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Constraints

BENOPT model

The blue lines identify the decision variables, and red sentences are our comments.

1. Objective functions

1.1. Objective function: cost

In the following paragraphs, red items specify variables and black items are parameters.

  • \(\pi_{t, i, s}=\) production of output from technology \(i\) at time \(t\) for market (sector) \(s\) \((PJ)\)
  • \(k^{endo}_{t, i}=\) endogenously net installed capacity (minus decommissioned capacity) at time \(t\) for technology \(i\) (GW)
  • \(\dot{m}_{t, i, f, c}=\) feed used (PJ)
  • \(\dot{m}^{imp}_{t, i, f}=\) imported feed (PJ)
  • \(mc_{t, i}=\) marginal cost of technology \(i\) at time \(t\) - OPEX excluding feedstock inputs (\(\frac{Mil\,€}{PJ}\))
  • \(\dot{m}^{el}_{i}=\) electricity input of technology \(i\) \((\frac{MWh_{el}}{PJ})\)
  • \(p^{el}_{t, i}=\) Power price of technology \(i\) at time \(t\) \((\frac{Mil\,€}{MWh_{el}})\)
  • \(\dot{m}^{th}_{i}=\) heat input of technology \(i\) \((\frac{MWh_{th}}{PJ})\)
  • \(p^{th}_{t}=\) Heat price at time \(t\) \((\frac{Mil\,€}{MWh_{th}})\)
  • \(\dot{m}^{CO_2}_{i}=\) Process \(CO_2\) feedstock input for technology \(i\) (PtX) \((\frac{Mil\,tCO2}{PJ})\)
  • \(p^{CO_2}_{t}=\) \(CO_2\) feedstock price \((\frac{Mil\,€}{MtCO_2})\)
  • \(I^{+}_{t,i}=\) Levelized investment cost (\(\frac{Mil\,€}{GW}\))
  • \(p_{t, f, c}=\) feed price at time \(t\) for feed \(f\) for category \(c\) \((\frac{Mil\,€}{PJ_{feed}})\)
  • \(p^{imp}_{t,f}=\) price of imported feed \(f\) at time \(t\), which is set to the maximum value (3rd cat.) from the feedstock prices (up) \((\frac{Mil\,€}{PJ_{feed}})\)

\(min\, cost = \overbrace{\sum_{t, i, s}(mc_{t, i} + \dot{m}^{el}_{i} \times p^{el}_{t, i} + \dot{m}^{th}_{i} \times p^{th}_{t} + \dot{m}^{CO_2}_{i} \times p^{CO_2}_{t}) \times \pi_{t, i, s}}^\text{production costs} + \underbrace{\sum_{t, i}(I^{+}_{t,i}\times k^{endo}_{t, i})}_\text{the investment cost} + \underbrace{\sum_{t, i, f, c}(p_{t, f, c} \times \dot{m}_{t,i,f,c})}_\text{the utilized domestic feedstock cost} + \overbrace{\sum_{t, i, f}(p^{imp}_{t,f} \times m^{imp}_{t, i, f})}^\text{the cost of imported feedstock}\)

