When microgrids are in islanded operation

there may be power imbalance at any time due to real-time fluctuations of both loads and renewable energy sources. The amount of power imbalance is a primary frequency regulation function of the frequency deviation and the system frequency regulation coefficient. Therefore

the traditional real-time active power balance equation constraint is extended to a real-time balancing equation constraint containing the primary frequency regulation function

and a coordinated optimization model of islanded microgrids with frequency deviation constraints is established with the objective of minimizing the operation cost and maximizing the renewable energy accommodation. It is found that the traditional real-time active power balance can be inferred as a special cointegration relationship with zero residuals in its time series; therefore

the extended real-time balance equation constraint is regarded as a potential cointegration relationship

and the power imbalance can be regarded as a non-zero residual series. Then the source-load-storage cointegration relationship is established by setting a smoothness constraint on the residual series

and a source-load-storage cointegration optimization model is established for the islanded microgrids with frequency deviation constraints. The simulation results show that both the coordinated and cointegration optimization models with frequency deviation expand the feasible solution range

and have higher economics; the cointegration model enhances frequency security during the operation of isolated microgrids compared with the coordinated model by the constraints of the source-load-storage cointegration relationship. Both models demonstrate that the power fluctuation results in frequency deviation

and reveal a new power balance in microgrids. Both models take the frequency deviation as the available frequency regulation resource

which expands their feasible solutions. The cointegration model is more suitable for operation and optimization modeling of the new-type power system.