  In a grid constrained transactive distribution system market, distribution
locational marginal pricing DLMP is influenced by the distance from the
substation to an energy user, thereby causing households that are further away
from the substation to be charged more. The Jain index of fairness, which has
been recently applied to alleviate this undesirable effect of inefficient
energy allocations, is used in this research to quantify fairness. It is shown
that the Jain index is strictly quasi-concave. A bilevel distributed mechanism
is proposed, where at the lower level, auction mechanisms are invoked
simultaneously at each aggregator to obtain energy costs under market
equilibrium conditions. A constrained multi gradient ascent algorithm,
Augmented Lagrangian Multigradient Approach ALMA, is proposed for
implementation at the upper level to attain energy allocations that represent
tradeoffs between efficiency and fairness. Theoretical issues pertaining to
ALMA as a generic algorithm for constrained vector optimization are considered.
It is shown that when the objectives are restricted to be strictly quasi
concave functions and if the feasible region is convex, ALMA converges towards
global Pareto optimality. The overall effectiveness of the proposed approach is
confirmed through a set of MATLAB simulations implemented on a modified IEEE
37-bus system platform.
