Holomorphic Embedding Load-flow method: Difference between revisions

Revision as of 11:54, 1 October 2012

Revision as of 14:07, 22 October 2012

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| invent1 = Antonio Trias

}}</ref> A detailed description was presented at the 2012 IEEE PES General Meeting, and published in <ref name="helmpaper">A. Trias, "The Holomorphic Embedding Load Flow Method", ''IEEE Power and Energy Society General Meeting 2011'', 22-26 July 2012.</ref>.

The method is founded on advanced concepts and results from [[Complex_analysis|complex analysis]], such as [[Holomorphic_function|holomorphicity]], the theory of [[Algebraic_curve|algebraic curves]], and [[Analytic_continuation|analytic continuation]]. However, the numerical implementation is rather straightforward as it uses standard linear algebra and [[Padé_approximant|Padé approximation]]. Additionally, since the limiting part of the computation is the factorization of the admittance matrix and this is done only once, its performance is competitive with established fast-decoupled loadflows. The method is currently implemented into industrial-strength real-time and off-line packaged EMS applications.

== Background ==

The [[Power_flow_study|load-flow]] calculation is one of the most fundamental components in the analysis of power systems and is the

cornerstone for almost all other tools used in [[Power_system_simulation|power system simulation]] and [[Energy_management_system|management]]. The load-flow equations can be written in the following general form:

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

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HELM provides a solution to a long-standing problem of all iterative load-flow methods, namely the unreliability of the iterations in finding the correct solution (or any solution at all).

This makes HELM particularly suited for real-time applications, and mandatory for any EMS software based on exploratory algorithms, such as contingency analysis, and under alert and emergency conditions solving operational limits violations and restoration providing guidance through action plans.

== Holomorphic Embedding ==

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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|1}}) 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 after order. Consider the power series expansion for <math>\textstyle V(s)=\sum_{n = 0}^\infty a[n] s^n</math> and <math>\textstyle 1/V(s)=\sum_{n = 0}^\infty b[n] s^n</math>. By substitution into equations ({{EquationNote|1}}) and identifying terms at each order in {{math|<var>sⁿ</var>}}, one obtains:

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== Analytic Continuation ==

Once the power series at {{math|<var>s</var>{{=}}0}} are calculated to the desired order, the problem of calculating them at {{math|<var>s</var>{{=}}1}} becomes one of [[Analytic_continuation|analytic continuation]]. It should be strongly remarked that this does not have anything in common with the techniques of [[Homotopy#Applications|homotopic continuation]]. Homotopy is powerful since it only makes use of the concept of continuity and thus it is applicable to general smooth nonlinear systems, but on the other hand it does not always provide a reliable method to approximate the functions (as it relies on iterative schemes such as Newton-Raphson).

It can be proven<ref>L. Ahlfors, ''Complex analysis (3rd ed.)'', McGraw Hill, 1979.</ref> that algebraic curves are complete [[Global_analytic_function|global analytic functions]], that is, knowledge of the power series expansion at one point (the so-called germ of the function) uniquely determines the function everywhere on the complex plane, except on a finite number of [[Branch_point#Branch_cuts|branch cuts]]. Stahl’s extremal ___domain theorem<ref>G. A. Baker Jr and P. Graves-Morris, ''Padé Approximants'' (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Second Ed. 2010, p. 326.</ref> further asserts that there exists a maximal ___domain for the analytic continuation of the function, which corresponds to the choice of branch cuts with minimal [[Conformal_radius#Version_from_infinity:_transfinite_diameter_and_logarithmic_capacity|logarithmic capacity]] measure. In the case of algebraic curves the number of cuts is finite, therefore it would be feasible to find maximal continuations by finding the combination of cuts with minimal capacity. For further improvements, Stahl’s theorem on the convergence of Padé Approximants<ref>H. Stahl, “The Convergence of Padé Approximants to Functions with Branch Points”, ''J. Approx. Theory'', '''91''' (1997), 139-204.

* G. A. Baker Jr and P. Graves-Morris, ''Padé Approximants'' (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Second Ed. 2010, p. 326-330.</ref> states that the diagonal and supra-diagonal Padé (or equivalently, the continued fraction approximants to the power series) converge to the maximal analytic continuation. The zeros and poles of the approximants remarkably accumulate on the set of [[Branch_point#Branch_cuts|branch cuts]] having minimal capacity.

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== See also ==

* [[Power_flow_study|Power flow study]]

* [[Power_system_simulation|Power system simulation]]