_i(p, t) \cdot \eta_i(t) \cdot \tau(t)

    Connect the invest variables of the input and the output flow.
        .. math::
          InvestmentFlowBlock.invest(source(n), n, p) + existing = \\
          (InvestmentFlowBlock.invest(n, target(n), p) + existing) \\
          * invest\_relation\_input\_output(n) \\
          \forall n \in \textrm{INVEST\_REL\_IN\_OUT} \\
          \forall p \in \textrm{PERIODS}



    =========================== ======================= =========
    symbol                      explanation             attribute
    =========================== ======================= =========
    :math:`E(t)`                energy currently stored `storage_content`
    :math:`E_{nom}`             nominal capacity of     `nominal_storage_capacity`
                                the energy storage
    :math:`c(-1)`               state before            `initial_storage_level`
                                initial time step
    :math:`c_{min}(t)`          minimum allowed storage `min_storage_level[t]`
    :math:`c_{max}(t)`          maximum allowed storage `max_storage_level[t]`
    :math:`\beta(t)`            fraction of lost energy `loss_rate[t]`
                                as share of
                                :math:`E(t)` per hour
    :math:`\gamma(t)`           fixed loss of energy    `fixed_losses_relative[t]`
                                per hour relative to
                                :math:`E_{nom}`
    :math:`\delta(t)`           absolute fixed loss     `fixed_losses_absolute[t]`
                                of energy per hour
    :math:`\dot{E}_i(t)`        energy flowing in       `inputs`
    :math:`\dot{E}_o(t)`        energy flowing out      `outputs`
    :math:`\eta_i(t)`           conversion factor       `inflow_conversion_factor[t]`
                                (i.e. efficiency)
                                when storing energy
    :math:`\eta_o(t)`           conversion factor when  `outflow_conversion_factor[t]`
                                (i.e. efficiency)
                                taking stored energy
    :math:`\tau(t)`             duration of time step
    :math:`t_u`                 time unit of losses
                                :math:`\beta(t)`,
                                :math:`\gamma(t)`
                                :math:`\delta(t)` and
                                timeincrement
                                :math:`\tau(t)`
    :math:`c_{storage}(t)`      costs of having         `storage_costs`
                                energy stored
    =========================== ======================= =========

    **The following parts of the objective function are created:**

    *Standard model*

    * :attr: `storage_costs` not 0

        .. math::
            \sum_{t \in \textrm{TIMESTEPS}} c_{storage}(t) \cdot E(t)


    *Multi-period model*

    * :attr:`fixed_costs` not None

        .. math::
            \displaystyle \sum_{pp=0}^{year_{max}} E_{nom}
            \cdot c_{fixed}(pp) \cdot DF^{-pp}

    where:

    * :math:`DF=(1+dr)` is the discount factor with discount rate :math:`dr`.
    * :math:`year_{max}` denotes the last year of the optimization
      horizon, i.e. at the end of the last period.

    """  # noqa: E501

    CONSTRAINT_GROUP = True

    def __init__(self, *args, **kwargs):
        super().__init__(*args, **kwargs)

    def _create(self, group=None):
        """
        Parameters
        ----------
        group : list
            List containing storage objects.
            e.g. groups=[storage1, storage2,..]
        """
        m = self.parent_block()

        if group is None:
            return None

        i = {n: [i for i in n.inputs][0] for n in group}
        o = {n: [o for o in n.outputs][0] for n in group}

        #  ************* SETS *********************************

        self.STORAGES = Set(initialize=[n for n in group])

        self.STORAGES_BALANCED = Set(
            initialize=[n for n in group if n.balanced is True]
        )

        self.STORAGES_INITITAL_LEVEL = Set(
            initialize=[
                n for n in group if n.initial_storage_level is not None
            ]
        )

        self.STORAGES_WITH_INVEST_FLOW_REL = Set(
            initialize=[
                n for n in group if n.invest_relation_input_output is not None
            ]
        )

        #  ************* VARIABLES *****************************

        def _storage_content_bound_rule(block, n, t):
            """
            Rule definition for bounds of storage_content variable of
            storage n in timestep t.
            """
            bounds = (
                n.nominal_storage_capacity * n.min_storage_level[t],
                n.nominal_storage_capacity * n.max_storage_level[t],
            )
            return bounds

        self.storage_content = Var(
            self.STORAGES, m.TIMEPOINTS, bounds=_storage_content_bound_rule
        )

        # set the initial storage content
        # ToDo: More elegant code possible?
        for n in group:
            if n.initial_storage_level is not None:
                self.storage_content[n, 0] = (
                    n.initial_storage_level * n.nominal_storage_capacity
                )
                self.storage_content[n, 0].fix()

        #  ************* Constraints ***************************

        def _storage_balance_rule(block, n, t):
            """
            Rule definition for the storage balance of every storage n and
            every timestep.
            """
            expr = 0
            expr += block.storage_content[n, t + 1]
            expr += (
                -block.storage_content[n, t]
                * (1 - n.loss_rate[t]) ** m.timeincrement[t]
            )
            expr += (
                n.fixed_losses_relative[t]
                * n.nominal_storage_capacity
                * m.timeincrement[t]
            )
            expr += n.fixed_losses_absolute[t] * m.timeincrement[t]
            expr += (
                -m.flow[i[n], n, t] * n.inflow_conversion_factor[t]
            ) * m.timeincrement[t]
            expr += (
                m.flow[n, o[n], t] / n.outflow_conversion_factor[t]
            ) * m.timeincrement[t]
            return expr == 0

        self.balance = Constraint(
            self.STORAGES, m.TIMESTEPS, rule=_storage_balance_rule
        )

        def _balanced_storage_rule(block, n):
            """
            Storage content of last time step == initial storage content
            if balanced.
            """
            return (
                block.storage_content[n, m.TIMEPOINTS.at(-1)]
                == block.storage_content[n, m.TIMEPOINTS.at(1)]
            )

        self.balanced_cstr = Constraint(
            self.STORAGES_BALANCED, rule=_balanced_storage_rule
        )

        def _power_coupled(block):
            """
            Rule definition for constraint to connect the input power
            and output power
            """
            for n in self.STORAGES_WITH_INVEST_FLOW_REL:
                for p in m.PERIODS:
                    expr = (
                        m.InvestmentFlowBlock.total[n, o[n], p]
                    ) * n.invest_relation_input_output == (
                        m.InvestmentFlowBlock.total[i[n], n, p]
                    )
                    self.power_coupled.add((n, p), expr)

        self.power_coupled = Constraint(
            self.STORAGES_WITH_INVEST_FLOW_REL, m.PERIODS, noruleinit=True
        )

        self.power_coupled_build = BuildAction(rule=_power_coupled)

    def _objective_expression(self):
        r"""
        Objective expression for storages with no investment.

