Dynamic Droop Control in Low-inertia Power Systems

Enrique Mallada; Richard Pates; Yan Jiang

REVIEW 3 major objections 6 minor 34 references

Dynamic Droop Control in Low-inertia Power Systems

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that a dynamic droop controller can tune low-inertia grids for noise rejection, fast synchronization, and no frequency Nadir without changing the steady-state control burden.

desk verdict Careful and useful theoretical analysis of iDroop; the results are conditional on a restrictive proportionality assumption, and the robustness evidence is thinner than the abstract suggests. read the letter →

arxiv 1908.10983 v1 pith:OO3URMBO submitted 2019-08-28 eess.SY cs.SYmath.OC

Yan Jiang , Richard Pates , Enrique Mallada This is my paper · ORCID

classification eess.SYcs.SYmath.OC

keywords low-inertiapowersystemsinverter-basedfrequencycontroldroopvirtualinertiadynamiccontrollerH2normNadirsynchronizationcost

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

The paper argues that the usual way of supporting low-inertia grids—using inverters to imitate synchronous generators through droop control or virtual inertia—is not the best use of fast power electronics. It claims that both standard strategies have built-in trade-offs: droop control cannot improve transient frequency behavior without increasing the inverters' steady-state share of regulation, and virtual inertia amplifies measurement noise so badly that frequency variance becomes unbounded. The authors propose iDroop, a first-order lead/lag controller, and show analytically that its extra tuning knobs decouple steady-state effort from dynamic performance. With the right settings, iDroop can reject noise, synchronize the network without transient cost, or eliminate the frequency Nadir entirely. If correct, this gives a principled alternative to synthetic inertia for operating grids with high renewable penetration.

What carries the argument

The enabling object is the iDroop transfer function, $\hat c_i(s) = -(\nu_i s + \delta_i r_{r,i}^{-1})/(s+\delta_i)$, a lead/lag compensator whose dc gain is fixed at $-r_{r,i}^{-1}$ while its high-frequency gain is $\nu_i$. The analysis runs through a modal decomposition: under the proportionality assumption $\hat G(s)=\hat g_o(s)F^{-1}$ and $\hat C(s)=\hat c_o(s)F$, the network dynamics diagonalize via the scaled Laplacian $L_F=F^{-1/2}L_B F^{-1/2}$, reducing every performance metric to sums of scalar transfer-function norms. The key cancellation is Theorem 9: with $\delta=\tau^{-1}$ and $\nu=r_r^{-1}+r_t^{-1}$, the representative loop gain becomes first-order, $\hat h_{p,1,T,\mathrm{iDroop}}(s)=1/(ms+\check d+r_t^{-1})$, so the step response has no overshoot and the Nadir vanishes. Theorem 7 and Theorem 8 use the same diagonalized formulas to tune noise rejection and zero synchronization cost.

What would settle it

Pick a network whose generators have strongly heterogeneous turbine time constants, violating the proportionality assumption, and apply the Theorem 9 tuning $\delta_i=\tau_i^{-1}$, $\nu_i=r_{r,i}^{-1}+r_{t,i}^{-1}$ at every bus. Simulate a step disturbance: if the system frequency retains a measurable Nadir (for instance, deeper than 5% of the droop-only Nadir), or if iDroop's H2 norm exceeds droop control's in a noise-dominated simulation, the central claim would be contradicted.

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Extended reading notes

Core claim

The paper's central claim is that a first-order dynamic droop controller, iDroop, can outperform both droop control and virtual inertia on dynamic performance metrics without changing the inverters' steady-state effort share. Under a proportionality assumption that diagonalizes the network, the paper derives closed-form expressions for the H2 norm, synchronization cost, and Nadir, and shows: choosing ν near the minimizer ν* = −d + $\sqrt$($d^{2}$ + (κ_p/κ_ω)^2) with δ small makes iDroop strictly better than droop control at rejecting power fluctuations and measurement noise; taking δ → 0 and ν → ∞ drives the synchronization cost to zero; and setting δ = $τ^{{-1}}$ and ν = $r_r^{{-1}}$ + $r_t^{{-1}}$ cancels the turbine lag, making the system frequency respond as a first-order system so the Nadir disappears. The same steady-state effort share as droop and virtual inertia is preserved throughout, because iDroop's dc gain is still −$r_r^{{-1}}$. The paper also proves that droop control cannot eliminate Nadir in low-inertia systems and that virtual inertia generically has infinite H2 norm, and it validates the iDroop tunings on a realistic non-proportional network.

