Power Factor Angle Droop Control-A General Decentralized Control of Cascaded inverters

Guangze Shi; Lang Li; Mei Su; Xiaochao Hou; Yao Sun

REVIEW 3 major objections 5 minor 11 references

Power Factor Angle Droop Control-A General Decentralized Control of Cascaded inverters

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A power-factor-angle droop law equalizes active and reactive power among cascaded (series-connected) inverter modules in both grid-connected and islanded modes, with a stability condition that does not involve the transmission line…

desk verdict A clean new droop law for cascaded inverters with a correct islanded-mode consensus proof, but the grid-connected line-impedance independence claim is overstated and needs qualification. read the letter →

arxiv 1908.08233 v1 pith:YGWRTL2S submitted 2019-08-22 eess.SY cs.SYmath.OC

Yao Sun , Lang Li , Guangze Shi , Xiaochao Hou , Mei Su This is my paper · ORCID

classification eess.SYcs.SYmath.OC

keywords powerfactorangledroopcontrolcascadedinvertersdecentralizedsharingsmall-signalstabilitygrid-connectedmodeislandedfour-quadrantoperation

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 proposes that a single decentralized law—droop each cascaded (series-connected) inverter module's frequency against its own power factor angle (the angle with tangent $Q_i/P_i$), while holding its voltage magnitude fixed at a common reference—can replace communication-based control of the whole string. The claimed result is that all modules converge to the same power factor angle, and because every module sits at the same voltage magnitude $V^*$, equal power angles translate into equal active and reactive powers: power is shared equally whether the string feeds a standalone load or is tied to a stiff grid. Because one law covers both modes, mode transitions need no reconfiguration; the paper also claims a unique operating point, suitability for any load type and any transmission line impedance, and four-quadrant operation, and supports these claims with small-signal stability proofs and four-module simulations.

What carries the argument

The central object is the power factor angle $\varphi_i = \arctan(Q_i/P_i)$, the phase of the module's complex power, used as the feedback signal in the frequency droop $\omega_i = \omega^* - m(\varphi_i - \varphi^*)$. Because all modules in a synchronized cascade share one frequency, the droop forces every $\varphi_i$ to the common reference $\varphi^*$; the companion law $V_i = V^*$ then factors a common voltage out of the power expressions (2)–(6), which is exactly what converts equal power factor angles into equal $P_i$ and equal $Q_i$. The supporting mechanism is the small-signal linearization of $\varphi_i$ in the module phase angles: in islanded mode it collapses to a complete-graph Laplacian consensus with eigenvalues $0$ and $-m$, and in grid-connected mode to a matrix whose eigenvalues isolate a voltage-magnitude stability condition that is independent of the transmission line impedance.

What would settle it

Operate a two-module islanded cascade under $\omega_i = \omega^* - m(\varphi_i - \varphi^*)$ but set the two voltage references unequal, say $V_1 = 0.9V^*$ and $V_2 = 1.1V^*$, with a resistive load. The equal-sharing claim predicts $P_1 = P_2$ and $Q_1 = Q_2$ from equal power factor angles; if direct measurement shows unequal module powers once the frequencies have synchronized, the theorem fails exactly where the proof divides out a common $V^*$, and the linearization (14) would acquire a $V_1 - V_2$ term that no longer vanishes.

Watch

Extended reading notes

Core claim

The paper's core claim is that the droop law $\omega_i = \omega^* - m(\varphi_i - \varphi^*)$ with $V_i = V^*$ for every module $i$ equalizes the power factor angles $\varphi_i = \arctan(Q_i/P_i)$ in steady state, and that equal power factor angles mean equal active and reactive powers because a common voltage magnitude $V^*$ factors out of the power expressions. In islanded operation, linearizing the angle dynamics around equilibrium gives a Laplacian consensus whose eigenvalues are $0$ and $-m$ (the latter with multiplicity $n-1$), so all modules synchronize for any load type and any load parameters. In grid-connected operation the linearized system has eigenvalues $-m$ and $-mV_g\bigl(nV^* - V_g\cos(\delta^* - \delta_g)\bigr)$, so stability holds whenever $V_g\bigl(nV^* - V_g\cos(\delta^* - \delta_g)\bigr) \ge 0$; this condition involves only the grid voltage magnitude, the common module voltage, and the number of modules, and contains no transmission line quantity. The paper also claims a unique equilibrium, four-quadrant operation, and that no earlier decentralized scheme combines all of these properties, and it verifies the claims by simulation of a four-module system.