        Note
        ----
        * For standard models, this adds nothing as variable costs are
          already added in the Block :py:class:`~.SimpleFlowBlock`.
        * For multi-period models, fixed costs may be introduced
          and added here.
        """
        m = self.parent_block()

        if not hasattr(self, "STORAGES"):
            return 0

        fixed_costs = 0

        if m.es.periods is not None:
            for n in self.STORAGES:
                if n.fixed_costs[0] is not None:
                    fixed_costs += sum(
                        n.nominal_storage_capacity
                        * n.fixed_costs[pp]
                        * (1 + m.discount_rate) ** (-pp)
                        for pp in range(m.es.end_year_of_optimization)
                    )
        self.fixed_costs = Expression(expr=fixed_costs)

        storage_costs = 0

        for n in self.STORAGES:
            if n.storage_costs[0] is not None:
                storage_costs += (
                    self.storage_content[n, 0] * n.storage_costs[0]
                )
                for t in m.TIMESTEPS:
                    storage_costs += (
                        self.storage_content[n, t + 1] * n.storage_costs[t + 1]
                    )

        self.storage_costs = Expression(expr=storage_costs)
        self.costs = Expression(expr=storage_costs + fixed_costs)

        return self.costs


class GenericInvestmentStorageBlock(ScalarBlock):
    r"""
    Block for all storages with :attr:`Investment` being not None.
    See :class:`.Investment` for all parameters of the
    Investment class.

    **Variables**

    All Storages are indexed by :math:`n` (denoting the respective storage
    unit), which is omitted in the following for the sake of convenience.
    The following variables are created as attributes of
    :attr:`om.GenericInvestmentStorageBlock`:

    * :math:`P_i(p, t)`

        Inflow of the storage
        (created in :class:`oemof.solph.models.BaseModel`).

    * :math:`P_o(p, t)`

        Outflow of the storage
        (created in :class:`oemof.solph.models.BaseModel`).

    * :math:`E(t)`

        Current storage content (Absolute level of stored energy).

    * :math:`E_{invest}(p)`

        Invested (nominal) capacity of the storage in period p.

    * :math:`E_{total}(p)`

        Total installed (nominal) capacity of the storage in period p.

    * :math:`E_{old}(p)`

        Old (nominal) capacity of the storage to be decommissioned in period p.

    * :math:`E_{old,exo}(p)`

        Exogenous old (nominal) capacity of the storage to be decommissioned
        in period p; existing capacity reaching its lifetime.

    * :math:`E_{old,endo}(p)`

        Endogenous old (nominal) capacity of the storage to be decommissioned
        in period p; endgenous investments reaching their lifetime.

    * :math:`E(-1)`

        Initial storage content (before timestep 0).
        Not applicable for a multi-period model.

    * :math:`b_{invest}(p)`

        Binary variable for the status of the investment, if
        :attr:`nonconvex` is `True`.

    **Constraints**

    The following constraints are created for all investment storages:

        Storage balance (Same as for :class:`.GenericStorageBlock`)

        .. math:: E(t) = &E(t-1) \cdot
            (1 - \beta(t)) ^{\tau(t)/(t_u)} \\
            &- \gamma(t)\cdot (E_{total}(p)) \cdot {\tau(t)/(t_u)}\\
            &- \delta(t) \cdot {\tau(t)/(t_u)}\\
            &- \frac{\dot{E}_o(p, t))}{\eta_o(t)} \cdot \tau(t)
            + \dot{E}_i(p, t) \cdot \eta_i(t) \cdot \tau(t)

        Total storage capacity (p > 0 for multi-period model only)

        .. math::
            &
            if \quad p=0:\\
            &
            E_{total}(p) = E_{exist} + E_{invest}(p)\\
            &\\
            &
            else:\\
            &
            E_{total}(p) = E_{total}(p-1) + E_{invest}(p) - E_{old}(p)\\
            &\\
            &
            \forall p \in \textrm{PERIODS}

        Old storage capacity (p > 0 for multi-period model only)

        .. math::
            &
            E_{old}(p) = E_{old,exo}(p) + E_{old,end}(p)\\
            &\\
            &
            if \quad p=0:\\
            &
            E_{old,end}(p) = 0\\
            &\\
            &
            else \quad if \quad l \leq year(p):\\
            &
            E_{old,end}(p) = E_{invest}(p_{comm})\\
            &\\
            &
            else:\\
            &
            E_{old,end}(p)\\
            &\\
            &
            if \quad p=0:\\
            &
            E_{old,exo}(p) = 0\\
            &\\
            &
            else \quad if \quad l - a \leq year(p):\\
            &
            E_{old,exo}(p) = E_{exist} (*)\\
            &\\
            &
            else:\\
            &
            E_{old,exo}(p) = 0\\
            &\\
            &
            \forall p \in \textrm{PERIODS}

        where:

        * (*) is only performed for the first period the condition is True.
          A decommissioning flag is then set to True to prevent having falsely
          added old capacity in future periods.
        * :math:`year(p)` is the year corresponding to period p
        * :math:`p_{comm}` is the commissioning period of the storage

    Depending on the attribute :attr:`nonconvex`, the constraints for the
    bounds of the decision variable :math:`E_{invest}(p)` are different:\

        * :attr:`nonconvex = False`

        .. math::
            &
            E_{invest, min}(p) \le E_{invest}(p) \le E_{invest, max}(p) \\
            &
            \forall p \in \textrm{PERIODS}

        * :attr:`nonconvex = True`

        .. math::
            &
            E_{invest, min}(p) \cdot b_{invest}(p) \le E_{invest}(p)\\
            &
            E_{invest}(p) \le E_{invest, max}(p) \cdot b_{invest}(p)\\
            &
            \forall p \in \textrm{PERIODS}

    The following constraints are created depending on the attributes of
    the :class:`.GenericStorage`:

        * :attr:`initial_storage_level is None`;
          not applicable for multi-period model

            Constraint for a variable initial storage content:

        .. math::
               E(-1) \le E_{exist} + E_{invest}(0)

        * :attr:`initial_storage_level is not None`;
          not applicable for multi-period model

            An initial value for the storage content is given:

        .. math::
               E(-1) = (E_{invest}(0) + E_{exist}) \cdot c(-1)

        * :attr:`balanced=True`;
          not applicable for multi-period model

            The energy content of storage of the first and the last timestep
            are set equal:

