Frequency control is vital for isolated microgrids. As a common regulation approach

controllable load is often utilized for microgrid frequency control. The optimal frequency control problems of isolated microgrids with individual regulation capacity constraints are addressed. Moreover

a fully distributed algorithm is derived

which guarantees that regulation capacity constraints can be satisfied either at the equilibrium point or during the entire transient process. In addition

the convergence and stability of the algorithm as well as the optimality of results are theoretically proved. Since communications are required only between one area and its neighboring areas

it can provide a feasible way to realize‘plug and play' operation of microgrids. Numerical experiments on a five-microgrid system demonstrate the effectiveness of the proposed method. This work is supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China(No. 51321005)and the Research Program of Tsinghua University(No. 20151080416).