Power grid expansion planning is a challenging problem that typically considers the facility location, sizing, and transmission line upgrade aspects, with the objective of minimizing the upgrade and operational costs. The consideration of fairness and equity between the populations being served by the power grid has not been addressed previously in the literature. These issues are of special interest regarding the power grid infrastructure in developing countries, where certain populations might be ``last in line'' to be connected to the grid. In this paper, we develop a power grid expansion optimization model that considers both effectiveness and equity, given a budget constraint on upgrade expenditure. Effectiveness is measured by the deprivation costs of all populations served by the power grid, while equity is measured by their Gini mean absolute difference. Node upgrade rules are applied, and the upgrade plan is provided over a given planning horizon. We optimally solved our model for small instances and performed sensitivity analysis on its parameters. We then developed an LNS (Large Neighborhood Search) heuristic for solving large instances and using publicly available data. Additional instances are generated based on the Myanmar power grid. Our analysis shows that the LNS can provide good solutions relative to a greedy approach. The approach taken in this paper can be applied to a wide range of infrastructure planning problems in which both effectiveness and equity should be considered.