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Stimulated Oscillations in Renewable Energy Integrated Power Systems - Part II: Methodology of Oscillation Mitigation

Peng Zhang

arXiv:2608.29518v1eess.SYcs.SY

TL;DR

Closely spaced poles can produce high-amplitude oscillations under small disturbances, so stability alone does not determine oscillation risk. The paper analyzes pole–zero relationships and develops a feedback-control mitigation methodology. It concludes that feedback can separate closely spaced poles and place zeros near them, with tuning of the feedback-path poles, zeros, and gain.

  • Problem

    Closely spaced poles can cause high-amplitude oscillations under small disturbances, while stability alone does not determine oscillation risk.

    Source: §I · p. 1, §II · p. 2

  • Method

    The paper analyzes pole–zero placement and proposes feedback control using a reduced-order model to configure feedback-path poles, zeros, and gain.

    Source: §I · p. 1, §VI · p. 7

  • Results

    The paper concludes that feedback control can separate closely spaced poles and introduce zeros near them.

    Source: §VI · p. 7

  • Takeaways & Limitations

    The feedback-based methodology is presented as a reference for designing oscillation suppression schemes across diverse devices.

    Source: §VI · p. 7

    The conventional dominant second-order poles theory is inapplicable to oscillations arising from closely spaced poles, for which a fourth-order system is needed for accurate characterization.

    Source: A. Order Reduction of High-Order… · p. 4

Abstract

from arXiv · show

As presented in Part I of this series, closely located poles can produce high-amplitude oscillations even under small perturba-tions, which are referred to as stimulated oscillations. As the second installment of this series, this paper develops a method-ology for stimulated oscillation mitigation. Firstly, the logical relationship between system stability and oscillation risk is in-vestigated, clarifying that stability is neither a necessary nor a sufficient condition for oscillation risk. Secondly, the dominant factors affecting the pole and zero position on the complex plane, as well as their effectiveness and limitations are analyzed. For feedback control systems in particular, the influence of feedback paths on the pole-zero distribution of closed-loop systems is ana-lyzed. On this basis, a methodology for stimulated oscillation mitigation is proposed. Combined with the reduced-order trans-fer function of systems with closely separated poles, the order relation and configuration of poles and zeros for the feedback path transfer function, together with the criteria for gain selec-tion, are elaborated. Finally, a parameter tuning scheme for the poles, zeros and gain of the feedback path transfer function is presented. The proposed methodology for stimulated oscillation mitigation based on feedback control is not restricted to specific devices and can serve as a methodological reference for the design of diverse oscillation suppression schemes.

I. INTRODUCTION

Closely spaced poles can produce high-amplitude oscillations under small disturbances, which stability-based methods may not suppress because they do not change pole–zero relationships. The section argues that stability alone does not determine disturbance-time oscillation risk and illustrates this with a system whose well-damped modes still produce large oscillations.

I. INTRODUCTION

  • Existing stability-based methods regulate pole positions but leave pole–zero relative positions unchanged, so they cannot effectively suppress stimulated oscillations.The paper identifies separating closely approaching poles and/or bringing zeros near neighboring poles as the mitigation objective.Source: §I · p. 1

II. THE RELATIONSHIP BETWEEN STABILITY AND OSCILLATION RISK: SUFFICIENT, NECESSARY, OR NEITHER?

  • Stability is neither a necessary nor a sufficient condition for determining oscillation risk, contrary to the traditional stability-based criterion.The analysis frames the question in terms of sufficiency and necessity.Source: §I · p. 1, §II · p. 2

A. Insufficiency of Stability for Determining Oscillation Risk

  • Disturbance responses depend on partial-fraction contributions, while the poles alone do not determine oscillation risk.The section introduces a two-pole-pair, one-zero-pair system to illustrate the distinction.Source: A. Insufficiency of Stability for… · p. 2
  • Closely spaced poles with favorable damping can still produce high-amplitude oscillations during small random disturbances, despite rapid post-disturbance convergence.The example uses disturbances with amplitude range ±0.01, duration 30 s, and pulse intervals of 1 s, 0.01 s, and 0.001 s.Source: A. Insufficiency of Stability for… · p. 2, Figure 1 · p. 2

B. Non-necessity of Stability for Determining Oscillation Risk

The example shows that worsening damping does not necessarily increase disturbance-time oscillation risk. Shifting the poles rightward while increasing their separation substantially suppresses oscillation amplitude.

