Its main features are that it is 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-flow methods (due to their convergence problems).
An example of this would be decision-support tools providing validated action plans in real time.
The HELM load-flow algorithm was invented by Antonio Trias and has been granted two US Patents.
[1][2] A detailed description was presented at the 2012 IEEE PES General Meeting and subsequently published.
[3] The method is founded on advanced concepts and results from complex analysis, such as holomorphicity, the theory of algebraic curves, and analytic continuation.
However, the numerical implementation is rather straightforward as it uses standard linear algebra and the 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.
The load-flow equations can be written in the following general form: where the given (complex) parameters are the admittance matrix Yik, the bus shunt admittances Yish, and the bus power injections Si representing constant-power loads and generators.
To solve this non-linear system of algebraic equations, traditional load-flow algorithms were developed based on three iterative techniques: the Gauss–Seidel method,[4] which has poor convergence properties but very little memory requirements and is straightforward to implement; the full Newton–Raphson method[5] which has fast (quadratic) iterative convergence properties, but it is computationally costly; and the Fast Decoupled Load-Flow (FDLF) method,[6] 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.
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.
[7] A simple illustration for the two-bus model is provided in[8] Although there exist homotopic continuation techniques that alleviate the problem to some degree,[9] the fractal nature of the basins of attraction precludes a 100% reliable method for all electrical scenarios.
It brings a solid mathematical treatment of the load-flow problem that provides new insights not previously available with the iterative numerical methods.
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.
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 analytic continuation which will allow the calculation of V(s) at s=1.
On the other hand, it is easy to see that the replacement V*(s*) does allow the equations to define a holomorphic function V(s).
It is this specific fact, which becomes true because the embedding is holomorphic that guarantees the uniqueness of the result.
Note that the coefficients of the expansions for V and 1/V are related by the simple convolution formulas derived from the following identity:
so that the right-hand side in (2) can always be calculated from the solution of the system at the previous order.
Note also how the procedure works by solving just linear systems, in which the matrix remains constant.
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).
Stahl's extremal domain theorem[12] 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.
For further improvements, Stahl's theorem on the convergence of Padé Approximants[13] 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.