        .. math::
            E(-1) = E(t_{last})

        * :attr:`invest_relation_input_capacity is not None`

            Connect the invest variables of the storage and the input flow:

        .. math::
            &
            P_{i,total}(p) =
            E_{total}(p) \cdot r_{cap,in} \\
            &
            \forall p \in \textrm{PERIODS}

        * :attr:`invest_relation_output_capacity is not None`

            Connect the invest variables of the storage and the output flow:

        .. math::
            &
            P_{o,total}(p) =
            E_{total}(p) \cdot r_{cap,out}\\
            &
            \forall p \in \textrm{PERIODS}

        * :attr:`invest_relation_input_output is not None`

            Connect the invest variables of the input and the output flow:

        .. math::
            &
            P_{i,total}(p) =
            P_{o,total}(p) \cdot r_{in,out}\\
            &
            \forall p \in \textrm{PERIODS}

        * :attr:`max_storage_level`

            Rule for upper bound constraint for the storage content:

        .. math::
            &
            E(t) \leq E_{total}(p) \cdot c_{max}(t)\\
            &
            \forall p, t \in \textrm{TIMEINDEX}

        * :attr:`min_storage_level`

            Rule for lower bound constraint for the storage content:

        .. math::
            &
            E(t) \geq E_{total}(p) \cdot c_{min}(t)\\
            &
            \forall p, t \in \textrm{TIMEINDEX}


    **Objective function**

    Objective terms for a standard model and a multi-period model differ
    quite strongly. Besides, the part of the objective function added by the
    investment storages also depends on whether a convex or nonconvex
    investment option is selected. The following parts of the objective
    function are created:

    *Standard model*

        * :attr:`nonconvex = False`

            .. math::
                E_{invest}(0) \cdot c_{invest,var}(0)

        * :attr:`nonconvex = True`

            .. math::
                E_{invest}(0) \cdot c_{invest,var}(0)
                + c_{invest,fix}(0) \cdot b_{invest}(0)\\

    Where 0 denotes the 0th (investment) period since
    in a standard model, there is only this one period.

    *Multi-period model*

        * :attr:`nonconvex = False`

            .. math::
                &
                E_{invest}(p) \cdot A(c_{invest,var}(p), l, ir)
                \cdot \frac {1}{ANF(d, ir)} \cdot DF^{-p}\\
                &
                \forall p \in \textrm{PERIODS}

        In case, the remaining lifetime of a storage is greater than 0 and
        attribute `use_remaining_value` of the energy system is True,
        the difference in value for the investment period compared to the
        last period of the optimization horizon is accounted for
        as an adder to the investment costs:

            .. math::
                &
                E_{invest}(p) \cdot (A(c_{invest,var}(p), l_{r}, ir) -
                A(c_{invest,var}(|P|), l_{r}, ir)\\
                & \cdot \frac {1}{ANF(l_{r}, ir)} \cdot DF^{-|P|}\\
                &\\
                &
                \forall p \in \textrm{PERIODS}

        * :attr:`nonconvex = True`

            .. math::
                &
                (E_{invest}(p) \cdot A(c_{invest,var}(p), l, ir)
                \cdot \frac {1}{ANF(d, ir)}\\
                &
                +  c_{invest,fix}(p) \cdot b_{invest}(p)) \cdot DF^{-p} \\
                &
                \forall p \in \textrm{PERIODS}

        In case, the remaining lifetime of a storage is greater than 0 and
        attribute `use_remaining_value` of the energy system is True,
        the difference in value for the investment period compared to the
        last period of the optimization horizon is accounted for
        as an adder to the investment costs:

            .. math::
                &
                (E_{invest}(p) \cdot (A(c_{invest,var}(p), l_{r}, ir) -
                A(c_{invest,var}(|P|), l_{r}, ir)\\
                & \cdot \frac {1}{ANF(l_{r}, ir)} \cdot DF^{-|P|}\\
                &
                +  (c_{invest,fix}(p) - c_{invest,fix}(|P|))
                \cdot b_{invest}(p)) \cdot DF^{-p}\\
                &\\
                &
                \forall p \in \textrm{PERIODS}

        * :attr:`fixed_costs` not None for investments

            .. math::
                &
                \sum_{pp=year(p)}^{limit_{end}}
                E_{invest}(p) \cdot c_{fixed}(pp) \cdot DF^{-pp})
                \cdot DF^{-p}\\
                &
                \forall p \in \textrm{PERIODS}

        * :attr:`fixed_costs` not None for existing capacity

            .. math::
                \sum_{pp=0}^{limit_{exo}} E_{exist} \cdot c_{fixed}(pp)
                \cdot DF^{-pp}

    where:

    * :math:`A(c_{invest,var}(p), l, ir)` A is the annuity for
      investment expenses :math:`c_{invest,var}(p)`, lifetime :math:`l`
      and interest rate :math:`ir`.
    * :math:`l_{r}` is the remaining lifetime at the end of the
      optimization horizon (in case it is greater than 0 and
      smaller than the actual lifetime).
    * :math:`ANF(d, ir)` is the annuity factor for duration :math:`d`
      and interest rate :math:`ir`.
    * :math:`d=min\{year_{max} - year(p), l\}` defines the
      number of years within the optimization horizon that investment
      annuities are accounted for.
    * :math:`year(p)` denotes the start year of period :math:`p`.
    * :math:`year_{max}` denotes the last year of the optimization
      horizon, i.e. at the end of the last period.
    * :math:`limit_{end}=min\{year_{max}, year(p) + l\}` is used as an
      upper bound to ensure fixed costs for endogenous investments
      to occur within the optimization horizon.
    * :math:`limit_{exo}=min\{year_{max}, l - a\}` is used as an
      upper bound to ensure fixed costs for existing capacities to occur
      within the optimization horizon. :math:`a` is the initial age
      of an asset.
    * :math:`DF=(1+dr)` is the discount factor.

    The annuity / annuity factor hereby is:

        .. math::
            &
            A(c_{invest,var}(p), l, ir) = c_{invest,var}(p) \cdot
                \frac {(1+ir)^l \cdot ir} {(1+ir)^l - 1}\\
            &\\
            &
            ANF(d, ir)=\frac {(1+ir)^d \cdot ir} {(1+ir)^d - 1}

    They are retrieved, using oemof.tools.economics annuity function. The
    interest rate :math:`ir` for the annuity is defined as weighted
    average costs of capital (wacc) and assumed constant over time.

