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This article addresses the development of the energy compensation method used for the design of hybrid energy storage systems—HBESS. The combination of two battery technologies offers better cost and performance when considering microgrid systems to provide uninterrupted power to sensitive loads (substation auxiliary system) and also provides greater energy security. In the event of a failure, the load needs to continue operating, and batteries such as lithium ions have a fast response, but are expensive for large-scale systems. However, some technologies offer low-cost and good availability of energy for long hours of discharge, such as lead–acid batteries. Consequently, different battery technologies can be used to meet all the needs of the sensitive loads. A specific method for sizing a HBESS was developed for islanded microgrids to support sensitive loads. This method was developed to meet the demand for substations outside the Brazilian standard of power systems that lack an uninterrupted and reliable energy source. The method is validated by designing a microgrid to support the auxiliary systems of a transmission substation in northeastern Brazil. The results showed a system with a capacity of 1215 kWh of lead-carbon and 242 kWh of lithium ions is necessary to maintain an islanded microgrid for at least 10 h. Furthermore, the microgrid comprises a PV plant with an AC output power of 700 kW in connected operation and 100 kW when islanded from the grid.

Using the notion of symplectic structure and Weyl product of non-commutative geometry, unitary representations for the Galilei group are construct, and the Schrödinger equation in phase space is derived. An approach for perturbation theory in phase space is derived. The phase space amplitude and the Wigner function are calculated for quartic potential in phase space.