Systematic SDP Search for Decentralized Stability Certificates

I. INTRODUCTION

• System-wide modal or Nyquist studies must be repeated after topology, device-placement, or dispatch changes, and operators often lack sufficiently detailed models.Practical methods must use terminal frequency responses because control algorithms are proprietary and often available only as black-box EMT models.Source: §I · p. 1
• Passivity-based decentralized certificates can verify stability from local terminal responses, but realistic converter controls, including droop-controlled GFMs, are not passive at low frequencies.Strict passivity consequently rejects designs that are stable in practice.Source: §I · p. 1

A. Main Contributions

• The paper formulates multiplier searches for several decentralized certificate families as SDPs using sampled frequency responses, with global optima within the admitted multiplier class and infeasibility verdicts.The paper also presents a systematic Chebyshev-basis search for the additive nodal-admittance certificate.Source: 1) Certificate search as a convex SDP · p. 2, A. Main Contributions · p. 2

2) Automatic synthesis of frequency-dependent multipliers:

• Chebyshev parameterization and certificate-driven frequency splitting produce piecewise-polynomial multipliers without manually chosen frequency boundaries or sampling points.The splitting is governed by a certificate tolerance and a maximum local degree.Source: 2) Automatic synthesis of… · p. 2

II. SDP FOR DECENTRALIZED FREQUENCY-DOMAIN STABILITY

• Frequency-dependent multiplier matrices enter the positive-definiteness conditions linearly, yielding LMIs that support convex optimization and can reduce the conservatism of fixed multipliers.A single fixed multiplier may be insufficient when practical inverter frequency responses vary substantially across frequency ranges.Source: §II · p. 3

III. SDP FOR ADDITIVE NODAL-ADMITTANCE CERTIFICATE

• The additive nodal-admittance formulation derives decentralized certificates from system-level stability conditions through a homotopy argument rather than the passivity framework.It also admits an equivalent frequency-domain interpretation.Source: §III · p. 3

A. Decentralized Admittance Decomposition Method

• The certificate follows a homotopy from a known stable reference system to the system of interest and requires regular interconnection, no imaginary-axis component poles, and no modes entering through infinity.Stability is preserved if no mode crosses the imaginary axis along the homotopy.Source: A. Decentralized Admittance… · p. 3
• A sufficient no-crossing condition is obtained by requiring the Hermitian part of the total admittance, transformed by a nonsingular multiplier, to be positive definite.When the multiplier does not depend on the homotopy parameter, it is sufficient and necessary to check the condition at the two endpoints.Source: A. Decentralized Admittance… · p. 3

B. SDP-Based Certificates

• Device and network admittances decompose into sums of local component contributions, enabling component-wise decentralized stability conditions.The network contributions correspond to individual lines and their incident buses.Source: B. SDP-Based Certificates · p. 3
• A common block-diagonal multiplier yields a fully decentralized certificate: compliant devices and lines may be arbitrarily interconnected with stability guaranteed regardless of topology or size.The certificate depends on the operating point through linearized device admittances; multiple operating points can be included as additional constraints.Source: B. SDP-Based Certificates · pp. 3–4
• At a fixed frequency, maximizing the smallest eigenvalue of the transformed Hermitian component admittance is a convex SDP.A positive optimized margin gives a strict certificate and can provide tolerance to bounded admittance uncertainty or operating-point variation.Source: B. SDP-Based Certificates · p. 4
• Enforcing the component constraints on a prescribed frequency set directly yields an SDP, while certifying the full continuous frequency interval is treated separately.Source: B. SDP-Based Certificates · p. 4

IV. SDP APPLIED TO OTHER PASSIVITY-BASED DECENTRALIZED METHODOLOGIES

• Representative decentralized stability conditions can be written as SDPs, with structural constraints and normalization needed to preserve local device-level tests.The full decentralized formulations are omitted for space reasons.Source: §IV · p. 4

A. Extended frequency-domain passivity

• Extended input-feedforward and output-feedback passivity conditions are linear in the weighting matrix and indices at fixed frequency, enabling an SDP search for the weighting matrix.The cited work suggested convex multiplier optimization but did not provide a detailed implementation.Source: A. Extended frequency-domain passivity · p. 4

