We consider single cell multi-user OFDMA downlink resource allocation on a flat-fading channel such that average supply power is minimized while fulfilling a set of target rates. Available degrees of freedom are transmission power and duration. This paper extends our previous work on power optimal resource allocation in the mobile downlink by detailing the optimal power control strategy investigation and extracting fundamental characteristics of power optimal operation in cellular downlink. We find that only a system wide allocation of transmit powers is optimal rather than on link level. The allocation strategy that minimizes overall power consumption requires the transmission power on all links to be increased if only one link degrades. Furthermore, we show that for mobile stations with equal channels but different rate requirements, it is power optimal to assign equal transmit powers with proportional transmit durations. To relate the effectiveness of power control to live operation, we take the power model into consideration which maps transmit power to supply power. We show that due to the affine mapping, the solution is independent of the power model. However, the effectiveness of power control measures is completely dependent on the underlying hardware and the load dependence factor of a base station (instead of absolute consumption values). Finally, we conclude that power control measures in base stations are most relevant in macro stations which have load dependence factor of more than 50%.