1.2. Objective function: Maximum abatement

  • \(\pi_{t, i, s}=\) production of output from technology \(i\) at time \(t\) for market (sector) \(s\) \((PJ)\)
  • \(\dot{m}_{t, i, f, c}=\) feed \(f\) (or \(r\) = residue) used (PJ)
  • \(\varepsilon_{sub, t, s}=\) GHG reference (fossil substitute) - for some sectors (e.g., CH4 market and H2Market) it is zero and 94.1 for others \((\frac{ktCO_2eq}{PJ})\)
  • \(w_{i, s}=\) relative fuel economy between fuels in passenger road sector \((\%) \text{ which is calculated based on } \frac{GJ}{vkm} \text { or } \frac{GJ}{tkm}\)
  • \(\varepsilon_{t, i}=\) Gate to tank \(CO_2\) emissions, which is equal to GHG emissions from cultivation, conversion and transportation combined - byproducts \((\frac{ktCO_2eq}{PJ})\)
  • \(\varepsilon_{t, f}=\) GHG emissions for feedstocks \((\frac{ktCO_2eq}{PJ_{feed}})\)
  • \(\varepsilon_{t}^{trans}=\) transport GHG emissions \((\frac{ktCO_2eq}{PJ})\) This paremeter has been calculated for various energy crops but they are all ignored and used one for all feedstock types.
  • \(\varepsilon^{elMix}_{t}=\) Electricity mix emissions \((\frac{ktCO_2eq}{PJ})\)
  • \(\dot{m}^{CO_2}_{i}=\) Process \(CO_2\) feedstock input for technology \(i\) (PtX) \((\frac{Mil\,tCO_2}{PJ})\)
  • \(\varepsilon^{CO_2}_{in}=\) GHG reference for input \(CO_2\). It is either 1000 or 0. It is currently set to zero, which means that we use \(CO_2\) from renewable sources. \((\frac{k\,tCO_2}{MtCO_2})\)

\(max\, ghgAbatement = \overbrace{\sum_{t, i, s}\left((\varepsilon_{sub, t,s} \times w_{i, s} - \varepsilon_{t, i}) \right)\times \pi_{t, i, s}}^{\varepsilon_{max} = \text{ avoided emissions}} - \underbrace{\sum_{t,i,f,c}\left((\varepsilon_{t, f} + \varepsilon_{t}^{trans}\right)\times \dot{m}_{t, i, f, c})}_\text{feedstock emissions} - \overbrace{\sum_{t,i,r,c}(\varepsilon_{t}^{trans} \times \dot{m}_{t, i, r, c})}^\text{rsidues emission - just transportation} - \underbrace{\sum_{t,i,el,c}(\varepsilon^{elMix}_{t} \times \dot{m}_{t, i, el, c})}_\text{emissions of utilized electricity} - \overbrace{\sum_{t,i,s}(\pi_{t, i, s} \times \dot{m}^{CO_2}_{i} \times \varepsilon^{CO_2}_{in})}^\text{additional emissions from utilized CO2}\)

2. Constraints

2.1. Capacity expansion:

  • \(k_{t, i}=\) capacity at time \(t\) for technology \(i\) (GW)
  • \(k^{+}_{t, i}=\) newly installed capacity at time \(t\) for technology \(i\) (GW)

  • \(k^{-}_{t, i}=\) decomissioned capacity of newly installed technologies \((i \in I)\) at time \(t\) (GW)

  • \(k^{0}_{t, i}=\) initial capacity (continously decommisioned) (GW)

\(k_{t + 1, i}=k_{t, i} + k^{+}_{t + 1, i} + \overbrace{(k^{0}_{t + 1, i}-k^{0}_{t, i})}^{\text{ decomissioned initial capacity at priod t + 1}} - k^{-}_{t + 1, i} \; \forall t \in T, i \in I\)

2.2. Newly expanded capacity

We need this decision variable (and its corresponding constraint) to calculate the investment costs in the cost objective function. * $k^{endo}_{t, i} = $ endogenously net installed capacity (minus decommissioned capacity) at time \(t\) for technology \(i\) (GW)

\(k^{endo}_{t + 1, i} = k^{endo}_{t, i} + k^{+}_{t + 1, i} - k^{-}_{t + 1, i} \; \forall t \in T, i \in I\)

2.3. Decomissioned capacity

Decomissioned capacity at the end of the lifetime for each technology \((\hat{t}_i)\) is equal to the installed capacity for that technology at time \(t\).

\(k^{-}_{t + \hat{t}_i, i}=k^{+}_{t, i}\; \forall t \in T, i \in I\)

2.4. Production bound

The capacity (GW) first converted to GWh, and then PJ to find the maximum production of the power plants.

  • \(Cf_{t,i}=\) the capacoty factor of technology (plant) \(i\) at time \(t\) (\%).

