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In this paper, the concepts related to loss partitioning among the phase currents in three-phase distribution systems are revisited in the light of new findings identified by the authors. In particular, the presence of a paradox in the classical loss partitioning approach, based on the use of the phase-by-phase difference between the input and output complex power, is highlighted. The conditions for performing effective loss partitioning without the occurrence of the paradox are thus established. The corresponding results are then used to extend the branch current decomposition loss allocation method for enabling its application to three-phase unbalanced distribution systems with distributed generation. Several numerical examples on a three-phase line with grounded neutral and on the modified IEEE 13-node test system are provided to assist the illustration and discussion of the novel conceptual framework. Index Terms--Distributed generation, distribution systems, loss allocation, loss partitioning, losses, paradox, unbalanced systems.

The allocation of the system losses to suppliers and consumers is a challenging issue for the restructured electricity business. Meaningful loss allocation techniques have to be adopted to set up appropriate economic penalties or rewards. The allocation factors should depend on size, location, and time evolution of the resources connected to the system. In the presence of distributed generation, the variety of the power flows in distribution systems calls for adopting mechanisms able to discriminate among the contributions that increase or reduce the total losses. Some loss allocation techniques already developed in the literature have shown consistent behavior. However, their application requires computing a set of additional quantities with respect to those provided by the distribution system power flow solved with the backward/forward sweep approach. This paper presents a new circuit-based loss allocation technique, based on the decomposition of the branch currents, specifically developed for radial distribution systems with distributed generation. The proposed technique is simple and effective and is only based on the information provided by the network data and by the power flow solution. Examples of application are shown to confirm its effectiveness. Index Terms--Branch current decomposition, distributed generation (DG), distribution systems, losses, loss allocation.

In reliability analysis of distribution systems, random events like the occurrence of a fault or the time to restore the service after a fault are represented by using random variables (RVs), so that the reliability indices built on the basis of these RVs also become RVs. Existing techniques for the evaluation of the probability distributions of reliability indices are typically based on Monte Carlo and analytical simulations. This paper presents a new method for computing the probability distribution of reliability indices. The random sums introduced by the randomness of the number of fault occurrences in the time interval of analysis are handled by using a characteristic functions-based approach. The direct convolution of the probability density functions is avoided by resorting to the properties of the compound Poisson process. In addition, the direct and inverse discrete Fourier transforms are used to allow for handling any type of probability distribution. The proposed method is an effective alternative to the existing methods, providing a fast and simple computation of probability distributions and moments for local and global reliability indices. Results obtained for large real urban distribution systems are presented. Index Terms--Characteristic functions, compound Poisson process, continuity of supply, energy not supplied, interruptions, probability distributions, reliability indices, service restoration times.