Interpolation-Free Hybrid Bergeron–π Line Model with Accurate Zero-Sequence Impedance

Abstract

Fixed-step real-time electromagnetic-transient (EMT) simulation of large power networks typically relies on parallel partitioning, where transmission-line elements serve as step-synchronous decoupling boundaries between subsystems. In distribution and subtransmission studies, however, many line sections are electrically short and have propagation delays smaller than the simulation step. Classical Bergeron models then lose their pure one-step delay structure and require interpolation or sub-stepping, which undermines step determinism and limits the availability of decoupling boundaries, thereby constraining partition quality and scalability. This paper proposes an interpolation-free hybrid Bergeron–π boundary-line model with zero-sequence impedance modification (HB-π-ZIM). A one-step uncoupled per-phase Bergeron section enforces a delay equal to the simulation step to provide a strictly step-synchronous interface. Shunt compensation removes the artificial shunt susceptance introduced by the enforced delay, and a passive RL two-port synthesis reconstructs the residual series impedance so that, at the fundamental frequency, the terminal positive- and zero-sequence series impedances and shunt admittances match the conventional lumped-pi model. Case studies show close agreement with the lumped-pi benchmark under representative balanced and unbalanced transients, while parallel tests on a 327-node network demonstrate near-linear speedup (9.31 times on 10 cores) when HB-π-ZIM is applied only to cut-set lines. The proposed model therefore enlarges the feasible set of decoupling boundaries in short-line-dominated networks and enables scalable fixed-step real-time EMT simulation.

1. Introduction

Electromagnetic-transient (EMT) simulation remains a cornerstone for analyzing fast phenomena in modern power systems, with its numerical foundation tracing back to Dommel’s seminal algorithm [1] and subsequent program-level developments [2]. In the past decade, the rapid growth of inverter-based resources (IBRs) and the increasing operational complexity of HV distribution and subtransmission networks have significantly expanded the scope of EMT studies from offline analysis to protection-oriented validation and closed-loop hardware/software-in-the-loop testing. Reliability organizations have also emphasized EMT modeling and verification for IBR interconnection studies (e.g., the NERC guideline) [3]. Along with this trend, multi-core parallel EMT execution and real-time simulation have become increasingly important because large-scale EMT models must satisfy not only accuracy requirements but also strict step-level determinism and repeatability under fixed-step execution [4].

A central bottleneck for fixed-step EMT simulation arises in the presence of electrically short-line sections whose physical propagation delay $\tau$ is smaller than the global simulation time step $\Delta t$. This “short-delay” regime ($\tau <\Delta t$) is common in HV distribution and subtransmission studies where line lengths are often a few kilometers and the simulation step is chosen for overall system stability and computational feasibility (e.g., tens of microseconds). Classical traveling-wave (Bergeron) line models are attractive because they naturally provide terminal decoupling via a delay interface [5]. However, when $\tau$ is not an integer multiple of $\Delta t$—in particular, when $\tau <\Delta t$—a pure one-step history structure is no longer available. Practical implementations typically resort to interpolation of history terms or sub-stepping inside the line element, which increases overhead and, more critically for real-time and parallel workflows, complicates deterministic step-time budgeting and weakens the clean “history-only” boundary semantics that make traveling-wave lines effective partition boundaries [6].

A variety of modeling strategies have been proposed to address short lines, each with specific trade-offs. One class of methods focuses on improving the fidelity of the physical line representation. Frequency-dependent (FD) models and the Universal Line Model (ULM) can provide high accuracy over a broad frequency range [7,8]. Yet for $\tau <\Delta t$, these models still face the fundamental issue of history terms evaluated at non-integer delay offsets, which again requires interpolation or smaller internal steps and thus conflicts with strict fixed-step determinism demanded by real-time workflows. Admittance-based alternatives such as the folded line equivalent (FLE) [9] and frequency-aware cascaded Bergeron-cell approaches [10] further improve accuracy and numerical robustness, but they are not explicitly designed to preserve a simple, step-synchronous decoupling interface for parallel boundary exchange under a single global time step. Therefore, while these high-fidelity lines are ideal for offline accuracy-centric EMT studies, their direct use as boundary-line interfaces in strictly fixed-step parallel/real-time contexts remains challenging.

Another class of strategies seeks computational structure rather than wideband physical fidelity. The lumped-parameter π model is widely used in HV distribution/subtransmission EMT studies because it is simple, stable, and naturally compatible with fixed-step stamping [11]. It also aligns with the type of engineering data typically available for such studies: sequence parameters at 50/60 Hz. However, lumped models do not provide traveling-wave terminal decoupling; thus, they are not directly suited as boundary elements for partitioned EMT execution when step-synchronous interface semantics are desired.

To recover boundary decoupling without relying on natural long lines, compensation-based decoupling constructs artificial interfaces by inserting compensation networks [12]. Such approaches can enable parallel execution even when the physical grid lacks sufficiently long lines. Nevertheless, the interface is “algorithmic” rather than “physical,” and the tuning of compensation networks can be case-dependent; moreover, additional interface states and coupling structures may still affect step-time determinism depending on implementation details.

