Utilities apply an additional fee for medium and large customers with low power factors. However, unfair financial charges may occur in installations subjected to voltage unbalance and harmonic distortion. The objective of this paper is to determine the fairest PF definitions and their corresponding measurement algorithms in the case in which a constant impedance load or an induction motor is supplied with unbalanced and nonsinusoidal voltages. Fairness is defined considering that the meter (built based on a particular definition and measurement method) under nonideal supply should lead to very close values as if it was submitted to an ideal balanced sinusoidal supply. We performed computational simulations to emulate several conditions in which a balanced customer (modeled as a constant impedance load or an induction motor) is charged due to a voltage supply no longer balanced and sinusoidal. We also performed experimental tests with an induction motor subjected to a wide range of unbalanced nonsinusoidal supply conditions to ratify the conclusions drawn from the simulations. Based on the simulation results and the experimental tests, we indicate some power factor definitions and measurement methods that are not significantly affected by voltage unbalance and harmonic distortions. These indicated PF definitions provide the fairest billing for conditions with unbalanced nonsinusoidal voltages.

Nowadays, reactive power and power factor in unbalanced circuits and/or with the presence of harmonics are still being investigated by several researchers. In this context, quaternions, whose use has recently been intensified in the electrical engineering field, is presented as an alternative tool to analyze electrical quantities in single and three-phase circuits. However, studies solving three-phase circuits by quaternions are not yet in literature, nor a complete electrical description with this tool. This article, therefore, presents single and balanced three-phase quantities expressed as quaternions. Results of an analysis of a series RLC single and balanced three-phase circuits are presented. It is important to notice that this tool can also be employed to represent three-phase power under unbalanced situations and with the presence of harmonics.

Electrical grid technical losses affects the electricity tariff and the planning of new lines. Nonetheless, their assessment by means of time-series or real-time simulations is computationally burdensome for large grids. In order to achieve simplified equivalent circuits, reduction techniques have been developed. The only one that is applicable for unbalanced grids is the multiphase reduction, but it does not preserve the losses. This paper proposes an algorithm that allows indirect evaluation of total losses, thus extending the multiphase reduction technique. In the validation process, IEEE 34-bus and a Brazilian real feeder are evaluated. A daily time-series simulation, with hourly loadshapes, is performed in OpenDSS. Using this novel method, the losses errors obtained are lower than 2%. The proposed algorithm, therefore, may be applied to speed up the definition of electricity tariff and the planning process of unbalanced grids.

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