Pith. sign in

REVIEW 4 major objections 4 minor 33 references

Compositional and Equilibrium-Free Conditions for Power System Stability -- Part I: Theory

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Equilibrium-free conditions certify power system stability

desk verdict A genuine local-DAE extension of delta dissipativity with a clean network LMI, but Theorem 1's convergence proof outsources to an unstated prior theorem and the example defers its verification to Part II. read the letter →

arxiv 2506.11406 v1 pith:UKPFEQTS submitted 2025-06-13 eess.SY cs.SY

Peng Yang , Xiaoyu Peng , Xi Ru , Hua Geng , Feng Liu This is my paper
classification eess.SYcs.SY MSC 93D2093C1093A14
keywords powersystemstabilityequilibrium-freeanalysiscompositionaldeltadissipativityequilibriasetdifferential-algebraicequationsstructure-preservingmodel
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

This paper proves that the stability of a large power system can be certified without ever computing the system-wide equilibrium point. The method breaks the system into individual buses, asks each device to pass a local delta dissipativity check that depends only on its own dynamics, and combines those checks with a single linear matrix inequality describing the network. If all checks pass, the entire equilibrium set is asymptotically stable, and every isolated equilibrium inside it is stable. This matters because equilibrium points couple all subsystems together, so traditional centralized analysis must be redone whenever operating conditions change; the proposed conditions are reusable across load variations and device types.

What carries the argument

The central object is delta dissipativity with a quadratic supply rate, denoted $\text{delta-}D(X_i,\mathcal{D}_i)$: a system whose storage function is bounded by class-$\mathcal{K}$ functions of the vector field norm and whose dissipation inequality involves only the time derivatives of input and output, holding on a region $\mathcal{D}$ rather than globally. For static buses the same notion reduces to a matrix inequality (13) on the Jacobian of the device map. The argument combines these local conditions through the network coupling matrix $C=(A_I+M_Y A_V)^{-1}(B_I+M_Y B_V)$, which is constant because the network is modeled by linear voltage-current equations, turning the coupling condition into the constant LMI (22). Stability of the equilibrium set follows by invoking the invariance and region-of-attraction result from [25, Theorem 6] on the sublevel sets of the weighted storage function.

What would settle it

On the SMSL benchmark, vary the load scaling factor $s$ and locate the first value where $\det(\partial g/\partial u)=0$ or where the algebraic manifold leaves the dissipative region; then simulate trajectories from a nearby initial state. If trajectories still converge to the equilibrium set despite the singularity, Assumption 2 is stronger than needed; if they do not, the theorem's conclusion is false without it.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes Theorem 2: for a structure-preserving differential-algebraic power system model (8), if every dynamic bus satisfies delta dissipativity on a region $\mathcal{D}_i$ with a quadratic supply rate, every static bus satisfies the static version of that condition, and the constant network LMI (22) holds for some positive weights $p_i$, then the equilibrium set $\mathcal{E}$ is asymptotically stable and every isolated equilibrium in $\mathcal{E}$ is asymptotically stable. The verification of the local conditions does not require knowing the equilibrium of the interconnected system, and the conclusion covers all equilibria inside the dissipative region rather than a single operating point. The paper also derives transient stability estimates from sublevel sets of the weighted storage function, showing convergence to the post-fault equilibrium set without knowing that equilibrium in advance.

Load-bearing premise

The proof's convergence conclusion rests on applying the invariance and region-of-attraction theorem from reference [25] to the differential-algebraic model, together with Assumption 2 that $\partial g/\partial u$ is nonsingular on the closure of the dissipative region; neither is verified for the single-machine single-load example, and if either fails the argument collapses.