Load-bearing premise

The load-bearing premise is that every module is held at the same voltage magnitude $V^*$; the proof converts equal power factor angles into equal powers only by dividing out this common $V^*$, so modules with mismatched DC-link voltages, filter gains, or voltage references would break the equal-sharing conclusion even though the droop law itself would still force the frequencies to synchronize.

Editorial extensions

If this is right

One control law and one parameter set cover both operating modes, so switching between grid-connected and islanded operation requires no reconfiguration of the local controllers.
In islanded mode, equal sharing holds for pure resistive, resistive-inductive, and resistive-capacitive loads with no retuning per load type.
In grid-connected mode, the small-signal stability margin depends on the grid voltage, the common module voltage, and the module count, not on the transmission line, so capacitive, inductive, and resistive lines behave the same.
The steady state is unique for given references and load, so the multiple-equilibrium states reported for earlier f-P/Q droop schemes do not arise.
Because the law does not presuppose the sign of $P_i$ or $Q_i$, the same controller covers all four quadrants, which includes battery charging, reactive compensation, and photovoltaic feed-in.

Reading between the lines

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

Editorial inference: the equal-sharing proof leaves the identical-voltage premise unexamined; a quantitative bound on the sharing error as a function of $|V_i - V_j|/V^*$ would tell a practitioner how tightly DC-link voltages must match, and would test the method's practical margin.
Editorial inference: in islanded mode the linearized dynamics are a phase-consensus protocol whose convergence rate scales as $m/n$, so the droop gain doubles as a consensus gain; the paper does not discuss the trade-off between synchronization speed and steady-state frequency deviation from $\omega^*$ under load.
Editorial inference: the seamless-transition claim rests on a simulation of the mode switch; the paper gives no analysis of the transient at the switching instant, so whether the grid-connected voltage condition must hold continuously along the switching trajectory is left open.
Editorial inference: since the grid-connected stability boundary is $V_g(nV^* - V_g\cos(\delta^* - \delta_g)) = 0$, stepping the grid voltage down experimentally is a cheap probe of the claimed margin—as the quantity approaches zero, the dominant eigenvalue should approach zero and the stability margin vanish.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 one thing to know: this is a compact, competent letter proposing a new droop law for cascaded inverters—frequency droops on the local power-factor angle—and proving equal power sharing and small-signal stability in islanded mode with a neat consensus argument. The equal-sharing result is correct under the stated assumption of identical voltage magnitudes. But the headline claim that grid-connected stability is independent of the transmission line impedance is oversold. A careful linearization gives a stability condition involving the equilibrium angle δ_g−δ_s, which itself depends on the line through the power-flow solution. The condition becomes line-independent only if V_g ≥ nV*, which the paper never states. The simulations pick V_g = nV*, so they cannot expose the issue.

What it does well: the control law is genuinely a new member of the droop family, unifying islanded and grid-connected operation. The islanded-mode stability proof is clean: the linearization collapses to a complete-graph Laplacian, with eigenvalues 0 and −m. The paper also demonstrates via simulation that the method handles RL, RC, and R loads, four-quadrant operation, and different line impedance types. For a letter, the technical core is mostly self-contained and no parameters are fitted to data.

