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Equivalent DQ Sequence-Domain Model of Unbalanced Three-Phase Passive Elements for Power Converter Controllers
Air\án Franc\és, Luis Saz, Rafael Castillo, Dionisio Ramirez, Javier Uceda
TL;DR
Balanced-impedance assumptions do not cover phase-unequal passive elements, and prior sequence-domain models omitted proper zero-sequence interaction. The paper derives an analytical and graphical dq sequence-domain model including all three sequences, then validates it experimentally on an unbalanced RL impedance. The model reproduces the resulting dq sequence-domain currents and keeps components constant in steady state for asymmetrical systems.
Problem
Existing controller models commonly assume balanced passive impedances, while prior dq sequence-domain models did not properly account for interactions involving the zero sequence.
Method
The paper derives an analytical and graphical dq sequence-domain model using positive, negative, and zero sequences for unbalanced passive impedances.
Results
The proposed model reproduces the dq sequence-domain components of measured currents and predicts zero-sequence voltage produced by an unbalanced impedance.
Takeaways & Limitations
The framework represents interactions among all sequences and keeps dq components constant in steady state even when impedances and signals are asymmetrical.
Takeaways & Limitations
The SOGI-QSG used to obtain symmetrical components filters the derivation, causing transient responses and an initial mismatch in current prediction.
Abstract
from arXiv · showhide
Equivalent dq0 models of three-phase passive impedances are widely used for the design of three-phase inverter and rectifier controllers in the synchronous frame. However, these equivalent models assume that the impedance in each phase are the same, which is not always applicable. The fact that the impedance in each phase is different, generates second order harmonic content in the dq0 components, which hinders the advantages of the dq0 transformation. To avoid the second harmonic in these scenarios, it is possible to use the Fortescue's theorem to represent any asymmetrical set of three-phase signals as a linear combination of three symmetrical sequences. In the literature, only a few works have attempted to derive the equivalent model of unbalanced three-phase passive elements in the sequence domain, and none is able to account for the interaction between the zero sequence and the positive and negative sequences. This work presents the derivation of the equivalent dq sequence-domain model of unbalanced three-phase passive elements both analytically and with an intuitive graphical approach. Experimental results are shown to validate the proposed models.
I. INTRODUCTION
Three-phase converter controllers often model passive impedances as balanced, but phase differences can invalidate that assumption and perturb feedback regulation. The paper addresses prior sequence-domain models' incomplete treatment of zero-sequence interactions.
- Most three-phase inverter and rectifier controllers use alpha-beta, dq synchronous-frame, or sequence-domain frameworks while assuming balanced passive impedances.
- Component tolerances, especially in magnetics, and intentional design choices can produce unequal phase impedances.
- Modeling errors from unequal impedances become feedback-loop perturbations that affect regulator performance.
- Earlier dq sequence-domain models did not properly account for zero-sequence coupling with positive and negative sequences.
- The paper derives a dq sequence-domain model for any unbalanced impedance that includes interactions among positive, negative, and zero sequences.
II. EQUIVALENT MODEL INFERENCE
The model decomposes asymmetrical three-phase signals into positive, negative, and zero symmetrical sequences, then expresses each sequence in a dq rotating frame. This representation keeps sequence dq components constant in steady state, enabling an equivalent sequence-domain circuit.
- Fortescue’s theorem represents an asymmetrical three-phase signal as positive, negative, and zero symmetrical sequences.
- The Lyon transformation derives the symmetrical sequences from asymmetrical time-domain signals.
- Clarke and Park transformations are applied separately to the positive, negative, and zero sequences to obtain dq rotating-frame representations.
- Even asymmetrical and unbalanced signals have constant sequence dq components in steady state.
A. Derivation of the Resistive Equivalent Impedance
The resistive-case derivation constructs a sequence-domain equivalent impedance by transforming phase quantities into positive, negative, and zero sequences and then into dq components. Superposition fills a 6x6 matrix that captures cross-sequence voltage-current interactions, including generation of zero sequence from balanced current.
- Derivation setup: A purely resistive impedance provides the starting case for constructing the equivalent impedance model.
