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A modified sequence domain impedance definition and its equivalence to the dq-domain impedance definition for the stability analysis of AC power electronic systems
Atle Rygg, Marta Molinas, Zhang Chen, Xu Cai
TL;DR
AC power-electronic systems are difficult to analyze, and prior dq-domain and sequence-domain impedance approaches lacked a rigorous comparison. The paper derives their mathematical relationship, proposes a modified sequence-domain matrix, and shows equivalent GNC stability estimates while defining MFD systems and the limits of the original sequence definition.
Problem
Prior AC-system impedance studies used dq-domain or sequence-domain approaches independently, without rigorously comparing or bridging their stability estimates.
Method
The paper derives a mathematical relationship between dq-domain and modified sequence-domain impedance matrices and defines Mirror Frequency Decoupled (MFD) systems.
Results
The dq-domain and modified sequence-domain impedances produce equivalent Generalized Nyquist Criterion stability estimates, while the original sequence-domain impedance can be ambiguous in general.
Takeaways & Limitations
Stability analysis is unaffected by choosing dq or modified sequence coordinates, whereas neglecting mirror-frequency coupling or assuming MFD incorrectly can produce ambiguous or erroneous impedance estimates.
Abstract
from arXiv · showhide
Representations of AC power systems by frequency dependent impedance equivalents is an emerging technique in the dynamic analysis of power systems including power electronic converters. The technique has been applied for decades in DC-power systems, and it was recently adopted to map the impedances in AC systems. Most of the work on AC systems can be categorized in two approaches. One is the analysis of the system in the \textit{dq}-domain, whereas the other applies harmonic linearization in the phase domain through symmetric components. Impedance models based on analytical calculations, numerical simulation and experimental studies have been previously developed and verified in both domains independently. The authors of previous studies discuss the advantages and disadvantages of each domain separately, but neither a rigorous comparison nor an attempt to bridge them has been conducted. The present paper attempts to close this gap by deriving the mathematical formulation that shows the equivalence between the \textit{dq}-domain and the sequence domain impedances. A modified form of the sequence domain impedance matrix is proposed, and with this definition the stability estimates obtained with the Generalized Nyquist Criterion (GNC) become equivalent in both domains. The second contribution of the paper is the definition of a \textit{Mirror Frequency Decoupled} (MFD) system. The analysis of MFD systems is less complex than that of non-MFD systems because the positive and negative sequences are decoupled. This paper shows that if a system is incorrectly assumed to be MFD, this will lead to an erroneous or ambiguous estimation of the equivalent impedance.
I. INTRODUCTION
The paper addresses the lack of a rigorous bridge between dq-domain and sequence-domain impedance analysis for AC power-electronic systems. It proposes modified sequence-domain definitions, establishes stability-equivalence results, and identifies when mirror-frequency coupling can be neglected.
- I. INTRODUCTION: AC power systems with high power-electronics penetration are difficult to analyze because controllers introduce multiple nonlinearities and fast dynamics.Impedance-based analysis reduces the system to interacting source and load equivalents.
- I. INTRODUCTION: Previous studies use either sequence-domain symmetric components or the synchronous dq reference frame, but neither approach had been rigorously compared with the other.The paper mathematically relates the two impedance domains and treats them as equivalent for stability analysis.
- I. INTRODUCTION: The proposed modified 2x2 sequence-domain impedance matrix shifts positive and negative sequences by twice the fundamental frequency to account for mirror-frequency coupling.The paper derives its equivalence to the established 2x2 dq-domain impedance matrix and proves equivalent GNC estimates.
- I. INTRODUCTION: Mirror Frequency Decoupled (MFD) systems are defined as systems that avoid the mirror-frequency effect, yielding reduced impedance matrices because positive and negative sequences are decoupled.The original sequence-domain impedance definition is ambiguous unless the system is MFD.
- I. INTRODUCTION: The dq-to-sequence derivation uses the Park transform and frequency-domain relations between dq phasors and sequence phasors shifted by the fundamental frequency.The derivation applies to both voltage and current components.
- I. INTRODUCTION: The sequence domain represents three-phase systems through positive, negative, and zero sequence subsystems, with zero-sequence effects disregarded in this paper.The analysis relates phasors at arbitrary frequencies rather than only fundamental-frequency components.
D. Illustration of harmonic phasor relations
A 80 Hz perturbation illustrates that one dq-domain frequency component maps to two abc-domain components with different sequence identities. FFT-based phasor calculations verify the predicted frequency and sequence relationships.
- D. Illustration of harmonic phasor relations: An 80 Hz dq-domain current perturbation is transformed into abc-domain harmonic components for comparison in both domains.The resulting harmonic phasors are calculated using FFT and plotted in the complex plane.
- D. Illustration of harmonic phasor relations: The single 80 Hz dq tone becomes 30 Hz and 130 Hz abc-domain components after applying the inverse Park transform.The 30 Hz component is pure negative sequence, while the 130 Hz component is pure positive sequence.