Soft spots, in order of concern: 1. Grid-connected stability. Re-deriving from (5)–(6) gives the common-mode eigenvalue proportional to V_g(V_g − nV* cos(δ_g−δ_s)). Stability requires V_g ≥ nV* cosΔ. Since Δ is the equilibrium power-flow angle, the condition is not line-independent in general. The paper should state the sufficient condition V_g ≥ nV* to justify the claim; as printed, (25) and the surrounding text appear garbled and the inequality sign may be mis-set. 2. The equal-power-sharing proof assumes perfectly equal module voltage magnitudes V_i = V*. In a real cascaded converter with unequal DC links or filter gains, φ_i−φ_j ≠ δ_i−δ_j, and the consensus argument weakens. This is not fatal, but it should be acknowledged. 3. The unique equilibrium claim is only validated by simulation (Case 3), not by analysis. 4. The islanded stability proof cites [12], which is not in the reference list.

The central method is plausible and useful for the cascaded-converter community. The paper deserves a serious referee. I would send it out, with a request that the authors fix the line-impedance claim, correct the missing reference, and ideally add a short discussion of voltage-magnitude mismatches. It is not a transformative contribution, but it is a solid incremental step from a group with prior work in this niche.

Referee Report

3 major / 5 minor

Summary. The paper proposes a decentralized control law for cascaded inverters, called power factor angle droop control, defined by (7)–(8): each inverter sets its frequency as ω_i = ω* − m(φ_i − φ*) and its voltage magnitude as V_i = V*. The authors claim that this scheme provides equal active and reactive power sharing in both islanded and grid-connected modes, is stable with a small-signal analysis, avoids multiple equilibria, suits all load types and all transmission line impedances, and allows four-quadrant operation. The islanded-mode stability proof reduces to a Laplacian consensus on phase angles, while the grid-connected proof gives an eigenvalue expression in (25). Simulations with four inverter modules illustrate mode transition, load variations, initial-angle variation, line impedance variation, and four-quadrant operation.

Significance. If the claims are correct, the proposed control is attractive because it unifies islanded and grid-connected operation with only local measurements and gives a simple, parameter-light stability proof. The islanded-mode power-sharing result follows directly from equal voltage magnitudes and equal power factor angles, and the small-signal consensus argument is clean. The paper also correctly identifies the need for decentralized series-connected inverter control. However, the central grid-connected stability claim is not established as stated, and the uniqueness-of-equilibrium claim is only simulated, not proven. The equal-power-sharing and islanded-stability contributions are solid within the equal-voltage-magnitude assumption, but the advertised independence of stability from transmission line impedance is a load-bearing claim that is not supported by the provided analysis.

major comments (3)

[§II-E, Eq. (25)] The stability condition reported immediately after (25), namely V_g ≥ nV* cos(δ_g − δ_s), is not independent of the transmission line impedance. The angle difference δ_g − δ_s is itself the equilibrium solution of the power-flow equations that contain Z_line, and the paper neither proves that the inequality holds at every admissible equilibrium nor states the required relation between V_g and nV*. Under the simulation parameters in Table I, V_g = 315 = nV*, which makes the condition automatically satisfied for any nonzero angle difference, so the simulations do not test the claimed independence. This is a load-bearing issue because benefit (3) in the abstract and introduction is the independence claim; the provided analysis supports at most a conditional stability statement for equilibria satisfying the inequality.
[§II-D and §III-C] The benefit of a unique equilibrium point is asserted in the abstract and conclusion but is never proven analytically. Section III-C shows only a single simulation with different initial phase angles, which demonstrates convergence in that case but does not establish uniqueness of the equilibrium across the state space. Because the paper explicitly distinguishes its method from the multi-equilibrium problem of [6], this claim should be supported by a proof or at least a formal argument based on the closed-loop equations, not solely by simulation.
[§II-D] In the grid-connected mode, the equal-power-sharing conclusion is stated without derivation ('the similar conclusions about the active and reactive power can be drawn as above'). The steady-state argument requires all δ_i to converge to a common synchronous angle δ_s, and the existence of such a synchronous solution (i.e., solvability of the grid-connected power-flow equations) is not discussed. The stability analysis therefore implicitly assumes an equilibrium exists without characterizing it, which also leaves the domain of validity of the stability condition in (25) unclear.

minor comments (5)

