Alternative PV Bus Modelling with the Holomorphic Embedding Load Flow Method

Abstract: The Holomorphic Embedding Load Flow Method (HELM) has been suggested as an alternative approach to solve load flow problems. However, the current literature does not provide any HELM models that can accurately handle general power networks containing PV and PQ buses of realistic sizes. The original HELM paper dealt only with PQ buses, while a second paper showed how to include PV buses but suffered from serious accuracy problems. This paper fills this gap by providing several models capable of solving general networks, with computational results for the standard IEEE test cases provided for comparison. In addition this paper also presents a new derivation of the theory behind the method and investigates some of the claims made in the original HELM paper.

• The paper introduces alternative holomorphic embeddings to overcome convergence issues in traditional HELM methods for mixed PQ/PV power networks.
• It utilizes adaptable parameters to control voltage paths and shunt compensation, eliminating double convolutions and improving Padé approximant stability.
• Robust performance is demonstrated on IEEE test cases, achieving residuals below 10⁻¹⁰ and voltage errors below 10⁻¹¹ even on large-scale networks.

Alternative PV Bus Modelling with the Holomorphic Embedding Load Flow Method

Background and Motivation

The Holomorphic Embedding Load Flow Method (HELM) represents a deterministic, non-iterative framework for power system load flow analysis. However, the initial HELM formulation was confined to networks exclusively composed of PQ buses, precluding direct application to practical networks with both PQ and PV buses. Extensions for handling PV buses, such as those by Subramanian et al., introduced holomorphic embeddings for PV conditions but suffered from significant numerical issues, including loss of accuracy caused by double convolutions and weak convergence on large-scale networks.

The work addresses these deficiencies by introducing a set of alternative holomorphic embeddings for general PQ/PV power networks. These models circumvent the numerical pitfalls of previous approaches via structurally distinct formulations, accompanied by a rigorous theoretical investigation of holomorphicity and the so-called "reflecting condition."

Theoretical Development

HOLM for PQ Bus Systems

The foundation of the method is the bus-power-equilibrium equations (BPEE), which for PQ bus $i$ can be written as:

$\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$

HELM introduces a complex parameter $z$ (demand scaling) and embeds bus voltages as holomorphic functions of $z$, creating the homotopy:

$$\sum_{k\in B} Y_{ik}V_k(z) = \frac{zS_i^*}{V_i^*(z^*)}$$

Trias' original work proves holomorphicity by embedding the conjugation into an augmented system with additional variables $\overline{V}_i(z)$, ultimately extracting physical solutions via a "reflecting condition." The current paper formalizes this with an explicit application of the Complex Implicit Function Theorem, ensuring local holomorphicity under mild topological assumptions (non-singular reduced $Y$).

Padé approximants are then utilized for analytic continuation, exploiting Stahl's theorems to accomplish maximal continuation and global solution construction—provided that $z=1$ is not separated by a branch cut.

Extensions for PV Buses

In practical power systems, PV buses (with fixed active power and voltage magnitude) are essential. Previous holomorphic embeddings introduced algebraic equations targeting fixed voltage magnitude, but introduced terms (such as $V^2$ times a sum over $\overline{V}$ variables) that resulted in problematic double convolutions, leading to large truncation errors for high-order terms and poor Padé approximant convergence, as evidenced in standard IEEE cases.

Generalized Holomorphic Embedding

The proposed method generalizes the holomorphic embedding for both PQ and PV buses:

• Introduces parameters $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$0 to control the voltage path and shunt compensation, allowing for systematic tuning of the homotopy and analytic properties.
• Constructs embeddings avoiding higher-order product terms to ensure single convolution in recurrence relations, critical for numerical stability in high-order series computation.
• For PV buses, the voltage magnitude constraint is enforced via

$\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$1

where $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$2 is a real polynomial enforcing the correct voltage at $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$3.

• Several canonical choices for embedding parameters are explored, each corresponding to a different interpolation in the $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$4 homotopy.

The holomorphicity and redundancy of the reflecting condition are proven via induction on the recurrence and use of the symmetry in polynomial coefficients, generalizing prior arguments to mixed PQ/PV networks.

Numerical Performance and Model Comparison

Benchmark Setup

Four alternative models are constructed, each defined by parameter choices for $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$5 and $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$6. These, together with two prior approaches by Subramanian, are benchmarked on the IEEE 9-, 14-, 30-, 39-, 57-, 118-, and 300-bus test systems. Key metrics include maximal equation residuals ($\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$7) and voltage deviations from MatPower reference solutions ($\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$8), using double-precision [15/15] Padé approximants for analytic continuation.

Results Summary

For small- to medium-scale cases (up to 118-bus), all new models (especially Models 1, 2, 4) deliver residuals below $\sum_{k\in B} Y_{ik}V_k = \frac{S_i^*}{V_i^*}$9 and voltage errors below $z$0. Subramanian's first method exhibits poor convergence, especially for larger systems, with both high residuals and solution deviation. Model 4 alone achieves convergence within double-precision for the 300-bus network.

Analysis of the series coefficients and singularity structure of Padé approximants highlights the source of instability in prior approaches—numerous branch points and rapid coefficient growth (see Figure 1 & 3 in the original paper)—contrasting with the well-behaved analytic structure for Model 4 (Figures 2 & 4).

Pathological Case Analysis

In networks artificially modified to contain only PQ buses, the original HELM method can fail to track the physical (stable) solution as load increases, a direct consequence of encountering singularities or instability in the analytic continuation. The work demonstrates, via solution tracking and application of the $z$1 stability criteria, that while HELM finds a stable solution, it is not always coincident with that found by traditional techniques, underlining practical subtleties in selecting the physically meaningful solution branch.

Implications and Future Directions

The paper presents alternative holomorphic embeddings that enable robust, convergent, and numerically stable HELM solutions for general PQ/PV power system networks. The flexible parameterization of embeddings provides resilience against the limitations of prior art; critically, the models eliminate double convolutions and maintain analytic structure compatible with reliable Padé continuation for large real-world systems.

Theoretically, the work strengthens the foundation of HELM by providing transparent criteria for holomorphic extension and clarifying when model-specific reflecting conditions are redundant, increasing confidence in the structured analytic continuation approach.

Practically, the improved numerical behavior on standardized IEEE test cases establishes these alternative models as operationally viable for real-world power grid analysis, with potential for further scaling and inclusion of non-standard elements (e.g., shunt compensation, more intricate bus behaviors).

Future work may extend the method to encompass dynamic security analysis (continuation towards voltage collapse), explore more sophisticated local parameter pathways for $z$2 to further optimize analytic domains, and integrate these models into fast contingency analysis and online decision support tools.

Conclusion

The work systematically resolves deficiencies in the holomorphic embedding of PV buses within the HELM framework. By introducing general, non-iterative holomorphic models with tunable homotopy parameters, it achieves robust, accurate, and scalable load flow solutions on arbitrary networks. Theoretical guarantees for holomorphic extendibility and reflecting condition satisfaction are rigorously established, with empirical validation across a hierarchy of IEEE test systems. These developments mark HELM as a practically viable and theoretically robust alternative to traditional iterative power flow solvers for modern power system analysis.