# Optimal Incentives for Electric Vehicles at e-Park & Ride Hub with Renewable Energy Source

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Methodologies

## 2. E-Park & Ride Hub Scenario

#### 2.1. Route Choice

#### 2.1.1. Model Assumptions

- Two types of vehicle are considered: an electric one (denoted EV and associated with subscript e) and a gasoline one (GV, associated with subscript g). Each commuter is associated with one of the two vehicles and the proportion of EV among them is denoted by ${X}_{e}$ in the model (in numerical tests, ${X}_{e}=50\%$ which is in line with 2035 predictions for France; see middle scenario of [1]). The proportion of GV is then given by ${X}_{g}=1-{X}_{e}$. The choice made by all commuters between the two transport modes of Figure 1 is represented by the two variables ${x}_{e,publ}$ and ${x}_{g,publ}$, which are respectively the proportions of EV and GV choosing the public transport mode. Note that the proportions of vehicles of type $s=e,g$ choosing the private transport mode may be easily deduced: ${x}_{s,priv}=1-{x}_{s,publ}$.
- The decision process of commuters is assumed rational, meaning that they choose the transport mode (publ or priv) with minimal cost. Here, the costs considered are travel duration (by private or public transport), energy consumption (electricity for EV and fuel for GV) and the ticket fare (for public transport only).

#### 2.1.2. Costs Functions

**(a) Travel Duration Costs**

- –
- For the private mode, it depends on the total proportion (here, proportion and number of vehicles are equivalent, as the total number of vehicles is fixed) of vehicles driving downtown ${x}_{priv}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{x}_{e,priv}{X}_{e}+{x}_{g,priv}{X}_{g}=(1-{x}_{e,publ}){X}_{e}+(1-{x}_{g,publ}){X}_{g}$ due to congestion effects [14] and is expressed as:$${\tau}_{priv}\times \underset{{d}_{priv}\left({x}_{priv}\right)}{\underset{\u23df}{\frac{l}{v}\left(1+2{\left(\frac{{x}_{priv}}{C}\right)}^{4}\right)}}\phantom{\rule{0.166667em}{0ex}},$$
- ${\tau}_{priv}=10$ €/h the value of time when driving, based on a French government report (http://www.strategie.gouv.fr/sites/strategie.gouv.fr/files/archives/Valeur-du-temps.pdf),
- $l=5$ km the length of the road, approximately the radius of Paris,
- $v=50$ km/h the speed limit, as in French urban areas,
- $C=1$ the capacity of the road, expressed in proportion of the total number of vehicles, like ${x}_{priv}$.

- –
- For the public transport mode linking the hub and the destination, the travel cost is assumed constant:$${\tau}_{publ}\times {d}_{publ}\phantom{\rule{0.166667em}{0ex}},$$
- ${\tau}_{publ}=12$ €/h value of time in public transport (http://www.strategie.gouv.fr/sites/strategie.gouv.fr/files/archives/Valeur-du-temps.pdf), which is perceived by commuters as less comfortable than personal vehicles,
- ${d}_{publ}=\frac{l}{v}=6$ min constant travel time of public transport, which was chosen equal to the free flow travel time of the private mode. Indeed, there exist reserved pathways for public transportation in several cities like Paris, so that congestion can be considered as marginal.

The duration cost of the public mode is then equal to the fixed value ${\tau}_{publ}\phantom{\rule{3.33333pt}{0ex}}{d}_{publ}=1.2$ € and is higher than the free flow cost of the private mode. This induces trade-off decisions for vehicles between both strategies.

**(b) Energy Consumption Cost**

- ${l}_{r}$ is the total distance driven by the vehicles which have chosen transport mode r, and is equal to:
- –
- ${l}_{publ}=10$ km distance between the origin and the hub, so that the two-way trip between origin and destination is 30 km, the daily average individual driving distance in France (following Enquête Nationale Transports et Déplacements: https://utp.fr/system/files/Publications/UTP_NoteInfo1103_Enseignements_ENTD2008.pdf), 2008, in French),
- –
- ${l}_{priv}={l}_{publ}+l=15$ km,

- ${m}_{s}$ is the electricity or fuel consumed per distance unit and is supposed constant (e.g., it does not depend on speed profiles):
- –
- ${m}_{e}=0.2$ kWh/km, following [16],
- –
- ${m}_{g}=0.06$ L/km (Liter/km),