\(k_{t, i} \times Cf_{t,i} \times \dfrac{8760 \times 3.6}{1000} \geq \sum_s \pi_{t, i, s} \; \forall t \in T, i \in I\)

2.5. \(CH_4\) production

This constraints asks for all technologies that require methane to purchase from the CH4 market.

  • \(\eta_{t, i}=\) Energetic conversion efficiency feed to main energy carrier \((\frac{PJ}{PJ_{feed}})\).
  • \(I_{CH4in}=\) set of technologies that require methane as input such as BioCH4_el, LCH4, CH4.
  • \(S_{CH4}=\) \(CH_4\) market

\(\sum_{s \in S_{CH4}, i}\pi_{t, i, s} = \sum_{s,i \in I_{CH4in}}\dfrac{\pi_{t, i, s}}{\eta_{t,i}} \; \forall t \in T\)

2.6. \(H_2\) production

The required hydrogen for technologies (FCVE, PtG-VH4, etc.) should be purchased from the \(H_2\) market.

  • $\dot{m}^{H_2}_{t, i} = $ H2 feedstock input (\(\frac{PJ_{H_2}}{PJ}\))

\(\sum_{s \in S_{H2},i} \pi_{t, i, s} = \sum_{s,i} (\pi_{t, i, s} \times \dot{m}^{H_2}_{t, i}) \; \forall t \in T\)

2.7. Capacity ramp-up

This constraint sets an upper bound for the capacity expansion.

  • $r_f^{cap} = $ capacity expansion ramp per year ( = 50\%)
  • $r_{min}^{GW} = $ minimum capacity expansion ramp per year, which is set to 0.03 GW (GW)

$ k^{+}{t+1, i} \leq r \; \forall t \in T, i \in I$}^{GW} + r_f^{cap} \times k_{t, i

2.8. Production ramp-up

These constraints prevent technologies from volatile adjustment of their production levels from one year to the next.

  • $r_{min} = $ minimum capacity expansion ramp per year, which is set to 1 (PJ)

\(\pi_{t + 1, i, s} \leq r_{min} + (1 + r_f^{cap}) \times \pi_{t, i, s} \; \forall t \in T, i \in I, s \in S\)

\(\sum_{s}(\pi_{t + 1, i, s}) \leq r_{min} + (1 + r_f^{cap}) \times \sum_s\pi_{t, i, s} \; \forall t \in T, i \in I\)

2.9. Calculating the required feed

Using this constrain, the model determines the required domestic feed \(\dot{m}_{t, i, f, c}\) and imported feed \(\dot{m}^{imp}_{t, i, f}\) to produce \(\pi_{t, i, s}\) for all technologies except those that need \(CH_4\) or \(H_2\).

  • \(\dot{m}_{t, i, f, c} =\) feed used (PJ)
  • \(\dot{m}^{imp}_{t, i, f} =\) imported feed (PJ)
  • \(\pi_{t, i, s} =\) production of output from technology \(i\) at time \(t\) for market (sector) \(s\) \((PJ)\)
  • \(\eta_{t,f,i} =\) energetic conversion efficiency feed to main energy carrier \((\frac{PJ}{PJ_{feed}})\).
  • \(I_{CH4+H2} =\) technologies that require intermediate \(CH_4\) or \(H_2\)

\(\sum_{s}\pi_{t, i, s}=\sum_{f,c}(\dot{m}_{t, i, f, c} \times \eta_{t,f,i}) +\sum_{f}(\dot{m}^{imp}_{t, i, f}\times \eta_{t,f,i}) \; \forall t \in T, i \in I \backslash \{I_{CH4+H2}\}\)

2.10. Residues potential in each price range

We have 3 categories \((c \in C)\) for different price ranges for each residue (\(r\)). In here, we assume that the residues used in each price range is at most \(33\%\) of the whole potential. Also, for the imported residues, we have only one price range. Since the first constraint is tighter, the second constraint for residues will be inactive.

