Non-Abelian Holonomy & Geometric Phase Shifts in Microgrid Reconfigurations

J. McKenney

This paper sits in the numbered WG-02-DT digital twin series alongside WG-02-DT-06's symplectic integrators for substation physics and WG-02-DT-09's treatment of asynchronous distributed consensus under Byzantine adversaries in substation automation. Its own mathematical result is that non-Abelian holonomy detects stealth topology poisoning.

Licence: CC BY 4.0. 17 September 2026.

Executive Abstract#

Distribution microgrids built on fast solid-state inverters behave differently from the high-inertia transmission systems classical tools were built for. Conventional monitoring treats each switching event, a feeder tie closing, a fault isolated, a transition into island mode, as a discrete change to the grid's admittance matrix, discarding the continuous physical path the grid takes between one configuration and the next.

This treatise shows that a microgrid which reconfigures and returns to its starting topology does not necessarily return to its starting electrical phase. The path taken through parameter space leaves a measurable geometric trace, independent of how long the reconfiguration took. Because that trace is a property of the physical switching path itself, an adversary who forges only the status messages a control system reads cannot reproduce it without reproducing the underlying physical trajectory.

The practical result is a real-time detector built on that trace. A monitoring system tracking this geometric quantity across a switching sequence flags a stealth topology-poisoning attack, one that falsifies breaker status and power-injection telemetry together to defeat conventional state estimation, within a few milliseconds, before an out-of-phase reclosure destroys switchgear or transformer windings.

Abstract#

Inverter-based resources in distribution microgrids have altered power system dynamics. Unlike bulk transmission driven by high-inertia synchronous machines, these microgrids run on fast solid-state switching, low line X/R ratios, and dynamic reconfiguration, switching feeder ties, isolating faulted segments, and transitioning between grid-connected and islanded modes. Conventional SCADA platforms and digital twins treat switching events as discrete algebraic updates to the nodal admittance matrix, discarding the continuous evolution of electromagnetic state trajectories across parameter space. McKenney and the Eigenia Cyber Digital Twin Working Group show that reconfigurations in balanced three-phase converter microgrids induce geometric phase shifts governed by non-Abelian holonomy on principal fiber bundles. We model the parameter space M of branch conductances and inverter setpoints under the gauge symmetry group SU(2), representing rotating reference frames and coupled active-reactive power dynamics, and define a non-Abelian connection form over the bundle P(M, SU(2)). Cyclic reconfigurations trace closed paths whose parallel transport yields path-ordered Wilson loop holonomies. This holonomy is gauge-invariant and physically unforgeable. When an adversary launches stealth topology poisoning through compromised IEC 61850 GOOSE messages or DNP3 commands, spoofing breaker statuses while manipulating bus injections to bypass linear state estimators, the actual electromagnetic path violates the holonomy condition. Our real-time geometric digital twin detects malicious switching sequences within 2.1 ms, under 0.13 cycles at 60 Hz, preventing inverter overcurrent tripping and mechanical shaft fatigue in hybrid diesel-BESS installations.

1. Introduction and Problem Formulation#

Microgrid architectures represent the operational backbone of localized electrical resilience, integrating distributed energy resources (DERs) such as rooftop solar photovoltaics, battery energy storage systems (BESS), and reciprocating combined heat and power (CHP) generators. To respond dynamically to fluctuating solar irradiance, industrial load spikes, or upstream utility outages, microgrids rely on automated distribution management systems (ADMS) and microgrid controllers ($\mu\text{GC}$). These controllers execute automated switching sequences that alter the network's topological connectivity.

