Mathematical equation
In the following, the mathematical description of the optimization model is explained, elaborating on all relevant functionalities.
Nomenclature
Table: Sets, decision variables and parameters
| Notation | Description | Unit |
|---|---|---|
| $y \in \mathcal{Y}$ | year | |
| $p \in \mathcal{M}$ | product type (incl. passengers) | |
| $m \in \mathcal{M}$ | mode | |
| $r \in \mathcal{R}$ | O-D pair | |
| $k \in \mathcal{K}$ | route | |
| $v \in \mathcal{V}$ | vehicle type | |
| $t \in \mathcal{T}$ | drive-train technology - fuel pair | |
| $l \in \mathcal{L}_t$ | fuel supply options for technology $t$ | |
| $e \in \mathcal{E}$ | location | |
| $ic \in \mathcal{IC}$ | income class | |
| $b \in \mathcal{B}_{kmtg}$ | subset for each route $k$ and technology $t$ for year $y$ | |
| $g \in \mathcal{G}$ | generation of vehicle fleet | |
| $\mathcal{V}_k$ | $(e_1, e_2, e_3, \dots, e_I)$ | |
| $\mathcal{U}_k$ | $\{(i, e_i) \mid e_i \in \mathcal{V}_k, 1 \leq i \leq I \}$ | |
| $\mathcal{Y}_y$ | $\{0, \dots, y\}$ | |
| $\mathcal{E}_{kmtgb}$ | subset of edges within the driving range of technology $t$ in year $y$ along route $k$ |
Decision variables
| Notation | Description | Unit |
|---|---|---|
| $f_{yprkmvtg}$ | transport volumes using tech $t$ on mode $m$, route $k$ in year $y$ | T |
| $h_{yprmtg}$ | vehicle fleet for mode $m$ with technology $t$ | # |
| $h^{+}_{yprmtg}$ | vehicle fleet growth for mode $m$ with technology $t$ | # |
| $h^{-}_{yprmtg}$ | vehicle fleet reduction for mode $m$ with technology $t$ | # |
| $h^{exist}_{yprmtg}$ | vehicle fleet existing at start of year $y$ | # |
| $s_{ypkmvtle}$ | fueling demand during peak hour at edge $e$ for tech $t$ via supply option $l$, route $k$ in year $y$ | kWh |
| $s_{ypkmtln}$ | fueling demand during peak hour at node $n$ for tech $t$ via supply option $l$, route $k$ in year $y$ | kWh |
| $q^{+, mode\_infr}_{yet}$ | installed mode infrastructure for tech $t$ on edge $e$ in year $y$ | kW |
| $q^{+, fuel\_infr}_{yet}$ | installed fueling infrastructure for tech $t$ on edge $e$ in year $y$ | kW |
| $q^{+, supply\_infr}_{yle}$ | capacity of supply infrastructure $l$ on edge $e$ in year $y$ | kW |
Parameters
| Notation | Description | Unit |
|---|---|---|
| $LoS_{yktv}$ | level of service | h |
| $F_{yrp}$ | transport demand between O-D pair $r$ for product $p$ in year $y$ | T |
| $D^{spec}_{yvtg}$ | specific energy consumption of tech $t$ in year $y$ | kWh/km |
| $W_{yvtg}$ | average load of a vehicle of tech $t$ bought in year $g$ | T |
| $L^a_{gmvt}$ | max annual mileage of vehicle | km |
| $L_k$ | length of path $k$ | km |
Objective function
\[\min_{x} Z\]
\[{C}^{infrastructure, total} = \sum_{t} \sum_{y} \left( \sum_{e} C^{fuel\_infr}_{yte} q^{+, fuel\_infr}_{yte} + \sum_{y' \in \mathcal{Y}^y} C_{yte}^{fuel\_infr, OM, fix}\left( Q^{fuel\_infr}_{te} + q^{+, fuel\_infr}_{y'te}\right) \right) \\ + \sum_{m} \sum_{y} \left( \sum_{e} C^{mode\_infr}_{yme} q^{+, mode\_infr}_{yme} + \sum_{y' \in \mathcal{Y}^y} C_{yme}^{mode\_infr, OM, fix}\left( Q^{fuel\_infr}_{me} + q^{+, fuel\_infr}_{y'me}\right) \right) \\ + \sum_{l} \sum_{y} \left( \sum_{e} C^{supply\_infr}_{yle} q^{+, supply\_infr}_{yle} + \sum_{y' \in \mathcal{Y}^y} C_{yle}^{supply\_infr, OM, fix, supply}\left( Q^{supply\_infr}_{te} + q^{+, supply\_infr}_{y'te}\right) \right) \\ \]
\[{C}^{vehiclestock, total} = \sum_y \sum_m \sum_v \sum_t \sum_g \left( C^{CAPEX}_{yvtg} h^{+}_{yprvtg} + C^{h, OM, fix}_{yvtg} h_{yprvtg} + \sum_l \sum_{e \in E^k} C^{fuelcosts}_{yle}* s_{ypkmvtle} \right)\]
\[{C}^{transportactivity, total} = \sum_y \left( C^{OM, fix, dist}_{mvt} f_{y,prkmvtg} + \sum_k C^{OM, var, dist}_{mvt} \sum_k L_k f_{y,prkmvtg} \right)\]
