This paper analyzes the performance of a real air heat pump plant via methods of finite-time thermodynamics. The analytical relations between heating load and pressure ratio, and between coefficient of performance (COP) and pressure ratio of real closed regenerated air Brayton heat pump cycles coupled to constant- and variable- temperature heat reservoirs are derived. In the analysis, the irreversibilities include heat transfer-irreversible losses in the hot- and cold-side of the heat exchangers and the regenerator, and the non-isentropic expansion and compression losses in the compressor and expander. The optimal performance characteristics of the cycle may be obtained by optimizing the distribution of heat conductances or heat transfer surface areas among the two heat exchangers and the regenerator, and the matching of the working fluid and the heat reservoirs. Detailed numerical examples are given to demonstrate the influence of various losses on the cycle performance characteristics.