Mathematical Inadequacy of Algebraic State Estimation#

Historically, distribution network reconfigurations have been treated as quasi-static transitions. Power flow solvers modify the system admittance matrix $Y_{\mathrm{bus}} \in \mathbb{C}^{N \times N}$ from configuration $A$ to configuration $B$ via discrete rank-one or rank-two updates:

where $\Delta Y$ accounts for line admittance removal or addition upon breaker actuation:

While this algebraic treatment is sufficient for slow, high-inertia grids, it introduces severe blind spots in inverter-dominated networks:

1. Continuous Transients in Parameter Space: Physical breakers do not switch instantaneously; vacuum interrupters, solid-state switches, and arc suppression dynamics produce a continuous trajectory through impedance space lasting several milliseconds.
2. Rotating Frame Symmetries: Inverter inner current loops track rotating synchronous reference frames ($dq0$) via Phase-Locked Loops (PLL). Non-linear coupling between active current ($i_d$) and reactive current ($i_q$) under cross-axis decoupling filters creates a dynamical system whose state space possesses non-Euclidean geometry.
3. Accumulation of Path-Dependent Phase: When a microgrid undergoes a cyclic sequence of reconfigurations, such as transferring a feeder from Feeder 1 to Feeder 2 and back to Feeder 1, the terminal voltage phase angle does not return to its initial value. Instead, it accumulates a geometric phase shift that depends strictly on the area enclosed in parameter space, entirely distinct from the dynamical phase resulting from time elapsed.

The Stealth Topology Poisoning Threat#

As microgrids become software-defined, communication protocols between digital substations and intelligent electronic devices (IEDs) present an expansive attack surface. Threat actors targeting operational technology (OT) protocols, specifically unauthenticated or weakly authenticated IEC 61850 GOOSE, MMS, or IEEE 2030.5 telecontrol channels, can falsify breaker status bits.

In a stealth topology poisoning attack, the adversary manipulates telecontrol payloads such that the central digital twin observes a legitimate sequence of breaker operations while the physical plant is driven into an unsynchronized parallel connection or an unmonitored loop flow. By concurrently injecting false power injection telemetry via compromised remote terminal units (RTUs), the attacker satisfies standard Weighted Least Squares (WLS) residual thresholds:

Because linear state estimators evaluate static algebraic snapshots without tracking geometric phase accumulation across the switching path, the attack proceeds undetected until inverter bridges trigger overcurrent trips, islanding the microgrid and damaging downstream industrial machinery.

I should say plainly where this paper sits against its companion in the Mathematical Physics working group. Non-Abelian Gauge Symmetries and Conserved Topological Currents in Interconnected OT Microgrids takes the same gauge group onto a discrete multigraph instead of a continuous parameter manifold, and derives from it a conserved Noether current whose divergence localizes an injected defect to a particular branch in under two milliseconds (reference 7). That paper answers where the anomaly is on a topology that is not moving; this one answers what a cyclic reconfiguration leaves behind after the topology returns to where it started, which a discrete current cannot see because it has no path through parameter space to enclose.

2. Mathematical Foundations & Physical Derivations#

To capture the true physical evolution of reconfiguring microgrids, we formulate the system dynamics using the differential geometry of fiber bundles and non-Abelian gauge theory.

The Configuration Bundle of an Inverter Microgrid#

Let the microgrid comprise $N$ three-phase electrical buses interconnected by dynamic transmission lines and solid-state switches. The operational configuration of the network is parameterized by a smooth, finite-dimensional manifold $M$. Coordinates on $M$ represent branch conductances $g_{ij}$, susceptances $b_{ij}$, and inverter droop control parameters:

At each fixed operating point $\lambda \in M$, the instantaneous electromagnetic state of the three-phase inverters is characterized by the internal converter voltage vector $\mathbf{v}(t) \in \mathbb{R}^{3N}$. In balanced three-phase systems, Clarke and Park transformations map instantaneous physical phase quantities $(a, b, c)$ into the orthogonal rotating frame $(d, q, 0)$. When active and reactive power balance is maintained, the dynamical state equations are invariant under global and local phase rotations.

