Compositional and Equilibrium-Free Conditions for Power System Stability -- Part II: Method and Application

Feng Liu; Hua Geng; Peng Yang; Xiaoyu Peng; Yifan Su

REVIEW 3 major objections 5 minor 25 references

Compositional and Equilibrium-Free Conditions for Power System Stability -- Part II: Method and Application

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.11411 v1 pith:E55UBJC5 submitted 2025-06-13 eess.SY cs.SY

Peng Yang , Yifan Su , Xiaoyu Peng , Hua Geng , Feng Liu This is my paper

classification eess.SYcs.SY

keywords powersystemstabilitydeltadissipativitycompositionaldistributedverificationADMMKrasovskii-typestoragefunctionmultipleequilibriaequilibrium-free

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

$\Delta$ 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%.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

A 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.

These 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section III-B, Step 2 (Algorithm 1, Eq. (31))] 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 III-A and Sections IV-A, IV-B case studies] 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 III-B, Step 6 (Eq. (36))] 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.

minor comments (5)

[Section II heading] The heading 'V erification of Local Conditions' contains a typo; it should read 'Verification of Local Conditions'.
[Section II-B, after Eq. (22)] The phrase 'a linear matrix equality' should be 'a linear matrix inequality'.
[Section IV-A3 and elsewhere] The notation 'R12' should be written as R^12 (or \mathbb{R}^{12}) to denote the 12-dimensional Euclidean space.
[References] Reference [24] contains an incomplete entry ('02102'); please correct the bibliographic details.
[Figure 1] The flow-chart text in Figure 1 appears garbled in the PDF; please check the rendering before final submission.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the local and coupling certificates are obtained from feasibility searches and independently checked by eigenvalue analysis and time-domain simulation; the only self-citation debt is the companion Part I, which is a normal two-part-paper reference rather than a circular reduction.

full rationale

Part II's derivation chain is: Part I supplies the stability theorem; Part II constructs local delta dissipativity certificates through Proposition 1 and 2 with the LMIs (19), (22), and (24), then verifies the coupling condition lc(X1,...,XN) <= 0 as a separate feasibility problem (25)-(28), or via Algorithm 1. The system-level stability assertions in Section IV are applications of Theorem 2 (Part I) to the found certificates. Nothing in this chain reduces the conclusion to the premises: the matrices Pi and Xi are free variables solved from the local LMIs, the coupling condition is an independently stated inequality, and the resulting stability claims are checked against centralized eigenvalue analysis and nonlinear time-domain simulations. The principal debt is the companion Part I by the same authors, which is not fully reproducible from Part II alone; however, relying on a companion theory in a two-part paper is standard practice and does not make the derivation circular. The p-check step in Eq. (31) is a correctness or implementation flaw (it certifies only the scaled matrices p_i X_i and yet returns the unscaled X_i as satisfying lc(X_i) <= 0), but this is not a circularity: the check is invalid, not tautological. Overall, the central claim has independent content and is externally checked, so the circularity score is low.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No physical entities are invented; the ledger records algorithmic and modeling assumptions. The ADMM hyperparameters and sampled verification grids are the main free choices, while the P_i/X_i certificates are decision variables rather than fitted data. The dominant assumptions are reliance on Part I theory and on sampling-based local verification.

free parameters (3)

ADMM hyperparameters in Section IV-C2 = epsilon=1e-4, eps_pri=eps_dual=0.0042, mu=10, tau_incr=tau_decr=2, tbar=1e-2, M=1000
Chosen by hand; they affect convergence and stopping behavior, and the p-check stopping uses epsilon, so they influence which certificates are returned.
Per-device certificate matrices P_i and X_i in case studies = e.g., P1 and X1 printed in Section IV-A2; solved numerically
These are synthesized by LMI solving rather than fitted to data, but the central stability claims depend on their existence and the paper gives no code to reproduce them.
Sampled verification grid for dissipative regions D_i = not specified
The regions D_i are inferred from pointwise or scenario checks; the grid density and sample size are not reported, so the certificate regions are not reproducible.

