Indices and Sets |

F | Set of phases {A, B, C}. |

L | Set of circuits. |

N | Set of nodes. |

V | Set of vehicles. |

n, m | Indices of buses. |

f | Indices of phases. |

t | Indices of time. |

v | Indices of vehicles. |

Parameters |

${\mathsf{\alpha}}_{\mathrm{t}}^{\mathrm{dg}}$ | Price of energy from DG units. |

${\mathsf{\alpha}}_{\mathrm{t}}^{\mathrm{Up}}$ | Price of up reserve supplied by aggregator. |

${\mathsf{\alpha}}_{\mathrm{t}}^{\mathrm{Dn}}$ | Price of down reserve supplied by aggregator |

${\mathsf{\alpha}}_{\mathrm{t}}^{\mathrm{pv}}$ | Price of energy from PV units. |

${\mathsf{\alpha}}_{\mathrm{t}}^{\mathrm{Cfix}}$ | Price for the charging control. |

${\mathsf{\gamma}}_{\mathrm{t}}^{\mathrm{cp}}$ | Cost of the charging power. |

${\mathrm{s}}_{\mathrm{v},\mathrm{t}}$ | Binary parameter corresponding to the state of EV v, 1 when the EV v is on trip and 0 otherwise. |

${\mathsf{\zeta}}_{\mathrm{v},\mathrm{t}}$ | Binary parameter corresponding to the availability for charging of EV v, 1 when the EV v can charge the battery and 0 otherwise. |

${\Delta}_{\mathrm{t}}$ | Duration of the time interval t. |

${\overline{\mathsf{\delta}}}_{\mathrm{mn}}$ | Discretization step for the current of circuit mn. |

$\overline{\mathsf{\lambda}}$ | Number of blocks of the square current piecewise linearization. |

$\mathsf{\theta}$ | Vector of reference phase angles. |

${\mathsf{\theta}}_{1}$ | Maximum negative deviation of phase angles. |

${\mathsf{\theta}}_{2}$ | Maximum positive deviation of phase angles. |

${\mathrm{B}}_{\mathrm{mn},\mathrm{f}}$ | Shunt susceptance of circuit mn for phase f. |

${\mathrm{X}}_{\mathrm{mn},\mathrm{f},\mathrm{h}}$ | Reactance of circuit mn between phases f and h. |

${\mathrm{R}}_{\mathrm{mn},\mathrm{f},\mathrm{h}}$ | Resistance of circuit mn between phases f and h. |

${\overline{\mathrm{I}}}_{\mathrm{mn}}$ | Upper current limit of circuit mn. |

$\overline{\mathrm{V}}/\underset{\_}{\mathrm{V}}$ | Upper and lower voltage limits. |

${\mathrm{P}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{D}}$ | Active power demand at node n for phase f in time interval t. |

${\mathrm{Q}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{D}}$ | Reactive power demand at node n for phase f in time interval t. |

${\overline{\mathrm{P}}}_{\mathrm{n}}^{\mathrm{DG}}$ | Active power capacity of DG unit at node n. |

${\overline{\mathrm{Q}}}_{\mathrm{n}}^{\mathrm{DG}}/{\underset{\_}{\mathrm{Q}}}_{\mathrm{n}}^{\mathrm{DG}}$ | Upper/lower reactive power capacities of DG unit at node n. |

${\mathrm{pf}}_{\mathrm{n}}$ | Lower power factor for the operation of DG unit at node n |

${\mathsf{\pi}}_{\mathrm{d}}$ | Probability of deployment for up and down reserve. |

${\mathrm{E}}_{\mathrm{v}}^{\mathrm{SOC}}$ | Initial state of charge of EV v. |

${\overline{\mathrm{E}}}_{\mathrm{v}}^{\mathrm{EV}}$ | Energy capacity of EV v. |

${\underset{\_}{\mathrm{E}}}_{\mathrm{v}}^{\mathrm{EV}}$ | Lower energy limit of EV v. |

${\underset{\_}{\mathrm{E}}}_{\mathrm{v}}^{\mathrm{Trip}}$ | Energy for motion required by the owner of EV v. |

${\overline{\mathrm{P}}}_{\mathrm{v}}^{\mathrm{CH}}$ | Charger rated power of the EV v. |

${\mathrm{V}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{re}\ast}$ | Real part of the estimated voltage at node n for phase f in time interval t. |

${\mathrm{V}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{im}\ast}$ | Imaginary part of the estimated voltage at node n for phase f in time interval t. |