More recently, “delay-unification” hybrid models have emerged as an effective compromise for the short-delay regime, aiming to avoid interpolation and preserve step-synchronous decoupling by enforcing a unified delay structure across modes. A representative example is the segmented transmission delay model (STDM) [13], which partitions a line into a delay (decoupling) section and a compensation/synthesis section. These hybrids are attractive for fixed-step parallel EMT because they restore a step-synchronous, history-based boundary interface without sub-stepping. However, a key limitation is that the unified delay is typically enforced using a reference mode (often the positive sequence), which can distort non-reference modal quantities. In protection-oriented studies, this limitation becomes particularly critical because asymmetrical faults—especially single-line-to-ground faults—depend strongly on the zero-sequence network. Even modest deviations in $Z_0$ can bias fault currents and voltage profiles, thereby affecting relay-relevant quantities and the credibility of HIL-oriented validation [14,15,16,17,18]. In other words, the very feature that makes unified-delay hybrids computationally attractive (strict step synchrony) can introduce a systematic modeling error precisely in the scenario class most common in HV distribution/subtransmission protection studies (asymmetrical faults).

While extensive research has focused on symmetrical operations, there is a notable lack of evaluated works addressing the fidelity of hybrid models under asymmetrical fault conditions in the short-delay regime. Most existing hybrid methods [13] validate performance primarily through positive-sequence-dominated scenarios, often overlooking the zero-sequence impedance deviations that critically affect ground-fault currents. This paper aims to bridge this gap.

Table 1. Comparison of representative options for fixed-step EMT boundary-line modeling in the short-delay regime (τ < Δt).

Approach	No Interpolation/Sub-Stepping Needed for τ < Δt ?	Step-Synchronous Decoupling Usable as Boundary?	Asym-Fault/Z0 Fidelity	Key Remark
Bergeron TW (classical) [5]	✗	△	✓	Needs fractional delay fix
Bergeron + interpolation [5]	✗	△	✓	Breaks “pure one-step”
Bergeron + sub-stepping [5]	✗	△	✓	Sub-step overhead
Folded Line Equivalent (FLE) [9]	✓	✗	△	Fit-dependent
Frequency-aware cascaded Bergeron cells [10]	△	△	△	Multi-cell overhead
Lumped-π [11]	✓	✗	△	50/60 Hz target
STDM/unified-delay [13]	✓	✓	✗ (Z0 may distort)	Z0 distortion risk
Compensated decoupling [12]	✓	✓	✗ (Z0 may distort)	Artificial interface tuning
Proposed HB-π-ZIM (this work)	✓	✓	✓	Z0-restored boundary

Legend: ✓ = yes/good; △ = partial/case-dependent; ✗ = no/poor.

summarizes these representative options in the short-delay fixed-step setting. It highlights a central gap: traveling-wave approaches preserve modal fidelity but typically require interpolation/sub-stepping when $\tau <\Delta t$, while unified-delay hybrids avoid interpolation and provide step-synchronous boundary decoupling but may compromise zero-sequence fidelity. This gap motivates a boundary-line model that simultaneously satisfies three requirements: (i) interpolation-free evaluation under $\tau <\Delta t$ with a fixed global $\Delta t$; (ii) step-synchronous terminal decoupling suitable for partition boundaries; and (iii) accurate fundamental-frequency sequence networks, including the zero sequence, which governs asymmetrical-fault behavior.

To address this gap, this paper proposes HB-π-ZIM, an interpolation-free hybrid Bergeron–π boundary-line model with accurate zero-sequence impedance. The proposed architecture separates “time-structure” from “frequency-calibrated parameter restoration.” First, a one-step uncoupled per-phase Bergeron (UPB) decoupling section enforces ${\tau }_b=\Delta t$ to provide a strictly step-synchronous interface without interpolation. Second, a dedicated shunt compensation network removes the artificial shunt introduced by enforcing ${\tau }_b=\Delta t$ and restores the physical phase-to-phase capacitance consistent with (C₁,C₀). Third, a passive RL two-port synthesis reconstructs the residual series impedance such that, at 50/60 Hz, the positive- and zero-sequence terminal networks match those of the conventional lumped-π model exactly. As a result, HB-π-ZIM provides the computational semantics required for parallel and real-time EMT boundary exchange while explicitly preventing the zero-sequence distortion that can occur in unified-delay hybrids during asymmetrical faults.

The main contributions of this paper are as follows:

• We introduce an interpolation-free boundary-line architecture for the short-delay regime $\tau <\Delta t$ that preserves a strictly step-synchronous, history-based terminal interface suitable for fixed-step parallel and real-time EMT simulation.
• We develop a sequence-parameter restoration method that matches both positive- and zero-sequence series impedances and shunt admittances of a lumped-π reference at the fundamental frequency, thereby addressing the zero-sequence sensitivity of asymmetrical-fault studies.
• We validate the proposed model in representative fault scenarios and in a partitioned multi-core execution context, demonstrating close agreement with a high-resolution time-domain reference and improved robustness of asymmetrical-fault results compared to a representative unified-delay hybrid.