We model this symmetry group as the special unitary Lie group $G = \mathrm{SU}(2)$, which is isomorphic to the double cover of $\mathrm{SO}(3)$ and represents rotational symmetry in two-dimensional complex phasor space. The total state space forms a principal fiber bundle:

where $M$ is the base manifold of topological parameters, $\pi: P \to M$ is the canonical projection, and the fiber $\pi^{-1}(\lambda) \cong \mathrm{SU}(2)$ represents the internal gauge degree of freedom of converter phase angles.

Derivation of the Non-Abelian Gauge Connection#

Let the continuous-time dynamics of the converter microgrid be governed by the differential equation:

where $x(t) \in \mathbb{C}^{N}$ denotes the concatenated complex phasor state, pairing the direct and quadrature axis currents and voltages at each of the $N$ buses into a single complex phasor $x_k = x_{d,k} + i\, x_{q,k}$, consistent with the two-dimensional complex phasor space introduced in Section 2. Under adiabatic or quasi-steady parameter variations $\dot{\lambda}(t)$, the instantaneous state $x(t)$ remains close to the instantaneous eigenspace of the linearized Jacobian operator $J(\lambda) = \left. \frac{\partial f}{\partial x} \right|_{x^*(\lambda)}$.

Let $\{ |\psi_a(\lambda)\rangle \}_{a=1}^k$ be a degenerate or near-degenerate $k$-dimensional subspace of oscillatory modes (e.g., cross-coupled inter-inverter modes) corresponding to an eigenvalue manifold $\sigma(\lambda)$. Following the Wilczek-Zee geometric formulation, the connection form $\mathcal{A}$ constructed from this subspace carries the non-Abelian holonomy this treatise studies.

Assumption (Symmetrizable Spectral Block). The construction below requires an orthonormal basis $\langle \psi_a | \psi_b \rangle = \delta_{ab}$ of the subspace, which in turn requires $J(\lambda)$ to be normal, or the subspace to be restricted to a block on which it is, so that left and right eigenvectors coincide. A converter-dominated microgrid is dissipative, and $J(\lambda)$ is not normal in general: line resistance and inverter damping break the symmetry that would guarantee orthogonal eigenvectors, so a generic non-normal $J(\lambda)$ gives distinct left and right eigenvectors. Two routes keep the construction well-posed. The symmetrized-block route, taken throughout the rest of this treatise, restricts attention to a spectral block of $\sigma(\lambda)$ on which the damping acts as a scalar multiple of the identity, so the restricted operator is normal and the orthonormal basis above exists; this is the condition that makes $\mathcal{A}_\mu$ anti-Hermitian. The biorthogonal route instead pairs each right eigenvector $|\psi_a\rangle$ with its left eigenvector $\langle \chi_a|$ under $\langle \chi_a | \psi_b \rangle = \delta_{ab}$ and defines $(\mathcal{A}_\mu)_{ab} = \langle \chi_a(\lambda) | \partial_{\lambda^\mu} \psi_b(\lambda) \rangle$; the resulting connection is valued in the general linear algebra $\mathfrak{gl}(k, \mathbb{C})$ and the structure group widens correspondingly from $\mathrm{SU}(2)$ to $\mathrm{GL}(2, \mathbb{C})$. Either route leaves the stealth-detection result of Section 3 intact: the discrepancy metric $\Delta\mathcal{W}$ is built from a trace, which is conjugation-invariant in $\mathrm{GL}(2, \mathbb{C})$ exactly as it is in $\mathrm{SU}(2)$, so a topology-poisoning attack still produces a non-zero discrepancy under the biorthogonal formulation. What the choice of route changes is the group label and the normalization of the connection, not whether the detector fires.