    The overall summed cost expressions for all *InvestmentFlowBlock* objects
    can be accessed by

    * :attr:`om.GenericInvestmentStorageBlock.investment_costs`,
    * :attr:`om.GenericInvestmentStorageBlock.fixed_costs` and
    * :attr:`om.GenericInvestmentStorageBlock.costs`.

    Their values  after optimization can be retrieved by

    * :meth:`om.GenericInvestmentStorageBlock.investment_costs`,
    * :attr:`om.GenericInvestmentStorageBlock.period_investment_costs`
      (yielding a dict keyed by periods); note: this is not a Pyomo expression,
      but calculated,
    * :meth:`om.GenericInvestmentStorageBlock.fixed_costs` and
    * :meth:`om.GenericInvestmentStorageBlock.costs`.

    .. csv-table:: List of Variables
        :header: "symbol", "attribute", "explanation"
        :widths: 1, 1, 1

        ":math:`P_i(p, t)`", ":attr:`flow[i[n], n, p, t]`", "Inflow
        of the storage"
        ":math:`P_o(p, t)`", ":attr:`flow[n, o[n], p, t]`", "Outflow
        of the storage"
        ":math:`E(t)`", ":attr:`storage_content[n, t]`", "Current storage
        content (current absolute stored energy)"
        ":math:`E_{invest}(p)`", ":attr:`invest[n, p]`", "Invested (nominal)
        capacity of the storage"
        ":math:`E_{old}(p)`", ":attr:`old[n, p]`", "
        | Old (nominal) capacity of the storage
        | to be decommissioned in period p"
        ":math:`E_{old,exo}(p)`", ":attr:`old_exo[n, p]`", "
        | Old (nominal) capacity of the storage
        | to be decommissioned in period p
        | which was exogenously given by :math:`E_{exist}`"
        ":math:`E_{old,end}(p)`", ":attr:`old_end[n, p]`", "
        | Old (nominal) capacity of the storage
        | to be decommissioned in period p
        | which was endogenously determined by :math:`E_{invest}(p_{comm})`
        | where :math:`p_{comm}` is the commissioning period"
        ":math:`E(-1)`", ":attr:`init_cap[n]`", "Initial storage capacity
        (before timestep 0)"
        ":math:`b_{invest}(p)`", ":attr:`invest_status[i, o, p]`", "Binary
        variable for the status of investment"
        ":math:`P_{i,invest}(p)`", "
        :attr:`InvestmentFlowBlock.invest[i[n], n, p]`", "
        Invested (nominal) inflow (InvestmentFlowBlock)"
        ":math:`P_{o,invest}`", "
        :attr:`InvestmentFlowBlock.invest[n, o[n]]`", "
        Invested (nominal) outflow (InvestmentFlowBlock)"

    .. csv-table:: List of Parameters
        :header: "symbol", "attribute", "explanation"
        :widths: 1, 1, 1

        ":math:`E_{exist}`", "`flows[i, o].investment.existing`", "
        Existing storage capacity"
        ":math:`E_{invest,min}`", "`flows[i, o].investment.minimum`", "
        Minimum investment value"
        ":math:`E_{invest,max}`", "`flows[i, o].investment.maximum`", "
        Maximum investment value"
        ":math:`P_{i,exist}`", "`flows[i[n], n].investment.existing`
        ", "Existing inflow capacity"
        ":math:`P_{o,exist}`", "`flows[n, o[n]].investment.existing`
        ", "Existing outflow capacity"
        ":math:`c_{invest,var}`", "`flows[i, o].investment.ep_costs`
        ", "Variable investment costs"
        ":math:`c_{invest,fix}`", "`flows[i, o].investment.offset`", "
        Fix investment costs"
        ":math:`c_{fixed}`", "`flows[i, o].investment.fixed_costs`", "
        Fixed costs; only allowed in multi-period model"
        ":math:`r_{cap,in}`", ":attr:`invest_relation_input_capacity`", "
        Relation of storage capacity and nominal inflow"
        ":math:`r_{cap,out}`", ":attr:`invest_relation_output_capacity`", "
        Relation of storage capacity and nominal outflow"
        ":math:`r_{in,out}`", ":attr:`invest_relation_input_output`", "
        Relation of nominal in- and outflow"
        ":math:`\beta(t)`", "`loss_rate[t]`", "Fraction of lost energy
        as share of :math:`E(t)` per hour"
        ":math:`\gamma(t)`", "`fixed_losses_relative[t]`", "Fixed loss
        of energy relative to :math:`E_{invest} + E_{exist}` per hour"
        ":math:`\delta(t)`", "`fixed_losses_absolute[t]`", "Absolute
        fixed loss of energy per hour"
        ":math:`\eta_i(t)`", "`inflow_conversion_factor[t]`", "
        Conversion factor (i.e. efficiency) when storing energy"
        ":math:`\eta_o(t)`", "`outflow_conversion_factor[t]`", "
        Conversion factor when (i.e. efficiency) taking stored energy"
        ":math:`c(-1)`", "`initial_storage_level`", "Initial relative
        storage content (before timestep 0)"
        ":math:`c_{max}`", "`flows[i, o].max[t]`", "Normed maximum
        value of storage content"
        ":math:`c_{min}`", "`flows[i, o].min[t]`", "Normed minimum
        value of storage content"
        ":math:`l`", "`flows[i, o].investment.lifetime`", "
        Lifetime for investments in storage capacity"
        ":math:`a`", "`flows[i, o].investment.age`", "
        Initial age of existing capacity / energy"
        ":math:`ir`", "`flows[i, o].investment.interest_rate`", "
        interest rate for investment"
        ":math:`\tau(t)`", "", "Duration of time step"
        ":math:`t_u`", "", "Time unit of losses :math:`\beta(t)`,
        :math:`\gamma(t)`, :math:`\delta(t)` and timeincrement :math:`\tau(t)`"

    """

    CONSTRAINT_GROUP = True

    def __init__(self, *args, **kwargs):
        super().__init__(*args, **kwargs)

    def _create(self, group=None):
        """Create a storage block for investment modeling"""
        m = self.parent_block()
        if group is None:
            return None

        # ########################## CHECKS ###################################
        if m.es.periods is not None:
            for n in group:
                error_fixed_absolute_losses = (
                    "For a multi-period investment model, fixed absolute"
                    " losses are not supported. Please remove parameter."
                )
                if n.fixed_losses_absolute.default != 0:
                    raise ValueError(error_fixed_absolute_losses)
                error_initial_storage_level = (
                    "For a multi-period model, initial_storage_level is"
                    " not supported.\nIt needs to be removed since it"
                    " has no effect.\nstorage_content will be zero,"
                    " until there is some usable storage capacity installed."
                )
                if n.initial_storage_level is not None:
                    raise ValueError(error_initial_storage_level)

