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Impedance-Based Stability Analysis for Interconnected Converter Systems with Open-Loop RHP Poles

Yicheng Liao, Xiongfei Wang

arXiv:1811.09237v2eess.SYcs.SYeess.SP

TL;DR

Impedance-based stability analysis faces a gap when the impedance ratio contains open-loop RHP poles, which complicates impedance specification using forbidden regions. The paper proposes a general Bode-plot rule equivalent to the Nyquist criterion, and experimental case studies are reported to validate it.

  • Problem

    Impedance-ratio practices based on converter source types or avoiding RHP poles do not apply to Z+Z systems, and forbidden-region methods fail when the ratio has RHP poles.

    Source: §I · p. 5

  • Method

    The paper proposes a general impedance-based stability rule on Bode plots that is equivalent to the Nyquist stability criterion and accommodates open-loop RHP poles.

    Source: §I · p. 6

  • Results

    Experimental case studies validate the proposed method, including analyses that identify an unstable Scenario I and a stable Scenario II.

    Source: B. Stability analysis of Scenario I · p. 26, C. Stability analysis of Scenario II · p. 28, §V · p. 30

  • Takeaways & Limitations

    The method can be used to formulate impedance specifications graphically even when open-loop RHP poles are present.

    Source: §I · p. 6, §V · p. 30

Abstract

from arXiv · show

Small-signal instability issues of interconnected converter systems can be addressed by the impedance-based stability analysis method, where the impedance ratio at the point of common connection of different subsystems can be regarded as the open-loop gain, and thus the stability of the system can be predicted by the Nyquist stability criterion. However, the right-half plan (RHP) poles may be present in the impedance ratio, which then prevents the direct use of Nyquist curves for defining stability margins or forbidden regions. To tackle this challenge, this paper proposes a general rule of impedance-based stability analysis with the aid of Bode plots. The method serves as a sufficient and necessary stability condition, and it can be readily used to formulate the impedance specifications graphically for various interconnected converter systems. Experimental case studies validate the correctness of the proposed method.

I. INTRODUCTION

Impedance ratios at the point of common connection can serve as open-loop gains for Nyquist stability analysis. The paper addresses cases where open-loop RHP poles undermine impedance-shaping methods by proposing a general Bode-plot rule equivalent to the Nyquist stability criterion.

I. INTRODUCTION

  • At the point of common connection, the impedance ratio can serve as the interconnected system’s open-loop gain for Nyquist stability analysis.The impedance-based method uses subsystem terminal impedances and can be applied in scenarios including black-box systems.Source: §I · p. 3
  • Bode-plot forbidden-region methods provide intuitive impedance specifications, but established criteria are generally sufficient rather than necessary conditions.A necessary-and-sufficient criterion had also been proposed for source-load impedance ratios.Source: §I · pp. 3–4
  • When an impedance ratio has open-loop RHP poles, forbidden-region design fails, while Nyquist stability analysis requires encirclements of the unity circle.The paper identifies this issue for source-load ratios and notes that the Nyquist plot makes impedance interactions and design difficult to assess.Source: §I · pp. 4, 6
  • Existing impedance-ratio formulation practices based on converter source types or avoiding RHP poles do not apply to Z+Z systems.The introduction also identifies the difficulty of seeing impedance interactions on a whole-ratio Nyquist plot and the lack of a general ratio-formulation approach.Source: §I · pp. 5–6
  • The paper proposes a general Bode-plot rule equivalent to the Nyquist criterion that supports impedance specifications even with open-loop RHP poles.The rule is presented as filling the gaps in general impedance-ratio formulation and design-oriented stability analysis.Source: §I · p. 6

II. REVIEW OF IMPEDANCE-BASED STABILITY ANALYSIS

The review represents an interconnected system as two subsystems joined at a point of common connection. Their impedances and impedance ratios provide a basis for stability analysis.

II. REVIEW OF IMPEDANCE-BASED STABILITY ANALYSIS

  • An interconnected system is represented as two subsystems connected at a point of common connection, using Thevenin or Norton equivalents.Thevenin’s form uses a voltage source in series with an impedance; Norton’s form uses a current source in parallel with an impedance.Source: §II · p. 6, Figure 1 · p. 7
  • The sum of the two subsystem impedances can act as the system characteristic equation, while either impedance ratio can act as open-loop gain.Stability therefore depends significantly on the terminal impedances of both subsystems.Source: §II · pp. 6–7
  • Conventional impedance-based stability analysis includes impedance-ratio, impedance-sum, and multi-loop Nyquist methods.The review identifies these as its three main categories.Source: §II · p. 7

A. Impedance-ratio-based method

The impedance ratio can serve as the open-loop gain, with stability assessed by Nyquist encirclements; right-half-plane poles and ratio formulation constrain direct use of the criterion. Forbidden-region criteria support impedance specifications, while Bode plots expose interactions between numerator and denominator impedances.