Editorial extensions

If this is right

  • Local device certificates can be issued without any grid-side equilibrium data, so the same certificate remains valid under load or generation changes as long as the new equilibrium stays inside the dissipative region.
  • The coupling condition is a constant linear matrix inequality that can be solved by convex optimization, either centrally or in a distributed way, and only needs to be rechecked when network topology or device dissipativity changes.
  • Transient stability can be assessed from the initial state alone: if the state lies in a sublevel set of the weighted storage function inside the dissipative region, it converges to the equilibrium set without knowing the post-fault equilibrium.
  • Enlarging the local dissipative region of a device, without changing its storage function value, directly enlarges the guaranteed region of attraction, giving a localized control design target for transient stability enhancement.

Reading between the lines

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

  • If the invariance theorem from [25] carries over to DAEs with singular algebraic Jacobians, the method might extend to devices whose static maps have degenerate points, but the paper does not establish that, since Assumption 2 is not verified on the SMSL example.
  • The method's conservativeness, visible in the load range $s\in(0.901,1.075)$ versus the true range $(0,1.564)$, could be reduced by optimizing storage functions; the paper flags this as future work.
  • The privacy-preserving property suggests a practical deployment pattern where device manufacturers publish only their dissipativity matrix $X_i$ rather than full models, which would let system operators compose certificates without knowing device internals, though privacy guarantees are not proven here.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a compositional, equilibrium-free stability certification framework for structure-preserving power systems described by differential-algebraic equations. The key object is a region-local variant of delta dissipativity: each dynamic bus is assumed to satisfy a quadratic dissipation inequality with storage bounded in terms of the local vector field, and each static bus is assumed to satisfy a corresponding algebraic dissipation inequality on a region D_i. The paper's main theoretical result, Theorem 1, states that if each subsystem is delta-D(X_i,D_i) and a network LMI (21) holds, then the equilibrium set E of the interconnected DAE is asymptotically stable, every isolated equilibrium is asymptotically stable, and sublevel sets of the weighted storage provide positively invariant regions of attraction. The proof combines Lemma A1, which derives a weighted dissipation inequality on the algebraic manifold, with an external theorem from the authors' prior work [25]. Theorem 2 specializes this to power systems with linear voltage-current network coupling, and Theorem 3 turns the sublevel-set statement into a transient-stability certificate. The theory is illustrated on a single-machine-single-load benchmark with a synchronous generator and a constant PQ load.

Significance. If the proof gaps are closed, this would be a useful contribution: the conditions are genuinely local and equilibrium-free, the network condition is a constant LMI for the linear voltage-current coupling, the structure-preserving DAE formulation covers lossy networks, and the equilibrium-set viewpoint is well motivated for systems with multiple or shifting operating points. The explicit benchmark comparison with eigenvalue analysis and the honest statement that the method is conservative and only sufficient are also strengths. However, the central convergence argument is not self-contained: Theorem 1 outsources the key invariance and convergence step to [25, Theorem 6], a prior result that is neither restated nor shown to apply to the semiexplicit DAE setting with local dissipativity and an algebraic-variable-dependent storage. The numerical verification in Section V also reports matrices and a region but does not supply the verification procedure. The central claim is plausible and likely fixable, but as written the paper does not fully support it.