Load-bearing premise

The closed-form guarantees all rely on the assumption that every bus's generator and inverter dynamics are proportional copies of one representative pair, so the whole network can be diagonalized by a single scaling matrix; if real machines differ strongly in time constants and ratings, the tuning formulas may not deliver the promised behavior.

Editorial extensions

If this is right

In a proportional network, iDroop can be tuned so that its H2 norm is strictly smaller than droop control's whenever $(\kappa_p/\kappa_\omega)^2 \neq 2r_r^{-1}d + r_r^{-2}$, with the best performance approached as $\delta\to 0$ and $\nu\to\nu^*$.
iDroop can drive the synchronization cost to zero by taking $\delta\to 0$ and $\nu\to\infty$, a capability the paper shows droop control and virtual inertia cannot match without changing the steady-state effort share.
Setting $\delta=\tau^{-1}$ and $\nu=r_r^{-1}+r_t^{-1}$ eliminates the frequency Nadir, and in the realistic regime $\kappa_p\gg\kappa_\omega$ this same tuning also improves frequency variance relative to droop control.
The steady-state effort share and synchronous frequency under iDroop are identical to those under droop and virtual inertia, so the dynamic improvements do not shift the regulation burden.
Simulations on the Icelandic grid suggest these tunings remain useful when the proportionality assumption is violated, including in scenarios with combined step and stochastic disturbances.

Reading between the lines

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

A testable extension the authors do not pursue is to treat the $\delta=0$ limit as a pure high-frequency gain with DC regulation preserved; this suggests iDroop's single pole-zero pair is close to the minimal dynamic structure needed to separate transient shaping from steady-state power sharing.
For deployment, one could compute $\delta$ and $\nu$ from fleet-averaged turbine time constants and droop coefficients, as the paper's numerical setup does, and then monitor how per-bus heterogeneity degrades the H2 guarantee; the paper shows the Nadir tuning works on one non-proportional network but does not quantify a general robustness margin.
The noise result points to a fairer benchmark for virtual synchronous machines: comparing iDroop against virtual inertia with an explicit measurement low-pass filter, since the paper's unbounded-variance result applies to the unfiltered VI law and a filtered version could close some of the gap.
Because the Nadir-eliminating choice $\nu=r_r^{-1}+r_t^{-1}$ always lies between $r_r^{-1}$ and $\nu^*$ when $\kappa_p\gg\kappa_\omega$, the same tuning that removes the Nadir should also reduce frequency variance; the paper observes this in simulation but leaves the analytical trade-off curve between the two objectives implicit.

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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

Here's the short version: this paper is a solid, careful control-theoretic analysis of an inverter controller called iDroop, and it delivers genuinely useful closed-form tuning rules. The central caveat is that those rules are proved only under a proportionality assumption on generator and inverter transfer functions, and the paper's robustness claim for non-proportional networks rests on a single simulation that only partially violates the assumption.

The iDroop controller itself is not new — it appeared in the authors' CDC 2016 and 2017 papers. What is new here is the systematic performance analysis: an explicit condition for when iDroop beats droop control in H2 frequency variance (Theorem 7), a zero-synchronization-cost limit (Theorem 8), and a simple parameter choice that eliminates frequency Nadir (Theorem 9). These are real design rules, with proofs, expressed directly in terms of system parameters. The paper also gives a clean explanation of why droop control cannot decouple dynamic performance from steady-state effort and why virtual inertia amplifies measurement noise. That comparison is worth reading on its own.

The math is mostly rigorous. The H2 norm computations are spelled out, and the diagonalization under Assumption 1 is standard and well executed. The derivation of the effort share and synchronous frequency is clean. The reliance on prior synchronization-cost results from Paganini and Mallada is appropriate and cited.

The soft spots are real but not fatal. Assumption 1 — proportionality of transfer functions to a common F — is the load-bearing element. Every closed-form theorem depends on it. The paper claims "extensive numerical experimentation" shows the tuning works when the assumption is violated, but the validation is one network (Icelandic Kron-reduced 35-bus) and one disturbance pattern, and damping is still set proportional to the ratings (d_i = f_i d). So the simulation doesn't really test strong violations of the assumption. That is a moderate gap, not a fatal one. The zero-sync-cost result (Theorem 8) is a limit as δ→0 and ν→∞, which the authors themselves note may be slow and non-robust. The unbounded variance of virtual inertia is derived for white-noise measurement; that is a standard model but worth remembering when interpreting it as a practical failure. No code is released, but the formulas are explicit enough to reproduce.