The remainder of this paper is organized as follows. Section 2 formulates the short-delay fixed-step problem and defines the equivalence target in terms of fundamental-frequency sequence networks. Section 3 presents the HB-π-ZIM architecture and its shunt/series synthesis. Section 4 describes a step-synchronous parallel implementation consistent with the proposed boundary interface. Section 5 reports case studies and verification results under representative fault scenarios, followed by conclusions and future work in Section 6.

2. Problem Formulation and Design Targets

2.1. Fixed-Step EMT Requirement for HV Distribution and Subtransmission Studies

Electromagnetic-transient (EMT) simulation with a fixed time step is widely used in high-voltage (HV) distribution and subtransmission studies, especially for protection-oriented fault transients, switching events, and closed-loop validation of control and protection functions. In such workflows, determinism and step-level repeatability are often as important as average runtime because timing jitter undermines the credibility of real-time or near-real-time testing and complicates event alignment in post-processing.

A fundamental computational bottleneck arises when a line section is electrically short, characterized by a physical propagation delay $\tau$ smaller than the solver time step $\Delta t$ (e.g., $\Delta t$ on the order of tens of microseconds). In this regime, a classical traveling-wave (Bergeron) line cannot be evaluated as a pure one-step delay without interpolation or sub-stepping. While interpolation/sub-stepping can restore numerical correctness for the line itself, they introduce non-negligible overhead and, more importantly, break the simple terminal-decoupling structure that makes Bergeron lines attractive as partition boundaries in parallel EMT execution.

This paper targets this short-delay regime precisely and seeks a line representation that is compatible with fixed-step EMT execution while maintaining the decoupling benefits of traveling-wave models.

2.2. Reference Model and Equivalence Definition

In this work, two “references” are used for different purposes.

(1)

Calibration reference for equivalence definition. The proposed HB-π-ZIM is frequency-calibrated at the system fundamental frequency. For protection-oriented HV feeder and subtransmission studies, the engineering data most commonly available are the line length l and the positive-/zero-sequence parameters at $f_0$. Although equivalent 50/60 Hz sequence parameters can, in principle, also be derived from geometry-based or traveling-wave (e.g., Bergeron/ULM) representations via frequency-domain reduction, the sequence form remains the standard interface used for utility data exchange and model validation in this application domain. Therefore, the conventional lumped-parameter π equivalent (denoted as “Lumped-π”) is adopted as the calibration reference to define the required terminal sequence networks at $f_0$. Specifically, HB-π-ZIM is designed to match the Lumped-π positive- and zero-sequence series impedances and shunt admittances at $f_0$.

(2)

Time-domain reference for waveform verification. To provide an accuracy-oriented time-domain baseline beyond the fundamental-frequency calibration, a small-step Bergeron traveling-wave simulation (e.g., $\Delta t=1 \mu \mathrm{s}$) is additionally used in Section 5 to compare transient waveforms under representative fault scenarios.

Based on the above, the equivalence definition of HB-π-ZIM is imposed with respect to the Lumped-π terminal sequence networks at $f_0$, while the small-step Bergeron case serves as a time-domain waveform reference in the verification section.

HB-π-ZIM’s inputs are the line length $l$ and the sequence parameters at the system fundamental frequency $f_0\in \{50,60\}\hspace{0.17em}\mathrm{Hz}$: $(R_1,L_1,C_1)$ and $(R_0,L_0,C_0)$ for the line section of length $l$. These parameters reflect the engineering data typically available to EMT users and protection engineers. In this paper, the line sequence parameters are treated as the effective 50/60 Hz values under nominal environmental conditions, which is consistent with typical utility-provided data for protection-oriented EMT studies. Temperature-dependent resistance (and, if needed, frequency-/soil-condition-dependent capacitance for cables) can be incorporated by updating these inputs prior to simulation or at slow time scales; the proposed HB-π-ZIM synthesis then follows the same workflow without structural changes. A systematic environmental-parameter sensitivity study is left as future work.

We adopt the following equivalence definition for the proposed model (“HB-π-ZIM”). At the fundamental frequency ${\omega }_0=2\pi f_0$, HB-π-ZIM must exhibit terminal characteristics identical to the positive- and zero-sequence series impedances and shunt admittances of the Lumped-π model, i.e.,

$$Z_1^{\mathrm{H}\mathrm{B}}\left(j{\omega }_0\right)=Z_1^{\pi }\left(j{\omega }_0\right), Z_0^{\mathrm{H}\mathrm{B}}\left(j{\omega }_0\right)=Z_0^{\pi }\left(j{\omega }_0\right)$$

$$Y_1^{\mathrm{H}\mathrm{B}}\left(j{\omega }_0\right)=Y_1^{\pi }\left(j{\omega }_0\right), Y_0^{\mathrm{H}\mathrm{B}}\left(j{\omega }_0\right)=Y_0^{\pi }\left(j{\omega }_0\right)$$

Here, $Z_{\{0,1\}}^{\pi }\left(j{\omega }_0\right)=R_{\{0,1\}}+j{\omega }_0{L}_{\{0,1\}}$ denotes the series branch of the lumped-$\pi$ model, and $Y_{\{0,1\}}^{\pi }\left(j{\omega }_0\right)=j{\omega }_0{C}_{\{0,1\}}$ denotes the total shunt admittance (i.e., $j{\omega }_0{C}_{\left\{\mathrm{0,1}\right\}}/2$ at each end).

