Holomorphic Embedding Load-flow method: Difference between revisions

Revision as of 14:07, 22 October 2012

Revision as of 22:54, 10 January 2013

Line 1:

The '''Holomorphic Embedding Load-flow Method (HELM<ref group="note">HELM is a trademark of Gridquant Inc.</ref>)''' is a solution method for the [[Power_flow_studyPower flow study|power flow]] equations of electrical power systems. Its main features are that it is

[[Direct_method_Direct method (computational_mathematicscomputational mathematics)|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). An example of this would be decision-support tools providing validated action plans in real time.

Line 19:

| gdate = 2011-07-12

| 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'', 22-2622–26 July 2012.</ref>.

The method is founded on advanced concepts and results from [[complex analysis]], such as [[Holomorphic_functionHolomorphic function|holomorphicity]], the theory of [[Algebraic_curve|algebraic curvescurve]]s, and [[analytic continuation]]. However, the numerical implementation is rather straightforward as it uses standard linear algebra and [[Padé_approximantPadé 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_studyPower 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]] and [[Energy_management_systemEnergy 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}}}}

Line 49:

Load-Flow (FDLF) method

<ref name="FDLF">B. Stott and O. Alsac, "Fast Decoupled Load Flow," ''IEEE Transactions on Power Apparatus and Systems'', vol. PAS-93, no.3, pp.859-869, May 1974.</ref>

, 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. Many other incremental improvements exist; however, the underlying technique in all of them is still an iterative solver, either of Gauss-Seidel or of Newton type. 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,<ref group="note" name="multsol">It is a well-known fact that the load flow equations for a power system have multiple solutions. For a network with {{math|<var>N</var>}} non-swing buses, the system may have up to {{math|2^<var>N</var>}} possible solutions, but only one is actually possible in the real electrical system. This fact is used in stability studies, see for instance: Y. Tamura, H. Mori, and S. Iwamoto,"Relationship Between Voltage Instability and Multiple Load Flow Solutions in Electric Power Systems", '' IEEE Transactions on Power Apparatus and Systems'', vol. PAS-102 , no.5, pp.1115-1125, 1983.</ref>, 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 behavior.<ref group="note">This is a general phenomenon affecting the Newton-Raphson method when applied to equations in

''complex'' variables. See for instance [[Newton's_method#Complex_functions]].</ref>. As a result, no matter how close the chosen initial point of the iterations (seed) is to the correct solution, there is always some non-zero chance of straying off to a different solution. These fundamental problems of iterative loadflows have been extensively documented

.<ref>R. Klump and T. Overbye, “A new method for finding low-voltage power flow solutions", ''in IEEE 2000 Power Engineering Society Summer Meeting,'', Vol. 1, pp. 593-–597, 2000.

* J. S. Thorp and S. A. Naqavi, "Load flow fractals", ''in Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 2, pp. 1822--1827, 1989.

* J. S. Thorp, S. A. Naqavi, and H. D. Chiang, "More load flow fractals", ''in Proceedings of the 29th IEEE Conference on Decision and Control, Vol. 6, pp. 3028--3030, 1990.

Line 57:

* J. S. Thorp, and S. A. Naqavi, S.A., "Load-flow fractals draw clues to erratic behaviour", IEEE Computer Applications in Power, Vol. 10, No. 1, pp. 59--62, 1997.

* H. Mori, "Chaotic behavior of the Newton-Raphson method with the optimal multiplier for ill-conditioned power systems", in ''The 2000 IEEE International Symposium on Circuits and Systems (ISCAS 2000 Geneva), Vol. 4, pp. 237--240, 2000.

</ref>. A simple

illustration for the two-bus model is provided in<ref>[http://www.elequant.com/products/agora/demo/iterativeloadflow/ Problems with Iterative Load Flow], Elequant, 2010.</ref> Although there exist [[Homotopy|homotopic]] [[Numerical_continuationNumerical continuation|continuation]] techniques that alleviate the problem to some degree,<ref>V. Ajjarapu and C. Christy, "The continuation power flow: A tool for steady state voltage

stability analysis", ''IEEE Trans. on Power Systems'', vol.7, no.1, pp. 416-423, Feb 1992.</ref>, the fractal nature of the basins of attraction precludes a 100% reliable method for all electrical scenarios.

The key differential advantage of the HELM is that it is fully deterministic and unambiguous: it guarantees that the solution always

Line 103:

</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_equationsSystem of linear equations|linear systems]], in which the matrix remains constant.

A more detailed discussion about this procedure is offered in Ref. <ref name="helmpaper" />.

== 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]]. 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 functionsfunction]]s, 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.

Line 124:

* [[Power flow study]]

* [[Power system simulation]]

{{DEFAULTSORT:Power Flow Study}}

[[Category:Electrical engineering]]

[[Category:Power engineering]]