For a large electricity grid comprising many units (plants) of various types, such as coal, oil, nuclear, hydro, etc., with known unreliabilities (outage rates) we study the optimal (i.e. the cheapest) total capacity, or numbers, of each type of unit. Existing treatments of the problem involve numerical methods and approximations of unknown accuracy. For a range of cases, we find explicit solutions. This extends the known explicit solutions, which are confined to completely reliable units. The cases we analyse are (I) a demand (load) which has a shifted Rayleigh distribution—a good approximation to the real load-duration curve—with some restriction on reliability (big units are more reliable) and (II) an exponential load distribution—which is unrealistic—with no restrictions on reliability. In both cases, the solutions reduce to transformed versions of the exact solutions for totally reliable units and, like the latter, can be exhibited by means of a cost polygon.