- ${\lambda}_{s}$ is the charging/fueling unit price:
- –
- For EV, the key distinction made here is that it depends on the transport mode chosen.
**Public mode:**At the hub, this charging unit price ${\lambda}_{e}$ will depend on the total charging need ${L}_{e}\left({x}_{e,publ}\right)$, proportional to the number of EV parked in the hub: for example, if there are few EV at the hub (${x}_{e,publ}$ close to 0), there is enough electricity produced at the hub to provide the charging need of these EV. This price is obtained by solving a charging problem, which is detailed in the next section.**Private mode:**Downtown, there is a standard constant electricity fare ${\lambda}_{e}^{0}=$ 40 c€/kWh, which corresponds to the electricity unit price in France (15 c€/kWh) with an additional cost (25 c€/kWh) meant for the charging operation. - –
- ${\lambda}_{g}=1.50$ €/L is considered constant.

**(c) Public Transport Ticket Fare**

#### 2.2. Hub Charging Operation

#### 2.2.1. Charging Scenario

#### 2.2.2. Modeling of Charging Problem

#### Aggregated Charging Need

#### Temporal Charging Scheduling

#### Photovoltaic Production

#### Cost/Impact on the Local Electrical Grid

#### Charging Problem and Solution

- If the aggregated charging need ${L}_{e}\left({x}_{e,publ}\right)$ verifies ${L}_{e}<E$, any charging profile below the PV production is optimal, since the associated cost is zero.
- If ${L}_{e}=E$ (which corresponds to the charging need of 29 EV), the optimal scheduling has to perfectly match the production.
- If ${L}_{e}>E$, all PV production is consumed and the remaining charging need has to be equally shared between all time slots such that the net load taken from the grid is constant.

## 3. Numerical Experiments

#### 3.1. Wardrop Equilibrium Representation

#### 3.2. Equilibrium Sensitivity to Parameters of the Problem

#### 3.3. Optimal Solar Panel Surface

- I the initial Investment cost in solar panels, with 750 €/kWp for a solar park of the order of magnitude of 1 MWp,

- C the daily grid Costs (associated with the electricity bought from the grid), defined in Equation (8),
- R the daily Revenues from EV charging at the hub which are, by definition of the charging unit price ${\lambda}_{e}$:$$R={\lambda}_{e}\times {L}_{e}\left({x}_{e,publ}^{*}\right)\phantom{\rule{0.166667em}{0ex}},$$

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the charging hub scenario: commuters can either choose to leave their vehicle at the hub and take public transport ($publ$), or drive all the way to their destination ($priv$). A local source of renewable energy is available at the hub.

**Figure 2.**Travel duration ${d}_{priv}(.)$ (and the associated cost ${\tau}_{priv}{d}_{priv}(.)$, along the right axis) for vehicles driving downtown depending on their proportion ${x}_{priv}$. If all vehicles choose the private transport mode, the associated duration will be three times higher than the free-flow case (when all vehicles choose public mode).

**Figure 3.**Energy produced by 1 m${}^{2}$ (equivalent to a nominal power of 175 Wp) of a solar panel in Paris during working hours (averaged over the year 2014).

**Figure 4.**Cost/impact on the electrical grid of electric vehicles (EV) charging and PhotoVoltai (PV) production (through the net load ${\ell}_{t}$) at time slot t. If PV production is higher (resp. smaller) than EV charging load, there is no (resp. a quadratic) impact.

**Figure 5.**Water-filling optimal scheduling of the charging operation with 125 m${}^{2}$ of solar panel (in black), for 15 EV (in red) and 45 EV (in blue). For 15 EV, any scheduling using only PV production is optimal, while for 45 EV, the only optimal scheduling uses the whole PV production plus the same amount from the electrical grid at each time slot.

**Figure 6.**EV costs as a function of the proportion ${x}_{e,publ}$ of EV choosing public transport. In blue (resp. red) is the cost for EV choosing public (resp. private) transport mode. The dotted lines refer to the monetary costs (consumption and ticket fare for public mode; only consumption for private mode) and the dashed lines refer to travel duration. The equilibrium (black star) happens when total costs are equal between the two transport modes, for ${x}_{e,publ}={x}_{e,publ}^{*}=0.53$.