This definition aligns with protection-oriented HV studies because fundamental-frequency sequence networks largely govern asymmetrical-fault behavior, particularly steady-state fault currents and relay reference quantities. In such studies, the primary need is a reliable fixed-step interface rather than a wideband frequency-dependent line model.

2.3. Design Targets

Given the equivalence definition above and the computational constraints of fixed-step parallel EMT, the design targets are as follows:

• Interpolation-free fixed-step operation for $\tau <\Delta t$. The model shall not require interpolation or sub-stepping to be evaluated under a fixed $\Delta t$.
• Terminal decoupling suitable for partition boundaries. The model shall preserve a terminal-decoupled structure so that each terminal can be solved within its local subsystem using historical information received from the remote terminal.
• Fundamental-frequency sequence fidelity, including zero sequence. The model shall preserve the 50/60 Hz positive- and zero-sequence networks of the Lumped-π $\pi$ equivalent. Zero-sequence fidelity is emphasized because single-line-to-ground faults are prevalent in HV distribution/subtransmission, and protection behavior is sensitive to $Z_0$.
• Passive realizability and robust EMT stamping. The internal realization shall admit standard EMTP companion-form discretization (e.g., trapezoidal rule) and avoid numerical pathologies under practical $\Delta t$ and line parameters.

3. HB-π-ZIM Model

3.1. Architecture Overview

The structural composition of the proposed HB-π-ZIM is shown in Figure 1. In all figures and derivations, A, B, and C denote the three-phase terminals of the proposed transmission line model.

HB-π-ZIM represents a short line as three series-connected functional sections:

• Decoupling section (one-step UPB cell): An uncoupled per-phase Bergeron (UPB) structure whose parameters are modified so that its travel time equals exactly one simulation step, ${\tau }_b=\Delta t$. This provides interpolation-free terminal decoupling.
• Shunt compensation section: A lumped network that cancels the artificial shunt susceptance introduced by enforcing ${\tau }_b=\Delta t$ and inserts the physical phase-to-phase capacitance consistent with $(C_1,C_0)$.
• Series-impedance synthesis section: A passive RL two-port equivalent that reconstructs the remaining series impedance so that the total 50/60 Hz sequence impedances match the Lumped-π $\pi$ reference in both $Z_1$ and $Z_0$.

The architecture is strategically “frequency-calibrated” at $f_0$ while being “time-structured” for fixed-step decoupling. This separation between (i) decoupling structure and (ii) sequence-parameter restoration is the key to avoiding the zero-sequence distortion found in many unified-delay hybrids.

3.2. One-Step UPB Decoupling Section

The decoupling section uses three independent single-phase Bergeron cells (uncoupled per-phase Bergeron, UPB), one per phase, intentionally avoiding modal-domain transformations. Its series inductance allocation is based on the positive-sequence inductance $L_1$ of the line section, which provides a well-defined reference for the one-step decoupling cell under the transposed-line assumption. To reserve a positive residual inductive margin for the series-impedance synthesis section, a coefficient $k\in \left(\mathrm{0,1}\right)$ (e.g., $k=0.99$) is introduced, and the decoupling section is assigned a per-phase series inductance $k{L}_1$. The detailed structure of the uncoupled per-phase Bergeron (UPB) cell and its equivalence to a lumped-parameter π-model are illustrated in Figure 2.

To enforce ${\tau }_b=\Delta t$, the per-phase shunt capacitance of the UPB section is increased accordingly. Using the travel-time relation ${\tau }_b=l\sqrt{L_b^{\prime }{C}_b^{\prime }}$, with $L_b^{\prime }=k{L}_1/l$, where $L_1$ denotes the total positive-sequence inductance of the line section of length $l$, the required $C_b^{\prime }$ is

$$\Delta t=l\sqrt{{\displaystyle \frac{k{L}_1}{l}}{C}_b^{\prime }} \Rightarrow C_b^{\prime }={\displaystyle \frac{\Delta t^2}{lk{L}_1}}$$

(1)

So, the total (phase-to-ground) self-capacitance of the UPB decoupling section becomes the following (which may exceed the physical self-capacitance for τ<Δt, motivating the shunt compensation in Section 3.3):

$$z_C=\sqrt{{\displaystyle \frac{L_b^{\prime }}{C_b^{\prime }}}}={\displaystyle \frac{k{L}_1}{\Delta t}}$$

(2)

Note that $z_C$ is a purely computational construct utilized to enforce ${\tau }_b=\Delta t$ and does not represent the physical characteristic impedance of the line.

3.3. Shunt Compensation and Capacitance Decomposition

Because the UPB section introduces an excess phase-to-ground capacitance relative to the physical line, a shunt compensation network is inserted. Under the transposed (or transposition-equivalent) symmetric three-phase line assumption, the standard capacitance decomposition applies:

$$C_0=C_s, C_1=C_s+3C_m \Rightarrow C_m={\displaystyle \frac{C_1-C_0}{3}}$$

(3)

where $C_s$ is the phase-to-ground (self) capacitance and $C_m$ is the phase-to-phase (mutual) capacitance.

