Model ============================ Sets ------- - :math:`T`: Ordered time steps. - :math:`Y`: Ordered years. - :math:`CO`: commodities. - :math:`CP`: conversion process. - :math:`CS`: conversion subprocess which is determined by the tuple :math:`(cp\in CP, cin \in CO, cout \in CO)`. Every conversion process has at least one conversion subprocess. - :math:`SCS`: is a subset of :math:`CS` that contains storage conversion subprocesses. Parameters ---------- - All parameters are non-negative unless specified otherwise, and names start with a lowercase letter. - Energy/Time is Power. Global ~~~~~~ - :math:`dt`: time step size. It shows how many hours each time step represents. Dimension: Time. Range: Positive. - :math:`w`: weight of each time step withim the whole year. It's equal to :math:`8760/|T|`. Range: Positive. default = 1. - :math:`discount\_rate`: The discount rate is the interest rate used to calculate the present value of future cash flows from a project or investment. For example, at an interest rate of 5%, the value of €100 will increase to €105 in one year. Dimension: -. Range: Non-negative. - :math:`discount\_factor(Y)`: Discount factor for year :math:`y` which is equal to :math:`discount\_factor(y)=(1+discount\_rate)^{y-Y[0]}`. Dimension: -. Range: Non-negative. Cost ~~~~ - :math:`opex\_cost\_energy(CS,Y)`: Operational cost per energy output of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Money/Energy. Range = Non-negative. default = 0. - :math:`opex\_cost\_power(CS,Y)`: Operational cost per active capacity of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Money/Power. Range = Non-negative. default = 0. - :math:`capex\_cost\_power(CS,Y)`: Capital cost per unit of new capacity of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Money/Power. Range = Non-negative. default = 0. CO2 ~~~ - :math:`spec\_co2(CS)`: Specific CO2 emission intensity per energy output of conversion subprocess :math:`cs`. Dimension: Mass/Energy. Range = Non-negative. default = 0. - :math:`annual\_co2\_limit(Y)`: Annual CO2 emission limit of the energy system in year :math:`y`. Dimension: Mass. Range = Non-negative. - :math:`co2\_price(Y)`: CO2 price for emission from the energy system in year :math:`y`. Dimension: Money/Mass. Range = Non-negative. default = 0. Energy ~~~~~~ - :math:`max\_eout(CS,Y)`: maximum energy output of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Energy. Range = Non-negative. default = :math:`\infty`. - :math:`min\_eout(CS,Y)`: minimum energy output of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Energy. Range = Non-negative. default = 0. Capacity ~~~~~~~~ - :math:`cap\_min(CS,Y)`: minimum allowed active capacity of the conversion subprocess :math:`cs` at year :math:`y`. Dimension: Power. Range = Non-negative. default = 0. - :math:`cap\_max(CS,Y)`: maximum allowed active capacity of the conversion subprocess :math:`cs` at year :math:`y`. Dimension: Power. Range = Non-negative. default = :math:`\infty` . - :math:`cap\_res\_max(CS,Y)`: maximum residual capacity of the conversion subprocess :math:`cs` at year :math:`y`. Dimension: Power. Range = Non-negative. default = 0. - :math:`cap\_res\_min(CS,Y)`: minimum residual capacity of the conversion subprocess :math:`cs` at year :math:`y`. Dimension: Power. Range = Non-negative. default = 0. Residual capacity is the remaining capacity from before the planning period. The parameters :math:`cap\_res\_max(CS,Y)` and :math:`cap\_res\_min(CS,Y)` define the upper and lower limit for the active capacity from this remaining capacity. Technology ~~~~~~~~~~ - :math:`efficiency(CS)`: output efficiency of conversion subprocess :math:`cs`. Dimension: -. Range: Non-negative. default = 1. - :math:`technical\_lifetime(CS)`: Technical lifetime of conversion subprocess :math:`cs`. Dimension: Time. default: 100. Availability ~~~~~~~~~~~~ - :math:`availability\_profile(CS,T)`: Availability profile of the subprocess :math:`cs` at time step :math:`t`. Range = [0,1]. default = 1. Dimension:-. - :math:`technical\_availability(CS)`: Technical Availability factor of the conversion subprocess :math:`cs`. Range = [0,1]. default = 1. - :math:`output\_profile(CS,T)`: Share of the annual energy output supplied of the conversion subprocess :math:`cs` at time step :math:`t` such that :math:`\sum_{t \in T}output\_profile[cs,t]=1 \quad \forall cs \in CS`. Dimension: -. Range: Non-negative. Fractions of generation and consumption ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - :math:`out\_frac\_min(CS,Y)`: minimum fraction of the output commodity(cout) generated by conversion subprocess :math:`cs` in year :math:`y`. Dimension: -. Range: [0,1]. default = 0. - :math:`out\_frac\_max(CS,Y)`: maximum fraction of the output commodity(cout) generated by conversion subprocess :math:`cs` in year :math:`y`. Dimension: -. Range= [0,1]. default = 1. - :math:`in\_frac\_min(CS,Y)`: minimum fraction of the input commodity(cin) consumed by conversion subprocess :math:`cs` in year :math:`y`. Dimension: -. Range= [0,1]. default = 0. - :math:`in\_frac\_max(CS,Y)`: maximum fraction of the input commodity(cout) consumed by conversion subprocess :math:`cs` in year :math:`y`. Dimension: -. Range= [0,1]. default = 1. For `in\_frac\_min(CS,Y)` and `in\_frac\_max(CS,Y)` see :ref:`decentralized heating ` example. For `out\_frac\_min(CS,Y)` and `out\_frac\_max(CS,Y)` see :ref:`CHP ` example. Storage ~~~~~~~ - :math:`c\_rate(CS)`: indicates the discharge and charging rate of the storage conversion subprocess :math:`cs`. 2C means that the full storage can be fully discharged in (1 hour)/2=30 minutes. Range: Positive. Dimension: 1/Time. - :math:`efficiency\_charge(CS)`: Storage charging efficiency of conversion subprocess :math:`cs`. Dimension: -. Range = (0,1]. default = 1. Variables --------- All variables are non-negative and names start with a capital letter. Costs ~~~~~ - :math:`TOTEX`: Total Expenditure. Dimension: Money. - :math:`CAPEX`: Capital Expenditure. Dimension: Money. - :math:`OPEX`: Operational Expenditure. Dimension: Money. - :math:`TotalSalvageValue`: Total Salvage Value. Dimension: Money. - :math:`DiscountedSalvageValue(CS,Y)`: Discounted Salvage Value of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Money. CO2 ~~~ - :math:`Total\_annual\_co2\_emission(Y)`: Total Annual CO2 emission in year :math:`y`. Dimension: Mass. Power ~~~~~ - :math:`Cap\_new(CS,Y)`: New Capacity of conversion subprocess :math:`cs` installed at the beginning of year :math:`y`. Dimension: Power. - :math:`Cap\_active(CS,Y)`: Active Capacity of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Power. - :math:`Cap\_res(CS,Y)`: residual Capacity of conversion subprocess :math:`cs` in year :math:`y`. Dimension: Power. - :math:`Pin(CS,Y,T)`: Power input of conversion subprocess :math:`cs` at time step :math:`t` in year :math:`y`. Dimension: Power. - :math:`Pout(CS,Y,T)`: Power output of conversion subprocess :math:`cs` at time step :math:`t` in year :math:`y`. Dimension: Power. Energy ~~~~~~ - :math:`Eouttot(CS,Y)`: Total energy output of the conversion subprocess :math:`cs` in year :math:`y`. Dimension: Energy. - :math:`Eintot(CS,Y)`: Total energy input of the conversion subprocess :math:`cs` in year :math:`y`. Dimension: Energy. - :math:`Eouttime(CS,Y,T)`: Total energy