  • $\dot{m}^{imp}_{t, i, f} = $ imported feed (PJ)
  • \(\phi_{t,r}=\) residue \(r\) potential at time \(t\) (PJ)
  • \(\phi^{imp}_{t,r}=\) imported residue \(r\) potential at year \(t\) (PJ)
  • \(\hat{\phi}^{imp}_{t}=\) maximal residual feed import at year \(t\) (PJ)
\[\begin{equation} \label{EQ2.10.1} \begin{split} &\dfrac{\phi_{t,r}}{|C|} &\geq \sum_i(\dot{m}_{t, i, r, c}) \; &\forall t \in T, c \in C, r \in R \\ &\phi^{imp}_{t,fr} &\geq \sum_i(\dot{m}^{imp}_{t, i, fr}) \; &\forall t \in T, fr \in F \cup R \\ &\hat{\phi}^{imp}_{t} &\geq \sum_{i,fr \in F \cup R} \dot{m}^{imp}_{t, i, fr} \; &\forall{t \in T} \end{split} \end{equation}\]

2.11. \(CO_2\) as input

In here, we set an upperbound over the consumed \(CO_2\) per year (60 MtCO2).

  • \(\dot{m}^{CO_2}_{i} =\) Process \(CO_2\) feedstock input for technology \(i\) (PtX) \((\frac{Mil\,tCO2}{PJ})\)
  • \(\hat{\dot{m}}^{CO_2}_{t} =\) \(CO_2\) feedstock maximum value \(({Mil\,tCO_2})\).
  • \(\dot{m}^{CO_2}_{t} =\) \(CO_2\) used at time \(t\) \(({Mil\,tCO_2})\)

\(\dot{m}^{CO_2}_{t} = \sum_{i,s}(\pi_{t, i, s} \times \dot{m}^{CO_2}_{i}) \; \forall t \in T\)

\(\hat{\dot{m}}^{CO_2}_{t} \geq \dot{m}^{CO_2}_{t} \; \forall t \in T\)

2.12. Residual load: excess of power

It sets the maximum excess power available in each time slice of each year.

  • \(\hat{E}_{j, t}=\) the residual load (excess of power) in timeslice \(j \in \{1 .. 50\}\) at year \(t\) (PJ)
  • \({E}_{j, t}=\) the residual load used (excess of power) in timeslice \(j \in \{1 .. 50\}\) at year \(t\) (PJ)

\(\hat{E}_{j, t} \geq E_{j, t} \; \forall t \in T, j \in D\)

The next constraint calculates the sum of all utilized surplus power in all timeslices to be equal to its value in the corresponding year.

  • \(\dot{m}_{t, i, P^{Res}, c}=\) the residual power used (PJ)
  • \(\dot{m}^{el}_{t, j, i, P^{Res}} =\) electricity used daily from excess power in timeslice \(j\) at year \(t\) (PJ)

\(\sum_{i,c} \dot{m}_{t, i, P^{Res}, c}=\sum_{j \in D, i} \dot{m}^{el}_{t,j,i,P^{Res}}\; \forall{t \in T}\)

Sum of all utilized surplus power in each time slice by all technologies equals the utilized surplus power in each time slice of each year.

  • \({E}_{j, t} =\) the residual load used (excess of power) in timeslice \(j \in \{1 .. 50\}\) at year \(t\) (PJ)
  • \(\dot{m}^{el}_{t, j, i, P^{Res}} =\) electricity used daily from excess power in timeslice \(j\) at year \(t\) (PJ)

\({E}_{j, t} = \sum_{i,P^{Res}}\overbrace{\dot{m}^{el}_{t, j, i, P^{Res}}}^{E_{i,j,t}} \; \forall{t \in T, j\in D}\)

Finally, the capacity limitation of excess power used in each time slice.