major comments (4)
  1. [Section III-C, proof of Theorem 1] The proof's step 'It follows from [25, Theorem 6] that S_b^{-1} is a positive invariant set and an estimate of the f-RoA' is load-bearing: this is where positive invariance and f(x(t),u(t))->0 are obtained. Theorem 6 of [25] is not stated in the manuscript, and the hypotheses required to apply it to the semiexplicit DAE (18) are not checked. In particular, [25, Theorem 6] may have been proved for an augmented ODE system or for global dissipativity, whereas here S depends on the algebraic variable u and the dissipation inequality holds only on D_G, not globally. The authors should either restate [25, Theorem 6], prove that all its hypotheses are satisfied by the DAE (18) under Assumptions 1-2 and Lemma A1, or replace the invocation with a self-contained convergence proof. This gap is inherited by Theorems 2 and 3, since both are direct corollaries of Theorem 1.
  2. [Section III-C, after Eq. (20) and proof of Theorem 1] The implication 'f(x(t),u(t))->0 implies dist((x(t),u(t)),E)->0' is not justified by boundedness of E alone. A trajectory could be unbounded while f(x(t),u(t)) tends to zero, in which case its distance to a bounded set E need not converge to zero. The storage bounds alpha(||f||)<=S<=beta(||f||) together with Sdot<=-gamma(||f||) do not yield boundedness of (x,u), because S bounds f, not the state. The proof needs an additional precompactness argument, for example boundedness of the relevant sublevel set S_l^{-1}, or a properness assumption relating S to distance from E. Without this, conclusion 1) and conclusion 3) of Theorem 1 are unsupported.
  3. [Section III-B, Assumption 2] Assumption 2 is stated only for (x,u) in D, but the proof and the invocation of [25, Theorem 6] require nonsingularity of the algebraic Jacobian on the closure of the relevant sublevel sets, not merely on D. If a trajectory approaches the boundary of D_G, differentiability of the algebraic variables and the DAE solution may fail before convergence is established. The assumption should be strengthened to hold on the closure of D (or on the closure of the sublevel set used in the proof), and the SMSL validation should report that this condition was checked. As written, the regularity hypothesis needed for the convergence argument is not verified in the example.
  4. [Section V.B and V.C] The SMSL example asserts that the synchronous generator is delta-D(X_1,D_1) with the displayed matrices P_1, X_1 and that the PQ load is delta-D(X_2,D_2), but the verification is not reported. The text says 'we found' and shows a projection of D_1, but it does not state which inequalities were checked, over which grid, or whether det(dg/du) != 0 was verified. Since the example is the only numerical support for the theory in this paper, the verification should be reproducible: either provide the defining inequalities and the numerical procedure, or refer to a detailed derivation in Part II in a way that does not leave the current paper's claim unsupported.
minor comments (4)
  1. [Section II-B, first paragraph] The sentence 'Define the collective input of all buses as u:=col({u_i}) in R^m, and the collective input of all buses as y:=col({y_i}) in R^m' should read 'collective output' for y; this is a typo but confusing in a definitional section.
  2. [Appendix A, proof of Lemma A1] The set D(r) is defined as the positive quadrant vectors with norm r, not a disk; the name D(r) is easy to confuse with the dissipativity region D. Consider renaming it, e.g., R(r), to avoid ambiguity.
  3. [Section V.C, Table II] The table would be more informative if it also reported the actual maximal real part of the eigenvalues for the two equilibria, since the text states the centralized result but the table only marks 'stable' or 'unstable'.
  4. [Section V.D, Fig. 5 caption] The caption's statement that D_G is independent of omega should be justified or rephrased: the figure is a projection/slice, and the dissipativity region D_G in the full state-input space may still depend on omega through the vector field; the text should make clear what exactly is plotted.

Circularity Check

0 steps flagged · score 2.0 of 10

No derivation step reduces by construction; Theorem 2 is a Lyapunov-style composition result, though the proof leans on a self-cited prior theorem and the example's certificates are numerically constructed.

full rationale

The central theorem (Theorem 2) derives system-wide stability from local delta dissipativity conditions and a network LMI, rather than from a fitted quantity or from a definition that contains the conclusion. The proof of Theorem 1 constructs Lemma A1, which establishes alpha(||f||) <= S <= beta(||f||) and Sdot <= -gamma(||f||) on D_G, and then invokes [25, Theorem 6] to obtain positive invariance and f -> 0. This is a load-bearing citation of prior work by the first author, but it is a mathematical theorem with stated hypotheses, not a fitted dataset or an equation identical to the target claim; no step in the paper sets a parameter equal to the predicted quantity or defines an input in terms of the output. The SMSL example uses numerically selected P1, X1, X2 and regions D1, D2 which feed the LMI (22); however, the verified object is the dissipativity property on the region, and the predicted stability of equilibria follows logically from Theorem 2 rather than being a restatement of the numerical fit. The main limitations, such as whether [25, Theorem 6] applies verbatim to the DAE (18), whether Assumption 2 holds on the needed sublevel-set closures, and the unshown construction of D1/D2, are correctness or verification concerns rather than circularity. Therefore no significant circularity is found; the score reflects only the self-citation and certificate-construction reliance.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central theorem relies on structural assumptions (nonempty bounded equilibrium set, nonsingular algebraic manifold, smoothness), on the authors' prior augmented synchronization theorem [25], and on numerically chosen matrices and regions in the example. No new physical entities are introduced.

