The transition to a sustainable and low-carbon energy landscape has created extensive interest in green hydrogen as a sustainable and environmentally friendly energy carrier. This paper presents a comprehensive review of green hydrogen applications and optimization strategies employed in these applications within the power sector. We categorize the hydrogen system into two groups: grid-dependent green hydrogen systems (GDGHS) and grid-independent green hydrogen systems (GIGHS), recognizing their distinct operational characteristics and challenges. The paper also assesses a range of optimization methods and models employed to maximize the efficiency and economic viability of green hydrogen production, storage, and utilization within GDGHS and GIGHS. Furthermore, we discuss the key challenges and opportunities in deploying green hydrogen solutions within the power sector. We highlight emerging trends and potential future developments. This review aims to provide a holistic understanding of green hydrogen's transformative potential in the power sector. It also sheds light on cutting-edge optimization methodologies that can unlock its full potential in a sustainable energy future.