A. Impedance-ratio-based method

  • The impedance ratio is treated as the open-loop transfer function, and stability requires its Nyquist plot to encircle (−1, j0) N times anticlockwise, where N is the ratio’s RHP-pole count.Source: A. Impedance-ratio-based method · p. 7

a) Stability analysis based on the NSC

b) Stability analysis based on forbidden regions

B. Impedance-sum-based method

The impedance-sum criterion avoids RHP poles introduced by impedance ratios in Z+Z systems and provides a necessary and sufficient stability test. Its use is constrained by limited subsystem interaction insight and unavailable endpoint frequency responses.

B. Impedance-sum-based method

  • For Z+Z systems, Z1+Z2 serves as the characteristic equation, and stability holds if and only if it has no RHP zeros.Stability can be checked from its zero map or by applying the argument principle.Source: B. Impedance-sum-based method · p. 13
  • The impedance-sum criterion is theoretically a sufficient and necessary stability condition.Each subsystem in the described HESS has a stable output impedance, making it a Z+Z system.Source: B. Impedance-sum-based method · p. 13
  • The impedance sum treats Z1 and Z2 as a whole, obscuring their interaction and offering little guidance for reshaping subsystem impedances.Source: B. Impedance-sum-based method · p. 13
  • For practical converter systems, impedance responses at zero and infinite frequency may be unavailable, leaving measured Nyquist plots unclosed and weakening argument-principle assessment.Analytical impedances may only be valid below half the switching frequency; frequency scanning also cannot obtain both endpoints.Source: B. Impedance-sum-based method · p. 14

C. NSC for multi-loop systems

The multi-loop NSC assesses stability by successively closing loops from a stable all-open system and counting encirclements. For paralleled converters, each converter connection is modeled as an additional loop.

C. NSC for multi-loop systems

III. GENERAL RULE FOR IMPEDANCE-BASED STABILITY ANALYSIS

The proposed impedance-ratio rule derives from the NSC and gives a necessary and sufficient stability condition while supporting graphical impedance specifications. It defines a proper impedance ratio to make the NSC applicable.

III. GENERAL RULE FOR IMPEDANCE-BASED STABILITY ANALYSIS

  • The general rule derives from the NSC, provides a sufficient and necessary stability condition, and enables design-oriented impedance-profile specifications.Its four stages are ratio formulation, identification of open-loop RHP poles, encirclement identification, and stability analysis.Source: §III · p. 15

A. Formulation of the impedance ratio

  • In Z+Z systems, denominator-impedance RHP zeros can produce open-loop RHP poles, so preventing them solely through ratio definition is ineffective.Source: A. Formulation of the impedance ratio · pp. 15–16
  • The ratio is defined as a proper transfer function, with its frequency response approaching the origin at infinite frequency.The paper calls this a proper impedance ratio and states the definition applies to Z+Z systems as well as other system types.Source: A. Formulation of the impedance ratio · p. 16
  • For practical selection when infinite-frequency responses are unavailable, choose the numerator impedance with the smaller high-frequency magnitude slope, or the smaller magnitude if slopes match.Source: A. Formulation of the impedance ratio · p. 16

B. Identification of the open-loop RHP poles

The method identifies open-loop RHP poles from impedance frequency responses, then determines encirclements using a four-step Bode-plot rule. Derivatives help classify crossings, and nonzero-frequency crossings are counted with their inverse-frequency counterparts.

B. Identification of the open-loop RHP poles

  • RHP poles and zeros can be identified directly from impedance frequency responses using the paper's Bode-plot rules, rather than fitting a transfer function and checking its pole-zero map.The number of ratio RHP poles is then calculated using identified numerator-impedance poles and denominator-impedance zeros.Source: B. Identification of the open-loop RHP… · p. 17

C. Identification of the encirclements

D. Stability analysis

The Bode-plot rule is equivalent to the Nyquist stability criterion and gives an iff stability condition. It accounts for open-loop RHP poles while supporting impedance analysis and specification across multiple system types.