assumptions (4)

domain assumption The delta dissipativity definitions, storage conditions, and Theorem 2 from Part I are correct and applicable to the device models used here.
Part II takes Part I's theory as given and does not restate or prove it; Section I and Section IV-A3 invoke Theorem 2.
domain assumption The device models in Section II-A (SG, PLL inverter, VSG, conventional droop, quadratic droop, ZIP) are valid reduced-order models for stability studies.
Stability certificates are computed for these models; if the models are inaccurate, the certificates apply to the models rather than to the physical grid.
ad hoc to paper In the case studies, verifying the local LMI (19) on sampled points of D is sufficient to certify delta dissipativity on the whole region D.
Section II-B suggests scenario sampling and 're-characterizing' D, but no global proof or sample-size guarantee is given; all three applications rely on accurate D_i.
standard math ADMM converges for (28) because the local feasible sets L_i are convex.
Section III-A invokes Boyd et al. [12] for convergence; the LMI-defined feasible sets are indeed convex.

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Cite this review

Pith. "Pith review of Compositional and Equilibrium-Free Conditions for Power System Stability -- Part II: Method and Application." pith.science (2026). https://pith.science/paper/E55UBJC5

@misc{pith2026250611411,
  author       = {Pith},
  title        = {Pith review of: Compositional and Equilibrium-Free Conditions for Power System Stability -- Part II: Method and Application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E55UBJC5}},
  note         = {Machine review of arXiv:2506.11411}
}

read the original abstract

This two-part paper proposes a compositional and equilibrium-free approach to analyzing power system stability. In Part I, we have established the stability theory and proposed stability conditions based on the delta dissipativity. In Part II, we focus on methods for applying our theory to complex power grids. We first propose a method to verify the local condition, i.e., delta dissipativity, for heterogeneous devices in power systems. Then, we propose a method to verify the coupling condition based on Alternating Direction Method of Multipliers (ADMM). Finally, we investigate three applications of our theory including stability assessment toward multiple equilibria, stability assessment under varying operating conditions, and a distributed computing framework. Case studies on modified IEEE 9-bus, 39-bus, and 118-bus benchmarks well verified our theory and methods.

Figures

Figures reproduced from arXiv: 2506.11411 by the authors.

**Figure 1.** The flow chart of Algorithm 1. A. Application 1: Stability Assessment for Multiple Equilibria Power systems often exhibit multiple equilibria under fixed parameters due to nonlinearity. Traditional equilibriumoriented methods, such as eigenvalue analysis, require individual equilibrium evaluations. In contrast, our approach certifies stability for all equilibria within a specified region via Theorem 2 (see Part I)… view at source ↗

**Figure 3.** The cross sections of Di, i = 1, 2, 3 on the 2-dimensional input plane. The red dots represent the nominal inputs u 0 i of each dynamic subsystem. 3) Multiple equilibria assessment: We identified five isolated equilibria under nominal parameters, as shown in Table II. Theorem 2 (Part I) ensures that any isolated equilibrium in D = D1 × D2 × D3 × R 12 is asymptotically stable. For any equilibrium (x 0 , u0 ) of the … view at source ↗

**Figure 2.** The modified IEEE 9-bus system [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The system equilibrium stably shifts from #1 to #2 after a disturbance [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 6.** Figure 6: The cross sections of Di, i = 33, 16 on the 2-dimensional input plane with xi = x 0 i . The red dots represent the nominal inputs u 0 i of each dynamic subsystem. To verify, we consider the equilibrium (x 0 , u0 ) of the system under the nominal operation condition. Ea…

**Figure 5.** Figure 5: The modified IEEE 39-bus system. 2) Verifying stability conditions: Using Proposition 1, we verify that all dynamic subsystems satisfy delta dissipativity with computed Pi , Xi , and Di . For example, the dissipative region Di of PLL1 and QD1 are 8- and 6-dimensional, …

**Figure 7.** Figure 7: The system load scaling factor s switches from 1 to 0.86 at t = 10s, from 0.86 to 1 at t = 40s, and from 1 to 1.16 at t = 70s. Fig.8. This approach reduces the computational load on the central coordinator while preserving the privacy of individual subsystems, as each …

**Figure 9.** Figure 9: (A) The modified IEEE 118-bus system. (B) The primal and dual residuals in the iteration process. (C) The minimal [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

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Works this paper leans on

25 extracted references · 22 canonical work pages

[12]

Distributed optimization and statistical learning via the alternating direction method of multipliers,

S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein et al., “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. , vol. 3, no. 1, pp. 1–122, 2011

work page 2011
[1]

