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The Holomorphic Embedded Load Flow Method(HELM)

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

The 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). All of these iterative methods suffer from a fundamental set of problems that limit their reliability. 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.

The patented HELM load-flow algorithm was invented by Antonio Trias. It is implemented as industrial-strength real time and off line packaged EMS applications for management and analysis.

Background
The load-flow problem, also known as power flow, computes the steady state of three-phase balanced AC power networks with complex injections expressed totally or partially in terms of power.

Traditional load-flow algorithms were developed based on three foundational approaches: the Gauss-Seidel method, which has poor convergence properties, but very little memory requirements and it is straightforward to implement; the full Newton-Raphson method, which has fast (quadratic) iterative convergence properties, but it is computationally costly; and the Fast Decoupled Load-Flow (FDLF) method, 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 have been published. The underlying technique in all of the existing methods remains 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, 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.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 have been illustrated for the two-bus model. Although there exist homotopic continuation techniques, which alleviate the problem to some degree, the fractal nature of the basins of attraction precludes a 100% reliable method.

The key differential advantage of the HELM is that it is fully deterministic and unambiguous: it guarantees that the solution always corresponds to the correct operative solution, when it exists; and it signals the non-existence of the solution when the conditions are such that there is not any solution (voltage collapse). Additionally, the method is competitive with the FDNR method in terms of computational cost. It brings a solid mathematical treatment of the load-flow problem that provides new insights not previously available with the iterative numerical methods.

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 s, such that for s=0 the system has an obvious correct solution, and for s=1 one recovers the original problem. 

 It is now possible to compute univocally power series for voltages as analytic functions of s. The correct load-flow solution at s=1 will be obtained by analytic continuation of the known correct solution at s=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). 



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

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


 * $$\Sigma_kY_{ik}V_k+Y_i^{sh}V_i=\frac{s_i^*}{v_i^*}$$

where the given (complex) parameters are the admittance matrix $$Y_{ik}$$￼, ￼the bus shunt admittances $$Y_i^{sh}$$, and the bus power injections $$S_i￼$$. 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 $$V(s)$$ 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 $$V(s)$$ at $$s=1$$. Looking at equations (1), a necessary condition for the embedding to be holomorphic is that $$V^*$$ is replaced under the embedding with $$V^* (s^*)$$, not $$V^* (s)$$. This is because complex conjugation itself is not a holomorphic function. On the other hand, it is easy to see that the replacement $$V^* (s^*)$$ does allow the equations to define a holomorphic function $$V$$. However, for a given arbitrary embedding, it remains to be proven that $$V(s)$$ is indeed holomorphic. Taking into account all these considerations, an embedding of this type is proposed:


 * $$\sum_kY_{ik}V_k(s)+Y_i^{sh}V_i(s)=s\frac{s_i^*}{v_i^*(s^*)}$$

With this choice, at $$s=0$$ 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 (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 $$s=0$$ 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 $$s=0$$) of voltages $$V$$ 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 $$1/V$$, since their coefficients are related by the convolution formulas derived from the following identity:


 * $$1=V(s)V^{-1}(s)=\left \{\sum_{n=0}^\infty a_bs^n\right \}\left \{\sum_{n=0}^\infty b_ns^n\right \}$$


 * $$=a_0b_0+\left(\sum_{k=0}^1a_{1-k}b_k\right)s+\left(\sum_{k=o}^2a_{2-k}b_k\right)s^2+\dots +\left(\sum_{k=0}^na_{n-k}b_k\right)s^n$$

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. Analytic Continuation Once the power series at $$s=0$$ are calculated to the desired order, the problem of calculating them at $$s=1$$ 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). 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. 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.