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Optimal Installation of the Power Transmitters in the Dynamic Wireless Charging for Electric Vehicles in a Multipath Network with the Round-Trip Case

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Abstract

With the emergence of environmental problems, the implementation of electric vehicles in the transport sector presents a solution that meets environmental and economic objectives. For this reason, electric vehicles (EVs) have become increasingly popular as a mainstream transportation solution, opportunities to recharge the vehicle away from home have become a critical issue, and it needs a long waiting time, with a risk of electrocution. One of the solutions to avoid these disadvantages is the wireless charging EVs. For that reason, the main contribution of this work is to propose the strategic location of inductive power transmitters especially when there are several routes between an origin and a destination. Our goal is to find a compromise between the cost of installing the power transmitters and the cost of the battery while maintaining the quality of the vehicle routing. To show the efficiency of our mathematical model and resolution method, we compare our results with the results found in the literature.

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Correspondence to Hassane Elbaz.

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Appendices

Appendix 1 Linearization of the constraints (17) and (18) using the indicator constraints

The following constraints replaces the constraint (17). For this, we integrate the following binary variable \( {z}_{ij}^{l_p} \)

$$ {\displaystyle \begin{array}{c}{z}_{ij\left(j+1\right)}^{l_p}=0\mathbf{\Rightarrow}I\left({t}_{i\left(j+1\right)}^{l_p}\right)={I}_h\\ {}\forall p\in \left\{0,1\right\}\kern0.5em \forall {l}_p\in L\kern0.5em \forall i\in I\kern0.75em \forall j\in \left\{1\dots {N}_i-1\right\}\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_{ij\left(j+1\right)}^{l_p}=1\Rightarrow I\left({t}_{i\left(j+1\right)}^{l_p}\right)=I\left({t}_{ij}^{l_p}\right)-{\int}_{t_{ij}^{l_p}}^{t_{i\left(j+1\right)}^{l_p}}{P}_b(t) dt+{P}_c\left({t}_{i\left(j+1\right)}^{l_p}\_{t}_{ij}^{l_p}\right){S}_{ij}^p\\ {}\forall p\in \left\{0,1\right\}\kern0.5em \forall {l}_p\in L\kern0.5em \forall i\in I\kern0.75em \forall j\in \left\{1\dots {N}_i-1\right\}\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_{ij\left(j+1\right)}^{l_p}=1\Rightarrow I\left({t}_{ij}^{l_p}\right)-{\int}_{t_{ij}^{l_p}}^{t_{i\left(j+1\right)}^{l_p}}{P}_b(t) dt+{P}_c\left({t_{i\left(j+1\right)}^{l_p}}_{t_{ij}^{l_p}}\right){S}_{ij}^p\le {I}_h\\ {}\forall p\in \left\{0,1\right\}\kern0.5em \forall {l}_p\in L\kern0.5em \forall i\in I\kern0.75em \forall j\in \left\{1\dots {N}_i-1\right\}\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_{ij\left(j+1\right)}^{l_p}=0\Rightarrow I\left({t}_{ij}^{l_p}\right)-{\int}_{t_{ij}^{l_p}}^{t_{i\left(j+1\right)}^{l_p}}{P}_b(t) dt+{P}_c\left({t}_{i\left(j+1\right)}^{l_p}\_{t}_{ij}^{l_p}\right){S}_{ij}^p>{I}_h\\ {}\forall p\in \left\{0,1\right\}\kern0.5em \forall {l}_p\in L\kern0.5em \forall i\in I\kern0.75em \forall j\in \left\{1\dots {N}_i-1\right\}\end{array}} $$

We apply the same procedure for the constraint (18)

$$ {\displaystyle \begin{array}{c}{z}_{n1}^{l_0}=0\mathbf{\Rightarrow}\mathrm{I}\left({t}_{n1}^{l_0}\right)={I}_h\\ {}\forall {l}_0\in L\kern0.75em \forall n\in {D}_0\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_{n1}^{l_0}=1\Rightarrow I\left({t}_{n1}^{l_0}\right)=\Delta T\times {P}_c^{\prime }-{\int}_0^{t_{n1}^{l_0}}{P}_b(t) dt+{P}_c\times {t}_{n1}^{l_0}\times {S}_{n1}^p\\ {}\forall {l}_0\in L\kern0.75em \forall n\in {D}_0\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_{n1}^{l_0}=1\Rightarrow \Delta T\times {P}_c^{\prime }-{\int}_0^{t_{n1}^{l_0}}{P}_b(t) dt+{P}_c\times {t}_{n1}^{l_0}\times {S}_{n1}^p\le {I}_h\\ {}\forall {l}_0\in L\kern0.75em \forall n\in {D}_0\end{array}} $$
$$ {\displaystyle \begin{array}{c}{z}_{n1}^{l_0}=0\Rightarrow \Delta T\times {P}_c^{\prime }-{\int}_0^{t_{n1}^{l_0}}{P}_b(t) dt+{P}_c\times {t}_{n1}^{l_0}\times {S}_{n1}^p>{I}_h\\ {}\forall {l}_0\in L\kern0.75em \forall n\in {D}_0\end{array}} $$

Appendix 2 Coding of the nonlinear constraint (12)

forall(i in 0..I, j in 0..N-Lmax)

(if) (aij = 1)

{

for (var k = 0; k < Lm; k++)

$$ {\displaystyle \begin{array}{c}\Big\{\kern1.25em {z}_{ij}^k=\mathrm{sum}\left(\mathrm{r}\ \mathrm{in}\ \mathrm{k}\right){S}_{ij}^0;\\ {}{w}_{ij}^k=\mathrm{sum}\left(\mathrm{r}\ \mathrm{in}\ \mathrm{k}\right){S}_{i\left(N-j-r\right)}^1;\end{array}} $$

(if) \( {\displaystyle \begin{array}{c}\left({z}_{ij}^k==k+1\right)\\ {}{y}_{ij}^{k0}=1-{S}_{i\left(j-1\right)}^0;\end{array}} \)

ilse

$$ {y}_{ij}^{k0}=0; $$

(if) \( {\displaystyle \begin{array}{c}\left({w}_{ij}^k==k+1\right)\\ {}{y}_{ij}^{k0}=1-{S}_{i\left(N-j+1\right)}^0;\end{array}} \)

else

$$ \kern1.75em {y}_{ij}^{k1}=0; $$

}

$$ \mathrm{sum}\left(\mathrm{r}\ \mathrm{in}\ \mathrm{Lm}\ \right)\kern0.5em \left(\ {y}_{ij}^{k0}+{y}_{ij}^{k1}\right)<=\mathrm{Lm}; $$

}

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Elbaz, ., Elhilali Alaoui, A. Optimal Installation of the Power Transmitters in the Dynamic Wireless Charging for Electric Vehicles in a Multipath Network with the Round-Trip Case. Int. J. ITS Res. 20, 46–63 (2022). https://doi.org/10.1007/s13177-021-00270-5

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