The UPB section contributes only phase-to-ground (self) capacitance and introduces no phase-to-phase capacitive coupling. Therefore, the excess self-capacitance that must be compensated is as follows (with $\Delta C_s>0$ ensured in the intended $\tau <\Delta t$ regime; see Appendix B):

$$\Delta C_s=C_{b,\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-C_s={\displaystyle \frac{\Delta t^2}{k{L}_1}}-C_0$$

(4)

To cancel the corresponding artificial susceptance at the fundamental frequency ${\omega }_0=2\pi f_0$ (with $f_0=50/60\hspace{0.25em}\mathrm{H}\mathrm{z}$), a shunt inductor $\Delta L$ is selected so that

$$j{\omega }_0\Delta C_s+{\displaystyle \frac{1}{j{\omega }_0\Delta L}}=0 \Rightarrow \Delta L={\displaystyle \frac{1}{{\omega }_0^2\Delta C_s}}$$

(5)

Simultaneously, the physical phase-to-phase capacitance $C_m$ is inserted in this shunt compensation section, restoring the correct 50/60 Hz shunt admittance of the Lumped-π $\pi$ model in both sequence networks.

The physical phase-to-phase capacitance $C_m$ is explicitly allocated to the shunt compensation section rather than the series synthesis section. This strategic placement preserves the series synthesis section as a strictly resistive-inductive (RL) network, which facilitates a robust EMTP-compatible realization and simplifies coefficient synthesis.

An optional small damping resistor in series with $\Delta L$ (or an $R\parallel L$ realization tuned at ${\omega }_0$) can be introduced to suppress numerical ringing away from the calibration frequency without affecting the matching at ${\omega }_0$.

3.4. Series-Impedance Synthesis Targets at 50/60 Hz

The decoupling section, being per-phase and uncoupled, contributes the same inductive reactance $j{\omega }_{0}k{L}_1$ to both positive and zero sequence. The synthesis section must therefore supply the remaining sequence impedances required to reach the Lumped-π targets at ${\omega }_0$. Define

$$Z_{S1}=R_{S1}+j{X}_{S1}=R_1+j{\omega }_0(1-k)L_1$$

(6)

$$Z_{S0}=R_{S0}+j{X}_{S0}=R_0+j{\omega }_0(L_0-k{L}_1)$$

(7)

The synthesis section is required to realize a three-phase series-impedance matrix $Z_S$ whose eigenvalues in symmetrical components are $(Z_{S0},Z_{S1},Z_{S1})$. In phase coordinates, this corresponds to

$$Z_S={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}2Z_{S1}+Z_{S0}& Z_{S0}-Z_{S1}& Z_{S0}-Z_{S1}\\ Z_{S0}-Z_{S1}& 2Z_{S1}+Z_{S0}& Z_{S0}-Z_{S1}\\ Z_{S0}-Z_{S1}& Z_{S0}-Z_{S1}& 2Z_{S1}+Z_{S0}\end{array}\right]$$

(8)

Its inverse $Y_S=Z_S^{-1}$ defines the required two-port nodal-admittance stamp for the synthesized series element:

$$Y_{2p}=\left[\begin{array}{cc}{Y}_S& -Y_S\\ -Y_S& Y_S\end{array}\right]$$

(9)

which can be implemented by a finite set of RL branches as described next.

3.5. RL Lattice Realization and EMT Discretization

To implement $Y_{2p}$ at ${\omega }_0$, HB-$\pi$-ZIM adopts a 15-branch $R\parallel L$ lattice as a computational realization for two-port matrix stamping. This realization is not a physical reconstruction of the line geometry; rather, it reproduces the terminal two-port admittance exactly at the fundamental frequency while remaining compatible with standard EMTP companion-form stamping.

The branch set comprises: three “self” branches (same phase across terminals), six cross-terminal coupling branches (different phases across terminals), and six within-terminal branches (phase-to-phase at each terminal). Closed-form coefficients $(A,B,C,D)$ for these branches are derived in Appendix A.

Instead of explicitly listing all 15 branches, we exploit the topological symmetry of the transposed line to group the lattice into three functional sets. The synthesis network is realized by connecting the six terminal nodes (phases A, B, C at terminals $i$ and $j$) according to the concise definitions in

Table 2. Structure and parameters of the 15-branch synthesis lattice.

Branch Set Type	$\mathbf{Topology} \mathbf{Connectivity} (\mathit{\varphi },\mathit{\psi }\in \{\mathit{A},\mathit{B},\mathit{C}\},\hspace{0.25em}\mathit{\varphi }\ne \mathit{\psi })$	Count	$\mathbf{Resistance} \left(\mathit{R}\right)$	$\mathbf{Inductance} \left(\mathit{L}\right)$
Longitudinal Self	Terminal $i_{\varphi }$—Terminal $j_{\varphi }$	3	$A$	$B$
Longitudinal Mutual	Terminal $i_{\varphi }$—Terminal $j_{\psi }$	6	$C$	$D$
Transverse Mutual	Terminal $i_{\varphi }$—Terminal $i_{\psi }$ Terminal $j_{\varphi }$—Terminal $j_{\psi }$	6	$-C$	$-D$

Note: The “transverse mutual” entries represent a signed-admittance decomposition used for two-port nodal-admittance assembly. They should not be interpreted as standalone physical components with negative parameters; in implementation, the lattice is stamped via an incidence-matrix (or direct $Y_{2p}$) assembly to preserve two-port passivity and stability at the calibration frequency.