Who should read this: anyone designing inverter frequency control in low-inertia systems, especially if they care about provable trade-offs rather than heuristic tuning. It deserves a serious peer review — the analysis is careful and the design principle is useful even if the robustness claim needs qualification. I would send it to review, and ask the authors to either broaden the non-proportional validation or soften the claim.

Referee Report

3 major / 6 minor

Summary. The paper studies inverter-based frequency control in low-inertia power systems. Under a proportionality assumption (Assumption 1) that makes the generator and inverter transfer-function matrices share a common diagonal factor F, the authors diagonalize the network and derive closed-form expressions for the steady-state effort share, the H2 frequency variance, bounds on the synchronization cost, and Nadir-elimination conditions. They use these expressions to show that conventional droop control cannot decouple dynamic performance from steady-state control effort, and that virtual inertia produces an unbounded H2 norm from white measurement noise. They then propose a dynamic droop (iDroop) controller and prove three tuning results: a variance-optimal tuning (Theorem 7), a zero-synchronization-cost limit (Theorem 8), and a Nadir-eliminating tuning (Theorem 9). The numerical section applies the Nadir-eliminating tuning to a Kron-reduced Icelandic network with heterogeneous parameters.

Significance. If the claims hold, the paper provides a genuinely useful control-theoretic contribution: explicit, parameter-free tuning formulas for iDroop that decouple steady-state effort share from dynamic performance, and a systematic comparison of droop, virtual inertia, and the proposed controller. The derivations are careful: the H2 formulas follow from a proved Lyapunov-based lemma (Lemma 2), and the key theorems (Theorems 7-9) come with proofs. The tuning rules are falsifiable and directly implementable from a small set of aggregate parameters. The main limitation is that the closed-form theory is proved only under Assumption 1, and the evidence that the tuning works when that assumption fails is much thinner than the abstract claims. A secondary limitation is that the zero-synchronization-cost result is an asymptotic limit rather than a finite tuning. The paper is significant for the control community even with these caveats, but the advertised generality needs to be reined in or supported by additional evidence.

major comments (3)

[Abstract and Section VI] The abstract states that 'extensive numerical experimentation shows that the proposed tuning is effective even when our proportionality assumptions are not valid,' but Section VI contains only one network (the Icelandic Kron-reduced 35-bus system), one disturbance pattern, and a parameter choice that retains partial proportionality: the manuscript sets d_i = f_i d with f_i = m_i/m, so damping is exactly proportional while inertia, turbine time constants, and turbine droop are taken from the dataset. This is a single heuristic test, not a sensitivity analysis, and it does not establish robustness as Assumption 1 is violated in a controlled or systematic way. Since all closed-form tuning formulas in Theorems 7-9 are derived under Assumption 1, the general decoupling claim outside that assumption is currently unsupported; the authors should either soften the abstract/conclusion claims or provide a systematic sensitivity study (e.g., random perturbations to F, independent variations of d_i, multiple test networks).
[Theorem 8 and Section V.C] The advertised feature of 'fast system-wide synchronization' is not established by Theorem 8. The zero-synchronization-cost result is obtained only in the double limit delta -> 0 and nu -> infinity, and the authors themselves note that delta near zero may lead to slow response and that nu -> infinity may hinder robustness. The theorem therefore provides an asymptotic idealization, not a finite parameter tuning, and it does not imply that the response is fast. Since the abstract and introduction list 'fast system-wide synchronization' as one of the three tunable iDroop properties, this is a load-bearing overstatement. The authors should state clearly that Theorem 8 only shows a limiting flexibility, and either provide finite-parameter bounds or withdraw the word 'fast' from the claims.
[Corollary 3 and Section IV.B] The claim that virtual inertia leads to unbounded frequency variance is a direct consequence of the white-noise measurement model: h_omega,k,VI(s) has a nonzero high-frequency gain, so ||T_omega dn,VI||_H2 = infinity under the stated H2 definition. This is internally consistent, but the paper should explicitly warn that the 'unbounded variance' conclusion is a white-noise-model phenomenon; with band-limited measurement noise the variance would be finite. As written, the abstract and Section IV.B present the unboundedness as a universal property of VI, which may mislead readers who do not work with ideal white-noise models.

minor comments (6)

