A Weakly Conditioned Imputation of an Impedance-Branch Dissipation Power

In a sinusoidal steady-state circuit, how much of an impedance-branch dissipation power should be assigned to each source? The answer is the basis of electricity transmission pricing, such as the pricing of line loss and line security due to thermal limit. However, it is difficult to obtain a unique and common consent answer because of the nonlinearity of power expression. This paper presents two weakly conditioned imputation formulas, respectively for dissipation active- and reactive-power, to answer the question. First, a detailed form of Shapley value formula and three weak conditions are proposed. On this basis, the two imputation formulas are then derived and proved mathematically. The imputation formulas are explicitly in terms of source currents/voltages, thus analytic and easy to compute. They possess eight mathematical and economic features that lay a good foundation for electricity transmission pricing. The two imputation formulas are also extended to actual power systems where trades are conducted in MWh/MVARh instead of amperes/volts. Numerical examples illustrate their computing process and features.