        # ########################## SETS #####################################

        self.INVESTSTORAGES = Set(initialize=[n for n in group])

        self.CONVEX_INVESTSTORAGES = Set(
            initialize=[n for n in group if n.investment.nonconvex is False]
        )

        self.NON_CONVEX_INVESTSTORAGES = Set(
            initialize=[n for n in group if n.investment.nonconvex is True]
        )

        self.INVESTSTORAGES_BALANCED = Set(
            initialize=[n for n in group if n.balanced is True]
        )

        self.INVESTSTORAGES_NO_INIT_CONTENT = Set(
            initialize=[n for n in group if n.initial_storage_level is None]
        )

        self.INVESTSTORAGES_INIT_CONTENT = Set(
            initialize=[
                n for n in group if n.initial_storage_level is not None
            ]
        )

        self.INVEST_REL_CAP_IN = Set(
            initialize=[
                n
                for n in group
                if n.invest_relation_input_capacity is not None
            ]
        )

        self.INVEST_REL_CAP_OUT = Set(
            initialize=[
                n
                for n in group
                if n.invest_relation_output_capacity is not None
            ]
        )

        self.INVEST_REL_IN_OUT = Set(
            initialize=[
                n for n in group if n.invest_relation_input_output is not None
            ]
        )

        # The storage content is a non-negative variable, therefore it makes no
        # sense to create an additional constraint if the lower bound is zero
        # for all time steps.
        self.MIN_INVESTSTORAGES = Set(
            initialize=[
                n
                for n in group
                if sum([n.min_storage_level[t] for t in m.TIMESTEPS]) > 0
            ]
        )

        self.OVERALL_MAXIMUM_INVESTSTORAGES = Set(
            initialize=[
                n for n in group if n.investment.overall_maximum is not None
            ]
        )

        self.OVERALL_MINIMUM_INVESTSTORAGES = Set(
            initialize=[
                n for n in group if n.investment.overall_minimum is not None
            ]
        )

        self.EXISTING_INVESTSTORAGES = Set(
            initialize=[n for n in group if n.investment.existing is not None]
        )

        # ######################### Variables  ################################
        self.storage_content = Var(
            self.INVESTSTORAGES, m.TIMESTEPS, within=NonNegativeReals
        )

        def _storage_investvar_bound_rule(block, n, p):
            """
            Rule definition to bound the invested storage capacity `invest`.
            """
            if n in self.CONVEX_INVESTSTORAGES:
                return n.investment.minimum[p], n.investment.maximum[p]
            elif n in self.NON_CONVEX_INVESTSTORAGES:
                return 0, n.investment.maximum[p]

        self.invest = Var(
            self.INVESTSTORAGES,
            m.PERIODS,
            within=NonNegativeReals,
            bounds=_storage_investvar_bound_rule,
        )

        # Total capacity
        self.total = Var(
            self.INVESTSTORAGES,
            m.PERIODS,
            within=NonNegativeReals,
            initialize=0,
        )

        if m.es.periods is not None:
            # Old capacity to be decommissioned (due to lifetime)
            self.old = Var(
                self.INVESTSTORAGES, m.PERIODS, within=NonNegativeReals
            )

            # Old endogenous capacity to be decommissioned (due to lifetime)
            self.old_end = Var(
                self.INVESTSTORAGES, m.PERIODS, within=NonNegativeReals
            )

            # Old exogenous capacity to be decommissioned (due to lifetime)
            self.old_exo = Var(
                self.INVESTSTORAGES, m.PERIODS, within=NonNegativeReals
            )

        else:
            self.init_content = Var(
                self.INVESTSTORAGES, within=NonNegativeReals
            )

        # create status variable for a non-convex investment storage
        self.invest_status = Var(
            self.NON_CONVEX_INVESTSTORAGES, m.PERIODS, within=Binary
        )

        # ######################### CONSTRAINTS ###############################
        i = {n: [i for i in n.inputs][0] for n in group}
        o = {n: [o for o in n.outputs][0] for n in group}

        reduced_periods_timeindex = [(p, t) for (p, t) in m.TIMEINDEX if t > 0]

        # Handle unit lifetimes
        def _total_storage_capacity_rule(block):
            """Rule definition for determining total installed
            capacity (taking decommissioning into account)
            """
            for n in self.INVESTSTORAGES:
                for p in m.PERIODS:
                    if p == 0:
                        expr = (
                            self.total[n, p]
                            == self.invest[n, p] + n.investment.existing
                        )
                        self.total_storage_rule.add((n, p), expr)
                    else:
                        expr = (
                            self.total[n, p]
                            == self.invest[n, p]
                            + self.total[n, p - 1]
                            - self.old[n, p]
                        )
                        self.total_storage_rule.add((n, p), expr)

        self.total_storage_rule = Constraint(
            self.INVESTSTORAGES, m.PERIODS, noruleinit=True
        )

        self.total_storage_rule_build = BuildAction(
            rule=_total_storage_capacity_rule
        )

        # multi-period storage implementation for time intervals
        if m.es.periods is not None:

            def _old_storage_capacity_rule_end(block):
                """Rule definition for determining old endogenously installed
                capacity to be decommissioned due to reaching its lifetime.
                Investment and decommissioning periods are linked within
                the constraint. The respective decommissioning period is
                determined for every investment period based on the components
                lifetime and a matrix describing its age of each endogenous
                investment. Decommissioning can only occur at the beginning of
                each period.

                Note
                ----
                For further information on the implementation check
                PR#957 https://github.com/oemof/oemof-solph/pull/957
                """
                for n in self.INVESTSTORAGES:
                    lifetime = n.investment.lifetime
                    if lifetime is None:
                        msg = (
                            "You have to specify a lifetime "
                            "for a Flow going into or out of "
                            "a GenericStorage unit "
                            "in a multi-period model!"
                            f" Value for {n} is missing."
                        )
                        raise ValueError(msg)
                    # get the period matrix describing the temporal distance
                    # between all period combinations.
                    periods_matrix = m.es.periods_matrix

                    # get the index of the minimum value in each row greater
                    # equal than the lifetime. This value equals the
                    # decommissioning period if not zero. The index of this
                    # value represents the investment period. If np.where
                    # condition is not met in any row, min value will be zero
                    decomm_periods = np.argmin(
                        np.where(
                            (periods_matrix >= lifetime),
                            periods_matrix,
                            np.inf,
                        ),
                        axis=1,
                    )

                    # no decommissioning in first period
                    expr = self.old_end[n, 0] == 0
                    self.old_rule_end.add((n, 0), expr)