[Equation (7)] The displayed formula for L_B,ij appears to have a typesetting error: the partial derivative with respect to theta_j is written as '∂θj' without a fraction or clear differentiation operator. Please correct this.
[Section VI, first paragraph] There is a typo in the phrase 'simultation results'; it should read 'simulation results'.
[Table I] The abbreviation 'o.w.' in the turbine droop row is undefined; please spell out 'otherwise' or define the abbreviation in the table caption.
[Theorem 7 statement] The interval notation in (51) is slightly ambiguous when nu* equals r_r^{-1}; the equality case is already excluded by the stated condition, but the phrase 'for any delta > 0 and nu such that nu in [nu*, r_r^{-1}) or nu in (r_r^{-1}, nu*]' would be clearer if the two cases were separated according to whether nu* < r_r^{-1} or nu* > r_r^{-1}.
[Section VI.C] The simulation in the combined step-and-noise scenario does not state whether the noise weighting matrices W_p(s) and W_omega(s) are chosen according to Assumption 2 (i.e., proportional to F^{1/2} and F^{-1/2}). Please state this explicitly, since the H2 comparisons depend on it.
[Lemma 3] Lemma 3 is stated as a 'direct extension' of [15, Proposition 2] without a proof. Since the synchronization-cost analysis in Corollaries 5-6 and Theorem 8 relies on it, a proof or a precise statement of the extension conditions would improve the paper's self-containedness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: iDroop's H2, synchronization, and Nadir results are explicit algebraic consequences of the stated model, not fits or self-citation loops.

full rationale

The derivation chain is self-contained. Theorem 2 decomposes the H2 norm by modal diagonalization under Assumption 1; Corollary 9 and Theorem 7 evaluate that norm for iDroop using Lemma 2, an analytic Lyapunov/H2 formula proved in Section III-B2. Theorem 9's Nadir elimination follows by substituting delta = tau^{-1} and nu = r_r^{-1} + r_t^{-1} into (46) and (19), producing a first-order scalar response; no parameter is fitted to the simulation output. Theorem 8's zero-synchronization-cost result is a limit of H2 norms of scalar modes, combined with the cost bounds in Theorem 3. The only prior-work overlap is Lemma 3, which invokes a direct extension of [15, Proposition 2] for the synchronization-cost expression, and the common-mode factorization cited to [17] in the proof of Theorem 9; both are general mathematical identities that do not assume the iDroop tuning and are not equivalent to the results derived here. The numerical section is a single validation on the Icelandic grid with damping kept proportional (d_i = f_i d), so it is weak evidence for robustness outside Assumption 1, and the footnote reassigns some turbine droops to create a deeper Nadir; these are generalization and scenario-selection concerns, not circularity. No prediction in the paper reduces by construction to a fitted input or to an author-imported uniqueness claim.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central analytical results rest on Assumption 1 (proportionality), Assumption 2 (proportional weighting), and Assumption 3 (step input with turbine deadband zero). The linearized network model assumptions in Remark 1 and the internal stability assumption in Remark 2 are also load-bearing. No new physical entities are invented; iDroop is a control law, not a particle or force. The only hand-chosen numerical values are simulation settings: representative damping d = 0.0014 s/rad (since the dataset lacks damping), the virtual inertia tuning mv = 0.022 s^2/rad for the comparison in the step scenario, and the noise weighting constants kappa_p = 1e-4 and kappa_omega = 1e-5. The tunings delta = tau^{-1}, nu = r_r^{-1} + r_t^{-1}, and nu_star are derived from system parameters, not fitted.

free parameters (3)

Representative generator damping d = 0.0014 s/rad
The Icelandic dataset does not include generator damping; the paper sets d_i = f_i d with d = 0.0014 s/rad. This hand-chosen value affects the simulation results and the numerical tuning values, but not the analytical formulas.
Virtual inertia mv in the step-input comparison = 0.022 s^2/rad
Chosen so that the VI system frequency is critically damped in the step simulation (Section VI-A). This is a comparison setting, not part of the iDroop tuning theory.
Noise weighting constants kappa_p and kappa_omega = kappa_p = 1e-4, kappa_omega = 1e-5 (simulation)
Assumption 2 defines the weighting functions up to constants kappa_p and kappa_omega, which are positive. In simulations they are chosen with kappa_p much larger than kappa_omega to reflect power-dominated disturbances. These are user-chosen weights, not fitted to data.