free parameters (5)
  • P1 (SG storage weighting matrix) = 3x3 numerical matrix in Section V-B, e.g. P1(1,1)=92.44, P1(3,3)=1000
    Chosen by numerical search to verify delta dissipativity of the synchronous generator; no analytical formula or uniqueness argument is given.
  • X1 (SG supply rate matrix) = 4x4 symmetric matrix in Section V-B
    Selected so that the SG satisfies delta-D(X1,D1); this matrix enters the network coupling LMI (22).
  • X2 (PQ load supply rate matrix) = 4x4 symmetric matrix in Section V-B
    Solved from inequality (13) for the constant PQ load; no derivation is shown.
  • p1, p2 (coupling weights) = p1=p2=1
    Chosen to satisfy Condition 3, the LMI (22); uniqueness is not discussed.
  • Dissipative region D1 for the SG = Nonlinear five-dimensional region, projection shown in Fig. 5
    Identified numerically; no analytic description is provided, and membership of a given equilibrium is checked offline rather than by a stated algorithm.
assumptions (5)
  • domain assumption Assumption 1: the equilibrium set E in the dissipative region is non-empty and bounded
    Stated in Section III-B and used in Theorem 1 to conclude that f->0 implies convergence to E. Not verified for the SMSL example.
  • domain assumption Assumption 2: det(∂g/∂u) != 0 on the closure of D, ensuring a well-posed DAE interconnection
    Standard index-1 DAE condition stated in Section III-B; needed to differentiate the algebraic constraint and to combine the dissipation inequalities in Lemma A1.
  • domain assumption Continuity and smoothness of f, h, and h_net (twice continuously differentiable)
    Assumed for all subsystems and the network in Section III; used for differentiation along trajectories and for Lyapunov arguments.
  • standard math Theorem 6 of [25] (augmented synchronization) is valid and applies to the DAE system (18)
    Invoked in the proof of Theorem 1 to conclude positive invariance and f(x,u)->0 from the Lyapunov inequality of Lemma A1. The theorem is not restated or proved in this paper, and it is the authors' own prior work.
  • domain assumption The network coupling h_net and matrix C are well-defined, requiring u_net to form a complete set of circuit variables
    Footnote in Section II-B; needed for the linear network equation (6) and the constant coupling matrix C in (7).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Compositional and Equilibrium-Free Conditions for Power System Stability -- Part I: Theory." pith.science (2026). https://pith.science/paper/UKPFEQTS

@misc{pith2026250611406,
  author       = {Pith},
  title        = {Pith review of: Compositional and Equilibrium-Free Conditions for Power System Stability -- Part I: Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UKPFEQTS}},
  note         = {Machine review of arXiv:2506.11406}
}
read the original abstract

Traditional centralized stability analysis struggles with scalability in large complex modern power grids. This two-part paper proposes a compositional and equilibrium-free approach to analyzing power system stability. In Part I, we prove that using equilibrium-free local conditions we can certificate system-wide stability of power systems with heterogeneous nonlinear devices and structure-preserving lossy networks. This is built on a recently developed notion of delta dissipativity, which yields local stability conditions without knowing the system-wide equilibrium. As a consequence, our proposed theory can certificate stability of equilibria set rather than single equilibrium. In Part I, we verify our theory and demonstrate promising implications by the single machine single load benchmark, which helps to better explain the compositional and equilibrium-set-oriented stability analysis. Part II of this paper will provide methods for applying our theory to complex power grids, together with case studies across a wide range of system scales. Our results enable a more scalable and adaptable approach to stability analysis. It also sheds light on how to regulate grid-connected devices to guarantee system-wide stability.

