As an effective emulator of ill-conditioned power flow

continuous Newton methods (CNMs) have been extensively investigated using explicit and implicit numerical integration algorithms. However

explicit CNMs often suffer from non-convergence due to their limited stability region

while implicit CNMs require additional iterative loops to solve nonlinear equations. To address this

we propose a semi-implicit version of CNM. We formulate the power flow equations as a set of differential algebraic equations (DAEs)

and solve the DAEs with the stiffly accurate Rosenbrock type method (SARM). The proposed method succeeds the numerical robustness from the implicit CNM framework while prevents the iterative solution of nonlinear systems

hence revealing higher convergence speed and computation efficiency. We develop a novel 4-stage

3rd-order hyper-stable SARM with an embedded 2nd-order formula for adaptive step size control. This design enhances convergence through damping adjustment. Case studies on ill-conditioned systems verify the alleged performance. An algorithm extension for MATPOWER is made available on Github for benchmarking.