The Holomorphic Embedding Load-flow Method (HELM) is a solution method for the power-flow equations of electrical power systems. 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. A detailed description was presented at the 2012 IEEE PES General Meeting and subsequently published. 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.
Background The 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 management. 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, which has poor convergence properties but very little memory requirements and 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 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, 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 of iterative loadflows have been extensively documented. A simple illustration for the two-bus model is provided in Although there exist homotopic continuation techniques that alleviate the problem to some degree, 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 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 no 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 a specific (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. Given this holomorphic embedding, 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.
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