## 2 Problem Statement
Here, \(SOC_{p,s}(t_{dp})\) shows SOC of \(p\)-th PEV battery at the departure time (\(t_{dp}\)) in scenario \(s\). The undermentioned constraint must be satisfied for all PEVs' battery. \[ \begin{array}{l}SOC_{p,s}(t)=SOC_{p,s}(t-1)+P_{p}^{dr}(t)\times\eta_{ddr}-P_ {p}^{dichr}(t)/\eta_{lidir}\\ p=1,2,\ldots,N_{PEV}\qquad t=1,2,\ldots,24\end{array}  \tag{9}\]  
where, \(P_{p}^{clr}(t),P_{p}^{dichr}(t)\) are charging and discharging rate of each PEVs in time interval \(t\), and \(\eta_{clir},\eta_{lidir}\) are the total efficiency of battery charging/discharging and converter, respectively [16]. In Equation (10), battery degradation cost has been calculated based on the PEVs total battery degradation at each time for each scenario [21]. \[ \begin{array}{l}DEC_{s}=\sum\limits_{s=1}^{10}\big{(}\sum\limits_{p=1}^{N_ {PEV}}\sum\limits_{n_{p}^{PEV}}^{n_{p}^{PEV}}[C_{p}^{Batt}\times\frac{D_{p,s}^{ TOT}\times\mathcal{T}_{c,s}^{ave}}{Q_{p}^{Datt}-Q_{p}^{Use}}]\big{)}\\ T_{c}^{cyc}=c\text{-th Charging cycle}\end{array}  \tag{10}\]  
where \(C_{p}^{Batt}\) is the \(p\)-th PEV battery cost and \(D_{p,s}^{TOT}\) represents summation of calendar (\(D_{p,s}^{CAL}\)) and cyclic (\(D_{p,s}^{CYc}\)) degradation of PEVs' battery in the cycle of \(T_{c,s}^{cyc}\) in scenario \(s\) and cycle \(c\). Also, \(Q_{p}^{0},Q_{p}^{Use}\) stand for the nominal and useful capacity of \(p\)-th PEV, respectively. \(Q_{p}^{Use}\) is extracted based on Equation (11). \[Q_{p}^{USE}=\rho\times Q_{p}^{0}\qquad 0<\rho<1 \tag{11}\]  
Stem from the United States Advanced Battery Consortium (USABC) PEVs' battery should be changed if the battery capacity is reduced to 80% of its total capacity [21], thus \(\rho=0.8\). \[D_{p,s}^{TOT}(T_{c,s}^{cyc})=D_{p,s}^{CAL}(T_{c,s}^{cyc})+D_{p,s}^{CYc}(T_{c,s }^{cyc}) \tag{12}\]  
Equations (13) and (14) illustrate the mechanism of the calculating calendar (\(D_{p,s}^{CAL}\)) and cyclic (\(D_{p,s}^{CYc}\)) degradation [21]. \[D_{p,s}^{CAL}(T_{c,s}^{cyc})=Q_{p}^{0}\times\exp(\frac{SOC_{p,s}(T_{c,s}^{cyc })-\overline{SOC}}{b})\times\exp(\frac{\psi_{B}-\psi_{0}}{a})\times\sqrt{t_{s}} \tag{13}\]  
\[D_{p,s}^{CYc}(T_{c,s}^{cyc})=\alpha_{1}\times\big{[}DOD_{p,s}(T_{c,s}^{cyc}) \big{]}^{2}+\alpha_{2}\times DOD_{p,s}(T_{c,s}^{cyc})+\alpha_{3} \tag{14}\]  
To model the degradation of the PEVs' batteries, two main criteria including calendar degradation (\(D_{p,s}^{CAL}\)) and cycling degradation are considered. Employing calendar degradation (\(D_{p,s}^{CAL}\)), which is a function of the SOC of \(p\)-th PEV over period \(T\) (\(SOC_{p,s}(T_{c,s}^{cyc})\)), nominal SOC of \(p\)-th PEV (\(\overline{SOC}\)), ambient temperature (\(\psi_{B}\)), nominal temperature (\(\psi_{0}\)), and charging time (\(t_{s}\)), is a vital part in battery degradation task. Also, \(a\) and \(b\) are calendar degradation's fitting parameter. The cycling degradation, which is a function of DOD over period \(T\) (\(DOD_{p,s}(T_{c,s}^{cyc})\)), has a significant effect on the total battery life. \(a\) and \(b\) in Equation (13) are fitting parameters for calendar degradation. \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are fitting parameters for cycling degradation related to DOD. In Equation (15), the benefits of PEV aggregator has been introduced. \[ \begin{array}{l}PEVs\text{ }benefit=\sum\limits_{s=1}^{10}\sum\limits_{t=1}^{24} \sum\limits_{p=1}^{n_{pev}}\big{[}OCa_{s}(t)\times Pr(s)\times[PEVa_{s}(t)|_{ m=UNC}-PEVa_{s}(t)|_{m=smart}]\big{]}+\\ \sum\limits_{s=1}^{10}\sum\limits_{t=1}^{24}\sum\limits_{p=1}^{n_{pev}}\big{[}OCr _{s}(t)\times Pr(s)\times[PEVr_{s}(t)|_{m=UNC}-PEVr_{s}(t)|_{m=smart}]\big{]} \end{array}  \tag{15}\]  
where the welfare is calculated as the difference between power consumed by charging and discharging of PEVs' batteries in two charging modes: uncoordinated and smart charging [21]. The power loss in the transmission line in the smart charging mode is different from the uncoordinated mode, and the difference between this two is considered as the network benefit. \[