assumptions (6)

domain assumption Assumption 1 (Proportionality): there exists F := diag(f_i) >= 0 such that G(s) = g_o(s) F^{-1} and C(s) = c_o(s) F
Used in Section III-A to diagonalize the closed-loop system and to compute every closed-form metric. If this fails, the tuning formulas are not guaranteed. The paper tests one non-proportional system numerically.
domain assumption Assumption 2 (Proportional weighting scenario): Wp(s) = kappa_p F^{1/2}, Womega(s) = kappa_omega F^{-1/2}, and |omega_i(t)| < omega_epsilon for all i
Used in Theorem 2 and Corollaries 3 and 9 to compute the H2 frequency variance. The weighting choice is justified by CLT and oscillator jitter, but is still an assumption about disturbance and noise scaling.
domain assumption Assumption 3 (Step input scenario): step power injection, dp = 0, n_omega = 0, and omega_epsilon = 0 so turbines are constantly triggered
Used for the steady-state effort share (Theorem 1), synchronization cost (Theorem 3), and Nadir elimination (Theorems 5-6, 8-9).
domain assumption Internal stability of the feedback interconnection (Remark 2)
Standing assumption that the closed loop is internally stable; this is a precondition for H2 and L2 metrics to be finite and meaningful. The paper cites positive-real arguments for stability.
domain assumption Standard transmission-network model assumptions (Remark 1): lossless lines, constant voltage magnitudes, no reactive-power coupling, small equilibrium angle differences
These justify the linearized swing-plus-network model used throughout; standard and well-justified for frequency control studies.
standard math Standard linear algebra and control facts: Laplacian diagonalization, interlacing theorem, Rayleigh quotient, Lyapunov equation, Routh-Hurwitz
Used in the proofs of Theorems 2, 3, 4 and Lemma 2; these are accepted background results.

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Pith. "Pith review of Dynamic Droop Control in Low-inertia Power Systems." pith.science (2026). https://pith.science/paper/OO3URMBO

@misc{pith2026190810983,
  author       = {Pith},
  title        = {Pith review of: Dynamic Droop Control in Low-inertia Power Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OO3URMBO}},
  note         = {Machine review of arXiv:1908.10983}
}

abstract

A widely embraced approach to mitigate the dynamic degradation in low-inertia power systems is to mimic generation response using grid-connected inverters to restore the grid's stiffness. In this paper, we seek to challenge this approach and advocate for a principled design based on a systematic analysis of the performance trade-offs of inverter-based frequency control. With this aim, we perform a qualitative and quantitative study comparing the effect of conventional control strategies --droop control (DC) and virtual inertia (VI)-- on several performance metrics induced by $\mathcal L_2$ and $\mathcal L_\infty$ signal norms. By extending a recently proposed modal decomposition method, we capture the effect of step and stochastic power disturbances, and frequency measurement noise, on the overall transient and steady-state behavior of the system. Our analysis unveils several limitations of these solutions, such as the inability of DC to improve dynamic frequency response without increasing steady-state control effort, or the large frequency variance that VI introduces in the presence of measurement noise. We further propose a novel dynam-i-c Droop controller (iDroop) that overcomes the limitations of DC and VI. More precisely, we show that iDroop can be tuned to achieve high noise rejection, fast system-wide synchronization, or frequency overshoot (Nadir) elimination without affecting the steady-state control effort share, and propose a tuning recommendation that strikes a balance among these objectives. Extensive numerical experimentation shows that the proposed tuning is effective even when our proportionality assumptions are not valid, and that the particular tuning used for Nadir elimination strikes a good trade-off among various performance metrics.

Figures

Figures reproduced from arXiv: 1908.10983 by the authors.

**Figure 1.** Block diagram of power network. with ϕωǫ (ωi) :=    −r −1 t,i (ωi + ωǫ) ωi ≤ −ωǫ 0 −ωǫ < ωi < ωǫ −r −1 t,i (ωi − ωǫ) ωi ≥ ωǫ , where τi > 0 represents the turbine time constant and rt,i > 0 the turbine droop coefficient. Two special cases of our interest are: Generator Dynamics 1 (Standard swing dynamics). When |ωi(t)| < ωǫ, the turbines are not triggered and the generator dynamics can be described by the trans… view at source ↗

**Figure 2.** Equivalent block diagrams of power network under pro [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Diagonalized block diagram of power network. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: shows how different controllers perform when the system suffers from a step drop of −0.3 p.u. in power injection at bus number 2 at time t = 1 s. As for the representative inverter, we turn δ = τ −1 = 0.218 s−1 and ν = r −1 r + r −1 t = 0.004 s rad−1 in iDroop such tha…

**Figure 5.** Figure 5: Comparison between controllers when power fluctua [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

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Works this paper leans on

34 extracted references · 32 canonical work pages

[1]

iDroop: A dynamic droop controller to decou ple power grid’s steady-state and dynamic performance,