B. Dynamic Loop Shifting

• Dynamic loop shifting admits an SDP search for the loop-shifting transfer matrix when frequency-response data are available and structural constraints such as block diagonality are imposed.Under the regularity assumptions in [15], stability of the loop-shifted interconnection implies stability of the original interconnection.Source: B. Dynamic Loop Shifting · pp. 4–5

C. Small-Gain Conditions

• Congruence transformations using P := DHD express the small-gain inequalities as LMIs that can be incorporated into an SDP.D is a nonsingular scaling matrix.Source: C. Small-Gain Conditions · p. 5

D. Small-Phase Conditions

• Small-phase conditions require a rotation angle that places both appropriately rotated numerical ranges in the right half-plane.Their weight matrices can be interpreted as particular multiplier matrices.Source: D. Small-Phase Conditions · p. 5

E. Generalized Gain–Phase Conditions

• Small-gain and small-phase conditions are particular choices of L, M, and N within a common family of positive-definiteness constraints.Table I summarizes the parameterizations.Source: E. Generalized Gain–Phase Conditions · p. 5
• Nonnegative weights recover pure small-gain or small-phase certificates at zero weight, while nonzero weights yield a mixed certificate.Treating L, M, and N as decision variables makes the frequency-wise generalized-certificate search an SDP; the DW-shell criterion is also in this family.Source: E. Generalized Gain–Phase Conditions · p. 5

V. SEARCH ALGORITHM FOR FREQUENCY-DEPENDENT MULTIPLIERS

• The plug-and-play certificate requires m(jω) over the full frequency range, and the proposed SDP search constructs it directly from sampled frequency responses.Samples may come from analytical models, linearizations, impedance scans, or measurements; the approach does not require explicit transfer-function models.Source: §V · pp. 5–6
• Analytical constructions and rational parameterizations can require explicit transfer-function models and may be restrictive or nonconvex.The SDP certificates of are verified only on a manually chosen frequency grid, without a guarantee between sampled points.Source: §V · pp. 5–6

A. Piecewise-Constant Multipliers

• Piecewise-constant multipliers can use different transformations across frequency regions, including conventional passivity at high frequencies and rotation at low frequencies.The low-frequency rotation targets a region where IBR admittances commonly violate passivity,.Source: A. Piecewise-Constant Multipliers · p. 6
• A region-specific constant multiplier can be computed by SDP, but piecewise-constant choices remain conservative and require manual frequency boundaries and sampling frequencies.The reported example uses 1000 samples over ω ∈ (0, 0.5) p.u. and combines the low-frequency multiplier with m2 = I2 at higher frequencies.Source: A. Piecewise-Constant Multipliers · p. 6
• Piecewise-constant multipliers do not exploit the full flexibility of a frequency-dependent certificate, motivating a systematic representation with coefficients optimized by SDP.The figure depicts eigenvalues of transformed Hermitian parts for two- and three-region multipliers.Source: A. Piecewise-Constant Multipliers · p. 6, Figure 1 · p. 6

B. Global Chebyshev Multiplier Search

• The admittance is approximated with a degree-d Chebyshev polynomial using d+1 Chebyshev–Lobatto samples, which are concentrated near band edges and support well-conditioned interpolation.For smooth admittances, the interpolation error decreases exponentially in d.Source: B. Global Chebyshev Multiplier Search · pp. 6–7
• Chebyshev multiplier coefficients can be optimized directly in one joint SDP, retaining the pointwise freedom of d+1 independent matrices at the interpolation frequencies.The admittance values are fixed data and the multiplier is affine in its coefficients.Source: B. Global Chebyshev Multiplier Search · p. 7
• Degrees greater than d = 200 are required for 1% relative Frobenius error in the global GFM-admittance approximation, motivating lower-degree local approximations.At d = 200, the search has more than 1600 direct optimization variables and K(d + 1) positive-semidefinite constraints when certifying K admittances simultaneously.Source: B. Global Chebyshev Multiplier Search · p. 7
• The Chebyshev parameterization extends to other decentralized certificates in the quadratically constrained class, with coefficient matrices optimized together in an SDP.Substitution at interpolation frequencies produces constraints linear in the multiplier coefficients.Source: B. Global Chebyshev Multiplier Search · p. 7

