Holomorphic Embedding Load-flow method: Difference between revisions

Revision as of 10:16, 1 October 2012

Revision as of 11:54, 1 October 2012

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The method uses an embedding technique by means of a complex parameter {{math|<var>s</var>}}.

The first key ingredient in the method lies in requiring the embedding to be holomorphic, that is, that the system of equations for voltages {{math|<var>V</var>}} is turned into a system of equations for functions {{math|<var>V(s)</var>}} in such a way that the new system defines {{math|<var>V(s)</var>}} as holomorphic functions (i.e. complex analytic) of the new complex variable {{math|<var>s</var>}}. The aim is to be able to use the process of analyticalanalytic continuation which will allow the calculation of {{math|<var>V(s)</var>}} at {{math|<var>s</var>{{=}}1}}. Looking at equations ({{EquationNote|1}}), a necessary condition for the embedding to be holomorphic is that {{math|<var>V^*</var>}} is replaced under the embedding with {{math|<var>V^*(s^*)</var>}}, not {{math|<var>V^*(s)</var>}}. This is because complex conjugation itself is not a holomorphic function. On the other hand, it is easy to see that the replacement {{math|<var>V^*(s^*)</var>}} does allow the equations to define a holomorphic function {{math|<var>V(s)</var>}}. However, for a given arbitrary embedding, it remains to be proven that {{math|<var>V(s)</var>}} is indeed holomorphic. Taking into account all these considerations, an embedding of this type is proposed:

{{NumBlk|:|<math>\sum_k Y_{ik} V_k(s) + Y_i^{\text{sh}} V_i(s) = s\frac{S_i^*}{V_i ^*(s^*)}</math>|{{EquationRef|1}}}}