{"id":"5e154a43-b3d4-4d9a-bc3c-74f882bdd2d4","arxiv_id":"2506.11411","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper proposes and tests local delta-dissipativity verification and an ADMM-based distributed coupling check for stability certificates that cover entire equilibrium sets.","lead":"This paper gives algorithms for certifying power system stability without computing equilibria: per-device delta-dissipativity certificates found with matrix inequalities, plus an ADMM-based distributed check of the interconnection. The methods are demonstrated on modified IEEE 9, 39, and 118 bus systems with mixed generators and inverters.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1's p-check (Eq. 31) certifies the coupling condition only for scaled matrices p_i X_i, yet the algorithm returns the unscaled X_i; because lc is linear in each X_i, a positive weighted sum can be negative while the true unweighted sum is positive, so early termination is not a valid…","rationale":"The reader's weakest assumption identifies the decisive flaw: the p-check in Algorithm 1 verifies a scaled version of the coupling condition but returns the unscaled matrices. The paper's strongest claim, repeated in Sections IV-A3 and IV-C2, is that Algorithm 1 yields Xi satisfying both local delta dissipativity and the coupling condition, and hence that the equilibrium set is asymptotically stable. The p-check is the only mechanism that allows early termination before ADMM residual convergence, and it is unsound because lc is linear in each Xi, so a positive weighted sum with sum p_i = N does not preserve the sign of the unweighted sum. The scalar counterexample is a direct falsification of the logical step. The secondary concern about pointwise sampling of the local dissipativity region is also real, but the p-check issue alone invalidates the central certificate as stated. I concur with the reader's REJECT verdict and recommend no adjustment.","tokens_in":16543,"tokens_out":6156,"duration_ms":69425,"concrete_test":"Implement Eq. (31) on a minimal instance with m = 1, L_1(X_1) = X_1, L_2(X_2) = 2 X_2, X_1 = -100, X_2 = 80, and epsilon = 1e-4. With p_1 = 1.9 and p_2 = 0.1, the p-check returns t_z = -174 < -epsilon, while the actual coupling value is lc(X_1, X_2) = 60 > 0. This directly falsifies the implication used in Algorithm 1 Step 2. If the authors supply code for the IEEE 118-bus case, an alternative check is to log p_i at iteration k = 259 and recompute lc using the returned, unscaled X_i; the p-check passing with p_i different from 1 will not imply that the unmixed condition is negative semidefinite.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Algorithm 1's p-check in Section III-B, Step 2. It solves Eq. (31), finding p_i > 0 with sum p_i = N such that lc(p_1 X_1^{k+1}, ..., p_N X_N^{k+1}) <= t_z I with t_z < -epsilon, and on success it returns the unscaled X_i^{k+1} as satisfying the coupling condition. This implication is invalid. The function lc is linear in each X_i (it is a constant congruence of blkdiag(X_i)), so lc(p_1 X_1, ..., p_N X_N) = sum_i p_i L_i(X_i) with each L_i linear in X_i. Positive weights p_i are not necessarily equal to 1, and a weighted sum can be negative even when the unweighted sum is positive. Scalar illustration: take L_1(X_1) = X_1, L_2(X_2) = 2 X_2, X_1 = -100, X_2 = 80. Then lc(X_1, X_2) = 60 > 0, but with p_1 = 1.9, p_2 = 0.1 the weighted sum is -174 < 0, so Eq. (31) passes. Thus the p-check termination at k = 259 in Section IV-C2 does not certify lc(X_1^{k+1}, ..., X_N^{k+1}) <= 0. The proposed repair of returning p_i X_i also requires scaling P_i, since the local LMI in Eq. (29) is not invariant under scaling X_i alone while leaving P_i fixed. As written, Algorithm 1's output does not satisfy the system-level condition that the paper claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper, Part II of a two-part work, develops computational methods for a compositional and equilibrium-free stability framework for power systems. The authors first propose a local verification method based on a Krasovskii-type storage function and characterize delta dissipativity for several device models. They then propose an ADMM-based distributed algorithm, including a relaxation variable, an adaptive penalty, and a 'p-check' early-termination step, to verify the coupling condition. Three applications are presented: stability assessment of multiple equilibria on an IEEE 9-bus system, stability assessment under varying operating conditions on an IEEE 39-bus system, and distributed stability assessment on an IEEE 118-bus system. The central claim is that local delta-dissipativity certificates and a coupling certificate produced by Algorithm 1 imply, via Theorem 2 of Part I, asymptotic stability of every equilibrium in a dissipative region D.","tokens_in":16937,"tokens_out":11270,"duration_ms":128096,"significance":"The intended contribution is significant if it can be made sound: an equilibrium-free, compositional certificate would remove the need to linearize and compute eigenvalues at each equilibrium, and the ADMM formulation addresses scalability and privacy. The local verification via Krasovskii-type