B. Non-necessity of Stability for Determining Oscillation Risk

  • Greater spacing between the two pole pairs substantially suppresses disturbance oscillations even as their damping ratios decrease.The comparison is between the responses in Fig. 2 and Fig. 1.Source: B. Non-necessity of Stability for… · pp. 2–3, Figure 2 · p. 2
  • The poles shift rightward and the damping ratios decrease, yet oscillation amplitude under disturbances is substantially suppressed.Fig. 2 shows responses to random pulse disturbances with amplitude range ±0.01, duration 30 s, and pulse intervals of 1 s, 0.01 s, and 0.001 s.Source: B. Non-necessity of Stability for… · pp. 2–3
  • Stability is not a necessary condition for determining oscillation risk because pole spacing can matter more to disturbance amplitude than damping alone.The paper concludes that pole locations and pole–zero relative positions are neither sufficient nor necessary conditions.Source: B. Non-necessity of Stability for… · pp. 2–3

C. Physical Implications of the Foregoing Conclusions

Disturbance-time oscillation amplitude is tied to modal excitation, not simply to pole-based stability. The paper accordingly focuses mitigation on pole–zero relationships, while noting that power-system poles depend on topology and equipment parameters and that changing them can be costly or infeasible.

C. Physical Implications of the Foregoing Conclusions

  • A poorly stable mode can pose negligible oscillation risk when disturbances excite it little, even if its post-disturbance convergence is slow.In renewable-energy integrated power systems, continuous disturbances make the disturbed oscillation process relevant to risk assessment.Source: C. Physical Implications of the… · p. 3

III. KEY FACTORS AFFECTING ZERO AND POLE POSITIONS ON THE COMPLEX PLANE

  • Mitigation targets separating closely approaching poles and/or bringing zeros near neighboring poles, thereby regulating their relative positions on the complex plane.The section poses the determination of pole and zero positions as the key question.Source: §III · p. 3

A. Key Factors for Pole Positions on the Complex Plane

  • Power-system poles are determined by topology and equipment parameters; changing operating modes or tuning parameters can be costly and sometimes infeasible.The poles do not depend on disturbance locations or selected output variables.Source: A. Key Factors for Pole Positions on… · p. 3

B. Key Factors for Zero Positions on the Complex Plane

Transfer-function zeros are jointly determined by the state, control, output, and feedforward matrices. Adjusting their positions is difficult because it requires simultaneous changes across operating modes, equipment or controller parameters, and system inputs and outputs.

B. Key Factors for Zero Positions on the Complex Plane

C. Key Factors for Pole and Zero Positions on the Complex Plane in Feedback Control Systems

In a feedback control system, the closed-loop zeros and poles depend on different combinations of the forward- and feedback-path transfer functions. This relationship provides a basis for using feedback-path parameters to change pole-zero placement.

C. Key Factors for Pole and Zero Positions on the Complex Plane in Feedback Control Systems

IV. METHODOLOGY FOR STIMULATED OSCILLATION MITIGATION BASED ON FEEDBACK CONTROL

The paper proposes feedback-path parameter tuning for mitigating stimulated oscillations, using a reduced-order transfer function to make high-order systems with closely separated poles more tractable. The approach is bounded by the need for an appropriate model order and challenging root assignment in high-order systems.

IV. METHODOLOGY FOR STIMULATED OSCILLATION MITIGATION BASED ON FEEDBACK CONTROL

  • High-order systems make root assignment through the feedback-path zeros and gain challenging, while tuning controller parameters or operating modes has limited mitigation effect and may be costly or infeasible.Source: §IV · p. 4
  • The method tunes the poles, zeros, and gain of H(s) to assign feedback-system poles and zeros and suppress stimulated oscillations.Source: §IV · p. 4

A. Order Reduction of High-Order Transfer Functions for Power Systems with Closely Separated Poles

  • For two pairs of closely spaced complex-conjugate poles, the high-order partial-fraction expansion is reduced by retaining only terms associated with those pairs.Equation (7) facilitates tuning H(s); its reduced-order transfer function's gain is determined using the final value theorem.Source: A. Order Reduction of High-Order… · p. 4
  • Conventional dominant second-order pole theory does not characterize oscillations from closely spaced poles accurately; a fourth-order system with two pole pairs is needed.Source: A. Order Reduction of High-Order… · p. 4

B. Mitigation of Stimulated Oscillation Based on Feedback Control

The method reduces the forward path to a transfer function centered on closely separated poles, then tunes the feedback path to reshape the closed-loop pole-zero distribution. Its design constraints balance pole separation and oscillation suppression against high-frequency amplification risk.