${\mathrm{V}}_{\mathrm{v},\mathrm{t}}^{\mathrm{re}\ast}/{\mathrm{V}}_{\mathrm{v},\mathrm{t}}^{\mathrm{im}\ast}$ | Real/imaginary part of the estimated voltage at the node and phase associated with EV v in time interval t. |

${\mathrm{f}}_{\mathrm{t}}^{\mathrm{pv}}$ | PV power factor available time interval t |

${\overline{\mathrm{P}}}_{\mathrm{n}}^{\mathrm{pv}}$ | Active power capacity of PV. |

${\overline{\mathrm{Q}}}_{\mathrm{n}}^{\mathrm{pv}}$ | Upper reactive power capacity of PV. |

${\underset{\_}{\mathrm{Q}}}_{\mathrm{n}}^{\mathrm{pv}}$ | Lower reactive power capacity of PV. |

${\varphi}^{\mathrm{pv}}$ | Minimum power factor for the operation of the PV |

Variables: |

${\mathrm{V}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{re}}$ | Real part of the voltage at node n for phase f in time interval t. |

${\mathrm{V}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{im}}$ | Imaginary part of the voltage at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{Dre}}$ | Real part of the current required by a load at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{Dim}}$ | Imaginary part of the current required by a conventional load at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{SEre}}$ | Real part of the current provided by substation at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{SEim}}$ | Imaginary part of the current provided by substation at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{re}}$ | Real part of the current in circuit mn for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{im}}$ | Imaginary part of the current in circuit mn for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{re}+}$ | Positive component of the current’s real part in circuit mn for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{re}-}$ | Negative component of the current’s real part in circuit mn for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{im}+}$ | Positive component of the current’s imaginary part in circuit mn for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{im}-}$ | Negative component of the current’s imaginary part in circuit mn for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{mn},\mathrm{f},\mathrm{t}}^{\mathrm{sqr}}$ | Square of the current in circuit mn for phase f in time interval t. |

${\mathsf{\delta}}_{\mathrm{mn},\mathrm{f},\mathrm{t},\mathsf{\lambda}}^{\mathrm{sqr}}$ | Value of the λth block of the piecewise linearization for the current of circuit mn for phase f in time interval t. |

${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{DG}}$ | Active power produced by DG units at node n in time interval t. |

${\mathrm{Q}}_{\mathrm{n},\mathrm{t}}^{\mathrm{DG}}$ | Reactive power produced by DG units at node n in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{DGre}}$ | Real part of the current provided by DG units at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{DGim}}$ | Imaginary part of the current provided by DG units at node n for phase f in time interval t. |

${\mathrm{P}}_{\mathrm{v},\mathrm{t}}^{\mathrm{EV}}$ | Active power consumption of EV v in time interval t. |

${\mathrm{I}}_{\mathrm{v},\mathrm{t}}^{\mathrm{EVre}}$ | Real part of the current required by the EV v in time interval t. |

${\mathrm{I}}_{\mathrm{v},\mathrm{t}}^{\mathrm{EVim}}$ | Imaginary part of the current required by the EV v in time interval t. |

${\mathrm{E}}_{\mathrm{v},\mathrm{t}}^{\mathrm{EV}}$ | Energy of EV v in each time interval t. |

${\mathrm{E}}_{\mathrm{v},\mathrm{t}}^{\mathrm{EVRdn}}$ | Down expected energy offered by EVs v in time interval t. |

${\mathrm{E}}_{\mathrm{v},\mathrm{t}}^{\mathrm{EVRup}}$ | Up expected energy offered by EV v in time interval t. |

${\mathrm{P}}_{\mathrm{t}}^{\mathrm{RUp}}$ | Up power reserve offered by the aggregator in the day-ahead reserve market. |

${\mathrm{P}}_{\mathrm{t}}^{\mathrm{RDn}}$ | Down power reserve offered by the aggregator in the day-ahead reserve market. |

${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{pv}}$ | Active power injected by PV units at node n in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{PVre}}$ | Real part of the current supplied by PV units at node n for phase f in time interval t. |

${\mathrm{I}}_{\mathrm{n},\mathrm{f},\mathrm{t}}^{\mathrm{PVim}}$ | Imaginary part of the current supplied by PV units at node n for phase f in time interval t. |

${\mathrm{Q}}_{\mathrm{n},\mathrm{t}}^{\mathrm{pv}}$ | Reactive power generated by PV units at node n in time interval t. |