III. MODIFIED SEQUENCE DOMAIN IMPEDANCE
The paper defines a modified 2×2 sequence-domain impedance matrix that represents positive- and negative-sequence quantities at shifted frequencies and captures mirror-frequency coupling. This matrix is derived from the dq-domain formulation through a linear transformation, with each element assigned a physical voltage-current interpretation.
- III. MODIFIED SEQUENCE DOMAIN IMPEDANCE: The proposed 2×2 matrix contains positive- and negative-sequence impedances at two different frequencies.It is introduced as the paper’s main contribution for relating current and voltage phasors through impedance.
- III. MODIFIED SEQUENCE DOMAIN IMPEDANCE: Any dq-domain phasor set decomposes into positive-sequence components at ωdq + ω1 and negative-sequence components at ωdq − ω1.This frequency mapping motivates the modified sequence-domain representation.
- III. MODIFIED SEQUENCE DOMAIN IMPEDANCE: The modified sequence-domain impedance matrix is obtained by rewriting the transformed dq-domain equations in matrix form.The derivation uses a unitary transformation and corresponding admittance equations interchange voltages and currents.
- III. MODIFIED SEQUENCE DOMAIN IMPEDANCE: Zpp measures positive-sequence voltage at ωp induced by positive-sequence current at ωp, while Zpn measures positive-sequence voltage at ωp induced by negative-sequence current at ωn.The off-diagonal terms represent cross-frequency sequence coupling.
- III. MODIFIED SEQUENCE DOMAIN IMPEDANCE: The dq-domain circuit represents d-q cross coupling through current-dependent voltage sources, while the sequence-domain equivalent represents mirror-frequency coupling.These circuit equivalents provide a physical interpretation of the two formulations.
B. Positive sequence impedances below the fundamental frequency
The modified sequence-domain impedance definition is extended to positive-sequence frequencies below the fundamental frequency by interpreting negative frequencies through phase-order reversal. The same impedance framework is then used in the dq- and sequence-domain GNC formulations.
- B. Positive sequence impedances below the fundamental frequency: For positive-sequence frequencies below the fundamental frequency, the corresponding dq-domain frequency becomes negative.This follows from ωp = ωdq + ω1.
- B. Positive sequence impedances below the fundamental frequency: Negative-frequency balanced three-phase signals are represented using positive frequency at the same absolute value with reversed phase order.This equivalence enables the impedance definition to cover positive-sequence frequencies below the fundamental frequency.
- C. Nyquist stability criterion equivalence: The GNC results are identical in the dq and modified sequence domains because their minor-loop gain eigenvalues are equal.The equality is stated as λdq = λpn.
- C. Nyquist stability criterion equivalence: Using the original sequence-domain definition instead can produce stability results different from dq-domain calculations when subsystems are not mirror-frequency decoupled.The discrepancy occurs when one or both subsystems lack the MFD property.
IV. THE MIRROR FREQUENCY EFFECT
The mirror frequency effect describes frequency-shifted responses that complicate impedance analysis. MFD systems avoid this effect and yield reduced, decoupled impedance matrices.
- A harmonic disturbance induces responses at its own frequency and at a mirror frequency shifted by twice the fundamental frequency.The shift direction depends on whether the disturbance is a positive or negative sequence.
- Applying the modified sequence-domain definition can restore linearity for sequence-domain balanced systems despite mirror-frequency coupling.
- Sequence-domain balance and absence of mirror-frequency coupling are independent properties.Thus, a sequence-domain balanced system may still contain the mirror frequency effect.
- A subsystem is MFD when a harmonic disturbance produces responses only at the same frequency, equivalently when Zpn = Znp = 0.Under this condition, the current-dependent voltage sources in the sequence domain are removed.
- For MFD systems, the sequence- and dq-domain impedance matrices are reduced, with a skew-symmetric dq matrix satisfying Zdd = Zqq and Zdq = −Zqd.The paper states that the proof appears in Appendix C.
- Because Zpn is diagonal in MFD systems, the original and modified sequence-domain impedance definitions are equivalent.
C. dq impedance extraction in MFD systems
The paper explains impedance extraction as an integrated simulation process for dq and sequence domains. In MFD systems, decoupling can reduce the measurements required.
- Claims that sequence impedances require only one measurement and no matrix inversion are valid only when the subsystem is MFD.
- Only a single measurement is needed to obtain the dq impedance matrix in an MFD subsystem.
- Mirror-frequency coupling can arise from unequal d- and q-axis controller structures or parameters, DC-link control, power controllers, and salient-pole machines.The subsystem is MFD only when all relevant d- and q-axis transfer functions are identical and cross-coupling conditions are satisfied.
- The simulation procedure begins by selecting dq-domain frequencies at which the impedances will be calculated.
- Impedances can be extracted using either shunt-current or series-voltage injection.The two injection methods differ in how the three-phase perturbation is applied.
