Recently, renewable wind-turbine and photovoltaic energy sources have become more popular all over the world as the main vectors of energy transition. However, their intermittent nature along with loading variations can adversely affect the grid stability in terms of active and reactive powers as well as voltage fluctuations, which have to be looked at carefully at the point of common coupling (PCC) with the network grid. This research introduces a novel approach to enhance the stability of grid-connected renewable energy sources (RESs) by employing a hybrid energy storage system (HESS) with optimal sizing and control parameters. This HESS combines the merits of energy-based dry-gravity energy storage (GES) and power-based supercapacitor energy storage (SCES), optimized using an innovative Artificial Protozoa Optimizer (APO). The HESS effectively regulates active and reactive power, as well as voltage at the PCC, contributing to the overall stability of the grid. This can be achieved by decreasing a multi-objective function (MOF) constructed by a weighted sum of active power and voltage oscillations at the PCC. The weight factors obtained by methods reported in the literature are compared with those predicted by the proposed method. To conduct a comprehensive techno-economic study, the HESS is compared with standalone GES, SCES, and superconducting magnetic energy storage (SMES). Furthermore, the performance of the proposed APO is evaluated against particle swarm optimization (PSO) and genetic algorithm (GA). Simulation results demonstrate the effectiveness of the proposed optimized HESS in decreasing power fluctuations under various case studies, including transient changes in wind speed, and/or solar irradiance, and/or load. The HESS significantly reduces the overshoot of active and reactive power peaks in the investigated four scenarios with a reduction in the ranges of 79.4 % to 99.4 % and 98.2 % to 99.5 %, respectively. The voltage profile at the point of common coupling (PCC) is also improved, reaching values within a range of 97.4 % to 100 %. Notably, the proposed HESS offers a cost-effective solution compared to standalone GES and SMES. The capital cost of the HESS is $1880, compared to $1667 for standalone GES and $8333 for SMES.