Holomorphic Embedding Load-flow method: Difference between revisions

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The [[Power_flow_study|load-flow]] calculation is one of the most fundamental components in the analysis of power systems and is the corner stonecornerstone for almost all other tools used in [[Power_system_simulation|power system managementsimulation]] and simulation[[Energy_management_system|management]].

The load-flow equations can be written in the following general form:

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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 analytical 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|21}}}}

With this choice, at {{math|<var>s</var>{{=}}0}} the right hand side terms become zero, (provided that the denominator is not zero), this corresponds to the case where all

the injections are zero and this case has a well known and simple operational solution: all voltages are equal and all flow intensiti es are zero. Therefore this choice for the embedding provides at s=0 a well known operational solution.

Now using classical techniques for variable elimination in polynomial systems<ref>B. Sturmfels, "Solving Systems of Polynomial Equations”, CBMS Regional Conference Series in Mathematics 97, AMS, 2002.</ref> (results from the theory of [[Resultants|Resultants]] and [[Groebner_basis#Elimination_property|Gröbner basis]] it can be proven that equations ({{EquationNote|21}}) do in fact define {{math|<var>V(s)</var>}} as holomorphic functions. More significantly, they define {{math|<var>V(s)</var>}} as [[Algebraic_curves|algebraic curves]]. It is this specific fact, which becomes true because the embedding is holomorphic that guarantees the uniqueness of the result. The solution at {{math|<var>s</var>{{=}}0}} determines uniquely the solution everywhere (except on a finite number of branch cuts), thus getting rid of the multi-valuedness of the load-flow problem.

The technique to obtain the coefficients for the power series expansion (on {{math|<var>s</var>{{=}}0}}) of voltages {{math|<var>V</var>}} is quite straightforward, once one realizes that equations ({{EquationNote|2}}) can be used to obtain them, order byafter order,. by usingConsider the power series expansion for <math>\textstyle V(s)=\sum_{{n = 0}^\infty a[n] s^n</math|> and <varmath>\textstyle 1/V(s)=\sum_{n = 0}^\infty b[n] s^n</varmath>}},. sinceBy theirsubstitution coefficientsinto areequations related({{EquationNote|1}}) byand theidentifying convolutionterms formulasat derivedeach fromorder thein following{{math|<var>sⁿ</var>}}, one identityobtains:

{{NumBlk|:|<math>\sum_k Y_{ik} a_k[n] + Y_i^{\text{sh}} a_i[n] = S_i^* b_i^*[n-1] \qquad (n=0, \ldots, \infty)</math>|{{EquationRef|2}}}}

It is then straightforward to solve the sequence of linear systems ({{EquationNote|2}}) successively order after order, starting from {{math|<var>n</var>{{=}}0}}. Note that the coefficients of the expansions for {{math|<var>V</var>}} and {{math|<var>1/V</var>}} are related by the simple convolution formulas derived from the following identity:

{{NumBlk|:|

<math>

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</math>

|{{EquationRef|3}}}}

so that the right-hand side in ({{EquationNote|2}}) can always be calculated from the solution of the system at the previous order. Note also how the procedure works by solving just [[System_of_linear_equations|linear systems]], in which the matrix remains constant.

A more detailed discussion about this procedure is included in the Holomorphic Embedding Wikipedia page, and a complete treatment is offered in Ref. <ref>A. Trias, "The Holomorphic Embedding LoadflowLoad Flow Method as an enabler of reliable real-time applications", ''IEEE Peprint'', 2011.</ref>.

The particular choice of the embedding then allows to successively obtain the coefficients of the voltages order by order, by solving linear systems (in which the matrix remains constant!) whose right-hand-sides are determined by the calculation of the coefficients for at the previous order.

A more detailed discussion about this procedure is included in the Holomorphic Embedding Wikipedia page, and a complete treatment is offered in Ref. <ref>A. Trias, "Holomorphic Embedding Loadflow", ''IEEE Peprint'', 2011.</ref>.