Holomorphic Embedding Load-flow method
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.