. This compact representation fully defines the $6\times 6$ admittance coupling.

Each parallel $R\parallel L$ branch is discretized using the trapezoidal rule into a Norton companion form, yielding an equivalent conductance and a history current source updated at each step. The equivalent resistance for a branch is

$$R_{\mathrm{e}\mathrm{q}}={\displaystyle \frac{1}{\frac{1}{R}+\frac{\Delta t}{2L}}}={\displaystyle \frac{2RL}{2L+R\Delta t}}$$

(10)

The overall two-port remains robust under practical parameter ranges when $k$ is chosen within the passivity constraints, summarized in Appendix B.

3.6. Workflow Summary and Parameter-Selection Procedure

Section 3.6 summarizes the end-to-end synthesis procedure to improve clarity and reproducibility. Figure 3 provides a step-by-step workflow covering boundary construction, shunt handling, key parameter selection with feasibility checks, and the final RL-lattice realization and validation.

4. Real-Time Parallel Implementation

4.1. Role of the Implementation in This Paper

HB-$\pi$-ZIM is intended to act as an interpolation-free boundary-line interface for fixed-step parallel EMT simulation in the short-delay regime $(\tau <\Delta t)$. In partitioned EMT execution, step-level timing determinism is frequently as critical as average throughput because excessive step-time jitter undermines event alignment, repeatability, and the credibility of real-time or near-real-time HIL testing.

Accordingly, this section summarizes an implementation approach that (i) preserves fixed-step, step-synchronous boundary coupling consistent with HB-$\pi$-ZIM’s one-step decoupling structure and (ii) reduces runtime jitter by eliminating unbounded blocking, dynamic allocation, and OS-side perturbations on solver cores. The implementation is presented as an enabling environment that matches the model’s computational interface; the primary contribution remains the line model and its terminal behavior.

4.2. RT-Linux Platform and DPDK-Based Low-Jitter Execution Substrate

Deterministic fixed-step execution in this work is achieved by combining (i) a real-time Linux platform (RT-Linux) with system-level isolation measures and (ii) a user-space runtime built on DPDK primitives. DPDK [19] is used here primarily as a low-jitter user-space execution and communication substrate (EAL core affinity and NUMA-aware hugepage memory, lockless rings, and fixed-size mempools), rather than for network I/O. The experimental software environment was based on Ubuntu 22.04 LTS with Linux kernel 5.15 LTS (PREEMPT_RT-enabled configuration), DPDK version 23.11.1, and GCC 11.4.0. All experiments were conducted under a fixed software configuration without runtime updates during testing.

The key platform measures are summarized below.

• Dedicated solver cores and OS isolation. Solver threads are affinitized to dedicated logical cores. Interrupt handling is isolated away from these solver cores via IRQ affinity configuration; housekeeping activities are confined to separate cores to prevent asynchronous interrupt servicing from perturbing the fixed-step loop. Where applicable, simultaneous multithreading (SMT) is avoided on solver cores (i.e., one solver thread per physical core) to reduce shared-resource contention and tail-latency outliers.
• Pinned CPU frequency and restricted power management. CPU frequency is pinned to avoid DVFS-induced variability in step execution time. Power-management features that introduce latency variability (e.g., aggressive idle states or dynamic boosting) are restricted as appropriate for the target real-time platform.
• NUMA-aware memory placement with hugepages. Subsystem-local data structures (state histories, stamping buffers, branch states, and boundary payloads) are allocated on the NUMA node local to the worker core whenever possible. Hugepage-backed memory is used to reduce TLB pressure and stabilize access latency.
• Preallocation of runtime objects. All boundary payload objects are preallocated at initialization from fixed-size object pools (DPDK mempools). This eliminates dynamic allocation inside the time-step loop and avoids allocator-induced jitter.
• Polling-based step loop with bounded synchronization. Each subsystem executes a polling-based fixed-step loop. The design avoids blocking synchronization primitives (e.g., mutexes/condition variables) in the step path; inter-core exchange is performed through bounded, non-blocking data structures described in Section 4.3.

These measures do not imply an unconditional guarantee of real-time deadlines on all platforms; rather, they establish an execution substrate with reduced jitter and improved timing predictability. Step-time instrumentation is employed to quantify deadline compliance and jitter statistics on the target RT-Linux system.

4.3. Lock-Free Boundary Exchange Matched to HB-π-ZIM

A partitioned EMT solver must exchange boundary quantities once per time step. The cost and determinism of this exchange depend on (a) payload size per boundary element and (b) synchronization overhead.

HB-π-ZIM minimizes (a) by enforcing a one-step travel time in its decoupling section (${\tau }_b=\Delta t$), making the boundary coupling strictly step-synchronous and interpolation-free. Each boundary line requires only a compact set of remote-terminal historical quantities from the previous step. Hence, communication scales with the number of boundary lines (cut-set size), not with the network size.