  • \(E_{i, j, t} =\) excess electrical energy (ERE) used by technology \(i\) at the same time point
  • \(k_{t, i} =\) capacity at time \(t\) for technology \(i\) (GW)
  • \(\dot{m}^{el}_{t, j, i, P^{Res}} =\) electricity used daily from excess power in timeslice \(j\) at year \(t\) (PJ)
  • \(\eta_{t, i} =\) Energetic conversion efficiency feed to main energy carrier \((\frac{PJ}{PJ_{feed}})\).
  • \(D^{max} =\) the number of timeslices = 50.

\(k_{t, i}\times \dfrac{\dfrac{8760 \times 3.6}{1000}}{D^{max}} \geq \underbrace{\dot{m}^{el}_{t, j, i, P^{Res}}}_{E_{i,j,t}} \times \eta_{t, i} \; \forall{t \in T, j \in D, i \in I}\)

2.13. Demand limit for dispatchable power

The dispachable power should be below demand for the electricity in each timeslice.

  • \(P_{j, t} =\) dispatchable power production (PJ)
  • \(\delta^{el}_{j, t} =\) positive residual load, which means the demand for dispatchable power (PJ)
\[\begin{equation}\label{EQ2.13.1} \delta^{el}_{j, t} \geq P_{j,t} \; \forall{j \in D, t \in T} \end{equation}\]

The dispached power from these technologies cannot surpass their theoretical production capacity. We should also add the availability factor here.

  • \(k_{t, i} =\) capacity at time \(t\) for technology \(i\) (GW)
  • \(E_{t,j,i,s}\) dispatchable power production from technology \(i\) for sector \(s\) in \(j\)th timeslice of year \(t\) (PJ)
\[\begin{equation}\label{EQ2.13.2} \dfrac{k_{t, i}\times 8760 \times \dfrac{ 3.6}{1000}}{D^{max}} \geq E_{t,j,i,s={CONVel}} \; \forall{t \in T, j \in D, i \in I} \end{equation}\]

Sum of the generated electricity from all technologies can be dispatched in timeslice \(j\)th of year \(t\).

\[\begin{equation}\label{EQ2.13.3} P_{j,t} = \sum_{i}E_{t,j,i,s={CONVel}} \;\forall{t \in T, j \in D} \end{equation}\]

The dispatched electricity in each year should not surpass the demand for electrity * $\delta_{t, s} = $ demand of sector \(s\) at year \(t\) (PJ) * \(E_{t,j,i,s}\) dispatchable power production from technology \(i\) for sector \(s\) in \(j\)th timeslice of year \(t\) (PJ)

\[\begin{equation}\label{EQ2.13.4} \delta_{t,s=CONVel} \geq \sum_{i, j \in D}E_{t,j,i,s=CONVel} \;\forall{t \in T} \end{equation}\]

The electricity production over all time slices are set to a single variable * $\pi_{t, i, s} = $ production of output from technology \(i\) at time \(t\) for market (sector) \(s\) \((PJ)\) * \(E_{t,j,i,s}\) dispatchable power production from technology \(i\) for sector \(s\) in \(j\)th timeslice of year \(t\) (PJ)

\[\begin{equation}\label{EQ2.13.5} \pi_{t,i, s={CONVel}} = \sum_{j \in D} E_{t,j,i,s=CONVel} \;\forall{i\in I, t \in T} \end{equation}\]

2.14. Maximum electricity usage from mix

The maximum permissible usage of grid electricity for EVs and H2 production.

  • \(\hat{E}_{el}^{mix} =\) Maximum electricity usage from mix for EVs and \(H_2\) production (277 PJ)
  • \(\dot{m}_{t, i, P^{Mix}, c} =\) the grid power (Mixed) used (PJ)
\[\begin{equation}\label{EQ2.14.1} \hat{E}_{el}^{mix} \geq \sum_{i, c} \dot{m}_{t, i, P^{Mix}, c} \; \forall{t \in T} \end{equation}\]

2.15. Land Used

The maximum land used at each year should not surpass \(\Lambda_t\). In the older documentation, we have it inversed and multiplied by the DM energy contents \(\eta_{f}\). In here, we have one important conflicting constraints (the second and the third constraints). The third constraint should be embedded in the second constraiant and the second constraint should be equality constraint.

