In solution of non-linear algebraic equations, now researchers suggest Holomorphic Embedding solutions. These are supposed to give a non-iterative unique solution. And reliable at the same time.
Homotopy methods using a real number for embedding (say Lamda) are also suggested. It would be interesting to me to see a simple tutorial on the issue. Let us take say, three non-linear equations in x1, x2 and x3. and solve by both the methods so we can know how each one id different from the other. Thanks to research gate..
The Holomorphic Embedding Load-flow Method is a solution method for the power flow equations of electrical power systems. Its main features are that it is direct (that is, non-iterative) and that it mathematically guarantees a consistent selection of the correct operative branch of the multivalued problem, also signalling the condition of voltage collapse when there is no solution. These properties are relevant not only for the reliability of existing off-line and real-time applications, but also because they enable new types of analytical tools that would be impossible to build with existing iterative load flows (due to their convergence problems).
traditional load-flow algorithms were developed based on three iterative techniques: the Gauss-Seidel method, which has poor convergence properties but very little memory requirements and is straightforward to implement; the full Newton-Raphson method, which has fast (quadratic) iterative convergence properties, but it is computationally costly; and the Fast Decoupled Load-Flow (FDLF) method, which is based on Newton-Raphson, but greatly reduces its computational cost by means of a decoupling approximation that is valid in most transmission networks.There are two fundamental problems with all iterative schemes of this type. On the one hand, there is no guarantee that the iteration will always converge to a solution; on the other, since the system has multiple solutions,[note 2] it is not possible to control which solution will be selected. As the power system approaches the point of voltage collapse, spurious solutions get closer to the correct one, and the iterative scheme may be easily attracted to one of them because of the phenomenon of Newton fractals: when the Newton method is applied to complex functions, the basins of attraction for the various solutions show fractal behaviour.
Right, Mr Ravi.Homotopy methods use a real valued embedding parameter on a convex trajectory and traverse from a easy solution to the real and more difficult solution.
The other similar method is holomorphic embedding with a complex parameter, also used foe loadflow studies.
My doubt is: Is the homotopy method a special case of holomorphic method. What special advantage, if any, does the holomorphic method has over homotopy method?
G K Rao
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