We analyse non-equilibrium Carnot-like cycles built with a colloidal particle in a harmonic trap, which is immersed in a fluid that acts as a heat bath. Our analysis is carried out in the overdamped regime. The cycle comprises four branches: two isothermal processes and two textit{locally} adiabatic ones. In the latter, both the temperature of the bath and the stiffness of the harmonic trap vary in time, but in such a way that the average heat vanishes for all times. All branches are swept at a finite rate and, therefore, the corresponding processes are irreversible, not quasi-static. Specifically, we are interested in optimising the heat engine to deliver the maximum power and characterising the corresponding values of the physical parameters. The efficiency at maximum power is shown to be very close to the Curzon-Ahlborn bound over the whole range of the ratio of temperatures of the two thermal baths, pointing to the near optimality of the proposed protocol.
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