Figures

Figures reproduced from arXiv: 2506.11406 by the authors.

Figure 1
Figure 1. The feedback interconnection between the bus subsystems and the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The input-output differential dissipativity. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Relations of the sets in the proof of Theorem 1. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The SMSL 2-bus system. A SG with frequency PI regulator is [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: The stability region estimation of the 2-bus system. (A) The dissipative region [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Dissipative region D eq 1 in the u1-plane and the equilibrium trajectory as the load factor s varies. For s ∈ [0.901, 1.075], the corresponding equilibria are inside the dissipative region and hence are stable. 0 150 1 1.025 0.09 0.23 t (s) (rad)  E q  E q   (p.u.)…
Figure 7
Figure 7. Figure 7: The load step changes from the nominal value [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

33 extracted references · 31 canonical work pages

  1. [25]

    Augmented synchronization of power systems,

    P. Yang, F. Liu, T. Liu, and D. J. Hill, “Augmented synchronization of power systems,”IEEE Trans. Autom. Control, vol. 69, no. 6, pp. 3673– 3688, 2024

    work page 2024

  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]

    Stability and control of power grids,

    T. Liu, Y . Song, L. Zhu, and D. J. Hill, “Stability and control of power grids,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 5, no. 1, pp. 689–716, 2022

    work page 2022

  6. [5]

    Dissipative dynamical systems part i: General theory,

    J. C. Willems, “Dissipative dynamical systems part i: General theory,” Arch. Ration. Mech. Anal., vol. 45, no. 5, pp. 321–351, 1972

    work page 1972

  7. [6]

    Bao and P

    J. Bao and P. L. Lee,Process control: the passive systems approach. Springer Science & Business Media, 2007

    work page 2007

  8. [7]

    A. J. van der Schaft and A. Van Der Schaft,L2-gain and passivity techniques in nonlinear control. Springer, 2000, vol. 2

    work page 2000

Show all 33 references
  1. [8]

    Arcak, C

    M. Arcak, C. Meissen, and A. Packard,Networks of dissipative sys- tems: compositional certification of stability, performance, and safety. Springer, 2016

    2016

  2. [9]

    Distributed optimal frequency control considering a nonlinear network-preserving model,

    Z. Wang, F. Liu, J. Z. F. Pang, S. H. Low, and S. Mei, “Distributed optimal frequency control considering a nonlinear network-preserving model,”IEEE Trans. Power Syst., vol. 34, no. 1, pp. 76–86, Jan 2019

    2019

  3. [10]

    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

    2017

  4. [11]

    Distributed load-side control: Coping with variation of renewable generations,

    Z. Wang, S. Mei, F. Liu, S. H. Low, and P. Yang, “Distributed load-side control: Coping with variation of renewable generations,”Automatica, vol. 109, p. 108556, 2019

    2019

  5. [12]

    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

  6. [13]

    Compositional Transient Stability Anal- ysis of Multimachine Power Networks,

    S. Y . Caliskan and P. Tabuada, “Compositional Transient Stability Anal- ysis of Multimachine Power Networks,”IEEE Trans. Control Network Syst., vol. 1, no. 1, pp. 4–14, 2014

    2014

  7. [14]

    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

  8. [15]

    Passivity and decentralized stability conditions for grid-forming converters,

    X. He and F. D ¨orfler, “Passivity and decentralized stability conditions for grid-forming converters,”IEEE Trans. Power Syst., vol. 39, no. 3, pp. 5447–5450, 2024

    2024

  9. [16]

    Gain and phase: Decentralized stability conditions for power electronics-dominated power systems,

    L. Huang, D. Wang, X. Wang, H. Xin, P. Ju, K. H. Johansson, and F. D¨orfler, “Gain and phase: Decentralized stability conditions for power electronics-dominated power systems,”IEEE Trans. Power Syst., 2024

