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Embedding Single-Phase Grid-Forming Inverters in Three-Phase Unbalanced Power Flow
Kamini Shahare, Fei Feng, Peng Zhang
TL;DR
GFM-3PF addresses the need to model single-phase grid-forming inverters in three-phase unbalanced networks. It combines phase-domain network and load modeling with droop-constrained Newton solution, and demonstrates accuracy, robustness, and scalability across three test systems.
Problem
Existing formulations do not provide a unified way to embed droop-controlled single-phase GFM behavior into three-phase unbalanced network equations.
Method
GFM-3PF uses augmented Newton equations with positive-sequence initialization, phase-domain unbalanced modeling, frequency-sensitive loads, and a common droop-governed frequency state.
Results
GFM-3PF demonstrates accuracy, robustness, and scalability on 3-bus, IEEE 33-bus, and IEEE 118-bus systems.
Takeaways & Limitations
GFM-3PF provides a practical steady-state analysis tool for inverter-dominated unbalanced power systems.
Abstract
from arXiv · showhide
This letter introduces Grid-Forming Three-Phase Power Flow (GFM-3PF), an augmented Newton power flow framework for modeling single-phase grid-forming inverters (GFMs) in three-phase unbalanced networks. The contributions of this work are threefold: 1) developing a two-stage solution strategy based on positive-sequence initialization followed by three-phase phase-domain lifting for robust large-scale computation; 2) incorporating phase-domain unbalanced network modeling with controlled mutual coupling together with phase-dependent, frequency-sensitive load representation; and 3) embedding a single-phase GFM bus model directly into the three-phase unbalanced Newton power flow equations through a common droop-governed frequency state. GFM-3PF is validated on 3-bus, 33-bus, and 118-bus test systems, demonstrating its accuracy, robustness, and scalability.
I. INTRODUCTION
Conventional power flow must account for inverter-based resources, three-phase unbalance, and reduced synchronous generation. GFM-3PF embeds single-phase GFM droop behavior into three-phase equations while solving phase voltages and common frequency together.
- Standard slack or PV buses do not correctly represent GFM steady-state operating points governed by droop control.
- GFM-3PF directly enforces single-phase GFM droop equations in the augmented Newton residual.
- The framework simultaneously solves phase-domain voltage profiles and droop-governed frequency under unbalanced loading and frequency-sensitive demand.
- The common system frequency is an additional unknown determined by participating GFMs' active-power/frequency droop relations.
A. Single-Phase GFM Bus Formulation
The single-phase GFM bus model determines its operating point through network equations, voltage regulation, and active-power/frequency droop rather than fixed frequency and active power.
- Unlike a conventional slack bus, a GFM determines its steady-state operating point through network equations and active-power/frequency droop.
- The aggregate droop model incorporates scheduled or nominal active-power setpoints to avoid tying frequency deviation to absolute injected power.
- The common per-unit frequency is governed by the aggregate droop relation, with ω0 = 1 pu and mp as the active-power/frequency droop coefficient.
- The droop relation enters the augmented Newton method through a dedicated frequency residual.
- The GFM-connected phase regulates voltage magnitude, while its phase angle is solved by the network equations except for one reference angle.
- Reactive-power injection is obtained from the network solution, while explicit reactive-power/voltage droop and inverter limits remain outside this formulation.
B. Three-Phase Unbalanced Network Modeling
The method lifts a positive-sequence initialization into phase-domain quantities and models unbalanced, mutually coupled three-phase networks with controlled coupling.
- A positive-sequence power flow is first solved to initialize the subsequent phase-domain calculation.
- The positive-sequence voltage is expanded as Va = V (+), Vb = a2V (+), and Vc = aV (+).
- The phase-domain admittance matrix combines self terms with a controlled mutual-coupling factor µ for synthetic unbalanced test systems.
- The same formulation can use measured or benchmark phase-domain line impedance matrices in practical feeder studies.
C. Unbalanced Load Modeling
The load model distributes demand across phases using participation coefficients and scales active and reactive power according to frequency-dependent functions.
- Total bus demand is distributed across phases using participation coefficients α = [αA, αB, αC].
- Active and reactive loads use frequency-scaling functions sP(ω) and sQ(ω), with derivatives included in the augmented Jacobian.
