## The Holomorphic Embedding Load-flow Method (HELM)

The keystone technology behind both AGORA and HELM-Flow is the Holomorphic Embedding Load-flow Method (HELM).  This is a novel method for solving the power flow equations of electrical networks. In contrast to all other power-flow methods, HELM does not rely on numerical iteration. HELM is constructive, direct, and fully reliable: it mathematically guarantees a consistent selection of the most desirable operating solution (the power-flow is a multivalued problem), and it also signals correctly that there is no solution when the case is unfeasible.

These properties are relevant not only for the reliability of existing off-line and real-time simulations, but also because they enable new types of analytical tools that would otherwise be impossible to build.  The prime example of this are decision-support tools for operators, based on extensive (intelligent) search in the state-space of steady states of the network.  Thanks to HELM, it is technically feasible to perform this exploration reliably—this is how AGORA’s Restoration solver was made possible.

By contrast, all iterative power-flows suffer from erratic convergence problems, to a greater or lesser extent.  Their operation is not fully reliable when run unattended: they always need human supervision, in order to tinker with the initial seed when problems appear.

The HELM  algorithm was invented by Dr. Antonio Trias and has been granted two US Patents. A detailed description was presented at the 2012 IEEE PES General Meeting (see the references on the side). The method is based 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 Padé approximants. Its computational performance is competitive with established fast-decoupled loadflows.

Convergence issues with iterative methods

The animation below is an example that graphically shows the issues with the Newton-Raphson method.  Other iterative methods show qualitatively the same problems.  We’ve chosen a two bus model primarily because the mathematical solutions can be analytically calculated by hand.  Even in this simple and idealized scenario we can see most of the key problems.  These include: the existence of both a high-voltage and low-voltage solution to the quadratic equation; the fractal nature of the basins of attraction to each solution; convergence to a solution with a very large phase angle (which needs normalization); and, particularly when close to voltage collapse, divergence or even spurious convergence to a non-solution. Our direct HELM™-based algorithms do not suffer from any of these problems.

Move your mouse along the voltage collapse curve. For each point along the curve, we’ve generated for over a million starting points, the load flow solution shown in the circular image.  No convergence The iterations have not convergend after the cut-off limit. Anomalous convergence The iterations have converged to a point that is neither one of the two known solutions. Unstable: wrong winding The iterations have converged to the unstable solution, but the angle is shifted by possibly large multiples of 2π. Unstable: right winding The iterations have converged to the unstable solution, and the angle is normalized within -π, π. Stable: wrong winding The iterations have converged to the correct operational solution, but the angle is shifted by possibly large multiples of 2π. Stable: right winding The iterations have converged to the correct operational solution, and the angle is normalized within -π, π.