We study the nonequilibrium thermodynamics of a heat engine operating between two finite-sized reservoirs with well-defined temperatures. Within the linear response regime, it is found that the uniform temperature of the two reservoirs at final time $\tau$ is bounded from below by the entropy production ${\sigma }_{\min }\propto 1/\tau$. We discover a general power-efficiency tradeoff depending on the ratio of heat capacities ($\gamma$) of the reservoirs for the engine, and a universal efficiency at maximum average power of the engine for arbitrary $\gamma$ is obtained. For practical purposes, the operation protocol of an ideal gas heat engine to achieve the optimal performance associated with ${\sigma }_{\min }$ is demonstrated. Our findings can be used to develop a general optimization scenario for thermodynamic cycles with finite-sized reservoirs in real-world circumstances.

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• Nonequilibrium & irreversible thermodynamics
• Nonequilibrium statistical mechanics
• Efficiency at maximum power
• Heat engines
• Linear response theory