  • $\lambda_{f, t} = $ land used by given crop type \(f\) at year \(t\) (ha)
  • $\dot{m}_{t, i, f, c} = $ feed used (PJ)
  • \(Y_{t,f} =\) land demand for energy crops f at year t. (\(\dfrac{ha}{PJ}\))
  • \(\Lambda_t =\) Maximal available land for cultivation of energy crops (ha)
  • \(\Lambda_{2020,f} =\) The available land for cultivation of energy crop \(f\) in the base year (2020) (ha)
  • \(r_{min}^{\Lambda} =\) The minimum use expansion per year (= 5000 ha)
  • \(r_{f}^{\Lambda} =\) land use expansion ramp per year (= 50\%).
\[\begin{equation}\label{EQ2.15.1} \begin{split} \Lambda_{t} &\geq \sum_{i,f,c} Y_{t,f} \times \dot{m}_{t, i, f, c} &\; \forall{t \in T} \\ \dfrac{\Lambda_{t}}{|C|} &\geq \sum_{i} Y_{t,f} \times \dot{m}_{t, i, f, c} &\; \forall{t \in T, f \in F, c \in C} \\ \Lambda_{2020,f} &\geq \sum_{i,c} Y_{2020,f} \times \dot{m}_{2020, i, f, c} &\; \forall{f \in F} \\ \underbrace{\sum_{i,c} Y_{t+1,f} \times \dot{m}_{t+1, i, f, c}}_{\lambda_{f, t+1}} &\leq r_{min}^{\Lambda} + (1 + r_{f}^{\Lambda}) \times \underbrace{\left(\sum_{i,c} Y_{t,f} \times \dot{m}_{t, i, f, c}\right)}_{\lambda_{f, t}} &\; \forall{t \in T, f \in F} \end{split} \end{equation}\]

2.16. Demand for fuel and heat

The multiplication of \(\dfrac{\eta^{th}_{t,i}}{\eta_{t,i}} \times w_{i,s}\) is mostly one. However, the places that one has a different value, the other one is again one to prevent double counting.

  • \(\pi_{t, i, s} =\) production of output from technology \(i\) at time \(t\) for market (sector) \(s\) \((PJ)\)
  • \(\delta_{t, s} =\) demand of sector \(s\) at year \(t\) (PJ).
  • \(w_{i,s} =\) relative fuel economy between fuels in passenger road sector.
  • \(\dfrac{\eta^{th}_{t,i}}{\eta_{t,i}} =\) fuel or heat by product per unit of main product (\(\dfrac{PJ_{th}}{PJ}\)).
\[\begin{equation}\label{EQ2.16.1} \begin{split} &\delta_{t, s} &\geq \sum_{i}\left(w_{i,s} \times \dfrac{\eta^{th}_{t,i}}{\eta_{t,i}}\times \pi_{t, i, s} \right) \; &\forall{t \in T, s \in S - \{CONVel, CONVth, CH4Market, H2Market\}} \end{split} \end{equation}\]

2.17. Demand limit for passenger road sector (vehicle-km)

The demand for the fuel in the road transportation sector should not surpass the demand for transportation. If we add aviation, we need to change this constraint such that it considers multiple sector competition to satisfy the demand for passenger transportation.