E. Mallada, “iDroop: A dynamic droop controller to decou ple power grid’s steady-state and dynamic performance,” in Proc. of IEEE Confer- ence on Decision and Control , Dec. 2016, pp. 4957–4964

work page 2016
[2]

Performance tradeof fs of dynami- cally controlled grid-connected inverters in low inertia p ower systems,

Y . Jiang, R. Pates, and E. Mallada, “Performance tradeof fs of dynami- cally controlled grid-connected inverters in low inertia p ower systems,” in Proc. of IEEE Conference on Decision and Control , Dec. 2017, pp. 5098–5105. 16

work page 2017
[3]

Achieving a 100% renewable grid: Operating electric power systems with extremely high level s of variable renewable energy,

B. Kroposki, B. Johnson, Y . Zhang, V . Gevorgian, P . Denho lm, B. Hodge, and B. Hannegan, “Achieving a 100% renewable grid: Operating electric power systems with extremely high level s of variable renewable energy,” IEEE Power and Energy Magazine , vol. 15, no. 2, pp. 61–73, Mar. 2017

work page 2017
[4]

Foundations and challenges of low-inertia systems (invited paper),

F. Milano, F. D¨ orﬂer, G. Hug, D. J. Hill, and G. V erbiˇ c, “ Foundations and challenges of low-inertia systems (invited paper),” in Proc. of Power Systems Computation Conference , June 2018, pp. 1–25

work page 2018
[5]

Paving the way: A future without inertia is close r than you think,

T. Ackermann, T. Prevost, V . Vittal, A. J. Roscoe, J. Mate vosyan, and N. Miller, “Paving the way: A future without inertia is close r than you think,” IEEE Power and Energy Magazine , vol. 15, no. 6, pp. 61–69, Nov. 2017

work page 2017
[6]

Impact of low r otational inertia on power system stability and operation,

A. Ulbig, T. S. Borsche, and G. Andersson, “Impact of low r otational inertia on power system stability and operation,” in Proc. of IF AC W orld Congress, Aug. 2014, pp. 7290–7297

work page 2014
[7]

A framework for assessing renewable integration limits with respect to frequency performance,

A. S. Ahmadyar, S. Riaz, G. V erbiˇ c, A. Chapman, and D. J. H ill, “A framework for assessing renewable integration limits with respect to frequency performance,” IEEE Transactions on Power Systems , vol. 33, no. 4, pp. 4444–4453, July 2018

work page 2018
[8]

Studying the maximum instantaneous non-sync hronous generation in an island system—Frequency stability challe nges in Ire- land,

J. O’Sullivan, A. Rogers, D. Flynn, P . Smith, A. Mullane, and M. O’Malley, “Studying the maximum instantaneous non-sync hronous generation in an island system—Frequency stability challe nges in Ire- land,” IEEE Transactions on Power Systems , vol. 29, no. 6, pp. 2943– 2951, Nov. 2014

work page 2014

Show all 34 references

[9]

Global energy scenario and impact of power el ectronics in 21st century,

B. K. Bose, “Global energy scenario and impact of power el ectronics in 21st century,” IEEE Transactions on Industrial Electronics , vol. 60, no. 7, pp. 2638–2651, July 2013

2013
[10]

Droop vs. virtual inertia: Comparison from the perspective of converter oper ation mode,

R. Oﬁr, U. Markovic, P . Aristidou, and G. Hug, “Droop vs. virtual inertia: Comparison from the perspective of converter oper ation mode,” in Proc. of IEEE International Energy Conference , June 2018, pp. 1–6

2018
[11]

Placement and imp lementation of grid-forming and grid-following virtual inertia and fas t frequency response,

B. K. Poolla, D. Groß, and F. D¨ orﬂer, “Placement and imp lementation of grid-forming and grid-following virtual inertia and fas t frequency response,” IEEE Transactions on Power Systems , vol. 34, no. 4, pp. 3035–3046, July 2019

2019
[12]

Optimizing DER participation in inertial and primary -frequency response,

S. S. Guggilam, C. Zhao, E. Dall’Anese, Y . C. Chen, and S. V . Dho- ple, “Optimizing DER participation in inertial and primary -frequency response,” IEEE Transactions on Power Systems , vol. 33, no. 5, pp. 5194–5205, Sept. 2018

2018
[13]

LQR-base d adaptive virtual synchronous machine for power systems with high inv erter penetration,