\[{C}^{intangiblecosts, total} = \sum_y \sum_m \sum_r \sum_{kvt} VoT_{ykvt, ic} * LoS^{f}_{ykvt} * f_{yprkmvtg} \]
\[LoS^f_{yk} = \frac{L_k}{Speed_yvmt} + Fueling\_time_{ykvmt} + Waiting\_time_{ykm}\]
\[{C}^{paneltycosts, total} = \sum_y \sum_p \sum_r penalty^{budget}_{pry} \]
\[\sum_{kmvtg} f_{yprkmvtg} = F_{yrp} \quad : \forall y \in \mathcal{Y}, r \in \mathcal{R}, p \in \mathcal{P} \]
Constraints
Vehicle stock modelling
\[h_{yprmvtg} = h^{exist}_{yprmvt(g-1)} + h^{+}_{yprmvtg} - h^{-}_{yprmvtg} : \forall y \in \mathcal{Y}\setminus \{y_0\}, r \in \mathcal{R}, p \in \mathcal{P}, m \in \mathcal{M}, v \in \mathcal{V}, t \in \mathcal{T}, g \in \mathcal{G}\]
\[h^{exist}_{yprmvt(g-1)} = h_{(y-1)prmvtg} \quad : y = y_0, r \in \mathcal{R}, p \in \mathcal{P}, m \in \mathcal{M}, v \in \mathcal{V}, t \in \mathcal{T}, g \in \mathcal{G}\]
Fueling demand
\[\sum_{l \in \mathcal{L}_t} \sum_{e \in \mathcal{E}_{k}} s_{ypkmvtle} = \sum_{g \in \mathcal{G}} \sum_{a \in \mathcal{A}^p} \sum_{e \in \mathcal{E}_{k}} \sum_{n \in \mathcal{N}_{k}} \gamma \frac{D^{spec}_{yt} L_{ke}}{W_{ymvt}} f_{ypakmvtg} : \forall y \in \mathcal{Y}, p \in \mathcal{P}, k \in \mathcal{K}, m \in \mathcal{M}, t \in \mathcal{T}_m\]
\[\sum_{l \in \mathcal{L}_t} \sum_{e \in \mathcal{U}_{ke}} s_{ypkmvtle} \leq \gamma \sum_{g \in \mathcal{G}} \frac{1}{W_{gmvt}} f_{ypkmvtg} * Q^{tank}_{gmvt} :\forall y, p, k, m, t\]
Vehicle stock shift
\[\pm \left( \sum_g h_{yprmvt} - \sum_g h_{(y-1)prmvt} \right) \leq \alpha \sum_{gvt} h_{yprmvt} + \beta \sum_g h_{(y-1)prmvtg} : \forall y \in \mathcal{Y}\setminus \{ y_0\}, r \in \mathcal{R}, m \in \mathcal{M}, v \in \mathcal{V}, t \in \mathcal{T}_m\]
\[\pm \left( \sum_g h_{yprmvtg} - \sum_g h^{exist}_{yprmvtg} \right) \leq \alpha \sum_{gvt} h_{yprmvt} + \beta \sum_g h^{exist}_{yprmvtg} : \forall y \in \{y_0\}, r \in \mathcal{R}, m \in \mathcal{M}, v \in \mathcal{V}, t \in \mathcal{T}_m\]
Mode shift
\[\left( \sum_{kg} f_{yprkmvtg} - \sum_{kg} f_{(y-1)prkmvtg} \right) \leq \alpha F_{yrp} + sum_{kg} f_{(y-1)prkmvtg } : \forall y \in \mathcal{Y}\setminus \{ y_0\}, r \in \mathcal{R}, m \in \mathcal{M}\]
\[\left( \sum_g h_{yprmvtg} - \sum_g h^{exist}_{yprmvtg} \right) \leq \alpha \sum_{gvt} h_{yprmvt} + \beta \sum_g h^{exist}_{yprmvtg} : \forall y \in \{y_0\}, r \in \mathcal{R}, m \in \mathcal{M}, v \in \mathcal{V}, t \in \mathcal{T}_m\]
Mode infrastructure expansion
\[Q^{mode\_infr}_{me} + \sum_{y \in \mathcal{Y}_y} q^{+, mode\_infr}_{yme} \geq \gamma \sum_{e \in \mathcal{K}_e} \sum_{p \in \mathcal{P}} \sum_{t \in T_{m}} f_{ypkmtg} : \forall y \in \mathcal{Y}, m \in \mathcal{M}, e \in \mathcal{E}\]
Fueling Infrastructure expansion
\[Q^{fuel\_infr}_{te} + \sum_{y \in \mathcal{Y}_y} q^{+, fuel\_infr}_{yte} \geq \sum_{k \in \mathcal{K}_e} \sum_{p \in \mathcal{P}} \sum_{m \in \mathcal{M}} \sum_{l \in \mathcal{L}_t} s_{ypkmtle} : \forall y \in \mathcal{Y}, t \in \mathcal{T}, e \in \mathcal{E}\]
\[ Q^{supply\_infr}_{le} + \sum_{y \in \mathcal{Y}_y} q^{+, supply\_infr}_{yle} \geq \sum_{e \in \mathcal{K}_e} \sum_{p \in \mathcal{P}} \sum_{m \in M} \sum_{t \in \mathcal{T}_l} s_{ypkmtle} : \forall y \in \mathcal{Y}, l \in \mathcal{L}, e \in \mathcal{E}\]
Monetary budget
\[\sum_y C^{CAPEX}_{yvtg} * h^{+}_{yr} \leq Budget_{ic} * f * |Y| + penalty^{+, invbudget}\]
\[\sum_y C^{CAPEX}_{yvtg} * h^{+}_{yr} \geq Budget_{ic} * f * |Y|- penalty^{-, invbudget}\]
\[\sum_{y'\in Y_i} C^{CAPEX}_{y'vtg} * h^{+}_{y'r} \leq Budget_{ic} * f * \tau^i + penalty^{+, invbudget} \]
\[\sum_{y'\in Y_i} C^{CAPEX}_{y'vtg} * h^{+}_{y'r} \geq Budget_{ic} * f * \tau^i - penalty^{+, invbudget}\]