C. Adaptive Frequency Splitting

• Each local interval is mapped to [-1, 1], approximated by a Chebyshev polynomial, and processed with the sampled-frequency SDP.The loop solves the local SDP and checks the discrepancy at additional validation frequencies.Source: C. Adaptive Frequency Splitting · pp. 7–8
• Intervals are accepted when the certificate discrepancy is within e_H and transformed admittances remain positive at validation frequencies; otherwise they are split and refined locally.Splitting targets the minimum smallest eigenvalue when positivity fails, or the largest discrepancy otherwise, with the split point clipped to the interval's central 60%.Source: C. Adaptive Frequency Splitting · p. 7
• The method determines both the frequency partition and the multiplier's frequency dependence within each region automatically, producing a piecewise-polynomial multiplier.Accepted local multipliers are assembled across the partition.Source: C. Adaptive Frequency Splitting · p. 8
• The tolerance e_H sets required certificate accuracy, while dmax limits local SDP size, trading more small SDPs against one high-dimensional optimization.The maximum degree can also be specified indirectly through a maximum number of optimization variables.Source: C. Adaptive Frequency Splitting · p. 8
• For admittances analytic near each interval, Chebyshev interpolation converges uniformly at a geometric rate; finite-degree error is assessed numerically through refinement and independent validation.Explicit analytical error bounds would require additional information such as pole locations and magnitude bounds.Source: C. Adaptive Frequency Splitting · p. 8

VI. CASE STUDY

• The case study replaces all ten synchronous generators in the IEEE 39-bus system with 200-MVAR GFMs and scales loading down accordingly.The GFM model is taken from.Source: §VI · p. 8, Figure 3 · p. 8
• For (mp, nq) = (2%, 1%), the minimum eigenvalue remains positive across voltage-loop bandwidths of 658.4–804.8 Hz in Fig. 4.The plotted black dashed curve is the minimum over all network lines.Source: Figure 4 · p. 9, §VI · p. 9
• The common multiplier accommodates heterogeneous converters by testing distinct admittances; the study varies voltage-loop bandwidth while keeping droop coefficients identical.No multiplier reoptimization is needed for the reported positive-eigenvalue bandwidth range.Source: §VI · pp. 8–9
• The Chebyshev-based frequency-dependent multiplier certifies the largest region, covering (mp, nq) = (2%, 1%), which neither piecewise-constant choice covers.It exceeds both fixed-matrix regions without manual frequency partitioning.Source: §VI · p. 9, Figure 5 · p. 9
• Two and three manually partitioned piecewise-constant transformation matrices yield valid certificates, but neither certified region includes (mp, nq) = (2%, 1%).The two-matrix and three-matrix regions are shown as light-blue and dark-blue curves, respectively.Source: §VI · p. 9, Figure 5 · p. 9
• Across three selected droop pairs, all 200 random inverter placements tested in each scenario were stable, consistent with the certificate.The Monte Carlo boundary is only an outer bound on conservatism because unstable placements may remain unsampled between it and the certified region.Source: §VI · p. 9

VII. CONCLUSION

• At fixed frequency, linear matrix inequalities in multiplier variables make the least-conservative search an SDP with a guaranteed optimum within the admitted multiplier class.Infeasibility establishes that no multiplier exists in that class.Source: §VII · p. 9
• For the additive nodal-admittance certificate, sampled frequency responses feed a joint SDP and adaptive splitting algorithm that constructs a piecewise-polynomial multiplier.The algorithm determines the partition and frequency dependence within each segment rather than leaving them to the user.Source: §VII · pp. 9–10
• In the IEEE 39-bus system with full-order droop-controlled GFMs, SDP-optimized multipliers certify a practical droop-gain region; simulations confirm every tested setting.Monte Carlo estimation of the stability boundary supplies an upper bound on remaining conservatism.Source: §VII · p. 10
• Joint searches may become nonconvex when certificate weights enter nonlinear combinations, such as when optimizing a coordinate transformation simultaneously.The same joint SDP can also include admittances at multiple prescribed parameter values as additional constraints, if feasible.Source: §VII · p. 10

• For the GFM control layout of, the appendix lists controller, filter, droop, and network parameters, including ω = 100π rad/s and Lc = 36 mH.Source: APPENDIX · p. 10