Power system dynamic response calculations,

B. Stott, “Power system dynamic response calculations,” Proc. IEEE , vol. 67, no. 2, pp. 219–241, 1979

work page 1979
[2]

Hierarchical stability and alert state steering control of interconnected power systems,

S. Sastry and P. Varaiya, “Hierarchical stability and alert state steering control of interconnected power systems,” IEEE Trans. Circuits Syst. , vol. 27, no. 11, pp. 1102–1112, 1980

work page 1980
[3]

Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective,

H.-D. Chang, C.-C. Chu, and G. Cauley, “Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective,” Proc. IEEE, vol. 83, no. 11, pp. 1497–1529, 1995

work page 1995
[4]

On relationships among passivity, positive realness, and dissipativity in linear systems,

N. Kottenstette, M. J. McCourt, M. Xia, V . Gupta, and P. J. Antsaklis, “On relationships among passivity, positive realness, and dissipativity in linear systems,” Automatica, vol. 50, no. 4, pp. 1003–1016, 2014

work page 2014
[5]

Kalman-popov-yakubovich lemma and the s-procedure: A historical essay,

S. V . Gusev and A. L. Likhtarnikov, “Kalman-popov-yakubovich lemma and the s-procedure: A historical essay,”Autom. Remote Control, vol. 67, pp. 1768–1810, 2006

work page 2006
[6]

A port-hamiltonian approach to power network modeling and analysis,

S. Fiaz, D. Zonetti, R. Ortega, J. M. Scherpen, and A. Van der Schaft, “A port-hamiltonian approach to power network modeling and analysis,” Eur. J. Control, vol. 19, no. 6, pp. 477–485, 2013

2013
[7]

A unifying energy- based approach to stability of power grids with market dynamics,

T. Stegink, C. De Persis, and A. van der Schaft, “A unifying energy- based approach to stability of power grids with market dynamics,” IEEE Trans. Autom. Control, vol. 62, no. 6, pp. 2612–2622, 2017

work page 2017

Show all 25 references

[8]

The stability of nonlinear dissipative systems,

D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Autom. Control , vol. 21, no. 5, pp. 708–711, 1976

1976
[9]

Equilibrium-independent dissipativity with quadratic supply rates,

J. W. Simpson-Porco, “Equilibrium-independent dissipativity with quadratic supply rates,” IEEE Trans. Autom. Control , vol. 64, no. 4, pp. 1440–1455, 2018

2018
[10]

A distributed framework for stability evaluation and enhancement of inverter-based microgrids,

Y . Song, D. J. Hill, T. Liu, and Y . Zheng, “A distributed framework for stability evaluation and enhancement of inverter-based microgrids,” IEEE Trans. Smart Grid , vol. 8, no. 6, pp. 3020–3034, 2017

2017
[11]

Distributed stability conditions for power systems with heterogeneous nonlinear bus dynamics,

P. Yang, F. Liu, Z. Wang, and C. Shen, “Distributed stability conditions for power systems with heterogeneous nonlinear bus dynamics,” IEEE Trans. Power Syst., vol. 35, no. 3, pp. 2313–2324, 2020

2020
[13]

A survey of distributed optimization and control algorithms for electric power systems,

D. K. Molzahn, F. D ¨orfler, H. Sandberg, S. H. Low, S. Chakrabarti, R. Baldick, and J. Lavaei, “A survey of distributed optimization and control algorithms for electric power systems,” IEEE Trans. Smart Grid, vol. 8, no. 6, pp. 2941–2962, 2017

2017
[14]

Compositional stability analysis based on dual decomposition,

U. Topcu, A. K. Packard, and R. M. Murray, “Compositional stability analysis based on dual decomposition,” in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. IEEE, 2009, pp. 1175–1180

2009
[15]

Compositional performance certification of interconnected systems using admm,

C. Meissen, L. Lessard, M. Arcak, and A. K. Packard, “Compositional performance certification of interconnected systems using admm,” Au- tomatica, vol. 61, pp. 55–63, 2015

2015
[16]

A structure preserving energy function for power system transient stability analysis,

N. Tsolas, A. Arapostathis, and P. Varaiya, “A structure preserving energy function for power system transient stability analysis,” IEEE Trans. Circuits Syst., vol. 32, no. 10, pp. 1041–1049, 1985

1985
[17]