[§II-E, Eq. (14)] The linearization leading to (14) from (13) is not shown; a brief derivation would help readers verify the factor 1/n and the Laplacian form in (17).
[§II-E, Eq. (25)] The eigenvalue expression in (25) and the verbal stability condition have inconsistent dimensions: λ_1 is given as a frequency, but the condition is written as V_g − nV* cos(...), which has units of volts. The condition should be stated as a dimensionless inequality on the ratio V_g/(nV*) or on cos(δ_g − δ_s).
[§III-D, Case 4] The simulations for capacitive, inductive, and resistive transmission lines show stable waveforms, but they do not quantify power-sharing accuracy or stability margin, so the claim of suitability for all transmission line impedances is only qualitatively supported.
[Throughout] The typesetting of the equations contains numerous artifacts (e.g., garbled subscripts, missing parentheses in (2)–(6), and unclear summation indices). A careful revision of the equations would substantially improve readability.
[§II-E, reference [11]] The islanded-mode stability proof cites [11] for the conclusion that the Laplacian system is stable, but the connection to the specific consensus result should be stated explicitly, since the matrix in (17) is a complete-graph Laplacian and the eigenvalue result is elementary.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equal-power-sharing and stability derivations follow directly from the circuit equations and the proposed control law, with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's central claims are derived from the series-connected circuit equations and the proposed control law (7)-(8), and no parameter is fitted to data. With V_i=V* enforced by (8), the complex power of module i is S_i=(V*^2/Z)e^{jθ}e^{jδ_i}Σ_j e^{-jδ_j} in islanded mode, and an analogous expression with a grid term in grid-connected mode; therefore equal power factor angles φ_i imply equal angles δ_i and hence equal active and reactive powers. This is a direct algebraic consequence, not a renaming of the conclusion. The small-signal stability analysis is a standard linearization (12)-(25) yielding eigenvalues in terms of V_g, V*, n, and the equilibrium angle; no parameter is back-fitted from the simulation results. The self-citations in the introduction ([5]-[9]) are used only to describe prior limitations and are not load-bearing for the proposed derivation. One advertised benefit, that the grid-connected stability condition is independent of the transmission line impedance, is not rigorously established because the eigenvalue condition in (25) contains the equilibrium angle δ_s, which itself depends on the line impedance and power-factor setpoint; however, that is an unproven or overbroad correctness claim, not a circular reduction of the derivation to its own inputs. Overall, no significant circularity is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The control law is a new algorithm with two tunable gains (m and φ*). The stability proof rests on standard linearization plus an ideal-voltage-source model of the inverters and a linear passive load model. No parameters are fitted to data and no new physical entities are introduced.

free parameters (2)

m (droop coefficient) = 0.5
Positive controller gain chosen for the simulation; stability requires m>0 only, so it is not fitted to data and does not affect the qualitative claims.
φ* (power factor angle reference) = 0.2 rad
Setpoint chosen for the simulation; it shifts the operating frequency but is not fitted to data and is not load-bearing for the main stability result.

assumptions (4)

domain assumption Each inverter behaves as an ideal voltage source with fixed magnitude V* and a frequency directly set by the droop law (7), with no inner-loop dynamics.
Used throughout Section II; the small-signal models (16) and (23) are phase-only, so the current and voltage control loops are assumed infinitely fast.
domain assumption The islanded load is a single linear passive impedance with constant angle θ'_load; in grid-connected mode the grid is an ideal voltage source V_g e^{jδ_g} behind a line impedance Z_line e^{jθ_line}.
Equations (2)-(6) are derived under this linear circuit model; the claim of suitability for all load types is limited to linear passive RLC loads.
standard math Small-signal linearization around the equilibrium is sufficient to conclude local stability of the nonlinear system.
Used in Section II-E to replace (11) with (12) and to derive eigenvalues (18) and (25).
ad hoc to paper A unique equilibrium exists with all phase angles equal in both modes; this is asserted for the uniqueness claim but not proven analytically.
Section II-D states P_i=P_j and Q_i=Q_j, and Case 3 simulates different initial phases, but no rigorous uniqueness proof is given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Power Factor Angle Droop Control-A General Decentralized Control of Cascaded inverters." pith.science (2026). https://pith.science/paper/YGWRTL2S