- The formulation supports linear, exponential, and other nonlinear load-frequency characteristics through the selected scaling functions.
- The case studies use a linear approximation around the nominal frequency ω0 = 1 pu.
III. AUGMENTED NEWTON SOLUTION
The augmented Newton formulation jointly solves phase-domain network equations, GFM voltage constraints, and a common droop-governed frequency state. Its residuals and Jacobian are dimensionally closed after selecting one GFM phase angle as the reference.
- Table I organizes known quantities, unknown variables, and residual equations for each bus/phase type in the proposed formulation.
- The unknown vector includes complex voltages for solved non-reference phase nodes, including PQ phases and GFM-connected phases with constrained voltage magnitude.
- One selected GFM phase angle is removed from the unknown vector and used only as the mathematical angular reference.
- The residual vector combines active-power mismatch, reactive-power or voltage-magnitude constraints, and the aggregate single-phase GFM droop-frequency residual.
- The augmented Jacobian adds derivatives for frequency-sensitive load scaling, GFM voltage-magnitude constraints, and the droop-frequency residual, updating voltages and frequency simultaneously.
IV. TEST CASE SCENARIOS
Three systems of different scales are tested to demonstrate the performance and scalability of GFM-3PF.
- The evaluation covers three systems of different scales to demonstrate GFM-3PF performance and scalability.
A. IEEE 3-bus Test Feeder
The IEEE 3-bus feeder verifies GFM-3PF against PSCAD under disturbance events and evaluates its three-phase voltage behavior. GFM-3PF reproduces operating-point transitions and closely matches pre-event and post-event voltage levels.
- The 3-bus system contains a source at Bus 1, a PQ load at Bus 2, and a single-phase GFM at Bus 3.
- VUF remains below 3% at all buses in the three-phase voltage-magnitude profile.
- At events occurring at 5 s, 10 s, and 15 s, GFM-3PF is compared with PSCAD using voltage profiles at Bus 2 and Bus 3 phase A.
- PSCAD captures sharper EMT transients, whereas GFM-3PF reproduces the associated operating-point transitions and closely matches pre-event and post-event voltage levels.
B. IEEE 33-bus System
The IEEE 33-bus case evaluates GFM-3PF on a medium-scale unbalanced radial feeder with a phase-selective single-phase GFM. The formulation converges while producing phase-dependent voltage trajectories and feeder voltage attenuation.
- The IEEE 33-bus case places a single-phase GFM at bus 4 on phase A in a medium-scale radial feeder.
- ω = 0.999807 pu is the solved common frequency after applying the phase-specific droop model with the active-power setpoint P ⋆_i.
- 4.24 × 10^-8 is the final mismatch, confirming convergence of the augmented Newton formulation.
- Phase-dependent separation among voltage trajectories reflects unbalanced loading, while nodal voltage attenuation follows radial distribution-system behavior.
C. IEEE 118-bus Meshed System
The IEEE 118-bus case demonstrates GFM-3PF on a large meshed network with phase-selective single-phase GFMs and three-phase voltage profiling. The solver converges within the prescribed mismatch tolerance.
- Two single-phase GFMs provide phase-selective support at Bus 69 phase A and Bus 1 phase B, rated 1.00 pu and 1.02 pu.The network includes phase mutual coupling and unevenly distributed, frequency-sensitive loads.
- The IEEE 118-bus case evaluates GFM-3PF scalability in a large-scale meshed network.
- Fig. 3 presents the three-phase voltage magnitude profile for the IEEE 118-bus meshed system.
- ω = 0.99861 pu, with a final mismatch of 4.854 × 10−7 below the prescribed tolerance of 10−6.The result satisfies the stated convergence criterion.
V. CONCLUSION
The conclusion presents GFM-3PF as a unified steady-state framework for single-phase GFMs in three-phase unbalanced power flow. Tests on 3-bus, IEEE 33-bus, and IEEE 118-bus systems demonstrate accuracy, robustness, and scalability.
- GFM-3PF embeds single-phase GFM behavior within a three-phase unbalanced power flow solver.
- The framework integrates positive-sequence initialization, unbalanced network modeling, frequency-sensitive loads, and a common droop-governed frequency state.
- Results on 3-bus, IEEE 33-bus, and IEEE 118-bus systems demonstrate the method's accuracy, robustness, and scalability.