Differentiating the orthonormality condition $\langle \psi_a | \psi_b \rangle = \delta_{ab}$ along $\lambda^\mu$ shows that the connection built from inner products of the symmetrized spectral block is anti-Hermitian, $\mathcal{A}_\mu^\dagger = -\mathcal{A}_\mu$, which places it in $\mathfrak{u}(2) = \mathfrak{u}(1) \oplus \mathfrak{su}(2)$, not in $\mathfrak{su}(2)$ directly: anti-Hermitian matrices need not be traceless. We take the same decomposition used for the discrete gauge group of the companion treatise on this microgrid class, where the full symmetry group is $G = \mathrm{U}(1) \times \mathrm{SU}(2)$ (reference 7): the trace part of $\mathcal{A}_\mu$ generates the $\mathrm{U}(1)$ sector, an overall phase common to every mode that carries no non-Abelian content, and the traceless part generates the $\mathrm{SU}(2)$ sector in which the holonomy studied here actually lives.

In local coordinates $\lambda^\mu$ on $M$, the components of the full connection are given by inner products over the complex Hilbert space:

and its traceless part, the $\mathfrak{su}(2)$-valued connection carried through the rest of this treatise, is

Because the trace part of $\mathcal{A}_\mu$ is proportional to the identity, it commutes with every other term in the connection and drops out of every commutator $[\mathcal{A}_\mu, \mathcal{A}_\nu]$, the Wilson loop's non-Abelian content, and the discrepancy metric $\Delta\mathcal{W}$ of Section 3; projecting it away changes none of those quantities. From here, $\mathcal{A}$ denotes $\mathcal{A}^{\mathrm{su}(2)}$, an $\mathfrak{su}(2)$-valued differential 1-form defined on $M$, and the connection 1-form is expressed as:

where $\tau_k = -i \frac{\sigma_k}{2}$ are the generators of the Lie algebra $\mathfrak{su}(2)$, and $\sigma_k$ are the standard Pauli spin matrices:

The Lie bracket on $\mathfrak{su}(2)$ satisfies the commutation relation:

where $\epsilon_{jkl}$ is the Levi-Civita permutation tensor.

Curvature Two-Form and Non-Abelian Field Strength#

The field strength tensor (curvature 2-form) $\mathcal{F}$ of the connection $\mathcal{A}$ on the base manifold $M$ is defined via the Cartan structural equation:

In component notation:

The non-vanishing Lie bracket $[\mathcal{A}_\mu, \mathcal{A}_\nu] \ne 0$ embodies the non-Abelian character of the system. This mathematical property implies that sequential reconfigurations do not commute in general: closing switch 1 then switch 2 produces a fundamentally different physical state than closing switch 2 then switch 1.

Wilson Loop Formulation and Geometric Phase Shift#

Consider a cyclic operational reconfiguration where the microgrid begins at parameter configuration $\lambda_0 \in M$, undergoes an operational cycle along a smooth closed curve $\mathcal{C}: [0, T] \to M$ with $\mathcal{C}(0) = \mathcal{C}(T) = \lambda_0$. Parallel transport of the converter state vector around $\mathcal{C}$ is governed by the matrix differential equation:

The terminal transformation $U(T) \in \mathrm{SU}(2)$ is the holonomy of the connection around $\mathcal{C}$, commonly termed the Wilson loop operator:

where $\mathcal{P}$ denotes the path-ordering operator. Using the non-Abelian Stokes theorem, the Wilson loop can equivalently be expressed as a surface-ordered integral over any smooth two-dimensional orientable surface $\Sigma \subset M$ bounded by $\partial \Sigma = \mathcal{C}$, with the field strength at each point $\lambda \in \Sigma$ parallel-transported back to the basepoint $\lambda_0$ along a path within $\Sigma$:

where $U(\lambda_0, \lambda) \in \mathrm{SU}(2)$ is the Wilson line from $\lambda_0$ to $\lambda$ along that path. The conjugation by $U(\lambda_0, \lambda)$ is what makes the right-hand side transform covariantly under a local gauge transformation; the bare field strength $\mathcal{F}_{\mu\nu}(\lambda)$ alone does not. Nothing later in this treatise depends on the surface-ordered form; the line-integral definition of $W(\mathcal{C})$ above is what the hardware-in-the-loop engine of Section 3 evaluates.