@misc{pith2026190808233,
  author       = {Pith},
  title        = {Pith review of: Power Factor Angle Droop Control-A General Decentralized Control of Cascaded inverters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGWRTL2S}},
  note         = {Machine review of arXiv:1908.08233}
}

read the original abstract

This letter proposes a general decentralized control of cascaded inverters-power factor angle droop control. Compared to the existing control strategies, it has the following attractive benefits: 1) it is suitable for both grid-connected and islanded modes; 2) Seamless transition between different modes can be obtained; 3) stability condition in the grid-connected mode is independent of the transmission line impedance; 4) it is suited for any types of loads in islanded modes; 5) multi-equilibrium point problem is avoided; 6) it is suitable for four quadrant operation. The small signal stability of the control is proved. And the feasibility of the proposed method is verified by simulation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[6]

An f -P/Q Droop Control in Cascaded-type Microgrid,

Y. Sun, G. Shi, X. Li, W. Yuan, M. Su, H. Han, and X. Hou, “An f -P/Q Droop Control in Cascaded-type Microgrid,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 1136–1138, 2018

work page 2018
[1]

A survey on cascaded multilevel inverters,

M. Malinowski, K. Gopakumar, J. Rodriguez, and M. P´ erez, "A survey on cascaded multilevel inverters," IEEE Trans. Ind. Electron., vol. 57, no. 7, pp. 2197–2206, Jul. 2010

work page 2010
[2]

Mul tifunctional Control Strategy for Asymmetrical Cascaded H -Bridge Inverter in Microgrid Applications,

A. Mortezaei, M. G. Simoes, A. S. Bubshait, T. D. C. Busarello, F. P. Marafao, and A. A. Durra, “Mul tifunctional Control Strategy for Asymmetrical Cascaded H -Bridge Inverter in Microgrid Applications,” IEEE Transactions on Industry Applications, vol. PP, no. 99, pp. 1-1, 2015

work page 2015
[3]

Power balance of cascaded H -bridge multilevel converters for large -scale photovoltaic grid integration,

Y. Yu, G. Konstantinou, B. Hredzak, and V. G. Agelidis, "Power balance of cascaded H -bridge multilevel converters for large -scale photovoltaic grid integration," IEEE Trans. Power Electron., vol. 31, no. 1, pp. 292–303, Jan. 2016

work page 2016
[4]

A comparison of grid -connected battery energy storage system desig ns,

E. Chatzinikolaou and D. J. Rogers, "A comparison of grid -connected battery energy storage system desig ns," IEEE Trans. Power Electron. , vol. 32, no. 9, pp. 6913-6923, Sep. 2017

work page 2017
[5]

Inverse Power Factor Droop Control for Decentralized Power Sharing in Series -Connected Micro -converters Based Islanding Microgrids,

J. He, Y. Li, B. Liang, and C. Wang, “Inverse Power Factor Droop Control for Decentralized Power Sharing in Series -Connected Micro -converters Based Islanding Microgrids,” IEEE Transactions on Industrial Electronics, vol. 64, no. 9, pp. 7444-7454, 2017

work page 2017
[7]

A Decentralized Control With Unique Equilibrium Point for Cascaded -Type Microgrid,

L. Li, Y. Sun, Z. Liu, X. Hou, G. Shi and M. Su, "A Decentralized Control With Unique Equilibrium Point for Cascaded -Type Microgrid," in IEEE Transactions on Sustainable Energy, vol. 10, no. 1, pp. 324-326, Jan. 2019

work page 2019
[8]

A fully decentralized control of grid -connected cascaded inverters,

X. Hou, Y. Sun, H. Han, Z. Liu, W. Yuan and M. Su, "A fully decentralized control of grid -connected cascaded inverters," in IEEE Transactions on Sustainable Energy, vol. 10, no. 1, pp. 315-317, Jan. 2019

work page 2019

Show all 11 references

[9]

A General Decentralized Control Scheme for Medium-/High-Voltage Cascade d STATCOM,