  • \(\pi_{t, i, s} =\) production of output from technology \(i\) at time \(t\) for market (sector) \(s\) \((PJ)\)
  • $\delta^{v-km}_{t} = total vehicle-km in passenger road sector (billion vehicle-km).
  • \(w_{i,s} =\) relative fuel economy between fuels in passenger road sector.
  • \(\tau_{t} =\) fuel economy baseline passenger road sector (\(\dfrac{PJ}{Bvkm}\)).
  • \(\dfrac{\eta^{th}_{t,i}}{\eta_{t,i}} =\) fuel or heat by product per unit of main product (\(\dfrac{PJ_{th}}{PJ}\)).
\[\begin{equation}\label{EQ.2.17.1} \delta^{v-km}_{t} \geq \dfrac{\sum_i w_{i,s} \times \dfrac{\eta^{th}_{t,i}}{\eta_{t,i}} \times \pi_{t, i, s}}{\tau_t} \, \forall{t \in T, s \in Passenger} \end{equation}\]

2.18. Demand limit for conventional thermal and power + H2, CH4 and LCH4

The first constraint says that the amount of heat produced as byproducts from CHP power plants should not surpass the heat requirement in each year. The second constraint says that except various technologies to satisfy electricity demand of CONVel, we should consider electricity produced by BioCH4_el for INDth and Gasif_s for HHth. $I^{CONVth} = $ {Biogas, GUD, KWK_l, KWK_m, KWK_HKW}, where GUD is Gas-and-steam turbine power plant, KWK_HKW is Wood-fired power plant with steam turbine, KWKs are CHP in general.

\[\begin{equation}\label{EQ.2.18.1} \begin{split} &\delta_{t,s} &\geq \sum_{i \in I^{CONVth}} \pi_{t,i,s} \times \dfrac{\eta^{th}_{t,i}}{\eta_{t,i}} &\, \forall{t \in T, s \in CONVth} \\ &\delta_{t,s} &\geq \sum_{i}(\pi_{t,i,s}) + \pi_{t,\text{BioCH4el},\text{INDth}} + \pi_{t,\text{Gasif_s},\text{HHth}} &\, \forall{t \in T, s \in CONVel} \\ &\delta_{t,s} \times \dfrac{\delta^{H2Max}_{t,s}}{\delta_{t,s}} &\geq \sum_{i \in I^{H2}} \pi_{t,i,s} &\;\forall{s \in S, t \in T} \\ &\delta_{t,s} \times \dfrac{\delta^{CH4Max}_{t,s}}{\delta_{t,s}} &\geq \sum_{i \in I^{CH4}} \pi_{t,i,s} &\;\forall{s \in S, t \in T} \\ &\delta_{t,s} \times \dfrac{\delta^{LCH4Max}_{t,s}}{\delta_{t,s}} &\geq \sum_{i \in I^{LNG}} \pi_{t,i,s} &\;\forall{s \in S, t \in T} \end{split} \end{equation}\]

Fuel (\(fu \in FU\)) consists of diesel (DSL), gasoline (EtOH), methane (CH4), hydrogen (H2) and electric (ELC). The first constraint is related to vehicle fuel type expansion ramp. The third constraint says that what we add will be decommisioned 14 years later (all at once). The fourth constraint sets an upper limit on the energy consumed by the new vehicles considering the energy efficiency at that year. * $r_{min}^{veh} = $ minimum vehicle expansion ramp per year (PJ). * $r_{f}^{veh} = $ vehicle fuel type expansion per year. * $\delta_{t, fu} = $ Fuel demand development constraint based on past fleet (PJ). * $\delta^{v-km}{t} = $ total vehicle-km in passenger road sector (billion vehicle-km). * $\hat{t} = $ vehicle life time defined in GAMS (14 years). * $S^{+v}{t} = $ share of new vehicles yearly in the passenger road sector (%). * $\tau = $ fuel economy baseline passenger road sector (\(\dfrac{PJ}{Bvkm}\)). * $\delta_{t,s,fu} = $ vehicle park demand per fuel (PJ). * $\delta^{+}{t,s,fu} = $ new vehicle fuel demand development (PJ). * $\delta^{-} = $ vehicle fuel type decommision (PJ).