storage functions (Proposition 1) is a valid sufficient condition, and the model-transformation examples and static-device propositions are useful building blocks. However, the current manuscript has a demonstrable flaw in the p-check step of Algorithm 1 and an unresolved gap between sampled verification and certification over a continuous region D. These issues affect the validity of the reported stability certificates and the claimed 42.83% iteration reduction, so the manuscript is not acceptable in its present form.","major_comments":[{"comment":"The p-check is unsound as written. The algorithm solves (31) for positive p_i summing to N such that lc(p1 X1^{k+1}, ..., pN XN^{k+1}) <= t_z I with t_z < -epsilon, and if this succeeds it returns the unscaled matrices X_i^{k+1} as a certificate of the coupling condition lc(X1^{k+1}, ..., XN^{k+1}) <= 0. Since lc is linear in each X_i, the weighted sum lc(p1 X1, ..., pN XN) = sum_i p_i L_i(X_i) can be negative even when the unweighted sum is positive; positive weights can cancel positive terms. A scalar illustration is L1(X1)=X1, L2(X2)=2 X2, X1=-100, X2=80, for which lc(X1,X2)=60>0 but p1=1.9, p2=0.1 gives lc(p1 X1, p2 X2)=-174<0. Thus the termination at k=259 in Section IV-C2 does not certify the coupling condition for the returned X_i. The repair of returning p_i X_i is not automatic because the local LMI (29) is not invariant under scaling X_i alone; scaling both P_i and X_i by p_i still leaves the epsilon I term unchanged, so the local certificate would have to be re-solved.","section":"Section III-B, Step 2 (Algorithm 1, Eq. (31))"},{"comment":"The local delta-dissipativity constraint (19) must hold for all (x,u) in the region D_i, but the paper verifies it only at sampled points or on two-dimensional cross-sections. The text after (28) states that one may use a scenario optimization program by sampling points in D_i and then re-characterize the dissipative region; however, no deterministic procedure or guarantee is given, and the figures show only cross-sections. Consequently, the statements in Section IV-A3 and IV-B2 that 'any isolated equilibrium in D' is asymptotically stable are not supported by the evidence presented. The authors need to provide a rigorous method (e.g., robust optimization or an explicit region-recharacterization with a certificate) that verifies (19) on the full D_i used in the theorem.","section":"Section III-A and Sections IV-A, IV-B case studies"},{"comment":"The paper claims a convergence guarantee for the ADMM scheme by citing [12], but the algorithm uses an adaptive penalty update (36) that is not covered by the standard ADMM convergence proof in [12], and no alternative proof is supplied. Since the distributed case study relies on convergence to a feasible point, the convergence claim is currently unsupported and should be either proved or stated as empirical.","section":"Section III-B, Step 6 (Eq. (36))"}],"minor_comments":[{"comment":"The heading 'V erification of Local Conditions' contains a typo; it should read 'Verification of Local Conditions'.","section":"Section II heading"},{"comment":"The phrase 'a linear matrix equality' should be 'a linear matrix inequality'.","section":"Section II-B, after Eq. (22)"},{"comment":"The notation 'R12' should be written as R^12 (or \\mathbb{R}^{12}) to denote the 12-dimensional Euclidean space.","section":"Section IV-A3 and elsewhere"},{"comment":"Reference [24] contains an incomplete entry ('02102'); please correct the bibliographic details.","section":"References"},{"comment":"The flow-chart text in Figure 1 appears garbled in the PDF; please check the rendering before final submission.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central theoretical debt is to Part I; the referee was not able to check Theorem 2 and Condition 3 in detail. If Part I has not yet appeared, the authors should ensure the definitions and theorem statements are self-contained. Given the p-check flaw, I would require a corrected algorithm or removal of the p-check, and a rigorous treatment of the local-region verification, before any acceptance decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper recombines known ingredients—equilibrium-independent dissipativity, Krasovskii-type storage functions, and ADMM—into a compositional, equilibrium-free stability certificate for power systems. The local verification part (Proposition 1) is a legitimate sufficient condition, and the worked examples showing how to cast synchronous generators and inverter-based devices into the required input-output form are genuinely useful. The three applications (multiple equilibria, varying operating conditions, distributed assessment) are well chosen and showcase what the framework could do if it worked.\n\nThe soft spot is in Algorithm 1's p-check. The function lc is linear in each X_i, so solving (31) finds positive weights p_i with sum N such that the weighted sum of the individual block contributions is negative definite. That does not imply the unweighted sum is negative definite. A scalar example (L1 = X1, L2 = 2X2, X1 = -100, X2 = 80; p1 = 1.9, p2 = 0.1) makes the weighted sum -174 while the unweighted sum is +60. The algorithm then returns the unscaled X_i as satisfying the coupling condition, which is exactly what happens at iteration 259 in the 118-bus case. The claimed 42.83% speedup from the p-check is therefore based on an invalid certificate. The fix is to return the scaled matrices p_i X_i (and p_i P_i, since the local LMI in (29) is not invariant under scaling X_i alone), but that is not what the paper does.