This paper proposes an optimization framework to address the component sizing and energy management problems in an electric-hydrogen hybrid energy storage system connected to a wind turbine. The total cost of the hybrid system is minimized using a particle swarm optimization (PSO) algorithm. In particular, four decision variables are optimized: the electrolyzer (EL) size, the supercapacitor (SC) size, and two parameters in the energy management strategy (EMS). To determine the power split factor for the wind power, the EMS introduces an artificial potential field (APF) and defines a virtual force based on the SC state of charge (SOC). Two APF parameters are optimized to tune the power allocation between the EL and the SC: the shaping parameter of the virtual force and the basis parameter of the power split factor. Since the cutoff frequency of the low pass filter (LPF) in the EMS is adaptively updated based on the optimized APF parameters, the proposed framework is referred to as the "OP-APF"framework. The effectiveness of the OP-APF framework is validated by performing MATLAB and real-time simulations. Compared to three baseline frameworks, OP-APF is more effective in reducing the system total cost, controlling the SC SOC, and alleviating the EL degradation.