- Series-voltage and shunt-current injections require different frequencies for linearly independent signals when solving the impedance matrices.For series injection, the voltage signal replaces the current signal.
- The required simulation outputs are terminal current and voltage signals, which are transformed to the frequency domain before calculating both impedance matrices.
- Once the dq- and sequence-domain matrices are established, the paper derives all other impedance expressions from them.
B. Case study description
The case studies simulate source and load converters under different mirror-frequency-coupling conditions. Impedances are obtained in dq and modified sequence domains using two calculation methods.
- Case study setup: Cases A and B use MATLAB/Simulink models containing a source converter and a load converter with dq-domain control systems.
- Case A: Case A uses voltage control for the source converter and DC-voltage plus reactive-current control for the load converter.The source uses a fixed clock, while the load is synchronized to the grid by a PLL.
- Impedance extraction: The simulation flowchart obtains dq- and sequence-domain impedances as functions of frequency and supports both shunt-current and series-voltage injection.
- Case A: Case A1 retains mirror-frequency coupling in both subsystems, whereas Case A2 removes it from the source subsystem, making only that subsystem MFD.
- Case B: Case B removes mirror-frequency coupling from both subsystems, producing a complete MFD system.Its load converter uses a constant DC-side voltage, eliminating the need for DC-voltage control.
- Results presentation: Figures 7–9 show dq-domain impedances, while Figures 10–12 show modified sequence-domain impedances for the three cases.The sequence-domain results are obtained by direct simulation and by transforming simulated dq impedances.
- Results presentation: The two methods for obtaining modified sequence-domain impedances produce identical results in all cases, confirming the dq-to-sequence transformation.
- Results presentation: Case A1 has no impedance-curve symmetries because both subsystems exhibit mirror-frequency coupling.
D. Simulation results - Modified vs. original
The paper compares original and modified sequence-domain impedances for the load subsystem across Cases A1, A2, and B using Figures 13–15.
- Figures 13–15 compare original positive- and negative-sequence impedances with the corresponding diagonal elements of the modified sequence-domain matrix.
- The comparison covers the three simulation cases and uses legend entries distinguishing equation notation and estimation method.
- The shunt-injection estimates are identified as direct simulations using the original sequence-domain definition.
E. Simulation results - Generalized Nyquist Criterion
The simulations show that dq-domain and modified sequence-domain impedance analyses produce identical GNC stability results, while original sequence impedances can differ unless systems are MFD. The discussion also identifies practical trade-offs between the domains and cautions against neglecting dq off-diagonal terms.
- Simulation comparison: The dq-domain and modified sequence-domain analyses produce exactly the same GNC result in all cases.
- Simulation comparison: The original sequence-domain impedances produce different Nyquist plots in Cases A1 and A2, but all methods agree in Case B.Only the most critical eigenvalue at isd = 1.1pu is plotted.
- Time-domain validation: The time-domain simulations increase Idc stepwise, with transient dq-current oscillations growing until instability occurs.
- Practical comparison: The sequence domain avoids dq transformations and reference-angle requirements when measured signals are used to obtain impedances.
- Practical comparison: Sequence-domain off-diagonal terms are often small and vanish for MFD systems, whereas neglecting off-diagonal terms is valid in the sequence domain but not the dq-domain.
- Practical comparison: When both subsystems are non-MFD, original sequence-domain impedances depend on injection type, with series injection expected to be more accurate when the load has stronger mirror-frequency coupling.
B. Proof of equal determinants
The paper derives original sequence-domain impedances for different injection choices and compares them with the modified definition. Because the resulting expressions differ, the original definition is not generally well defined and depends on injection type.
- Proof: The determinant of the modified sequence-domain impedance matrix equals the determinant of the dq-domain impedance matrix.The equality is established mathematically and supports equivalent stability estimates.
- General case: The derivation relates modified and original sequence impedances by solving source and load subsystem equations simultaneously.
- General case: Original sequence impedances are derived under shunt current or series voltage injection and positive or negative sequence perturbations.
- Conclusion: The resulting expressions differ between injection types, showing that the original sequence-domain impedance is not well defined in the general case.The paper validates this conclusion through simulations in Figure 13.
2) Special case with one MFD subsystem:
When one subsystem is MFD, the impedance expressions simplify: the MFD subsystem’s original and modified impedances coincide, and the other subsystem’s original impedance no longer depends on injection type. If both are MFD, the two definitions are equal.
- One MFD subsystem: If the source subsystem is MFD, its original and modified sequence-domain impedances are equal.
- One MFD subsystem: When the source is MFD, the load subsystem’s original sequence-domain impedance no longer depends on injection type.
- One MFD subsystem: The difference between the load impedances is proportional to ZL_np and also depends on the source impedance ZS_nn.
- Both subsystems MFD: When both subsystems are MFD, the original and modified sequence-domain impedances are equal, making MFD sufficient for unique original impedance definitions.The corresponding result is shown in Figure 15.