A method for approximating an optimal power flow of a smart electric power grid includes providing a cost function that models a smart electric power grid having buses connected by branches, deriving a set of linear equations that minimize the cost function subject to constraints from an expression of an extremum of the cost function with respect to all arguments, reducing a dimension of the linear equations by solving for a subset of the linear equations, re-organizing the reduced dimension linear equations into primal and dual parts, and decomposing the re-organized reduced dimensional linear equations into two systems of block matrix equations which can be solved by a series of back substitutions. A solution of the two systems of block matrix equations yields conditions for a lowest cost per kilowatthour delivered through the smart electric power grid.