output of the conversion subprocess :math:`cs` at time step :math:`t` in year :math:`y`. Dimension: Energy. - :math:`Eintime(CS,Y,T)`: Total energy input of the conversion subprocess :math:`cs` at time step :math:`t` in year :math:`y`. Dimension: Energy. - :math:`Enetgen(CO,Y,T)`: Net energy generation of commodity :math:`co` at time step :math:`t` in year :math:`y`. Dimension: Energy. - :math:`Enetcons(CO,Y,T)`: Net energy consumption of commodity :math:`co` at time step :math:`t` in year :math:`y`. Dimension: Energy. Storage ~~~~~~~ - :math:`E\_storage\_level(CS,Y,T)`:Storage Energy level of storage conversion subprocess :math:`cs` at time step :math:`t` in year :math:`y`. Dimension: Energy. - :math:`E\_storage\_level\_max(CS,Y)`: Maximum Energy stored in the storage conversion subprocess :math:`cs` in year :math:`y`. Dimension: Energy. Constraints ----------- Costs ~~~~~ .. math:: TOTEX = CAPEX + OPEX :label: totex_eq .. math:: CAPEX = \sum_{y \in Y} \left(co2\_price[y] * Total\_annual\_co2\_emission[y] + discount\_factor[y] * \sum_{cs \in CS} \left(Cap\_new[cs, y] * capex\_cost\_power[cs,y]\right)\right) - TotalSalvageValue :label: capex_eq :eq:`capex_eq` capital cost consists of CO2 cost and capital investment. .. math:: OPEX = \sum_{cs\in CS}\sum_{i \in (1,\ldots,|Y|)} discount\_factor[Y[i]] * \left( Cap\_active[cs,Y[i]] * opex\_cost\_power[cs,Y[i]] + Eouttot[cs,Y[i]] * opex\_cost\_energy[cs,Y[i]] \right) * \left( Y[i+1]-Y[i]\quad \text{if} \quad i<|Y| \quad \text{else} \quad 1 \right) :label: opex_eq :eq:`opex_eq` operational cost consists of cost per active unit of capacity and cost per unit of generation. .. math:: DiscountedSalvageValue[cs,y] = Cap\_new[cs,y]* capex\_cost\_power[cs,y]*(1-(Y[-1]-y+1)/technical\_lifetime[cs]) * discount\_factor[y] \quad \forall y\in Y,\forall cs\in CS .. math:: TotalSalvageValue = \sum_{cs\in CS} \sum_{y \in Y} DiscountedSalvageValue[cs,y] \quad \text{if} \quad Y[|Y|] - y < technical\_lifetime[cs] :label: total_salvage Power Balance ~~~~~~~~~~~~~ .. math:: \sum_{cs \in CS| cs.cin = co} Pin[cs, t , y] = \sum_{cs \in CS| cs.cout = co} Pout[cs, t , y] \quad \forall t\in T, \forall y\in Y, \forall co\in CO\setminus \{Dummy\} :label: power_balance_eq :eq:`power_balance_eq` At time step :math:`t` in year :math:`y` the total output and input of the commodity :math:`co` by all conversion processes should be equal. CO2 ~~~ .. math:: Total\_annual\_co2\_emission[y] = \sum_{cs \in CS} spec\_co2[cs] * Eouttot[cs,y] \quad \forall y \in Y :label: annual_co2_emission_eq :eq:`annual_co2_emission_eq` total annual CO2 emission is equal to the sum of energy produced by each conversion subprocess multiplied by its specific CO2 emission. .. math:: Total\_annual\_co2\_emission[y] \leq annual\_co2\_limit[y] \quad \forall y \in Y :label: annual_co2_emission_limit_eq :eq:`annual_co2_emission_limit_eq` The Annual CO2 emission is limited. Power output ~~~~~~~~~~~~~~~~~~~~~~~~ .. math:: Pout[cs,y,t] = Pin[cs,y,t] * efficiency[cs] \quad \forall y\in Y, \forall t\in T, \forall cs\in CS :label: efficiency_eq :eq:`efficiency_eq` the ratio of output to input is equal to efficiency for each converssion subprocess. .. math:: Pout[cs,y,t] \leq Cap\_active[cs,y] \quad \forall y\in Y, \forall t\in T, \forall cs\in CS :label: max_power_out_eq :eq:`max_power_out_eq` The output is limited by the capacity of the conversion subprocess. .. math:: Pout[cs,y,t] \leq Cap\_active[cs,y] * technical\_availability[cs] \quad \forall y\in Y,\forall t\in T, \forall cs\in CS :label: technical_availability_eq :eq:`technical_availability_eq` Energy generation is limited by the technical