Static analysis and dynamic simulation play vital roles in the planning

operation

and stability assessment of power systems. Traditional iterative power flow algorithms suffer from issues including dependence on initial values

singular matrices

and potential convergence to non-operational solutions. To address these challenges

the holomorphic embedding method (HEM)

based on complex analysis theory

has been proposed. This method transforms power flow equations into complex equations with respect to a complex variable

enabling a recursive

non-iterative power flow solution that is independent from initial values. As a result

it has attracted growing attention from both academia and industry worldwide. In addition

the HEM can also be applied to solve differential-algebraic equation systems

supporting fast electromechanical transient simulation of power systems. To this end

this paper reviews the theoretical foundations and recent advances of HEM in power systems

focusing on both static analysis and dynamic simulation. First

the basic principles and analytic continuation techniques of HEM are briefly introduced

along with a summary of the mainstream holomorphic embedding formulations. Then

the latest research progress is systematically reviewed across various application scenarios

including power flow analysis

voltage stability margin assessment

probabilistic power flow analysis

optimal power flow

and dynamic simulation. Finally

the key challenges related to convergence and computational efficiency are discussed

and future research directions are proposed in light of emerging requirements of new-type power systems.