This paper presents a hybrid renewable energy system (RES) including wind and photovoltaic (PV) power sources. The wind energy subsystem (WES) consists of a squirrel-cage induction generator (SCIG) driven by a variable-speed wind turbine (WT) and corresponding power electronic converter, by means of which a speed-sensorless indirect-rotor-field-oriented control of the SCIG is implemented. The outputs of both the WES and PV power source rated 1.5 kW and 3.5 kW, respectively, are connected to the DC bus, with the quasi-Z-source inverter (qZSI) acting as an interlinking converter between the DC bus and the AC grid/load. An advanced pulse-width-modulation scheme is applied to reduce the qZSI switching losses. The considered RES can operate both in grid-tie and island operation, whereas the battery storage system—integrated within the qZSI impedance network—enables more efficient energy management. The proposed control scheme includes successively executed algorithms for the optimization of the WES and PV power outputs under varying atmospheric conditions. A perturb-and-observe PV optimization algorithm is executed first due to the significantly faster dynamics and higher-rated power of the PV source compared to the WES. The WES optimization algorithm includes two distinct fuzzy logic optimizations: one for extraction of the maximum wind power and the other for minimization of the SCIG losses. To reduce the number of the required sensors, all three MPPT algorithms utilize the same input variable—the qZSI’s input power—thus increasing the system’s reliability and reducing the cost of implementation. The performance of the proposed hybrid RES was experimentally evaluated over wide ranges of simulated atmospheric conditions in both the island and grid-tie operation.