Fossil fuel consumption remains a significant contributor to global greenhouse gas emissions, accounting for approximately 90% of the overall emissions, with fossil fuel power generation systems identified as a major source of CO2 emissions. Given the ongoing industrial activities and increasing energy demand, completely discontinuing the use of nonrenewable resources for power production is not feasible in the near future. CO2 capture and storage (CCS) technologies offer a promising option for continuing to utilize fossil fuels in a cleaner and more sustainable manner. The wide deployment of carbon capture technologies alone has the potential to decrease power plant emissions by as much as 90%. However, the current CCS technologies face several challenges for broad implementation, specifically significant energy requirements, high capital cost, and flexible operation. The current CCS technologies lead to a decrease in the net power output of the plant by approximately 25-40% and result in a substantial increase in power generation costs, potentially up to 70%. Another challenge is the requirement for flexible operation of CCS, as with the increasing penetration of renewable energy sources in the power grid, fossil fuel-fired power plants need to operate in a load-following manner to ease the integration of intermittent renewable sources. Consequently, significant fluctuations in the power plant flue gas necessitate flexible operation of the downstream carbon capture system to adapt to these changes. The above challenges necessitate implementing innovative solutions in the operation and design of carbon capture systems to reduce the energy penalty and cost of CO2 capturing and improve the flexible operation of CCS to accommodate both base-load and load-following operating of the power plants. Membrane systems offer promising advantages for separating CO2 from other components of power plant flue gas, although the process encounters several technical and economic challenges. These challenges must be addressed to optimize their design and integration with fossil-fueled power plants and enhance the feasibility of this environmentally-friendly technology for extensive adoption. This dissertation is focused on the development of flexible and efficient membrane-based carbon capture technologies for large-scale implementation and integration with both base-load and load-following fossil-fueled power plants under high renewable energy integration. This dissertation aims to address and provide insights into the current challenges by employing advanced modeling, simulation, and optimization techniques. An efficient and flexible multi-stage membrane-based CCS process is developed and optimized to address the challenge of energy requirements and cost penalties of the system. In this context, a comprehensive techno-economic model for the possible designs and operating strategies of the membrane separation process is developed in order to investigate the potential and viability of the membrane-based CCS system. Furthermore, the optimal process design and the possible trade-offs between performance indicators of the membrane-based CCS are presented with the aim of reducing energy and cost penalties. Finally, the transient behavior of the membrane-based process is further investigated at different disturbances and variations in the power plant operation imposed by the plant load-following behavior to address the required flexibility of the carbon capture system. The results substantiate that the proposed system could be an optimal and flexible option for the decarbonization of power plants operating in a load-following manner. The best possible trade-offs between objective functions show that the CO2 capture cost and energy penalty of the process could be as low as 13.1 $/tCO2 and 10% at optimal design and operating conditions. Also, the results show that the response of the membrane module to step-change in feed flowrate conditions is much faster than the conventional CO2 capture process, making this technology promising for flexible integration. This study provides valuable insight into membrane separation and can be used by decision-makers for the sustainable development of fossil-fueled power plants. Moreover, to address CCS integration challenges for the power plant operating at base-load mode, a hybrid solar-assisted membrane-amine CCS equipped with thermal energy storage is developed in order to improve the power plant operation and reduce the low-carbon electricity cost and the energy penalty associated with the CO2 capturing. In this regard, the conventional amine-based CCS is hybridized with a multi-stage membrane process for selective CO2 recirculation to improve the separation driving force and decrease both CCS equipment size and energy penalty. Furthermore, the solar energy field with 4-hour thermal energy storage is integrated with the developed CCS to provide the required thermal energy and flexibility for capturing 90% of released CO2. The results proved that the specific reboiler duty and total packing volume in the case of the proposed design could be significantly reduced by 4.3-6.9% and 39-44%, respectively, compared to the baseline case. Also, due to the change in the inlet air properties and integration of the solar field to the CO2 capture plant, the output power of the system in the proposed designs can be increased by 13.8-19.4% in comparison to the conventional case. Moreover, the results revealed that the developed design represents the lowest levelized cost of electricity and CO2 avoided cost, 81.43 $/MWh and 101.66 $/tonneCO2, among the other CCS-equipped power plant. The proposed designs and system investigation conducted in this dissertation and for addressing the technology challenges of the CO2 capture process hold considerable promise in facilitating the ideal reduction of carbon emissions from fossil-fueled power plants and promoting sustainability within the power sector. These advancements and developments, along with appropriate governmental policies and incentive programs, can potentially enhance the economic viability and desirability of CO2 capture systems, making them increasingly favorable for widespread implementation.