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This study investigates the detection and localisation of faults provoking short-duration voltage variations - sags (dips) and swells - in small power distribution networks. It aims to accomplish those tasks by capturing fault records, voltage and current waveforms, at just one point in the system, the substation. The main objective is to classify the fault type and locate the fault origin occurring in a given region of a power delivery network. For that purpose, fault inception is triggered by a sensitive phase-locked loop. Then, the captured signals are decomposed using damped sinusoids of arbitrary temporal support by means of an adaptive decomposition algorithm. Subsets of the parameters defining the damped sinusoids are used for classifying the fault type and indicating the fault location. The fault-type classification is obtained by using support vector machines, whereas the fault location is obtained by means of an artificial neural network. The simulation results for a simple but actual power distribution system with three possible places for fault occurrence are presented. The exact fault-type classifications were obtained while a correct localisation of 85- of the faults was accomplished.