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QSVT-Based Three-Phase Unbalanced Power Flow

Kamini Shahare, Peng Zhang

arXiv:2609.03165v1cs.ET

TL;DR

Three-phase unbalanced power flow in larger, more inverter-dominated distribution networks motivates scalable alternatives, while existing quantum studies have primarily used HHL-type or variational solvers. QSVT-3PF recasts the Newton correction as a filtered Jacobian inverse problem, and the reported tests show close agreement with classical Newton solutions on the IEEE 5-bus and 123-node systems.

  • Problem

    Existing quantum power-flow studies have primarily used HHL-type or variational linear solvers, while larger, more unbalanced, inverter-dominated distribution networks motivate scalable alternatives.

    Source: §I · p. 1

  • Method

    QSVT-3PF applies regularized QSVT updates to a Newton correction reformulated as a matrix-inversion problem with a normalized block-encoded Jacobian.

    Source: §I · p. 1, §II · p. 1

  • Results

    On the IEEE 123-node feeder, Quantum QSVT PF and Classical Newton PF produce identical values at every iteration and converge at the fourth iteration.

    Source: B. IEEE 123-Node Test Feeder · p. 3

  • Takeaways & Limitations

    The formulation replaces only the Jacobian solve step and is presented as a reusable computational primitive for repeated grid analytics.

    Source: §VI · p. 3

Abstract

from arXiv · show

This letter introduces QSVT-3PF, a quantum singular value transformation (QSVT) based solver for three-phase unbalanced power flow with embedded single-phase grid-forming inverter (GFM) operation. The contributions are threefold: 1) reformulating the Newton correction as a QSVT-compatible inverse problem using a normalized block-encoded phase-domain Jacobian; 2) introducing a regularized singular-value filter to improve robustness under ill-conditioned and stressed operating conditions; and 3) validating the proposed solver on an IEEE 5-bus system and the IEEE 123-node test feeder with single-phase GFM integration. Test results show that QSVT-3PF matches the classical Newton benchmark in residual convergence, voltage profile, and final operating point, demonstrating the feasibility of QSVT-3 for large, unbalanced distribution systems.

I. INTRODUCTION

The paper presents QSVT-3PF for three-phase unbalanced power flow with embedded single-phase GFM operation. It targets the Jacobian inverse step in Newton power flow and tests the solver on two unbalanced-system benchmarks.

I. INTRODUCTION

  • Earlier quantum power-flow studies primarily used HHL-type or variational linear solvers, while the paper motivates QSVT as an alternative for Jacobian inversion.The passage notes demanding state-preparation and circuit-depth assumptions for HHL, and possible trainability limitations and classical optimization dependence for variational methods.Source: §I · p. 1
  • The method reformulates the Newton correction as a QSVT-compatible matrix-inversion problem with a normalized block-encoded Jacobian, then applies regularized QSVT updates.The updates use a polynomial inverse singular-value filter to compute Newton corrections without variational optimization.Source: §I · p. 1
  • QSVT-3PF is validated on the IEEE 5-bus system and IEEE 123-node feeder with single-phase GFM integration.The stated aim is to demonstrate applicability to unbalanced inverter-dominated distribution networks.Source: §I · p. 1

II. THREE-PHASE UNBALANCED POWER FLOW PROBLEM

  • In the embedded-GFM formulation, the residual includes a droop-frequency term and the common frequency is solved alongside phase-domain voltages.The GFM is represented through its phase-specific network injection and active-power/frequency droop behavior.Source: §II · p. 1
  • At each Newton iteration, the correction solves J(x_k)Δx_k = −r(x_k), whose augmented three-phase Jacobian inverse action is the targeted computational bottleneck.Source: §II · p. 1

III. QSVT-BASED NEWTON CORRECTION

The stated objective is to use quantum singular value transformation to approximate the Newton correction.

III. QSVT-BASED NEWTON CORRECTION

  • QSVT-3PF aims to approximate the Newton correction using quantum singular value transformation.Source: §III · p. 1

A. Jacobian Normalization

The section normalizes the Jacobian before expressing the Newton correction through its singular-value decomposition and inverse action.

A. Jacobian Normalization

B. QSVT Inverse Polynomial

The QSVT inverse polynomial approximates the inverse singular-value map over a restricted interval to avoid ill-conditioning near zero.

B. QSVT Inverse Polynomial

  • Because direct inversion becomes ill-conditioned near zero, the QSVT polynomial approximates f(σ) = 1/σ only on the well-conditioned interval.Source: B. QSVT Inverse Polynomial · p. 2

C. Regularized QSVT Filter

The section introduces a regularized inverse filter for ill-conditioned cases and uses it in the QSVT Newton correction.