availability. .. math:: Pout[cs,y,t] \leq Cap\_active[cs,y] * availability\_profile[cs,t] \quad \forall y\in Y,\forall t\in T, \forall cs\in CS \setminus SCS :label: re_availability_eq :eq:`re_availability_eq` The Generation of renewable energy is limited by the availability profile. Power-Energy ~~~~~~~~~~~~~~~~~~~~~~~~ .. math:: Eouttime[cs,y,t] = Pout[cs,y,t]*dt*w \quad \forall y\in Y,\forall t\in T, \forall cs\in CS :label: eouttime_eq :eq:`eouttime_eq` The energy output of converssion subprocess :math:`cs` at time step :math:`t` in year :math:`y`. .. math:: Eintime[cs,y,t] = Pin[cs,y,t]*dt*w \quad \forall y\in Y,\forall t\in T, \forall cs\in CS :label: eintime_eq Fractions ~~~~~~~~~~~~~~~~~~~ .. math:: Eouttime[cs,t,y] \geq out\_frac\_min[cs,y] * Enetgen[cs.cout,y,t] \quad \forall y\in Y,\forall t\in T, \forall cs\in CS :label: min_cosupply_eq :eq:`min_cosupply_eq` .. math:: Eouttime[cs,t,y] \leq out\_frac\_max[cs,y]*Enetgen[cs.cout,y,t] \quad \forall y\in Y,\forall t\in T, \forall cs\in CS :label: max_cosupply_eq .. math:: Eintime[cs,t,y] \geq in\_frac\_min[cs, y]*Enetcons[cs.cin,y,t] \quad \forall y\in Y, \forall t\in T, \forall cs\in CS :label: min_couse_eq .. math:: Eintime[cs,t,y] \leq in\_frac\_max[cs, y] * Enetcons[cs.cin,y,t] \quad \forall y\in Y,\forall t\in T, \forall cs\in CS :label: max_couse_eq Capacity ~~~~~~~~~~~~~~~~~~ .. math:: Cap\_res[cs, y] \leq cap\_res\_max[cs, y] \quad \forall y\in Y, \forall cs\in CS :label: max_cap_res_eq .. math:: Cap\_res[cs, y] \geq cap\_res\_min[cs, y] \quad \forall y\in Y, \forall cs\in CS :label: min_cap_res_eq .. math:: Cap\_active[cs, y] = Cap\_res[cs, y] + \sum_{yy\in Y|y-technical\_lifetime[cs] < yy \leq y} Cap\_new[cs, yy] \quad \forall y\in Y, \forall cs\in CS :label: cap_active_eq .. math:: Cap\_active[cs,y] \leq cap\_max[cs,y] \quad \forall y\in Y, \forall cs\in CS :label: max_active_cap_eq .. math:: Cap\_active[cs,y] \geq cap\_min[cs,y] \quad \forall y\in Y, \forall cs\in CS :label: min_active_capacity_eq Auxiliary Linking Variables ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. math:: Eouttot[cs,y] = \sum_{t \in T} Eouttime[cs,t,y] \quad \forall y\in Y, \forall cs\in CS :label: energy_powerout_eq .. math:: Eintot[cs, y] = \sum_{t \in T} Eintime[cs, t, y] \quad \forall y\in Y, \forall cs\in CS :label: energy_powerin_eq .. math:: Enetgen[co,t,y] = \sum_{cs\in CS|cs.cout=co} Eouttime[cs,t,y] \quad \forall y\in Y, \forall t\in T,\forall co\in CO :label: nettogen_eq .. math:: Enetcons[co,t,y] = \sum_{cs\in CS|cs.cin=co} Eintime[cs,t,y] \quad \forall y\in Y, \forall t\in T, \forall co\in CO :label: nettocon_eq Generation ~~~~~~~~~~ .. math:: Eouttot[cs,y] \leq max\_eout[cs,y] \quad \forall y\in Y, \forall cs\in CS :label: max_energyout_eq .. math:: Eouttot[cs,y] \geq min\_eout[cs,y] \quad \forall y\in Y, \forall cs\in CS :label: min_energyout_eq .. math:: Eouttime[cs,t,y] = output\_profile[cs,t] * Eouttot[cs,y] \quad \forall y\in Y, \forall t\in T,\forall cs\in CS :label: loadshape_eq Storage ~~~~~~~ .. math:: E\_storage\_level[cs,t,y] \leq E\_storage\_level\_max[cs,y] \quad \forall y\in Y, \forall t\in T, \forall cs\in SCS :label: strorage_energy_limit .. math:: Pin[cs,t,y] \leq Cap\_active[cs, y] \quad \forall y\in Y, \forall t\in T, \forall cs\in SCS :label: charge_power_limit .. math:: E\_storage\_level[cs,t,y] = E\_storage\_level[cs, t-1, y] + efficiency\_charge[cs] * Pin[cs, t,y] * dt - (Pout[cs,t,y]*dt)/(efficiency[cs]) \quad \forall y\in Y, \forall t\in T, \forall cs\in SCS :label: storage_energy_balance .. math:: E\_storage\_level\_max[cs, y] = Cap\_active[cs, y]/c\_rate[cs] \quad \forall y\in Y, \forall cs\in SCS :label: c_rate_relation