  Theoretical treatments of periodically-driven quantum thermal machines
(PD-QTMs) are largely focused on the limit-cycle stage of operation
characterized by a periodic state of the system. Yet, this regime is not
immediately accessible for experimental verification. Here, we present a
general thermodynamic framework that can handle the performance of PD-QTMs both
before and during the limit-cycle stage of operation. It is achieved by
observing that periodicity may break down at the ensemble average level, even
in the limit-cycle phase. With this observation, and using conventional
thermodynamic expressions for work and heat, we find that a complete
description of the first law of thermodynamics for PD-QTMs requires a new
contribution, which vanishes only in the limit-cycle phase under rather weak
system-bath couplings. Significantly, this contribution is substantial at
strong couplings even at limit cycle, thus largely affecting the behavior of
the thermodynamic efficiency. We demonstrate our framework by simulating a
quantum Otto engine building upon a driven resonant level model. Our results
provide new insights towards a complete description of PD-QTMs, from turn-on to
the limit-cycle stage and, particularly, shed light on the development of
quantum thermodynamics at strong coupling.