B. Mitigation of Stimulated Oscillation Based on Feedback Control

  • The forward path is decomposed into a reduced-order transfer function G´(s) and parallel remaining partial fractions Gi(s), with H(s) tuned first for G´(s).The influence of Gi(s) on mitigation performance is analyzed afterward.Source: B. Mitigation of Stimulated… · p. 5
  • H(s) must have at least four more zeros than poles; for two closely spaced pole pairs, its numerator order must satisfy mH≥6.At mH=6, the feedback design has seven undetermined parameters: b1 through b6 and KH.Source: B. Mitigation of Stimulated… · p. 5
  • The desired closed-loop characteristic polynomial separates the two close pole pairs, while placing the remaining two roots far from them; coefficient matching yields seven equations.The targeted separated pairs are σ´1G±jω´1G and σ´2G±jω´2G.Source: B. Mitigation of Stimulated… · p. 5
  • When K_G´KH>>1, the closed-loop poles approximate H(s)'s zeros, but increasing KH raises high-frequency amplification risk, requiring a gain trade-off.The forward-path gain K_G´ generally takes a large value.Source: B. Mitigation of Stimulated… · p. 5

A. Parameter Tuning of Feedback Path

The tuning example places feedback-path zeros near the closely spaced pole pairs and uses a further zero pair to attenuate specific high-frequency signals. Gain selection trades oscillation suppression against high-frequency amplification.

A. Parameter Tuning of Feedback Path

  • The example system has pole pairs at −0.51±j30.0×2π and −0.5±j30.1×2π, plus −0.1±j500×2π, and zeros at −3.0±j200×2π.Its DC gain is set to unity with kG = 7.9428×109.Source: A. Parameter Tuning of Feedback Path · p. 6
  • Fig. 5 responses use random pulse disturbances of ±0.01 amplitude over 30 s, with pulse intervals of 1 s and 0.001 s.The feedback-path poles are selected as −0.5±j30.1×2π to introduce zeros near the closely spaced pole pairs.Source: A. Parameter Tuning of Feedback Path · p. 6, Figure 5 · p. 6
  • The feedback-path zeros are assigned −0.2±j29×2π and −0.6±j31×2π to separate the close pole pairs, with −0.4±j300.0×2π as a high-frequency pair.The high-frequency pair is intended to attenuate potential signals entering the feedback path.Source: A. Parameter Tuning of Feedback Path · p. 6
  • Two zero pairs increase spacing between the close conjugate pole pairs, while the remaining pair creates a high-frequency magnitude dip to suppress signals at specific frequencies.A smaller kH is preferred from the perspective of limiting high-frequency signal amplification in the feedback path.Source: Figure 6 · p. 6
  • A larger kH lets closed-loop poles approach H(s)'s zeros and improves suppression, but its value must trade off that performance against high-frequency amplification.Source: Figure 8 · p. 7

B. Influence of Feedback Control on Risk-Free Oscillatory Modes

For stimulated oscillations caused by closely spaced poles, risk-free modes have negligible associated residues. Consequently, the feedback system formed by such a mode and H(s) approximates an open circuit.

B. Influence of Feedback Control on Risk-Free Oscillatory Modes

  • Risk-free modes have negligible residues for stimulated oscillations caused by closely spaced poles, so the feedback system comprising Gi(s) and H(s) approximates an open circuit.Source: B. Influence of Feedback Control on… · p. 7

VI. CONCLUSION

The paper argues that stability alone does not determine stimulated-oscillation risk and that mitigation should address large oscillations during disturbances. It develops feedback-control pole-zero placement and parameter-tuning methods, including a sixth-order-system example.

VI. CONCLUSION

  • Stability is neither a necessary nor sufficient condition for oscillation risk, so mitigation should prioritize avoiding large oscillations during disturbances over post-disturbance convergence rate.Source: §VI · p. 7
  • Pole-only adjustments can be costly or infeasible under technical and economic constraints, whereas feedback control can separate close poles and place zeros nearby.Source: §VI · p. 7
  • The proposed method models stimulated-oscillation systems with a fourth-order transfer function containing two pairs of closely separated poles and analyzes feedback-path pole-zero order, placement, and gain criteria.Source: §VI · p. 7
  • A sixth-order-system example demonstrates tuning feedback-path zeros, poles, and gain while examining high-frequency amplification risk and effects on risk-free modes.Source: §VI · p. 7
  • The feedback-based methodology is not limited to particular devices and is offered as a reference for designing diverse oscillation-suppression schemes.Source: §VI · p. 7
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