To minimize (b), inter-core exchange uses DPDK lock-free rings configured as single-producer/single-consumer (SP/SC) channels. After completing step $n$, a subsystem packages the required boundary payload into a preallocated object and enqueues a pointer; the receiving subsystem dequeues at the beginning of step $n+1$, updates the boundary history terms, and returns the object to the pool. This avoids blocking locks and reduces copying, keeping the exchange latency bounded and predictable.

In summary, the implementation harmonizes runtime communication semantics with the strictly step-synchronous decoupling of the HB-π-ZIM: boundary exchange is step-synchronous, bounded, and non-blocking in the steady state, which is essential for scalable parallel EMT execution and real-time HIL-oriented workflows on an RT-Linux platform.

5. Case Study Verification

The proposed model was implemented in C++ within a proprietary high-performance real-time simulation environment designed for large-scale EMT studies. Supplementary File S1 provides the supplementary code and data package used in this study, including the parameter calculator script, example C++ implementation files, plotting scripts, and raw data used for figure generation and comparison. The solver utilizes the parallel architecture described in Section 4 to distribute computational load across multiple cores. The verification compares the proposed HB-π-ZIM against the segmented transmission delay model (STDM) [14] and the benchmark small-step Bergeron traveling-wave model.

5.1. Simple Case Study

The structure of the simple case study is shown in Figure 4. To verify the model, both single-line-to-ground and three-phase-to-ground faults are simulated.

The main system parameters are as follows: The source line voltage is 110 kV with a solidly grounded neutral. The line length is 5 km (shorter than the approximately 15 km required for the conventional Bergeron model). The resulting total (5 km) sequence quantities used in the simulations (at 50 Hz) are as follows: positive-sequence resistance $R_1=1.00 \Omega$, zero-sequence resistance $R_0=3.00 \Omega$, positive-sequence reactance $X_1=1.30 \Omega$, zero-sequence reactance $X_0=5.65 \Omega$, positive-sequence inductance $L_1=X_1/\omega \approx 4.14 \mathrm{mH}$, zero-sequence inductance $L_0=X_0/\omega \approx 17.98 \mathrm{mH}$ with ($\omega =2\pi \times 50 \mathrm{rad}/\mathrm{s}$, positive-sequence capacitance $C_1=0.06 \mu \mathrm{F}$, and zero-sequence capacitance $C_0=0.03 \mu \mathrm{F}$. The load is 60 MW.

Simulation Scenario 1: At 4.0 s, a single-line-to-ground fault occurs in Phase A and lasts for 0.3 s. The Phase A short-circuit currents from the different models are shown in Figure 5.

Simulation Scenario 2: At 4.0 s, a three-phase-to-ground fault occurs and lasts for 0.3 s. The Phase A short-circuit currents from the different models are shown in Figure 6.

Figure 5 compares the Phase A fault current under a single-line-to-ground (SLG) fault. The Bergeron model simulated with a small step (Δt = 1 μs) is taken as the time-domain reference. HB-π-ZIM (Δt = 50 μs) closely tracks the reference waveform over both the initial transient and the subsequent quasi-steady fault interval. In contrast, STDM exhibits a noticeable amplitude bias and waveform deviation in the asymmetric-fault case, which is consistent with the expected sensitivity to zero-sequence parameter distortion when a unified delay is enforced across modes. The inset around the first crest further shows that the proposed model reproduces the crest magnitude and phase with only a marginal discrepancy relative to the 1 μs Bergeron reference.

Figure 6 shows the Phase A current under a three-phase-to-ground fault. Since the fault is symmetrical, the response is dominated by the positive-sequence network, and all models are therefore much closer than in the SLG case. HB-π-ZIM remains in excellent agreement with the 1 μs Bergeron reference. STDM also approaches the reference in this symmetric condition, indicating that the major discrepancy observed in Figure 5 is primarily associated with the zero-sequence sensitivity of asymmetrical faults rather than a generic transient-fidelity issue.

5.2. Typical Distribution System Case Study

Figure 7 shows the modified IEEE 33-bus test system, where distributed photovoltaics (PV) with a capacity of 100 kW each are connected at nodes 2, 4, 5, 8, and 26. During the simulation, all PV units operate at full power output and do not inject reactive power. The system frequency is 50 Hz, the connected source has a line voltage of 10 kV, and the short-circuit capacity is 122 MVA.

To verify the decoupling performance of the proposed model in multi-core CPU parallel computing, the modified IEEE 33-bus system is partitioned into three sub-regions, each assigned to a different CPU core. Consequently, two lines—the line between node 7 and node 8, and the line between node 5 and node 25—are modeled using the HB-π-ZIM. The 1 μs Bergeron case serves as a high-resolution time-domain reference for the boundary-line behavior (the two boundary lines are modeled by Bergeron, while internal lines remain lumped-π), whereas HB-π-ZIM and STDM operate at Δt = 50 μs.

The voltage at bus 5 is selected for detailed observation because it has a connected PV unit and is electrically close to the other two sub-regions simulated on separate CPU cores. The simulated Phase A voltage at bus 5 is shown in Figure 8. For current comparison, the Phase A current of the line connecting bus 0 and bus 1 is observed, as shown in Figure 9. During and after the fault, the distributed PVs remain connected to the grid, operating under a constant power and grid-following control strategy.