Under a local gauge transformation $g(\lambda) \in \mathrm{SU}(2)$, the connection transforms as:

and the holonomy transforms by conjugation at the basepoint $\lambda_0$:

Therefore, the trace of the Wilson loop, defined as the Wilson loop invariant:

is strictly gauge-invariant and independent of local coordinate choices.

3. Empirical Benchmarks & Cyber-Physical Validation#

To validate the non-Abelian holonomy digital twin, we established an experimental co-simulation testbed coupling real-time electromagnetic transient (EMT) solvers with a cyber-attack emulation network.

Hardware-in-the-Loop Testbed Setup#

The experimental microgrid testbed models an industrial facility comprising four distributed microgrid segments:

• Substation Bus 1: $2.5\text{ MVA}$ Grid-Forming Battery Energy Storage System (BESS) running virtual synchronous machine (VSM) inertia synthesis.
• Feeder Bus 2: $1.8\text{ MW}$ Grid-Following Solar Photovoltaic Inverter array with MPPT and reactive power droop.
• Industrial Feeder Bus 3: Variable-speed induction motor loads ($1.2\text{ MVA}$, $0.85$ power factor).
• Tie-Lines and Switches: Three vacuum circuit breakers ($CB_{12}$, $CB_{23}$, $CB_{31}$) controlled via IEC 61850-8-1 GOOSE messaging with high-speed sampled value streams (IEC 61850-9-2 LE, 80 samples per cycle at $4.8\text{ kHz}$).
• Digital Twin Engine: Implemented on an embedded industrial server computing the discrete Wilson loop integral in $C++20$ via matrix Lie-algebra exponential Taylor expansions.

Quantitative Experimental Results#

We evaluated three scenarios:

1. Legitimate Operational Reconfiguration (Baseline): Sequential transfer of industrial load from Feeder 1 to Feeder 2 during maintenance, closing $CB_{23}$ before opening $CB_{12}$ (make-before-break).
2. Stealth Topology Poisoning (Cyber Attack): Adversary manipulates GOOSE breaker telemetry to report $CB_{12}$ open when it is actually closed, while injecting false power flow values into SCADA to simulate normal islanded operation.
3. Severe Physical Transients (False Alarm Stress Test): Three-phase line-to-ground fault on an adjacent non-critical feeder, causing severe voltage depression ($0.45\text{ p.u.}$) and rapid inverter current limiting.

The observed performance metrics are documented below:

Empirical Analysis of the Phase Drift#

In Scenario 2, while the conventional state estimator's normalized residual remained well below the $\chi^2$ significance threshold of $7.81$, the geometric holonomy invariant:

spiked from its nominal baseline of $0.003$ to $0.438$ within $2.1\text{ ms}$. This dramatic signal-to-noise ratio occurs because the physical trajectory traced an unannounced loop in the non-Abelian curvature field $\mathcal{F}_{\mu\nu}$, accumulating a non-commutative phase shift that cannot be neutralized by linear scaling of bus telemetry.

Conversely, in Scenario 3, the external electrical fault induced severe voltage distortion that caused classical residual estimators to trigger false alarms. However, because the external fault did not alter the internal switching loop in parameter space $M$, the non-Abelian connection form remained localized, and $\Delta \mathcal{W}$ remained safely below the alert threshold of $0.05$.

4. Regulatory Mapping & Residual Exposure#

Deploying geometric digital twin observers directly addresses emerging cybersecurity and critical infrastructure regulations.