X. Hou et al. , "A General Decentralized Control Scheme for Medium-/High-Voltage Cascade d STATCOM," in IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 7296-7300, Nov. 2018

2018
[10]

A Distributed Power Control of Series -Connected Module -Integrated Inverters for PV Grid -Tied Applications,

L. Zhang, K. Sun, Y. W. Li, X. Lu and J. Zhao, "A Distributed Power Control of Series -Connected Module -Integrated Inverters for PV Grid -Tied Applications," in IEEE Transactions on Power Electronics, vol. 33, no. 9, pp. 7698-7707, Sept. 2018

2018
[11]

Synchronization and power sharing for droop-controlled inverters in islanded microgrids,

J. W. Simpson-Porco, F. Dö rfler, and F. Bullo, “Synchronization and power sharing for droop-controlled inverters in islanded microgrids,” Automatica, vol. 49, no. 9, pp. 2603–2611, Sep. 2013

2013

Pith tools

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

[1] [6]

An f -P/Q Droop Control in Cascaded-type Microgrid,

Y. Sun, G. Shi, X. Li, W. Yuan, M. Su, H. Han, and X. Hou, “An f -P/Q Droop Control in Cascaded-type Microgrid,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 1136–1138, 2018

work page 2018

[2] [1]

A survey on cascaded multilevel inverters,

M. Malinowski, K. Gopakumar, J. Rodriguez, and M. P´ erez, "A survey on cascaded multilevel inverters," IEEE Trans. Ind. Electron., vol. 57, no. 7, pp. 2197–2206, Jul. 2010

work page 2010

[3] [2]

Mul tifunctional Control Strategy for Asymmetrical Cascaded H -Bridge Inverter in Microgrid Applications,

A. Mortezaei, M. G. Simoes, A. S. Bubshait, T. D. C. Busarello, F. P. Marafao, and A. A. Durra, “Mul tifunctional Control Strategy for Asymmetrical Cascaded H -Bridge Inverter in Microgrid Applications,” IEEE Transactions on Industry Applications, vol. PP, no. 99, pp. 1-1, 2015

work page 2015

[4] [3]

Power balance of cascaded H -bridge multilevel converters for large -scale photovoltaic grid integration,

Y. Yu, G. Konstantinou, B. Hredzak, and V. G. Agelidis, "Power balance of cascaded H -bridge multilevel converters for large -scale photovoltaic grid integration," IEEE Trans. Power Electron., vol. 31, no. 1, pp. 292–303, Jan. 2016

work page 2016

[5] [4]

A comparison of grid -connected battery energy storage system desig ns,

E. Chatzinikolaou and D. J. Rogers, "A comparison of grid -connected battery energy storage system desig ns," IEEE Trans. Power Electron. , vol. 32, no. 9, pp. 6913-6923, Sep. 2017

work page 2017

[6] [5]

Inverse Power Factor Droop Control for Decentralized Power Sharing in Series -Connected Micro -converters Based Islanding Microgrids,

J. He, Y. Li, B. Liang, and C. Wang, “Inverse Power Factor Droop Control for Decentralized Power Sharing in Series -Connected Micro -converters Based Islanding Microgrids,” IEEE Transactions on Industrial Electronics, vol. 64, no. 9, pp. 7444-7454, 2017

work page 2017

[7] [7]

A Decentralized Control With Unique Equilibrium Point for Cascaded -Type Microgrid,

L. Li, Y. Sun, Z. Liu, X. Hou, G. Shi and M. Su, "A Decentralized Control With Unique Equilibrium Point for Cascaded -Type Microgrid," in IEEE Transactions on Sustainable Energy, vol. 10, no. 1, pp. 324-326, Jan. 2019

work page 2019

[8] [8]

A fully decentralized control of grid -connected cascaded inverters,

X. Hou, Y. Sun, H. Han, Z. Liu, W. Yuan and M. Su, "A fully decentralized control of grid -connected cascaded inverters," in IEEE Transactions on Sustainable Energy, vol. 10, no. 1, pp. 315-317, Jan. 2019

work page 2019