This paper introduces an approach for the real-time extraction of the frequency, phase angle, and symmetrical components of the grid signal, which is of great importance for many applications in power systems such as power quality and protection. The proposed method is based on the concept of the adaptive notch filter that provides a fast and accurate estimation of the symmetrical components in the presence of frequency and amplitude variations. In addition, the system offers a high degree of immunity and insensitivity to power system disturbances, harmonics, and other types of pollutions that exist in the grid signal. The simplicity of the structure makes the method suitable for both software and hardware implementations. Moreover, this very simple and very powerful tool can be used as a synchronization technique, which further simplifies the control issues currently challenging the integration of distributed energy technologies into the electricity grid. Mathematical derivations are presented to describe the principles of operation, and experimental results confirm the validity of the analytical work.

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We present a time complexity analysis of the Opt-IA arti cial immune system (AIS). We rst highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential and ageing) by analysing them in isolation. Recent work has shown that ageing combined with local mutations can help escape local optima on a dynamic optimisation benchmark function. We generalise this result by rigorously proving that ageing leads to considerable speed-ups (compared to evolutionary algorithms (EAs)) on the standard C benchmark function both when using local and global mutations. Unless the stop at rst constructive mutation (FCM) mechanism is applied, we show that hypermutations require exponential expected runtime to optimise any function with a polynomial number of optima. If instead FCM is used, the expected runtime is at most a linear factor larger than the upper bound achieved for any random local search algorithm using the arti cial tness levels method. Nevertheless, we prove that algorithms using hypermutations can be considerably faster than EAs at escaping local optima. An analysis of the complete Opt-IA reveals that it is e cient on the previously considered functions and highlights problems where the use of the full algorithm is crucial.

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We present a time complexity analysis of the Opt-IA artificial immune system (AIS). We first highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential and ageing) by analysing them in isolation. Recent work has shown that ageing combined with local mutations can help escape local optima on a dynamic optimisation benchmark function. We generalise this result by rigorously proving that, compared to evolutionary algorithms (EAs), ageing leads to impressive speed-ups on the standard Cliff d benchmark function both when using local and global mutations. Unless the stop at first constructive mutation (FCM) mechanism is applied, we show that hypermutations require exponential expected runtime to optimise any function with a polynomial number of optima. If instead FCM is used, the expected runtime is at most a linear factor larger than the upper bound achieved for any random local search algorithm using the artificial fitness levels method. Nevertheless, we prove that algorithms using hypermutations can be considerably faster than EAs at escaping local optima. An analysis of the complete Opt-IA reveals that it is efficient on the previously considered functions and highlights problems where the use of the full algorithm is crucial. We complete the picture by presenting a class of functions for which Opt-IA fails with overwhelming probability while standard EAs are efficient.