\[\begin{equation}\label{EQ.2.19.1} \begin{split} &\delta^{+}_{t+1,s,fu} &\leq r_{min}^{veh} + r_{f}^{veh} \times \delta_{t,s,fu} &\;\forall{s = \text{Passenger}, t \in T, fu \in FU}\\ &\delta_{t+1,s,fu} &= \delta_{t,s,fu} + \delta^{+}_{t+1,s,fu} - \delta_{t,fu} + \delta_{t+1,fu} - \delta^{-}_{t+1,s,fu} &\;\forall{s = \text{Passenger}, t \in T, fu \in FU}\\ & \delta^{-}_{t+\hat{t}_{veh},s,fu} &= \delta^{+}_{t,s,fu} &\;\forall{s = \text{Passenger}, t \in T, fu \in FU}\\ \sum_{fu \in FU} &\delta^{+}_{t,s,fu} &\leq S^{+v}_{t} \times \tau_{t} \times \delta^{v-km}_{t} &\;\forall{s = \text{Passenger}, t \in T, fu \in FU} \end{split} \end{equation}\]

2.20. Fuel production for road vehicles

Fuel production for road vehicles should not surpass the demand! In here, the summation over FU in the right-hand side is redundent. Also, regarding the sign of the inequality, it should be otherway around.

\[\begin{equation}\label{EQ.2.20.1} \begin{split} &\sum_{i \in I^{DSL}} \pi_{t,i,s} &\leq \sum_{fu \in Fu} \delta_{t,s, fu = {DSL}} &\;\forall{s = \text{Passenger}, t \in T}\\ &\sum_{i \in I^{EtOH}} \pi_{t,i,s} &\leq \sum_{fu \in Fu} \delta_{t,s, fu = {EtOH}}&\;\forall{s = \text{Passenger}, t \in T}\\ &\sum_{i \in I^{CH4}} \pi_{t,i,s} &\leq \sum_{fu \in Fu} \delta_{t,s, fu = {CH4}} &\;\forall{s = \text{Passenger}, t \in T}\\ &\sum_{i \in I^{H2}} \pi_{t,i,s} &\leq \sum_{fu \in Fu} \delta_{t,s, fu = {H2}} &\;\forall{s = \text{Passenger}, t \in T}\\ &\sum_{i \in I^{ELC}} \pi_{t,i,s} &\leq \sum_{fu \in Fu} \delta_{t,s, fu = {ELC}} &\;\forall{s = \text{Passenger}, t \in T} \end{split} \end{equation}\]

2.21. Regulating the expansion

  • \(r_{min} =\) minimum capacity expansion ramp per year, which is set to 1 (PJ)
  • \(r_{f}^{veh} =\) vehicle fuel type expansion per year.
\[\begin{equation}\label{EQ.2.21.1} \begin{split} \sum_{i \in I^{H2}} \pi_{t+1,i,s} &\leq r_{min} + (1 + r_{f}^{veh}) \times \sum_{i \in I^{H2}} \pi_{t,i,s} &\;\forall{s \in S \backslash \{H2Market\}, t \in T}\\ \sum_{i \in I^{LNG}}\pi_{t+1,i,s} &\leq r_{min} + (1 + r_{f}^{veh}) \times \sum_{i \in I^{LNG}} \pi_{t,i,s} &\;\forall{s \in S, t \in T}\\ \sum_{i \in I^{CH4}}\pi_{t+1,i,s} &\leq r_{min} + (1 + r_{f}^{veh}) \times \sum_{i \in I^{CH4}} \pi_{t,i,s} &\;\forall{s \in S \backslash \{Passenger, CH4Market\}, t \in T} \end{split} \end{equation}\]

Nomenclature

| Symbol | Description | Unit |

|--------|-------------|------|

| \(p\_{g,t}\) | Power output of generator \(g\) at time \(t\) | MW |

| \(\\bar{P}\_g\) | Installed capacity of generator \(g\) | MW |

| \(f\_{l,t}\) | Power flow on line \(l\) at time \(t\) | MW |

| \(d\_{n,t}\) | Demand at node \(n\), time \(t\) | MW |

| \(c\_g^{var}\) | Variable cost of generator \(g\) | €/MWh |