U. Markovic, Z. Chu, P . Aristidou, and G. Hug, “LQR-base d adaptive virtual synchronous machine for power systems with high inv erter penetration,” IEEE Transactions on Sustainable Energy , vol. 10, no. 3, pp. 1501–1512, July 2019

2019
[14]

Graph laplacian spectrum and primary frequency regulation,

L. Guo, C. Zhao, and S. H. Low, “Graph laplacian spectrum and primary frequency regulation,” in Proc. of IEEE Conference on Decision and Control, Dec. 2018, pp. 158–165

2018
[15]

Global analysis of synchro nization per- formance for power systems: bridging the theory-practice g ap,

F. Paganini and E. Mallada, “Global analysis of synchro nization per- formance for power systems: bridging the theory-practice g ap,” arXiv preprint:1905.06948, May 2019

1905 arXiv
[16]

Optimal placement of inerti a and pri- mary control: A matrix perturbation theory approach,

L. Pagnier and P . Jacquod, “Optimal placement of inerti a and pri- mary control: A matrix perturbation theory approach,” arXiv preprint: 1906.06922, June 2019

1906 arXiv
[17]

Global performance metric s for synchro- nization of heterogeneously rated power systems: The role o f machine models and inertia,

F. Paganini and E. Mallada, “Global performance metric s for synchro- nization of heterogeneously rated power systems: The role o f machine models and inertia,” in Proc. of Allerton Conference on Communication, Control, and Computing , Oct. 2017, pp. 324–331

2017
[18]

A voltage and frequency droop control metho d for parallel inverters,

K. D. Brabandere, B. Bolsens, J. V . den Keybus, A. Woyte, J. Driesen, and R. Belmans, “A voltage and frequency droop control metho d for parallel inverters,” IEEE Transactions on Power Electronics , vol. 22, no. 4, pp. 1107–1115, July 2007

2007
[19]

Virtual synchronous machine,

H. Beck and R. Hesse, “Virtual synchronous machine,” in Proc. of International Conference on Electrical Power Quality and U tilisation, Oct. 2007, pp. 1–6

2007
[20]

The price of sync hrony: Evaluating the resistive losses in synchronizing power net works,

E. Tegling, B. Bamieh, and D. F. Gayme, “The price of sync hrony: Evaluating the resistive losses in synchronizing power net works,” IEEE Transactions on Control of Network Systems , vol. 2, no. 3, pp. 254–266, Sept. 2015

2015
[21]

Usefulness of DC power ﬂow for active power ﬂow analysis,

K. Purchala, L. Meeus, D. V an Dommelen, and R. Belmans, “ Usefulness of DC power ﬂow for active power ﬂow analysis,” in Proc. of IEEE Power Engineering Society General Meeting , June 2005, pp. 454–459

2005
[22]

Kundur, Power System Stability and Control

P . Kundur, Power System Stability and Control . McGraw-Hill, 1994

1994
[23]

Power system dy- namics as primal-dual algorithm for optimal load control,

C. Zhao, U. Topcu, N. Li, and S. H. Low, “Power system dy- namics as primal-dual algorithm for optimal load control,” arXiv preprint:1305.0585, May 2013

2013 arXiv
[24]

Design and stabilit y of load- side primary frequency control in power systems,

C. Zhao, U. Topcu, N. Li, and S. Low, “Design and stabilit y of load- side primary frequency control in power systems,” IEEE Transactions on Automatic Control , vol. 59, no. 5, pp. 1177–1189, May 2014

2014
[25]

Connecting automatic gener ation control and economic dispatch from an optimization view,

N. Li, C. Zhao, and L. Chen, “Connecting automatic gener ation control and economic dispatch from an optimization view,” IEEE Transactions on Control of Network Systems , vol. 3, no. 3, pp. 254–264, Sept. 2016

2016
[26]

Optimal load-side cont rol for frequency regulation in smart grids,

E. Mallada, C. Zhao, and S. Low, “Optimal load-side cont rol for frequency regulation in smart grids,” IEEE Transactions on Automatic Control, vol. 62, no. 12, pp. 6294–6309, Dec. 2017

2017
[27]

H. K. Khalil, Nonlinear Systems , 3rd ed. Prentice Hall, 2002

2002
[28]

Robust scale free synthesis fo r frequency regulation in power systems,

R. Pates and E. Mallada, “Robust scale free synthesis fo r frequency regulation in power systems,” IEEE Transactions on Control of Network Systems, 2019

2019
[29]

