Holomorphic Embedding Load-flow method: Difference between revisions

Revision as of 13:44, 11 January 2012

Revision as of 16:02, 11 January 2012

Line 5:

Based on a holomorphic embedding technique, HELM provides the operational solution to the (multi-valued) load-flow problem in real time.

The [[Power_flow_study|load-flow]] calculation is one of the most fundamental components in the analysis of power systems and is the corner stone for almost all other tools used in power system management and simulation. Prior to the development of HELM, previous methods were based on numerical iterative schemes such as [[Gauss-Seidel]], [[Newton-Raphson]], or variants (such as homotopic continuation methods<ref>E. L. Allgower and K. Georg, "Introduction to Numerical Continuation Methods", SIAM Classics in Applied Mathematics 45, 2003.</ref>). All of these iterative methods suffer from a fundamental set of problems that limit their reliability<ref>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>. Iterative Load-flow methods become increasingly unreliable as the electrical grid progresses towards voltage collapse as visualized by the position on the PV/QV curves. By contrast, HELM is non-iterative, deterministic, and non-equivocal. This breakthrough was possible through the understanding of the ability to apply techniques associated with [[Algebraic Curves]] to the load-flow problem. The resultant HELM always ensures the computation of the operational solution to the (multi-valued) load-flow problem. HELM’s equation structure allows competitive computational costs. It is scalable for all sizes and complexities of grids. The HELM algorithm provides new capabilities such as model-based scenarios, assessment of data quality and guided restoration. HELM allows the utilization of more accurate state estimation, solving some of the fundamental problems of data reliability in electrical grids.<br /><br />

The patented HELM load-flow algorithm was invented by Antonio Trias<ref>

Line 40:

= Methodology and Applications =

HELM is grounded on a rigorous mathematical theory, and in practical terms it could be summarized as follows:

# Define an (holomorphic) embedding for the equations in terms of a complex parameter {{math|<var>s</var>}}, such that for {{math|<var>s</var>{{=}}0}} the system has an obvious correct solution, and for {{math|<var>s</var>{{=}}1}} one recovers the original problem.

# It is now possible to compute univocally power series for voltages as analytic functions of {{math|<var>s</var>}}. The correct load-flow solution at {{math|<var>s</var>{{=}}1}} will be obtained by analytic continuation of the known correct solution at {{math|<var>s</var>{{=}}0}}.

# Perform the analytic continuation using algebraic approximants, which in this case are guaranteed to either converge to the solution if it exists, or not converge if the solution does not exist (voltage collapse).

Line 51:

= Holomorphic Embedding =

Consider the following general form for the load-flow equations:

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

where the given (complex) parameters are the admittance matrix <{{math|<var>Y<sub>Y_{ik}</mathsub></var>}}, thethe bus shunt admittances <{{math|<var>Y_i</var><sup>Y_i^{sh}</mathsup>}}, and the bus power injections <{{math|<var>S<sub>S_ii</mathsub></var>}}. For the purposes of the discussion, we will omit the treatment of controls, but the method can accommodate all types of controls. 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)</mathvar>}} 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|<mathvar>V(s)</mathvar>}} at <{{math|<var>s=1</mathvar> {{=}}1}}. Looking at equations ({{EquationNote|1}}), a necessary condition for the embedding to be holomorphic is that <{{math|<var>V^<sup>*</mathsup></var>}} is replaced under the embedding with <{{math|<var>V^^*(s^^*)</mathvar>}}, not <{{math|<var>V^^*(s)</mathvar>}}. 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^^*)</mathvar>}} does allow the equations to define a holomorphic function <{{math|<var>V(s)</mathvar>}}. However, for a given arbitrary embedding, it remains to be proven that <{{math|<var>V(s)</mathvar>}} 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|2}}}}

With this choice, at {{math|<mathvar>s=0</mathvar>{{=}}0}} the system becomes the load-flow equations for the case where all intensities are zero. The operational solution for this case being, as is well known, that all the voltages are equal.

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|2}}) do in fact define ???{{math|<var>V(s)</var>}} as holomorphic functions. More significantly, they define {{math|<var>V(s)</var>}} as '''algebraic curves'''. The solution atat {{math|<mathvar>s=0</mathvar>{{=}}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>s=0</math>) of voltages <math>V</math> is quite straightforward once one realizes that Eqs. [eqref:lf_embedded] can be used to obtain them, order by order, by using the power series expansion for <math>1/V</math>, since their coefficients are related by the convolution formulas derived from the following identity:

::<math>1=V(s)V^{-1}(s)=\left \{\sum_{n=0}^\infty a_bs^n\right \}\left \{\sum_{n=0}^\infty b_ns^n\right \}</math>

The technique to obtain the coefficients for the power series expansion (on {{math|<mathvar>s=0</mathvar>{{=}}0}}) of voltages <{{math|<var>V</mathvar>}} is quite straightforward once one realizes that Eqs.equations [eqref:lf_embedded]({{EquationNote|2}}) can be used to obtain them, order by order, by using the power series expansion for {{math|<mathvar>1/V</mathvar>}}, since their coefficients are related by the convolution formulas derived from the following identity:

::<math>=a_0b_0+\left(\sum_{k=0}^1a_{1-k}b_k\right)s+\left(\sum_{k=o}^2a_{2-k}b_k\right)s^2+\dots +\left(\sum_{k=0}^na_{n-k}b_k\right)s^n</math>

{{NumBlk|:|

<math>

\begin{align}

1 & = V(s)V^{- 1} (s) \\

::<math>1=V(s)V^{-1}(s) & = \left \{(\sum_{n=0}^\infty a_bsa_n s^n\right) \}\left \{(\sum_{n = 0}^\infty b_nsb_n s^n\right) \}</math>\

::<math> & =a_0b_0 a_0 b_0 + \left(\sum_{k=0}^1a_1 a_{1 - k} b_k\right) s + \left(\sum_{k=o0}^2a_{2 a_{1 - k} b_k\right) s^2 + \dotscdots + \left(\sum_{k = 0}^na_n a_{n - k} b_k\right) s^n</math> + \ldots

\end{align}

</math>

|{{EquationRef|3}}}}

The particular choice of the embedding then allows to successively obtain the coefficients of the voltages orderorder 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 atat 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>.

= Analytic Continuation =

Once the power series at {{math|<mathvar>s=0</mathvar>{{=}}0}} are calculated to the desired order, the problem of calculating them at <{{math|<var>s=1</mathvar>{{=}}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 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.