- Balanced reference case: For balanced impedance, Ra = Rb = Rc = R and the abc impedance matrix is diagonal, allowing Clarke transformation into the alpha-beta frame.
- Coordinate transformations: The alpha-beta components are rotated by θ = ωt to express them in the dq framework.
- Sequence-domain formulation: A 6x6 equivalent impedance matrix models voltage-current interactions across positive, negative, and zero sequences.
- Sequence-domain formulation: The extended impedance represents how each voltage sequence is affected by each current sequence, including a dq representation of zero sequence.
- Zero-sequence representation: Zero sequence receives an artificial quadrature signal so its phase relative to the positive sequence can be represented in dq coordinates.
- Coordinate transformations: The derivation applies Clarke and Park transformations before obtaining the equivalent circuit’s phasor and dq-frame equations.
- Graphical and superposition derivation: Superposition applies unit d and q currents for each sequence, calculates their voltage contributions, and uses those relationships to fill the matrix columns.
B. Derivation of the Inductive Equivalent Impedance
The inductive equivalent model extends the impedance representation to unbalanced phase inductances and transforms it into positive, negative, and zero dq sequences. The negative sequence uses an oppositely rotating reference frame.
- Balanced reference case: For balanced inductance, the phase-inductance matrix is diagonal with La = Lb = Lc = L.
- Transformations: The derivation applies Clarke and Park transformations to obtain the inductive impedance in the sequence-domain dq frame.
- Phasor formulation: The derivative term is represented in matrix form before obtaining the final phasor expression for the equivalent circuit.
- Unbalanced inductive model: The unbalanced inductive model extends the impedance matrix to represent each voltage sequence’s contribution from each current sequence.
- Sequence reference frames: The negative sequence uses Tp(−ω), whereas the positive and zero sequences use Tp(ω), reflecting opposite reference-frame rotation.
- Equivalent circuit: The resulting sequence-domain equivalent circuit is depicted for the unbalanced three-phase inductive impedance.
C. Derivation of the Capacitive Equivalent Impedance
The capacitive equivalent model follows the inductive derivation while relating current to the voltage derivative. Its sequence-domain phasor equation and equivalent circuit are then obtained.
- Capacitive relationship: For a capacitive three-phase impedance, the current–voltage relationship is established through the voltage derivative.
- Phasor formulation: Following the established derivation method, the capacitive sequence-domain phasor equation is obtained.
- Rotation matrix: The rotation matrix used in the capacitive model is the one expressed in equation (33).
- Equivalent circuit: The sequence-domain equivalent circuit of an unbalanced three-phase capacitive impedance is depicted in Fig. 5.
III. EXPERIMENTAL VALIDATION
The proposed model was tested on an experimentally imposed asymmetric voltage applied to an unbalanced RL impedance. Measured and predicted dq sequence currents agreed, while SOGI-QSG filtering caused transient mismatch.
- Experimental setup: The validation used a four-quadrant grid simulator to apply an asymmetric three-phase voltage to an unbalanced RL impedance.
- Experimental setup: The tested impedance had Ra = 50 Ω, Rb = 270 Ω, Rc = 270 Ω, La = 32 mH, Lb = 32 mH, and Lc = 16 mH.
- Voltage validation: The applied voltage was balanced, yet the model predicted the zero-sequence voltage appearing between neutral points because of impedance unbalance.
- Current validation: The measured and predicted currents were compared in abc and positive-, negative-, and zero-sequence dq components.
- Current validation: The equivalent model reproduced the dq sequence-domain components of the resulting current seamlessly.
- Validation caveat: SOGI-QSG filtering of abc signals caused transient responses and an initial mismatch in current prediction.
IV. CONCLUSIONS
The paper presents a dq sequence-domain model for unbalanced three-phase passive impedances, with constant steady-state components and explicit interaction among all sequences. Experimental results validate the model.
- Conclusion: The framework models unbalanced three-phase passive impedances in the dq sequence domain for three-phase power-converter controller design.
- Conclusion: All components remain constant in steady state even when the impedance and three-phase voltage or current are asymmetrical and unbalanced.
- Conclusion: The model explicitly represents interactions among positive, negative, and zero sequences caused by impedance unbalance.
- Conclusion: Experimental results validate the proposed equivalent model.