Impacts of grid structure on pll-synchronization stability of converter-integrated power systems,

L. Huang, H. Xin, W. Dong, and F. D ¨orfler, “Impacts of grid structure on pll-synchronization stability of converter-integrated power systems,” IFAC-PapersOnLine, vol. 55, no. 13, pp. 264–269, 2022

2022
[18]

Virtual synchronous generators: a survey and new perspectives,

H. Bevrani, T. Ise, and Y . Miura, “Virtual synchronous generators: a survey and new perspectives,” Int. J. Electr. Power Energy Syst., vol. 54, pp. 244–254, 2014

2014
[19]

Online dynamic security assessment of microgrid interconnections in smart distribution systems,

Y . Zhang and L. Xie, “Online dynamic security assessment of microgrid interconnections in smart distribution systems,”IEEE Trans. Power Syst., vol. 30, no. 6, pp. 3246–3254, 2015

2015
[20]

V oltage stabilization in microgrids via quadratic droop control,

J. W. Simpson-Porco, F. D ¨orfler, and F. Bullo, “V oltage stabilization in microgrids via quadratic droop control,” IEEE Trans. Autom. Control , vol. 62, no. 3, pp. 1239–1253, 2017

2017
[21]

Kundur, N

P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability and control. McGraw-hill New York, 1994, vol. 7

1994
[22]

Robust optimization–methodology and applications,

A. Ben-Tal and A. Nemirovski, “Robust optimization–methodology and applications,” Math. Program., vol. 92, pp. 453–480, 2002

2002
[23]

Anderson and A

P. Anderson and A. Fouad, POWER SYSTEM CONTROL AND STABIL- ITY, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2008

2008
[24]

Pai, Energy Function Analysis for Power System Stability

A. Pai, Energy Function Analysis for Power System Stability . Boston: Springer, 1989, 02102

1989
[25]

A practical method for the direct analysis of transient stability,

T. Athay, R. Podmore, and S. Virmani, “A practical method for the direct analysis of transient stability,” IEEE Trans. Power Apparatus and Systems, no. 2, pp. 573–584, 1979

1979

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [12]

Distributed optimization and statistical learning via the alternating direction method of multipliers,

S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein et al., “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. , vol. 3, no. 1, pp. 1–122, 2011

work page 2011

[2] [1]

Power system dynamic response calculations,

B. Stott, “Power system dynamic response calculations,” Proc. IEEE , vol. 67, no. 2, pp. 219–241, 1979

work page 1979

[3] [2]

Hierarchical stability and alert state steering control of interconnected power systems,

S. Sastry and P. Varaiya, “Hierarchical stability and alert state steering control of interconnected power systems,” IEEE Trans. Circuits Syst. , vol. 27, no. 11, pp. 1102–1112, 1980

work page 1980

[4] [3]

Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective,

H.-D. Chang, C.-C. Chu, and G. Cauley, “Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective,” Proc. IEEE, vol. 83, no. 11, pp. 1497–1529, 1995

work page 1995

[5] [4]

On relationships among passivity, positive realness, and dissipativity in linear systems,

N. Kottenstette, M. J. McCourt, M. Xia, V . Gupta, and P. J. Antsaklis, “On relationships among passivity, positive realness, and dissipativity in linear systems,” Automatica, vol. 50, no. 4, pp. 1003–1016, 2014

work page 2014

[6] [5]

Kalman-popov-yakubovich lemma and the s-procedure: A historical essay,

S. V . Gusev and A. L. Likhtarnikov, “Kalman-popov-yakubovich lemma and the s-procedure: A historical essay,”Autom. Remote Control, vol. 67, pp. 1768–1810, 2006

work page 2006

[7] [6]

A port-hamiltonian approach to power network modeling and analysis,

S. Fiaz, D. Zonetti, R. Ortega, J. M. Scherpen, and A. Van der Schaft, “A port-hamiltonian approach to power network modeling and analysis,” Eur. J. Control, vol. 19, no. 6, pp. 477–485, 2013

2013

[8] [7]

A unifying energy- based approach to stability of power grids with market dynamics,

T. Stegink, C. De Persis, and A. van der Schaft, “A unifying energy- based approach to stability of power grids with market dynamics,” IEEE Trans. Autom. Control, vol. 62, no. 6, pp. 2612–2622, 2017

work page 2017