\n\nA second, quieter issue: local delta dissipativity is verified on sampled points in D_i, not on the entire region. The paper acknowledges this and suggests re-characterizing D_i afterward, but it never does so. The stability claims for any equilibrium in D = product of the D_i require the dissipation inequality on the whole region, so the examples as presented are not fully rigorous. The numerical results also lack code or data, so they are not independently reproducible.\n\nThese are load-bearing flaws, but they are also clearly repairable. Proposition 1 is sound, the model transformations are solid, and the distributed architecture is sensible once the p-check is either corrected or removed in favor of the residual-based termination. The paper deserves a serious referee, but the current version should not be accepted. Send it out with a pointed request to fix Section III-B and to replace the sampled local verification with a rigorous region argument.","headline":"Useful compositional framework for power-system stability, but the ADMM p-check that terminates the 118-bus certificate is invalid as written, so the headline application is not supported.","tokens_in":17467,"tokens_out":2984,"would_cite":false,"duration_ms":35113,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A new method certifies stability of entire equilibrium sets in power grids from per-device dissipativity checks and a distributed coupling check, without per-equilibrium linearization.","keywords":["power system stability","delta dissipativity","compositional stability","distributed verification","ADMM","Krasovskii-type storage function","multiple equilibria","equilibrium-free"],"falsifier":"Construct a two-subsystem network with explicitly chosen local matrices X1 and X2 such that lc(X1,X2) has a strictly positive eigenvalue, then run the p-check problem (31) with N=2; if the optimal tz is negative while the raw lc remains positive, Algorithm 1 would wrongly certify the coupling condition, and the raw returned certificates must be rechecked.","tokens_in":16305,"feed_emoji":"⚡","tokens_out":6677,"duration_ms":63154,"temperature":0.7,"pith_summary":"Even when a grid has several possible operating points, a stability verdict can be issued for the whole set at once. This paper argues that if each device is delta-dissipative on a region and the interconnection satisfies a coupling condition, every isolated equilibrium in that region is asymptotically stable, so no eigenvalue analysis per equilibrium is needed. To make the conditions usable, it proposes a systematic verification of local delta dissipativity via Krasovskii-type storage functions, and a distributed ADMM-based algorithm for the coupling condition. Applying the machinery to modified 9-bus, 39-bus, and 118-bus benchmarks, the authors show the certificate extends across varying load conditions and multiple equilibria, and can be computed in a privacy-preserving, scalable way.","feed_headline":"One certificate proves stability for a whole equilibrium set","feed_subtitle":"Per-device dissipativity checks and a distributed ADMM coupling check replace repeated eigenvalue analysis.","key_machinery":"The central objects are the delta-dissipativity supply rate matrix Xi for each device, the matrix-valued coupling function lc(X1,...,XN), and the Krasovskii-type storage function S(x,u)=f(x,u)^T P f(x,u). The storage function turns the dissipation inequality into the algebraic matrix inequality in Proposition 1, so local verification reduces to an LMI. The coupling function lc is fed the Xi through a linear map involving the interconnection matrix C and permutation P_pi as defined in Part I, and the ADMM algorithm with relaxation variables ti and an adaptive penalty rho_k drives Xi toward Z_i while a p-check searches for positive weights pi that make the weighted coupling condition hold. The paper uses this machinery to certify that the equilibrium set inside the dissipative region is asymptotically stable.","core_discovery":"$\\Delta$ dissipativity with quadratic supply rates can be certified locally for heterogeneous dynamic and static power devices by a matrix inequality involving a Krasovskii-type storage function, and the network-wide coupling condition can be verified in a distributed manner. When both hold on a region D = D1 x ... x DN, the theory from Part I asserts that any isolated equilibrium inside D is asymptotically stable. The paper constructs this certificate on three benchmark systems: five equilibria are found in the 9-bus case, two of them inside D and therefore provably stable; load scaling in the 39-bus case allows reusing the same Xi matrices over s in (0.86, 1.16); and the 118-bus distributed computation terminates in 259 iterations via a p-check that rescales the local certificates, reducing