Statutory Compliance Mapping#

1. EU Cyber Resilience Act (CRA, Regulation 2024/2847):
   • Annex I, Section 1 (Essential Cybersecurity Requirements): Requires products with digital elements to maintain integrity of control systems and prevent unauthorized access or state manipulation. The geometric holonomy engine provides hardware-level verification of control integrity, satisfying Section 1(3)(e) requirements for tamper-evident operational states.
2. NIS2 Directive (Directive 2022/2555):
   • Article 21 (Cybersecurity Risk-Management Measures): Demands that energy utilities implement continuous operational monitoring, incident handling, and supply chain security. The sub-cycle detection latency ($2.1\text{ ms}$) satisfies Article 23 mandatory early-warning criteria for high-impact industrial disruptions.
3. IEEE 1547-2018 Standard for Interconnection of Distributed Energy Resources:
   • Mandates rigorous anti-islanding detection within $2.0\text{ seconds}$ under unintentional separation while preventing nuisance trips during ride-through events. The geometric observer decouples physical topology verification from voltage depressions, eliminating ride-through false disconnects.

Residual Exposure and Detection Speed#

Once detection is fast relative to the physical damage timescale, as it is here ($\tau_{\mathrm{detect}} = 2.1\text{ ms}$ against a thermal and electromechanical destruction threshold of $\tau_{\mathrm{damage}} \ge 16.6\text{ ms}$, one 60 Hz cycle), the residual risk that matters for actuarial purposes stops being dominated by detection speed and becomes dominated by the detector's miss rate instead: how often a genuine stealth topology attack fails to trigger $\Delta\mathcal{W}$ above threshold at all. Turning that observation into a premium discount factor requires a validated miss rate for the geometric holonomy detector under adversarial conditions, a figure this treatise does not derive from the Section 3 benchmarks and does not assign a value to here.

5. Conclusion & Implementation Roadmap#

The incorporation of non-Abelian differential geometry into cyber-physical digital twins bridges the longstanding divide between continuous electromagnetic transient physics and discrete operational technology cybersecurity. By treating microgrid operational reconfigurations as smooth trajectories over principal fiber bundles, asset owners can extract unforgeable topological invariants that expose malicious telecontrol injections in under three milliseconds.

Architectural Deployment Phasing#

1. Phase 1 (Parameter Manifold Characterization): Construct the offline differential model of the microgrid base manifold $M$, identifying all valid operational switching curves $\mathcal{C}_k$ and computing the non-Abelian connection matrices $(\mathcal{A}_\mu)_{ab}$ from electromagnetic inverter parameters.
2. Phase 2 (Edge Execution & Stream Ingestion): Compile the Wilson loop evaluation kernel into real-time C++20 microservices running on IEC 61850-3 certified substation edge computers, ingesting 80-sample/cycle PMU telemetry via multicast network taps.
3. Phase 3 (Protective Interlocking & Actuarial Binding): Wire observer alert outputs directly to master trip interlocks via hardware contact outputs or high-priority GOOSE publisher queues, locking microgrid breakers into failsafe configurations upon detection of holonomy discrepancies.

6. References#

1. Wilczek, F., & Zee, A. (1984). Appearance of gauge structure in simple dynamical systems. Physical Review Letters, 52(24), 2111-2114.
2. Berry, M. V. (1984). Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802), 45-57.
3. Liu, Y., Ning, P., & Reiter, M. K. (2011). False data injection attacks against state estimation in electric power grids. ACM Transactions on Information and System Security, 14(1), 1-33.
4. CIGRE Working Group C4.56. (2022). Electromagnetic transient simulation models for large-scale system impact studies in power systems having a high penetration of inverter-connected generation (Technical Brochure 881). CIGRE.
5. European Parliament & Council. (2024). Regulation (EU) 2024/2847 on horizontal cybersecurity requirements for products with digital elements (Cyber Resilience Act). Official Journal of the European Union.
6. Institute of Electrical and Electronics Engineers. (2018). IEEE Standard for Interconnection and Interoperability of Distributed Energy Resources with Associated Electric Power Systems Interfaces (IEEE Std 1547-2018). IEEE.
7. McKenney, J. (2026). Non-Abelian Gauge Symmetries & Conserved Topological Currents in Interconnected OT Microgrids. Eigenia Research Working Group MP-MATH Treatise MP-MATH-05. Cited for the discrete multigraph formulation of the same gauge group and for the conserved Noether current, neither of which is derived here.