Figure 8 reports the Phase A voltage at bus 5 in the modified IEEE-33 system under a staged grounding fault applied at bus 17 (A-G at 4.0 s, AB-G at 4.1 s, and ABC-G at 4.2 s; all cleared at 4.3 s). HB-π-ZIM reproduces the 1 μs Bergeron reference waveform with nearly indistinguishable crest magnitude and phase in the zoomed window, demonstrating that the proposed zero-sequence-aware compensation remains effective in a multi-line network environment. By contrast, STDM yields a systematically lower crest in the zoomed interval (on the order of a few 10 mV per volt, i.e., ≈0.4% relative), which is consistent with the accumulated impact of zero-sequence mismatch during asymmetrical stages of the fault.

Figure 9 compares the Phase A current of the feeder section between bus 1 and bus 2 (cf. the line (1–2) in the standard IEEE-33 dataset). HB-π-ZIM matches the 1 μs Bergeron reference closely in both amplitude and phase. STDM, however, overestimates the first crest in the zoomed window by roughly 0.01 kA (≈2%), again indicating that the unified-delay treatment can introduce non-negligible errors when the network response involves zero-sequence components.

Notably, the HB-π-ZIM replaces a single line with a three-section equivalent circuit (containing 15 internal branches), which increases the model’s complexity compared to a simple lumped-π model. However, this complexity is a deliberate and strategic design choice. It is only applied to the handful of lines that cross subsystem boundaries, which are the bottlenecks in parallel simulation. For the vast majority of lines within each subsystem, simpler models can still be used. Therefore, this trade-off proves highly effective: the localized increase in complexity for a few boundary lines enables the entire system to be partitioned and solved in parallel, unlocking a significant gain in overall simulation speed. This design exchanges a minor, targeted local computational cost for a substantial global performance benefit.

5.3. Parallel Case

The experimental platform is hosted on a server equipped with an Intel Xeon Silver 4314 CPU (2.40 GHz, 32 physical cores). The operating system is Ubuntu 22.04, patched with PREEMPT_RT to ensure real-time determinism. Parallel scalability was evaluated on a county-level network with 327 nodes using a fixed EMT time step of $\Delta t=50\hspace{0.25em} \mu \mathrm{s}$. The single-core run (all lines modeled by the Lumped-π model) was used as the baseline. For the parallel runs, the network was partitioned into $p$ subsystems mapped to $p$ CPU cores; only the cut-set (boundary) lines were replaced by HB-$\pi$-ZIM to provide step-synchronous decoupling, while all internal lines remained Lumped-π. Speedup is defined as $S_p=T_1/T_p$, where $T_p$ is the measured wall-clock time per EMT step under $p$ cores. The boundary-line count $N_b$ denotes the number of cut-set lines (HB-$\pi$-ZIM instances) required by the partition.

Figure 10 provides an implementation-level view of the real-time prototype used in this study and highlights the main components involved in fixed-step parallel execution and runtime monitoring. Under the HIL timing measurement, the observed step-time jitter exhibits a standard deviation of 0.42 μs and a peak-to-peak deviation below 2 μs, indicating highly deterministic fixed-step execution for the proposed parallel real-time setup. This timing performance and provides sufficient margin for process-bus protection testing, for which timing deviations on the order of a few microseconds are generally expected (with many practical test setups adopting a ≤10 μs jitter budget).

Table 3. Parallel speedup and boundary-line count.

$\mathbf{CPU} \mathbf{Cores} \mathit{p}$	$\mathbf{Speedup} {\mathit{S}}_{\mathit{p}}={\mathit{T}}_1/{\mathit{T}}_{\mathit{p}}$	$\mathbf{Boundary} \mathbf{Lines} {\mathit{N}}_{\mathit{b}}$$(\mathbf{HB}$-π-ZIM Instances)
1	1.00	0
2	1.99	4
3	2.96	6
4	3.94	7
5	4.88	10
6	5.86	11
7	6.78	11
8	7.61	13
9	8.50	15
10	9.31	16

shows that the proposed boundary-line strategy achieves near-linear speedup over 2 to 10 cores in this test case. This scaling behavior is enabled by (i) restricting HB-π-ZIM to the cut-set only (keeping small relative to the total number of lines) and (ii) maintaining strictly step-synchronous boundary coupling via $\tau =\Delta t$, which bounds boundary payload size and avoids interpolation/sub-stepping overhead. The additional internal states introduced by HB-π-ZIM therefore affect only a small fraction of the network elements and do not dominate runtime.

6. Conclusions and Future Work

This paper proposed HB-π-ZIM, an interpolation-free hybrid Bergeron–π boundary-line model for real-time fixed-step EMT simulation when τ < Δt. A one-step uncoupled per-phase Bergeron section enforces a step-synchronous interface, and shunt compensation plus a passive RL two-port synthesis restores the lumped-π terminal sequence networks at 50/60 Hz, including the zero-sequence behavior. Case studies show close agreement with the lumped-π benchmark under balanced and unbalanced transients, and parallel results on a 327-node network demonstrate near-linear scaling (9.31× on 10 cores) when HB-π-ZIM is applied only to cut-set lines. Future work will extend the equivalence bandwidth and generalize the formulation to untransposed/asymmetrical lines and cable geometries.