**Figure 7.**Proportion of vehicles at the hub at WE in function of … (

**a**) PT ticket fare ${t}_{publ}$; (

**b**) value of time on the road ${\tau}_{priv}$; (

**c**) unit consumption of EV ${m}_{e}$; (

**d**) unit consumption of GV ${m}_{g}$; (

**e**) penetration of EV ${X}_{e}$.

**Figure 8.**Impact of the charging unit price ${\lambda}_{\mathrm{cst}}$ on the Wardrop equilibrium (WE) and the charge point operator (CPO) revenues R. (

**a**) Number of vehicles at the hub at WE in function of ${\lambda}_{\mathrm{cst}}$; (

**b**) CPO daily revenues R in function of ${\lambda}_{\mathrm{cst}}$. As ${\lambda}_{\mathrm{cst}}$ increases, fewer EV choose the hub while R increases, up to a threshold ${\lambda}_{\mathrm{cst}}^{*}=37.5$ c€/kWh beyond which all EV drive downtown.

**Figure 9.**Daily payoff $R-C$ in function of the charging unit price ${\lambda}_{\mathrm{cst}}$ at the hub, for different daily PV productions E. No matter E, ${\lambda}_{\mathrm{cst}}={\lambda}_{\mathrm{cst}}^{*}$ maximizes $R-C$.

**Figure 10.**Payoff F over 20 years (with ${\lambda}_{\mathrm{cst}}={\lambda}_{\mathrm{cst}}^{*}$) in function of the PV nominal power. The CPO can make more profits by installing (the right amount of) PV.

**Figure 11.**Optimal CPO variables to maximize F in function of nominal power. (

**a**) Optimal fixed part ${\lambda}_{\mathrm{cst}}^{*}$) in function of nominal power; (

**b**) Payoff F (with ${\lambda}_{\mathrm{cst}}={\lambda}_{\mathrm{cst}}^{*}$) in function of nominal power. The variable charging unit price ${\lambda}_{e}$ offers a little more benefit than a fixed one (${\lambda}_{\mathrm{cst}}$, Figure 10).

Total Costs | Public Transport Mode | Private Transport Mode |
---|---|---|

EV | $\begin{array}{c}{c}_{e,publ}\left({x}_{e,publ}\right)={\tau}_{publ}\phantom{\rule{3.33333pt}{0ex}}{d}_{publ}+{t}_{publ}\\ {}_{\phantom{\rule{3.33333pt}{0ex}}}+{l}_{publ}\phantom{\rule{3.33333pt}{0ex}}{m}_{e}{\lambda}_{e}\left({L}_{e}\left({x}_{e,publ}\right)\right)\end{array}$ | $\begin{array}{c}{c}_{e,priv}\left(\mathbf{x}\right)={\tau}_{priv}\phantom{\rule{3.33333pt}{0ex}}{d}_{priv}\left({x}_{priv}\right)\\ {}_{\phantom{\rule{3.33333pt}{0ex}}}+{l}_{priv}\phantom{\rule{3.33333pt}{0ex}}{m}_{e}{\lambda}_{e}^{0}\end{array}$ |

GV | $\begin{array}{c}{c}_{g,publ}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}={\tau}_{publ}\phantom{\rule{3.33333pt}{0ex}}{d}_{publ}+{t}_{publ}\\ \phantom{\rule{3.em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}+{l}_{publ}\phantom{\rule{3.33333pt}{0ex}}{m}_{g}{\lambda}_{g}\end{array}$ | $\begin{array}{c}{c}_{g,priv}\left(\mathbf{x}\right)={\tau}_{priv}\phantom{\rule{3.33333pt}{0ex}}{d}_{priv}\left({x}_{priv}\right)\\ {}_{\phantom{\rule{3.33333pt}{0ex}}}+{l}_{priv}\phantom{\rule{3.33333pt}{0ex}}{m}_{g}{\lambda}_{g}\end{array}$ |

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**MDPI and ACS Style**

Sohet, B.; Beaude, O.; Hayel, Y.; Jeandin, A.
Optimal Incentives for Electric Vehicles at e-Park & Ride Hub with Renewable Energy Source. *World Electr. Veh. J.* **2019**, *10*, 70.
https://doi.org/10.3390/wevj10040070

**AMA Style**

Sohet B, Beaude O, Hayel Y, Jeandin A.
Optimal Incentives for Electric Vehicles at e-Park & Ride Hub with Renewable Energy Source. *World Electric Vehicle Journal*. 2019; 10(4):70.
https://doi.org/10.3390/wevj10040070

**Chicago/Turabian Style**

Sohet, Benoît, Olivier Beaude, Yezekael Hayel, and Alban Jeandin.
2019. "Optimal Incentives for Electric Vehicles at e-Park & Ride Hub with Renewable Energy Source" *World Electric Vehicle Journal* 10, no. 4: 70.
https://doi.org/10.3390/wevj10040070