Developing generic dynamic models for the 2030 eastern interconnection grid,

G. Kou, S. W. Hadley, P . Markham, and Y . Liu, “Developing generic dynamic models for the 2030 eastern interconnection grid,” Oak Ridge National Laboratory, Tech. Rep., Dec. 2013. [Online] . Available: http://www.osti.gov/scitech/

2013
[30]

R. A. Horn and C. R. Johnson, Matrix Analysis , 2nd ed. Cambridge University Press, 2012

2012
[31]

Analysis of timin g jitter in CMOS ring oscillators,

T. C. Weigandt, B. Kim, and P . R. Gray, “Analysis of timin g jitter in CMOS ring oscillators,” in Proc. of IEEE International Symposium on Circuits and Systems , May 1994, pp. 27–30

1994
[32]

Automatic generation control part II—Digital control techniques,

F. P . deMello, R. J. Mills, and W. F. B’Rells, “Automatic generation control part II—Digital control techniques,” IEEE Transactions on Power Apparatus and Systems , vol. PAS-92, no. 2, pp. 716–724, Mar. 1973

1973
[33]

of Edinburgh

U. of Edinburgh. Power systems test case archive. [Onli ne]. Available: https://www.maths.ed.ac.uk/optenergy/NetworkData/icelandDyn/
[34]

Kron reduction of graphs with a pplications to electrical networks,

F. D¨ orﬂer and F. Bullo, “Kron reduction of graphs with a pplications to electrical networks,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 1, pp. 150–163, Jan. 2013

2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

iDroop: A dynamic droop controller to decou ple power grid’s steady-state and dynamic performance,

E. Mallada, “iDroop: A dynamic droop controller to decou ple power grid’s steady-state and dynamic performance,” in Proc. of IEEE Confer- ence on Decision and Control , Dec. 2016, pp. 4957–4964

work page 2016

[2] [2]

Performance tradeof fs of dynami- cally controlled grid-connected inverters in low inertia p ower systems,

Y . Jiang, R. Pates, and E. Mallada, “Performance tradeof fs of dynami- cally controlled grid-connected inverters in low inertia p ower systems,” in Proc. of IEEE Conference on Decision and Control , Dec. 2017, pp. 5098–5105. 16

work page 2017

[3] [3]

Achieving a 100% renewable grid: Operating electric power systems with extremely high level s of variable renewable energy,

B. Kroposki, B. Johnson, Y . Zhang, V . Gevorgian, P . Denho lm, B. Hodge, and B. Hannegan, “Achieving a 100% renewable grid: Operating electric power systems with extremely high level s of variable renewable energy,” IEEE Power and Energy Magazine , vol. 15, no. 2, pp. 61–73, Mar. 2017

work page 2017

[4] [4]

Foundations and challenges of low-inertia systems (invited paper),

F. Milano, F. D¨ orﬂer, G. Hug, D. J. Hill, and G. V erbiˇ c, “ Foundations and challenges of low-inertia systems (invited paper),” in Proc. of Power Systems Computation Conference , June 2018, pp. 1–25

work page 2018

[5] [5]

Paving the way: A future without inertia is close r than you think,

T. Ackermann, T. Prevost, V . Vittal, A. J. Roscoe, J. Mate vosyan, and N. Miller, “Paving the way: A future without inertia is close r than you think,” IEEE Power and Energy Magazine , vol. 15, no. 6, pp. 61–69, Nov. 2017

work page 2017

[6] [6]

Impact of low r otational inertia on power system stability and operation,

A. Ulbig, T. S. Borsche, and G. Andersson, “Impact of low r otational inertia on power system stability and operation,” in Proc. of IF AC W orld Congress, Aug. 2014, pp. 7290–7297

work page 2014

[7] [7]

A framework for assessing renewable integration limits with respect to frequency performance,

A. S. Ahmadyar, S. Riaz, G. V erbiˇ c, A. Chapman, and D. J. H ill, “A framework for assessing renewable integration limits with respect to frequency performance,” IEEE Transactions on Power Systems , vol. 33, no. 4, pp. 4444–4453, July 2018

work page 2018

[8] [8]

Studying the maximum instantaneous non-sync hronous generation in an island system—Frequency stability challe nges in Ire- land,

J. O’Sullivan, A. Rogers, D. Flynn, P . Smith, A. Mullane, and M. O’Malley, “Studying the maximum instantaneous non-sync hronous generation in an island system—Frequency stability challe nges in Ire- land,” IEEE Transactions on Power Systems , vol. 29, no. 6, pp. 2943– 2951, Nov. 2014

work page 2014