Matrix balancing with unknown total costs: preserving economic relationships in the electric power sector

Many preferred matrix balancing methods suffer limitations when total costs (i.e. column sums) are unknown or uncertain. If the total cost (column) constraint is relaxed, economic relationships both between inputs to and/or amongst the sub-sectors may not be preserved (i.e. cost structure and row share, respectively). These relationships are significant in modeling, where production behavior depends on relative costs. This paper presents a share preserving cross-entropy (SPCE) approach which targets economic relationships directly and allows the column constraint to be relaxed. Further, the SPCE solution is identical to the RAS solution when the column constraint is imposed. This cross-entropy formulation complements an existing sum squared error-type approach. The two matrix balancing methods are demonstrated with a disaggregation of the electric power industry where only unit input costs are known with greater certainty. There is a clear trade-off between preserving economic relationships versus the column totals when compared to their column-constrained counterparts.