In this paper, the power density, defined as the ratio of power output to the maximum specific volume in the cycle, is taken as the objective for performance analysis and optimization of an irreversible regenerated closed Brayton cycle coupled to variable-temperature heat reservoirs from the viewpoint of finite time thermodynamics (FTT) or entropy generation minimization (EGM). The analytical formulae about the relations between power density and pressure ratio are derived with the heat resistance losses in the hot- and cold-side heat exchangers and the regenerator, the irreversible compression and expansion losses in the compressor and turbine, the pressure drop losses at the heater, cooler and regenerator as well as in the piping, and the effect of the finite thermal capacity rate of the heat reservoirs. The obtained results are compared with those results obtained by using the maximum power criterion, and the advantages and disadvantages of maximum power density design are analysed. The maximum power density optimization is performed in two stages. The first is to search the optimum heat conductance distribution corresponding to the optimum power density among the hot- and cold-side heat exchangers and the regenerator for a fixed total heat exchanger inventory. The second is to search the optimum thermal capacitance rate matching corresponding to the optimum power density between the working fluid and the high-temperature heat source for a fixed ratio of the thermal capacitance rates of two heat reservoirs. The influences of some design parameters, including the effectiveness of the regenerator, the inlet temperature ratio of the heat reservoirs, the effectiveness of the heat exchangers between the working fluid and the heat reservoirs, the efficiencies of the compressor and the turbine, and the pressure recovery coefficient, on the optimum heat conductance distribution, the optimum thermal capacitance rate matching, and the maximum power density are provided by numerical examples. The power plant design with optimization leads to a smaller size including the compressor, turbine, and the hot- and cold-side heat exchangers and the regenerator. When the heat transfers between the working fluid and the heat reservoirs are carried out ideally, the pressure drop loss may be neglected, and the thermal capacity rates of the heat reservoirs are infinite, the results of this paper then replicate those obtained in recent literature.