                    # all periods not in decomm_periods have no decommissioning
                    # zero is excluded
                    for p in m.PERIODS:
                        if p not in decomm_periods and p != 0:
                            expr = self.old_end[n, p] == 0
                            self.old_rule_end.add((n, p), expr)

                    # multiple invests can be decommissioned in the same period
                    # but only sequential ones, thus a bookkeeping is
                    # introduced andconstraints are added to equation one
                    # iteration later.
                    last_decomm_p = np.nan
                    # loop over invest periods (values are decomm_periods)
                    for invest_p, decomm_p in enumerate(decomm_periods):
                        # Add constraint of iteration before
                        # (skipped in first iteration by last_decomm_p = nan)
                        if (decomm_p != last_decomm_p) and (
                            last_decomm_p is not np.nan
                        ):
                            expr = self.old_end[n, last_decomm_p] == expr
                            self.old_rule_end.add((n, last_decomm_p), expr)

                        # no decommissioning if decomm_p is zero
                        if decomm_p == 0:
                            # overwrite decomm_p with zero to avoid
                            # chaining invest periods in next iteration
                            last_decomm_p = 0

                        # if decomm_p is the same as the last one chain invest
                        # period
                        elif decomm_p == last_decomm_p:
                            expr += self.invest[n, invest_p]
                            # overwrite decomm_p
                            last_decomm_p = decomm_p

                        # if decomm_p is not zero, not the same as the last one
                        # and it's not the first period
                        else:
                            expr = self.invest[n, invest_p]
                            # overwrite decomm_p
                            last_decomm_p = decomm_p

                    # Add constraint of very last iteration
                    if last_decomm_p != 0:
                        expr = self.old_end[n, last_decomm_p] == expr
                        self.old_rule_end.add((n, last_decomm_p), expr)

            self.old_rule_end = Constraint(
                self.INVESTSTORAGES, m.PERIODS, noruleinit=True
            )

            self.old_rule_end_build = BuildAction(
                rule=_old_storage_capacity_rule_end
            )

            def _old_storage_capacity_rule_exo(block):
                """Rule definition for determining old exogenously given
                capacity to be decommissioned due to reaching its lifetime
                """
                for n in self.INVESTSTORAGES:
                    age = n.investment.age
                    lifetime = n.investment.lifetime
                    is_decommissioned = False
                    for p in m.PERIODS:
                        # No shutdown in first period
                        if p == 0:
                            expr = self.old_exo[n, p] == 0
                            self.old_rule_exo.add((n, p), expr)
                        elif lifetime - age <= m.es.periods_years[p]:
                            # Track decommissioning status
                            if not is_decommissioned:
                                expr = (
                                    self.old_exo[n, p] == n.investment.existing
                                )
                                is_decommissioned = True
                            else:
                                expr = self.old_exo[n, p] == 0
                            self.old_rule_exo.add((n, p), expr)
                        else:
                            expr = self.old_exo[n, p] == 0
                            self.old_rule_exo.add((n, p), expr)

            self.old_rule_exo = Constraint(
                self.INVESTSTORAGES, m.PERIODS, noruleinit=True
            )

            self.old_rule_exo_build = BuildAction(
                rule=_old_storage_capacity_rule_exo
            )

            def _old_storage_capacity_rule(block):
                """Rule definition for determining (overall) old capacity
                to be decommissioned due to reaching its lifetime
                """
                for n in self.INVESTSTORAGES:
                    for p in m.PERIODS:
                        expr = (
                            self.old[n, p]
                            == self.old_end[n, p] + self.old_exo[n, p]
                        )
                        self.old_rule.add((n, p), expr)

            self.old_rule = Constraint(
                self.INVESTSTORAGES, m.PERIODS, noruleinit=True
            )

            self.old_rule_build = BuildAction(rule=_old_storage_capacity_rule)

            def _initially_empty_rule(block):
                """Ensure storage to be empty initially"""
                for n in self.INVESTSTORAGES:
                    expr = self.storage_content[n, 0] == 0
                    self.initially_empty.add((n, 0), expr)

            self.initially_empty = Constraint(
                self.INVESTSTORAGES, m.TIMESTEPS, noruleinit=True
            )

            self.initially_empty_build = BuildAction(
                rule=_initially_empty_rule
            )

        # Standard storage implementation for discrete time points
        else:

            def _inv_storage_init_content_max_rule(block, n):
                """Constraint for a variable initial storage capacity."""
                return (
                    block.init_content[n]
                    <= n.investment.existing + block.invest[n, 0]
                )

            self.init_content_limit = Constraint(
                self.INVESTSTORAGES_NO_INIT_CONTENT,
                rule=_inv_storage_init_content_max_rule,
            )

            def _inv_storage_init_content_fix_rule(block, n):
                """Constraint for a fixed initial storage capacity."""
                return block.init_content[n] == n.initial_storage_level * (
                    n.investment.existing + block.invest[n, 0]
                )

            self.init_content_fix = Constraint(
                self.INVESTSTORAGES_INIT_CONTENT,
                rule=_inv_storage_init_content_fix_rule,
            )

            def _storage_balance_first_rule(block, n):
                """
                Rule definition for the storage balance of every storage n
                for the first time step.
                """
                expr = 0
                expr += block.storage_content[n, 0]
                expr += (
                    -block.init_content[n]
                    * (1 - n.loss_rate[0]) ** m.timeincrement[0]
                )
                expr += (
                    n.fixed_losses_relative[0]
                    * (n.investment.existing + self.invest[n, 0])
                    * m.timeincrement[0]
                )
                expr += n.fixed_losses_absolute[0] * m.timeincrement[0]
                expr += (
                    -m.flow[i[n], n, 0] * n.inflow_conversion_factor[0]
                ) * m.timeincrement[0]
                expr += (
                    m.flow[n, o[n], 0] / n.outflow_conversion_factor[0]
                ) * m.timeincrement[0]
                return expr == 0

            self.balance_first = Constraint(
                self.INVESTSTORAGES, rule=_storage_balance_first_rule
            )

        def _storage_balance_rule(block, n, p, t):
            """
            Rule definition for the storage balance of every storage n
            for every time step but the first.
            """
            expr = 0
            expr += block.storage_content[n, t]
            expr += (
                -block.storage_content[n, t - 1]
                * (1 - n.loss_rate[t]) ** m.timeincrement[t]
            )
            expr += (
                n.fixed_losses_relative[t]
                * self.total[n, p]
                * m.timeincrement[t]
            )
            expr += n.fixed_losses_absolute[t] * m.timeincrement[t]
            expr += (
                -m.flow[i[n], n, t] * n.inflow_conversion_factor[t]
            ) * m.timeincrement[t]
            expr += (
                m.flow[n, o[n], t] / n.outflow_conversion_factor[t]
            ) * m.timeincrement[t]
            return expr == 0