C. Regularized QSVT Filter

IV. QUANTUM BLOCK-ENCODING FORMULATION

The formulation block-encodes the normalized Jacobian and applies QSVT polynomial filtering to a residual state to approximate a Newton correction.

IV. QUANTUM BLOCK-ENCODING FORMULATION

  • The normalized Jacobian is embedded in a larger unitary through a block encoding.The encoding unitary is denoted UJ, and a is the number of ancilla qubits.Source: §IV · p. 2
  • QSVT alternates applications of UJ, U†J, and single-qubit phase rotations, with phase angles chosen to implement the polynomial pd(σ).Source: §IV · p. 2
  • After rescaling by ∥r∥/α, the resulting vector approximates the Newton correction.The QSVT circuit is applied to the encoding of the residual vector.Source: §IV · p. 2
  • The solver normalizes each iteration’s Jacobian and applies either an inverse or regularized inverse filter to compute the QSVT Newton correction.Algorithm 1 identifies the inputs as network admittance, load data, GFM droop, initial voltage, tolerance, and iteration limit; its outputs are voltage and residual history.Source: §IV · p. 2

V. TEST CASE SCENARIOS

The evaluation uses a compact IEEE 5-bus system and the IEEE 123-node test feeder to assess QSVT-3PF in three-phase unbalanced settings.

V. TEST CASE SCENARIOS

  • QSVT-3PF is validated on the IEEE 5-bus system and the IEEE 123-node test feeder.Source: §V · p. 2
  • The IEEE 5-bus system verifies the QSVT inverse approximation in a compact network.Source: §V · p. 2
  • The IEEE 123-node feeder evaluates the method under feeder-scale three-phase unbalanced conditions with single-phase GFM.Source: §V · p. 2

A. IEEE 5-Bus System

On the IEEE 5-bus system, QSVT-3PF closely matches classical Newton convergence and reaches the same final operating point.

A. IEEE 5-Bus System

  • Classical Newton PF and Quantum QSVT PF exhibit nearly identical convergence behavior and reach the same final voltage profile and droop-regulated frequency.Source: A. IEEE 5-Bus System · p. 2
  • The close overlap of voltage magnitudes verifies that QSVT-3PF preserves the classical steady-state operating point.Source: A. IEEE 5-Bus System · p. 2
  • The QSVT singular-value filtering step produces an equivalent Newton correction for the augmented three-phase Jacobian.Source: A. IEEE 5-Bus System · p. 2

B. IEEE 123-Node Test Feeder

On the IEEE 123-node feeder, Quantum QSVT PF and Classical Newton PF follow identical iteration-wise trajectories and converge to the same operating point. The feeder-scale test includes phase-specific laterals and highly unbalanced loading.

B. IEEE 123-Node Test Feeder

  • The IEEE 123-node feeder provides a large-scale unbalanced benchmark with more voltage nodes, phase-specific laterals, and highly unbalanced loading.Fig. 2 presents the feederwide voltage-magnitude profile produced by QSVT-3PF.Source: B. IEEE 123-Node Test Feeder · pp. 2–3, Figure 2 · p. 3
  • Both solvers produce identical iteration-wise values and converge at iteration four to V40A = 0.9979 pu, V82B = 1.0395 pu, V114A = 1.0254 pu, and ω = 0.9998 pu.The comparison covers three GFM-connected voltage quantities and frequency.Source: B. IEEE 123-Node Test Feeder · p. 3, Table I · p. 3
  • The voltage-magnitude distributions at GFM buses closely agree between classical Newton and QSVT, indicating preserved voltage solutions without noticeable numerical dispersion.The histograms show concentration around the converged operating point.Source: B. IEEE 123-Node Test Feeder · p. 3, Figure 3 · p. 3

VI. CONCLUSION

QSVT-3PF recasts three-phase unbalanced power flow as singular-value filtering, replacing the Jacobian solve while retaining classical network-equation formation. Its regularization addresses weak Jacobian singular directions, and the formulation is proposed as a reusable primitive for repeated grid analytics.

VI. CONCLUSION

  • QSVT-3PF recasts three-phase unbalanced power flow from a direct Newton solve into a singular-value filtering problem.The network equations and three-phase Jacobian are formed as in a classical Newton power-flow program; QSVT replaces the Jacobian-solve step for the voltage correction.Source: §VI · p. 3
  • Regulating weak Jacobian singular directions makes the formulation well suited for stressed, unbalanced, and inverter-rich feeders.Source: §VI · p. 3
  • QSVT-3PF offers a reusable computational primitive for repeated grid analytics, including feasible operating-region analysis, DER hosting studies, and future quantum-enabled distribution-grid digital twins.Source: §VI · p. 3
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