“…where Losses is the objective function that minimizes losses; Pg ph,t is the vector of active power generation of each phase ph and scenario t with dimension (3.nb.np × 1); Pd ph,t is the vector of active power load of each phase ph and scenario t with dimension (3.nb.np × 1); Qg ph,t is the vector of reactive power generation of each phase ph and scenario t with dimension (3.nb.np × 1); Qd ph,t is the vector of reactive power load of each phase ph and scenario t with dimension (3.nb.np × 1); Pgd ph,t is the vector of active power generation of distributed generation of each phase ph and scenario t with dimension (3.nb.np × 1); Qgd ph,t is the vector of reactive power generation of distributed generation of each phase ph and scenario t with dimension (3.nb.np × 1); a ph,t is the ratio voltage magnitudes of voltage regulators of each phase ph and scenario t with dimension (3.nreg.np × 1); nreg is the number of voltage regulators; a min and a max are the minimum and maximum voltage magnitude ratio of the voltage regulators with dimension (3.nreg.np × 1); c ph,t is the capacitive susceptance of capacitor banks installed at nc buses with dimension (3.nb.np × 1); Pg min and Pg max are the minimum and maximum limits of the active generation with dimension (3.nb.np × 1); Qg min and Qg max are the minimum and maximum limits of the reactive generation with dimension (3.nb.np × 1); V min and V max are the minimum and maximum limits of voltage magnitude phasor with dimension (3.nb.np × 1); α is the vector of taps adjusted on the primary of DT with dimension (ndt × 1), where ndt is the number of distributer transformers; and α min and α max are the minimum and maximum tap position of the DTs with dimension (ndt × 1). The functions P ph,t x, a ph,t , α .x and Q ph,t x, a ph,t , α .x, which represent the active and reactive power injections of each bus, respectively, are quadratic equations due to the rectangular representation of the voltage phasor [31]. To illustrate this, the vector Pd ph,t (active power load) and the vector Qd ph,t (reactive power load) have the following layout: where Pd k,t i represents the active power load at bus i, phase k, and scenario t, and Qd k,t i represents the reactive power load at bus i, phase k, and scenario t.…”