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Good morning to you from India Ms Shruti Rao and many thanks for your answer, an exhaustive one. Your answer motivated me to study your paper once again, for, my doubts are now clear. First thing I should do is to expand the complex node voltage  in power series.

Thanks again,

G Kesava Rao 

Shruti provided very good answers.  Let me add my 2 cents.  The Holomorphic Embedding method gets typically confused with homotopy and similar continuation methods, so let me clarify the key differences.

Yes, you could easily devise a homotopy that replicates the same effect as a given holomorphic embedding scheme, but you would be missing the point completely. Homotopy only exploits the implicit function theorem, continuity, and first derivatives, all of this in the framework of real function analysis.  By contrast, HELM exploits holomorphicity of complex functions, and more specifically, algebraic-geometric aspects of plane algebraic curves.  In a way, you could say that Homotopy only exploits the first derivative, while HELM exploits the infinite derivatives (if this view helps).  It is thanks to these techniques, grounded in Complex Analysis, that the method is able to turn the nonlinear powerflow problem into a sequence of linear problems, thus providing a constructive method to calculate the solution.

And yes, Newton-Raphson or Fast-Decoupled iterative methods (or even Homotopy methods) could be faster, but all of these methods are prone to convergence failure, as everybody knows. For some use-cases (example: automated loadflows, as part of a larger algorithm) this unpredictable behavior is unacceptable, and this is where HELM shines.

Thanks Mr. Marin for the answer.If there is a bifurcation point in the physical system, the multiple solutions around that critical point may oscillate and give no feasible solution. In such case holomorphic method is the only way.  In most engineering problems, the initial solution point may not be a problem, the previous state can be taken as initial solution for the current state. Unique solution and non-divergence are the special advantages. It is non-iterative alright, but I am afraid that the pre-processing of the nodal equations may take time and therefore for problems in normal region of operation, holomorphics may not be any better than Newton's method from computational point of view. Thanks again.

Yes, it is true that HELM is computationally slower than NR or Fast Decoupled methods, mainly due to the overhead involved in two things: (1) calculating the successive "right-hand-sides" (which involve convolutions among power series coefficients); and (2) the calculation of Padé approximants.  Note, however, that these two computational tasks are just O(n) in network size, since they are performed independently by bus.  Although the work on each bus grows roughly as O(M^2), M being the max order of the power series, this value M is capped at around 60 of less.  Long story short, HELM is asymptotically as performant as a Fast Decoupled method, since it only needs to factor the admittance matrix once.

Have a look at https://doi.org/10.1109/TCSI.2015.2512723 if you want to see some benchmarks.  But bear in mind that MATLAB is an unfair environment for HELM, because the two items cited above have to be implemented in slow interpreted code, while the linear system solving takes place in ultra-efficient compiled code.

But the main point of HELM is not performance; it is reliability and the ability to run completely unattended (i.e. you don't need to choose an initial seed, and you don't need to inspect the results to see if it diverged or converged to a bad solution).  Think about its usage in applications for automated and autonomous control of (large) power systems.

Many thanks Mr Marin for your clarifications on my doubts. I will see the site you recommended. 

Thanks again.

 G Kesava Rao