        self.balance = Constraint(
            self.INVESTSTORAGES,
            reduced_periods_timeindex,
            rule=_storage_balance_rule,
        )

        if m.es.periods is None:

            def _balanced_storage_rule(block, n):
                return (
                    block.storage_content[n, m.TIMESTEPS.at(-1)]
                    == block.init_content[n]
                )

            self.balanced_cstr = Constraint(
                self.INVESTSTORAGES_BALANCED, rule=_balanced_storage_rule
            )

        def _power_coupled(block):
            """
            Rule definition for constraint to connect the input power
            and output power
            """
            for n in self.INVEST_REL_IN_OUT:
                for p in m.PERIODS:
                    expr = (
                        m.InvestmentFlowBlock.total[n, o[n], p]
                    ) * n.invest_relation_input_output == (
                        m.InvestmentFlowBlock.total[i[n], n, p]
                    )
                    self.power_coupled.add((n, p), expr)

        self.power_coupled = Constraint(
            self.INVEST_REL_IN_OUT, m.PERIODS, noruleinit=True
        )

        self.power_coupled_build = BuildAction(rule=_power_coupled)

        def _storage_capacity_inflow_invest_rule(block):
            """
            Rule definition of constraint connecting the inflow
            `InvestmentFlowBlock.invest of storage with invested capacity
            `invest` by nominal_storage_capacity__inflow_ratio
            """
            for n in self.INVEST_REL_CAP_IN:
                for p in m.PERIODS:
                    expr = (
                        m.InvestmentFlowBlock.total[i[n], n, p]
                        == self.total[n, p] * n.invest_relation_input_capacity
                    )
                    self.storage_capacity_inflow.add((n, p), expr)

        self.storage_capacity_inflow = Constraint(
            self.INVEST_REL_CAP_IN, m.PERIODS, noruleinit=True
        )

        self.storage_capacity_inflow_build = BuildAction(
            rule=_storage_capacity_inflow_invest_rule
        )

        def _storage_capacity_outflow_invest_rule(block):
            """
            Rule definition of constraint connecting outflow
            `InvestmentFlowBlock.invest` of storage and invested capacity
            `invest` by nominal_storage_capacity__outflow_ratio
            """
            for n in self.INVEST_REL_CAP_OUT:
                for p in m.PERIODS:
                    expr = (
                        m.InvestmentFlowBlock.total[n, o[n], p]
                        == self.total[n, p] * n.invest_relation_output_capacity
                    )
                    self.storage_capacity_outflow.add((n, p), expr)

        self.storage_capacity_outflow = Constraint(
            self.INVEST_REL_CAP_OUT, m.PERIODS, noruleinit=True
        )

        self.storage_capacity_outflow_build = BuildAction(
            rule=_storage_capacity_outflow_invest_rule
        )

        def _max_storage_content_invest_rule(block, n, p, t):
            """
            Rule definition for upper bound constraint for the
            storage content.
            """
            expr = (
                self.storage_content[n, t]
                <= self.total[n, p] * n.max_storage_level[t]
            )
            return expr

        self.max_storage_content = Constraint(
            self.INVESTSTORAGES,
            m.TIMEINDEX,
            rule=_max_storage_content_invest_rule,
        )

        def _min_storage_content_invest_rule(block, n, p, t):
            """
            Rule definition of lower bound constraint for the
            storage content.
            """
            expr = (
                self.storage_content[n, t]
                >= self.total[n, p] * n.min_storage_level[t]
            )
            return expr

        # Set the lower bound of the storage content if the attribute exists
        self.min_storage_content = Constraint(
            self.MIN_INVESTSTORAGES,
            m.TIMEINDEX,
            rule=_min_storage_content_invest_rule,
        )

        def maximum_invest_limit(block, n, p):
            """
            Constraint for the maximal investment in non convex investment
            storage.
            """
            return (
                n.investment.maximum[p] * self.invest_status[n, p]
                - self.invest[n, p]
            ) >= 0

        self.limit_max = Constraint(
            self.NON_CONVEX_INVESTSTORAGES,
            m.PERIODS,
            rule=maximum_invest_limit,
        )

        def smallest_invest(block, n, p):
            """
            Constraint for the minimal investment in non convex investment
            storage if the invest is greater than 0. So the invest variable
            can be either 0 or greater than the minimum.
            """
            return (
                self.invest[n, p]
                - n.investment.minimum[p] * self.invest_status[n, p]
                >= 0
            )

        self.limit_min = Constraint(
            self.NON_CONVEX_INVESTSTORAGES, m.PERIODS, rule=smallest_invest
        )

        if m.es.periods is not None:

            def _overall_storage_maximum_investflow_rule(block):
                """Rule definition for maximum overall investment
                in investment case.
                """
                for n in self.OVERALL_MAXIMUM_INVESTSTORAGES:
                    for p in m.PERIODS:
                        expr = self.total[n, p] <= n.investment.overall_maximum
                        self.overall_storage_maximum.add((n, p), expr)

            self.overall_storage_maximum = Constraint(
                self.OVERALL_MAXIMUM_INVESTSTORAGES, m.PERIODS, noruleinit=True
            )

            self.overall_maximum_build = BuildAction(
                rule=_overall_storage_maximum_investflow_rule
            )

            def _overall_minimum_investflow_rule(block):
                """Rule definition for minimum overall investment
                in investment case.

