Solid oxide fuel cell combined heat and power (CHP) is a promising technology to serve electricity and heat demands. In order to analyse the potential of the technology, a detailed techno-economic energy-cost minimisation model of a micro-CHP system is developed drawing on steady-state and dynamic SOFC stack models and power converter design. This model is applied it to identify minimum costs and optimum stack capacities under various current density change constraints. Firstly, a characterisation of the system electrical efficiency is developed through the combination of stack efficiency profiles and power converter efficiency profiles. Optimisation model constraints are then developed, including a limitation in the change of current density (Acm−2) per minute in the stack. The optimisation model is then presented and further expanded to account for the inability of a stack to respond instantaneously to load changes, resulting in a penalty function being applied to the objective function proportional to the size of load changes being serviced by the stack. Finally, the optimisation model is applied to examine the relative importance, in terms of minimum cost and optimum stack maximum electrical power output capacity, of the limitation on rate of current density change for a UK residential micro-CHP application. It is found that constraints on the rate of change in current density are not an important design parameter from an economic perspective.