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HELM is grounded on a rigorous mathematical theory, and in practical terms it could be summarized as follows:<br />
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#3 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).<br />▼
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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).<br />
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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.<br />
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== Holomorphic Embedding == <br />
Consider the following general form for the load-flow equations:<br />
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::<math>\Sigma_kY_{ik}V_k+Y_i^{sh}V_i=\frac{s_i^*}{v_i^*}</math><br />
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where the given (complex) parameters are the admittance matrix <math>Y_{ik}</math>, the bus shunt admittances <math>Y_i^{sh}</math>, and the bus power injections <math>S_i</math>. 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 ''s''.
The first key ingredient in the method lies in requiring the embedding to be holomorphic, that is, that the system of equations for voltages V is turned into a system of equations for functions <math>V(s)</math> in such a way that the new system defines V(s) as holomorphic functions (i.e. complex analytic) of the new complex variable ''s''. The aim is to be able to use the process of analytical continuation which will allow the calculation of <math>V(s)</math> at <math>s=1</math> . Looking at equations (1), a necessary condition for the embedding to be holomorphic is that <math>V^*</math> is replaced under the embedding with <math>V^* (s^*)</math>, not <math>V^* (s)</math>. This is because complex conjugation itself is not a holomorphic function. On the other hand, it is easy to see that the replacement <math>V^* (s^*)</math> does allow the equations to define a holomorphic function <math>V()</math>. However, for a given arbitrary embedding, it remains to be proven that <math>V(s)</math> is indeed holomorphic. Taking into account all these considerations, an embedding of this type is proposed:<br />
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::<math>\sum_kY_{ik}V_k(s)+Y_i^{sh}V_i(s)=s\frac{s_i^*}{v_i^*(s^*)}</math><br />
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With this choice, at <math>s=0</math> 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.<br />
Now using classical techniques for variable elimination in polynomial systems (results from the theory of Resultants and Gröbner basis it can be proven that equations (2) do in fact define  as holomorphic functions. More significantly, they define  as algebraic curves. The solution at <math>s=0</math> 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:<br />
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::<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><br />
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::<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> <br/>
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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.<br/><br/>
A more detailed discussion about this procedure is included in the Holomorphic Embedding Wikipedia page, and a complete treatment is offered in Ref.<br/>
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== Analytic Continuation ==<br/>
Once the power series at <math>s=0</math> are calculated to the desired order, the problem of calculating them at <math>s=1</math> becomes one of [[analytic continuation]]. It should be strongly remarked that this does not have anything in common with the techniques of 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).<br/><br/>
It can be proven that algebraic curves are complete 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 cuts. Stahl’s extremal ___domain theorem 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 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 cuts having minimal capacity.<br/><br/>
These properties confer the load-flow method with the ability to unequivocally detect the condition of voltage collapse: the algebraic approximations are guaranteed to either converge to the solution if it exists, or not converge if the solution does not exist.
== References ==
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