D. Stability analysis

  • The system is stable if and only if Eq. (13) holds, making the proposed Bode-plot rule equivalent to the Nyquist stability criterion.The rule uses the number of open-loop RHP poles and encirclements to apply the NSC.Source: D. Stability analysis · p. 21
  • Steps 3–4 use frequency-response derivatives to determine Nyquist-trajectory direction and identify crossing numbers and types on Bode plots.Steps 1–2 use frequency-response magnitude and phase; the derivatives are considered in Step 3.Source: D. Stability analysis · p. 21

E. Advantages of the proposed stability analysis method

  • The proper impedance-ratio method applies to both Z+Y and Z+Z systems because it accounts for open-loop RHP poles.The comparison is with conventional methods, as summarized in Table V.Source: E. Advantages of the proposed… · pp. 21–22, Table V · p. 22
  • Given impedance frequency responses, the method can identify the number of RHP poles or zeros.Source: E. Advantages of the proposed… · p. 22
  • The four-stage Bode-plot rule predicts stability without calculating the impedance ratio or drawing a pole-zero map.The passage also states that neither the impedance ratio nor impedance sum is required for calculation.Source: E. Advantages of the proposed… · p. 22
  • Bode-plot analysis supports design-oriented impedance specifications, including for systems with open-loop RHP poles.The impedance can be specified so characteristic boundaries cross within ERs as many times as the number of open-loop RHP poles.Source: E. Advantages of the proposed… · p. 22

IV. EXPERIMENTAL CASE STUDIES

The case studies use two grid-connected inverters and two switch-selected operating scenarios. The system is modeled through inverter and grid admittances, with the analysis requiring a subsystem partition at the PCC.

IV. EXPERIMENTAL CASE STUDIES

  • Two grid-connected inverters with PR current controllers are tested with the load switch off and on, representing grid-tied inverter and distributed-power-system scenarios, respectively.The two inverters have identical parameters.Source: §IV · p. 23

A. System modeling

  • The inverter is modeled from its small-signal model, and its output admittance is derived with a third-order Padé delay approximation and a low-bandwidth PLL assumption.The inverter output admittance and grid admittance are derived for the system model.Source: A. System modeling · p. 24, Figure 7 · p. 24
  • Stability analysis requires dividing the system into two subsystems at the PCC; the study aggregates Inverter 1 and the grid into one admittance, intentionally introducing RHP zeros.The remaining subsystem admittance is denoted Yto1, while the aggregated admittance is Yto2.Source: A. System modeling · pp. 23, 25

B. Stability analysis of Scenario I

In Scenario I, the Bode-plot analysis predicts instability, which is confirmed by experimental waveforms. It also identifies four RHP poles in the admittance ratio and explains the stability result without relying on the Nyquist plot alone.

B. Stability analysis of Scenario I

  • The Bode plot permits direct stability prediction without calculating the admittance ratio, while the Nyquist plot alone gives little insight into subsystem interactions or impedance shaping.The case study reports that the Bode method also makes RHP pole and zero identification more intuitive and requires less calculation than drawing a Nyquist plot.Source: B. Stability analysis of Scenario I · p. 27
  • Two anti-resonances in Yto2, each with a corresponding −180° phase change, indicate two pairs of conjugate RHP zeros and four RHP poles in Yto1/Yto2.The pole-zero map agrees with the four-pole count inferred from the Bode plot.Source: B. Stability analysis of Scenario I · pp. 25, 27
  • There is no crossing within the ERs, and the total anticlockwise encirclement count does not equal the open-loop RHP-pole count, indicating instability.The Bode-plot result agrees with the Nyquist-plot result.Source: B. Stability analysis of Scenario I · p. 27
  • The experimental waveforms confirm that Scenario I is unstable, as predicted by the stability analysis.The waveforms include the PCC line-to-line voltage and inverter output currents.Source: B. Stability analysis of Scenario I · p. 27

C. Stability analysis of Scenario II

In Scenario II, two anticlockwise crossings within the ER yield four ACCs including negative frequencies, matching the four open-loop RHP poles; stable experimental waveforms agree with this prediction. The conclusion emphasizes derivative-informed Bode analysis and its use in impedance specification.

C. Stability analysis of Scenario II

V. CONCLUSION

  • The proposed method considers frequency-response derivatives that conventional analysis ignored, and the experiments demonstrate its effectiveness.The conclusion highlights easier implementation without drawing a pole-zero map or calculating the impedance ratio.Source: §V · p. 30
  • Bode-plot analysis supports impedance specification to enhance system stability, including when open-loop RHP poles are present.Source: §V · p. 30
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