iterations by 42.83%.","pith_inferences":["We infer that the same coupling-verification scheme could certify stability in other networked systems whose interconnection admits a similar quadratic supply-rate coupling, such as multi-agent or traffic networks.","We infer that replacing the p-check's final output by the weighted certificates pi Xi, rather than the raw Xi, would make the termination criterion valid, since the coupling condition is linear in each Xi.","We conjecture that a scenario-optimization variant of the local LMI verification, sampling points in Di, will inherit the same theoretical guarantee if the sample count is chosen by standard scenario-optimization bounds; the paper uses such sampling but does not quantify the robustness gap."],"forward_implications":["One stability certificate can cover all equilibria lying in a precomputed dissipative region, so shifting operating points do not force fresh eigenvalue computations.","Because the Xi matrices are equilibrium-independent, they can be reused as loads vary within a rated range, turning stability assessment into a simple local membership test.","The distributed ADMM formulation lets each device keep its full model private, sharing only an exchange matrix with a coordinator.","The method scales to the 118-bus test case with 188 state and 236 algebraic variables, where a single Jacobian/eigenvalue evaluation takes over 147 seconds but the distributed certificate is reusable.","Certification remains sufficient rather than necessary, so equilibria outside the certified region are not necessarily unstable."],"supporting_citations":[{"why":"Supplies the equilibrium-independent dissipativity notion that the paper's delta-dissipativity conditions are adapted from.","marker":"[9]"},{"why":"Provides the ADMM convergence framework that the distributed coupling-condition verification builds on.","marker":"[12]"},{"why":"The prior ADMM method for compositional performance certification that Algorithm 1 extends with relaxation and p-check.","marker":"[15]"},{"why":"Gives the classical dissipativity condition for nonlinear systems that the Krasovskii-type local verification generalizes.","marker":"[8]"},{"why":"Earlier distributed stability conditions for heterogeneous nonlinear bus dynamics that this paper extends to delta dissipativity.","marker":"[11]"}],"fun_headline_variants":["One delta-dissipativity certificate covers many equilibria","Distributed ADMM check certifies stability without equilibria","One inequality certifies multiple equilibria","Equilibrium-free grid stability via distributed ADMM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The final p-check assumes that positively reweighting each device's local condition cannot turn a failing coupling condition into a passing one, yet such a cancellation is possible in general.","fun_headline_variants_meta":{"raw":{"variants":["One delta-dissipativity certificate covers many equilibria","Distributed ADMM check certifies stability without equilibria","One inequality certifies multiple equilibria","Equilibrium-free grid stability via distributed ADMM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000855,"raw_usage":{"total_tokens":3664,"prompt_tokens":844,"completion_tokens":2820,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2759}},"tokens_in":460,"tokens_out":2820,"duration_ms":23948,"temperature":1.0,"reasoning_tokens":2759,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:11:12.943957+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a two-subsystem network with explicitly chosen local matrices X1 and X2 such that lc(X1,X2) has a strictly positive eigenvalue, then run the p-check problem (31) with N=2; if the optimal tz is negative while the raw lc remains positive, Algorithm 1 would wrongly certify the coupling condition, and the raw returned certificates must be rechecked.","supporting_citations":[{"cited_title":"Equilibrium-independent dissipativity with quadratic supply rates,","cited_arxiv_id":null,"evidence_quote":"Supplies the equilibrium-independent dissipativity notion that the paper's delta-dissipativity conditions are adapted from."},{"cited_title":"Distributed optimization and statistical learning via the alternating direction method of multipliers,","cited_arxiv_id":null,"evidence_quote":"Provides the ADMM convergence framework that the distributed coupling-condition verification builds on."},{"cited_title":"Compositional performance certification of interconnected systems using admm,","cited_arxiv_id":null,"evidence_quote":"The prior ADMM method for compositional performance certification that Algorithm 1 extends with relaxation and p-check."},{"cited_title":"The stability of nonlinear dissipative systems,","cited_arxiv_id":null,"evidence_quote":"Gives the classical dissipativity condition for nonlinear systems that the Krasovskii-type local verification generalizes."},{"cited_title":"Distributed stability conditions for power systems with heterogeneous nonlinear bus dynamics,","cited_arxiv_id":null,"evidence_quote":"Earlier distributed stability conditions for heterogeneous nonlinear bus dynamics that this paper extends to delta dissipativity."}],"review_version":1}