                Note: This is only applicable for the last period
                """
                for n in self.OVERALL_MINIMUM_INVESTSTORAGES:
                    expr = (
                        n.investment.overall_minimum
                        <= self.total[n, m.PERIODS[-1]]
                    )
                    self.overall_minimum.add(n, expr)

            self.overall_minimum = Constraint(
                self.OVERALL_MINIMUM_INVESTSTORAGES, noruleinit=True
            )

            self.overall_minimum_build = BuildAction(
                rule=_overall_minimum_investflow_rule
            )

    def _objective_expression(self):
        """Objective expression with fixed and investment costs."""
        m = self.parent_block()

        if not hasattr(self, "INVESTSTORAGES"):
            return 0

        investment_costs = 0
        period_investment_costs = {p: 0 for p in m.PERIODS}
        fixed_costs = 0

        if m.es.periods is None:
            for n in self.CONVEX_INVESTSTORAGES:
                for p in m.PERIODS:
                    investment_costs += (
                        self.invest[n, p] * n.investment.ep_costs[p]
                    )
            for n in self.NON_CONVEX_INVESTSTORAGES:
                for p in m.PERIODS:
                    investment_costs += (
                        self.invest[n, p] * n.investment.ep_costs[p]
                        + self.invest_status[n, p] * n.investment.offset[p]
                    )

        else:
            msg = (
                "You did not specify an interest rate.\n"
                "It will be set equal to the discount_rate of {} "
                "of the model as a default.\nThis corresponds to a "
                "social planner point of view and does not reflect "
                "microeconomic interest requirements."
            )
            for n in self.CONVEX_INVESTSTORAGES:
                lifetime = n.investment.lifetime
                interest = n.investment.interest_rate
                if interest == 0:
                    warn(
                        msg.format(m.discount_rate),
                        debugging.SuspiciousUsageWarning,
                    )
                    interest = m.discount_rate
                for p in m.PERIODS:
                    annuity = economics.annuity(
                        capex=n.investment.ep_costs[p],
                        n=lifetime,
                        wacc=interest,
                    )
                    duration = min(
                        m.es.end_year_of_optimization - m.es.periods_years[p],
                        lifetime,
                    )
                    present_value_factor = 1 / economics.annuity(
                        capex=1, n=duration, wacc=interest
                    )
                    investment_costs_increment = (
                        self.invest[n, p] * annuity * present_value_factor
                    ) * (1 + m.discount_rate) ** (-m.es.periods_years[p])
                    remaining_value_difference = (
                        self._evaluate_remaining_value_difference(
                            m,
                            p,
                            n,
                            m.es.end_year_of_optimization,
                            lifetime,
                            interest,
                        )
                    )
                    investment_costs += (
                        investment_costs_increment + remaining_value_difference
                    )
                    period_investment_costs[p] += investment_costs_increment

            for n in self.NON_CONVEX_INVESTSTORAGES:
                lifetime = n.investment.lifetime
                interest = n.investment.interest_rate
                if interest == 0:
                    warn(
                        msg.format(m.discount_rate),
                        debugging.SuspiciousUsageWarning,
                    )
                    interest = m.discount_rate
                for p in m.PERIODS:
                    annuity = economics.annuity(
                        capex=n.investment.ep_costs[p],
                        n=lifetime,
                        wacc=interest,
                    )
                    duration = min(
                        m.es.end_year_of_optimization - m.es.periods_years[p],
                        lifetime,
                    )
                    present_value_factor = 1 / economics.annuity(
                        capex=1, n=duration, wacc=interest
                    )
                    investment_costs_increment = (
                        self.invest[n, p] * annuity * present_value_factor
                        + self.invest_status[n, p] * n.investment.offset[p]
                    ) * (1 + m.discount_rate) ** (-m.es.periods_years[p])
                    remaining_value_difference = (
                        self._evaluate_remaining_value_difference(
                            m,
                            p,
                            n,
                            m.es.end_year_of_optimization,
                            lifetime,
                            interest,
                            nonconvex=True,
                        )
                    )
                    investment_costs += (
                        investment_costs_increment + remaining_value_difference
                    )
                    period_investment_costs[p] += investment_costs_increment

            for n in self.INVESTSTORAGES:
                if n.investment.fixed_costs[0] is not None:
                    lifetime = n.investment.lifetime
                    for p in m.PERIODS:
                        range_limit = min(
                            m.es.end_year_of_optimization,
                            m.es.periods_years[p] + lifetime,
                        )
                        fixed_costs += sum(
                            self.invest[n, p]
                            * n.investment.fixed_costs[pp]
                            * (1 + m.discount_rate) ** (-pp)
                            for pp in range(
                                m.es.periods_years[p],
                                range_limit,
                            )
                        )

            for n in self.EXISTING_INVESTSTORAGES:
                if n.investment.fixed_costs[0] is not None:
                    lifetime = n.investment.lifetime
                    age = n.investment.age
                    range_limit = min(
                        m.es.end_year_of_optimization, lifetime - age
                    )
                    fixed_costs += sum(
                        n.investment.existing
                        * n.investment.fixed_costs[pp]
                        * (1 + m.discount_rate) ** (-pp)
                        for pp in range(range_limit)
                    )

        self.investment_costs = Expression(expr=investment_costs)
        self.period_investment_costs = period_investment_costs
        self.fixed_costs = Expression(expr=fixed_costs)
        self.costs = Expression(expr=investment_costs + fixed_costs)

        return self.costs

    def _evaluate_remaining_value_difference(
        self,
        m,
        p,
        n,
        end_year_of_optimization,
        lifetime,
        interest,
        nonconvex=False,
    ):
        """Evaluate and return the remaining value difference of an investment

        The remaining value difference in the net present values if the asset
        was to be liquidated at the end of the optimization horizon and the
        net present value using the original investment expenses.

        Parameters
        ----------
        m : oemof.solph.models.Model
            Optimization model

        p : int
            Period in which investment occurs

        n : oemof.solph.components.GenericStorage
            storage unit

        end_year_of_optimization : int
            Last year of the optimization horizon

        lifetime : int
            lifetime of investment considered

        interest : float
            Demanded interest rate for investment

        nonconvex : bool
            Indicating whether considered flow is nonconvex.
        """
        if m.es.use_remaining_value:
            if end_year_of_optimization - m.es.periods_years[p] < lifetime:
                remaining_lifetime = lifetime - (
                    end_year_of_optimization - m.es.periods_years[p]
                )
                remaining_annuity = economics.annuity(
                    capex=n.investment.ep_costs[-1],
                    n=remaining_lifetime,
                    wacc=interest,
                )
                original_annuity = economics.annuity(
                    capex=n.investment.ep_costs[p],
                    n=remaining_lifetime,
                    wacc=interest,
                )
                present_value_factor_remaining = 1 / economics.annuity(
                    capex=1, n=remaining_lifetime, wacc=interest
                )
                convex_investment_costs = (
                    self.invest[n, p]
                    * (remaining_annuity - original_annuity)
                    * present_value_factor_remaining
                ) * (1 + m.discount_rate) ** (-end_year_of_optimization)
                if nonconvex:
                    return convex_investment_costs + self.invest_status[
                        n, p
                    ] * (n.investment.offset[-1] - n.investment.offset[p]) * (
                        1 + m.discount_rate
                    ) ** (
                        -end_year_of_optimization
                    )
                else:
                    return convex_investment_costs
            else:
                return 0
        else:
            return 0

oemof / oemof-solph / 9842218777

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