    2024

  10. [17]

    Distributed conditions for small-signal stability of power grids and local control design,

    S. Baros, A. Bernstein, and N. D. Hatziargyriou, “Distributed conditions for small-signal stability of power grids and local control design,”IEEE Trans. Power Syst., vol. 36, no. 3, pp. 2058–2067, 2020. 10

    2020

  11. [18]

    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

  12. [19]

    Compositional analysis of intercon- nected systems using delta dissipativity,

    K. S. Schweidel and M. Arcak, “Compositional analysis of intercon- nected systems using delta dissipativity,”IEEE Control Syst. Lett., vol. 6, pp. 662–667, 2021

    2021

  13. [20]

    Krasovskii and shifted passivity-based control,

    Y . Kawano, K. C. Kosaraju, and J. M. Scherpen, “Krasovskii and shifted passivity-based control,”IEEE Trans. Autom. Control, vol. 66, no. 10, pp. 4926–4932, 2020

    2020

  14. [21]

    Krasovskii and shifted passivity based output consensus,

    Y . Kawano, M. Cucuzzella, S. Feng, and J. M. Scherpen, “Krasovskii and shifted passivity based output consensus,”Automatica, vol. 155, p. 111167, 2023

    2023

  15. [22]

    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

  16. [23]

    Incremental passivity and output regulation,

    A. Pavlov and L. Marconi, “Incremental passivity and output regulation,” Systems & Control Letters, vol. 57, no. 5, pp. 400–409, 2008

    2008

  17. [24]

    On differential passivity of physical systems,

    F. Forni, R. Sepulchre, and A. J. van der Schaft, “On differential passivity of physical systems,” in52nd IEEE Conference on Decision and Control, Dec 2013, pp. 6580–6585

    2013

  18. [26]

    A structure preserving model for power system stability analysis,

    A. R. Bergen and D. J. Hill, “A structure preserving model for power system stability analysis,”IEEE transactions on power apparatus and systems, no. 1, pp. 25–35, 1981

    1981

  19. [27]

    A system reference frame approach for stability analysis and control of power grids,

    C. Spanias and I. Lestas, “A system reference frame approach for stability analysis and control of power grids,”IEEE Trans. Power Syst., vol. 34, no. 2, pp. 1105–1115, 2018

    2018

  20. [28]

    Passivity and evolutionary game dynamics,

    S. Park, J. S. Shamma, and N. C. Martins, “Passivity and evolutionary game dynamics,” in2018 IEEE Conference on Decision and Control (CDC). IEEE, 2018, pp. 3553–3560

    2018

  21. [29]

    Population games, stable games, and passivity,

    M. J. Fox and J. S. Shamma, “Population games, stable games, and passivity,”Games, vol. 4, no. 4, pp. 561–583, 2013

    2013

  22. [30]

    Dissipativity tools for convergence to Nash equilibria in population games,

    M. Arcak and N. C. Martins, “Dissipativity tools for convergence to Nash equilibria in population games,”IEEE Trans. Control Network Syst., vol. 8, no. 1, pp. 39–50, 2020

    2020

  23. [31]

    H. K. Khalil and J. W. Grizzle,Nonlinear systems. Prentice hall Upper Saddle River, NJ, 2002, vol. 3

    2002

  24. [32]

    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. APPENDIXA PROOF OFTHEOREM1 Lemma...

    2017

  25. [33]

    By Definition 1 and 2, there exists classKfunctionsγ i, i= 1, . . . , Nd such that for any(x, u)∈ DG we have ˙S(x, u)≤ NX i=1 pi ˙ui ˙yi T Xi ˙ui ˙yi − NdX i=1 piγi (∥fi∥) It follows from the definition ofP π that NX i=1 pi ˙ui ˙yi T Xi ˙ui ˙yi = ˙u ˙y T P T π blkdiag(p1X1, . ...

Pith tools

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